Optimization With Finite Element Analysis

Abstract

Particular methods are called for when structural optimization is coupled with finite element analysis. A sensitivity analysis can be employed to evaluate the constraint gradient data needed in a conventional, gradient-based optimization. This is to avoid the repeated finite element analysis required if constraint gradients were to be obtained by finite difference. By setting up a matrix of derivatives of constraints with respect to the displacements of the finite element model, the inverted stiffness matrix obtained in the normal course of finite element analysis is reused in a sensitivity analysis. In one method, the individual columns of this matrix of derivatives are treated as a set of ‘dummy loads’ with which the constraint gradients can be calculated. The other method is the so-called direct method. The computation can be further reduced by an active constraint strategy, in which constraint gradients are evaluated only for those constraints that are active or near-active at any stage. The number of variables involved in the optimization may be reduced by design variable linking. This is by defining some variables as slave variables that are then related to the remaining master variables for optimization, while for accuracy all variables are retained in the finite element analysis.

Exercises

• 9.1 Derive the stiffness matrix for the truss in Example 9.1.

  The stiffness matrix for a bar with pinned ends is given in any textbook on finite element methods. Assemble first the complete stiffness matrix, then delete rows and columns corresponding to displacements at the two supports.

• 9.2 Calculate the sensitivity of the constraint in Example 9.1 to the cross-sectional area  $A{\kern 1pt}_{2}$ of the upper bar. Verify the result by the analytical formula in Example 9.1.

  Follow the method in Example 9.1.

• 9.3 Consider why, in a large problem, it would be more economical to use the dummy load method when the number of constraints is less than the number of design variables.

  Examine the number of individual calculations in successive matrix multiplications when the number of constraints is less than the number of design variables, and when the number of constraints is greater than the number of design variables.

• 9.4 Repeat the optimization of the beam in the spreadsheet ‘Design Variable Linking’, selecting a different set of master and slave variables.

  Modify the linking matrix in the spreadsheet. If the number of master variables remains the same, use the same cell ranges in the spreadsheet for the linking matrix and master variables. If a different number of master variables are chosen see Table 9.1 . The total number of variables should remain 24 if further changes to the original spreadsheet are to be avoided.