A unified parametric design approach to structural shape optimization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN

ENGINEERING, VOL. 38, 2283-2292 (1995)

A UNIFIED PARAMETRIC DESIGN APPROACH TO STRUCTURAL SHAPE OPTIMIZATION WEI-HONG ZHANG, PIERRE BECKERS AND CLAUDE FLEURY

LTAS Aerospace Laboratory, University of Liige Rue Ernest-Solvay 21. 4000 Liige, Belgium

SUMMARY

This paper presents a general parametric design approach for 2-D shape optimization problems. This approach has been achieved by integrating practical design methodologies into numerical procedures. It is characterized by three features: (i) automatic selection of a minimum number of shape design variables based on the CAD geometric model; (ii) integration of sequential convex programming algorithms to solve equality constrained optimization problems; (iii) efficient sensitivity analysis by means of the improved semi-analytical method. It is shown that shape design variables can be either manually or systematically identified with the help of equality constraints describing the relationship between geometric entities. Numerical solutions are performed to demonstrate the applicability of the proposed approach. A discussion of the results is also given: KEY WORDS: design variable; shape optimization; sensitivity analysis

1. INTRODUCTION The development of CAD oriented shape optimization techniques is one of the most important subjects in computational mechanics. Their applications have been recognized particularly in aerospace and automobile industries. The key idea is to improve mechanical behaviours by modifying structural boundaries. As pointed out by Haftka and Grandhi,' a number of issues involved in this process such as geometric modelling, identification of shape design variables are crucial to give rise to satisfactory solutions. Until now, the usual approach as utilized by Luchi,' Braibant and Fleury3 is to represent the whole design boundary in terms of one flexible curve such as cubic spline, Bezier and B-spline curve. Because continuity requirements (e.g. C2 for cubic splines) are ensured in the curve's formulation, shape design variables associated with the co-ordinates of control points can be independently modified. However, it should be noted that different curve formulations have a great influence upon the quality of design results. In some cases, the optimum shape is not practicable from the engineering point of view. For example, considering the fillet design problem in Figure 1 described by Seong and Choi," it is demonstrated by the first author' that Bkzier curve exhibits a reliability superior to the Lagrangian curve. The latter provides an oscillating solution although stress constraints are satisfied. In this work, efforts are devoted to shape optimization of 2-D structures which are represented in terms of line segments and circular arcs. As shown in Figure 2, this type of representations are very popular in industrial designs. The advantage of doing so is that local modifications can be easily realized during the design process. However, due to equality constraints which must be imposed to maintain the geometric smoothness and the regularity (e.g. tangent conditions) along 0029-5981/95/132283-10 0 1995 by John Wiley & Sons, Ltd.

Received 21 February 1994 Revised 12 September 1994

W.-H. ZHANG, P.BECKERS AND C. FLEURY

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F

A

F

1 Fillet design example

Geometric model B - Bezier 1: 2 variables pa point 2 1 variable per point ( n2 direction) 3: 1 variable per point ( % direction)

Figure 1. Effect of the curve formulation upon the optimum solution

:

L5

,

-;

.

.Figure 2. Parametric design annotation of mechanical drawings

the whole boundary, shape design variables are no longer independent and implicitly coupled. In the meantime, the non-linearity of equality constraints makes it impossible to apply directly sequential convex programming algorithms. In this paper, parametric design methodologies popularly used in mechanical drawings are firstly reviewed to interpret the physical meanings of shape design variables. It is shown that the basic feature lies in their independence. Based on this property, an approach is developed to identify automatically independent shape design variables for numerical shape optimization.This approach is essential to reduce every convex optimization subproblem into a compact form without equality constraints. Consequently, the simplified formulation is generally easy to solve. Secondly, an improved semi-analytical sensitivity analysis method is established. It is shown that its efficient implementation depends upon the velocity field that characterizes the F.E. mesh deformation. The relationship between velocity field computations and automatic mesh

STRUCTURAL SHAPE OPTIMIZATION

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generators is also addressed. Finally, numerical applications are provided to illustrate the proposed approach.

