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On the importance of smoothness, interface resolution and numerical sensitivities in shape and topological sensitivity analysis

M. H. Gfrerer, P. Gangl

TL;DR

The study addresses how discretizing the PDE constraint affects shape and topological derivatives in a 1D two-material Poisson tracking problem. It compares standard B-spline discretizations with an interface-enriched discretization (XFEM) under a discretize-then-optimize framework and contrasts numerical sensitivities with continuous sensitivities. Key findings show that the smoothness of the shape derivative depends on basis regularity, and without interface inclusion the derivative can be discontinuous or oscillatory, while enrichment yields convergent topological derivatives and smoother sensitivities. Numerical experiments confirm that higher-order smooth discretizations and interface enrichment reduce oscillations and improve agreement with analytic derivatives, providing practical guidance for discretization choices in multi-material gradient-based optimization.

Abstract

In this paper we investigate the influence of the discretization of PDE constraints on shape and topological derivatives. To this end, we study a tracking-type functional and a two-material Poisson problem in one spatial dimension. We consider the discretization by a standard method and an enriched method. In the standard method we use splines of degree $p$ such that we can control the smoothness of the basis functions easily, but do not take any interface location into consideration. This includes for p=1 the usual hat basis functions. In the enriched method we additionally capture the interface locations in the ansatz space by enrichment functions. For both discretization methods shape and topological sensitivity analysis is performed. It turns out that the regularity of the shape derivative depends on the regularity of the basis functions. Furthermore, for point-wise convergence of the shape derivative the interface has to be considered in the ansatz space. For the topological derivative we show that only the enriched method converges.

On the importance of smoothness, interface resolution and numerical sensitivities in shape and topological sensitivity analysis

TL;DR

The study addresses how discretizing the PDE constraint affects shape and topological derivatives in a 1D two-material Poisson tracking problem. It compares standard B-spline discretizations with an interface-enriched discretization (XFEM) under a discretize-then-optimize framework and contrasts numerical sensitivities with continuous sensitivities. Key findings show that the smoothness of the shape derivative depends on basis regularity, and without interface inclusion the derivative can be discontinuous or oscillatory, while enrichment yields convergent topological derivatives and smoother sensitivities. Numerical experiments confirm that higher-order smooth discretizations and interface enrichment reduce oscillations and improve agreement with analytic derivatives, providing practical guidance for discretization choices in multi-material gradient-based optimization.

Abstract

In this paper we investigate the influence of the discretization of PDE constraints on shape and topological derivatives. To this end, we study a tracking-type functional and a two-material Poisson problem in one spatial dimension. We consider the discretization by a standard method and an enriched method. In the standard method we use splines of degree such that we can control the smoothness of the basis functions easily, but do not take any interface location into consideration. This includes for p=1 the usual hat basis functions. In the enriched method we additionally capture the interface locations in the ansatz space by enrichment functions. For both discretization methods shape and topological sensitivity analysis is performed. It turns out that the regularity of the shape derivative depends on the regularity of the basis functions. Furthermore, for point-wise convergence of the shape derivative the interface has to be considered in the ansatz space. For the topological derivative we show that only the enriched method converges.
Paper Structure (22 sections, 4 theorems, 107 equations, 10 figures, 1 table)

This paper contains 22 sections, 4 theorems, 107 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem settings and continuous shape and topological derivative
  3. Continuous shape derivative
  4. Continuous topological derivative
  5. Discretization
  6. Sensitivity analysis of the discretized problem
  7. Numerical shape derivative
  8. Shape derivative for the problem discretized by the standard method
  9. Shape derivative for the problem discretized by the enriched method
  10. Numerical topological derivative
  11. Topological derivative for the standard method
  12. Topological derivative for the enriched method
  13. Impact of the discretization on smoothness of the shape derivative
  14. Numerical results
  15. Discretization error for the direct state
  16. ...and 7 more sections

Key Result

Lemma 4.4

Let $m,n \in \mathbb{N}$ and $\overline \varepsilon>0$. Let $\varepsilon \mapsto \mathbf A(\varepsilon):\mathbb R \rightarrow\mathbb R^{n\times n}$ invertible on $(0, \overline \varepsilon)$ and $\varepsilon \mapsto \mathbf b(\varepsilon):\mathbb R \rightarrow\mathbb R^{n\times m}$ be given and such

Figures (10)

  • Figure 1: 1D problem setups
  • Figure 2: Discretization without enrichment functions
  • Figure 3: Enriched discretization with one interface at $x=0.4$. The enrichment function is denoted by $N^E$.
  • Figure 4: Convergence of the $L_2$ and $H^1$ errors for different discretizations
  • Figure 5: The objective function in dependency of the interface position for different discretizations
  • ...and 5 more figures

Theorems & Definitions (12)

  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • proof
  • Remark 4.6
  • Theorem 4.7
  • proof
  • ...and 2 more