Support structure constrained topology optimization for additive manufacturing

There is significant interest today in integrating additive manufacturing (AM) and topology optimization (TO). However, TO often leads to designs that are not AM friendly. For example, topologically optimized designs may require significant amount of support structures before they can be additively manufactured, resulting in increased fabrication and clean-up costs.In this paper, we propose a TO methodology that will lead to designs requiring significantly reduced support structures. Towards this end, the concept of ‘support structure topological sensitivity’ is introduced. This is combined with performance sensitivity to result in a TO framework that maximizes performance, subject to support structure constraints. The robustness and efficiency of the proposed method is demonstrated through numerical experiments, and validated through fused deposition modeling, a popular AM process.     

Large deformations and stability in topology optimization

The present contribution focuses on the influence of geometrical nonlinearities on the structural behavior in the design process. The notion of the stiffest structure loses its clear definition in the case of nonlinear kinematics; here we will discuss this concept on the basis of different objectives. Apparently topology optimization is often a generator of slender struts, which tend to buckle before the structure is completely loaded. To include the instability phenomena into the design process, the critical load level will be determined directly; this condition will be included as an inequality constraint. Further on, to reduce the imperfection sensitivity, a geometrically modified structure including the imperfection shape is also introduced. The present optimization procedures are demonstrated by examples showing rather the principal effects of the enhancements than real practical design problems.     

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.     

Piezoresistive sensor design using topology optimization

Piezoresistive materials, materials whose resistivity properties change when subjected to mechanical stresses, are widely utilized in many industries as sensors, including pressure sensors, accelerometers, inclinometers, and load cells. Basic piezoresistive sensors consist of piezoresistive devices bonded to a flexible structure, such as a cantilever or a membrane, where the flexible structure transmits pressure, force, or inertial force due to acceleration, thereby causing a stress that changes the resistivity of the piezoresistive devices. By applying a voltage to a piezoresistive device, its resistivity can be measured and correlated with the amplitude of an applied pressure or force. The performance of a piezoresistive sensor is closely related to the design of its flexible structure. In this research, we propose a generic topology optimization formulation for the design of piezoresistive sensors where the primary aim is high response. First, the concept of topology optimization is briefly discussed. Next, design requirements are clarified, and corresponding objective functions and the optimization problem are formulated. An optimization algorithm is constructed based on these formulations. Finally, several design examples of piezoresistive sensors are presented to confirm the usefulness of the proposed method.     

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 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.     

Application of spectral level set methodology in topology optimization

In this paper, two benchmark problems in structural boundary design are solved using the spectral level set methodology, which is a new approach to topology optimization of interfaces. This methodology is an extension of the level set methods, in which the interface is represented as the zero level set of a function. According to the proposed formulation, the Fourier coefficients of that function are the design variables describing the interface during the topology optimization. An advantage of the spectral level set methodology, in the case of a sufficiently regular interface, is to admit an upper bound error which is asymptotically smaller than the one for nonadaptive spacial discretizations of the level set function. Other advantages include the nucleation of holes in the interior of the interface and the avoidance of checkerboard-like designs. The theoretical framework of the methodology is presented and estimates on its convergence rate are discussed. The numerical applications consist in the design of short and long cantilevers subject to a vertical concentrated load opposite to the fixed end. The goal is to maximize the structural stiffness subject to a solid volume constraint. The zero level set of the level set function defines the structural boundary.     

Note on topology optimization of continuum structures including self-weight

This paper proposes to investigate topology optimization with density-dependent body forces and especially self-weight loading. Surprisingly the solution of such problems cannot be based on a direct extension of the solution procedure used for minimum-compliance topology optimization with fixed external loads. At first the particular difficulties arising in the considered topology problems are pointed out: non-monotonous behaviour of the compliance, possible unconstrained character of the optimum and the parasitic effect for low densities when using the power model (SIMP). To get rid of the last problem requires the modification of the power law model for low densities. The other problems require that the solution procedure and the selection of appropriate structural approximations be revisited. Numerical applications compare the efficiency of different approximation schemes of the MMA family. It is shown that important improvements are achieved when the solution is carried out using the gradient-based method of moving asymptotes (GBMMA) approximations. Criteria for selecting the approximations are suggested. In addition, the applications also provide the opportunity to illustrate the strong influence of the ratio between the applied loads and the structural weight on the optimal structural topology.     

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.     

