A study on X-FEM in continuum structural optimization using a level set model

In this paper, we implement the extended finite element method (X-FEM) combined with the level set method to solve structural shape and topology optimization problems. Numerical comparisons with the conventional finite element method in a fixed grid show that the X-FEM leads to more accurate results without increasing the mesh density and the degrees of freedom. Furthermore, the mesh in X-FEM is independent of the physical boundary of the design, so there is no need for remeshing during the optimization process. Numerical examples of mean compliance minimization in 2D are studied in regard to efficiency, convergence and accuracy. The results suggest that combining the X-FEM for structural analysis with the level set based boundary representation is a promising approach for continuum structural optimization.     

Evolutionary topology optimization of continuum structures with smooth boundary representation

This paper develops an extended bi-directional evolutionary structural optimization (BESO) method for topology optimization of continuum structures with smoothed boundary representation. In contrast to conventional zigzag BESO designs and removal/addition of elements, the newly proposed evolutionary topology optimization (ETO) method, determines implicitly the smooth structural topology by a level-set function (LSF) constructed by nodal sensitivity numbers. The projection relationship between the design model and the finite element analysis (FEA) model is established. The analysis of the design model is replaced by the FEA model with various elemental volume fractions, which are determined by the auxiliary LSF. The introduction of sensitivity LSF results in intermediate volume elements along the solid-void interface of the FEA model, thus contributing to the better convergence of the optimized topology for the design model. The effectiveness and robustness of the proposed method are verified by a series of 2D and 3D topology optimization design problems including compliance minimization and natural frequency maximization. It has been shown that the developed ETO method is capable of generating a clear and smooth boundary representation; meanwhile the resultant designs are less dependent on the initial guess design and the finite element mesh resolution.     

A level-set-based topology and shape optimization method for continuum structure under geometric constraints

Recent advances in level-set-based shape and topology optimization rely on free-form implicit representations to support boundary deformations and topological changes. In practice, a continuum structure is usually designed to meet parametric shape optimization, which is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. In order to solve the disadvantage of traditional step-type structural optimization, a unified optimization method which can fulfill the structural topology, shape, and sizing optimization at the same time is presented. The unified structural optimization model is described by a parameterized level set function that applies compactly supported radial basis functions (CS-RBFs) with favorable smoothness and accuracy for interpolation. The expansion coefficients of the interpolation function are treated as the design variables, which reflect the structural performance impacts of the topology, shape, and geometric constraints. Accordingly, the original topological shape optimization problem under geometric constraint is fully transformed into a simple parameter optimization problem; in other words, the optimization contains the expansion coefficients of the interpolation function in terms of limited design variables. This parameterization transforms the difficult shape and topology optimization problems with geometric constraints into a relatively straightforward parameterized problem to which many gradient-based optimization techniques can be applied. More specifically, the extended finite element method (XFEM) is adopted to improve the accuracy of boundary resolution. At last, combined with the optimality criteria method, several numerical examples are presented to demonstrate the applicability and potential of the presented method.     

Polygonal multiresolution topology optimization (PolyMTOP) for structural dynamics

We use versatile polygonal elements along with a multiresolution scheme for topology optimization to achieve computationally efficient and high resolution designs for structural dynamics problems. The multiresolution scheme uses a coarse finite element mesh to perform the analysis, a fine design variable mesh for the optimization and a fine density variable mesh to represent the material distribution. The finite element discretization employs a conforming finite element mesh. The design variable and density discretizations employ either matching or non-matching grids to provide a finer discretization for the density and design variables. Examples are shown for the optimization of structural eigenfrequencies and forced vibration problems.     

