Onate et al 2006a - Revision history

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← Older revision		Revision as of 13:05, 9 October 2018
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−	Published in ''Comput. Meth. Appl. Mech. Engng. '', Vol. 195, pp. 339-~~200~~, 2006<br />	+	Published in ''Comput. Meth. Appl. Mech. Engng. '', Vol. 195, pp. 339-362, 2006<br />
	doi: 10.1016/j.cma.2004.07.054		doi: 10.1016/j.cma.2004.07.054

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-<!-- metadata commented in wiki content
+Published in ''Comput. Meth. Appl. Mech. Engng. '', Vol. 195, pp. 339-200, 2006<br />
-==ERROR ESTIMATION AND MESH ADAPTIVITY IN INCOMPRESSIBLE VISCOUS FLOWS USING  A RESIDUAL POWER APPROACH ==
+doi: 10.1016/j.cma.2004.07.054
 ==Abstract==

@@ Line 327: / Line 327: @@
 where <math display="inline">N</math> is the number of nodes in the mesh.
-The element contributions to <math display="inline">{M}</math> and '''f''' are
+The element contributions to <math display="inline">{\bf M}</math> and <math display="inline">{\bf f}</math> are
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 341: / Line 341: @@
 We recall that in above <math display="inline">\displaystyle{\partial \hat u_i\over \partial x_j}</math> is the (discontinuous) derivative field obtained from the finite element solution and <math display="inline">\left(\displaystyle \overline{\partial \hat u_i\over \partial x_j}\right)_k</math> are the recovered derivative values at node <math display="inline">k</math>. The process is repeated for <math display="inline">i,j=1,n_d</math>.
-The full form of matrix <math display="inline">{M}</math> is used for the solution of Eq.(19) using a standard Jacobian iterative scheme.
+The full form of matrix <math display="inline">{\bf M}</math> is used for the solution of Eq.(19) using a standard Jacobian iterative scheme.
 The same procedure is followed in order to recover the nodal derivatives of the pressure and the nodal second derivatives of the velocity field <math display="inline">\left(\displaystyle{\partial ^2 \hat u_k\over \partial x_i\partial x_j}\right)</math> needed for computation of all the terms in the recovered momentum equations <math display="inline">R_{m_i}</math>.

@@ Line 308: / Line 308: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{M} \bar{\hat {d}}_{ij}= {f}_{ij} </math>
+| style="text-align: center;" | <math>{\bf M} \bar{\hat {\bf d}}_{ij}= {\bf f}_{ij} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (19)
@@ Line 320: / Line 320: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\bar{\hat {d}}_{ij}= \left[\left(\overline{\partial \hat u_i\over \partial x_j}\right)_1,\left(\overline{\partial \hat u_i\over \partial x_j}\right)_2,\cdots , \left(\overline{\partial \hat u_i\over \partial x_j}\right)_N\right]^T </math>
+| style="text-align: center;" | <math>\bar{\hat {\bf d}}_{ij}= \left[\left(\overline{\partial \hat u_i\over \partial x_j}\right)_1,\left(\overline{\partial \hat u_i\over \partial x_j}\right)_2,\cdots , \left(\overline{\partial \hat u_i\over \partial x_j}\right)_N\right]^T </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (20)

@@ Line 129: / Line 129: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>u_i\simeq \hat u_i =  \sum \limits _{j=1}^n N_j (\hat u_i)_j = {N}\hat{u}_i </math>
+| style="text-align: center;" | <math>u_i\simeq \hat u_i =  \sum \limits _{j=1}^n N_j (\hat u_i)_j = {\bf N}\hat{\bf u}_i </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" |  (8a)
@@ Line 139: / Line 139: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>p\simeq \hat p =   \sum \limits _{j=1}^n N_j \hat p_j = {N}\hat{p}</math>
+| style="text-align: center;" | <math>p\simeq \hat p =   \sum \limits _{j=1}^n N_j \hat p_j = {\bf N}\hat{\bf p}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" |  (8b)

