A Particle Finite Element Method (PFEM) for coupled thermal analysis of quasi and fully incompressible flows and fluid-structure interaction

Revision as of 13:50, 9 April 2018

Published in "Numerical Simulations of Coupled problems in Engineering", S.R. Idelsohn (Ed.). Computational Methods in Applied Sciences 33, pp. 129-156, Springer 2014

Abstract

We present a Lagrangian formulation for coupled thermal analysis of quasi and fully incompressible flows and fluid-structure interaction (FSI) problems that has excellent mass preservation features. The success of the formulation lays on a residual-based stabilized expression of the mass balance equation obtained using the Finite Calculus (FIC) method. The governing equations are discretized with the FEM using simplicial elements with equal linear interpolation for the velocities, the pressure and the temperature. The merits of the formulation in terms of reduced mass loss and overall accuracy are verified in the solution of 2D and 3D adiabatic and thermally-coupled quasi-incompressible free-surface flow problems using the Particle Finite Element Method (PFEM). Examples include the sloshing of water in a tank and the falling of a water sphere and a cylinder into a tank containing water.

keywords Particle Finite Element Method, Coupled Thermal Analysis, Quasi and Fully Incompressible

1 INTRODUCTION

The analysis of thermally coupled flows and their interaction with structures is relevant in many fields of engineering. In this work we present a Lagrangian numerical technique for solving this kind of problems for quasi and fully incompressible fluids using the Particle Finite Element Method (PFEM, www.cimne.com/pfem).

The PFEM treats the mesh nodes in the analysis domain as particles which can freely move and even separate from the domain representing, for instance, the effect of water drops or cutting particles in drilling problems. A mesh connects the nodes discretizing the domain where the governing equations are solved using a stabilized FEM. Examples of application of PFEM to problems in fluid and solid mechanics including fluid-structure interaction (FSI) situations can be found in [4,5,6,8,9,10,11,12,13,14,15,16,17,18,19,28,29,35,36,39,40,41,42,43]. Early attempts of the PFEM for solving thermally coupled flows were reported in [1,2].

In Lagrangian analysis procedures (such as PFEM) the motion of the fluid particles is tracked during the transient solution. Hence, the convective terms vanish in the momentum and heat transfer equations and no numerical stabilization is needed for treating those terms. Two other sources of mass loss, however, remain in the numerical solution of Lagrangian flows, i.e. that due to the treatment of the incompressibility constraint by a stabilized numerical method, and that induced by the inaccuracies in tracking the flow particles and, in particular, the free surface.

In this work the PFEM equations for analysis of thermally coupled flows and FSI problems are derived using the stabilized formulation based in the Finite Calculus (FIC) method proposed by Oñate et al. [20,21,22,23,24,25,26,27,30,31,32,37,38,39] that has excellent mass preservation features.

The lay-out of the paper is the following. In the next section we present the basic equations for conservation of linear momentum, mass and heat transfer for a quasi-incompressible fluid in a Lagrangian framework. A full incompressible fluid can be considered as a particular limit case of the former. Next we derive the stabilized FIC form of the mass balance equation. Then the finite element discretization using simplicial element with equal order approximation for the velocity, the pressure and the temperature is presented and the relevant matrices and vectors of the discretized problem are given. Details of the implicit solution of the Lagrangian FEM equations in time using a Newton iterative scheme are presented. The relevance of the bulk stiffness terms in the tangent matrix for enhancing the convergence and accuracy of the iterative solution scheme is discussed. The basic steps of the PFEM for solving coupled free-surface FSI problems are described.

The efficiency and accuracy of the PFEM technique are verified by solving a set of adiabatic and thermally coupled quasi-incompressible free surface flow problems in two (2D) and three (3D) dimensions with the PFEM. The adiabatic problems are the sloshing of water in a tank and the penetration of a water sphere into a cylindrical tank containing water. The thermally coupled problems considered are the extended 2D version of the adiabatic cases. The excellent performance of the numerical method proposed in terms of mass conservation and general accuracy is highlighted.

2 GOVERNING EQUATIONS

We write the governing equations for a quasi-incompressible Newtonian flow problem in the Lagrangian description as follows [3,46].

Momentum equations

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(1)

In Eq.(1), Failed to parse (MathML with SVG or PNG fallback (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 }

is the analysis domain with 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 }

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

is the density, Failed to parse (MathML with SVG or PNG fallback (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 (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_s=3}
for 3D problems) 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 \sigma _{ij}}
are the Cauchy stresses that are split in the deviatoric (Failed to parse (MathML with SVG or PNG fallback (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_{ij}}

) and 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} ) components 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} = s_{ij}+ p \delta _{ij}

(2)

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 _{ij}}

is the Kronecker delta. Note that the pressure is assumed to be positive for a tension state. Summation of terms with repeated indices  is assumed in Eq.(1) and in  the following, unless otherwise specified.

The relationship between the deviatoric 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 s_{ij}}

and the strain rates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varepsilon _{ij}}
has the standard form for a Newtonian 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/":): s_{ij} = 2\mu \left(\varepsilon _{ij} - {1 \over 3}\varepsilon _v \delta _{ij}\right)\quad \hbox{with}\quad \varepsilon _{ij} = {1 \over 2}\left({\partial v_i \over \partial x_j} + {\partial v_j \over \partial x_i} \right)

(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 \mu }

is the viscosity  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 \varepsilon _v}
is the volumetric strain rate 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 \varepsilon _v= \varepsilon _{ii} = {\partial v_i \over \partial x_i} }

.

Mass balance equation

The standard mass balance equation for a quasi-incompressible fluid can be written as [3,7,46]

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

(4a)

with

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(4b)

In Eq.(4b) Failed to parse (MathML with SVG or PNG fallback (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.(4a) simplifies 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 \varepsilon _v =0}

. In our work we will retain the quasi-incompressible 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 r_v}

of Eq.(4b) for convenience.

Thermal balance

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho c \frac{DT}{Dt} - {\partial \over \partial x_i} \left(k {\partial T \over \partial x_i}\right)+ Q =0 \quad i=1,n_s \quad \hbox{in }\Omega

(5)

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}

is the temperature, Failed to parse (MathML with SVG or PNG fallback (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 thermal capacity, Failed to parse (MathML with SVG or PNG fallback (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}
is the heat conductivity 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}
is the heat source.

