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Minimisation of peak stresses with the shape derivative

Phillip Baumann, Kevin Sturm

Abstract

This paper is concerned with the minimisation of peak stresses occurring in linear elasticity. We propose to minimise the maximal von Mises stress of the elastic body. This leads to a nonsmooth shape functional. We derive the shape derivative and associate it with the Clarke sub-differential. Using a steepest descent algorithm we present numerical simulations. We compare our results to the usual $p$-norm regularisation and show that our algorithm performs better in the presented tests.

Minimisation of peak stresses with the shape derivative

Abstract

This paper is concerned with the minimisation of peak stresses occurring in linear elasticity. We propose to minimise the maximal von Mises stress of the elastic body. This leads to a nonsmooth shape functional. We derive the shape derivative and associate it with the Clarke sub-differential. Using a steepest descent algorithm we present numerical simulations. We compare our results to the usual -norm regularisation and show that our algorithm performs better in the presented tests.
Paper Structure (16 sections, 16 theorems, 95 equations, 6 figures, 2 tables, 2 algorithms)

This paper contains 16 sections, 16 theorems, 95 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Shape sensitivity analysis and material derivative
  4. Shape derivative
  5. Hilbert space setting
  6. Clarke subgradient
  7. Numerical implementation
  8. Shape derivative and shape gradient
  9. Moving mesh method and mesh quality
  10. Differences to the analytic setting
  11. max-norm approach
  12. $p$-norm approach
  13. Numerical results
  14. L-bracket
  15. Bridge
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $\Omega\subset{\mathsf{D}}$ a Lipschitz set. Furthermore, let $f\in H^1({\mathsf{D}})^d$, $g\in H^2({\mathsf{D}})^d$, $X\in C^1(\bar{{\mathsf{D}}})^d$ with $\text{supp} (X)\subset ({\mathsf{D}}\setminus\bar{\omega})$ and define for $t>0$ sufficiently small and $Y\in C^1(\bar{{\mathsf{D}}})^d$ wi

Figures (6)

  • Figure 1: Visualisation of the L-bracket
  • Figure 2: Deformation of the L-bracket
  • Figure 3: Cost evolution for the L-bracket problem
  • Figure 4: Visualisation of the bridge
  • Figure 5: Deformation of the bridge
  • ...and 1 more figures

Theorems & Definitions (39)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 29 more