GRADIENT-BASED TWO-SCALE TOPOLOGY OPTIMIZATION WITH B-SPLINES ON SPARSE GRIDS

Structural optimization searches for the optimal shape and topology of components such that specific physical quantities are optimized, for instance, the stability of the resulting structure. Problems involving multiple scales, i.e., structures on a microscopic and a macroscopic level, can be efficiently solved by homogenization-based two-scale approaches. In each optimization iteration, many computationally expensive tensors E describing the macroscopic behavior of a given microstructure have to be calculated, implying that the solution of one optimization problem can take weeks. The computational complexity can be greatly reduced with surrogates E that are constructed in advance in an offline phase via interpolation and that can be reused for different scenarios. Three main issues arise in this context: First, the curse of dimensionality renders conventional interpolation schemes infeasible even for moderate dimensionalities > 4. Therefore, we use sparse grid interpolation combined with a novel problem-tailored boundary treatment to drastically reduce the necessary grid size with only slightly higher approximation errors. Second, common sparse grid bases are not continuously differentiable. Hierarchical B-splines achieve lower approximation errors and supply exact continuous gradients of (E) over tilde, which enables gradient-based optimization without approximating gradients of E. Third, the interpolated tensors are usually required to be positive definite, which is not fulfilled by common interpolation methods. We are able to preserve positive definiteness of the interpolated tensors by interpolating Cholesky factors instead. Combining these three contributions allows computing optimized structures for two- and three-dimensional optimization scenarios with speedups of up to 86 when compared to non-surrogate-based solutions.