← Older revision	Revision as of 14:14, 10 October 2018
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 ==1 INTRODUCTION==
-The estimation of the error in the finite element solution of the Navier-Stokes equations for an incompressible fluid is a difficult problem <span id='citeF-1'></span>[[#cite-1|[1]]]. The presence of convective terms in the momentum equations prevents the use of standard error estimators based on the energy norm of the error developed for elliptic type problems typical of solid mechanics and heat transfer analysis <span id='citeF-2'></span>[[#cite-2|[2]]]. Extension of these methods to fluid mechanics problems have been reported in <span id='citeF-5'></span>[[#cite-5|[5]]]. A different class of mesh refinement procedures for fluid flow problems based on error indicators using approximations of the Hessian operator has been reported in <span id='citeF-6'></span>[[#cite-6|[6]]]. Alternative error estimation approaches for fluid flow problems  can be found in <span id='citeF-11'></span>[[#cite-11|[11]]].
+The estimation of the error in the finite element solution of the Navier-Stokes equations for an incompressible fluid is a difficult problem <span id='citeF-1'></span>[[#cite-1|[1]]]. The presence of convective terms in the momentum equations prevents the use of standard error estimators based on the energy norm of the error developed for elliptic type problems typical of solid mechanics and heat transfer analysis <span id='citeF-2'></span>[[#cite-2|[2-4]]]. Extension of these methods to fluid mechanics problems have been reported in <span id='citeF-5'></span>[[#cite-5|[5]]]. A different class of mesh refinement procedures for fluid flow problems based on error indicators using approximations of the Hessian operator has been reported in <span id='citeF-6'></span>[[#cite-6|[6-10]]]. Alternative error estimation approaches for fluid flow problems  can be found in <span id='citeF-11'></span>[[#cite-11|[11-16]]].
-The error estimation method presented in this paper is based on the evaluation of a particular form of the energy rate (the power) of the finite element residuals of the momentum and mass balance (incompressibility) equations. The residuals are computed using recovered values of the derivatives of the velocity and pressure variable obtained via a nodal derivative recovery technique. Two mesh adaptation processes based on: a)  the equi-distribution of the residual power  among the elements in the mesh and b) the equi-distribution of the density of the total residual power are presented. The stabilized form of the Navier-Stokes equations using a finite calculus (FIC) procedure developed by the authors <span id='citeF-17'></span>[[#cite-17|[17]]] is used for the FE solution of the examples presented in the paper using a fractional step scheme. This allows the use of linear triangles and tetrahedra with an equal order interpolation for the velocity and pressure variables. A nodal-based approach is used for computation of the residual power error and the total power generated by the viscous stresses and the pressure terms in the mesh. The residual power form derived from the FIC approach includes power terms emanating from the velocity field and from the gradient of the velocity field. It is shown that accounting for the terms involving the velocity gradient provides an effective  error estimation and mesh adaptation procedure for CFD problems. Also the FIC form of the residual power expression can be effectively extended for error estimation and mesh adaptivity of other problems in computational mechanics <span id='citeF-22'></span>[[#cite-22|[22]]].
+The error estimation method presented in this paper is based on the evaluation of a particular form of the energy rate (the power) of the finite element residuals of the momentum and mass balance (incompressibility) equations. The residuals are computed using recovered values of the derivatives of the velocity and pressure variable obtained via a nodal derivative recovery technique. Two mesh adaptation processes based on: a)  the equi-distribution of the residual power  among the elements in the mesh and b) the equi-distribution of the density of the total residual power are presented. The stabilized form of the Navier-Stokes equations using a finite calculus (FIC) procedure developed by the authors <span id='citeF-17'></span>[[#cite-17|[17-21]]] is used for the FE solution of the examples presented in the paper using a fractional step scheme. This allows the use of linear triangles and tetrahedra with an equal order interpolation for the velocity and pressure variables. A nodal-based approach is used for computation of the residual power error and the total power generated by the viscous stresses and the pressure terms in the mesh. The residual power form derived from the FIC approach includes power terms emanating from the velocity field and from the gradient of the velocity field. It is shown that accounting for the terms involving the velocity gradient provides an effective  error estimation and mesh adaptation procedure for CFD problems. Also the FIC form of the residual power expression can be effectively extended for error estimation and mesh adaptivity of other problems in computational mechanics <span id='citeF-22'></span>[[#cite-22|[22]]].
 The examples show the ability of the error estimation  and the mesh adaptation process to capture the high gradients of the solution in the vecinity of boundary layers and in zones of high vorticity of the fluid for both steady state and transient flow situations.
@@ Line 170: / Line 170: @@
 In above <math display="inline">\hat r_{m_i}\equiv r_{m_i}(\hat u_j, \hat p)</math> and <math display="inline">\hat r_v =r_v(\hat u_j, \hat p)</math> denote the approximate finite element residuals for the momentum and mass balance equations, respectively, <math display="inline"> P_{m_i} (N_k)</math> and <math display="inline">P_v (N_k)</math> are appropriate functions of the shape functions <math display="inline">N_k</math> and <math display="inline">\tau _{m_i}^e</math> and <math display="inline">\tau _v ^e</math> are stabilization parameters which depend on the element dimensions and the physical variables of the problem.
-Eqs.(9) and (10) can be interpreted as  extended  Galerkin expressions where the last integrals over the element interiors has been added to the standard Galerkin forms of the momentum and mass balance equations in order to get a physically meanful (stabilized) numerical solution. Many popular stabilized methods (SUPG, GLS, SGS, etc.) can be written in the form of Eqs.(9) and (10) and they just differ in the choice of the weighting functions <math display="inline">P_{m_i}</math> and <math display="inline">P_v</math> and in the values of the stabilization parameters <span id='citeF-1'></span>[[#cite-1|[1]]].
