A structure-preserving immersed finite element method for the dynamics of multiphase continua with thermomechanical coupling

关于运动连续体的有限元模拟，有许多例子，其中所考虑的连续体由不同的嵌入相组成。这里，相可以是柔性固体、流体以及刚性体。它们包括牛顿流体中的转子和被认为是双相材料的纤维增强材料。在第一个例子中，通过众所周知的有限元方法的流体-结构-相互作用（FSI）的配方导致的缺点，在考虑到计算效率和稳定性，如果嵌入相的大旋转出现。众所周知的原因是对流项的难以近似，表面接触的网格划分不足以及相的频繁重新网格划分。在第二个例子中的固-固相互作用的模拟导致长的计算时间，因为嵌入相的网格决定了周围相的元素的数量。原因也是表面接触的必要的充分近似。浸入式有限元法（IFEM）可以避免这些缺点。本研究的目的是发展和实现一种新的结构保持IFEM的动态，非等温多相连续。这里，流体以及固体都被考虑在内。为了也考虑具有刚性截面的固体，直接基于有限元法，发展了一种新的非等温刚体列式。因此，微极旋转自由度保证了属于刚性部分的有限元网格的刚性。一种特殊的变分方法避免了欧拉张量的引入，并且柔性固体的刚性截面可以通过在总网格中的简单声明来定义。通过这种方式，可以很容易地模拟非等温刚体、柔性固体和流体之间的热力耦合。著名的IFEM假设一个固定的欧拉网格的总连续，并考虑拉格朗日网格只嵌入相。在目前的项目中，这一限制将被取消，以便通过IFEM模拟非等温多相固体的大变形。但是，也FSI模拟与欧拉网格周围的流体将改善结构保持IFEM。出现了新的时空近似，这导致增加的数值稳定性，而无需用户定义的稳定性参数。

Concerning finite element simulations of moving continua, there are many examples in which a considered continuum consists of different embedded phases. Here, the phases can be flexible solids, fluids as well as rigid bodies. They include a rotor in a Newtonian fluid and a fiber-reinforced material considered as a biphase material. The formulation of the fluid-structure-interaction (FSI) in the first example by means of well-known finite element methods leads to disadvantages in view of computational efficiency and stability if large rotations of embedded phases arise. Well-known reasons are the difficult approximation of convective terms, insufficient meshings of surface contacts as well as frequent remeshings of the phases. The simulation of the solid-solid-interactions in the second example leads to long computing times, because the mesh of the embedded phase determines the number of elements of the surrounding phase. The reason is also the necessary sufficient approximation of surface contacts. These disadvantages can be avoided by an immersed finite element method (IFEM). Aims of this research project is the development and implementation of a new structure-preserving IFEM for dynamic, non-isothermal multiphase continua. Here, fluids as well as solids are taken into account. In order to consider also solids with rigid sections, a noval non-isothermal rigid body formulation is developed, which is directly based on a finite element method. Thereby, micropolar rotational degrees of freedom guarantee the rigidity of the finite element meshes pertaining to the rigid sections. A special variational approach avoids the introduction of an Euler tensor, and rigid sections of flexible solids can be defined by a simple declaration in the total mesh. In this way, thermomechanical couplings between non-isothermal rigid bodies, flexible solids and fluids can be simulated easily. Well-known IFEM assume a fixed Eulerian mesh for the total continuum, and consider Lagrangian meshes only for embedded phases. This restriction will be abolished in the current project, in order to simulate large deformations of non-isothermal multiphase solids by the IFEM. But, also FSI simulations with an Eulerian mesh for a surrounding fluid will be improved by the structure-preserving IFEM. There emerge new space-time approximations, which lead to an increasing numerical stability without user-defined stability parameters.