Boundary conditions

Mechanical problem

The boundary conditions at the 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/":): {\textstyle \Gamma _v} ) and 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/":): {\textstyle \Gamma _t} ) boundaries 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 \Gamma = \Gamma _v \cup \Gamma _t}

are

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(6)

Failed to parse (MathML with SVG or PNG 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 -t_i^p =0 \quad \hbox{on }\Gamma _t \quad i,j=1,n_s

(7)

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 v_i^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 t_i^p}
are the prescribed velocities and prescribed 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 \Gamma _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 \Gamma _t}

, 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_j}

are the components of the unit normal vector to the boundary [3,7,46].

Thermal problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \phi - \phi ^p =0 \quad \hbox{on }\Gamma _\phi	(8)
Failed to parse (MathML with SVG or PNG 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 {\partial \phi \over \partial n} + q_n^p =0 \quad \hbox{on }\Gamma _q	(9)

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 \phi ^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 q_n^p}
are the prescribed temperature and the prescribed normal heat flux at the boundaries Failed to parse (MathML with SVG or PNG fallback (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 _\phi }
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 \Gamma _q}

, 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}

is the direction normal to the boundary.

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

in Eq.(1) is the material derivative 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 Failed to parse (MathML with SVG or PNG fallback (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_i} . This term is typically computed in a Lagrangian framework 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/":): \frac{Dv_i}{Dt} = \frac{{}^{n+1} v_i - {}^n v_i }{\Delta t}

(10)

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} v_i:=v_i ({}^{n+1} {\bf x},{}^{n+1} t)\quad ,\quad {}^n v_i:=v_i ({}^n{\bf x},{}^n t)

(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 v_i}

is the velocity of the material point that has the position Failed to parse (MathML with SVG or PNG fallback (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{\bf x}}
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={}^nt}

, 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 {\bf x}}

is the coordinates vector in a fixed Cartesian  system [3,7,46].

3 STABILIZED MASS BALANCE EQUATION

In this work we will use the second order FIC form of the mass balance equation in space for a quasi-incompressible fluid [37,38], as well as the first order FIC form of the mass balance equation in time. These forms have the following expressions:

Second order FIC mass balance equation in space

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(12a)

First order FIC mass balance equation in 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/":): r_v + \frac{\delta }{2} \frac{D r_v}{D t}=0 \qquad \hbox{in }\Omega

(12b)

Eq.(12a) is obtained by expressing the balance of mass in a rectangular domain of finite size with 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 h_1\times h_2}

(for 2D problems), 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 h_i}
are arbitrary distances, and retaining up to third order terms in the Taylor series expansions used for expressing the change of mass within the balance domain.

Eq.(12b), on the other hand, is obtained by expressing the balance of mass in a space-time domain of infinitesimal length in space and finite dimension Failed to parse (MathML with SVG or PNG fallback (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 }

in time [20]. The derivation of Eqs.(12) for a 1D problem are shown in [41].

The FIC terms in Eqs.(12) play the role of space and time stabilization terms respectively. In the discretized problem, the 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 h_i}

and the time dimension Failed to parse (MathML with SVG or PNG fallback (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 }
are related to characteristic element dimensions and the time step increment, respectively as it will be explained later. Note that 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 h_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 \delta =0}
the standard infinitesimal form of the mass balance 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 r_v =0}

, is obtained.

After some transformations the stabilized mass balance equation (12a) is written as [41]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): -\frac{1}{\kappa }\frac{Dp}{Dt}+\varepsilon _v - \frac{\tau }{c^2} \frac{D^2p}{Dt^2} +\tau {\partial \hat r_{m_i} \over \partial x_i} =0

(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 \tau }

is a stabilization parameter given by [41]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tau = \left(\frac{8\mu }{h^2}+ \frac{2\rho }{\delta } \right)^{-1}

(14)

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 r_{m_i}}

is a static momentum term 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/":): \hat r_{m_i}= {\partial \over \partial x_j} (2\mu \varepsilon _{ij}) + \frac{\partial p}{\partial x_i} + b_i

(15)

Eq.(13) is used as the starting point for deriving the stabilized FEM formulation as explained in the following sections.

4 VARIATIONAL EQUATIONS

4.1 Momentum equations

Multiplying Eq.(1) by arbitrary 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 w_i}

with dimensions of velocity and integrating over the analysis 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 }
gives the weighted residual form of the momentum equations as [3,7,46]

Failed to parse (MathML with SVG or PNG 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 _\Omega w_i \left(\rho \frac{Dv_i}{Dt}- {\partial \sigma _{ij} \over \partial x_j}-b_i\right)d\Omega =0

(16)

Integrating by parts the term involving Failed to parse (MathML with SVG or PNG fallback (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 using the Neumann boundary conditions (7) yields the weak variational form of the 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/":): \int _\Omega w_i \rho \frac{Dv_i}{Dt} d\Omega + \int _\Omega \delta \varepsilon _{ij} \sigma _{ij} d\Omega - \int _\Omega w_i b_i d\Omega - \int _{\Gamma _t} w_i t_i^p d\Gamma =0

(17)

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 \varepsilon _{ij}= {\partial w_i \over \partial x_j}+{\partial w_j \over \partial x_i}}

is an arbitrary (virtual) strain rate field. Eq.(17) is the standard form of the principle of virtual power  [3,7,46].

