Optimization of a Shell Structure

Colaboration:

• TU Delft
• burgeon

This example concerns the shape optimization of a shell which is described by a square reference plan. The out-of-plane location of its mid-surface is described using square patches. At the keypoints (corner nodes of the patches) both the out-of-plane coordinate and its derivatives with respect to the in-plane coordinates can be specified. The patches and the keypoints are depicted in Fig. 1. The geometry is assumed to be symmetric with respect to the diagonals and the lines A-A and B-B. The design variables are the out-of-plane coordinates of the keypoints and the corresponding derivatives. The out-of-plane coordinates of the corners are fixed. Furthermore, the thickness of the shell is taken as a design variable. The in-plane dimensions in millimetres are depicted in Fig. 1. The shell is supported at its corner nodes, for which all displacement components are prescribed. The shell is loaded by a uniform out-of-plane load with an intensity of 0.01 N/mm². Homogeneous and isotropic linear elastic material behavior is assumed with a Young's modulus 2.1^.10⁵ N/mm² and a Poisson's ratio 0.3. The optimization problem is formulated as minimization of the maximum displacement, whereas the volume should remain less than 40^.10³ mm³. The thickness can vary from 0.1 mm to 2.0 mm. All lower bounds on the locations of the keypoints are given by 0 mm, whereas the upper bounds are selected as 50 mm, 80 mm, 100 mm or 150 mm, depending on their distance to the corner nodes. The derivatives should be in the range [-1;1]. In total, 12 design variables are used. For all studies the entire search domain was selected as the search sub-domain of the first step of the optimization. The first plan of experiments has always been enhanced by means of 80 additional design evaluations. No design sensitivities have been used.

Numerical studies showed that this optimization problem has several local optima. In the course of the study two almost equally good optima have been found. These design alternatives and corresponding values of the objective and constraint functions are given in Fig. 2.

Figure 1. Rectangular patches and keypoints being used for the description of the geometry of the shell.

a)

b)

Figure 2. Nearly equivalent design alternatives. a) Maximum displacement is 1.6556.10-2mm, normalized constraint equals 0.993. This design has been found using linear approximation functions. During each step a large number (80) of additional designs has been added in the search domain. b) Maximum displacement is 1.6578 10-2 mm, normalized constraint equals 0.998. The design has been obtained using quadratic approximation functions. Only a minimum number of points has been added to the plan of experiments.