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Topology optimization and boundary observation for clamped plates

Cornel Marius Murea, Dan Tiba

TL;DR

The paper develops a Hamiltonian-based implicit boundary representation for topology optimization of clamped plates with holes, addressing the limitations of state penalization under Dirichlet conditions by working on a fixed domain $D$ and parametrizing the moving boundary via a Hamiltonian system with main period $T_g$. It recasts the problem as a fixed-domain optimal control in $D$ with a penalized boundary-observation cost, derives directional derivatives through state and trajectory variations, and implements a finite element scheme (with $g_h$ in $\mathbb{P}_3$, $u_h$ in $\mathbb{P}_1$, and $y_h$ in HCT) coupled with a line-search descent-direction algorithm. The discrete framework computes boundary-integral costs on $\partial\Omega_h$ and uses auxiliary PDEs to obtain descent directions, enabling simultaneous shape and topology changes without remeshing of the domain during iterations. Numerical experiments demonstrate opening and closing holes in the plate while satisfying the clamped boundary conditions, illustrating the approach’s capacity to perform boundary observation-based topology optimization in a robust fixed-domain setting.

Abstract

We indicate a new approach to the optimization of the clamped plates with holes. It is based on the use of Hamiltonian systems and the penalization of the performance index. The alternative technique employing the penalization of the state system, cannot be applied in this case due to the (two) Dirichlet boundary conditions. We also include numerical tests exhibiting both shape and topological modifications, both creating and closing holes.

Topology optimization and boundary observation for clamped plates

TL;DR

The paper develops a Hamiltonian-based implicit boundary representation for topology optimization of clamped plates with holes, addressing the limitations of state penalization under Dirichlet conditions by working on a fixed domain and parametrizing the moving boundary via a Hamiltonian system with main period . It recasts the problem as a fixed-domain optimal control in with a penalized boundary-observation cost, derives directional derivatives through state and trajectory variations, and implements a finite element scheme (with in , in , and in HCT) coupled with a line-search descent-direction algorithm. The discrete framework computes boundary-integral costs on and uses auxiliary PDEs to obtain descent directions, enabling simultaneous shape and topology changes without remeshing of the domain during iterations. Numerical experiments demonstrate opening and closing holes in the plate while satisfying the clamped boundary conditions, illustrating the approach’s capacity to perform boundary observation-based topology optimization in a robust fixed-domain setting.

Abstract

We indicate a new approach to the optimization of the clamped plates with holes. It is based on the use of Hamiltonian systems and the penalization of the performance index. The alternative technique employing the penalization of the state system, cannot be applied in this case due to the (two) Dirichlet boundary conditions. We also include numerical tests exhibiting both shape and topological modifications, both creating and closing holes.
Paper Structure (5 sections, 8 theorems, 65 equations, 14 figures, 1 table)

This paper contains 5 sections, 8 theorems, 65 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. The equivalent control problem and its approximation
  4. Finite element approximation and descent directions algorithm
  5. Numerical results

Key Result

Theorem 2.1

Under hypotheses (2.5), (2.6), (2.7), (2.8), $\partial\Omega_g$ has a finite number of connected components, for any $g\in \mathcal{F}$. Moreover, the connected component containing $\mathbf{x}_0\in D\subset \mathbb{R}^2$, $g(\mathbf{x}_0)=0$, for all $g\in \mathcal{F}$, is globally parametrized by that has a unique periodic solution and $I_g=[0,T_g]$ is the main period.

Figures (14)

  • Figure 1: Test 1. The penalized cost function for iterations $k\geq 1$. At $k=0$ the cost function is $1310.71$ and the last value is $1.25341$.
  • Figure 2: Test 1. The domains $\Omega_k$ for k=0, 1, 2, final.
  • Figure 3: Test 1. Final $y_h$ and $\partial\Omega_h$ (top). Final $g_h$ (left, bottom) and $u_h$ (right, bottom).
  • Figure 4: Test 2, case a). The penalized cost function for iterations $k\geq 4$. The first values are: $\mathcal{J}_{0}=1344.08$, $\mathcal{J}_{1}=724.423$, $\mathcal{J}_{2}=184.156$, $\mathcal{J}_{3}=74.8713$ and the last value is $1.09214$.
  • Figure 5: Test 2, case a). The domains $\Omega_k$ for k=0, 1, 2, 155.
  • ...and 9 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • Proposition 2.2: MT2022
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Remark 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Theorem 3.7
  • Remark 3.8
  • ...and 2 more