Rheinische Friedrich-Wilhelms-Universität Bonn

Institut für Angewandte Mathematik

Abteilung Wissenschaftliches Rechnen und Numerische Simulation

Wegelerstrasse 6

53115 Bonn

Germany

Michael Griebel

Abstract

Topology optimization is a relatively new field of structural mechanics
which tries to find an optimal design of a structure for a given load. One flexible and
advantageous FE based method is the so called SIMP approach (Solid Isotropic Microstructures with
Penalization for intermediate densities, cf. [1], [2]). In this approach the problem is reformulated
to find an optimal density distribution of the material in the domain. Therefore
the optimization process leads to elasticity problems with large jumps in the material coefficients
and linear systems which become nearly singular so that classical iteration schemes for solving the
linear systems provide only very poor convergence. Since the accuracy of the approximate solution
plays an essential role for the optimized design (solving the linear system with low accuracy leads
only to suboptimal designs) direct methods are often used during the optimization process which
strictly limits the problem size.

To overcome this problem and accelerate the convergence of classical iteration schemes we propose an algebraic multigrid preconditioner. Since algebraic multigrid methods can deal with large jumps in the coefficient functions and complex geometries, they seem to be well fit for the linear systems occuring in the SIMP approach. However, due to the fact that we have linear systems stemming from the discretization of a system of PDEs and classical AMG methods are not well suited to solve such systems efficiently, we use the so-called point-block approach [3]. We will describe the modifications of this method to obtain an efficient preconditioner for the conjugate gradient method in our optimization process. At the end we will present some numerical results which underline the efficiency of the method for topology optimization.

[1] Bendsoe MP "Optimal Shape design as a material distribution problem", Struct. Optim, 1989,1,193-202.

[2] Rozvany GIN and Zhou M "Applications of COC method in layout optimization". In: Eschenauer H, Mattheck C and Olhoff N (eds.) Proc. Conf. "eng. Opt. in Design Processes", Berlin, Springer-Verlag, 1991, pp 59-70.

[3] Griebel M, Oeltz D and Schweitzer, MA "An Algebraic Multigrid Method for Linear Elasticity" SIAM J. Sci. Comp., to appear.

To overcome this problem and accelerate the convergence of classical iteration schemes we propose an algebraic multigrid preconditioner. Since algebraic multigrid methods can deal with large jumps in the coefficient functions and complex geometries, they seem to be well fit for the linear systems occuring in the SIMP approach. However, due to the fact that we have linear systems stemming from the discretization of a system of PDEs and classical AMG methods are not well suited to solve such systems efficiently, we use the so-called point-block approach [3]. We will describe the modifications of this method to obtain an efficient preconditioner for the conjugate gradient method in our optimization process. At the end we will present some numerical results which underline the efficiency of the method for topology optimization.

[1] Bendsoe MP "Optimal Shape design as a material distribution problem", Struct. Optim, 1989,1,193-202.

[2] Rozvany GIN and Zhou M "Applications of COC method in layout optimization". In: Eschenauer H, Mattheck C and Olhoff N (eds.) Proc. Conf. "eng. Opt. in Design Processes", Berlin, Springer-Verlag, 1991, pp 59-70.

[3] Griebel M, Oeltz D and Schweitzer, MA "An Algebraic Multigrid Method for Linear Elasticity" SIAM J. Sci. Comp., to appear.