2. SELECTION OF INDEPENDENT SHAPE DESIGN VARIABLES 2.1. Review of parametric design methodology As described by Light and Gossard6 and Aldefeld,' the parametric design methodology, also called variational geometry theory, was originally developed to generate variants of the design. This process is realized by modifying a set of independent dimension parameters called high-level variables. For example, nine independent dimension parameters ( L , , L 1 , L 3 ,L4, L5,R 1 ,R2,R 3 , R4),invariant with respect to the co-ordinate system, are sufficient to describe the exterior boundary of the structure in Figure 2. Equality constraints related to the C' continuity of the boundary will be satisfied at each variant design. Therefore, this approach provides an efficient way to combine, modem dimension-driven CAD systems with numerical optimization methods. Independent dimension parameters correspond to exactly our desired shape design variables. However, the construction of a generalized dimensioning scheme is not simply a geometric problem. As pointed out by Hillyard and Braid,* it depends much on how the structure will be fabricated, measured, toleranced and assembled. This means that it is impossible to identify automatically independent high-level design variables in general cases. In fact, it is also possible to define line segments and circular arcs in terms of co-ordinates of characteristic points which are called low-level variables. For example, a 2-D line segment needs four independent variables: the co-ordinates of the starting point and the ending point. A circular arc needs five independent variables; the opening angle and the co-ordinates of the two extremity points. This description has the advantage of automatically ensuring a Co continuity when they are jointly used, although other definitions can be also adopted. Low-level variables can be firstly used to check the validity of high-level design variables when they are manually chosen. By writing geometric equality constraints managing the relationship between low-level and highlevel variables hk(X, D) = 0, k = 1, 1

(1)

where X = (XI, X2,. . . , X,)denotes the vector of low-level variables of related characteristic points; D = ( D , , D 2 , . . . , D,) the vector of high-level design variables (i.e. dimension parameters). It can be understood that the valid design vector D should be such that low-level variables can be uniquely determined by them. This implies that the number of equality constraints should be equal to the number of low-level variables. Under the condition r = I, the differentiation of (1) corresponds to the linear system: T6X

=

- B6D

(2)

the Jacobian matrices T and B are defined, respectively, by

Hence, the dimensioning scheme using the design vector D is considered to be valid for the determination of the geometric variation 6X only if the Jacobian matrix Tis non-singular. In this case, the system (2) can be expressed as

6x= - T - ~ B G D

(3)

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2.2 Automatic selection of independent shape design variables In fact, when the boundary shape is under-constrained, there is no unique rule to define design variables. The automatic approach proposed here was based on the original work of the first two authors.' The idea is to select independent design variables among only low-level variables. From this point of view, equality constraints will be expressed only in terms of low-level variables as

h k ( X )= 0, k = 1, 1 - n

(4)

and the corresponding linear system is

A 6 X = 0 with Ag-,),r = {aki}=

{2} -

(5)

By splitting the vector X = [Y, Z] with Y and Z two subvectors of (1 - n) and n dimensions respectively, the system ( 5 ) becomes

A16Y

+ A262 = 0

(6)

to which the solution exists only if det (A,) # 0:

6Y =

- A;'A26Z

= - A"6Z

(2= AT'A2)

(7)

As a result, Z is identified as the vector of independent design variables and Y the vector of dependent variables because the latter is able to be expressed as a function of the former. Numerically, to get a non-singular and well-conditioned square matrix A 1, the Gauss-Jordan elimination procedure based on pivoting is proved to be very effective.' In reality, both high-level independent variables D and low-level independent variables Z can be used in shape optimization; even a combination of them is also acceptable. The major difference is that the former is more suitable to industrial design and manufacturing procedure while the latter is easily adapted to automatic numerical design cycle. Mathematically, they can be transformed from one to the other. Their relationship can be obtained by decomposing the system (2) into

T i d y + T26Z + B6D = 0

(8)

3. MATHEMATICAL FORMULATION OF THE PROBLEM Shape optimization problems can be stated as follows: Min f(X)

(94

c j ( X )< Ej, j = 1, m

(9b)

hk(X) = 0, k = 1, 1

(94