Surrogate modeling in design optimization of structures with discontinuous responses: A new approach for ill-posed problems in crashworthiness design

Advances in computational technology have resulted in the dramatic reduction of computational time for crashworthiness analysis, hence enabling its structural design optimization. Surrogate modeling has been shown to further reduce computational effort as well as to smooth noisy responses. Crashworthiness optimization problems are, though, ill posed as they include nonlinear, noncontinuous and noisy responses. This violates the Hadamard conditions for well-posed problems and therefore the applicability of gradient-based algorithms is limited. Here, discontinuities in the responses with respect to the design variables will be handled that result in large changes in the system functions with only small changes in the design variables using a novel surrogate modeling technique. The applicability of typical global surrogate models is limited when critical discontinuities are present. An efficient method has been developed here to identify the number of discontinuities and their position in the design domain. Previous works assume a said number of discontinuities; here though, the number of discontinuities is not given a priori. The discontinuities are identified by examining the relative difference in the response value of samples in immediate proximity of each other. Samples in the same continuous subdomain are clustered and a support vector machine for classification is exploited to locate discontinuities. Local approximations are then used for the continuous subspaces between the discontinuities. Lastly, a surrogate-based design optimization is carried out. Starting with a two-bar truss, demonstrating a snap-through discontinuity, this method is shown to account for such discontinuities. This is then integrated into an optimization framework. Further academic example, namely a six-bar truss is modeled using the open-source framework Kratos Multiphysics and then optimized, showing the applicability of the method to problems with multiple discontinuities. Finally, a crash-absorbing tube is optimized that is impacted with an angle resulting in a noncontinuous design space: desired axial crushing and undesirable global buckling. After summarizing the results, advantages and possible limitations are discussed.