Onate Rojek et al 2006a - Revision history

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 <div id="cite-58"></div>
-'''[[#citeF-58|[58]]]''' J. Rojek, E. Oñate and R.L. Taylor, “CBS based stabilization in explicit solid mechanics”. ''Int. J. Num. Mech. Engng.'' Accepted for publication. October 2005.
+'''[[#citeF-58|[58]]]''' J. Rojek, E. Oñate and R.L. Taylor, “CBS based stabilization in explicit solid mechanics”. ''Int. J. Num. Mech. Engng.'', '''66'''(10), 1547-1568, 2006.
 ==APPENDIX==

← Older revision	Revision as of 13:26, 29 October 2018
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	+	Published in ''Comput. Methods Appl. Mech. Engrg.'' Vol. 195, pp. 6750-6777, 2006<br />
	+	doi: 10.1016/j.cma.2004.10.018

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 In Eqs.(24)
-<span id="eq-78"></span>
+<span id="eq-A.1"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 1,939: / Line 1,939: @@
 | style="text-align: center;" | <math>{\bf H}={\bf A}+{\bf K}+\hat {\bf K}  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (78)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (A.1)
 |}
@@ Line 1,972: / Line 1,972: @@
 It is understood that all the arrays are matrices (except <math display="inline">{f}</math>, which is a vector) whose components are obtained by grouping together the left indexes in the previous expressions (<math display="inline">a</math> and possibly <math display="inline">i</math>) and the right indexes (<math display="inline">b</math> and possibly <math display="inline">j</math>).
-Note that the stabilization matrix <math display="inline">\hat{K}</math> in Eq.([[#eq-78|78]]) adds additional orthotropic diffusivity terms of value <math display="inline">\rho \displaystyle{h_ku_l\over 2}</math>.
+Note that the stabilization matrix <math display="inline">\hat{K}</math> in Eq.([[#eq-A.1|A.1]]) adds additional orthotropic diffusivity terms of value <math display="inline">\rho \displaystyle{h_ku_l\over 2}</math>.

@@ Line 1,519: / Line 1,519: @@
 |}
-'''Remark'''. The examples presented show that the FIC/FEM formulation is an effective procedure for solving bulk metal forming problems involving full or quasi-incompressible situations. The key advantage of the FIC approach versus more standard mixed FEM formulations is that it provides a natural theoretical framework for equal order finite element interpolations for the velocity and pressure variables, both in the context of implicit and explicit solution schemes. We note the simplicity and effectiveness of the full explicit algorithm as demonstrated in the examples presented. The FIC formulation reproduces also the best feature of the so called stabilized FEM method for incompressible problems, such as the CBS scheme <span id='citeF-20'></span>[[#cite-20|[20,21,25,26,58]]], the pressure gradient operator method <span id='citeF-22'></span>[[#cite-22|22]] and the subgrid scale method <span id='citeF-23'></span>[[#cite-23|[23]]] among others.
+'''Remark'''. The examples presented show that the FIC/FEM formulation is an effective procedure for solving bulk metal forming problems involving full or quasi-incompressible situations. The key advantage of the FIC approach versus more standard mixed FEM formulations is that it provides a natural theoretical framework for equal order finite element interpolations for the velocity and pressure variables, both in the context of implicit and explicit solution schemes. We note the simplicity and effectiveness of the full explicit algorithm as demonstrated in the examples presented. The FIC formulation reproduces also the best feature of the so called stabilized FEM method for incompressible problems, such as the CBS scheme <span id='citeF-20'></span>[[#cite-20|[20,21,25,26,58]]], the pressure gradient operator method <span id='citeF-22'></span>[[#cite-22|[22]]] and the subgrid scale method <span id='citeF-23'></span>[[#cite-23|[23]]] among others.
 ==8 LAGRANGIAN FLOWS. THE PARTICLE FINITE ELEMENT  METHOD==

