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Free material design in linear elasticity

Sławomir Czarnecki, Grzegorz Dzierżanowski,Tomasz Lewiński, Tomasz Sokół
Warsaw University of Technology, Faculty of Civil Engineering,
Institute of Building Engineering, Department of Structural Mechanics and Computer Aided Engineering

 

Free material design in linear elasticity


Abstract. The problem: find 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 norm of the stress tensor 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. An extension to the multiple load cases will also be discussed.

References

  • S.Czarnecki, T.Lewiński,The stiffest designs of elastic plates. Vector optimization for two loading conditions. Computer Methods in Applied Mechanics and Engineering 200(2011) 1708-1728.
  • G. Dzierżanowski, T. Lewiński, Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part I. Solving the local optimization problem, Arch.Mech. 64(2012) no 1, 21-40.
  • G. Dzierżanowski, T. Lewiński, Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part II. The effective boundary value problem and exemplary solutions, Arch.Mech. 64(2012), no 2, 111-135.
  • S. Czarnecki, T.Lewiński, A stress-based formulation of the free material design problem with the trace constraint and single loading condition. Bull.Polish Ac. Sciences. Tech.Sci. no 2., 60(2012) , pp.191-204.
   
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