Structure-preserving time integrators for thermodynamics of nonlinear continua.

随着计算机性能的稳步提高，如今可以在充分考虑耦合物理场相互作用的情况下对热力学过程进行计算分析。因此，这些系统可以更快、更便宜地进行优化。因此，在热力系统的数值模拟中，应用时间积分器是很重要的，它在数值上准确地再现了相互作用，而且使模拟更加稳健。如果时间离散化算法复制物理结构，这是有保证的，这意味着结构特征如平衡定律和本构属性在独立于时间离散化的离散设置中被准确地填充。因此，与标准时间积分器相比，保结构时间积分器能够以较粗的时间离散化来模拟过程。提交的研究项目的目的是使所设计的保结构时间积分器更加人性化。新算法在保持结构的情况下，对每个耦合场的时间离散化进行单独的自适应，同时对空间有限元网格进行结构保持的自适应。时间离散化的自适应是通过确定时间形状函数的插值点在给定时间步长内的分布来获得的，但这种p-自适应不会像通常那样受数值误差的控制，而是受结构保持的控制。有限元在空间上的自适应是基于节点位置的计算和连续介质的物质力平衡方程(r-自适应)。

The steady increase of the performance of computers allows nowadays a computational analysis of thermodynamic processes with a full consideration of mutual interactions of coupled physical fields. Hence, these systems can be optimized much faster and cheaper. In numerical simulations of thermodynamic systems, it is therefore important to apply time integrators, which numerically exactly reproduce mutual interactions and, moreover, make the simulation more robust. This is guaranteed if a time discretisation algorithmically reproduce the physical structure, which means structural characteristics as balance laws and constitutive properties are fullfilled exactly in a discrete setting independent of the time discretisation. In comparison to standard time integrators, structure preserving time integrators therefore are able to simulate processes also with a coarse time discretisation.The aim of the submitted research project is to make the designed structure preserving time integrators more user-friendly. The new algorithms should perform an individual adaption of the time discretisation in each coupled field subject to the structure preservation, and at the same time a structure preserving adaption of the spatial finite element mesh. The adaption of the time discretisation is obtained by determining the distribution of interpolation points of temporal shape functions in a given time step.But this p-adaption will not be controlled by numerical errors as usual, but controlled by structure preservation. The finite element adaption in space is based on a calculation of the nodal positions with the balance equation of material forces of the continuum (r-adaption).