Substituting the expression of the stresses from Eq.(2) into (17) 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/":): \int _\Omega \! w_i \rho \frac{Dv_i}{Dt} d\Omega + \!\int _\Omega \left[\delta \varepsilon _{ij} 2\mu \left(\!\varepsilon _{ij} - \frac{1}{3}\varepsilon _{v} \delta _{ij}\!\right)+ \delta \varepsilon _{v}p\right]d\Omega - \!\int _\Omega w_i b_i d\Omega - \!\int _{\Gamma _t} w_i t_i^p d\Gamma =0

(18)

Eq.(18) can be written in matrix 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 \int _\Omega {\bf w}^T \rho \frac{D{\bf v}}{Dt} d\Omega + \!\int _\Omega \delta {\boldsymbol \varepsilon }^T {\bf D} {\boldsymbol \varepsilon } d\Omega + \!\int _\Omega \delta {\boldsymbol \varepsilon }^T {\bf m} p d\Omega - \int _\Omega {\bf w}^T {\bf b}d\Omega - \!\int _{\Gamma _t} {\bf w}^T {\bf t}^p d\Gamma =0

(19)

In Eq.(19) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\bf w},{\bf v},{\boldsymbol \varepsilon }}

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 {\boldsymbol \varepsilon }}
are vectors containing the test functions, the velocities, the strain rates and the virtual strain rates respectively; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\bf 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 {\bf t}^p}
are body force and surface traction vectors, respectively; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\bf D}}
is the viscous constitutive 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 {\bf m}}
is an auxiliary vector. These vectors are defined as (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/":): \begin{array}{l}\textbf{w} =[w_1,w_2,w_3]^T\quad ,\quad \textbf{v} =[v_1,v_2,v_3]^T \quad ,\quad \textbf{b} =[b_1,b_2,b_3]^T \quad ,\quad \textbf{t}^p =[t_1^p,t_2^p,t_3^p]^T\\[.5cm] {\boldsymbol \varepsilon }= [\varepsilon _{11},\varepsilon _{22}, \varepsilon _{33},\varepsilon _{12},\varepsilon _{13}, \varepsilon _{23}]^T \quad ,\quad \delta {\boldsymbol \varepsilon }= [\delta \varepsilon _{11},\delta \varepsilon _{22}, \delta \varepsilon _{33},\delta \varepsilon _{12},\delta \varepsilon _{13}, \delta \varepsilon _{23}]^T\\[.5cm] {\bf D}=\mu \left[ \begin{array}{cccccc}4/3 & -2/3 & -2/3 & 0 & 0 & 0 \\ & 4/3 & -2/3 &0 &0 & 0 \\ & & 4/3 & 0 & 0 & 0 \\ & & & 2 & 0 & 0 \\ {Sym.} & & & & 2 & 0 \\ & & & & & 2 \\ \end{array} \right]\qquad ,\qquad {\bf m}=[1,1,1,0,0,0]^T \end{array}

(20)

4.2 Mass balance equations

We multiply Eq.(13) by arbitrary (continuous) 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}

(with dimensions of pressure) defined over the analysis 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 }

. Integrating over Failed to parse (MathML with SVG or PNG fallback (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 }

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/":): \int _\Omega -\frac{q}{\kappa } \frac{Dp}{Dt}d\Omega -\int _\Omega q \frac{\tau }{c^2} \frac{D^2p}{Dt^2}d\Omega + \int _\Omega q \varepsilon _v d\Omega + \int _\Omega q \tau {\partial \hat r_{m_i} \over \partial x_i} d\Omega =0

(21)

Integrating by parts the last integral in Eq.(21) and using (15) gives after some transformations [39,40]

Failed to parse (MathML with SVG or PNG 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 _\Omega \frac{q}{\kappa } \frac{Dp}{Dt}d\Omega + \int _\Omega q \frac{\tau }{c^2} \frac{D^2p}{Dt^2}d\Omega - \int _\Omega q \varepsilon _v d\Omega + \int _\Omega \tau {\partial q \over \partial x_i} \left({\partial \over \partial x_i} (2\mu \varepsilon _{ij})+ {\partial p \over \partial x_i}+b_i\right)d\Omega \\ \displaystyle - \int _{\Gamma _t} q \tau \left[\rho \frac{Dv_n}{Dt}-\frac{2}{h_n} \left(2\mu {\partial v_n \over \partial n} + p -t_n\right)\right]d\Gamma =0 \end{array}

(22)

Expression (24) holds for 2D and 3D problems. The terms involving the first and second material time derivative of the pressure and the boundary term in Eq.(24) are important for preserving the conservation of mass in free-surface flow problems [10,41].

4.3 Thermal balance equation

Application of the standard weighted residual method to the heat balance equations (5) and (9) leads, after standard operations, to [7,44]

Failed to parse (MathML with SVG or PNG 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 _\Omega \hat w \rho c {\partial T \over \partial t} d\Omega + \int _\Omega {\partial \hat w \over \partial x_i} k {\partial T \over \partial x_i}d\Omega - \int _\Omega \hat w Q d\Omega + \int _{\Gamma _q} \hat w q_n^p d\Gamma =0

(23)

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 \hat w}

are the space weighting functions for the temperature.

5 FEM DISCRETIZATION

We discretize the analysis domain into finite elements 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}

nodes in the standard manner leading to a mesh with a total number 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_e}
elements 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}
nodes. In our work we will choose simple 3-noded linear triangles (Failed to parse (MathML with SVG or PNG fallback (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=3}

) for 2D problems and 4-noded tetrahedra (Failed to parse (MathML with SVG or PNG fallback (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=4} ) for 3D problems with local linear shape 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 N_i^e}

defined for each 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/":): {\textstyle i=1,n}

) of 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 e}

[34,44]. The velocity components, the  pressure and the temperature are interpolated over the mesh in terms of their nodal values in the same manner using the  global linear shape 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 N_j}
spanning over the elements sharing 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 j}
(Failed to parse (MathML with SVG or PNG fallback (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=1,N}

) [34,44,46]. In matrix 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/":): {\bf v} = {\bf N}_v\bar {\bf v} ~,~p = {\bf N}_p\bar {\bf p}\quad , \quad T = {\bf N}_T \bar {\bf T}

(24)

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}{c}\displaystyle \bar {\bf v} = \left\{\begin{matrix}\bar{\bf v}^1\\\bar{\bf v}^2\\\vdots \\ \bar{\bf v}^N\end{matrix} \right\}\quad \hbox{with}~ \bar{\bf v}^i = \left\{\begin{matrix}\bar{v}^i_1\\\bar{v}^i_2\\\bar{v}^i_3\end{matrix} \right\}~, ~ \bar {\bf p} =\left\{\begin{matrix}\bar{p}^1\\\bar{p}^2\\\vdots \\ \bar{p}^{N}\end{matrix} \right\}~, ~ {\bf T}= \left\{\begin{matrix}\bar{T}_1\\\bar{T}_2\\\vdots \\ \bar{T}^N\end{matrix} \right\}\\ \displaystyle {\bf N}_v = [{\bf N}_1, {\bf N}_2,\cdots , {\bf N}_N ]^T ~,~ {\bf N}_p = {\bf N}_T = [{N}_1, { N}_2,\cdots , {N}_N ] \end{array}

(25)

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 {\bf N}_j = N_j {\bf I}_n}

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 {\bf I}_n}
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 n\times n}
unit matrix.

