Topology optimization of continuum structures subjected to pressure loading

This paper presents a generalization of topology optimization of linearly elastic continuum structures to problems involving loadings that depend on the design. Minimum compliance is chosen as the design objective, assuming the boundary conditions and the total volume within the admissible design domain to be given. The topology optimization is based on the usage of a SIMP material model. The type of loading considered in this paper occurs if free structural surface domains are subjected to static pressure, in which case both the direction and location of the loading change with the structural design. The presentation of the material is given in a 2D context, but an extension to 3D is straightforward. The robustness of the optimization method is illustrated by some numerical examples in the end of the paper. Received August 3, 1999     

Topology optimization of continuum structures under hybrid uncertainties

The aim of this paper is to study the topology optimization for mechanical systems with hybrid material and geometric uncertainties. The random variations are modeled by a memory-less transformation of random fields which ensures their physical admissibility. The stochastic collocation method combined with the proposed material and geometry uncertainty models provides robust designs by utilizing already developed deterministic solvers. The computational cost is decreased by using of sparse grids and discretization refinement that are proposed and demonstrated as well. The method is utilized in the design of minimum compliance structure. The proposed algorithm provides a computationally cheap alternative to previously introduced stochastic optimization methods based on Monte Carlo sampling by using adaptive sparse grids method.     

Topology optimization of truss-like continuum structures for natural frequencies

A method to maximize the natural frequencies of vibration of truss-like continua with the constraint of material volume is presented. Truss-like is a kind of particular anisotropic continuum, in which there are finite numbers of members with infinitesimal spaces. Structures are analyzed by finite element method. The densities and orientations of members at nodes are taken as design variables. The densities and orientations of members in elements are interpolated by these values at nodes; therefore they vary continuously in design domain. For no intermediate densities being suppressed, there is no numerical instability, such as checkerboard patterns and one-node connected hinges. The natural frequency and its sensitivities of truss-like continuum are derived. Optimization is achieved by the techniques of moving asymptotes and steepest descent. Several numerical examples are provided to demonstrate this optimization method.     

Topology optimization of continuum structures with material failure constraints

This work presents an efficient strategy for dealing with topology optimization associated with the problem of mass minimization under material failure constraints. Although this problem characterizes one of the oldest mechanical requirements in structural design, only a few works dealing with this subject are found in the literature. Several reasons explain this situation, among them the numerical difficulties introduced by the usually large number of stress constraints. The original formulation of the topological problem (existence/non-existence of material) is partially relaxed by following the SIMP (Solid Isotropic Microstructure with Penalization) approach and using a continuous density field as the design variable. The finite element approximation is used to solve the equilibrium problem, as well as to control through nodal parameters. The formulation accepts any failure criterion written in terms of stress and/or strain invariants. The whole minimization problem is solved by combining an augmented Lagrangian technique for the stress constraints and a trust-region box-type algorithm for dealing with side constraints (0<_min1) . Numerical results show the efficiency of the proposed approach in terms of computational costs as well as satisfaction of material failure constraints. It is also possible to see that the final designs define quite different shapes from the ones obtained in classical compliance problems.     

Topology optimization of freely vibrating continuum structures based on nonsmooth optimization

The non-differentiability of repeated eigenvalues is one of the key difficulties to obtain the optimal solution in the topology optimization of freely vibrating continuum structures. In this paper, the bundle method, which is a very promising one in the nonsmooth optimization algorithm family, is proposed and implemented to solve the problem of eigenfrequency optimization of continuum. The bundle method is well-known in the mathematical programming community, but has never been used to solve the problems of topology optimization of continuum structures with respect to simple or multiple eigenfrequencies. The advantage of this method is that the specified information of iteration history may be collected and utilized in a very efficient manner to ensure that the next stability center is closer to the optimal solution, so as to avoid the numerical oscillation in the iteration history. Moreover, in the present method, both the simple and multiple eigenfrequencies can be managed within a unified computational scheme. Several numerical examples are tested to validate the proposed method. Comparisons with nonlinear semidefinite programming method and 0–1 formulation based heuristic method show the advantages of the proposed method. It is showed that, the method can deal with the nonsmoothness of the repeated eigenvalues in topology optimization in a very effective and efficient manner without evaluating the multiplicity of the eigenvalues.     

