Porous structures such as trabecular bone are widely seen in nature. These structures are lightweight and exhibit strong mechanical properties. In this paper, we present a method to generate bone-like porous structures as lightweight infill for additive manufacturing. Our method builds upon and extends voxel-wise topology optimization. In particular, for the purpose of generating sparse yet stable structures distributed in the interior of a given shape, we propose upper bounds on the localized material volume in the proximity of each voxel in the design domain. We then aggregate the local per-voxel constraints by their p -norm into an equivalent global constraint, in order to facilitate an efficient optimization process. Implemented on a high-resolution topology optimization framework, our results demonstrate mechanically optimized, detailed porous structures which mimic those found in nature. We further show variants of the optimized structures subject to different design specifications, and we analyze the optimality and robustness of the obtained structures.
Topology optimization is based on a volumetric element-wise parametrization of the design domain. This general formulation does not prescribe the topology , but allows structures to appear and adapt during the iterative optimization process. For a thorough review of topology optimization techniques, let us refer to recent survey articles @cite_28 @cite_22 . Our work is based on the density approach, which is known as Solid Isotropic Material with Penalization (SIMP) @cite_41 . The method is related to the length scale problem in the literature of topology optimization, where the interest is to control the minimal and or maximal structure size for manufacturability @cite_33 @cite_14 . In particular, we follow the idea of the projection filter @cite_13 @cite_48 in our implementation to impose local volume constraints. Different from exact length scale control, we propose a projection method in an approximate manner which facilitates fast numerical solution. In our work we employ the potential of numerical multigrid schemes to enable topology optimization at high resolution and efficiency @cite_1 .
@cite_14 @cite_22 @cite_33 @cite_28 @cite_41 @cite_48 @cite_1 @cite_13
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Since its original introduction in structural design, density-based topology optimization has been applied to a number of other fields such as microelectromechanical systems, photonics, acoustics and fluid mechanics. The methodology has been well accepted in industrial design processes where it can provide competitive designs in terms of cost, materials and functionality under a wide set of constraints. However, the optimized topologies are often considered as conceptual due to loosely defined topologies and the need of postprocessing. Subsequent amendments can affect the optimized design performance and in many cases can completely destroy the optimality of the solution. Therefore, the goal of this paper is to review recent advancements in obtaining manufacturable topology-optimized designs. The focus is on methods for imposing minimum and maximum length scales, and ensuring manufacturable, well-defined designs with robust performances. The overview discusses the limitations, the advantages and the associated computational costs. The review is completed with optimized designs for minimum compliance, mechanism design and heat transfer. Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain. This paper surveys topology optimization of continuum structures from the year 2000 to 2012. It focuses on new developments, improvements, and applications of finite element-based topology optimization, which include a maturation of classical methods, a broadening in the scope of the field, and the introduction of new methods for multiphysics problems. Four different types of topology optimization are reviewed: (1) density-based methods, which include the popular Solid Isotropic Material with Penalization (SIMP) technique, (2) hard-kill methods, including Evolutionary Structural Optimization (ESO), (3) boundary variation methods (level set and phase field), and (4) a new biologically inspired method based on cellular division rules. We hope that this survey will provide an update of the recent advances and novel applications of popular methods, provide exposure to lesser known, yet promising, techniques, and serve as a resource for those new to the field. The presentation of each method's focuses on new developments and novel applications. A new scheme for imposing a minimum length scale in topology optimization is presented. It guarantees the existence of an optimal design for a large class of topology optimization problems of practical interest. It is formulated as one constraint that is computationally cheap and for which sensitivities are also cheap to compute. The constraint value is ideally zero, but it can be relaxed to a positive value. The effect of the method is illustrated in topology optimization for minimum compliance and design of compliant mechanisms. Notably, the method produces compliant mechanisms with distributed flexibility, something that has previously been difficult to obtain using topology optimization for the design of compliant mechanisms. The term ‘MOLE method’ is suggested for the method. Copyright © 2003 John Wiley & Sons, Ltd. Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsoe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research. The paper presents a compact Matlab implementation of a topology optimization code for compliance minimization of statically loaded structures. The total number of Matlab input lines is 99 including optimizer and Finite Element subroutine. The 99 lines are divided into 36 lines for the main program, 12 lines for the Optimality Criteria based optimizer, 16 lines for a mesh-independency filter and 35 lines for the finite element code. In fact, excluding comment lines and lines associated with output and finite element analysis, it is shown that only 49 Matlab input lines are required for solving a well-posed topology optimization problem. By adding three additional lines, the program can solve problems with multiple load cases. The code is intended for educational purposes. The complete Matlab code is given in the Appendix and can be down-loaded from the web-site http: www.topopt.dtu.dk. This paper presents a technique for imposing maximum length scale on features in continuum topology optimization. The design domain is searched and local constraints prevent the formation of features that are larger than the prescribed maximum length scale. The technique is demonstrated in the context of structural and fluid topology optimization. Specifically, maximum length scale criterion is applied to (a) the solid phase in minimum compliance design to restrict the size of structural (load-carrying) members, and (b) the fluid (void) phase in minimum dissipated power problems to limit the size of flow channels. Solutions are shown to be near 0 1 (void solid) topologies that satisfy the maximum length scale criterion. When combined with an existing minimum length scale methodology, the designer gains complete control over member sizes that can influence cost and manufacturability. Further, results suggest restricting maximum length scale may provide a means for influencing performance characteristics, such as redundancy in structural design. A key requirement in 3D fabrication is to generate objects with individual exterior shapes and their interior being optimized to application-specific force constraints and low material consumption. Accomplishing this task is challenging on desktop computers, due to the extreme model resolutions that are required to accurately predict the physical shape properties, requiring memory and computational capacities going beyond what is currently available. Moreover, fabrication-specific constraints need to be considered to enable printability. To address these challenges, we present a scalable system for generating 3D objects using topology optimization, which allows to efficiently evolve the topology of high-resolution solids towards printable and light-weight-high-resistance structures. To achieve this, the system is equipped with a high-performance GPU solver which can efficiently handle models comprising several millions of elements. A minimum thickness constraint is built into the optimization process to automatically enforce printability of the resulting shapes. We further shed light on the question how to incorporate geometric shape constraints, such as symmetry and pattern repetition, in the optimization process. We analyze the performance of the system and demonstrate its potential by a variety of different shapes such as interior structures within closed surfaces, exposed support structures, and surface models. A methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described. Nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology. The projection is made via mesh independent functions that are based upon the minimum length scale. A simple linear projection scheme and a non-linear scheme using a regularized Heaviside step function to achieve nearly 0–1 solutions are examined. The new approach is demonstrated on the minimum compliance problem and the popular SIMP method is used to penalize the stiffness of intermediate volume fraction elements. Solutions are shown to meet user-defined length scale criterion without additional constraints, penalty functions or sensitivity filters. No instances of mesh dependence or checkerboard patterns have been observed. Copyright © 2004 John Wiley & Sons, Ltd.