Numerical method for shape optimization using T-spline based isogeometric method

Numerical methods for shape design sensitivity analysis and optimization have been developed for several decades. However, the finite-element-based shape design sensitivity analysis and optimization have experienced some bottleneck problems such as design parameterization and design remodeling during optimization. In this paper, as a remedy for these problems, an isogeometric-based shape design sensitivity analysis and optimization methods are developed incorporating with T-spline basis. In the shape design sensitivity analysis and optimization procedure using a standard finite element approach, the design boundary should be parameterized for the smooth variation of the boundary using a separate geometric modeler, such as a CAD system. Otherwise, the optimal design usually tends to fall into an undesirable irregular shape. In an isogeometric approach, the NURBS basis function that is used in representing the geometric model in the CAD system is directly used in the response analysis, and the design boundary is expressed by the same NURBS function as used in the analysis. Moreover, the smoothness of the NURBS can allow the large perturbation of the design boundary without a severe mesh distortion. Thus, the isogeometric shape design sensitivity analysis is free from remeshing during the optimization process. In addition, the use of T-spline basis instead of NURBS can reduce the number of degrees of freedom, so that the optimal solution can be obtained more efficiently while yielding the same optimum design shape.     

Simultaneous design of components layout and supporting structures using coupled shape and topology optimization technique

The purpose of this paper was to study the layout design of the components and their supporting structures in a finite packing space. A coupled shape and topology optimization (CSTO) technique is proposed. On one hand, by defining the location and orientation of each component as geometric design variables, shape optimization is carried out to find the optimal layout of these components and a finite-circle method (FCM) is used to avoid the overlap between the components. On the other hand, the material configuration of the supporting structures that interconnect components is optimized simultaneously based on topology optimization method. As the FE mesh discretizing the packing space, i.e., design domain, has to be updated itertively to accommodate the layout variation of involved components, topology design variables, i.e., density variables assigned to density points that are distributed regularly in the entire design domain will be introduced in this paper instead of using traditional pseudo-density variables associated with finite elements as in standard topology optimization procedures. These points will thus dominate the pseudo-densities of the surrounding elements. Besides, in the CSTO, the technique of embedded mesh is used to save the computing time of the remeshing procedure, and design sensitivities are calculated w.r.t both geometric variables and density variables. In this paper, several design problems maximizing structural stiffness are considered subject to the material volume constraint. Reasonable designs of components layout and supporting structures are obtained numerically.     

A new level-set based approach to shape and topology optimization under geometric uncertainty

Geometric uncertainty refers to the deviation of the geometric boundary from its ideal position, which may have a non-trivial impact on design performance. Since geometric uncertainty is embedded in the boundary which is dynamic and changes continuously in the optimization process, topology optimization under geometric uncertainty (TOGU) poses extreme difficulty to the already challenging topology optimization problems. This paper aims to solve this cutting-edge problem by integrating the latest developments in level set methods, design under uncertainty, and a newly developed mathematical framework for solving variational problems and partial differential equations that define mappings between different manifolds. There are several contributions of this work. First, geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity field characterized with a reduced set of random variables using the Karhunen–Loeve expansion. Multivariate Gauss quadrature is employed to propagate the geometric uncertainty, which also facilitates shape sensitivity analysis by transforming a TOGU problem into a weighted summation of deterministic topology optimization problems. Second, a PDE-based approach is employed to overcome the deficiency of conventional level set model which cannot explicitly maintain the point correspondences between the current and the perturbed boundaries. With the explicit point correspondences, shape sensitivity defined on different perturbed designs can be mapped back to the current design. The proposed method is demonstrated with a bench mark structural design. Robust designs achieved with the proposed TOGU method are compared with their deterministic counterparts.     

Topology optimization using the p-version of the finite element method

A multiresolution topology optimization approach is proposed using the p-version finite element method (p-version FEM). Traditional topology optimization, where a density design variable is assigned to each element, is suitable for low-order h-version FEM. However, it cannot take advantage of the higher accuracy of higher-order p-version FEM analysis for generating results with higher resolution. In contrast, the proposed method separates density variables and finite elements so that the resolution of the density field, which defines the structure, can be higher than that of the finite element mesh. Thus, the method can take full advantage of the higher accuracy of p-version FEM.     

PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes

We present an efficient Matlab code for structural topology optimization that includes a general finite element routine based on isoparametric polygonal elements which can be viewed as the extension of linear triangles and bilinear quads. The code also features a modular structure in which the analysis routine and the optimization algorithm are separated from the specific choice of topology optimization formulation. Within this framework, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. We address issues pertaining to the use of unstructured meshes and arbitrary design domains in topology optimization that have received little attention in the literature. Also, as part of our examination of the topology optimization problem, we review the various steps taken in casting the optimal shape problem as a sizing optimization problem. This endeavor allows us to isolate the finite element and geometric analysis parameters and how they are related to the design variables of the discrete optimization problem. The Matlab code is explained in detail and numerical examples are presented to illustrate the capabilities of the code.     

Integration of topology and shape optimization for design of structural components

This paper presents an integrated approach that supports the topology optimization and CAD-based shape optimization. The main contribution of the paper is using the geometric reconstruction technique that is mathematically sound and error bounded for creating solid models of the topologically optimized structures with smooth geometric boundary. This geometric reconstruction method extends the integration to 3-D applications. In addition, commercial Computer-Aided Design (CAD), finite element analysis (FEA), optimization, and application software tools are incorporated to support the integrated optimization process. The integration is carried out by first converting the geometry of the topologically optimized structure into smooth and parametric B-spline curves and surfaces. The B-spline curves and surfaces are then imported into a parametric CAD environment to build solid models of the structure. The control point movements of the B-spline curves or surfaces are defined as design variables for shape optimization, in which CAD-based design velocity field computations, design sensitivity analysis (DSA), and nonlinear programming are performed. Both 2-D plane stress and 3-D solid examples are presented to demonstrate the proposed approach. Received January 27, 2000 Communicated by J. Sobieski     

Constrained mesh optimization on boundary

This paper presents a new mesh optimization approach aiming to improve the mesh quality on the boundary. The existing mesh untangling and smoothing algorithms (Vachal et al. in J Comput Phys 196: 627–644, 2004; Knupp in J Numer Methods Eng 48: 1165–1185, 2002), which have been proved to work well to interior mesh optimization, are enhanced by adding constrains of surface and curve shape functions that approximate the boundary geometry from the finite element mesh. The enhanced constrained optimization guarantees that the boundary nodes to be optimized always move on the approximated boundary. A dual-grid hexahedral meshing method is used to generate sample meshes for testing the proposed mesh optimization approach. As complementary treatments to the mesh optimization, appropriate mesh topology modifications, including buffering element insertion and local mesh refinement, are performed in order to eliminate concave and distorted elements on the boundary. Finally, the optimization results of some examples are given to demonstrate the effectivity of the proposed approach.     

Topology optimization using an explicit interface representation

We introduce the Deformable Simplicial Complex method to topology optimization as a way to represent the interface explicitly yet being able to handle topology changes. Topology changes are handled by a series of mesh operations, which also ensures a well-formed mesh. The same mesh is therefore used for both finite element calculations and shape representation. In addition, the approach unifies shape and topology optimization in a complementary optimization strategy. The shape is optimized on the basis of the gradient-based optimization algorithm MMA whereas holes are introduced using topological derivatives. The presented method is tested on two standard minimum compliance problems which demonstrates that it is both simple to apply, robust and efficient.     

An immersed boundary approach for shape and topology optimization of stationary fluid-structure interaction problems

This paper presents an approach to shape and topology optimization of fluid-structure interaction (FSI) problems at steady state. The overall approach builds on an immersed boundary method that couples a Lagrangian formulation of the structure to an Eulerian fluid model, discretized on a deforming mesh. The geometry of the fluid-structure boundary is manipulated by varying the nodal parameters of a discretized level set field. This approach allows for topological changes of the fluid-structure interface, but free-floating volumes of solid material can emerge in the course of the optimization process. The free-floating volumes are tracked and modeled as fluid in the FSI analysis. To sense the isolated solid volumes, an indicator field described by linear, isotropic diffusion is computed prior to analyzing the FSI response of a design. The fluid is modeled with the incompressible Navier-Stokes equations, and the structure is assumed linear elastic. The FSI model is discretized by an extended finite element method, and the fluid-structure coupling conditions are enforced weakly. The resulting nonlinear system of equations is solved monolithically with Newton’s method. The design sensitivities are computed by the adjoint method and the optimization problem is solved by a gradient-based algorithm. The characteristics of this optimization framework are studied with two-dimensional problems at steady state. Numerical results indicate that the proposed treatment of free-floating volumes introduces a discontinuity in the design evolution, yet the method is still successful in converging to meaningful designs.     