@@ Line 1,322: / Line 1,322: @@
 A numerical simulation of an aluminium casting process is presented as a demonstration of  the  accuracy  of the stabilized formulation. The computations are performed with the finite element code VULCAN where the stabilized FEM presented has been implemented <span id='citeF-56'></span>[[#cite-56|[56]]].
-The analysis simulates the casting process of an aluminium (AlSi7Mg) specimen in a steel (X40CrMoV5) mould. Material behavior of aluminium casting has been modeled by a fully coupled thermo-viscoplastic model, while the steel mould  has been modeled by a simpler thermo-elastic model <span id='citeF-51'></span>[[#cite-51|[51]]. Geometrical and material data were provided by the foundry RUFFINI. Figure 1  shows the finite element mesh used for the part and the cooling system. The full mesh, including the mould has  380.000 four node tetrahedra. The pouring  temperature is 650<math display="inline">^\circ </math>C. Initial temperature for the mould is obtained through a thermal die-cycling simulation. Figure 2 shows the evolution of the mould temperature after 6 cycles. The cooling system has been kept at 20<math display="inline">^\circ </math>C. Filling evolution has been simulated as  in a pressure die-casting process using the stabilized VOF technique described in <span id='citeF-49'></span>[[#cite-49|[47,48]]]. Figure 3 shows different time steps of the simulation.
+The analysis simulates the casting process of an aluminium (AlSi7Mg) specimen in a steel (X40CrMoV5) mould. Material behavior of aluminium casting has been modeled by a fully coupled thermo-viscoplastic model, while the steel mould  has been modeled by a simpler thermo-elastic model <span id='citeF-51'></span>[[#cite-51|[51]]]. Geometrical and material data were provided by the foundry RUFFINI. Figure 1  shows the finite element mesh used for the part and the cooling system. The full mesh, including the mould has  380.000 four node tetrahedra. The pouring  temperature is 650<math display="inline">^\circ </math>C. Initial temperature for the mould is obtained through a thermal die-cycling simulation. Figure 2 shows the evolution of the mould temperature after 6 cycles. The cooling system has been kept at 20<math display="inline">^\circ </math>C. Filling evolution has been simulated as  in a pressure die-casting process using the stabilized VOF technique described in <span id='citeF-49'></span>[[#cite-49|[47,48]]]. Figure 3 shows different time steps of the simulation.
 <div id='img-1'></div>

← Older revision	Revision as of 10:15, 12 November 2018
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@@ Line 668: / Line 668: @@
 In above <math display="inline">\phi </math> is the temperature, <math display="inline">c</math> and <math display="inline">k</math> are the specific heat and the thermal conductivity of the material, respectively, <math display="inline">Q</math> is the heat source and <math display="inline">h_j</math> are the characteristic length distances which are typical of the FIC formulation <span id='citeF-30'></span>[[#cite-30|[30]]].
-Eq.(39a) is completed with the Dirichlet and Neuman boundary conditions for the heat problem. For details see <span id='citeF-35'></span>[[#cite-35|35]].
+Eq.(39a) is completed with the Dirichlet and Neuman boundary conditions for the heat problem. For details see <span id='citeF-35'></span>[[#cite-35|[35]]].
 The convective velocities <math display="inline">v_i</math> in Eq.(39b) are provided by the solution of the fluid flow problem. As usual in metal forming processes, the heat source <math display="inline">Q</math> is a function of the mechanical work generated in the flow of the material during the forming process. The temperature field affects in turn the flow viscosity via its dependence with the yield stress which is very sensitive to the temperature changes. The solution of the heat transfer equation is therefore fully coupled with that of the fluid flow problem <span id='citeF-30'></span>[[#cite-30|[30,46]]].

