Updated Lagrangian mixed finite element formulation for quasi and fully incompressible fluids

Revision as of 10:17, 7 June 2019

Published in Computational Mechanics, Vol. 54 (6), pp. 1583-1596, 2014
DOI: 10.1007/s00466-014-1078-1

Abstract

We present a mixed velocity-pressure finite element formulation for solving the updated Lagrangian equations for quasi and fully incompressible fluids. Details of the governing equations for the conservation of momentum and mass are given in both differential and variational form. The finite element interpolation uses an equal order approximation for the velocity and pressure unknowns. The procedure for obtaining stabilized FEM solutions is outlined. The solution in time of the discretized governing conservation equations using an incremental iterative segregated scheme is described. The linearization of these equations and the derivation of the corresponding tangent stiffness matrices is detailed. Other iterative schemes for the direct computation of the nodal velocities and pressures at the updated configuration are presented. The advantages and disadvantages of choosing the current or the updated configuration as the reference configuration in the Lagrangian formulation are discussed.

Keywords: Updated Lagrangian formulation, incompressible fluids, finite element method, incremental solution, tangent matrix, mixed formulation, stabilized method

1 INTRODUCTION

In Lagrangian analysis procedures, the motion of the particles of a deforming body is followed in time. In Eulerian formulations, on the other hand, attention is focused in the motion of the material through a stationary control volume. Lagrangian methods have been traditionally used for the numerical analysis of solids and structures, while Eulerian methods are typical in computational fluid dynamics [2,4,5,36,40,41].

Despite this evidence, the use of a Lagrangian description for solving fluid dynamics problems using the finite element method (FEM) [10,41] and different meshless and mesh-based particle-based numerical techniques [6,7,12],[16]–[20],[25,26] [29]–[35],[37,38] has received much attention in recent years. Of particular interest are numerical procedures, such as the Particle Finite Element Method (PFEM) [16,25,26,29,31], that combine the advantage of particle-based procedures with the formalism and accuracy of the FEM.

Despite the increasing interest in the Lagrangian approach for solving the equations of fluid mechanics using the FEM, there are few references to the derivation of incremental iterative solution schemes using linearized forms of the discretized Lagrangian equations for fluid flow problems. An early attempt in this direction was reported by Radovitzky and Ortiz [34] who derived the tangent matrix for the FEM Lagrangian analysis of compressible flows using a Newton-Rapsohn type iterative scheme.

The goal of this paper is to present a mixed velocity-pressure formulation for the finite element analysis of quasi and fully incompressible fluids using an updated Lagrangian formulation. We advocate an equal order interpolation for the velocity and pressure variables which invariably requires using an adequate stabilization scheme for the mass balance equation in order to obtain accurate numerical results. In the paper we derive in some detail both the continuum and discretized (FEM) forms of the equations governing the motion of the fluid in the updated Lagrangian description. An incremental Newton-Raphson type iterative staggered scheme for the solution in time of the non linear discretized equations is detailed. The derivations are carried out first using the current configuration as the reference configuration in the Lagrangian description of the motion. The expression of the different matrices and vectors involved in the incremental iterative scheme is given. The particular form of the FEM equations when the updated configuration is taken as the reference configuration is presented. The direct transient solution of the nodal velocities and pressures using monolithic and staggered schemes is also presented for completeness.

The chapter concludes with a discussion of the computational advantages and disadvantages of choosing the current or the the updated configuration as the reference configuration in the Lagrangian description.

In the next section the basic concepts of the motion of a fluid are briefly revisited. These concepts are standard and can be found in many books on computational solid and fluid mechanics and fluid-structure interaction, among others [2,3,4,5,13,36]. This introductory section aims to set up the updated Lagrangian framework where the governing equations for the fluid are written and subsequently solved with the FEM using a mixed velocity-pressure formulation.

2 MOTION. DISPLACEMENT, VELOCITY AND ACCELERATION

We will consider a domain containing a fluid which evolves in time due to external and internal forces and prescribed velocities from an initial configuration at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t={}^0 t}

(typically Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^0t =0}

) to a current configuration at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t={}^n t} . The fluid volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V}

and its boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma }
at the initial and current configurations are denoted as (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^0 V, {}^0 \Gamma }

) and (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^n V, {}^n \Gamma } ), respectively. The goal is to find the domain that the fluid occupies, as well as the velocities, strain rates and stresses (the deviatoric stresses and the pressure) in the so-called updated configuration at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} t= {}^n t+\Delta t}

(Figure 1). In the following a left superindex denotes the configuration of the fluid domain where the variable is computed.

Figure 1: Initial (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t ={}^0 t

), current (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t={}^n t ) and updated (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t={}^{n+1} t ) configurations of a fluid domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): V

with Neumann (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Gamma _t

) and Dirichlet (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Gamma _v ) boundaries. Trajectory of a material point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i

and velocity  (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {v}_i

) and displacement (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {u}_i ) vectors of the point at each configuration. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^n {u} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1}, {u}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta {u}
denote schematically the trajectories of the overall domain between two configurations.

The motion of the fluid domain is described by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^t{x} = \boldsymbol{\phi } ({}^0 {x} ,t) \quad \hbox{or}\quad {}^t x_i = \phi _i ({}^0 {x} ,t)

(1)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^t{x}}

is the position of the material point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^0 {x}}
laying on the initial configuration at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}

. The coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^t{x}}

give the spatial position of a point and are called spatial or Eulerian coordinates. The function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\phi }}
maps the initial configuration into the current configuration at  time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}

. The position of the points in the current and initial configurations at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t={}^n t}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t={}^0  t}

, respectively are expressed by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l}{}^n {x} = \boldsymbol{\phi }({}^0 {x} ,{}^n t) \quad \hbox{or}\quad {}^n x_i = \phi _i ({}^n {x} ,{}^n t)\\ {}^0 {x} = \boldsymbol{\phi } ({}^0 {x} ,{}^0 t) \quad \hbox{or}\quad {}^0 x_i = \phi _i ({}^0 {x} ,{}^0 t) \end{array}

(2)

Thus the mapping Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\phi }({}^0 {x} , {}^0t)}

is the identity mapping.

In the Lagrangian description (also called the material description) the independent variables are the material coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^0 {x}}

of the point on the initial configuration and the time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}

. In the Eulerian description, on the other hand, the independent coordinates are the spatial coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^t{x}}

and the time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}
[4,5].

The displacement of a material point is given by the difference between its position at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}

and its original position, so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {u} ({}^0 {x} ,t) = \boldsymbol{\phi } ({}^0 {x} ,t) - \boldsymbol{\phi }({}^0 {x} ,{}^0 t) = {}^t{x} - {}^0 {x}

(3)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u} = [u_1,u_2,u_3]^T}

is the displacement vector of a point.

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t = {}^n t}

we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^n{u} \equiv {u} ({}^0{x} ,{}^n t) = {}^n {x} - {}^0 {x}

(4)

The velocity vector is the rate of change of the position vector for a material point, i.e. the time derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi ({}^0 {x} ,t)}

 with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^0 {x}}
held constant (also called the material time derivative or total derivative). The velocity vector is usually written as (using Eq.(3) and noting that the coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^0 {x}}
are fixed)

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(5)

The velocity vector of a material point in the current configuration is written as

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(6)

The acceleration vector is the rate of change of the velocity vector of a material point (i.e. the material time derivative of the velocity vector) and it can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^t {a} = {a} ({}^0{x} ,t) \equiv \frac{\partial {v}({}^0 {x} ,t) }{\partial t}= \frac{\partial ^2 {u} ({}^0 {x} ,t) }{\partial t^2} = \ddot {u}

(7)

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^n{a} =[{}^n a_1,{}^n a_2,{}^n a_3]^T = {a} ({}^0{x} ,{}^n t)

(8)

Eq.(7) is the material form of the acceleration. Note the difference of Eq.(7) with the expression of the acceleration written in the Eulerian description which involves the convective terms [2,4,5,13,36,41].

In the total Lagrangian description the various equations and variables are referred to the initial configuration which is taken as the reference configuration. In the updated Lagrangian description, either the current or the updated configurations can be taken as the reference configuration [2,4,5,13,36].

For simplicity, in the first part of this work the current configuration will be taken as the reference configuration for the derivation of the incremental FEM equations. The reason is that the current configuration remain fixed during the iterative solution process. The particularization for the case when the updated configuration is taken as the reference configuration is presented in the last part of the paper.

