Approximate Riesz Representatives of Shape Gradients

We study finite element approximations of Riesz representatives of shape gradients. First, we provide a general perspective on its error analysis. Then, we focus on shape functionals constrained by elliptic boundary value problems and \(H^1\)-representatives of shape gradients. We prove linear convergence in the energy norm for linear Lagrangian finite element approximations. This theoretical result is confirmed by several numerical experiments.

[1]  M. C. Delfour,et al.  Shapes and Geometries - Metrics, Analysis, Differential Calculus, and Optimization, Second Edition , 2011, Advances in design and control.

[2]  Helmut Harbrecht,et al.  Shape Optimization for Free Boundary Problems - Analysis and Numerics , 2012, Constrained Optimization and Optimal Control for Partial Differential Equations.

[3]  K. Sturm,et al.  Domain expression of the shape derivative and application to electrical impedance tomography , 2013 .

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

[5]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[6]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[7]  Jürgen Roßmann,et al.  Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods , 2009, Numerische Mathematik.

[8]  Grégoire Allaire.  Conception optimale de structures , 2007 .

[9]  Rolf Rannacher,et al.  Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations , 1982 .

[10]  Ralf Hiptmair,et al.  Shape Optimization by Pursuing Diffeomorphisms , 2015, Comput. Methods Appl. Math..

[11]  A. Paganini.  Approximate Shape Gradients for Interface Problems , 2015 .

[12]  Martin Siebenborn,et al.  Efficient PDE Constrained Shape Optimization Based on Steklov-Poincaré-Type Metrics , 2015, SIAM J. Optim..

[13]  Vladimir Maz’ya,et al.  Weighted Lp estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains , 2003 .

[14]  Hideyuki Azegami,et al.  Error analysis of the H1 gradient method for shape-optimization problems of continua , 2013, JSIAM Lett..

[15]  R. Hiptmair,et al.  Comparison of approximate shape gradients , 2014, BIT Numerical Mathematics.