Onate et al 2003c - Revision history

Move page script: Move page script moved page Draft Samper 187067199 to Onate et al 2003c over redirect

2019-05-27T11:34:56Z

Move page script moved page Draft Samper 187067199 to Onate et al 2003c over redirect

Cinmemj at 08:45, 21 May 2019

2019-05-21T08:45:03Z

Cinmemj at 10:47, 20 May 2019

2019-05-20T10:47:41Z

Cinmemj at 10:39, 20 May 2019

2019-05-20T10:39:35Z

Cinmemj at 10:36, 20 May 2019

2019-05-20T10:36:06Z

Cinmemj at 10:33, 20 May 2019

2019-05-20T10:33:49Z

Show changes

Cinmemj at 10:27, 20 May 2019

2019-05-20T10:27:52Z

Cinmemj at 10:24, 20 May 2019

2019-05-20T10:24:17Z

Cinmemj at 10:23, 20 May 2019

2019-05-20T10:23:25Z

Cinmemj at 10:22, 20 May 2019

2019-05-20T10:22:03Z

@@ Line 1,424: / Line 1,424: @@
 |}
-<div id='img-19'></div>
+<div id='img-18b'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
 |[[Image:Draft_Samper_187067199-Fig1contBathe.png|451px|cont.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 19:''' cont.
+| colspan="1" | '''Figure 18:''' cont.
 |}
-<div id='img-20'></div>
+<div id='img-19'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
 |[[Image:Draft_Samper_187067199-Fig4ArtBathe.png|451px|Rotating water mill.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 20:''' Rotating water mill.
+| colspan="1" | '''Figure 19:''' Rotating water mill.
 |}
-<div id='img-21'></div>
+<div id='img-20'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
 |[[Image:Draft_Samper_187067199-Fig5ArtBathe.png|451px|Solid cube falling into a recipient with water.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 21:''' Solid cube falling into a recipient with water.
+| colspan="1" | '''Figure 20:''' Solid cube falling into a recipient with water.
 |}

@@ Line 1: / Line 1: @@
-<!-- metadata commented in wiki content
+Published in ''Fluid Dynamics and Aeronautics New Challenges'', Périaux J., Champion M., Gagnepain J.-J., Pironneau O., Stoufflet B. and Thomas Ph.(Eds.), CIMNE, Barcelona, Spain, 2003
-==A GENERAL STABILIZED FORMULATION FOR INCOMPRESSIBLE FLUID FLOW USING FINITE CALCULUS AND THE FINITE ELEMENT METHOD==
-−
-'''E. Oñate, J. García, G. Bugeda '''and''' S.R. Idelsohn'''
-−
-{|
-|-
-|International Centre for Numerical Methods in Engineering
-|-
-| Universidad Politécnica de Cataluña
-|-
-| Gran Capitán s/n, 08034 Barcelona, Spain
-|-
-| E&#8211;mail: [mailto:onate@cimne.upc.es onate@cimne.upc.es]
-|-
-| Web page: [http://www.cimne.upc.es www.cimne.upc.es]
-|}
--->
-−
 ==Abstract==

@@ Line 1,190: / Line 1,190: @@
 The examples chosen show the applicability of the Eulerian, ALE and lagrangian formulations presented to solve fluid flow problems. Linear tetrahedra and triangles have been used in the 3D and 2D analysis shown. The fractional step algorithm of Section [[#4.2 Fractional step method|4.2]] for <math display="inline">\theta _1 = \theta _3=\theta _4=0</math> and <math display="inline">\alpha =1</math> has been used in all cases. The first example is the standard square cavity problem solved in 3D using an Euler formulation (<math display="inline">\mu =0</math>). The second example is the flow past a submerged NACA 0012 profile. Here the free surface equation is solved together with the flow equations.
-The next four examples fall within the category of fluid-structure interaction problems. The first is the analysis of a sphere falling in a tube filled with liquid where the mesh updating procedure of Section 5.2 has been used. Then three of ship hydrodynamics problems are solved including the analysis of a Wigley hull, a scale model of a commercial ship and an American Cup racing sail boat. Numerical results are compared with experimental data in all cases.
+The next four examples fall within the category of fluid-structure interaction problems. The first is the analysis of a sphere falling in a tube filled with liquid where the mesh updating procedure of Section [[#5.2 A simple algorithm for updating the mesh nodes|5.2]] has been used. Then three of ship hydrodynamics problems are solved including the analysis of a Wigley hull, a scale model of a commercial ship and an American Cup racing sail boat. Numerical results are compared with experimental data in all cases.
 The last series of examples show applications of the Lagrangian formulation to the simulation of the collapse of a water column, a semi-submerged rotating water mill and a solid cube falling into a water recipient.