The displacement increment vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta {u}({}^0 {x} ,t) = [\Delta u_1,\Delta u_2,\Delta u_3]^T}

 of a material point between the updated and the current configurations is defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta {u} \equiv \Delta {u}({}^0 {x} ,t) = \boldsymbol{\phi } ({}^0 {x} , {}^{n+1}t) - \boldsymbol{\phi }({}^0 {x} ,{}^n t) = {}^{n+1}{x} - {}^n {x}

(9)

The displacement increment of a material point can be computed from the velocity as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta {u} = \int _{{}^n t}^{{}^{n+1}t} {}^\tau {v}d\tau

(10)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^\tau {v}}

is the velocity vector of the points laying on the trajectory followed by the material point between times Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^n t}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}t}
(Figure 1). The integral of Eq.(10) can be approximated in a number of ways (see Remark 4).

3 MOMENTUM EQUATIONS AND BOUNDARY CONDITIONS

Let us assume that at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} t}

the fluid occupies a volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} V}
with a boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\Gamma }

. The fluid is subjected to body forces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}b_i}

acting over the volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} V}
and surface tractions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} t_i^p}
acting on the part of the boundary termed as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\Gamma _t}

, where index Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

denotes the component of the force along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

th cartesian axis (Figure 1).

The equations of internal equilibrium between the applied body forces and the Cauchy stresses Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{ij}}

in the fluid are expressed by the momentum equations written in the updated configuration as [2,4,5,13,36]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} r_{m_i} :={}^{n+1} \rho ~ {}^{n+1} a_i-{\partial \sigma _{ij} \over \partial x_j}- {}^{n+1}b_i=0 \quad \hbox{in }{}^{n+1} V\quad ,\quad i,j = 1,\cdots ,n_s

(11)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} r_{m_i}}

is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

th momentum equation, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \rho } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} v_i}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} a_i}
are the fluid density and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

th component of the velocity and the acceleration at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} t}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_s}
is the number of space dimensions (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_s=3}
for three dimensional (3D) problems). Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{ij}}
is always referred to the updated configuration, i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{ij} \equiv {}^{n+1}\sigma _{ij}}

.

In Eq.(11) and in the following, the standard summation convention for repeated indices is adopted, unless otherwise specified.

The boundary conditions are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} v_i-{}^{n+1} v_i^p =0 \quad \hbox{on }{}^{n+1} \Gamma _v

(12)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma _{ij}n_j - {}^{n+1} t_i^p =0 \quad \hbox{on }{}^{n+1} \Gamma _t

(13)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}v_i^p}

 is the prescribed value of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

th velocity component at the external boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \Gamma _v}

 with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \Gamma _v \cup {}^{n+1} \Gamma _t = {}^{n+1} \Gamma }

. In Eq.(13) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_j}

is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j}

th component of the unit normal to the boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\Gamma _t}

where the prescribed surface tractions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} t_i^p}
are applied.

The goal is to obtain the geometry of the updated configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}V} , as well as the velocities and stresses in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}V}

that satisfy Eqs.(11)–(13).

4 PRINCIPLE OF VIRTUAL POWER IN THE UPDATED CONFIGURATION

The principle of virtual power (PVP) can be written in the updated configuration as [2,4,5]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} \delta A + {}^{n+1} \delta U - {}^{n+1}\delta W=0

(14)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \delta A} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\delta U}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\delta W}
are the virtual powers due to the acceleration terms, the stresses and the external loads acting on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}V}

, respectively given by [2,4,5]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1}\delta A = \int _{{}^{n+1}V}\delta{v}^T {~}^{n+1}\rho ~ {}^{n+1} {a} ~ d ~{}^{n+1}V	(15)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1}\delta U = \int _{{}^{n+1}V}\left\{\delta {D}\right\}^T \left\{{\boldsymbol \sigma }\right\}d ~ {}^{n+1}V	(16)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1}\delta W = \int _{{}^{n+1}V} \delta{v}^T {~}^{n+1} {b} ~ d{~}^{n+1}V + \int _{{}^{n+1}\Gamma _t} \delta{v}^T {~}^{n+1}{t}~ d ~ {}^{n+1}\Gamma	(17)

In Eq.(15)–(16) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta{v}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {b}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {t}}
are the virtual velocity vector, the body forces vector and the surface tractions vector, respectively defined for 3D problems as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta{v} = [\delta v_1,\delta v_2,\delta v_3]^T ~~,~~ {b}= [b_1,b_2,b_3]^T ~~,~~ {t}= [t_1,t_2,t_3]^T

(18)

In Eq.(16) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \sigma }}

is the Cauchy stress tensor and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta {D}}
is the virtual rate of deformation tensor defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta {D}_{ij}= \frac{1}{2} \left({\partial \delta v_i \over \partial {~}^{n+1} x_j} + {\partial \delta v_j \over \partial {~}^{n+1} x_i}\right)

(19)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta v_i}

is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

th component of the virtual velocity.

In the following Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left\{{A} \right\}} , where A is a symmetric tensor, will denote the vector representation of A. Hence, in Eq.(16) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left\{{\boldsymbol \sigma }\right\}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left\{\delta {D}\right\}}
are the Cauchy stress vector and the rate of deformation vector, respectively defined in Voigt notation [4,5] as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{{\boldsymbol \sigma }\right\}= \left[\sigma _{11},\sigma _{22},\sigma _{33},\sigma _{12},\sigma _{13},\sigma _{23} \right]^T	(20.a)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{\delta {D}\right\}= \left[\delta D_{11},\delta D_{22}, \delta D_{33},\delta D_{12},\delta D_{13},\delta D_{23} \right]^T	(20.b)

Similarly, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[{A} \right]}

will denote hereonwards the matrix form of tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A}

.

Remark 1. The PVP can be obtained by applying the standard weighted residual method to the governing equations (11) and (13) using the virtual velocities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta v_i}

as weighting functions [4].

Remark 2. Tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta {D}}

can be interpreted as the variation of the rate of deformation tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D}
defined in terms of the velocities at the updated configuration as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {D}_{ij} = \frac{1}{2} \left({\partial {~}^{n+1} v_i \over \partial {~}^{n+1}x_j}+{\partial {~}^{n+1} v_j \over \partial {~}^{n+1}x_i} \right)

(21)

5 TRANSFORMATION TO THE CURRENT CONFIGURATION. LAGRANGIAN STRESS AND STRAIN MEASURES

In the following sections we will use the current configuration as the reference configuration for computing the internal virtual due to the acceleration terms and the stresses, as well as for subsequently performing the linearization of its discretized form via the FEM. The alternative of using the updated configuration as the reference configuration is presented in Section 12.

After the standard transformations of continuum mechanics [2,4,5,13,36] the internal virtual power at the updated configuration due to the acceleration terms and the stresses can be expressed in terms of material parameters, integrals, strain rates and stress measures evaluated at the current configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n}V}

as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1}\delta A = \int _{{}^{n}V} \delta{v}^T {~}^n \rho {~}^{n+1} {a} ~d {~}^{n}V

(22.a)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1}\delta U = \int _{{}^{n}V} \left\{\delta \dot{E} \right\}^T \left\{{S}\right\}d {}^{n}V

(22.b)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \dot{E}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {S}}
are the material virtual strain rate tensor and the second Piola-Kirchhoff stress tensor, respectively. The relationship between the material strain rate tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{E}}
and the rate of deformation tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {D}}
and between the stress tensors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \sigma }}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {S}}
is [2,4,5,13,36]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot{E}= {F}^T {D}{F} \quad , \quad {S} =J {F}^{-1} {\boldsymbol \sigma } {F}^{-T}

(23)

where F is the deformation gradient and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J}

is the Jacobian, respectively defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): F_{ij}= \frac{\partial {~}^{n+1}{x}_i}{\partial {~}^n {x}_j}\quad ,\quad J=\vert {F}\vert

(24)

From the expression of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{E}}

of Eq.(23) we can deduce

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \dot{E}= {F}^T \delta {D}{F}

(25)

Remark 3. The material strain rate tensor can also be obtained from the time derivative of the Green-Lagrange strain tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {E}}

as [2,4,5]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot{E} = \frac{d}{dt}({E}) = \frac{d}{dt} \left(\frac{1}{2}\left({C}-{I} \right)\right)=\frac{1}{2} \frac{d}{dt}{C}= \frac{1}{2} \frac{d}{dt} ({F}^T {F}) = \frac{1}{2} ({F}^T \dot {F} + \dot {F}^T {F} )

(26)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {C}= {F}^T {F}}

 is the right Cauchy-Green tensor and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot {F} = \frac{d}{dt}({F})}