+Eqs.(9) and (10) can be interpreted as  extended  Galerkin expressions where the last integrals over the element interiors has been added to the standard Galerkin forms of the momentum and mass balance equations in order to get a physically meanful (stabilized) numerical solution. Many popular stabilized methods (SUPG, GLS, SGS, etc.) can be written in the form of Eqs.(9) and (10) and they just differ in the choice of the weighting functions <math display="inline">P_{m_i}</math> and <math display="inline">P_v</math> and in the values of the stabilization parameters <span id='citeF-1'></span>[[#cite-1|[1,24-30]]].
-A somehow different class of stabilized method developed by the authors is based on a finite calculus (FIC) formulation <span id='citeF-17'></span>[[#cite-17|[17]]]. Here the standard momentum and mass balance equations are modified by enforcing the balance laws in a domain of finite size. The modified governing equations contain additional terms which play the role of stabilizing terms in the numerical solution. The Galerkin FE form of the FIC equations of an incompressible fluid can also be written by integral forms analogous to Eqs.(9) and (10) and hence we will retain these equations as general expressions applicable to any stabilized method. Details on the FIC formulation used for the solution of the examples presented in the paper are given in a later section.
+A somehow different class of stabilized method developed by the authors is based on a finite calculus (FIC) formulation <span id='citeF-17'></span>[[#cite-17|[17-21]]]. Here the standard momentum and mass balance equations are modified by enforcing the balance laws in a domain of finite size. The modified governing equations contain additional terms which play the role of stabilizing terms in the numerical solution. The Galerkin FE form of the FIC equations of an incompressible fluid can also be written by integral forms analogous to Eqs.(9) and (10) and hence we will retain these equations as general expressions applicable to any stabilized method. Details on the FIC formulation used for the solution of the examples presented in the paper are given in a later section.
 ==3 RESIDUAL POWER FORMS BASED ON THE RECOVERED BALANCE EQUATIONS==
@@ Line 469: / Line 469: @@
 The mesh refinement strategy is as follows.
-'''Step 1''' Compute the nodal recovered values of the derivatives of the velocity and pressure fields using the algorithm described in  Section 4. '''Step 2''' Compute the residual power error for each element in the mesh and the total residual power error using Eqs.(15) and (16). '''Step 3''' Compute the global and local (element) error parameters <math display="inline">\xi _g</math> and <math display="inline">\xi ^e</math>. '''Step 4''' Compute the new element sizes using Eq.(27). Perform a smoothing of the new element sizes to avoid problems during the regeneration of the mesh. '''Step 5''' Regenerate a new mesh. Minimum and maximum sizes of the elements in the mesh are prescribed in order to have a better control of the mesh generation process. The mesh generation  is carried out using the GiD pre/postprocessing system <span id='citeF-23'></span>[[#cite-23|[23]]].
+'''Step 1''' Compute the nodal recovered values of the derivatives of the velocity and pressure fields using the algorithm described in  Section 4.
 Above mesh refinement strategy is equally applicable for the two mesh adaption procedures explained in the previous sections.
@@ Line 479: / Line 487: @@
 ===6.1 Finite calculus form of the Navier-Stokes equations===
-The FIC governing equations for incompressible viscous flows are obtained by applying the standard conservation laws expressing balance of momentum and mass over a control domain. Assuming that the control domain has finite dimensions and expressing the variation of mass and momentum over the domain using Taylor series expansions of one order higher than those used in the standard infinitesimal  theory, the following expressions are found <span id='citeF-17'></span>[[#cite-17|[17]]]:<br/>
+The FIC governing equations for incompressible viscous flows are obtained by applying the standard conservation laws expressing balance of momentum and mass over a control domain. Assuming that the control domain has finite dimensions and expressing the variation of mass and momentum over the domain using Taylor series expansions of one order higher than those used in the standard infinitesimal  theory, the following expressions are found <span id='citeF-17'></span>[[#cite-17|[17-21]]]:<br/>
 ''Momentum''
@@ Line 519: / Line 527: @@
 |}
-where <math display="inline">h_{1}</math> and <math display="inline">h_{2}</math> are characteristic dimensions of the finite control domain where balance of momentum and mass conservation is enforced. The components of vector <math display="inline">{h}</math> introduce the necessary stabilization along the streamline and transverse directions to the flow in the discrete problem <span id='citeF-17'></span>[[#cite-17|[17]]].
+where <math display="inline">h_{1}</math> and <math display="inline">h_{2}</math> are characteristic dimensions of the finite control domain where balance of momentum and mass conservation is enforced. The components of vector <math display="inline">{h}</math> introduce the necessary stabilization along the streamline and transverse directions to the flow in the discrete problem <span id='citeF-17'></span>[[#cite-17|[17-21]]].
 The method to derive the modified differential equations (30) and (31) incorporating the stabilization terms is termed ''finite calculus'' (FIC) as a reference to the standard infinitesimal calculus  where the size of the domain where balance of mass and momentum is enforced is assumed to be negligible. Note that for <math display="inline">{h}\to 0</math> the standard infinitesimal form of the momentum and mass balance equations is recovered <span id='citeF-1'></span>[[#cite-1|[1]]].
@@ Line 573: / Line 581: @@
 We will choose again an equal order finite element approximation for the velocity and pressure variables given by Eqs.(8)
-Again we will assume that a finite element solution has been found for the velocity and the pressure variables. The finite element algorithm used for the examples solved in this paper is based on a ''fractional step scheme'' <span id='citeF-19'></span>[[#cite-19|[19]]]. The next step is to obtain enhanced recovered values of the derivatives of the velocity and the pressure at the element nodes as explained in previouss section.
+Again we will assume that a finite element solution has been found for the velocity and the pressure variables. The finite element algorithm used for the examples solved in this paper is based on a ''fractional step scheme'' <span id='citeF-19'></span>[[#cite-19|[19-21,30]]]. The next step is to obtain enhanced recovered values of the derivatives of the velocity and the pressure at the element nodes as explained in previouss section.
 The recovered nodal derivative fields of the velocities and the pressure are used now to compute the FIC residuals of the momentum and mass balance equations (1) and (2):

