The modelling of the boundary conditions is crucial in topology optimisation. Small changes in these conditions will typically imply new optimal layouts of material. In many situations the design domain is connected to an assembly of components using contact interfaces. In order to generate proper layouts for this type of design domain one must treat these contact interfaces accurately in the topology optimisation procedure.
The bottle-neck in topology optimisation of non-linear structural problems, such as contact problems, is to solve the state equations and the adjoint equations. By choosing the potential energy as the objective the latter equations are not needed in the sensitivity analysis. In this paper topology optimization of contact problems including non-zero prescribed displacements by maximizing the potential energy is performed. For contact problems with zero initial contact gaps and zero prescribed displacements this is equivalent to minimising the compliance, which is the standard approach in topology optimisation. However, when the compliance is used as objective in topology optimisation of contact problems an extra adjoint equation must be solved. This is not needed in the formulation presented in this paper.
The efficiency of the approach is demonstrated by studying three different problems, two problems in two-dimensions and one in a three-dimensional setting. The method is implemented by using Matlab and Intel Fortran, where the Fortran code is linked to Matlab as mex-files. The implementation can be downloaded as a toolbox (Topo4abq). The three problems have beensolved using this toolbox on a laptop with an Intel Core i7 2.67 GHz processor and a 64 bit version of Windows. It is shown that the CPU-time is reduced as much as 25% compared to an adjoint approach developed in previous works.
Deterministic successive response surface optimization is a most efficient tool for improving machine components. A possible drawback might be that the optimal design proposal is non-robust. For instance, a constraint on the von Mises stress mightn be violated for small changes in the optimal values of the design parameters. This might be checked after the optimization by performing robustness analysis. Another approach would be to replace the deterministic constraint on the von Mises stress with a probabilistic reliability constraint. This is the scope of the following paper. A sequential linear programming approach for reliability based design optimization is developed and implemented. The design variables are assumed to be normally distributed, where the standard deviations of the design variables can be given as coefficients of variation. A deterministic LP-problem is derived by performing a Taylor expansion of the constraint at the most probable point (MPP). The MPP is found by solving the optimality conditions using Newton’s method and the LP-problem is solved by an interior point method. The approach is efficient and robust. This is demonstrated by performing reliability based shape optimization of a real knuckle component to a heavy truck.