@@ Line 668: / Line 668: @@
 where <math display="inline">0\le \theta _i\le 1</math>.
-In above <math display="inline">\bar{(\cdot )}^{n,i}</math> denote nodal values at the <math display="inline">n</math>th time step and the ith iteration. Note that <math display="inline">{\boldsymbol A}^{n+\theta _1,i-1}\equiv {\boldsymbol A}(\bar {\boldsymbol u}^{n+\theta _1,i-1})</math> etc. Also <math display="inline">(\cdot )^{n+\theta _i,0}\equiv (\cdot )^n</math> for the computations in step 1 at the onset of the iterations.
+In above <math display="inline">\bar{(\cdot )}^{n,i}</math> denote nodal values at the <math display="inline">n</math>th time step and the ith iteration. Note that <math display="inline">{\boldsymbol A}^{n+\theta _1,i-1}\equiv {\boldsymbol A}(\bar {\boldsymbol u}^{n+\theta _1,i-1})</math> etc. Also <math display="inline">(\cdot )^{n+\theta _i,0}\equiv (\cdot )^n</math> for the computations in step [[#step-1|1]] at the onset of the iterations.
 Steps [[#step-1|1]], [[#step-3|3]] and [[#step-4|4]] can be solved explicitely by choosing a ''lumped (diagonal) form'' of matrices <math display="inline">{\boldsymbol M}</math> and <math display="inline">\hat {\boldsymbol M}</math>. In this manner the main computational cost is the solution of step [[#step-2|2]] involving the inverse of a Laplacian matrix. This can be solved very effectively using an iterative method such as the conjugate gradient method or similar.