A method for shape and topology optimization of truss-like structure

We present a method for the shape and topology optimization of truss-like structure. First, an initial design of a truss-like structure is constructed by a mesh generator of the finite element method because a truss-like structure can be described by a finite element mesh. Then, the shape and topology of the initial structure is optimized. In order to ensure a truss-like structure can be easily manufactured via intended techniques, we assume the beams have the same size of cross-section, and a method based on the concept of the SIMP method is used for the topology optimization. In addition, in order to prevent intersection of beams and zero-length beams, a geometric constraint based on the signed area of triangle is introduced to the shape optimization. The optimization method is verified by several 2D examples. Influence on compliance of the representative length of beams is investigated.     

A level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration

We present a level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration. Considering that a simple eigenvalue is Fréchet differentiable with respect to the boundary of a structure but a repeated eigenvalue is only Gateaux or directionally differentiable, we take different approaches to derive the boundary variation that maximizes the first eigenvalue. In the case of simple eigenvalue, material derivative is obtained via adjoint method, and variation of boundary shape is specified according to the steepest descent method. In the case of N-fold repeated eigenvalue, variation of boundary shape is obtained as a result of a N-dimensional algebraic eigenvalue problem. Constraint of a structure’s volume is dealt with via the augmented Lagrange multiplier method. Boundary variation is treated as an advection velocity in the Hamilton–Jacobi equation of the level set method for changing the shape and topology of a structure. The finite element analysis of eigenvalues of structure vibration is accomplished by using an Eulerian method that employs a fixed mesh and ersatz material. Application of the method is demonstrated by several numerical examples of optimizing 2D structures.     

A computational paradigm for multiresolution topology optimization (MTOP)

This paper presents a multiresolution topology optimization (MTOP) scheme to obtain high resolution designs with relatively low computational cost. We employ three distinct discretization levels for the topology optimization procedure: the displacement mesh (or finite element mesh) to perform the analysis, the design variable mesh to perform the optimization, and the density mesh (or density element mesh) to represent material distribution and compute the stiffness matrices. We employ a coarser discretization for finite elements and finer discretization for both density elements and design variables. A projection scheme is employed to compute the element densities from design variables and control the length scale of the material density. We demonstrate via various two- and three-dimensional numerical examples that the resolution of the design can be significantly improved without refining the finite element mesh.     

Numerical comparison between two cost functions in contact shape optimization

A finite element approach to shape optimization in a 2D frictionless contact problem for two different cost functions is presented in this work. The goal is to find an appropriate shape for the contact boundary, performing an almost constant contact-stress distribution. The whole formulation, including the mathematical model for the unilateral problem, sensitivity analysis and geometry definition is treated in a continuous form, independently of the discretization in finite elements. Shape optimization is performed by a direct modification of the geometry throughB-spline curves and an automatic mesh generator is used at each new configuration to provide the finite element input data. Augmented-Lagrangian techniques (to solve the contact problem) and an interior-point mathematical-programming algorithm (for shape optimization) are used to obtain numerical results.     

基于无网格径向点插值方法的简谐激励下的连续体结构拓扑优化

针对用有限元法进行连续体结构拓扑优化时需不断重构网格来处理网格畸变和网格移动,且存在数值计算不稳定等问题,基于无网格径向点插值方法(Radial Point Interpolation Method,RPIM)对简谐激励下的连续体结构进行拓扑优化.选取节点的相对密度作为设计变量,以结构动柔度最小化为目标函数,基于带惩罚的各向同性固体微结构(Solid Isotropic Microstructure with Penalization,SIMP)模型建立简谐激励下的优化模型;采用伴随法求解得到目标函数的敏度分析公式;利用优化准则法求解优化模型.经典的二维连续体结构拓扑优化算例证明该方法的可行性和有效性.