@@ Line 1,286: / Line 1,286: @@
 The numerical results were obtained using the finite element code Tdyn provided by the company COMPASS Ingeniería y Sistemas SA <span id='citeF-31'></span>[[#cite-31|31]]. Thanks are also given to Dassault Aviation for providing the multiple airfoil geometry for the last example.
-===BIBLIOGRAPHY===
+==REFERENCES==
 <div id="cite-1"></div>

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 <div id='img-26'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-|[[Image:Draft_Samper_249588187-fig24.png|480px|]]
+|[[Image:Draft_Samper_249588187-fig24.png|460px|]]
 |style="padding-left:10px;"|[[File:Draft_Samper_249588187_3185_fig26b.png|500px|Mesh obtained after two refinement steps using equiditribution of global error (μ=0.01)]]
 |- style="text-align: center; font-size: 75%;"
@@ Line 1,186: / Line 1,186: @@
 <div id='img-27'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-|[[Image:Draft_Samper_249588187-fig25.png|480px|]]
+|[[Image:Draft_Samper_249588187-fig25.png|460px|]]
 |style="padding-left:10px;"|[[File:Draft_Samper_249588187_1872_fig27b.png|500px|Mesh obtained in the fith after changing μ=0.01 to μ= 0.005 in the fourth step]]
 |- style="text-align: center; font-size: 75%;"