On a two-scale phasefield model for topology optimization

Moritz Ebeling-Rump; Dietmar Hömberg; Robert Lasarzik

doi:10.3934/dcdss.2023206

Article Contents

2024, Volume 17, Issue 1: 326-361. Doi: 10.3934/dcdss.2023206

This issue Previous Article Second order doubly nonlinear evolution inclusions –quasi-variational approach– Next Article Structure-preserving schemes for Cahn–Hilliard equations with dynamic boundary conditions

On a two-scale phasefield model for topology optimization

1.
Endless Industries GmbH, Germany
2.
Weierstrass Institute Berlin, Germany
3.
Department of Mathematical Sciences NTNU, Norway
4.
Technische Universität Berlin, Germany
5.
Freie Universität Berlin, Germany

^*Corresponding author: Robert Lasarzik
^*Corresponding author: Robert Lasarzik

Dedicated to Pierluigi Colli on the occasion of his 65th birthday

Received: June 2023

Revised: October 2023

Early access: November 2023

Published: January 2024

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Abstract

In this article, we consider a gradient flow stemming from a problem in two-scale topology optimization. We use the phase-field method, where a Ginzburg–Landau term with obstacle potential is added to the cost functional, which contains the usual compliance but also an additional contribution including a local volume constraint in a penalty term. The minimization of such an energy by its gradient-flow is analyzed in this paper. We use an regularization and discretization of the associated state-variable to show the existence of weak solutions to the considered system.

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Mathematics Subject Classification: Primary: 35K61, 35M33, 35Q93, 49Q10 74P05, 74P10.

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Figure 1. Result of a pseudo-time stepping scheme for two-scale topology optimization for an MBB beam. The iteration numbers 1, 10, 20, 30, 40, 50, and 100 are displayed

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Figure 2. Approximation $ \psi_\beta $ of the double obstacle potential $ \psi $ for three different values of $ \beta $

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