Beyond isogeometric and stochastic collocation: maximizing efficiency in stochastic non-linear computational solid mechanics

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Actual computations or simulations of stochastic non-linear solid mechanics models can be very costly. This project will contribute to a reduction of the computational cost of assembling and solving the governing equations. For the spatial description isogeometric analysis with NURBS bases from CAD is used, which offers high order of convergence and accuracy on a per-unknown (degree of freedom) accounting. In a similar vein, the stochastic description will adaptively choose the basis and multi-element segmentation for a high per-unknown accuracy. Furthermore, the terms in the governing equations have to be computed through numerical integration, sampling at evaluation points . This is a considerable part of the total cost, and “collocation” -- the focus of the previous project phase -- uses the minimum possible number of evaluation points, but can be unstable. In the present phase we want to proceed beyond collocation and achieve stability and fast convergence (i.e. efficiency) with as few as possible evaluation points. To this purpose, a variational framework will be used to understand and analyse the respective computations as numerically perturbed variational terms, resp. in the light of mixed variational formulations. The variational framework allows to directly estimate the stability and accuracy of the computations. This becomes especially important when computing irreversible material models such as plasticity, which have internal phenomenological variables in their descriptions. A further reduction of the number of evaluation points is to be achieved through the use of “Bayesian integration”, which uses ideas from "probabilistic numerics".