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

实际计算或模拟随机非线性固体力学模型是非常昂贵的。这个项目将有助于减少装配和求解控制方程的计算成本。对于空间描述，使用来自CAD的NURBS基等几何分析，它在每个未知（自由度）会计上提供了高阶收敛和精度。同样，随机描述将自适应地选择基和多元素分割，以获得较高的每未知精度。此外，控制方程中的项必须通过数值积分计算，在评估点采样。这是总成本中相当大的一部分，并且“搭配”——前一个项目阶段的重点——使用尽可能少的评估点，但是可能不稳定。在当前阶段，我们想要超越搭配，以尽可能少的评估点实现稳定性和快速收敛（即效率）。为此目的，将使用变分框架来理解和分析作为数值扰动变分项的各自计算，例如：根据混合变分公式。变分框架允许直接估计计算的稳定性和准确性。在计算不可逆材料模型（如塑性）时，这一点尤为重要，因为塑性模型在其描述中具有内部现象学变量。进一步减少评估点的数量是通过使用“贝叶斯积分”来实现的，它使用了“概率数值”的思想。

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".