Isogeometric topology optimization using trimmed spline surfaces

Abstract

In the present work, a new spline based topology optimization using trimmed spline surfaces and the isogeometric analysis is proposed. In the proposed approach, the trimmed surface analysis which can treat topologically complex spline surfaces using trimming information provided by CAD systems is employed for structural response analysis and sensitivity calculation in the topology optimization. The outer and inner boundaries of design models are represented by a spline surface and trimming curves. Design variables used in this approach are the coordinates of control points of a spline surface and those of trimming curves. New sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified. The creation of new inner fronts during optimization is allowed for the topological flexibility. An inner front merging algorithm is also presented. The proposed spline based topology optimization is used to solve some benchmarking problems. Design space dependency which is one of serious shortcomings in conventional topology optimization approaches is naturally eliminated by the proposed spline based optimization. Design dependent load problems which are difficult to treat with conventional grid based topology optimization methods are easily dealt with by the proposed one. It is also shown that post-processing effort for converting to CAD model is eliminated by using the same spline information in numerical analysis and design optimization.

Introduction

Since the pioneering work of Bendsøe and Kichucki [1], topology optimization has received great deal of attentions by numerous researchers. The earliest method for topology optimization is the homogenization method [1], [2] which represents material layouts with the microstructures defined at each element. It was simplified to the density based topology optimization using the relationship between the elastic modulus of a material and element densities to be optimized [3], [4], [5]. The density based topology optimization is generally called SIMP (Solid Isotropic Material with Penalization) method. SIMP method is easy to implement and very efficient due to its good harmony with finite element method (FEM) and thus has been widely used in a broad range of engineering design problems. Also, a number of commercial tools for topology optimization based on SIMP method have been developed. The well-known numerical instabilities in SIMP method such as checkerboard patterns, mesh dependency and minimum member size control, etc. have been successfully overcome by several numerical techniques: filtering of sensitivity [6], [7], [8], [9] or density [10], the perimeter control [11], the slope constraint [12], [13], [14] and density redistribution [15].

Various new attempts for topology optimization have been proposed during the past two decades. Xie and Steven [16], [17], [18] proposed the evolutionary structural optimization (ESO) based on element-wise stress level without sensitivity calculation. Yoon and Kim [19] proposed a new topology optimization methodology named element connectivity parameterization (ECP). In the ECP, the elastic links are assumed to exist at vertices of elements and employed as design variables to determine existence of elements connected by the links. The material cloud (MC) method was proposed by Chang and Youn [20] for design space expansion. In the MC method, topology is represented by means of the material clouds and the size and/or position of them are design variables.

The above approaches represent an optimal topology with cells or elements. However, the cell-based representation of topology may lead irregular and vague boundary layouts. Thus new attempts for smooth and definite material boundaries have been studied. In recent years, level set based topology optimization which was first proposed by Sethian and Wiegmann [21] has been extensively investigated. In level set based topology optimization, material boundaries are represented by a level set function whose evolution is governed by Hamilton–Jacobi equation and the shape velocity computed from design sensitivity analysis. Generally, level set based topology optimization has a serious shortcoming – new inner fronts cannot be created naturally during the optimization process. Therefore, an initial level set may include a number of user defined inner fronts. The problem with this approach is that the optimal topology is strongly dependent on the initial number and positions of inner fronts. The reasonable strategies based on the topological sensitivity [22], [23], [24], [25] and strain energy density [26] have been proposed to introduce new inner fronts during optimization. As a similar approach, topology optimization using a nodal implicit level set function and extended finite element method (XFEM) was also proposed [27].

Besides the level set based approaches, several works on spline based topology optimization have been also proposed for smooth boundary representation. The earliest spline based topology optimization is the bubble method [28]. In the method, topological changes of design model and shape optimization of boundaries represented by splines are sequentially repeated for an optimal design. Topological changes are achieved by inserting bubbles based on the characteristic function defined by local stresses. Unstructured computational mesh for numerical analysis is constructed by remeshing scheme at every iteration. Cervera and Trevelyan [29] proposed a spline based topology optimization using boundary element method (BEM) with the evolutionary concept. In their work, boundaries are represented by NURBS and the coordinates of the control points of boundary NURBS curves are design variables. Then, the direction and amount of movements of the control points are determined based on stress levels of the boundary elements. New inner fronts are created at the lowest stress region by taking the internal evaluation points surrounding the region as control points of the inner fronts. Another spline based approach was made by Lee et al. [30] using a fixed FE grid. They proposed the selection criteria, which are defined in the ratio of sensitivity of an objective function to that of a constraint, for theoretically reasonable changes in structural topology.

Most of topology optimization methods developed so far employ the conventional FEM for structural response analysis and sensitivity calculation. However, they have two serious drawbacks due to the fixed computational FE grid used for material representation and numerical analysis. The first one is that a design space is highly dependent on the initial fixed FE grid. Since topologies are represented using cells or elements of the fixed grid in the conventional approaches, a design space is restricted on the fixed grid and design results are dependent on the design space. Different design results corresponding to different initial fixed grids are shown in Fig. 1. Unless the initial fixed grid includes wide range of design space, one cannot obtain near-optimal solution due to the restriction of design space. On the other hand, a superfluously wide fixed grid requires high computational costs. Level set based approaches also have the drawback since they employ fixed FE grid to define level set values at nodes and to analyze design models. The second drawback is high post-processing effort in converting the optimized result to the CAD model. Such high post-processing effort is required since analysis and design models in the conventional topology optimization approaches do not use spline data which is commonly used in CAD systems.

The first drawback can be overcome by using an alternative analysis method without fixed grid. And the second one can be eliminated by unifying analysis and design models which are defined with CAD data. In these regards, the isogeometric analysis [31] is a promising alternative to FEM in topology optimization and may provide powerful advantages in design optimization field.

There have been many attempts to use splines in numerical analysis for high accuracy and integration of CAD/CAE [32], [33], [34], [35], [36], [37], [38]. The framework of the isogeometric analysis using NURBS was recently proposed [31], [39], [40], [41] and the extended works using T-splines were also presented [42], [43], [44]. The trimmed surface analysis [45] which uses trimming information for analysis was proposed to overcome the difficulty in the treatment of topologically complex spline surfaces in the conventional isogeometric analysis. Works on shape optimization using the isogeometric analysis were recently presented [46], [47], [48], [49] but isogeometric topology optimization has been never tried. The authors also have studied design optimization based on the isogeometric analysis and first proposed the new framework for topology optimization with the isogeometric concept in our previous work [50].

In the present work, a spline based topology optimization using the isogeometric analysis and the trimmed spline surfaces is proposed to overcome the drawbacks in the conventional FEM based topology optimization approaches. A trimmed spline surface and trimming curves are employed to represent a design model. The trimmed surface analysis is used to analyze topologically complex spline surfaces which may be encountered during optimization process. Since the same spline data is used in modeling, analysis and design optimization in the proposed approach, integration of CAD/CAE can be easily achieved with the present approach. Moreover, design dependent load problems in which the applied loads depend on the design of the structure itself can be readily treated without any additional numerical effort and a complicated mixed formulation.

This paper is organized as follows. In Section 2, the trimmed surface analysis is briefly reviewed. The methods and strategies used in the present work are explained in Section 3. In Section 4, new sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified. Numerical examples are shown in Section 5. Conclusions and comments on further studies are addressed in Section 6.