Keywords: membrane, finite element simulation, wrinkling, shape optimization, p-mean criterion, sequential quadratic programming.

This paper reviews several possible formulations of an optimisation problem where the nodal out-of-plane displacements arising from wrinkling are minimised using sequential quadratic programming. A comparison of several problem formulations was performed on a benchmark problem of finding the optimum shape and thickness of the corner reinforcement patches at which the tension forces are applied to a square membrane modelled using shell elements. These problem formulations include the root mean square of the nodal out-of-plane displacement W, a generalized mean power of the exponent p of W for a range of values of parameter p, the maximum of nodal values of W, and the maximum of |W|. Another possible formulation of minmax problem where the local maxima of W in different zones are minimized using Olhoff's bound formulation was also performed. The design variables were parameters defining the geometry of the patches as well as their thickness, and the volume of the patch material was constrained. For the obtained designs a sensitivity study was performed to check whether a design is robust, i.e. whether it is affected by small variations of the design variables.

Even though the minmax formulation produced the best solution in terms of maximum of |W|, the performance of the solution deteriorated quite dramatically when the design variables varied by a small amount indicating the non-robustness of the obtained solution and hence is not recommended. This and other formulations that include operations of maximum and, or absolute value of the out-of-plane displacement should be used with care as convergence of a gradient-based technique can suffer in such cases.

The p-mean formulation where the parameter p was taken as 6 produced the maximum of |W| of 29% lower than that of the baseline design. This formulation was found to be superior to other formulations, it provides an optimization problem with a smooth objective function, a steady convergence history when solved by sequential quadratic programming and a robust properties of the obtained design with respect to the small perturbations of the design variables.

In addition, the morphing technique using the HyperMorph module in Altair HyperMesh 11.0 was found capable of introducing rapid changes in the mesh shapes while preserving the mesh quality.

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