Topology optimization of structures under variable loading using a damage superposition approach

We present an original algorithm and accompanying mathematical formulation for topology optimization of structures that can sustain material damage and are subject to multiple load cases with varying configurations. Damage accumulation is simulated using a coupled, non-linear brittle damage model. The structures are optimized for minimum mass subject to stiffness constraints defined as the compliance evaluated at the end of each loading sequence. To achieve robustness of the optimized structures, the respective damage fields caused by each individual load case are computed and combined using superposition to simulate a worst-case damage field. All load cases are then run a second time using the worst-case damage distribution as a starting point. In this way, one effectively accounts for the spectrum of possible load sequences to which the structure may be subjected. Results from this method are compared with an exhaustive, brute-force approach in which all non-repeating load sequences are analyzed individually. For each method, the corresponding sensitivities are derived and implemented analytically using a path-dependent adjoint method. The two approaches are implemented on a series of numerical examples, which demonstrate that the superposition method produces structures that are as robust as those obtained using the exhaustive method but require significantly less computational effort. Copyright © 2014 John Wiley & Sons, Ltd.