Nora Lüthen, Mika Juntunen, Rolf Stenberg
In a previous paper [6] we have extended Nitsche's method [8] for the Poisson equation with general Robin boundary conditions. The analysis required that the solution is in H^s, with s > 3/2. Here we give an improved error analysis using a technique proposed by Gudi [5].
Nora Lüthen, Martin Rumpf, Sascha Tölkes et al.
Fine scale elastic structures are widespread in nature, for instances in plants or bones, whenever stiffness and low weight are required. These patterns frequently refine towards a Dirichlet boundary to ensure an effective load transfer. The paper discusses the optimization of such supporting structures in a specific class of domain patterns in 2D, which composes of periodic and branching period transitions on subdomain facets. These investigations can be considered as a case study to display examples of optimal branching domain patterns. In explicit, a rectangular domain is decomposed into rectangular subdomains, which share facets with neighbouring subdomains or with facets which split on one side into equally sized facets of two different subdomains. On each subdomain one considers an elastic material phase with stiff elasticity coefficients and an approximate void phase with orders of magnitude softer material. For given load on the outer domain boundary, which is distributed on a prescribed fine scale pattern representing the contact area of the shape, the interior elastic phase is optimized with respect to the compliance cost. The elastic stress is supposed to be continuous on the domain and a stress based finite volume discretization is used for the optimization. If in one direction equally sized subdomains with equal adjacent subdomain topology line up, these subdomains are consider as equal copies including the enforced boundary conditions for the stress and form a locally periodic substructure. An alternating descent algorithm is employed for a discrete characteristic function describing the stiff elastic subset on the subdomains and the solution of the elastic state equation. Numerical experiments are shown for compression and shear load on the boundary of a quadratic domain.
Nora Lüthen, Stefano Marelli, Bruno Sudret
Sparse polynomial chaos expansions (PCE) are an efficient and widely used surrogate modeling method in uncertainty quantification for engineering problems with computationally expensive models. To make use of the available information in the most efficient way, several approaches for so-called basis-adaptive sparse PCE have been proposed to determine the set of polynomial regressors ("basis") for PCE adaptively. The goal of this paper is to help practitioners identify the most suitable methods for constructing a surrogate PCE for their model. We describe three state-of-the-art basis-adaptive approaches from the recent sparse PCE literature and conduct an extensive benchmark in terms of global approximation accuracy on a large set of computational models. Investigating the synergies between sparse regression solvers and basis adaptivity schemes, we find that the choice of the proper solver and basis-adaptive scheme is very important, as it can result in more than one order of magnitude difference in performance. No single method significantly outperforms the others, but dividing the analysis into classes (regarding input dimension and experimental design size), we are able to identify specific sparse solver and basis adaptivity combinations for each class that show comparatively good performance. To further improve on these findings, we introduce a novel solver and basis adaptivity selection scheme guided by cross-validation error. We demonstrate that this automatic selection procedure provides close-to-optimal results in terms of accuracy, and significantly more robust solutions, while being more general than the case-by-case recommendations obtained by the benchmark.