In Eq.(25) 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 \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/":): {\textstyle \bar {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 \bar {T}}
 contain the nodal velocities, the nodal pressures and the nodal temperatures for the whole mesh, respectively and the upperindex denotes the nodal value for each vector or scalar magnitude.

Substituting Eqs.(24) into Eqs.(17), (22) and (25) and choosing a Galerkin formulation 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 w_i = q = \hat w_i =N_i}

leads to the following system of algebraic 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/":): {\bf M}_0 {\dot{\bar{\bf v}}} + {\bf K}\bar{\bf v}+{\bf Q}\bar {\bf p}- {\bf f}_v={\bf 0}

(26a)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \textbf{M}_1 {\dot{\bar{\bf p}}}+\textbf{M}_2 {\ddot{\bar{\bf p}}}-{\bf Q}^T \bar{\bf v} + ({\bf L}+{\bf M}_b)\bar {\bf p}- {\bf f}_p={\bf 0}

(26b)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\bf C} {\dot{\bar{\bf T}}}+\hat {\bf L} \bar{\bf T} - {\bf f}_T={\bf 0}

(26c)

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{\bar{\bf a}}}}

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 {\ddot{\bar{\bf a}}}}
denote the first and second material time derivatives of the components of  a 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 {a}}

. The different matrices and vectors in Eqs.(26) are assembled from the element contributions given in Box 1.

Remark 2. The boundary terms 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 {\bf f}_p}

can be incorporated in the matrices of Eq.(26b). This leads to a non symmetrical set of equations. These boundary terms are computed here iteratively within the incremental solution scheme.

Remark 3. The presence 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 {\bf M}_b}

in Eq.(26b) allows us to compute the pressure without the need of prescribing its value at the free surface. This eliminates the error introduced when the pressure is prescribed to zero in free boundaries, which leads to considerable mass losses in viscous flows [15].

Remark 4. For transient problems the stabilization parameter Failed to parse (MathML with SVG or PNG fallback (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 }

of Eq.(14) is computed for each 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 e}
using Failed to parse (MathML with SVG or PNG fallback (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=l^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 \delta = \Delta t}
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/":): \tau = \left(\frac{8\mu }{(l^e)^2} + \frac{2\rho }{\Delta t} \right)^{-1}

(27)

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 t}

is the time step used for the transient solution 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^e}
is a characteristic element length 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/":): {\textstyle l^e = 2(\Omega ^e)^{1/n_s}}
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 \Omega ^e}
is the element area (for 3-noded triangles) or volume (for 4-noded tetrahedra). For fluids with heterogeneous material 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 \mu }
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 \rho }
 are computed at the element center.

For steady state problems the stabilization parameter is computed with Eq.(27) substituting Failed to parse (MathML with SVG or PNG fallback (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 t}

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/":): {\textstyle \frac{l^e}{\vert {\bf v}^e\vert }}

. The characteristic boundary length Failed to parse (MathML with SVG or PNG fallback (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_n}

in 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 {\bf f}_p}
(Box 1) has been taken equal to Failed to parse (MathML with SVG or PNG fallback (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^e}
in our computations.

Failed to parse (MathML with SVG or PNG 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} \displaystyle \mathbf{M}_{0_{ij}}^e =\int _{\Omega ^e}\rho {N}_{i}^e {N}_{j} {\bf I}_3 d\Omega ~,~\mathbf{K}^e_{ij} =\int _{\Omega ^e} \mathbf{B}_i^{eT} \mathbf{D}\mathbf{B}_j^e d\Omega ~,~\mathbf{Q}^e_{ij} =\int _{\Omega ^e} \mathbf{B}_i^{eT} \mathbf{m} {N}_j^e d\Omega \\[.4cm] \displaystyle {M}_{1_{ij}}^e =\int _{\Omega ^e} \frac{1}{\kappa }{N}_i^e {N}_j^e d\Omega ~,~ {M}_{2_{ij}}^e =\int _{\Omega ^e} \frac{\tau }{ c^2 }{N}_i^e{N}_j^e d\Omega ~,~ {M}_{b_{ij}}^e = \int _{\Gamma _t} \frac{2\tau }{h_n} {N}_i^e{N}_j^e d\Gamma \\[.4cm] \displaystyle {L}^e_{ij}= \int _{\Omega ^e} \tau ({\boldsymbol \nabla }^T {N}_i^e) {\boldsymbol \nabla } N_j^e d\Omega ~,~ \mathbf{f}^e_{v_i}= \int _{\Omega ^e}\mathbf{N}_{i}^e \mathbf{b} d\Omega + \int _{\Gamma _t} \mathbf{N}_{i}^e {\bf t}^p d\Gamma \\[.4cm] \displaystyle {f}^e_{p_i}=\int _{\Gamma _t}\tau N_i^e \left[\rho \frac{Dv_n}{Dt}-\frac{2}{h_n} (2\mu \varepsilon _n -t_n)\right]d\Gamma - \int _{\Omega ^e} \tau {\boldsymbol \nabla }^T {N}_i^e {b} d\Omega \\[.4cm] \displaystyle {C}^e_{ij}= \int _{\Omega ^e} \rho c N_i^e N_j^e d\Omega ~,~ \hat L^e_{ij}= \int _{\Omega ^e} k ({\boldsymbol \nabla }^T {N}_i^e) {\boldsymbol \nabla } N_j^e d\Omega ~,~ \displaystyle {f}^e_{T_i}= \int _{\Omega ^e} N_i^e Q d\Omega - \int _{\Gamma _q} N_i^e q^p_n d\Gamma \\[.4cm] \hbox{with } i,j=1,n.\\ \hbox{For 3D problems}\\[.4cm] \displaystyle{\bf B}_i^e = \left[\begin{matrix} \displaystyle {\partial N_i^e \over \partial x_1} &0&0\\ \displaystyle{0}& \displaystyle {\partial N_i^e \over \partial x_2}&0\\ \displaystyle{0}&0&\displaystyle {\partial N_i^e \over \partial x_3}\\ \displaystyle {\partial N_i^e \over \partial x_2}&\displaystyle {\partial N_i^e \over \partial x_1}&0\\[.25cm] \displaystyle {\partial N_i^e \over \partial x_3}&0&\displaystyle {\partial N_i^e \over \partial x_1}\\[.25cm] \displaystyle{0}&\displaystyle {\partial N_i^e \over \partial x_3}&\displaystyle {\partial N_i^e \over \partial x_2} \end{matrix} \right]~,~ \mathbf{\bf N}_{i}^e = N_i^e {\bf I}_3 \quad \hbox{and} \quad {\boldsymbol \nabla } = \left\{\begin{matrix}\displaystyle {\partial \over \partial x_1} \\ \displaystyle {\partial \over \partial x_2}\\\displaystyle {\partial \over \partial x_3} \end{matrix} \right\} \end{array}
Failed to parse (MathML with SVG or PNG 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_i^e Local shape function of 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} of 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 e} [34,44]

