An efficient approach for shape optimization of components

Abstract

The optimization problem of finding the optimal shape of a mechanical component is investigated with the aim of presenting a simple and efficient numerical approach for minimizing stress concentration factor. The proposed approach is based on finite element method in conjunction with the widely used fully stressed design criterion (or the axiom of uniform stress), i.e. for structural shape optimization, an essential requirement for optimality is the achievement of constant tangential stresses along a section of the boundary to be optimized. The design boundary is modeled by using cubic splines, which are determined by a number of control points. The optimal shape of the design boundary with constant stress is achieved iteratively by adjusting the design boundary shape based on a simple logic and algorithm. The result quality in terms of accuracy and efficiency are tested and discussed with finite element analysis examples. The approach presented has the attractive properties that it can be very simply implanted into standard finite element codes.

Introduction

In the past decades, there have been dramatic improvements in the engineering design process due to the widespread use of finite element analysis (FEA) as the computational tool and the increases in computer speed. Structural optimization is an important field of research due to its contribution to cost, material and time saving in engineering design. Among various structural optimization problems, shape optimization has deserved great attention from the numerical analysis community due to its highly non-linear character.

Generally, the shape optimization problem consists in finding out the best profile of a component that improves its mechanical properties and minimizes some properties, for example, to minimize the weight of the body or reduce high stress concentrations near corner. In recent years, researchers have carried out many investigations into shape optimization problems and various structural optimization methods have been proposed [1], which are proving useful for different optimization problems.

As for the optimal shape design of a component, the fully stressed design criterion is widely used as the optimality criterion to minimize the stress concentration factor. The criterion is based on the works of Baud [2] and Neuber [3] as well as extended statements of Schnack [4]. The objective is to achieve constant, or near constant, tangential stresses along a section of the design boundary that is being optimized. Based on the energy theorem, Pedersen [5] has recently provided the theoretical treatment of the optimality criterion, which shows that the strongest shape design will have uniform energy density along the designed shape, as far as the geometrical constraint makes this possible. Cherkaev et al. [6] have also discussed the necessary conditions of optimality using the classical variational scheme. They concluded that an optimal cavity under the shear stresses was necessarily bounded by a non-smooth curve, and the stress along the design boundary is piece-wise constant. Similarly, Mattheck and his co-workers have introduced the axiom of uniform stress based on the interesting examination of the growth behavior of biological structures [7], [8], [9]. They found that biological structures appear to be able to add material in regions of high stresses and to reduce material in regions of low stresses to bring about an optimal shape that produces a constant Von-Mises stress distribution on the free surface.

Various finite element based methods have been presented in the literature, including mainly sensitivity analysis methods [10], [11] and gradientless methods [4], [7], [8], [9], [12], [13], [14], [15], [16], with the aim of achieving constant tangential stresses along a design boundary (in whole or in part). Sensitivity analysis methods are based on sensitivity analysis and require stress gradient information to be determined. While sensitivity analysis can be a powerful general method for structural optimization, one feature is the fact that calculating sensitivity is computationally expensive when the number of design variables is large. On the other hand, analytical methods for sensitivity analysis are computationally more efficient, but considerable coding is required by experts in the field to implement them. Gradientless methods take advantage of the knowledge on the physics and mechanics of the respective problem set, and do not use stress derivatives to determine optimal boundary shape, hence they are typically much simpler to implement with standard finite element codes. They are particularly well proven for shape optimization where a large number of design variables are required since the convergence speed is independent of the number of design variables. One of the strategies for gradientless based methods to achieve a constant stressed design boundary is to add material where stresses are ‘high’ and remove it where stresses are ‘low’ [7], [8], [9], [12], [13], [14], [15]. The other one for minimizing the peak stress is to adjust the curvature of a point on the design boundary [4], [16], i.e. if the stress at a point is high (low), then the corresponding curvature at this point should be reduced (increased). In fact, these two strategies (albeit of differing effectiveness) have the same effect in minimizing the peak stress in the sense of achieving a constant stressed boundary by correctly adjusting boundary point positions. One deficiency in implementing the gradientless methods is that a criterion to determine “high stress” and “low stress” should be specified clearly by the user. The essential difference among various gradientless methods based on the above idea is actually in this criterion. This implies that an arbitrary threshold stress is required while implementing these methods, and apparently, the selection of the threshold stress will have great influence upon the efficiency and even upon the determination of initial boundary shape.

Another deficiency in the gradientless methods mentioned above may be that most authors did not use geometric modeling in the shape optimization problems dealt with by them. Instead, they defined the nodal coordinates of the discrete finite element model as design variables. This approach requires a large number of design variables and a large number of constraints [13], [14], [15], which complicates the design task. Generally, the choice of any parametric curve to represent the design boundary will result in a certain degree of restriction of an optimization problem. The objective of shape optimization of an engineering component is to search for a feasible solution within a prescribed tolerance. Therefore, an adequate selection of a geometric representation method for the design boundary and the minimum number of appropriate design variables is of fundamental importance in order to achieve an automatic design cycle during the shape optimization process only if the result quality satisfies the prescribed requirement.

In this study, an efficient gradientless approach for shape optimization of components is developed to overcome the drawbacks in the gradientless methods mentioned above. In current method, the shape of the design boundaries is modeled by using cubic splines. The optimal shape of the design boundary with constant stress is achieved iteratively by adjusting the design boundary shape based on a simple logic and algorithm. To demonstrate this approach, two numerical examples are presented and discussed.