. The expression of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \dot{E}}

 is obtained by writing the variation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \dot{E}}
with respect to the velocities as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \dot{E} =\frac{1}{2} \left({F}^T \delta \dot {F}+ \delta \dot {F}^T {F} \right)

(27)

A useful explicit expression of the material strain rate tensor components in terms of the velocities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}v_i}

and the displacement increments Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta u_i}
is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot E_{ij}= \frac{1}{2} \left({}^{n+1}_n v_{i,j} +{}^{n+1}_n v_{j,i} + {}^{n+1}_n v_{k,i} ~{}_n \Delta u_{k,j} + {}^{n+1}_n v_{k,j} ~{}_n \Delta u_{k,i} \right)

(28.a)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1}_n v_{i,j} = {\partial {~}^{n+1} v_i \over \partial {~}^n x_j}\quad ,\quad ~{}_n \Delta u_{i,j} = {\partial \Delta u_i \over \partial {~}^n x_j}

(28.b)

From Eqs.(5) we deduce

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \dot{E}_{ij}= \frac{1}{2} \left({}_n \delta v_{i,j}~ + {}_n \delta v_{j,i}+{}_n \delta v_{k,i}~ {}_n \Delta u_ {k,j}+ {}_n \delta v_{k,j} ~{}_n \Delta u_{k,i} \right)

(29.a)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}_n \delta v_{i,j} = \frac{\partial \delta v_i}{\partial {}^n x_j}

(29.b)

6 SPLIT OF THE INTERNAL VIRTUAL POWER INTO DEVIATORIC AND VOLUMETRIC COMPONENTS

The Cauchy stress tensor can be split in its deviatoric component Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol \sigma }'}

and the hydrostatic pressure component Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}
as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol \sigma } ={\boldsymbol \sigma }' + p {I}_3

(30)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {I}_3}

 is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3\times 3}
identity matrix.

Note that in incompressible fluid mechanics the pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}

is an independent variable. Also, unless otherwise specified we will assume that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}
is the pressure at the updated configuration (i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p= {~}^{n+1}p}

).

Substituting Eq.(30) into the internal virtual power expression in Eq.(16) gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} \delta U = \int _{{}^{n+1}V}\left\{\delta {D}\right\}^T \left(\left\{{\boldsymbol \sigma }'\right\}+{m} p \right)d{~}^{n+1}V = \left(\int _{{}^{n+1}V}\left\{\delta {D}\right\}^T \left\{{\boldsymbol \sigma }'\right\}+ \delta D_v p \right)d{~}^{n+1}V

(31)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_v}

is the volumetric strain rate given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): D_v = D_{ii}= {m}^T \left\{{D}\right\}\quad \hbox{with}\quad {m}= [1,1,1,0,0,0]^T

(32)

It can be proven that [4,5,13,36]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): D_v = \dot E_v \quad \hbox{with}\quad \dot E_v= \left\{\dot {E}\right\}^T \left\{{C}^{-1} \right\}

(33)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot E_v}

is the volumetric material strain rate,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{{E}' \right\}= \left[\dot E_{11}, \dot E_{22},\dot E_{33},2\dot E_{12},2\dot E_{13},2\dot E_{23} \right]^T

(34.a)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{{C}^{-1} \right\}=\left[C^{-1}_{11}, C^{-1}_{22},C^{-1}_{33},C^{-1}_{12},C^{-1}_{13},C^{-1}_{23} \right]^T

(34.b)

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_{ij}^{-1}}

is the element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ij}
of tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {C}^{-1}}

.

The internal virtual power expression of Eq.(31) can be written in the current configuration as follows.

Substituting the Cauchy stress split of Eq.(30) into the expression of the second Piola-Kirchhoff stress tensor of Eq.(23) gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {S} = J {F}^{-1} {\boldsymbol \sigma }' {F}^{-T} + p J {C}^{-1} = {S}' + p J {C}^{-1}

(35)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {S}' = J {F}^{-1} {\boldsymbol \sigma }' {F}^{-T}

(36)

is the deviatoric second Piola-Kirchhoff stress tensor. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {S}'}

is usually called the (true) deviatoric component of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S}
[4,5,13,36].

Substituting Eq.(35) into (22.b) gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1}\delta U = \int _{{}^{n}V} \left(\left\{\delta \dot {E}\right\}^T \left\{{S}'\right\}+ J\delta \dot E_v p \right)d{}^{n}V

(37)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{{S}'\right\}{= [S'}_{11}{,S'}_{22}{,S'}_{33}{,S'}_{12}{,S'}_{13}{,S'}_{23}]^T\quad , \quad \delta \dot E_v = \left\{\delta \dot {E}\right\}^T \left\{{C}^{-1}\right\}

(38)

Eq.(37) can also be obtained from Eq.(31) using the relationship between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {D},~{\boldsymbol \sigma }'}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_v}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot {E}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {S}'}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot E_v}

, respectively and the expression Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d^{n+1}V =Jd^nV}

[2,4,5].

Substituting Eqs.(15), (22.a) and (37) into (14), the PVP in the updated configuration can be written in terms of the pressure, the deviatoric second Piola-Kirchhoff stresses and the virtual Green-Lagrange strains computed in the current configuration as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{r} \displaystyle \int _{{}^{n}V} \delta{v}^T {~}^n \rho {~}^{n+1} {a} d {~}^{n}V +\int _{{}^{n}V} \left(\left\{\delta \dot {E}\right\}^T \left\{{S}'\right\}+ J\delta \dot E_v p \right)d{~}^{n}V -\\ - \displaystyle \int _{{}^{n+1}V} \delta{v}^T {~}^{n+1}{b}~ d{~}^{n+1} V - \int _{{}^{n+1}\Gamma _t} \delta{v}^T {~}^{n+1} {t}~d {~}^{n+1} \Gamma = 0\end{array}

(39)

The vector form of tensors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \dot {E}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {S}}
in Eq.(39) is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{\delta \dot {E}\right\}= [\delta \dot E_{11}, \delta \dot E_{22},\delta \dot E_{33},2\delta \dot E_{12},2\delta \dot E_{13},2\delta \dot E_{23}]^T

(40.a)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{{S}\right\}= [S_{11},S_{22},S_{33},S_{12},S_{13},S_{23}]^T

(40.b)

Note that the contribution of the external forces in Eq.(39) is computed at the updated configuration and this requires using the correct expression for the density and the surface tractions on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}V}

(see also Remark 5).

7 CONSTITUTIVE RELATIONSHIP FOR THE DEVIATORIC STRESSES

The relationship between the deviatoric Cauchy stresses Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sigma '}_{ij}}

and the rates of deformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_{ij}}
for a Newtonian fluid is written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\sigma '}_{ij} ={c}_{ijkl}{ D'}_{kl} = {c}_{ijkl} \left(D_{kl}- \frac{1}{3} D_v \delta _{kl}\right)

(41)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {D'}_{kl}}

are the deviatoric rates of deformation. The rates of deformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_{ij}}
are obtained in terms of the velocities by Eq.(21).

The components of the fourth order constitutive tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {c}}

in the updated configuration are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {c}_{ijkl} =\mu \left(\delta _{ik} \delta _{jl} + \delta _{il} \delta _{jk}\right)

(42)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu }

is the viscosity of the fluid.

Eq.(41) can be written in vector form as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{{\boldsymbol \sigma }'\right\}= \left[{c}\right]\left\{{D}\right\}

(43)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l}\left\{{\boldsymbol \sigma }'\right\}= \left[\sigma' _{11}, \sigma' _{22},\sigma' _{33},\sigma' _{12},\sigma' _{13},\sigma' _{23} \right]^T\\ \left\{{D}\right\}= \left[D_{11}, D_{22},D_{33},2D_{12},2D_{13},2D_{23} \right]^T \end{array}

(44)

and the constitutive matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [{c}]}

is given by (for 3D problems)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[{c}\right]=\mu \left[\displaystyle \begin{matrix}\displaystyle \frac{4}{3} & \displaystyle -\frac{2}{3} & \displaystyle -\frac{2}{3} &0&0&0\\[.25cm] & \displaystyle \frac{4}{3} & \displaystyle -\frac{2}{3} &0&0&0\\[.25cm] & &\displaystyle \frac{4}{3} &0&0&0\\ \hbox{ Sym.} &&& 1 &0&0\\ &&&&1&0\\ &&&&&1 \end{matrix} \right]

(45)

The constitutive equation (41) can be transformed to the current configuration to yield the relationship between the deviatoric second Piola-Kirchhoff stresses and the material strain rates as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {S'}_{ij} = {\cal C}_{ijkl} \dot E_{kl}

(46)

The constitutive tensor in the current configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[\boldsymbol{\cal C}\right]}

is obtained from its counterpart in the updated configuration as [36]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\cal C}_{ijkl} = F^{-1}_{Ai} F^{-1}_{Bj}F^{-1}_{Ck}F^{-1}_{Dl} c_{ABCD}

(47)

The vector form of Eq.(46) is written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{\mathbf{S}' \right\}= \left[\boldsymbol{\cal C}\right]\left\{\dot {E}\right\}

(48)

Matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[\boldsymbol{\cal C}\right]}

is obtained by applying Voigt rule to the terms of tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}}
[4,5].

