Isotropic Material Design

Abstract

The paper deals with optimal distribution of the bulk and shear moduli minimizing the compliance of an inhomogeneous isotropic elastic 3D body transmitting a given surface loading to a given support. The isoperimetric condition is expressed by the integral of the trace of the Hooke tensor being a linear combination of both moduli. The problem thus formulated is reduced to an auxiliary 3D problem of minimization of a certain stress functional over the stresses being statically admissible. The integrand of the auxiliary functional is a linear combination of the absolute value of the trace and norm of the deviator of the stress field. Thus the integrand is of linear growth. The auxiliary problem is solved numerically by introducing element-wise polynomial approximations of the components of the trial stress fields and imposing satisfaction of the variational equilibrium equations. The under-determinate system of these equations is solved numerically thus reducing the auxiliary problem to an unconstrained problem of nonlinear programming.

Optimal internal architectures of femoral bone based on relaxation by homogenization and isotropic material design

“…A second class of topology optimization methods, called IMD (Isotropic Material Design, a variant of the Free Material Design strategy) approach will be further applied to determine the optimal elasticity tensor in terms of the bulk and shear moduli for the present loading applied to the femoral bone sample. The IMD method has been recently developed in [9,10]; it is the isotropic version of the Free Material Design (FMD), originally proposed in [11], and then developed for various branches of optimization problems, see the review papers [12,13].…”

Material orientation design of planar structures with prescribed anisotropy classes. Study of rhombic systems

“…The kinematic formulation is under development. So far only the static problem has been the subject of the numerical treatment to solve material design problems see [2], [3]. This choice has been dictated by a direct link between the minimizers of the static problem and the optimum values of the design variables.…”

“…Because of that the algorithm that will solve OOD problem will be also based on the static formulation described in this paper. That will be done by an adaptation of the existing numerical algorithm used to solve IMD problem and described in [2]. The main change between both methods is the objective function.…”

“…A natural extension of FMD is a priori imposing certain material symmetries. The strongest assumption is isotropy -this modification has been proposed by Czarnecki [2] and it is called the Isotropic Material Design (IMD). In this formulation the only design variables are the bulk k and shear µ moduli at each point of the feasible domain.…”

“…Due to the similarity of CMD and IMD methods a stress field which minimizes (2) can be found by using the numerical algorithm described in Czarnecki [2] and Czarnecki and Wawruch [5]. The minimal compliance is expressed with the formula: Y = Z 2 /Λ in which Z is a number expressed by the minimization problem (2) and Λ is the integral of the trace of the Hooke tensor.…”