A surface reconstruction algorithm for topology optimization

For mechanical structural design, topology optimization is often utilized. During this process, a topologically optimized model must be converted into a parametric CAD solid model. The key point of conversion is that a discretized shape of a topologically optimized model must be smoothed, but features such as creases and corners must be retained. Thus, a surface reconstruction algorithm to produce the parametric CAD solid model from a topologically optimized model is proposed in this paper. Our presented algorithm consists of three parts: (1) an enclosed isosurface geometry from which the topologically optimized model is generated, (2) features detected and (3) the parametric CAD solid model reconstructed as biquartic surface splines. In order to generate an enclosed isosurface model effectively, we propose an algorithm based upon the marching cubes method to detect elements intersected by an isosurface. After generating an enclosed isosurface model, we produce biquartic surface splines. By applying our algorithm to an enclosed isosurface model, it is possible to produce smoothed biquartic surface splines with features retained. Some examples are shown and the effectiveness of our algorithm is discussed in this paper.     

A review of optimization of cast parts using topology optimization

In recent years, there has been considerable progress in the optimization of cast parts with respect to strength, stiffness, and frequency. Here, topology optimization has been the most important tool in finding the optimal features of a cast part, such as optimal cross-section or number and arrangement of ribs. An optimization process with integrated topology optimization has been used very successfully at Adam Opel AG in recent years, and many components have been optimized. This two-paper review gives an overview of the application and experience in this area. This is the first part of a two-paper review of optimization of cast parts.Here, we want to focus on the application of the original topology optimization codes, which do not take manufacturing constraints for cast parts into account. Additionally, the role of shape optimization as a fine-tuning tool will be briefly analyzed and discussed.     

A review of homogenization and topology optimization III-topology optimization using optimality criteria

This is the first part of a three-paper review of homogenization and topology optimization, viewed from an engineering standpoint and with the ultimate aim of clarifying the ideas so that interested researchers can easily implement the concepts described. In the first paper we focus on the theory of the homogenization method where we are concerned with the main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with a periodic micro structure. Such materials are described by the base cell, which is the smallest repetitive unit of material, and the evaluation of the effective constitutive parameters may be carried out by analysing the base cell alone. For simple microstructures this may be achieved analytically, whereas for more complicated systems numerical methods such as the finite element method must be employed. In the second paper, we consider numerical and analytical solutions of the homogenization equations. Topology optimization of structures is a rapidly growing research area, and as opposed to shape optimization allows the introduction of holes in structures, with consequent savings in weight and improved structural characteristics. The homogenization approach, with an emphasis on the optimality criteria method, will be the topic of the third paper in this review.     

Maximization of eigenvalues using topology optimization

Topology optimization is used to optimize the eigenvalues of plates. The results are intended especially for MicroElectroMechanical Systems (MEMS) but can be seen as more general. The problem is not formulated as a case of reinforcement of an existing structure, so there is a problem related to localized modes in low density areas. The topology optimization problem is formulated using the SIMP method. Special attention is paid to a numerical method for removing localized eigenmodes in low density areas. The method is applied to numerical examples of maximizing the first eigenfrequency. One example is a practical MEMS application; a probe used in an Atomic Force Microscope (AFM). For the AFM probe the optimization is complicated by a constraint on the stiffness and constraints on higher order eigenvalues. Received June 10, 1999     

An approach of topology optimization of multi-rigid-body mechanism

Topology synthesis of multi-rigid-body mechanisms has always been a very important stage in the mechanism design process. In most cases, the topology of the multi-rigid-body mechanism for particular task is obtained by designers’ experience and ingenuity, rather than automatic approach. In this work, an approach of topology optimization of multi-rigid-body mechanisms is investigated. The core process of the approach is an automatic optimization design process. In this approach, we construct kinematics mapping from truss structures to the joint-linked mechanisms, which transforms the topology optimization problem of multi-body system into the truss structure optimization problem. We also develop a new strategy for topology optimization of statically determinate truss, the advantage of which lies in the ability dealing with statically determinate truss topology optimization problem compared to the existing methods. By automatically optimizing the topology of the truss structure, the topology of the multi-rigid-body mechanism is optimized automatically, accordingly. Here, we utilize the investigated approach to design suitable layout for multi-rigid-body micro-displacement amplifying mechanisms (MMAMs) with a large amplification ratio (>50). The layout consists of not only the topology information of the mechanism, but also the dimension parameters of the mechanism. The procedure of the approach is carried out in steps, and a human–computer interaction program has been developed for it. Using the developed program, different MMAMs are achieved. Meanwhile, the direct kinematics analysis of the MMAMs is achieved automatically, the existence of dead point position in the mechanism within movement range is checked and the micro-displacement amplification ratio is calculated out. The computing results are validated by the ADAMS^® motion simulation, which proves that the achieved MMAMs fully fulfill the functional requirement. Along with two of the achieved MMAMs, the approach is explained, its functionality is shown, its advantages, limitations, some open problems and future works are discussed.     