8 THE MASS BALANCE EQUATION

The mass balance equation in the updated configuration is written for a quasi-incompressible fluid as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} r_v:= - \frac{1}{{}^{n+1}\rho c^2}\frac{\partial p}{\partial t}+D_v =0 \quad \hbox{in } {}^{n+1}V

(49)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c}

is the speed of sound in the fluid. For a fully incompressible fluid Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c=\infty }
and Eq.(49) reduces to the standard form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_v =0}

. The quasi-incompressible form will be retained here and this will allow us to account for the effect of the (small) compressibility in most fluids.

Eq.(49) can be written in terms of the bulk modulus of the fluid Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \kappa }

defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \kappa = \rho c^2}

. For convenience we will retain the form of Eq.(49).

The variational form of the mass balance equation is obtained via the standard weighted residual method [41] as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int _{{}^{n+1}V} q \left(- \frac{1}{{}^{n+1}\rho c^2}\frac{\partial p}{\partial t}+D_v \right)d {~}^{n+1}V=0

(50)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q}

are appropriate  test functions with dimensions of pressure [4,5,41].

The integral expression (50) can be written in the current configuration using Eq.(33) as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle \int _{{}^{n}V} q \left(- \frac{J^2}{{}^{n}\rho c^2}\frac{\partial p}{\partial t}+J\dot E_v \right)d{~}^{n}V=0

(51)

In the derivation of Eq.(51) we have used the standard expressions [4,5]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n}\rho = {}^{n+1}\rho J \quad \hbox{and}\quad d {~}^{n+1}V = J d {~}^{n}V

(52)

Eqs.(39) and (51) together with the constitutive relationship (47) and the boundary conditions (12) complete the set of governing variational equations for a fluid in the updated Lagrangian description. The solution of these equations with the FEM is described in the next section.

9 FINITE ELEMENT DISCRETIZATION

We interpolate the velocities and the pressure in terms of their nodal values in the standard finite element fashion [28,39,41]. For a mesh of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} -noded Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C^\circ }

continuous elements we can write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} {v}= {N}_v {}^{n+1} \bar{v}\quad ,\quad p = {N}_p {}^{n+1}\bar{p}

(53)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \bar{v} = \left\{\begin{matrix}\bar{v}^1\\ \bar{v}^2\\\vdots \\ \bar{v}^N \end{matrix} \right\}\quad \hbox{with } \bar{v}^i = [\bar v^i_1,\bar v^i_2, \bar v^i_3]^T\quad ,\quad \bar{p} = [\bar p_1,\bar p_2,\cdots ,\bar p_N]^T

(54.a)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}

is the total number of nodes in the mesh, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar v^i_j}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{p}^i}
are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j}

th velocity component and the pressure unknowns for node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l}{N}_v = [{N}^v_1,{N}^v_2,\cdots ,{N}^v_N ]\quad ,\quad {N}^v_i= N^v_i{I}_3\\ {N}_p = [{N}^p_1,{N}^p_2,\cdots ,{N}^p_N ]\end{array}

(54.b)

In Eq.(54.b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {N}^v_i}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N^p_i}
are the global  shape functions [28,39] for the velocity and pressure interpolations for node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

, respectively.

The velocity and pressure increments and the virtual velocities are interpolated in terms of their nodal values in the same fashion as in Eq.(53).

The strain rate vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left\{\dot {E}\right\}}

and its virtual expression Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left\{\delta \dot {E}\right\}}
are respectively expressed in terms of the nodal velocities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{v}}
and their virtual values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \bar{v}}
via Eqs.(28.a), (29.a) and (53) as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{\dot {E}\right\}= {B} {}^{n+1} \bar{v} \quad ,\quad \left\{\delta \dot {E}\right\}= {B} \delta \bar {v}

(55)

The actual and virtual volumetric material strain rates are related to the virtual velocities as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot {E}_v = \left\{{C}^{-T}\right\}{B}{~}^{n+1} \bar{v} \quad ,\quad \delta \dot {E}_v =\delta \bar{v}^T {B}^T\left\{{C}^{-1}\right\}

(56)

In the above equations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {B}}

is the material strain rate matrix given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {B} = [{B}_1,{B}_2,\cdots , {B}_N]^T \quad , \quad {B}_i = {}_n{B}_i^0+{B}_i^L

(57)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l}{}_n{B}_i^0 = \left[ \begin{matrix}{}_nN^v_{i,1} &0&0\\ 0&{}_nN^v_{i,2} &0\\ 0&0&{}_nN^v_{i,3}\\ {}_nN^v_{i,2} &{}_nN^v_{i,1} &0\\ {}_nN^v_{i,3}&0&{}_nN^v_{i,1}\\ 0&{}_nN^v_{i,3}&{}_nN^v_{i,2} \end{matrix}\right]~~\hbox{and}~~ {B}_i^L ={}_n{B}_i^0 {L}^T ~~\hbox{with}~~ {L}= \left[\begin{matrix}l_{11} & l_{12} & l_{13}\\ l_{21} & l_{22} & l_{23}\\ l_{31} & l_{32} & l_{33} \end{matrix}\right] \end{array}

(58.a)

are the linear and non linear counterparts of the material strain rate matrix, respectively. The components of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}_n{B}_i^0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {L}}
are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}_nN^v_{i,j}= {\partial N^v_i \over \partial {}^n x_j} \quad , \quad l_{ij} =\sum \limits _{k=1}^n {}_nN^v_{k,j} \Delta u_k

(58.b)

The deviatoric second Piola-Kirchhoff stresses are related to the nodal velocities via Eqs.(47) and (55) as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{\mathbf{S}' \right\}=\left[\boldsymbol{\cal C}\right]\left\{\dot {E}\right\}= \left[\boldsymbol{\cal C}\right]{B} {~}^{n+1} \bar{v}

(59)

Remark 4. The displacement increment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta u_i}

can be computed in terms of the velocity in a  number of ways. A popular choice is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta u_i = \Delta t {~}^{n+\alpha } v_i

(60.a)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {~}^{n+\alpha } v_i }

is the velocity at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=t_n + \alpha \Delta t}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha }
is a positive number (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\le \alpha \le 1}

). The value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {~}^{n+\alpha } v_i }

is typically computed as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {~}^{n+\alpha } v_i = (1-\alpha )^n v_i + \alpha {}^{n+1} v_i

(60.b)

9.1 Discretized form of the PVP

Substituting Eqs.(53), (55) and (56) into the PVP (Eq.(39)) we obtain, after simplifying the virtual velocities

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} \bar {r}_m := \int _{{}^{n}V} {}^n\rho {N}_v^T {N}_v {\dot{\bar {v}}}d {~}^{n}V +\int _{{}^{n}V} {B}^T \left[\left\{\mathbf{S}' \right\}+ J \left\{{C}^{-1}\right\}{p} \right]d {~}^{n}V -

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int _{{}^{n+1}V} {N}_v^T {~}^{n+1}{b} ~ d {~}^{n+1} V - \int _{{}^{n+1}\Gamma } {N}_v^T {~}^{n+1} {t}~d{~}^{n+1}\Gamma ={0}

(61)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \bar {r}_m}

is the residual vector of the discretized momentum equations in the updated configuration and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  {\dot{\bar {v}}} \equiv {\partial {}^{n+1} \bar {v} \over \partial t}}
is the nodal acceleration vector.

