Optimality criteria for the components of anisotropic constitutive matrix

• Published in: Structural and multidisciplinary optimization Vol. 50; no. 2; pp. 181 - 191
• Main Authors: Pedersen, Pauli, Pedersen, Niels L.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2014; Springer Nature B.V
• Subjects: Computational Mathematics and Numerical Analysis; Engineering; Engineering Design; Modulus of elasticity; Optimality criteria; Research Paper; Theoretical and Applied Mechanics; Positive definite; Uniform weighted elastic energy density; Free material optimization; Multiple load cases; Anisotropic constitutive components; Optimality criteria
• ISSN: 1615-147X; 1615-1488
• DOI: 10.1007/s00158-014-1060-8

In a free material formulation, the problem of minimizing a weighted sum of compliance’s from multiple load cases, subject to an active constraint on material volume, is solved in a formulation with two optimality criteria. The first optimality criterion for the distribution of material volume densities is equal value for the weighted elastic energy densities, as a natural extension of the optimality criterion for a single load case. The second optimality criterion for the components of a constitutive matrix (of unit norm) is proportionality to corresponding weighted strain components with the same proportionality factor λ ̂ for all the components, as shortly specified by C ijkl = λ ̂ − n η n ( 𝜖 ij ) n ( 𝜖 kl ) n , in traditional notation (n indicate load case). These simple analytical results should be communicated, in spite of the practical objection against design for weighted sum of compliance’s, as compared to worst case design and design considering strength. The application of the approach of the two optimality criteria is illustrated by a 2D example with 8 load cases. Stable and fast convergence is shown.