10 th World Congress on Structural and Multidisciplinary Optimization May 19 - 24, 2013, Orlando, Florida, USA Multiscale Topology Optimization of Structures and Non-Periodic Cellular Materials Kai Liu 1 , Kapil Khandelwal 2 , Andr´ es Tovar 1 1 Department of Mechanical Engineering Indiana University-Purdue University Indianapolis, Indianapolis, IN 46202, USA Emails: kailiu@iupui.edu,tovara@iupui.edu 2 Department of Civil & Environmental Engineering & Earth Sciences University of Notre Dame, Notre Dame, IN 46556, USA Email: kapil.khandelwal@nd.edu 1. Abstract Topology optimization allows designers to obtaining lightweight structures considering the binary distri- bution of a solid material. Further material savings and increased performance may be achieved if the material and the structure topologies are concurrently optimized. The use of homogenization methods promotes the introduction of material-scale parameters in the problem’s formulation. While some re- search has been focused on material parameters and periodic topology optimization, this work deals with non-periodic material topologies. Since no preconceived material and structure geometries are considered, the multiscale approach is capable of driving the design to innovative and potentially better configura- tions at both length scales. The proposed methodology is applied to minimum compliance problems and compliant mechanism synthesis. The multiscale results are compared with the traditional structural-level designs in the context of Pareto solutions, demonstrating benefits of ultra-lightweight configurations. 2. Keywords: hierarchical, homogenization, multiscale, non-periodic, topology optimization 3. Introduction Topology optimization is a design method used to find optimal material distribution within a design domain. For a given set of boundary and loading conditions, topology optimization drives the material distribution process to a structural layout that maximizes performance objectives and satisfies design constraints. Traditionally, lightweight concept designs have been obtained using homogeneous materials. However, this approach is challenged by the use of cellular materials. The use of cellular materials in topology optimization results on multiscale arrangements are referred to as ultra-lightweight structures. Ultra-lightweight structures are characterized by a high strength-to-weight ratio, and are desired in automobile, aerospace, and aircraft design due to their high performance and reduced energy consumption involved. Cellular materials are commonly selected from a set of commercially available layouts, e.g., triangular, cubic, and honeycomb among others. Topology optimization in cellular materials design acts for an important alternative to classic materials design procedures. The structural design is closely related to the design of meso-structural cellular ma- terials via the direct homogenization theory [1,2]. In contrast, the inverse homogenization is a topology optimization procedure, which denotes to obtain material distribution of micro-/meso-scale with desired homogenized or effective material properties. Since it has been proposed by Sigmund [3], the inverse homogenization method has been applied for numerous different applications, such as material with neg- ative Poisson’s ration [3], Functionally Graded Materials (FGM), which are continuously graded in one or more specified directions [4], and has been adapted for multi-functional composites design (c.f. [5], [6] and references therein). Most recent works as Schury et al. [7], an efficient hierarchical topology optimization procedure with manufacturable constraint was presented. A design methodology for optimal poroelastic actuators was proposed by Andreasen et al. [8]. One should notice that the multiscale optimization meth- ods mentioned above are actually a micro-/meso-scale material structure design optimization problem using macro-scale objective function. However, higher material savings and increased performance may be achieved if the structure and the cellular topologies are concurrently optimized. Multi-scale design of structural and material has been considered with the use of porous materials in the seminal work by Bendsøe et al [9]. In their work, the recommended strategy was to consider simple square voids at the micro-scale in the context of minimum compliance design. Similar approach was extended to compliant mechanism design by Nishiwaki et al [10]. Rodrigues et al. [11] proposed a hierarchical 1