Eq.(61) can be written in a more compact form as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle {}^{n+1} \bar {r}_m := {M}_v {\dot{\bar{v}}} + {}^{n+1} {g}_v + {}^{n+1} {g}_p - {}^{n+1} {f}_m

(62)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {M}_v = \!\!\int _{{}^{n}V} {}^n\rho {N}_v^T {N}_v d ~{}^nV ~ , ~ {}^{n+1}{g}_v = \!\!\int _{{}^{n}V} {B}^T {S}' d {~}^{n} V ~,~ {}^{n+1}{g}_p= \!\!\int _{{}^{n}V} {B}^T J \left\{{C}^{-1}\right\}p d {~}^{n} V

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} {f}_m = \!\!\int _{{}^{n+1}V} {N}_v^T {~}^{n+1}~{b} ~d{~}^{n+1} V + \!\!\int _{{}^{n+1}\Gamma } {N}_v^T {~}^{n+1}{t}~ d{~}^{n+1} V

(63)

In Eq.(62) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} {g}_v}

 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}{g}_p}
are internal force vectors contributed by the deviatoric second Piola-Kirchhoff stresses and the pressure, respectively and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} {f}_m}
is the external force vector.

Recall that the internal force vectors at the current configuration are obtained in terms of expressions at the updated configuration.

Remark 5. The computation of the body force vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}{b}}

in Eq.(63) for the case of selfweight requires computing the density at the updated configuration as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\rho = \frac{1}{J}{}^n\rho }

. We also note that the surface tractions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}{t}}

are applied on the boundary of the updated configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\Gamma }

, which requires the accurate identification of this boundary.

The acceleration term in Eq.(62) can be approximated in a number of manners. A backward Euler scheme gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} \bar {r}_m := {M}_v \frac{{}^{n+1} \bar{v}-{}^{n} \bar{v} }{\Delta t}+ {}^{n+1} {g}_v + {}^{n+1} {g}_p - {}^{n+1} {f}_m =0

(64)

9.2 Discretization of the mass conservation equation

The arbitrary pressure test functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q}

are interpolated in the same fashion as the pressure as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): q= {N}_p \bar{q}= \bar {q}^T{N}_p^T

(65)

where vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{q}}

contains the nodal values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q}

.

Substituting the expression of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot {E}_v}

in term of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \bar{v}}
from Eq.(56) and Eq.(65) in the variational form of the mass balance equation (Eq.(51)) we obtain, after eliminating the pressure test functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{q}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle {}^{n+1} \bar {r}_v : = - {M}_p {\dot{\bar{p}}} + {Q}^T {~}^{n+1} \bar {v}={0}

(66)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \bar {r}_v}

is the residual vector of the discretized mass conservation equation, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  {\dot{\bar {p}}} = {\partial {~}^{n+1} \bar {p} \over \partial t}}
and the terms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_p}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n}{Q}}
are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): M_{p_{ij}}=\int _{{}^{n}V} \frac{J^2}{^n\rho c^2}N^p_i N^p_j d{~}^{n}V ~~,~~{Q}= \int _{{}^{n}V} {B}^T \left\{{C}^{-1}\right\}{N}_p J d {~}^{n} V

(67)

9.3 Stabilization of the mass conservation equation

It is well known that the FEM solutions for the fully incompressible limit are unstable for some particular forms of the approximation for the velocities and the pressure [10,39,41]. This is the case, for instance, when an equal order interpolation is chosen for the velocity components and the pressure (i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N^v_i = N^p_i} ). This problem can be overcome by using finite element approximations for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {v}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}
satisfying the so-called LBB conditions [10,39,41], or else by introducing stabilization terms into the discretized mass balance equation [10,41]. In this work the later approach is chosen for obtaining stabilized numerical solutions.

In essence all stabilized formulations modify the discretized form of the mass balance equation as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle {}^{n+1} \hat {r}_v := {}^{n+1} \bar {r}_v + ^{n+1} \bar {r}_s =0

(68)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \bar {r}_v}

was defined in Eq.(66)  and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \bar {r}_s}
is a stabilization mass balance vector which expression can be typically written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} \bar {r}_s = - \boldsymbol{\cal S} {~}^{n+1} \bar {p}+ {}^{n+1} {f}_s

(69)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal S}}

is a stabilization matrix and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}{f}_s}
is a force vector that depends on the nodal velocities and pressures. The form of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal S}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {f}_s}
is different for each stabilization method. Typically, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal S}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}{f}_s}
contain integrals  over the volume and the Neumann boundary of the analysis domain and are both a function of the so-called stabilization parameters which expression also depends on the particular stabilization method chosen [3,8,9,10,15,31,37,41].

The derivation of a stabilized mixed velocity-pressure formulation for incompressible fluids exceeds the objectives of this paper. Details can be obtained in the references cited in the previous paragraphs.

A particular stabilized Lagrangian formulation with excellent mass conservation features based on the Finite Calculus (FIC) theory [21]–[24],[27,30] can be found in [31].

10 INCREMENTAL SOLUTION FOR THE NODAL VELOCITIES AND PRESSURES

We will derive next an incremental iterative procedure for solving the discretized form of the equations for conservation of linear momentum and mass conservation in a segregated form. This requires the linearization of Eqs.(61) and (68) with respect to the nodal velocity and pressure unknowns, respectively. The linearization procedure takes advantage from the fact that the reference configuration (i.e. the current configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n}V} ) remains fixed during the linearization process.

10.1 Linearization of the discretized momentum equations with respect to the nodal velocities

We linearize the residual vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \bar {r}_m}

of Eq.(61) with respect to the nodal velocities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {v}}
as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _{\bar v} {}^{n+1} \bar {r}_m ({x},\bar {v},\bar {p}) = \frac{d}{d\epsilon }\Bigg|_{\epsilon =0} {}^{n+1} \bar {r}_m ({x},\bar {v}+\epsilon d\bar{v},\bar {p})

(70)

Expression (70) is the directional derivative of the residual vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \bar {r}_m}

at a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {x}}
in the direction of the nodal velocity vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d\bar {v}}
(hereafter called nodal velocity pseudo-increment vector). The same definition applies to the directional derivative of a matrix or a scalar depending on the space coordinate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {x}}

, the nodal velocities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {v}}

and the nodal pressures Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar {p}}
[4,5,36].

Using the expression of vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \bar {r}_m}

of Eq.(62) in Eq.(70) and neglecting the changes of vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} {f}_m}
with the  velocity,  gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _{\bar v} {}^{n+1} \bar {r}_m = \frac{1}{\Delta t}{M}_v d \bar {v}+ \int _{{}^{n}V} \Big\{{B}^T \Big[\delta _{\bar v}\left\{\mathbf{S}' \right\}+ \left(\delta _{\bar v} J\left\{{C}^{-1}\right\}+ {J} \delta _{\bar v} \left\{{C}^{-1}\right\}\right)p +

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): + J \left\{{C}^{-1}\right\}\delta _{\bar v}p\Big]+ \delta _{\bar v} {B}^T \left\{\mathbf{S}' \right\}\Big\}d{~}^{n} V

(71)

After some algebra detailed in the Appendix, the following linearized form of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\bar{r}_m}

with respect to the nodal velocities is found

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _v {}^{n+1}\bar{r}_m = \left(\frac{1}{\Delta t}{M}_v + {K}_c + {K}_\sigma \right) d \bar {v}+ {Q} {\delta }_{\bar v}^{n+1}\bar {p}

(72)

where matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_v}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n} {Q}}
were given in Eq.(67) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^n{K}_c}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_\sigma }
are the constitutive and initial stress matrices, respectively. The expression of these matrices is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {K}_c = \!\!\int _{{}^{n}V}{B}^T [\boldsymbol{\cal C}_T] {B} d {~}^{n}V ~~ ,~~ {K}_\sigma = \Delta t \!\int _{{}^{n}V}{G}^T \hat{S}' {G}d {~}^{n}V

(73)

The form of matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {G}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{S}'}
is given in the Appendix.

The expression of the tangent constitutive tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}_T}

is (see Appendix)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{\cal C}_T = \boldsymbol{\cal C} + J \Delta t ~p ({C}^{-1}\otimes {C}^{-1} - 2 \boldsymbol{\cal I})

(74)

Tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}_T}

contains contributions from the  constitutive tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}}
of Eq.(46), the time step and  the pressure. This form of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}_T}
is very similar to that used for incompressible continua [4,5,13,36].  It can be shown that tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal I}}
is symmetric (see Appendix) and, hence, the tangent constitutive tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}_T}
is symmetric [4,5,13,36].

10.2 Linearization of the nodal pressures with respect to the nodal velocities. Derivation of the tangent matrix

From Eq.(66) we can obtain the directional derivative of the nodal pressure variables in the direction of the nodal velocity increments. Using a trapezoidal rule for approximating the time derivative term gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {M}_p\frac{1}{\theta \Delta t} ({~}^{n+1}\bar {p} - {}^{n+\theta }\bar {p}) + {Q}^T \bar {v} = {0}

(75)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta }

is a time parameter such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0<\theta \le 1}

. A value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta =1}

defines a backward Euler scheme [39,41].

