MIXED FINITE ELEMENT METHOD FOR A DEGENERATE CONVEX VARIATIONAL PROBLEM FROM TOPOLOGY OPTIMIZATION
The optimal design task of this paper seeks the distribution of two materials of prescribed amounts for maximal torsion stiffness of an infinite bar of a given cross section. This example of relaxation in topology optimization leads to a degenerate convex minimization problem $E(v): = \int_\Omega {{...
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description | The optimal design task of this paper seeks the distribution of two materials of prescribed amounts for maximal torsion stiffness of an infinite bar of a given cross section. This example of relaxation in topology optimization leads to a degenerate convex minimization problem $E(v): = \int_\Omega {{\varphi _0}} (|\nabla v|)dx - \int_\Omega {fvdx} $ for $v \in V: = H_0^1\left( \Omega \right)$ with possibly multiple primal solutions u, but with unique stress $\sigma : = \varphi _0^1\left( {|\nabla u|} \right)$ sign ∇u. The mixed finite element method is motivated by the smoothness of the stress variable $\sigma \in H_{loc}^1\left( {\Omega ;{\mathbb{R}^2}} \right)$ , while the primal variables are uncontrollable and possibly nonunique. The corresponding nonlinear mixed finite element method is introduced, analyzed, and implemented. The striking result of this paper is a sharp a posteriori error estimation in the dual formulation, while the a posteriori error analysis in the primal problem suffers from the reliability-efficiency gap. An empirical comparison of that primal formulation with the new mixed discretization schemes is intended for uniform and adaptive mesh refinements. |
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This example of relaxation in topology optimization leads to a degenerate convex minimization problem $E(v): = \int_\Omega {{\varphi _0}} (|\nabla v|)dx - \int_\Omega {fvdx} $ for $v \in V: = H_0^1\left( \Omega \right)$ with possibly multiple primal solutions u, but with unique stress $\sigma : = \varphi _0^1\left( {|\nabla u|} \right)$ sign ∇u. The mixed finite element method is motivated by the smoothness of the stress variable $\sigma \in H_{loc}^1\left( {\Omega ;{\mathbb{R}^2}} \right)$ , while the primal variables are uncontrollable and possibly nonunique. The corresponding nonlinear mixed finite element method is introduced, analyzed, and implemented. The striking result of this paper is a sharp a posteriori error estimation in the dual formulation, while the a posteriori error analysis in the primal problem suffers from the reliability-efficiency gap. An empirical comparison of that primal formulation with the new mixed discretization schemes is intended for uniform and adaptive mesh refinements.</description><subject>A posteriori knowledge</subject><subject>Applied mathematics</subject><subject>Convexity</subject><subject>Curl</subject><subject>Design optimization</subject><subject>Error analysis</subject><subject>Estimators</subject><subject>Finite element analysis</subject><subject>Finite element method</subject><subject>Flux density</subject><subject>Material concentration</subject><subject>Mathematical 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CARSTEN</au><au>GÜNTHER, DAVID</au><au>RABUS, HELLA</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>MIXED FINITE ELEMENT METHOD FOR A DEGENERATE CONVEX VARIATIONAL PROBLEM FROM TOPOLOGY OPTIMIZATION</atitle><jtitle>SIAM journal on numerical analysis</jtitle><date>2012-01-01</date><risdate>2012</risdate><volume>50</volume><issue>2</issue><spage>522</spage><epage>543</epage><pages>522-543</pages><issn>0036-1429</issn><eissn>1095-7170</eissn><abstract>The optimal design task of this paper seeks the distribution of two materials of prescribed amounts for maximal torsion stiffness of an infinite bar of a given cross section. This example of relaxation in topology optimization leads to a degenerate convex minimization problem $E(v): = \int_\Omega {{\varphi _0}} (|\nabla v|)dx - \int_\Omega {fvdx} $ for $v \in V: = H_0^1\left( \Omega \right)$ with possibly multiple primal solutions u, but with unique stress $\sigma : = \varphi _0^1\left( {|\nabla u|} \right)$ sign ∇u. The mixed finite element method is motivated by the smoothness of the stress variable $\sigma \in H_{loc}^1\left( {\Omega ;{\mathbb{R}^2}} \right)$ , while the primal variables are uncontrollable and possibly nonunique. The corresponding nonlinear mixed finite element method is introduced, analyzed, and implemented. The striking result of this paper is a sharp a posteriori error estimation in the dual formulation, while the a posteriori error analysis in the primal problem suffers from the reliability-efficiency gap. An empirical comparison of that primal formulation with the new mixed discretization schemes is intended for uniform and adaptive mesh refinements.</abstract><cop>Philadelphia</cop><pub>Society for Industrial and Applied Mathematics</pub><doi>10.1137/100806837</doi><tpages>22</tpages></addata></record> |
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subjects | A posteriori knowledge Applied mathematics Convexity Curl Design optimization Error analysis Estimators Finite element analysis Finite element method Flux density Material concentration Mathematical functions Variables |
title | MIXED FINITE ELEMENT METHOD FOR A DEGENERATE CONVEX VARIATIONAL PROBLEM FROM TOPOLOGY OPTIMIZATION |
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