A stress-based formulation of the free material design problem with the trace constraint and single loading condition

Abstract

Abstract. The problem to find an optimal distribution of elastic moduli within a given plane domain to make the compliance minimal under the condition of a prescribed value of the integral of the trace of the elastic moduli tensor is called the free material design with the trace constraint. The present paper shows that this problem can be reduced to a new problem of minimization of the integral of the stress tensor norm over stresses being statically admissible. The eigenstates and Kelvin’s moduli of the optimal Hooke tensor are determined by the stress state being the minimizer of this problem. This new problem can be directly treated numerically by using the Singular Value Decomposition (SVD) method to represent the statically admissible stress fields, along with any unconstrained optimization tool, e.g.: Conjugate Gradient (CG) or Variable Metric (VM) method in multidimensions.

A stress-based formulation of the free material design problem with the trace constraint and multiple load conditions

“…The results (50)- (53) have been for the first time published in Czarnecki and Lewiński (2012). These results are compatible with the results published in Bendsøe et al (1994) and in the next papers on FMD.…”

“…the singular value decomposition algorithm (SVD) and just this method has been applied in the present paper. The short description of the SVD algorithm has been outlined in Czarnecki and Lewiński (2012). Let us mention that the QR decomposition of the matrix makes it possible to find the representation (79) more effectively than the SVD decomposition, due to a smaller computational complexity.…”

“…7c) is subject simultaneously to a load T e 1 + T e 2 at its bottom edge and to a load −T e 1 + T e 2 at its upper edge, where the function T = T (ζ ) has the form (92). For this single load condition, optimal layout of the one Kelvin modulusλ 1 and componentsC ij kl of the Hooke tensor are found on the basis of the algorithm presented in Czarnecki and Lewiński (2012) (see Section 3.2). If we e.g.…”

“…The first step is to solve an auxiliary minimization problem over the statically admissible stress fields. Having found the minimizer one can determine all the optimal elastic moduli, see Czarnecki and Lewiński (2012).…”

“…The method is an extension of the numerical approach proposed in Czarnecki and Lewiński (2012) for the single load case. Although the meshing of the design domain is a starting point, the numerical method proposed does not belong to the finite element (FE) methods, since the integrand of the functional has a linear growth, while most FEM codes are applicable for the integrands of quadratic growth.…”