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1

Isotropic Material Design

Czarnecki S.

Computational Methods in Science and Technology

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2015

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Vol. 21, No. 2

49--64

EN

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.

2

Topology optimization in structural mechanics

Lewiński T., Czarnecki S., Dzierżanowski G., Sokół T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2013

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Vol. 61, nr 1

23-37

EN

Optimization of structural topology, called briefly: topology optimization, is a relatively new branch of structural optimization. Its aim is to create optimal structures, instead of correcting the dimensions or changing the shapes of initial designs. For being able to create the structure, one should have a possibility to handle the members of zero stiffness or admit the material of singular constitutive properties, i.e. void. In the present paper, four fundamental problems of topology optimization are discussed: Michell’s structures, two-material layout problem in light of the relaxation by homogenization theory, optimal shape design and the free material design. Their features are disclosed by presenting results for selected problems concerning the same feasible domain, boundary conditions and applied loading. This discussion provides a short introduction into current topics of topology optimization.

3

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

Czarnecki S., Lewiński T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2012

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Vol. 60, nr 2

191-204

EN

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.

4

Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part II. The effective boundary value problem and exemplary solutions

Dzierżanowski G., Lewiński T.

Archives of Mechanics

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2012

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Vol. 64, nr 2

137-137

EN

The compliance minimization of transversely homogeneous plates with predefined Kelvin moduli leads to the equilibrium problem of an effective hyperelastic plate with the hyperelastic potential expressed explicitly in terms of both the membrane and bending strain measures, as derived in Part I of the present paper. The aim of this second part of the paper is to show convexity of this potential and, consequently, uniqueness of solutions of the minimum compliance problem considered. Theoretical results are illustrated by numerically calculated optimal trajectories of the eigenstate corresponding to the largest Kelvin modulus.

5

Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part I. Solving the local optimization problem

Dzierżanowski G., Lewiński T.

Archives of Mechanics

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2012

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Vol. 64, nr 1

21-21

EN

The paper deals with compliance minimization of a transversely homogeneous plate, subjected to the in-plane and transverse loadings acting simultaneously. The set of design variables includes the eigenstates of Hooke’s tensor whose eigenvalues, i.e. Kelvin moduli fields, are assumed to be fixed on the middle plane of the plate, but no isoperimetric condition is imposed. The optimization task reduces to an equilibrium problem of an effective hyperelastic plate. The effective potential is explicitly expressed in terms of the invariants of both the strain fields involved.

6

On the fundamental problem of optimum design of anisotropic plates

Lewiński T.

Foundations of Civil and Environmental Engineering

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2006

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No. 7

215-226

EN

The paper deals with the free material design problem of minimum compliance of an anisotropic clastic plate loaded in-plane. All the characteristics of the plate stiffness tensor or of the form of the Hookc tensor for plane case, are treated as design variables. The cost function is expressed in terms of the Kelvin moduli. The necessary conditions of optimality are discussed. They imply that the deformation state within the optimal plate must satisfy the condition of colinearity of stress and strain tensors.