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

TL;DR

This paper extends the isotropic material design method by incorporating $L^p$-norm cost constraints on elastic moduli, providing new dual problem formulations and upper compliance estimates for optimized linear elastic structures.

Contribution

It introduces a generalized IMD framework with $L^p$-norm constraints, expanding the original method and deriving dual problems with non-linear power-law constitutive equations.

Findings

Provides upper bounds for optimal compliance.

Extends IMD method to $L^p$-norm cost conditions.

Formulates dual problems with non-linear elasticity models.

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.