From Eq.(75) it is straightforward to obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\delta }_{\bar v} {~}^{n+1}\bar {p}= \theta \Delta t {M}_p^{-1} {Q}^T d \bar {v}

(76)

Substituting Eq.(76) into (72) gives the final linearized form of the momentum equations in terms of the nodal velocities increments as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\delta }_{\bar v} {}^{n+1}\bar {r}_m = {K}_T^i d \bar {v}

(77)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\cdot )^i}

denotes values at the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

th iteration and the tangent matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^n {K}_T}

is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle {K}_T = \frac{1}{\Delta t} {M}_v + {K}_c + {K}_\sigma + {K}_p

(78)

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_v} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_c}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_\sigma }
 defined in Eqs.(63) and (72) and the tangent “bulk” matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_p}
is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {K}_p = \theta \Delta t {Q} {M}_p^{-1} {Q}^T

(79)

In practice, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_p}

is computed using the diagonal form of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_p}

, i.e.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {K}_p = \theta \Delta t {Q} {M}_{pD}^{-1} \bar{Q}^T

(80)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_{pD} ={diag}({M}_p)} .

The incremental form of the momentum equations can written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1}\bar {r}_{m} \simeq {}^{n+1}\bar {r}_{m}^i + {\delta }_{\bar v} {~}^{n+1}\bar {r}_m = {}^{n+1}\bar {r}_m^i + {K}_T^i d \bar {v} =0

(81)

Solution of Eq.(81) yields the nodal velocity increments Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d \bar {v}}

for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

th iteration.

Remark 6. For fully incompressible fluids (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c=\infty } ) a large but finite value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c}

is used in practice. This allows to eliminate the pressure DOFs in the momentum equations via Eq.((76)).

Remark 7. The tangent “bulk” stiffness matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{K}_p}

in the tangent matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{K}_T}
 accounts for the changes of the pressure due to the velocity. Including matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{K}_p}
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{K}_T}
has proven to be essential for the fast convergence, mass preservation and overall accuracy of the iterative solution in all cases [11].      The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta }
parameter in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_p}
(Eq.(79))  has the role of preventing the ill-conditioning of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{K}_T}
for very large values of the speed of sound in the fluid that lead to a dominant role of the terms of the tangent bulk stiffness matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_p}

. Clearly, the value of the terms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_p}

can also be limited by reducing the time step size. This, however, leads to an increase in the overall cost of the computations [11]. An adequate selection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta }
 improves the convergence of the iterative process and leads to a more accurate numerical solution with reduced mass loss, even for large time steps [11]. These considerations, however, do not affect the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c}
within matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_p}
in Eq.(67) that vanishes for the fully incompressible case (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c =\infty }

).

10.3 Linearization of the stabilized mass conservation equation with respect to the nodal pressures

The stabilized mass conservation equation (68) is linearized with respect to the nodal pressure pseudo-increment vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d \bar {p}}

using Eqs.(66), (68) and (69)  as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _{\bar p} {}^{n+1} \hat {r}_v ({x},\bar {v},\bar {p})= \frac{d}{d\epsilon }\Bigg|_{\epsilon =0} {}^{n+1} \hat{r}_v ({x},\bar {v},\bar {p}+\epsilon d \bar {p})

(82)

Using Eqs.(67)–(68) and a backward Euler scheme for approximating the time derivative of the nodal pressure in Eq.(67) gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _{\bar p} {~}^{n+1} \hat {r}_v = - \left(\frac{1}{\Delta t} {M}_p + \boldsymbol{\cal S}\right)d \bar {p}

(83)

In the derivation of Eq.(83) we have neglected the pressure dependence of the terms of matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal S}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} {f}_s}
in Eq.(69).

The incremental form of the mass balance equation is therefore

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1}{\hat{r}}_v \simeq {}^{n+1}{\hat{r}}_v^i + \delta _{\bar p} {~}^{n+1}{\hat{r}}_v = {}^{n+1}{\hat{r}}_v^i - \left(\frac{1}{\Delta t}{M}_p + \boldsymbol{\cal S}^i \right)d \bar {p}=0

(84)

Solution of Eq.(84) yields the nodal pressure pseudo-increment vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d \bar {p}}

at the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

th iteration.

10.4 Incremental solution of the discretized equations

An incremental Newton-Raphson type iterative solution scheme for the stabilized velocity-pressure formulation is as follows.

Within a time increment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [n,n+1]}

Initialize variables: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ({}^{n+1} {x}, {}^{n+1}\bar {v}^{1}, {}^{n+1}\bar {p}^{1}, {}^{n+1}\bar {r}_m, {}^{n+1}\hat {r}_v) \equiv ({}^{n} {x},{}^{n}\bar {v},{}^{n}\bar {p},{}^{n}\bar {r}_m, {}^{n}\hat{r}_v)}

.

Iteration loop = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1,\cdots , NITER}

For each iteration

Compute the nodal velocity pseudo-increments , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d \bar {v}} (from Eq.(81)) from

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {K}_T^i d \bar {v}= - {}^{n+1}\bar {r}_m^i ({}^{n+1}{\bar{v}}^i, {}^{n+1}{\bar{p}}^i) </li>

(85)

Update the nodal velocities :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^{n+1} \bar {v}^{i+1}=^{n+1} \bar {v}^i + d \bar {v} </li>

(86)

Compute the nodal pressure pseudo-increments , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d\bar {p}} . From Eq.(84),

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left(\frac{1}{\Delta t} {M}_p +\boldsymbol{\cal S}^i\right)d \bar {p} = ^{n+1}\hat {r}_v^i ({}^{n+1} \bar {v}^{i+1},{}^{n+1} \bar {p}^{i}) </li>

(87)

Update the nodal pressures :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^{n+1} \bar {p}^{i+1}= ^{n+1} \bar {p}^{i} + d\bar {p} </li>

(88)

Update the nodal displacement increments Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta {u}} using Eq.(60.a) and the approximate value for the nodal velocities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \bar {v}^{i+1}} .

Update the nodal coordinates in the updated configuration as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {}^{n+1} {x}^{i+1} = {}^{n+1} {x}^i + \Delta {u} </li>

(89)

Compute the material derivative of the Green strains Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot E_{ij} and the deviatoric second Piola-Kirchhoff stresses Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {S'}_{ij} (from Eqs.(55) and (45)).
Compute the momentum and mass balance residuals : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^{n+1}\bar {r}_m^{i+1}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^{n+1}\hat {r}_v^{i+1}}

Check convergence

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l} \Vert ^{n+1}\bar {r}_m^{i+1} \Vert \le e_m \Vert {}^n{f}_v\Vert \\[.5cm] \Vert ^{n+1}\hat {r}_v^{i+1} \Vert \le e_v {~}^{n}V </li> \end{array}

(90)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert \cdot \Vert }

denotes the quadratic norm and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e_m}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e_v}
are prescribed error tolerances for the discretized residuals of the momentum and mass balance equations, respectively.

If both conditions (90) are satisfied then make Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n\leftarrow n+1}

and proceed to the next time increment. Otherwise, make the iteration counter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i\leftarrow i+1}
and repeat Steps 1–9.

Remark 8. An alternative convergence criteria based on the nodal velocities and pressures can be defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert ^{n+1}\bar {v}^{i+1} - {}^{n+1}\bar {v}^i\Vert \le e_{\bar v} \Vert ^{n+1}\bar {v}^i\Vert	(91.a)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert ^{n+1}\bar {p}^{i+1} - ^{n+1}\bar {p}^i\Vert \le e_{\bar p} \Vert ^{n+1}\bar {p}^i\Vert	(91.b)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e_{\bar v}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e_{\bar p}}
are error tolerances.

Remark 9. The nodal velocity and pressure increment vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \bar {v}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \bar {p}}
can be computed at the end of each time step as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \bar {v} = {}^{n+1}\bar {v}- {}^{n}\bar {v}\quad \hbox{and}\quad \Delta \bar {p} = {}^{n+1}\bar {p} - {}^{n}\bar {p}

(92)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\bar {v}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\bar {p}}
denote the converged values at the end of the iteration loop. Clearly, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \bar {v}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \bar {p}}
can also be obtained by adding up the pseudo-increment vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d\bar {v}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d\bar {p}}
within the iterative solution.

