Collocation Methods and Beyond in Non-linear Mechanics

Within the realm of isogeometric analysis, isogeometric collocation has been driven by the attempt to minimize the cost of quadrature associated with higher-order discretizations, with the goal of achieving higher-order accuracy at low computational cost. While the first applications of isogeometric collocation have mainly concerned linear problems, here the focus is on non-linear mechanics formulations including hyperelasticity, elastoplasticity, contact and geometrically non-linear structural elements. We also address the treatment of locking issues as well as the establishment of a bridge between Galerkin and collocation schemes leading to a new reduced quadrature technique for isogeometric analysis. In stochastic uncertainty computations, the evaluation of full-scale deterministic models is the main computational burden, which may be avoided with cheap to evaluate proxy-models. Their construction is a kind of regression, which, when reduced to the minimum number of samples, turns into collocation or interpolation. It is possible to go well beyond that minimum using ideas from probabilistic numerics and Bayesian updating, which is shown both for constructing proxy-models and for upscaling (coarsening) of highly nonlinear material laws. Another way to reduce costly full-scale model evaluations is to use multi-level hierarchies of models, leading to multi-level Monte Carlo methods. In this chapter, we present the main achievements obtained on the above topics within the DFG Priority Program 1748, Reliable Simulation Techniques in Solid Mechanics.