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Systematic design of compliant morphing structures: a phase-field approach

Jamal Shabani, Kaushik Bhattacharya, Blaise Bourdin

Abstract

We investigate the systematic design of compliant morphing structures composed of materials reacting to an external stimulus. We add a perimeter penalty term to ensure existence of solutions. We propose a phase-field approximation of this sharp interface problem, prove its convergence as the regularization length approaches 0 and present an efficient numerical implementation. We illustrate the strengths of our approach through a series of numerical examples.

Systematic design of compliant morphing structures: a phase-field approach

Abstract

We investigate the systematic design of compliant morphing structures composed of materials reacting to an external stimulus. We add a perimeter penalty term to ensure existence of solutions. We propose a phase-field approximation of this sharp interface problem, prove its convergence as the regularization length approaches 0 and present an efficient numerical implementation. We illustrate the strengths of our approach through a series of numerical examples.

Paper Structure

This paper contains 17 sections, 4 theorems, 46 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Phase-field regularization
  4. Existence of solutions
  5. Numerical implementation
  6. Sensitivity analysis
  7. Minimization with respect to $\mathrm{s}$
  8. Minimization algorithm
  9. Numerical results
  10. Cantilever beam
  11. Cantilever beam with two target displacements
  12. Hexagonal domain with three prescribed displacements
  13. Conclusion
  14. Funding
  15. Competing Interests
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathbb{C}_1,\dots,\mathbb{C}_m$ be symmetric definite linear operators over $\mathrm{M}^{d\times d}_\mathrm{sym}$ and assume that $\beta_i \ge 0$ for any $1 \le j \le m$. For any given $\varepsilon>0$, the problem admits a solution $(\rho_\varepsilon,\mathrm{s}_\varepsilon)$. Furthermore, there exists $(\rho,\mathrm{s}) \in \widetilde{\mathcal{D}} \times \mathcal{S}$ such that a subsequence

Figures (8)

  • Figure 1: Monolithic (left) vs. staggered (right) scheme. Responsive material density (top), non-responsive material density (middle), and composite plot of both material density and the stimulus in the deformed configuration.
  • Figure 2: Optimized structure with a ratio $E_2/E_3 = 2$ in the reference (top) and deformed configuration (bottom) with $\nu_2 = 0.24$ and $\nu_3 = 0.12$ (left) and $\nu_2 = 0.18$ and $\nu_3 = 0.12$ (right).
  • Figure 3: Optimal design of a beam with two target displacements. (left) Material distribution and stimulus for a target displacement of $(0,1)$ in the reference (top) and deformed (bottom) configuration. (right) Material distribution and stimulus for a target displacement of $(0,2)$ in the reference (top) and deformed (bottom) configuration.
  • Figure 4: Optimal design of a beam with two target displacements. (left) Material distribution and stimulus for a target displacement of $(1,0)$ in the reference (top) and deformed (bottom) configuration. (right) Material distribution and stimulus for a target displacement of $(0,1)$ in the reference (top) and deformed (bottom) configuration.
  • Figure 5: Hexagonal domain clamped at three sides $\Gamma_D$
  • ...and 3 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 1: Equi-coercivity of the displacements
  • proof
  • Lemma 2: Continuity of displacements
  • proof
  • Lemma 3: Compactness
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 1
  • Remark 2