Remark 10. The nodal pressures Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^{n+1}\bar {p}^{i+1}}

can be directly obtained from Eq.(68), after substitution of Eqs.(66) and (69), as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left(\frac{1}{\Delta t} {M}_p + \boldsymbol{\cal S}^i \right){}^{n+1} \bar {p}^{i+1}= {Q}^T {~}^{n+1} \bar {v}^{i+1} + \frac{1}{\Delta t} {M}_p {~}^{n} \bar{p} + {}^{n+1} {f}_s

(93)

The rest of the iterative solution scheme is similar to that described above. Eq.(93) substitutes Eqs.(87) and (88) and the convergence of the nodal pressures is verified by Eq.(91.b).

Remark 11. For a fully incompressible fluid Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c = \infty }

and matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_p=0}
in Eqs.(67), (87) and (93).

The problem can still be accurately solved in this case using an adequate expression for the stabilization matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal S}} . The form of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal S}}

presented in [31]  includes a Laplacian term over the whole domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }
and an integral along the Neumann boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _t}

. The boundary term in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal S}}

avoids the need for prescribing the pressure on the domain boundary. If a standard Laplacian form is chosen for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal S}}

, then the value of the pressure has to be prescribed in strong form at some boundary points in order to obtain good results [41].

11 DIRECT ITERATIVE SOLUTION OF THE NODAL VELOCITIES AND PRESSURES

Substituting the FEM approximation for the velocities and the pressure (53) into Eq.(64) and assuming a Newtonian fluid with a constitutive equation given by Eq.(47), the following matrix expression of the discretized momentum equations can be obtained

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle {}^{n+1} \bar {r}_m := {M}_v {\dot{\bar{v}}} + {K}_v {~}^{n+1} \bar{v} + {Q}{~}^{n+1} \bar {p}-{}^{n+1} {f}_m=0

(94)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_v} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n}{Q}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} {f}_m}
have been defined previously and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {K}_v = \int _{{}^{n}V} {B}^T \left[ \boldsymbol{\cal C}\right]{B} d {~}^{n} V

(95)

Combining Eq.(94) with the stabilized mass balance equation (68) and using Eq.(66) gives the following matrix system for the nodal velocities and pressures

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[\begin{matrix}{M}_v & {0} \\ {0}& - {M}_p \end{matrix} \right]\left\{\begin{matrix}{\dot{\bar{v}}}\\ {\dot{\bar{p}}} \end{matrix} \right\}+ \left[\begin{matrix}{K}_v & {Q}\\ {Q}^T & - \boldsymbol{\cal S} \end{matrix} \right]\left\{\begin{matrix}{}^{n+1} \bar{v}\\ {}^{n+1} \bar{p} \end{matrix} \right\}- \left\{\begin{matrix}{}^{n+1} {f}_m\\ {}^{n+1} {f}_s \end{matrix} \right\}={0}

(96)

Eq.(96) can be written in a compact (monolithic) form as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {M} {\dot{\bar{a}}} + {K} {~}^{n+1} \bar{a} -{}^{n+1} {f}=0

(97)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {M}=\left[\begin{matrix}{M}_v & {0}\\ {0} & -{M}_p \end{matrix} \right]~~ ,~~ {K} = \left[\begin{matrix}{K}_v & {Q}\\ {Q}^T & - \boldsymbol{\cal S}\end{matrix} \right]~~ ,~~ {~}^{n+1} \bar{a} = \left\{\begin{matrix}{}^{n+1} \bar{v}\\ {}^{n+1} \bar{p} \end{matrix} \right\}~~ ,~~ {}^{n+1} {f} = \left\{\begin{matrix}{}^{n+1} {f}_m\\ {}^{n+1} {f}_s \end{matrix} \right\}

(98)

11.1 Monolithic solution scheme

Eq.(98) is the basis for deriving monolithic iterative time integration schemes for directly computing the nodal velocities and pressure at the updated configuration. For instance, the standard backward Euler scheme gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {H}^i {~}^{n+1} \bar{a}^{i+1} = {}^{n+1} {f}^i + \frac{1}{\Delta t}{M} {}^{n} \bar{a}

(99)

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {H}^i = \frac{1}{\Delta t} {M} + {K}_v^i} .

The values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1} \bar{a}}

can be directly found by solving Eq.(99) iterative.

The non linearity of matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n}{K}_v}

emanates from the non linear  terms in the material strain rate matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {B}^L}
involving the displacement increments (Eqs.(9)).

11.2 Segregated solution scheme

The nodal velocities and pressures at the updated configuration can be also computed starting from Eq.(97) using a segregated iterative scheme as follows

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[\frac{1}{\Delta t} {M}_v + {K}_v^i \right]{~}^{n+1} \bar{v}^{i+1} = {}^{n+1} {f}_m - {Q} {~}^{n+1} {p}^i

(100)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[\frac{1}{\Delta t} {M}_p + \boldsymbol{\cal S}^i \right]{~}^{n+1} \bar{p}^{i+1} = {~}^{n+1} {f}_s + {Q}^T {~}^{n+1} \bar{v}^{i+1} + \frac{1}{\Delta t} {M}_p {~}^{n} \bar{p}

(101)

Eqs.(100) and (101) are solved sequentially and iteratively until convergence for the nodal velocities and pressures is found.

The above iterative scheme can be considered as a simplification of the more accurate incremental iterative segregated scheme described in Section 10.4.

Remark 12. A variant of the iteration matrix in the left hand side of Eq.(100) can be used by adding the bulk stiffness matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_p}

to matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {H}^i}
[11,31].

12 PARTICULARIZATION FOR THE UPDATED CONFIGURATION

The finite element formulation presented in the previous sections can be particularized for the case when the updated configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {~}^{n+1} {V}}

is chosen as the reference configuration.

The particularization is simple if we note that now Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta u_i=0} . Hence, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {L}=0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {B}^L ={0}}
(see Eq.(58.a)). Also all the space derivatives are taken with respect to the updated coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {~}^{n+1}x_i}
and the integrals are carried out in the updated configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {~}^{n+1}V}

. In addition, the tangent constitutive tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}_T ={c}} . The expressions of the relevant matrices and vectors for this case are given in Box 1.

13 PROS AND CONS OF USING THE CURRENT OR THE UPDATED CONFIGURATION AS THE REFERENCE CONFIGURATION

We have shown in the previous sections that either the current or the updated configuration can be indistinctly used as the reference configuration for the finite element analysis of quasi and fully incompressible fluids using a Lagrangian formulation. Both choices imply solving a non linear system of equations. However, the nature of the non-linearity is different for each case.

If the current configuration is chosen as the reference configuration, all integrals in the tangent matrix are performed on the known configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^nV}

which remains fixed during the iterative solution process. The non-linearity affects, however, the non linear terms of the material strain rate matrix (i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {B}^L}

) and also the constitutive tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}}

that involves the deformation gradient. A simplification of the tangent stiffness matrix  for this case can be introduced by neglecting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {B}^L}
in the material strain rate matrix and assuming that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {F}={C}={I}_3}
and, hence, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}_T ={c}}

.

If, on the other hand, the updated configuration is chosen as the reference configuration, then all the integrals in the tangent matrix are computed in the unknown configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}V} , which should be updated at each iteration. On the other hand, the expression for the material strain rate matrix is linear and also Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}_T ={c}} . A simplification of the iterative process can be introduced by keeping the tangent matrix constant for a fixed number of iterations.

Indeed the above mentioned simplifications can affect the convergence rate of the iterative solution and should be implemented with care.

Which reference configuration should be chosen can be problem dependent and, certainly, the choice will affect the organization of the computer program and its efficiency. What should be kept in mind is that the final solution, i.e. the geometry of the updated configuration and the velocities and stresses on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}V} , should be identical in both cases.

In previous works of the authors with the Particle Finite Element Method (PFEM) the current configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^nV}

was typically chosen as the reference configuration with the simplified form for the tangent matrix as explained above and also neglecting the initial stress matrix terms in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_T}
[16,17,25,26,29,31]. Recent experiences indicate that using the updated configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}V}
as the reference configuration can be advantageous in many Lagrangian flow problems. The topic is still open for research  and hopefully  this paper will be useful for choosing  the adequate FEM expressions for each case.