Topology optimization framework for structures subjected to stationary stochastic dynamic loads

The field of topology optimization has progressed substantially in recent years, with applications varying in terms of the type of structures, boundary conditions, loadings, and materials. Nevertheless, topology optimization of stochastically excited structures has received relatively little attention. Most current approaches replace the dynamic loads with either equivalent static or harmonic loads. In this study, a direct approach to problem is pursued, where the excitation is modeled as a stationary zero-mean filtered white noise. The excitation model is combined with the structural model to form an augmented representation, and the stationary covariances of the structural responses of interest are obtained by solving a Lyapunov equation. An objective function of the optimization scheme is then defined in terms of these stationary covariances. A fast large-scale solver of the Lyapunov equation is implemented for sparse matrices, and an efficient adjoint method is proposed to obtain the sensitivities of the objective function. The proposed topology optimization framework is illustrated for four examples: (i) minimization of the displacement of a mass at the free end of a cantilever beam subjected to a stochastic dynamic base excitation, (ii) minimization of tip displacement of a cantilever beam subjected to a stochastic dynamic tip load, (iii) minimization of tip displacement and acceleration of a cantilever beam subjected to a stochastic dynamic tip load, and (iv) minimization of a plate subjected to multiple stochastic dynamic loads. The results presented herein demonstrate the efficacy of the proposed approach for efficient multi-objective topology optimization of stochastically excited structures, as well as multiple input-multiple output systems.

     

Topology optimization of continuum structures under hybrid additive-subtractive manufacturing constraints

Structural and Multidisciplinary Optimization - Additive manufacturing (AM) makes it possible to fabricate complicated parts that are otherwise difficult to manufacture by subtractive machining....     

Evolutionary topology optimization of continuum structures with an additional displacement constraint

Evolutionary structural optimization (ESO) and its later version Bi-directional ESO (BESO) have been successfully applied to optimum material distribution problems for continuum structures. However, the existing ESO/BESO methods are limited to the topology optimization of an objective function such as mean compliance with a single constraint e.g. structural volume. The present work extends the BESO method to the stiffness optimization with a material volume constraint and a local displacement constraint. As a result, one will obtain a structure with the highest stiffness for a given volume while the displacement of a certain node does not exceed a prescribed limit. Several examples are presented to demonstrate the effectiveness of the proposed method.     

Topology and shape optimization of continuum structures using GA and BEM

In a previous study, the authors presented a shape optimization scheme for continuum structures by a genetic algorithm and a boundary element method. In this paper, the study is extended to topology and shape optimization problems of the continuum structures.Boundary profiles are expressed by spline functions. The chromosomes for the profiles are defined by a gene related to the topology (the number of internal boundaries) and genes related to the control points of the spline functions. The population is constructed by individuals with such chromosomes. The genetic opertors such as selection, crossover and mutation are applied to the population for searching the profile satisfying the design objectives. In the case of the objects with internal boundaries, intersection of the boundaries very often occurs and thus, the computational cost may become high. Therefore, we also discuss a scheme for increasing the computational efficiency in this case. Finally, the present scheme is applied to the topology and shape optimization of a plate in order to confirm its validity.     

Topology optimization of composite structures with data-driven resin filling time manufacturing constraint

This paper presents a framework for modeling the data-driven manufacturing constraints and integrating them into the structural topology optimization. The empirical surrogate model of manufacturing constraints is constructed by mining the results of numerical process simulations of massively sampled topologies, using statistical learning. During optimization, the manufacturing constraints are modeled as additional objectives to the structural performance objective, and the multi-objective topology optimization is solved by the Kriging-interpolated level-set (KLS) approach and the multi-objective genetic algorithm (MOGA). The resulting Pareto frontiers offer opportunities to select the designs with some sacrifice in structural performance, yet improved manufacturability. An example of topology optimization of composite structures considering resin filling time showed the process of the proposed framework, and demonstrated its feasibility. Due to the proposed abstract topology features inspired by underlying physics of the filling process, the surrogate model of resin filling time is far more generalizable than the traditional surrogate models based on, e.g., bitmap and local feature representation. In particular, the model can reasonably be applied to the situations with the different inlet gate locations and initial bounding boxes from the training set, while the traditional surrogate models fail in such situations. Three case studies for composite structure topology optimization are discussed with different inlet gate locations and initial bounding boxes in order to show the robustness of the data-driven resin filling time predictive model.