Box 1. Element form of the matrices and vectors in Eqs.(26)

6 TRANSIENT SOLUTION OF THE DISCRETIZED EQUATIONS

Eqs.(26) are solved in time with an implicit Newton-Raphson type iterative scheme [3,7,44,46]. The basic steps 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]}

are:

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}{\bf x}^1,{}^{n+1}\bar {\bf v}^1,{}^{n+1}\bar{\bf p}^1, {}^{n+1}\bar{\bf T}^1, {}^{n+1}\bar{\bf r}^1_m)\equiv \left\{{}^{n}{\bf x},{}^{n}\bar{\bf v},{}^{n}\bar{\bf p},{}^{n}\bar{\bf T}, {}^{n}\bar{\bf r}_m \right\}}

.

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,NITER}

.

For each iteration.

Step 1. Compute 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/":): \Delta \bar {v}

From Eq.(26a), 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/":): {}^{n+1}{\bf H}_v^i \Delta \bar {\bf v} = - {}^{n+1}\bar{{\bf r}}_m^i \rightarrow \Delta \bar{\bf v}

(28a)

with the momentum residual Failed to parse (MathML with SVG or PNG fallback (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{\bf r}_m}

and the iteration 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 {\bf H}_v}
 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/":): \bar{\textbf{r}}_m = \textbf{M}_0 {\dot{\bar{\bf v}}} + \textbf{K}\bar{\bf v}+\textbf{Q}\bar {\bf p}-\textbf{f}_v\quad , \quad \textbf{H}_v = \frac{1}{\Delta t} \textbf{M}_0 + \textbf{K} + \textbf{K}_v

(28b)

Step 2. 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{\bf v}^{i+1}= {}^{n+1}\bar{\bf v}^i + \Delta \bar{\bf v}

(29)

Step 3. Compute 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{\bf p}^{i+1}

From Eq.(26b) we 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/":): {}^{n+1} \textbf{H}_p^i {}^{n+1}\bar{\bf p}^{i+1}= \frac{1}{\Delta t} \textbf{M}_1 {}^{n+1}\bar{\bf p}^i + \frac{1}{\Delta t^2} \textbf{M}_2 (2 ^n \bar{\bf p} - ^{n-1} \bar{\bf p}) + {\bf Q}^T {}^{n+1}\bar{\bf v}^{i+1}+ {}^{n+1}\bar{\bf f}^{i}_p \rightarrow {}^{n+1}\bar{\bf p}^{i+1}

(30a)

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/":): \textbf{H}_p = \frac{1}{\Delta t} \textbf{M}_1+ \frac{1}{\Delta t^2} \textbf{M}_2+\textbf{L} + \textbf{M}_b

(30b)

Step 4. Update the nodal 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/":): {}^{n+1}{\bf x}^{i+1}= {}^{n+1}{\bf x}^i + \frac{1}{2} ({}^{n+1}\bar{\bf v}^{i+1} + {}^{n}\bar{\bf v})\Delta t

(31)

A more accurate expression for computing Failed to parse (MathML with SVG or PNG fallback (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}{\bf x}^{i+1}}

can be used involving the  nodal accelerations [40].

Step 5.Compute the nodal temperatures

From Eq.(26c) we 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/":): \left[\frac{1}{\Delta t}{\bf C} + \hat{\bf L} \right]\Delta \bar{\bf T}=- {}^{n+1} \bar{\bf r}^i_T \quad ,\quad {}^{n+1}\bar{\bf T}^{i+1}= {}^{n}\bar{\bf T}^i + \Delta \bar{\bf T}

(32)

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/":): {\bf r}_T= {\bf C} {\dot{\bar{\bf T}}} + \hat{\bf L} \bar{\bf T} - {\bf f}_T

(33)

Step 6. Check convergence

Verify the following conditions:

Failed to parse (MathML with SVG or PNG 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}{c}\displaystyle \Vert {}^{n+1}\bar {\bf v}^{i+1}- {}^{n+1}\bar{\bf v}^{i}\Vert \le e_v \Vert {}^{n}\bar {\bf v}\Vert \\ \displaystyle \Vert {}^{n+1}\bar {\bf p}^{i+1}-{}^{n+1}\bar {\bf p}^i\Vert \le e_p \Vert {}^{n}\bar{\bf p}\Vert \\ \displaystyle \Vert {}^{n+1}\bar {\bf T}^{i+1} - {}^{n+1}\bar {\bf T}^i\Vert \le e_T \Vert {}^{n}\bar{\bf T}\Vert \end{array}

(34)

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_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 e_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 e_T}
are prescribed error norms. In our examples we have set Failed to parse (MathML with SVG or PNG fallback (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 = e_p=e_T = 10^{-3}}

.

