Nonconforming FEMs for an Optimal Design Problem

Some optimal design problems in topology optimization eventually lead to a degenerate convex minimization problem $E(v):=\int_{\Omega}W(\nabla v)dx-\int_{\Omega}f\,v\,dx$ for $v \in H_0^1(\Omega)$ with possibly multiple minimizers $u$, but with a unique stress $\sigma:=DW(\nabla u)$. This paper proposes the discrete Raviart--Thomas mixed finite element method (dRT-MFEM) and establishes its equivalence with the Crouzeix--Raviart nonconforming finite element method. The convergence analysis combines the a priori convergence rate of the conforming FEM with the efficient a posterior error control of MFEM. Numerical experiments provide empirical evidence that the proposed dRT-MFEM overcomes the reliability-efficiency gap for the first time.

[1]  Zhangxin Chen,et al.  On the implementation of mixed methods as nonconforming methods for second-order elliptic problems , 1995 .

[2]  CARSTEN CARSTENSEN,et al.  MIXED FINITE ELEMENT METHOD FOR A DEGENERATE CONVEX VARIATIONAL PROBLEM FROM TOPOLOGY OPTIMISATION , 2012 .

[3]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[4]  Donald A. French,et al.  On the convergence of finite-element approximations of a relaxed variational problem , 1990 .

[5]  R. Kohn,et al.  Numerical study of a relaxed variational problem from optimal design , 1986 .

[6]  Carsten Carstensen,et al.  Mixed Finite Element Method for a Degenerate Convex Variational Problem from Topology Optimization , 2012, SIAM J. Numer. Anal..

[7]  Carsten Carstensen,et al.  Guaranteed lower eigenvalue bounds for the biharmonic equation , 2014, Numerische Mathematik.

[8]  A. Cherkaev.  Variational Methods for Structural Optimization , 2000 .

[9]  Carsten Carstensen,et al.  Adaptive Finite Element Methods for Microstructures? Numerical Experiments for a 2-Well Benchmark , 2003, Computing.

[10]  Christoph Ortner,et al.  On the Convergence of Adaptive Nonconforming Finite Element Methods for a Class of Convex Variational Problems , 2011, SIAM J. Numer. Anal..

[11]  Carsten Carstensen,et al.  A convergent adaptive finite element method for an optimal design problem , 2007, Numerische Mathematik.

[12]  Weimin Han,et al.  A Posteriori Error Analysis Via Duality Theory: With Applications in Modeling and Numerical Approximations , 2004 .

[13]  Dan Givoli,et al.  Variational Methods for Structural Optimization. Applied Mathematical Sciences, Vol. 140 , 2002 .

[14]  Douglas N. Arnold,et al.  Analysis of a Linear–Linear Finite Element for the Reissner–Mindlin Plate Model , 1997 .

[15]  Carsten Carstensen,et al.  Axioms of adaptivity , 2013, Comput. Math. Appl..

[16]  Dorothee Knees,et al.  Global stress regularity of convex and some nonconvex variational problems , 2008 .

[17]  R. Laugesen,et al.  Minimizing Neumann fundamental tones of triangles: An optimal Poincaré inequality , 2009, 0907.1552.

[18]  Carsten Carstensen,et al.  Local Stress Regularity in Scalar Nonconvex Variational Problems , 2002, SIAM J. Math. Anal..

[19]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[20]  Susanne C. Brenner,et al.  Two-level additive Schwarz preconditioners for nonconforming finite element methods , 1996, Math. Comput..

[21]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[22]  Carsten Carstensen,et al.  Explicit Error Estimates for Courant, Crouzeix-Raviart and Raviart-Thomas Finite Element Methods , 2012 .

[23]  C. Bahriawati,et al.  Three Matlab Implementations of the Lowest-order Raviart-Thomas Mfem with a Posteriori Error Control , 2005 .

[24]  L. D. Marini.  An Inexpensive Method for the Evaluation of the Solution of the Lowest Order Raviart–Thomas Mixed Method , 1985 .

[25]  D. Arnold,et al.  Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates , 1985 .

[26]  John A. Trangenstein,et al.  Mixed and Hybrid Finite Elements , 2013 .

[27]  Carsten Carstensen,et al.  Numerical solution of the scalar double-well problem allowing microstructure , 1997, Math. Comput..

[28]  Carsten Carstensen.  Convergence of adaptive FEM for a class of degenerate convex minimization problems , 2007 .