14 CONCLUDING REMARKS

We have presented a mixed velocity-pressure finite element formulation for solving the updated Lagrangian equations for quasi and fully incompressible fluids. The finite element interpolation uses an equal order approximation for the velocity and pressure unknowns. We have detailed a number of iterative algorithms for solving the non-linear stabilized FEM equations for the velocities and the pressure at the nodes using incremental and direct solution schemes. The algorithms presented are useful for the study of Lagrangian flows, as well as for solving fluid-structure-interaction problems using a unified Lagrangian finite element formulation for modelling both the fluid and the structure [11,17].

The choice of the current or the updated configurations as the reference Lagrangian configuration is still an open topic. Researchers interested in Lagrangian CFD procedures will find in this paper the equations to be used for each case, from which simplifications or further computational refinements can be made.

ACKNOWLEDGEMENTS

This work was partially supported by the Advanced Grant project SAFECON of the European Research Council (ERC).

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APPENDIX A. LINEARIZATION OF THE MOMENTUM EQUATIONS WITH RESPECT TO THE NODAL VELOCITIES

Using the expression of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\bar{r}_m}

of Eq.(62) and neglecting the changes of the external vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}^{n+1}\bar{f}_m}
 with the velocity (accounting for these changes is possible and will lead to additional terms in the tangent matrix), we can write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _{\bar v} {}^{n+1} \bar {r}_m = \frac{1}{\Delta t}{M}_v d \bar {v}+ \int _{{}^{n}V} \Big\{{B}^T \Big[\delta _{\bar v}\left\{\mathbf{S}' \right\}+ \left(\delta _{\bar v} J\left\{{C}^{-1}\right\}+ {J} \delta _{\bar v} \left\{{C}^{-1}\right\}\right)p +

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): + J \left\{{C}^{-1}\right\}\delta _{\bar v}p\Big]+ \delta _{\bar v} {B}^T \left\{\mathbf{S}' \right\}\Big\}d{~}^{n} V

(A.1)

Introducing Eqs.(55) and (59) into (A.1) gives after some algebra [4,5,36]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _{\bar v} \left\{\mathbf{S}' \right\}= [\boldsymbol{\cal C}]\delta _{\bar v} \left\{\dot {E}\right\} = [\boldsymbol{\cal C}] {B} d \bar {v}

(A.2)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _{\bar v} J \left\{{C}^{-1}\right\}= J {C}^{-1} \otimes {C}^{-1} \delta _{\bar v} \left\{{E}\right\} = J \Delta t{C}^{-1} \otimes {C}^{-1} {B} d \bar {v}

(A.3)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): J \delta _{\bar v} \left\{{C}^{-1}\right\}= - 2 J \boldsymbol{\cal I} \delta _{\bar v} \left\{{E}\right\}= -2 J \Delta t \boldsymbol{\cal I} {B} d \bar {v}

(A.4)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{\cal I}_{ijkl}= \frac{1}{2} \left[({C}^{-1})_{ik} ({C}^{-1})_{jl}- ({C}^{-1})_{il}({C}^{-1})_{jk}\right]

(A.5)

It can be shown that tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal I}}

is symmetric [4,5].

On the other hand,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _{\bar v} {B}^T \left\{{S}' \right\}= \Delta t {G}^T \hat{S}'{G} d \bar {v}

(A.6a)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{l}{G}=\left[ \begin{matrix}\bar {G} & \bar {0} & \bar {0}\\ \bar {0} &\bar {G}&\bar {0}\\ \bar {0} & \bar {0}& \bar {G} \end{matrix}\right] ~~,~~ \bar {G} = \left[ \begin{matrix}{}_nN^v_{1,1} &0&0& {}_nN^v_{2,1}&0&0& \cdots & {}_nN^v_{n,1}\\ {}_nN^v_{1,2} &0&0& {}_nN^v_{2,2}&0&0& \cdots & {}_nN^v_{n,2}\\ {}_nN^v_{1,3} &0&0& {}_nN^v_{2,3}&0&0& \cdots & {}_nN^v_{n,3} \end{matrix}\right]\\[1cm] \hat {S}' = \left[ \begin{matrix}{S}' & {0} & {0}\\ {0} & {S}'& {0}\\ {0} & {0}& {S}' \end{matrix}\right]~~,~~ {0} = \left[ \begin{matrix}0&0&0\\ 0&0&0\\ 0&0&0 \end{matrix}\right]~~,~~ \bar {0} = \left\{ \begin{matrix}0\\0\\0 \end{matrix}\right\} \end{array}

(A.6b)

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {}_nN^v_{i,j}}

defined in Eq.(58.b).

In the derivation of Eqs.(A.3), (A.4) and (A.6a) we have assumed that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d(\Delta u_i)= du_i =\Delta t ~d ({}^{n+\alpha }v_i) = \Delta t ~dv_i} . This relationship follows from Eq.(60.a).

Substituting Eqs.(A.2)–(A.4) and (A.6a) and the interpolation for the pressure (Eq.(53)) into (A.1) yields the linearized form of the residual vector of the discretized momentum equations as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta _v {}^{n+1}\bar{r}_m =\frac{1}{\Delta t}{M}_v d \bar {v} + \left\{\int _{{}^{n}V}\left[{B}^T [\boldsymbol{\cal C}_T] {B}+ {G}^T \hat{S}{G}\right]d {~}^{n}V \right\}d \bar {v} +

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): +\left\{\int _{{}^{n}V} {B}^T \left\{{C}^{-1}\right\}{N}_p J d {~}^{n}V \right\}{\delta }_{\bar v}^{n+1}\bar {p}= \left(\frac{1}{\Delta t}{M}_v + {K}_c + {K}_\sigma \right) d \bar {v}+ {Q} {\delta }_{\bar v}^{n+1}\bar {p}

(A.7)

where matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_v}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {Q}}
were given in Eq.(67) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_c}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {K}_\sigma }
are the constitutive and initial stress matrices, respectively. The expression of these matrices is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {K}_c = \!\!\int _{{}^{n}V}{B}^T [\boldsymbol{\cal C}_T] {B} d {~}^{n}V ~~ ,~~ {K}_\sigma =\!\!\int _{{}^{n}V}{G}^T \hat{S}' {G}d {~}^{n}V

(A.8)

The tangent constitutive tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}_T}

is deduced from Eqs.(A.1)-(A.4) as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{\cal C}_T = \boldsymbol{\cal C} + J \Delta t p ({C}^{-1}\otimes {C}^{-1} - 2 \boldsymbol{\cal I})

(A.9)

It is straightforward to show that tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\cal C}_T}

is symmetric.

@@ Line 1: / Line 1: @@
+Published in ''Computational Mechanics'', Vol. 54 (6), pp. 1583-1596, 2014<br />
+DOI: 10.1007/s00466-014-1078-1
 ==Abstract==

Revision as of 10:17, 7 June 2019

Abstract

1 INTRODUCTION

2 MOTION. DISPLACEMENT, VELOCITY AND ACCELERATION

3 MOMENTUM EQUATIONS AND BOUNDARY CONDITIONS

4 PRINCIPLE OF VIRTUAL POWER IN THE UPDATED CONFIGURATION

5 TRANSFORMATION TO THE CURRENT CONFIGURATION. LAGRANGIAN STRESS AND STRAIN MEASURES

6 SPLIT OF THE INTERNAL VIRTUAL POWER INTO DEVIATORIC AND VOLUMETRIC COMPONENTS

7 CONSTITUTIVE RELATIONSHIP FOR THE DEVIATORIC STRESSES

8 THE MASS BALANCE EQUATION

9 FINITE ELEMENT DISCRETIZATION

9.1 Discretized form of the PVP

9.2 Discretization of the mass conservation equation

9.3 Stabilization of the mass conservation equation

10 INCREMENTAL SOLUTION FOR THE NODAL VELOCITIES AND PRESSURES

10.1 Linearization of the discretized momentum equations with respect to the nodal velocities

10.2 Linearization of the nodal pressures with respect to the nodal velocities. Derivation of the tangent matrix

10.3 Linearization of the stabilized mass conservation equation with respect to the nodal pressures

10.4 Incremental solution of the discretized equations

11 DIRECT ITERATIVE SOLUTION OF THE NODAL VELOCITIES AND PRESSURES

11.1 Monolithic solution scheme

11.2 Segregated solution scheme

12 PARTICULARIZATION FOR THE UPDATED CONFIGURATION

13 PROS AND CONS OF USING THE CURRENT OR THE UPDATED CONFIGURATION AS THE REFERENCE CONFIGURATION

14 CONCLUDING REMARKS

ACKNOWLEDGEMENTS

References

APPENDIX A. LINEARIZATION OF THE MOMENTUM EQUATIONS WITH RESPECT TO THE NODAL VELOCITIES

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