AFEM adaptive mixed finite element method optimal design degenerate convex minimisation Mixed Finite Element Method for a Degenerate Convex Variational Problem from Topology Optimisation Carsten Carstensen Carstensen Carsten David Günther Günther David Hella Rabus Rabus Hella

Mixed Finite Element Method for a Degenerate Convex Variational Problem from Topology Optimisation

Carsten Carstensen , David Günther , Hella Rabus

MSC 2000

65K10 Optimization and variational techniques
65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

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 given cross section. This example of relaxation in topology optimisation leads to a degenerate convex minimisation problem

E(v) := ∫Ω φ0(|∇v|) dx - ∫ fv dx for v∈V:=H01(Ω)

with possibly multiple primal solutions u, but with unique stress σ := φ0(|∇u|) sign ∇u The mixed finite element method is motivated by the smoothness of the stress variable σ ∈ H1loc(Ω;R2) while the primal variables are un-controllable and possibly non-unique. The corresponding nonlinear mixed finite element method is introduced, analysed, 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 with the new mixed discretisation schemes is intended for uniform and adaptive mesh-refinements.


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