Multi-scale design of composite materials and structures for maximum natural frequencies

Abstract This paper introduces a hierarchical concurrent design approach to maximizing the natural frequency of a structure. Multiple material phases are considered in the topology optimization performed on both the macro and micro scales. A general problem for composite structure and material design is formulated that contains the cellular design problem as a special case. The design of the macro structure and material micro structure is coupled. The designed material properties are applied to the analysis of the macro structure, while the macro structure displacement field is considered in the sensitivity analysis on the micro scale. The material edistribution is controlled by an optimality criterion for frequency maximization. Convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) algorithms are employed to obtain the final optimal solution. Several numerical examples of composite structures and materials are presented to demonstrate the capability and effectiveness of the proposed approach. Results include various orthotropic or anisotropic composite materials, as well as vibration-resisting layouts of the macro structure. In-depth discussions are also given on the effects of the base material phases and the assignment of the volume fractions on each scale.

[1]  Weihong Zhang,et al.  Scale‐related topology optimization of cellular materials and structures , 2006 .

[2]  M. Zhou,et al.  Generalized shape optimization without homogenization , 1992 .

[3]  M. Zhou,et al.  The COC algorithm, Part II: Topological, geometrical and generalized shape optimization , 1991 .

[4]  E. Hinton,et al.  A review of homogenization and topology opimization II—analytical and numerical solution of homogenization equations , 1998 .

[5]  Xiaoming Wang,et al.  A level set method for structural topology optimization , 2003 .

[6]  Yi Min Xie,et al.  Evolutionary topological optimization of vibrating continuum structures for natural frequencies , 2010 .

[7]  George I. N. Rozvany,et al.  DCOC: An optimality criteria method for large systems Part I: theory , 1992 .

[8]  N. Olhoff,et al.  Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps , 2007 .

[9]  Yi Min Xie,et al.  Evolutionary Structural Optimization , 1997 .

[10]  Z. Ishak,et al.  Kenaf fiber reinforced composites: A review , 2011 .

[11]  Behrooz Hassani,et al.  A direct method to derive the boundary conditions of the homogenization equation for symmetric cells , 1996 .

[12]  Shiwei Zhou,et al.  Topological design of structures and composite materials with multiobjectives , 2007 .

[13]  Yi Min Xie,et al.  Evolutionary Topology Optimization of Continuum Structures: Methods and Applications , 2010 .

[14]  J. Petersson,et al.  Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima , 1998 .

[15]  D. Chakraborty,et al.  Multi-scale modeling of carbon nanotube reinforced composites with a fiber break , 2012 .

[16]  Y. Xie,et al.  A simple evolutionary procedure for structural optimization , 1993 .

[17]  H. Rodrigues,et al.  Hierarchical optimization of material and structure , 2002 .

[18]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[19]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[20]  E. Hinton,et al.  A review of homogenization and topology optimization I- homogenization theory for media with periodic structure , 1998 .

[21]  Pedro G. Coelho,et al.  Hierarchical Topology Optimization of Structures Subjected to Constraints on Material Design , 2008 .

[22]  N. L. Pedersen.  Maximization of eigenvalues using topology optimization , 2000 .

[23]  Shiwei Zhou,et al.  Topology optimization of microstructures of cellular materials and composites for macrostructures , 2013 .

[24]  D. Fang,et al.  Nonlinear mechanical properties of lattice truss materials , 2009 .

[25]  Thiru Aravinthan,et al.  A review of optimization techniques used in the design of fibre composite structures for civil engineering applications , 2012 .

[26]  James K. Guest,et al.  Design of maximum permeability material structures , 2007 .

[27]  M. M. Neves,et al.  Optimal design of periodic linear elastic microstructures , 2000 .

[28]  S. Vel,et al.  Multi-objective optimization of functionally graded materials with temperature-dependent material properties , 2007 .

[29]  O. Sigmund.  Tailoring materials with prescribed elastic properties , 1995 .

[30]  Y. Xie,et al.  AN IMPROVED BI-DIRECTIONAL EVOLUTIONARY TOPOLOGY OPTIMIZATION METHOD FOR FREQUENCIES , 2010 .

[31]  Gengdong Cheng,et al.  Optimum structure with homogeneous optimum truss-like material , 2008 .

[32]  George I. N. Rozvany,et al.  Structural Design via Optimality Criteria , 1989 .

[33]  O. Sigmund.  A new class of extremal composites , 2000 .

[34]  N. Olhoff,et al.  Multiple eigenvalues in structural optimization problems , 1994 .

[35]  R. Lakes.  Materials with structural hierarchy , 1993, Nature.

[36]  Jun Yan,et al.  Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency , 2009 .

[37]  Y. Xie,et al.  Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials , 2009 .

[38]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[39]  M. Bendsøe.  Optimal shape design as a material distribution problem , 1989 .