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Hybrid Optimization Techniques for Multi-State Optimal Design Problems

Marko Erceg, Petar Kunštek, Marko Vrdoljak

TL;DR

This work tackles joint optimization of a domain and a two-material distribution under multi-state diffusion, addressing the nonexistence of classical optima by a hybrid relaxation that blends homogenization-based interior relaxation with restricted admissible domains. The interior relaxation introduces a local fraction $\theta\in[0,1]$ and effective conductivities ${\bf A}\in\mathcal{K}(\theta)$ via the G-closure, while the boundary is handled with a level-set representation and a shape-derivative-driven update. Existence results are established for Dirichlet-type, Neumann-type, and general boundary conditions through compactness arguments (Hausdorff convergence, H-convergence) and appropriate extensions, ensuring convergence of state solutions and the relaxed objective. The numerical algorithm combines an optimality-criteria update for the interior material distribution with a boundary evolution driven by the shape derivative, and a representative 2-material example confirms convergence to the predicted annular design and demonstrates practical effectiveness of the hybrid approach.

Abstract

This paper addresses optimal design problems governed by multi-state stationary diffusion equations, aiming at the simultaneous optimization of the domain shape and the distribution of two isotropic materials in prescribed proportions. Existence of generalized solutions is established via a hybrid approach combining homogenization-based relaxation in the interior with suitable restrictions on admissible domains. Based on this framework, we propose a numerical method that integrates homogenization and shape optimization. The domain boundary is evolved using a level set method driven by the shape derivative, while the interior material distribution is updated via an optimality criteria algorithm. The approach is demonstrated on a representative example.

Hybrid Optimization Techniques for Multi-State Optimal Design Problems

TL;DR

This work tackles joint optimization of a domain and a two-material distribution under multi-state diffusion, addressing the nonexistence of classical optima by a hybrid relaxation that blends homogenization-based interior relaxation with restricted admissible domains. The interior relaxation introduces a local fraction and effective conductivities via the G-closure, while the boundary is handled with a level-set representation and a shape-derivative-driven update. Existence results are established for Dirichlet-type, Neumann-type, and general boundary conditions through compactness arguments (Hausdorff convergence, H-convergence) and appropriate extensions, ensuring convergence of state solutions and the relaxed objective. The numerical algorithm combines an optimality-criteria update for the interior material distribution with a boundary evolution driven by the shape derivative, and a representative 2-material example confirms convergence to the predicted annular design and demonstrates practical effectiveness of the hybrid approach.

Abstract

This paper addresses optimal design problems governed by multi-state stationary diffusion equations, aiming at the simultaneous optimization of the domain shape and the distribution of two isotropic materials in prescribed proportions. Existence of generalized solutions is established via a hybrid approach combining homogenization-based relaxation in the interior with suitable restrictions on admissible domains. Based on this framework, we propose a numerical method that integrates homogenization and shape optimization. The domain boundary is evolved using a level set method driven by the shape derivative, while the interior material distribution is updated via an optimality criteria algorithm. The approach is demonstrated on a representative example.
Paper Structure (8 sections, 8 theorems, 92 equations, 3 figures)

This paper contains 8 sections, 8 theorems, 92 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Relaxation
  3. Preparation
  4. Dirichlet-type free boundary
  5. Neumann-type free boundary
  6. General boundary conditions
  7. Hybrid method - algorithm overview
  8. Numerical results

Key Result

Lemma 2.1

Let us assume (A1)-- (A4). There exist $N$ open and bounded disjoint subsets $O^i\subset \mathbb R^d$, $i=1,2,\dots, N$, such that for every $i\in\{1,2,\dots,N\}$ the following hold:

Figures (3)

  • Figure 1: The $\varepsilon$-cone property of a set
  • Figure 2: Results of the hybrid algorithm: $\Omega$ and local fraction $\theta$ at the $k$-th iteration ($k=1$, left; $k=200$, middle; $k=400$, right). The higher-conductivity phase is shown in black, and the lower-conductivity phase in grey.
  • Figure 3: Value of the objective functional. Note the significant jump at the end, which occurs after refinement and 30 additional iterations of the homogenization process.

Theorems & Definitions (22)

  • Remark 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Lemma 2.6
  • ...and 12 more