POD-assisted strategies for structural topology optimization

We propose a new numerical tool for structural optimization design. To cut down the computational burden typical of the Solid Isotropic Material with Penalization (SIMP) method, we apply Proper Orthogonal Decomposition on SIMP snapshots computed on a fixed grid to construct a rough structure (predictor) which becomes the input of a SIMP procedure performed on an anisotropic adapted mesh (corrector). The benefit of the proposed design tool is to deliver smooth and sharp layouts which require a contained computational effort before moving to the 3D printing production phase.     

A critical review of established methods of structural topology optimization

The aim of this article is to evaluate and compare established numerical methods of structural topology optimization that have reached the stage of application in industrial software. It is hoped that our text will spark off a fruitful and constructive debate on this important topic. This article is an extended version of a paper presented at the WCSMO-7 in Seoul in 2007.     

On topology optimization of linear and nonlinear plate problems

In this paper we propose a new restriction method based on employing C ⁰-continuous fields of density defined on a set of meshes different from the one used for the finite element analysis. The optimization procedure starts with using a coarse density-mesh compared to the finite element one. Once the convergence is obtained in the optimization steps, a finer density-mesh is nominated for the further steps. Linear and nonlinear plate behaviors are considered and formulated by Kirchhoff or Mindlin–Reissner hypothesis. Comparison is made with element/nodal based approaches using filter. The results show excellent and robust performance of the proposed method.     

Structural optimization based on topology optimization techniques using frame elements considering cross-sectional properties

This paper discusses a new structural optimization method, based on topology optimization techniques, using frame elements where the cross-sectional properties can be treated as design variables. For each of the frame elements, the rotational angle denoting the principal direction of the second moment of inertia is included as a design variable, and a procedure to obtain the optimal angle is derived from Karush–Kuhn–Tucker (KKT) conditions and a complementary strain energy-based approach. Based on the above, the optimal rotational angle of each frame element is obtained as a function of the balance of the internal moments. The above methodologies are applied to problems of minimizing the mean compliance and maximizing the eigen frequencies. Several examples are provided to show the utility of the presented methodology.     

Design of an energy-absorbing structure using topology optimization with a multimaterial model

To accommodate the dual objectives of many engineering applications, one objective to minimize the mean compliance for the stiffest structure under normal service conditions and the other objective to maximize the strain energy for energy absorption during excessive loadings, topology optimization with a multimaterial model is applied to the design of an energy-absorbing structure in this paper. The effective properties of the three-phase material are derived using a spherical microinclusion model. The dual objectives are combined in a ratio formation. Numerical examples from the proposed method are presented and discussed.     

A hierarchical neighbourhood search method for topology optimization

This paper presents a hierarchical neighbourhood search method for solving topology optimization problems defined on discretized linearly elastic continuum structures. The design of the structure is represented by binary design variables indicating material or void in the various finite elements.Two different designs are called neighbours if they differ in only one single element, in which one of them has material while the other has void. The proposed neighbourhood search method repeatedly jumps to the best neighbour of the current design until a local optimum has been found, where no further improvement can be made. The engine of the method is an efficient exploitation of the fact that if only one element is changed (from material to void or from void to material) then the new global stiffness matrix is just a low-rank modification of the old one. To further speed up the process, the method is implemented in a hierarchical way. Starting from a coarse finite element mesh, the neighbourhood search is repeatedly applied on finer and finer meshes.Numerical results are presented for minimum-weight problems with constraints on respectively compliance, strain energy densities in all non-void elements, and von Mises stresses in all non-void elements.     

Deposition path planning-integrated structural topology optimization for 3D additive manufacturing subject to self-support constraint

This paper presents a novel level set-based topology optimization implementation, which addresses two main problems of design-for-additive manufacturing (AM): the material anisotropy and the self-support manufacturability constraint. AM material anisotropy is widely recognized and taking it into account while performing structural topology optimization could more realistically evaluate the structural performance. Therefore, both build direction and in-plane raster directions are considered by the topology optimization algorithm, especially for the latter, which is calculated through deposition path planning. The self-support manufacturability constraint is addressed through a novel multi-level set modeling. The related optimization problem formulation and solution process are demonstrated in detail. It is proved by several numerical examples that the manufacturability constraints are always strictly satisfied. Marginally, the recently popular structural skeleton-based deposition paths are also employed to assist the structural topology optimization, and its characteristics are discussed.     

A topological derivative method for topology optimization

We propose a fictitious domain method for topology optimization in which a level set of the topological derivative field for the cost function identifies the boundary of the optimal design. We describe a fixed-point iteration scheme that implements this optimality criterion subject to a volumetric resource constraint. A smooth and consistent projection of the region bounded by the level set onto the fictitious analysis domain simplifies the response analysis and enhances the convergence of the optimization algorithm. Moreover, the projection supports the reintroduction of solid material in void regions, a critical requirement for robust topology optimization. We present several numerical examples that demonstrate compliance minimization of fixed-volume, linearly elastic structures.     

Sequential integer programming methods for stress constrained topology optimization

This paper deals with topology optimization of load carrying structures defined on a discretized design domain where binary design variables are used to indicate material or void in the various finite elements. The main contribution is the development of two iterative methods which are guaranteed to find a local optimum with respect to a 1-neighbourhood. Each new iteration point is obtained as the optimal solution to an integer linear programming problem which is an approximation of the original problem at the previous iteration point. The proposed methods are quite general and can be applied to a variety of topology optimization problems defined by 0-1 design variables. Most of the presented numerical examples are devoted to problems involving stresses which can be handled in a natural way since the design variables are kept binary in the subproblems.