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The Isotropic Material Design methods with the cost expressed by the $L^p$-norm

Karol Bołbotowski, Sławomir Czarnecki, Tomasz Lewiński

TL;DR

This work introduces two L^p-cost isotropic material design variants, vp-IMD and sp-IMD, to minimize structural compliance by optimizing isotropic moduli k and μ under new cost bounds expressed as L^p norms. By deriving stress-based and displacement-based dual formulations and recasting the problems as non-linear elasticity with power-law constitutive equations, the authors establish existence (vp-IMD) and, in general, existence without uniqueness (sp-IMD) and connect these to the original IMD as p→1 while analyzing p→∞ limits. Numerical implementations on 2D L-shaped and cantilever domains demonstrate p-dependent behavior: concentration of moduli near singular features for small p and more diffuse but stable distributions for large p, with monotonic trends in minimal compliance. The methods allow solving via standard finite element nonlinear elasticity tools, avoiding measure-valued formulations and offering a practical, robust alternative to classical IMD, albeit with trade-offs such as loss of cutting-out and potential moduli concentration.

Abstract

The paper concerns the problem of minimization of the compliance of linear elastic structures made of an isotropic material. The bulk and shear moduli are the design variables, both viewed as non-negative fields on the design domain. The design variables are subject to the isoperimetric condition which is the upper bound on the two kinds of $L^p$-norms of the elastic moduli. The case of $p=1$ corresponds to the original concept of the Isotropic Material Design (IMD) method proposed in the paper: S. Czarnecki, Isotropic material design, Computational Methods in Science and Technology, 21 (2), 49-64, 2015. In the present paper the IMD method will be extended by assuming the $L^p$-norms-based cost conditions. In each case the optimum design problem is reduced to the pair of mutually dual problems of the mathematical structure of a theory of elasticity of an isotropic body with non-linear power-law type constitutive equations. The state of stress determines the optimal layouts of the bulk and shear moduli of the least compliant structure. The new methods proposed deliver the upper estimates for the optimal compliance predicted by the original IMD method.

The Isotropic Material Design methods with the cost expressed by the $L^p$-norm

TL;DR

This work introduces two L^p-cost isotropic material design variants, vp-IMD and sp-IMD, to minimize structural compliance by optimizing isotropic moduli k and μ under new cost bounds expressed as L^p norms. By deriving stress-based and displacement-based dual formulations and recasting the problems as non-linear elasticity with power-law constitutive equations, the authors establish existence (vp-IMD) and, in general, existence without uniqueness (sp-IMD) and connect these to the original IMD as p→1 while analyzing p→∞ limits. Numerical implementations on 2D L-shaped and cantilever domains demonstrate p-dependent behavior: concentration of moduli near singular features for small p and more diffuse but stable distributions for large p, with monotonic trends in minimal compliance. The methods allow solving via standard finite element nonlinear elasticity tools, avoiding measure-valued formulations and offering a practical, robust alternative to classical IMD, albeit with trade-offs such as loss of cutting-out and potential moduli concentration.

Abstract

The paper concerns the problem of minimization of the compliance of linear elastic structures made of an isotropic material. The bulk and shear moduli are the design variables, both viewed as non-negative fields on the design domain. The design variables are subject to the isoperimetric condition which is the upper bound on the two kinds of -norms of the elastic moduli. The case of corresponds to the original concept of the Isotropic Material Design (IMD) method proposed in the paper: S. Czarnecki, Isotropic material design, Computational Methods in Science and Technology, 21 (2), 49-64, 2015. In the present paper the IMD method will be extended by assuming the -norms-based cost conditions. In each case the optimum design problem is reduced to the pair of mutually dual problems of the mathematical structure of a theory of elasticity of an isotropic body with non-linear power-law type constitutive equations. The state of stress determines the optimal layouts of the bulk and shear moduli of the least compliant structure. The new methods proposed deliver the upper estimates for the optimal compliance predicted by the original IMD method.

Paper Structure

This paper contains 35 sections, 12 theorems, 121 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. On designing isotropic inhomogeneity in stiffness optimization
  3. On the Free Material -- and Isotropic Material Design methods
  4. Two modifications of the cost conditions of the IMD method
  5. Adopted conventions and notation
  6. Useful analytical results
  7. An auxiliary minimization result for the vp-IMD method
  8. An auxiliary minimization result for the sp-IMD method
  9. The problem of compliance minimization and its stress based reformulation
  10. Preliminaries on linear elasticity
  11. The vp-IMD method
  12. The sp-IMD method
  13. Displacement-based dual problems and optimality conditions
  14. The general duality result
  15. Duality in the vp-IMD variant
  16. ...and 20 more sections

Key Result

Lemma 2.1

For $p\in [1,\infty)$ and a vector ${\mathbf{a}}={\mathbf{a}}(x)$ of non-negative measurable functions $a_i(x) \geq 0$, $i=1,\ldots,n$, the following variational equality holds true: where $\sqrt{{\mathbf{a}}}(x) = (\sqrt{a_1(x)}, \ldots , \sqrt{a_n(x)} )$, and Moreover, assuming that $\sqrt{a_i} \in L^r(\Omega)$ for every $i$ and that at least one function $a_i$ is not a.e. zero, the problem eq

Figures (8)

  • Figure 1: The problem data: (a) the L-shaped domain problem; (b) the cantilever problem; $L=1.0\,\mathrm{m}$, $g_x = g_y= 5\, \mathrm{N/m}$.
  • Figure 2: L-shaped domain problem, optimal bulk modulus $\hat{k}$ for various $p\in[1,\infty)$.
  • Figure 3: L-shaped domain problem, optimal shear modulus $\hat{\mu}$ for various $p\in[1,\infty)$.
  • Figure 4: L-shaped domain problem, optimal Young modulus $\hat{E}$ for various $p\in[1,\infty)$.
  • Figure 5: L-shaped domain problem, optimal Poisson ratio $\hat{\nu}$ for various $p\in[1,\infty)$.
  • ...and 3 more figures

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • proof
  • ...and 12 more