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 ==Abstract==
@@ Line 22: / Line 5: @@
 The paper describes some recent developments  in finite element and particle methods for analysis  of a wide range of bulk forming  processes. The developments include new stabilized linear triangles and tetrahedra  using finite calculus and a new procedure combining particle methods and finite element methods. Applications of the new numerical methods  to casting, forging and  other bulk metal forming problems and mixing processes are shown.
-'''keywords''' Bulk forming processes, stabilized finite element method, particle method, particle finite element method, mixing processes.
+'''Keywords''': Bulk forming processes, stabilized finite element method, particle method, particle finite element method, mixing processes.
 ==1 INTRODUCTION==
-The development of efficient and robust numerical methods for analysis of bulk forming problems  has been a subject of intensive research in recent years <span id='citeF-1'></span>[[#cite-1|1]]. Many of these problems require the solution of incompressible fluid flow situations (such as in mould filling problems) whereas in other cases (such as forging, rolling, extrusion, etc.) the numerical method must be able to account for the quasi/fully incompressible behaviour induced by the large plastic deformation. The solution of these problems has motivated the development of the so called ''stabilized numerical methods'' overcoming the two main sources of instability in the analysis of incompressible  continua, namely those originated by the high values of the convective terms in fluid flow situtions and those induced by the difficulty in satisfying the incompressibility condition.
+The development of efficient and robust numerical methods for analysis of bulk forming problems  has been a subject of intensive research in recent years <span id=‘citeF-1’></span>[[#cite-1|[1-7]]]. Many of these problems require the solution of incompressible fluid flow situations (such as in mould filling problems) whereas in other cases (such as forging, rolling, extrusion, etc.) the numerical method must be able to account for the quasi/fully incompressible behaviour induced by the large plastic deformation. The solution of these problems has motivated the development of the so called ''stabilized numerical methods'' overcoming the two main sources of instability in the analysis of incompressible  continua, namely those originated by the high values of the convective terms in fluid flow situtions and those induced by the difficulty in satisfying the incompressibility condition.
-Different approaches to solve both type of  problems in the context of the finite element method (FEM) have been recently developed <span id='citeF-8'></span>[[#cite-8|8]]. Traditionally, the underdiffusive character of the Galerkin FEM for high convection flows  has been corrected by adding some kind of artificial viscosity terms to the standard Galerkin equations <span id='citeF-8'></span>[[#cite-8|8]].
+Different approaches to solve both type of  problems in the context of the finite element method (FEM) have been recently developed <span id=‘citeF-8’></span>[[#cite-8|[8]]]. Traditionally, the underdiffusive character of the Galerkin FEM for high convection flows  has been corrected by adding some kind of artificial viscosity terms to the standard Galerkin equations <span id=‘citeF-8’></span>[[#cite-8|[8,9]]].
-A popular way to overcome the problems with the incompressibility constraint in the FEM is by introducing a pseudo-compressibility in the continuum  and using implicit and explicit algorithms ''ad hoc'' such as artificial compressibility schemes <span id='citeF-10'></span>[[#cite-10|10]] and preconditioning techniques <span id='citeF-11'></span>[[#cite-11|11]]. Other FEM schemes with good stabilization properties for the convective and incompressibility terms  in fluid flows 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-8'></span>[[#cite-8|8]]. 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 FEM of this kind  we can name the Streamline Upwind Petrov Galerkin (SUPG) method <span id='citeF-8'></span>[[#cite-8|8]], the Galerkin Least Square (GLS) method <span id='citeF-8'></span>[[#cite-8|8]], the Taylor-Galerkin method <span id='citeF-18'></span>[[#cite-18|18]], the Characteristic Galerkin method <span id='citeF-19'></span>[[#cite-19|19]] and its variant the Characteristic Based Split (CBS) method <span id='citeF-20'></span>[[#cite-20|20]], the pressure gradient operator method <span id='citeF-22'></span>[[#cite-22|22]] and the Subgrid Scale (SS) method <span id='citeF-23'></span>[[#cite-23|23]]. A good review of these methods can be found in <span id='citeF-24'></span>[[#cite-24|24]]. Extensions of the CBS and SS methods to treat incompressible problems in solid mechanics are reported in <span id='citeF-25'></span>[[#cite-25|25]] and <span id='citeF-32'></span>[[#cite-32|32]], respectively.
+A popular way to overcome the problems with the incompressibility constraint in the FEM is by introducing a pseudo-compressibility in the continuum  and using implicit and explicit algorithms ''ad hoc'' such as artificial compressibility schemes <span id=‘citeF-10’></span>[[#cite-10|[10]]] and preconditioning techniques <span id=‘citeF-11’></span>[[#cite-11|[11]]]. Other FEM schemes with good stabilization properties for the convective and incompressibility terms  in fluid flows 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-8’></span>[[#cite-8|[8,9,12]]]. 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 FEM of this kind  we can name the Streamline Upwind Petrov Galerkin (SUPG) method <span id=‘citeF-8’></span>[[#cite-8|[8,13-16]]], the Galerkin Least Square (GLS) method <span id=‘citeF-8’></span>[[#cite-8|[8,17]]], the Taylor-Galerkin method <span id='citeF-18'></span>[[#cite-18|[18]]], the Characteristic Galerkin method <span id='citeF-19'></span>[[#cite-19|[19]]] and its variant the Characteristic Based Split (CBS) method <span id='citeF-20'></span>[[#cite-20|[20,21]]], the pressure gradient operator method <span id='citeF-22'></span>[[#cite-22|[22]]] and the Subgrid Scale (SS) method <span id='citeF-23'></span>[[#cite-23|[23]]]. A good review of these methods can be found in <span id='citeF-24'></span>[[#cite-24|[24]]]. Extensions of the CBS and SS methods to treat incompressible problems in solid mechanics are reported in <span id='citeF-25'></span>[[#cite-25|[25,26,58]]] and <span id='citeF-32'></span>[[#cite-32|[27-29]]], respectively.
 In this paper a different class of stabilized FEM for quasi and fully incompressible fluid and solid materials applicable to a wide range of bulk forming problems is presented. The starting point is  the modified governing differential equations of the continuous  problem formulated via a finite calculus (FIC) approach <span id='citeF-35'></span>[[#cite-35|35]]. The FIC method is based in invoking the classical balance (or equilibrium) laws in a  domain of ''finite size''. This introduces naturally additional terms in the  differential equations of infinitesimal continuum  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 FEM. One of the main advantages of the FIC formulation versus other alternative approaches (such as mixed FEM, etc.) is that it allows  to solve incompressible fluid problems using  low order finite elements (such as linear triangles and tetrahedra) with equal order approximations for the velocity and pressure variables <span id='citeF-37'></span>[[#cite-37|37]]. The FIC formulation has been successfully used for analysis of fully or quasi incompressible  solids <span id='citeF-41'></span>[[#cite-41|41]].

