Optimization of Bodies with Locally Periodic Microstructure

This article describes a numerical study of the optimization of elastic bodies featuring a locally periodic microscopic pattern. The authors’ approach makes the link between the microscopic level and the macroscopic one. Two-dimensional linearly elastic bodies are considered; the same techniques can be applied to three-dimensional bodies. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body. The macroscopic domain is divided in (rectangular) finite elements and in each of them the microstructure is supposed to be periodic; the periodic pattern is allowed to vary from element to element. Each periodic microstructure is discretized using a finite element mesh on the periodicity cell, by identifying the opposite sides of the cell in order to handle the periodicity conditions in the cellular problem. Shape optimization and topology optimization are used at the microscopic level, following an alternate directions algorithm. Numerical examples are presented, in which a cantilever is optimized for different load cases, one of them being multi-load. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.

[1]  Qatu,et al.  Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, Vol 146 , 2003 .

[2]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[3]  Martin P. Bendsøe,et al.  Optimization of Structural Topology, Shape, And Material , 1995 .

[4]  G. Allaire,et al.  Structural optimization using topological and shape sensitivity via a level set method , 2005 .

[5]  Cristian Barbarosie,et al.  Shape optimization of periodic structures , 2003 .

[6]  Shmuel Vigdergauz,et al.  Two-dimensional grained composites of minimum stress concentration , 1997 .

[7]  G. Allaire,et al.  Optimal design for minimum weight and compliance in plane stress using extremal microstructures , 1993 .

[8]  Cristian Barbarosie.  Shape and Topology Optimization for Periodic Problems Part II: Optimization algorithm and numerical examples , 2008 .

[9]  Cristian Barbarosie,et al.  Optimization of bodies with locally periodic microstructure by varying the periodicity pattern , 2014, Networks Heterog. Media.

[10]  Cristian Barbarosie,et al.  Optimization of perforated domains through homogenization , 1997 .

[11]  D. Cioranescu,et al.  Homogenization in open sets with holes , 1979 .

[12]  Robert V. Kohn,et al.  Topics in the Mathematical Modelling of Composite Materials , 1997 .

[13]  OPTIMIZATION OF PERIODIC MICROSTRUCTURES USING SHAPE AND TOPOLOGY DERIVATIVES , 2008 .

[14]  Cristian Barbarosie,et al.  Shape and topology optimization for periodic problems , 2009 .

[15]  Vladimir Kobelev.  Comments on “Shape and topology optimization for periodic problems part I: the shape and the topological derivative” , 2010 .

[16]  Helder C. Rodrigues,et al.  A hierarchical model for concurrent material and topology optimisation of three-dimensional structures , 2008 .