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Phase field model for multi-material shape optimization of inextensible rods

Patrick Dondl, Alberto Maione, Steve Wolff-Vorbeck

TL;DR

The paper develops a rigorous framework for optimizing bending and torsional rigidities of non-homogeneous, inextensible rods by coupling a sharp-interface perimeter-regularized shape optimization with a phase-field (diffuse-interface) approximation. It provides a Γ-convergence analysis showing that diffuse-interface minimizers converge to sharp-interface minimizers as the interface thickness vanishes, and it proves existence of solutions for both formulations. Numerically, it implements a time-discrete $L^2$-gradient flow with P1 finite elements to compute optimal two-material distributions inside a fixed cross-section, revealing shapes that mimic plant-stem morphologies such as fibre reinforcement and boundary grooving. The approach yields a practical, rigorous link between multi-material rod design and biological morphogenesis, quantified through $D_{ m mean}$, $D_T$, and $RM$ and demonstrated via phase-field-based optimization and gradient-descent computations.

Abstract

We derive a model for the optimization of the bending and torsional rigidities of non-homogeneous elastic rods. This is achieved by studying a sharp interface shape optimization problem with perimeter penalization, that treats both rigidities as objectives. We then formulate a phase field approximation of the optimization problem and show the convergence to the aforementioned sharp interface model via $Γ$-convergence. In the final part of this work we numerically approximate minimizers of the phase field problem by using a steepest descent approach and relate the resulting optimal shapes to the development of the morphology of plant stems.

Phase field model for multi-material shape optimization of inextensible rods

TL;DR

The paper develops a rigorous framework for optimizing bending and torsional rigidities of non-homogeneous, inextensible rods by coupling a sharp-interface perimeter-regularized shape optimization with a phase-field (diffuse-interface) approximation. It provides a Γ-convergence analysis showing that diffuse-interface minimizers converge to sharp-interface minimizers as the interface thickness vanishes, and it proves existence of solutions for both formulations. Numerically, it implements a time-discrete -gradient flow with P1 finite elements to compute optimal two-material distributions inside a fixed cross-section, revealing shapes that mimic plant-stem morphologies such as fibre reinforcement and boundary grooving. The approach yields a practical, rigorous link between multi-material rod design and biological morphogenesis, quantified through , , and and demonstrated via phase-field-based optimization and gradient-descent computations.

Abstract

We derive a model for the optimization of the bending and torsional rigidities of non-homogeneous elastic rods. This is achieved by studying a sharp interface shape optimization problem with perimeter penalization, that treats both rigidities as objectives. We then formulate a phase field approximation of the optimization problem and show the convergence to the aforementioned sharp interface model via -convergence. In the final part of this work we numerically approximate minimizers of the phase field problem by using a steepest descent approach and relate the resulting optimal shapes to the development of the morphology of plant stems.
Paper Structure (13 sections, 5 theorems, 102 equations, 4 figures, 1 table)

This paper contains 13 sections, 5 theorems, 102 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The mathematical model of non-homogeneous elastic rods
  3. Nonlinear bending-torsion theory and dimension reduction
  4. The case of isotropic materials
  5. Derivation of the torsional rigidity by stress functions
  6. Optimization of the bending and torsional rigidities
  7. Perimeter penalized shape optimization
  8. The phase field approach
  9. Sharp interface limit
  10. Numerical implementation
  11. Gradient flow dynamics
  12. P1-finite element approximation
  13. Numerical experiments

Key Result

Theorem 2.2

Assume that the stored energy $W$ satisfies hypotheses $1.,2.,3.$ given in the Introduction, let $Q_3$ be twice the quadratic form of linearized elasticity and denote $Q_2:\mathbb{M}^{3\times 3}_{\rm skew}\to[0,+\infty)$ the quadratic form defined through the minimization problem for any $A\in\mathbb{M}^{3\times 3}_{\rm skew}$, where the density function $u\in L^\infty(S)$ satisfies condu. Then,

Figures (4)

  • Figure 1: Inextensible rod subject to bending moments $M_{x_2},M_{x_3}$ and torsional moment $T$.
  • Figure 2: Cross-sections of liana Condylocarpon Guianense in the non-self-supporting old ontogenetic stage after attachment to a support. The secondary xylem is marked with (1) and the cortex is marked with (2). © Plant Biomechanics Group Freiburg, annotated and used with permission.
  • Figure 3: Convergence history of the rigidities $D_{T}$ and $D_{\rm mean}$ with respect to pseudo-time $t$ during a gradient descent.
  • Figure 4: Local solutions of \ref{['min:diff']} for different weighting factors $\sigma_1,\sigma_3,\gamma$ corresponding to a maximization of rigidities in (b), a sole minimization of torsional rigidity in (c), a minimization of torsional and a maximization of bending rigidity in (d) and (e) and a minimization of both bending and torsional rigidity in (f) and (g). In experiments (b), (c) and (f) the weighting factor $\gamma$ for the perimeter penalization is set to $\gamma=0.5$, where in (e) and (g) the weighting factor is $\gamma=0.25$. In experiment (d) the weighting factor is set to $\gamma=1.0$. The stiffer material ($u=1$) is depicted in red where the softer material ($u=0.1$) is depicted in blue.

Theorems & Definitions (14)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 4 more