If conditions (34) 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 step. 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 1–5.

Remark 5. In Eqs.(28)–(34) Failed to parse (MathML with SVG or PNG fallback (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}(\cdot )}

denotes the values of a matrix or a vector computed using the nodal unknowns 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}

. In this work the derivatives and integrals in all the matrices and the residual 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 \bar{r}_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 \bar{\bf r}_T}
are computed on the discretized geometry 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}
 while the nodal force 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 {\bf f}_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 {\bf f}_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 {\bf f}_v}
are computed on 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 n+1}

. This is equivalent to using an updated Lagrangian formulation [3,45,46].

Remark 6. Including 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 \textbf{K}_v}

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 \textbf{H}_v}
has proven to be essential for the fast convergence, mass preservation and overall accuracy of the iterative solution [10,41].     The element 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 \textbf{K}_v}
can be obtained as [41]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {bf K}_v^e = \int _{\Omega ^e} \textbf{B}^T \textbf{m} \theta \Delta t\kappa \textbf{m}^T \textbf{B}d\Omega

(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/":): {\textstyle \theta }

is a positive number 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}
that has the role of preventing the ill-conditioning of the iteration 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 {\bf H}_v}
for highly incompressible fluids. 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 }
also improves the overall accuracy of the numerical solution and the preservation of mass for large time steps [10]. 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}
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 \kappa =\infty }

), a 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 \kappa }

is used in practice 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 {\bf K}_v}
as this helps to obtaining an accurate solution for velocities and pressure  with reduced mass loss in few iterations per time step [10]. 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 \kappa }
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 {\bf M}_1}
in Eq.(26b) that vanishes for a fully incompressible fluid. 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 {\bf K}_v^e}
can also be limited by reducing the time step size. This, however, leads to an increase in the cost of the computations. A similar approach for improving mass conservation in incompressible flows was proposed  in [42].

Remark 7. The iteration 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 \textbf{H}_v}

in Eq.(28a) is an approximation of the exact tangent matrix in the updated Lagrangian formulation for a quasi-incompressible fluid [40]. The simplified 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 \textbf{H}_v}
 used in this work has yielded good results  with convergence achieved for the nodal velocities,  the pressure and the temperature in 3–4 iterations in all the problems analyzed.

Remark 8. The time step within a time interval Failed to parse (MathML with SVG or PNG fallback (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]}

is chosen 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 \Delta t =\min \left(\frac{^n l_{\min }^e}{\vert {}^n{\bf v}\vert _{\max }},\Delta t_b\right)}
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 l_{\min }^e}
is the minimum characteristic distance  of all elements in the mesh, 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 l^e}
computed as explained in Remark 4, Failed to parse (MathML with SVG or PNG fallback (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 ^n{\bf v}\vert _{\max }}
is the maximum value of the modulus of the velocity of all nodes in the mesh  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 t_b}
is the critical time step of all nodes approaching a solid boundary 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 \Delta t_b = \min \left(\frac{^n l_b}{\vert ^n{\bf v}_b\vert _{\max }}\right)}
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 l_b}
is the distance from the node to the boundary 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{\bf v}_b}
is the velocity of the node. This definition 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 t}
intends that no node crosses a solid boundary during a time step.

A method that allows using large time steps in the integration of the PFEM equations can be found in [16].

7 ABOUT THE PARTICLE FINITE ELEMENT METHOD (PFEM)

7.1 The basis of the PFEM

Let us consider a 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 V}

containing  fluid and solid subdomains. Each subdomain is characterized by a set of points, hereafter termed particles. The particles contain all the information  for defining the geometry and the material and mechanical properties of the underlying subdomain. In the PFEM both subdomains are modelled using an updated Lagrangian formulation [3,45].

The solution steps within a time step in the PFEM are as follows:

Figure 1: Sequence of steps to update a “cloud” of nodes representing a domain containing a fluid and a solid part from 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/":): n

  (Failed to parse (MathML with SVG or PNG 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

) to 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/":): n+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/":): t=^n t +2\Delta t

)

The starting point at each time step is the cloud of points Failed to parse (MathML with SVG or PNG fallback (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 fluid and solid domains. For instance Failed to parse (MathML with SVG or PNG fallback (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} C} denotes the cloud 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 } (Figure 1).
Identify the boundaries defining the analysis 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 ^{n} V} , as well as the subdomains in the fluid and the solid. This is an essential step as some boundaries (such as the free surface in fluids) may be severely distorted during the solution, including separation and re-entering of nodes. The Alpha Shape method [8] is used for the boundary definition. Clearly, the accuracy in the reconstruction of the boundaries depends on the number of points in the vicinity of each boundary and on the Alpha Shape parameter. In the problems solved in this work the Alpha Shape method has been implementation as described in [12,28].
Discretize the the analysis 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 ^{n} V} with a finite element 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 {}^{n} M.} We use an efficient mesh generation scheme based on an enhanced Delaunay tesselation [11,12].
Solve the Lagrangian equations of motion for the overall continuum using the standard FEM. Compute the state variables in at the next (updated) configuration 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 ^n t+\Delta t}
velocities, pressure and viscous stresses in the fluid and displacements, stresses and strains in the solid.
Move the mesh nodes to a new position Failed to parse (MathML with SVG or PNG fallback (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} C} where n+1 denotes 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 {}^n t +\Delta t} , in terms of the time increment size.
Go back to step 1 and repeat the solution for the next time step 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/":): {\textstyle {}^{n+2} C} .

Note that the key differences between the PFEM and the classical FEM are the remeshing technique and the identification of the domain boundary at each time step.

The CPU time required for meshing grows linearly with the number of nodes. As a general rule, meshing consumes for 3D problems around 15% of the total CPU time per time step, while the solution of the equations (with typically 3 iterations per time step) and the system assembly consume approximately 70% and 15% of the CPU time per time step, respectively. These figures refer to analyses in a single processor Pentium IV PC [36]. Considerable speed can be gained using parallel computing techniques.