@@ Line 30: / Line 30: @@
 The solution of above problems in the context of the finite element method (FEM) has been attempted in a number of ways <span id='citeF-1'></span>[[#cite-1|[1]]]. The underdiffusive character of the Galerkin FEM for high convection flows (which incidentaly also occurs for the central finite difference (FD) and finite volume (FV) methods <span id='citeF-2'></span>[[#cite-2|[2]]]) has been corrected by adding some kind of artificial viscosity terms to the standard Galerkin equations.
-A popular way to overcome the problems with the incompressibility constraint is by introducing a pseudo-compressibility in the flow and using implicit and explicit algorithms developed for this kind of problems such as artificial compressibility schemes <span id='citeF-4'></span>[[#cite-4|[4]]-<span id='citeF-6'></span>[[#cite-6|6]]] and preconditioning techniques <span id='citeF-7'></span>[[#cite-7|[7]]]. Other FEM schemes with good stabilization properties for the convective and incompressibility terms  are based in Petrov-Galerkin (PG) techniques. The background of PG methods are the non-centred (upwind) schemes for computing the first derivatives of the convective operator in FD and FV methods <span id='citeF-2'></span>[[#cite-2|[2]],<span id='citeF-8'></span>[[#cite-8|8]]]. More recently a general class of Galerkin FEM has been developed where the standard Galerkin variational form is extended with adequate residual-based terms in order to achieve a stabilized numerical scheme. Among the many methods of this kind in the finite element universe we can name the Streamline Upwind Petrov Galerkin (SUPG) method <span id='citeF-1'></span>[[#cite-1|[1]],<span id='citeF-29'></span>[[#cite-9|9]]-<span id='citeF-18'></span>[[#cite-18|18]]] the Galerkin Least Square (GLS) method <span id='citeF-19'></span>[[#cite-19|[19]],<span id='citeF-20'></span>[[#cite-20|20]]], the Taylor-Galerkin method <span id='citeF-21'></span>[[#cite-21|[21]]], the Characteristic Galerkin method <span id='citeF-22'></span>[[#cite-22|[22]]-<span id='citeF-24'></span>[[#cite-24|24]]] and its variant the Characteristic Based Split (CBS) method <span id='citeF-25'></span>[[#cite-25|[25]],<span id='citeF-26'></span>[[#cite-26|26]]], pressure gradient operator methods <span id='citeF-27'></span>[[#cite-27|[27]]] and the Subgrid Scale (SS) method <span id='citeF-28'></span>[[#cite-28|[28]]-<span id='citeF-30'></span>[[#cite-30|30]]]. A good review of these methods can be found in <span id='citeF-31'></span>[[#cite-31|[31]]].
+A popular way to overcome the problems with the incompressibility constraint is by introducing a pseudo-compressibility in the flow and using implicit and explicit algorithms developed for this kind of problems such as artificial compressibility schemes <span id='citeF-4'></span><span id='citeF-5'></span>[[#cite-4|[4]]-<span id='citeF-6'></span>[[#cite-6|6]]] and preconditioning techniques <span id='citeF-7'></span>[[#cite-7|[7]]]. Other FEM schemes with good stabilization properties for the convective and incompressibility terms  are based in Petrov-Galerkin (PG) techniques. The background of PG methods are the non-centred (upwind) schemes for computing the first derivatives of the convective operator in FD and FV methods <span id='citeF-2'></span>[[#cite-2|[2]],<span id='citeF-8'></span>[[#cite-8|8]]]. More recently a general class of Galerkin FEM has been developed where the standard Galerkin variational form is extended with adequate residual-based terms in order to achieve a stabilized numerical scheme. Among the many methods of this kind in the finite element universe we can name the Streamline Upwind Petrov Galerkin (SUPG) method <span id='citeF-1'></span>[[#cite-1|[1]],<span id='citeF-9'></span><span id='citeF-10'></span><span id='citeF-11'></span><span id='citeF-12'></span><span id='citeF-13'></span><span id='citeF-14'></span><span id='citeF-15'></span><span id='citeF-16'></span><span id='citeF-17'></span>[[#cite-9|9]]-<span id='citeF-18'></span>[[#cite-18|18]]] the Galerkin Least Square (GLS) method <span id='citeF-19'></span>[[#cite-19|[19]],<span id='citeF-20'></span>[[#cite-20|20]]], the Taylor-Galerkin method <span id='citeF-21'></span>[[#cite-21|[21]]], the Characteristic Galerkin method <span id='citeF-22'></span>[[#cite-22|[22]]-<span id='citeF-23'></span><span id='citeF-24'></span>[[#cite-24|24]]] and its variant the Characteristic Based Split (CBS) method <span id='citeF-25'></span>[[#cite-25|[25]],<span id='citeF-26'></span>[[#cite-26|26]]], pressure gradient operator methods <span id='citeF-27'></span>[[#cite-27|[27]]] and the Subgrid Scale (SS) method <span id='citeF-28'></span>[[#cite-28|[28]]-<span id='citeF-29'></span><span id='citeF-30'></span>[[#cite-30|30]]]. A good review of these methods can be found in <span id='citeF-31'></span>[[#cite-31|[31]]].
-In this paper a stabilized finite element formulation for incompressible flows is derived in a different manner. The starting point are the modified governing differential equations of the fluid flow problem formulated via a finite calculus (FIC) approach <span id='citeF-32'></span>[[#cite-32|[32]]]. The FIC method is based in invoking the balance of fluxes in a fluid domain of finite size. This introduces naturally additional terms in the classical differential equations of infinitesimal fluid mechanics which are a function of the balance domain dimensions. The new terms in the modified governing equations provide the necessary stabilization to the discrete equations obtained via the standard Galerkin finite element method <span id='citeF-33'></span>[[#cite-33|[33]]-<span id='citeF-39'></span>[[#cite-39|39]]].
+In this paper a stabilized finite element formulation for incompressible flows is derived in a different manner. The starting point are the modified governing differential equations of the fluid flow problem formulated via a finite calculus (FIC) approach <span id='citeF-32'></span>[[#cite-32|[32]]]. The FIC method is based in invoking the balance of fluxes in a fluid domain of finite size. This introduces naturally additional terms in the classical differential equations of infinitesimal fluid mechanics which are a function of the balance domain dimensions. The new terms in the modified governing equations provide the necessary stabilization to the discrete equations obtained via the standard Galerkin finite element method <span id='citeF-33'></span><span id='citeF-34'></span><span id='citeF-35'></span><span id='citeF-36'></span><span id='citeF-37'></span><span id='citeF-38'></span>[[#cite-33|[33]]-<span id='citeF-39'></span>[[#cite-39|39]]].
 The layout of the chapter is the following. In the next section, the main concepts of the FIC approach are introduced via a simple 1D convection-diffusion model problem. Then the basic FIC equations for incompressible flow problems are presented. The finite element discretization is introduced and the resulting matrix formulation is detailed. Both monolithic and fractional step schemes for the transient solution are presented.

← Older revision	Revision as of 11:34, 27 May 2019
(No difference)

@@ Line 1,480: / Line 1,480: @@
 The authors are grateful to Copa America Desafio Español SA for providing the geometry and experimental data of the racing boat analyzed.
-Examples [[#|10.1]]-[[#10.7 Lagrangian flow|10.7]] were analyzed with the finite element code Tdyn based on the FEM formulation here presented <span id='citeF-66'></span>[[#cite-66|[66]]].
+Examples [[#10.1 Square cavity problem|10.1]]-[[#10.7 Lagrangian flow|10.7]] were analyzed with the finite element code Tdyn based on the FEM formulation here presented <span id='citeF-66'></span>[[#cite-66|[66]]].
 Thanks are also given to Dr. Roberto Flores for many useful discussions.