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 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{H}={A}+{K}+\hat {K}  </math>
+| style="text-align: center;" | <math>{\bf H}={\bf A}+{\bf K}+\hat {\bf K}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (78)
@@ Line 1,966: / Line 1,966: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{array}{l} \displaystyle M_{ij}^{ab}= \left(\int _{\Omega ^e} \rho N^a N^b d\Omega \right)\delta _{ij} \quad ,\quad A_{ij}^{ab}= \left(\int _{\Omega ^e} \rho N^a ({u}^T {\nabla } N^b) d\Omega \right)\delta _{ij}\\ \\ \displaystyle {K}_{ij}^{ab} = \left(\int _{\Omega ^e} \mu {\boldsymbol \nabla }^T N^a{\boldsymbol \nabla }N^b  d\Omega \right)\delta _{ij} \quad ,\quad {\boldsymbol \nabla } = \left[{\partial  \over \partial x_1},{\partial  \over \partial x_2},{\partial  \over \partial x_3}\right]^T\\ \\ \displaystyle \hat{K}_{ij}^{ab} = \left({1\over 2} \int _{\Omega ^e} ({h}^T {\boldsymbol \nabla } N^a)(\rho {u}^T {\boldsymbol \nabla } N^b)d\Omega \right)\delta _{ij}\quad ,\quad {G}_{i}^{ab}=\int _{\Omega ^e} {\partial N^a \over \partial x_i}N^b d\Omega \\ \\ \displaystyle {C}= \left[\begin{matrix}{C}_1\\ {C}_2\\ {C}_3\\\end{matrix}\right]\quad ,\quad {C}_{ij}^{ab}=\left({1\over 2} \int _{\Omega ^e} [{h}^T {\boldsymbol \nabla } N^a]N^bd\Omega \right)\delta _{ij}\\ \\ \displaystyle \hat L^{ab}= \int _{\Omega ^e} ({\boldsymbol \nabla }^T N^a) [\tau ] {\boldsymbol \nabla } N^b d\Omega \quad ,\quad [\tau ]= \left[\begin{matrix}\tau _1 &0 &0 \\ 0 & \tau _2&0\\ 0&0& \tau _3\\\end{matrix}\right]\\ \\ \displaystyle {Q}= [{Q}_1,{Q}_2,{Q}_3] \quad ,\quad  \displaystyle Q_{i}^{ab} = \int _{\Omega ^e}\tau _i {\partial N^a \over \partial x_i} N^b d\Omega \quad \quad \hbox{no sum in }i\\ \\ \displaystyle \hat{C}= [\hat{C}_1,\hat{C}_2,\hat{C}_3] \quad ,\quad  \displaystyle \hat{C}_1^{ab}=\hat{C}_2^{ab}=\hat{C}_3^{ab} = \int _{\Omega ^e} \rho ^2 N^a ({u}^T {\boldsymbol \nabla }N^b)d\Omega \\ \\ \displaystyle  \hat {M}^{ab}_{ij}= \left(\int _{\Omega ^e} \tau _i N^a N^b d\Omega \right)\delta _{ij}\quad ,\quad  \displaystyle {f}_i^a = \int _{\Omega ^e} N^a f_i d\Omega + \int _{\Gamma ^e} N^a t_i d\Gamma  \end{array}</math>
+| style="text-align: center;" | <math>\begin{array}{l} \displaystyle M_{ij}^{ab}= \left(\int _{\Omega ^e} \rho N^a N^b d\Omega \right)\delta _{ij} \quad ,\quad A_{ij}^{ab}= \left(\int _{\Omega ^e} \rho N^a ({\textbf u}^T {\nabla } N^b) d\Omega \right)\delta _{ij}\\ \\
 |}
 |}