In this work we will apply the PFEM to problems involving a rigid domain containing fluid particles only. Application of the PFEM in fluid and solid mechanics and in fluid-structure interaction problems can be found in [4,5,6,8,9,10,11,12,13,14,15,16,17,18,19,28,29,35,36,39,40,41,42,43], as well in www.cimne.com/pfem.

8 EXAMPLES

8.1 Sloshing of water in prismatic tank

The problem has been solved first in 2D. Figure 2 shows the analysis data. The fluid oscillates due to the hydrostatic forces induced by its original position.

Figure 2: 2D analysis of sloshing of water in rectangular tank. Initial geometry, analysis data and mesh of 5064 3-noded triangles discretizing the water in the tank

The problem has been run using different values of the parameter Failed to parse (MathML with SVG or PNG fallback (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 }

in 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 {\bf K}_v^e}
(Eq.(41)). The first set of results (Figures 3 and 4) were obtained 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 \theta =1}

. The problem was then solved 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 \theta = 0.08} , thereby, reducing in one order the magnitude the diagonal 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 {\bf K}_v^e} .

Figure 3 shows snapshots of the water geometry at different times. Pressure contours are superposed to the deformed geometry of the fluid in the figures.

Figure 4 shows the evolution of the percentage of water volume (i.e. mass) loss introduced by the numerical solution scheme. The accumulated volume loss (in percentage versus the initial volume) for the method proposed 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 \theta =1}

is approximately  1.33% over 20 seconds of simulation time (Figure 4a). The average volume variation in absolute value per time step 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/":): {\textstyle 1.09 \times 10^{-4}%}
(Figure 4b). The total water volume loss is the sum of the losses induced by the numerical scheme and the losses due to the updating of the free surface  using the PFEM. No correction of mass was introduced at the end of each time step. Taking all this into account, the fluid volume loss over the analysis period is remarkably low.

The volume losses induced by the free surface updating can be reduced using a finer mesh in that region in conjunction with an enhanced alpha shape technique.

The total fluid volume loss can be reduced to almost zero by introducing a small correction in the free surface at the end of each time step [41].

The fluid volume losses obtained using a standard first order fractional step method [41] and the PFEM are shown in Figure 4a for comparison. Clearly the method proposed in this work leads to a reduction in the overall fluid volume loss, as well as in the volume loss per time step.


(a) t=5.7s	(b) t=7.4s

(c) t=13.3s	(d) t=18.6s
Figure 3: 2D sloshing of water in rectangular tank. Snapshots of water geometry at two different 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/":): \theta = 1 ). Colours indicate pressure contours

(a) Accumulated volume loss over 20 seconds of analysis

(b) Volume variation (in %) per time step over 20 seconds of analysis (Current method 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/":): \theta = 1

)

Figure 4: 2D sloshing of water in rectangular tank. (a) Time evolution of the percentage of water volume loss due to the numerical algorithm. (b) Average volume variation per time step. Current method. 1.09Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \times 10^{-4}

% Fractional step: 2.07Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \times 10^{-4} %

Figure 5: 2D sloshing of water in rectangular tank. Time evolution of percentage of water volume loss obtained using the current method 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/":): \theta =0.08

(curve A) 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/":): \theta = 1
(curve 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/":): \Delta t = 10^{-3}s

Figure 6: 2D sloshing of water in rectangular tank. Time evolution of percentage of water volume loss obtained with the current method. Curve 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/":): \theta =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/":): \Delta t = 10^{-4}s

. Curve 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/":): \theta = 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/":): \Delta t = 10^{-3}s

		]]

(a) t=5.7s	(b) t=7.4s
Figure 7: 3D analysis of sloshing of water in prismatic tank (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \theta = 1 ). Analysis data and snapshots of water geometry 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/":): t=5.7 s (a) 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/":): t=7.4 s (b)

(a)

(b)

Figure 8: 3D analysis of sloshing of water in prismatic tank (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \theta = 1

). (a) Time evolution of accumulated water volume loss (in percentage) due to the numerical algorithm. (b) Volume loss (in %) per time step over 2 seconds of analysis. Average volume loss per time step: 1.64Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \times 10^{-4} %

Figure 9: Water sphere falling in a tank filled with water. Analysis data and initial discretization of the sphere and the water in the tank with 88892 4-noded tetrahedra


(a) t=0.175s	(b) t=0.275s

(c) t=0.5s	(d) t=0.9s
Figure 10: Water sphere falling in tank containing water. Evolution of the impact and mixing of the two liquids at different times. Results 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/":): \theta = 1

(a)

(b)

Figure 11: Water sphere falling in a tank containing water. (a) Accumulated volume over three seconds of analysis due to the numerical algorithm (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \theta =1

). (b) Volume loss (in %) per time step. Average volume variation per time step: 2.54Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \times 10^{-4} %


Figure 12: 2D sloshing of a fluid in a heated tank. Initial geometry, problem data, thermal boundary and initial conditions

Figure 13: 2D sloshing of a fluid in a heated tank. Snapshots of fluid geometry at six different times. Colours indicate temperature contours

Figure 14: 2D sloshing of a fluid in a heated tank. Evolution of temperature with time at the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A, 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/":): C
of  Figure 12

Figure 5 shows a comparison between the fluid volume loss 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 \theta =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 \theta = 0.08}
using the same time step in both cases (Failed to parse (MathML with SVG or PNG fallback (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 t = 10^{-3}s}

). Results show that the reduction of the tangent bulk stiffness matrix terms leads to an improvement in the preservation of the initial volume of the fluid. It is noted that the convergence of the iterative solution 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 \theta = 0.08}

was the same as 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 \theta =1}

.

Figure 6 shows that a similar improvement in the volume preservation can be obtained using Failed to parse (MathML with SVG or PNG fallback (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}

and reducing the time step to Failed to parse (MathML with SVG or PNG fallback (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 t = 10^{-4}s}

. This, however, increases the cost of the computations.

These results indicate that accurate numerical results with reduced volume losses can be obtained by appropriately adjusting the parameter Failed to parse (MathML with SVG or PNG fallback (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 }

in the tangent bulk modulus matrix while keeping the time step size to competitive values in terms of CPU cost. A study of the influence 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 }
in the numerical solution for quasi-incompressible free surface fluids in terms of volume preservation and overall accuracy using the formulation here presented can be found in [10].