Globally optimal benchmark solutions to some small-scale discretized continuum topology optimization problems

In this note, globally optimal solutions to three sets of small-scale discretized continuum topology optimization problems are presented. All the problems were discretized by the use of nine-node isoparametric finite elements. The idea is that these solutions can be used as benchmark problems when testing new algorithms for finding pure 0–1 solutions to topology optimization problems defined on discretized ground structures.     

Target-matching test problem for multiobjective topology optimization using genetic algorithms

This paper describes the multiobjective topology optimization of continuum structures solved as a discrete optimization problem using a multiobjective genetic algorithm (GA) with proficient constraint handling. Crucial to the effectiveness of the methodology is the use of a morphological geometry representation that defines valid structural geometries that are inherently free from checkerboard patterns, disconnected segments, or poor connectivity. A graph- theoretic chromosome encoding, together with compatible reproduction operators, helps facilitate the transmission of topological/shape characteristics across generations in the evolutionary process. A multicriterion target-matching problem developed here is a novel test problem, where a predefined target geometry is the known optimum solution, and the good results obtained in solving this problem provide a convincing demonstration and a quantitative measure of how close to the true optimum the solutions achieved by GA methods can be. The methodology is then used to successfully design a path-generating compliant mechanism by solving a multicriterion structural topology optimization problem.     

Combining genetic algorithms with BESO for topology optimization

This paper proposes a new algorithm for topology optimization by combining the features of genetic algorithms (GAs) and bi-directional evolutionary structural optimization (BESO). An efficient treatment of individuals and population for finite element models is presented which is different from traditional GAs application in structural design. GAs operators of crossover and mutation suitable for topology optimization problems are developed. The effects of various parameters used in the proposed GA on the optimization speed and performance are examined. Several 2D and 3D examples of compliance minimization problems are provided to demonstrate the efficiency of the proposed new approach and its capability of obtaining convergent solutions. Wherever possible, the numerical results of the proposed algorithm are compared with the solutions of other GA methods and the SIMP method.     

Adhesive surface design using topology optimization

Tailoring adhesive properties between surfaces is of great importance for micro-scale systems, ranging from managing stiction in MEMS devices to designing wall-scaling gecko-like robots. A methodology is introduced for designing adhesive interfaces between structures using topology optimization. Structures subjected to external loads that lead to delamination are studied for situations where displacements and deformations are small. Only the effects of adhesive forces acting normal to the surfaces are considered. An interface finite element is presented that couples a penalty contact formulation and a Lennard–Jones model of van der Waals adhesive forces. Two- and three dimensional design optimization problems are presented in which adhesive force distributions are designed such that load-displacement curves of delaminating structures match target responses. The design variables describe the adhesive energy per area of the interface between the surfaces, as well as the geometry of the delaminating structure. A built-in length scale in the formulation of the adhesion forces eliminates the need for filtering to achieve comparable optimal adhesive designs over a range of mesh densities. The resulting design problem is solved by gradient based optimization algorithms evaluating the design sensitivities by the adjoint method. Results show that the delamination response can be effectively manipulated by the method presented. Varying simultaneously both adhesive and geometric parameters yields a wider range of reachable target load-displacement curves than in the case varying adhesive energy alone.     

Reliability optimization of topology communication network design using an improved ant colony optimization

Network design problem is a well-known NP-hard problem which involves the selection of a subset of possible links or a network topology in order to minimize the network cost subjected to the reliability constraint. To overcome the problem, this paper proposes a new efficiency algorithm based on the conventional ant colony optimization (ACO) to solve the communication network design when considering both economics and reliability. The proposed method is called improved ant colony optimizations (IACO) which introduces two addition techniques in order to improve the search process, i.e. neighborhood search and re-initialization process. To show its efficiency, IACO is applied to test with three different topology network systems and its results are compared with those obtained results from the conventional approaches, i.e. genetic algorithm (GA), tabu search algorithm (TSA) and ACO. Simulation results, obtained these test problems with various constraints, shown that the proposed approach is superior to the conventional algorithms both solution quality and computational time.     

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.