More results for this example can be found in [41].

Figures 7 and 8 show a similar set of results for the 3D analysis of the same sloshing problem using a relative coarse initial mesh of 106771 4-noded tetrahedra 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 \theta = 1} . It is remarkable that the percentage of total fluid volume loss due to the numerical scheme after 10 seconds of analysis is approximately 1%

8.2 Falling of a water sphere in a cylindrical tank containing water

This example is the 3D analysis of the impact of a sphere made of water as it falls in a cylindrical tank containing water. Both the water in the sphere and in the tank mix in a single fluid after the impact. Figure 9 shows the material and analysis data and the initial discretization of the sphere and the water in the tank in 88892 4-noded tetrahedra. The problem was solved with the new stabilized method presented in the paper 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 \theta = 1} . Figure 10 shows snapshots of the mixing process at different times. An average of four iterations for convergence of the velocity and the pressure were needed during all the steps of the analysis. The total water mass lost in the sphere and the tank due to the numerical algorithm was Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \simeq }

2% after 3 seconds of analysis (Figure 11a).

8.3 Sloshing of a fluid in a heated tank

A fluid at initial temperature Failed to parse (MathML with SVG or PNG fallback (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=20\,^{\circ } C}

oscillates due to the hydrostatic forces induced by its initial position in a rectangular tank heated to a uniform and constant temperature 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 T=75\,^{\circ } C}

. The geometry and the problem data of the 2D simulation are shown in Figure 12. The fluid, with a very high thermal conductivity, changes its temperature only due to the contact with the hotter tank walls. The heat flux along the free surface has been considered null. The fluid domain has been initially discretized with 2828 3-noded triangles. The coupled thermal-fluid dynamics simulation has been run for 100 s using a time step increment 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 t} =Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0.005 s} .

Figure 13 shows some snapshots of the numerical simulation. The temperature contours have been superposed on the fluid domain at the different time instants.

In Figure 14 the evolution of temperature with time at the points Failed to parse (MathML with SVG or PNG fallback (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, 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}
of  Figure 12 is plotted. The coordinates of these sample points 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 m,0.1m}

), (Failed to parse (MathML with SVG or PNG fallback (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,0.4m} ) 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 m,0.1m} ), respectively. Figures 13 and 14 show that the fluid does not heat uniformly because of the convection effect automatically captured by the Lagrangian technique here presented.

8.4 Falling of a cylindrical object in a heated tank filled with fluid

An elastic object falls in a tank containing a fluid at rest. The tank walls are maintained at temperature Failed to parse (MathML with SVG or PNG fallback (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} =Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 75\,^{\circ } C}

during the whole analysis, while the fluid and the solid object have an initial temperature 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 T=20\,^{\circ } C}

. The geometry and the problem data of the 2D simulation as well the thermal initial and boundary conditions, are shown in Figure 15. Both the fluid and the solid have a high thermal conductivity. The heat flux along the fluid and solid surfaces in contact with the air has been considered null. The fluid and the solid domains have been discretized with 1986 and 108 3-noded triangular finite elements, respectively. The duration of the simulation is 10 s and the time step increment chosen 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/":): {\textstyle \Delta t} =Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0.0005 s} .

Figure 16 collects some representative snapshots of the numerical simulation with the temperature results plotted over the fluid and the solid domains.

The graph of Figure 17 is the evolution of temperature at the central point of the solid object. As expected, its temperature tends to Failed to parse (MathML with SVG or PNG fallback (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} =Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 75\,^{\circ } C} .

Figure 15: Falling of a solid object in a heated tank filled with fluid. Initial geometry, problem data, thermal boundary and initial conditions

Figure 16: Falling of a solid object in a heated tank filled with fluid. Snapshots at six different times. Colours indicate temperature contours

Figure 17: Falling of a solid object in a heated tank filled with fluid. Time evolution of the temperature at the center of the solid.

9 CONCLUDING REMARKS

We have presented a new FIC-based stabilized Lagrangian finite element method for thermal-mechanical analysis of quasi and fully incompressible flows and FSI problems that has excellent mass preservation properties. The method has been successfully applied to the adiabatic and thermal-mechanical analysis of free-surface quasi-incompressible flows using the PFEM and an updated Lagrangian formulation. These problems are more demanding in terms of the mass preservation features of the numerical algorithm. The method proposed has yielded excellent results for 2D and 3D adiabatic and thermally-coupled free surface flow problems involving surface waves, water splashing, violent impact of flows with containment walls and FSI situations.

ACKNOWLEDGEMENTS

This research was partially supported by the Advanced Grant project SAFECON of the European Research Council.

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Revision as of 13:50, 9 April 2018

Abstract

1 INTRODUCTION

2 GOVERNING EQUATIONS

Momentum equations

Mass balance equation

Thermal balance

Boundary conditions

Mechanical problem

Thermal problem

3 STABILIZED MASS BALANCE EQUATION

Second order FIC mass balance equation in space

First order FIC mass balance equation in time

4 VARIATIONAL EQUATIONS

4.1 Momentum equations

4.2 Mass balance equations

4.3 Thermal balance equation

5 FEM DISCRETIZATION

6 TRANSIENT SOLUTION OF THE DISCRETIZED EQUATIONS

Step 1. Compute 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/":): \Delta \bar {v}

Step 2. Update the nodal velocities

Step 3. Compute 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{\bf p}^{i+1}

Step 4. Update the nodal coordinates

Step 5.Compute the nodal temperatures

Step 6. Check convergence

7 ABOUT THE PARTICLE FINITE ELEMENT METHOD (PFEM)

7.1 The basis of the PFEM

8 EXAMPLES

8.1 Sloshing of water in prismatic tank

8.2 Falling of a water sphere in a cylindrical tank containing water

8.3 Sloshing of a fluid in a heated tank

8.4 Falling of a cylindrical object in a heated tank filled with fluid

9 CONCLUDING REMARKS

ACKNOWLEDGEMENTS

Bibliography

Document information

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