膜和绳的统一有限应变理论
A Unified Finite Strain Theory for Membranes and Ropes
摘要：
在切向微分（TDC）的框架中重新表述了有限应变理论，从而在三重意义上实现了统一。首先，绳索，薄膜和三维连续体处理的一组控制方程。其次，重新制定的边值问题适用于参数化和隐式几何。因此，该公式比经典公式更一般，因为它不依赖于隐含曲线坐标系的参数化以及协变和逆变基向量的概念。这导致了第三个统一：基于TDC的模型适用于两种根本不同的数值方法。一方面，可以使用经典的表面FEM，其中几何形状由用于绳的弯曲的一维单元和用于膜的二维表面单元离散化。另一方面，它也适用于最近的跟踪FEM方法，其中的几何形状是沉浸在一个更高维度的背景网格。然后，形状函数的背景网格上的浸入几何形状的轨迹进行评估，并用于近似。因此，迹有限元法是流形上偏微分方程的虚拟区域方法。数值结果表明，所提出的有限应变理论产生高阶收敛率的数值方法，流形的尺寸，和几何表示类型无关。
英文摘要：
The finite strain theory is reformulated in the frame of the Tangential Differential Calculus (TDC) resulting in a unification in a threefold sense. Firstly, ropes, membranes and three-dimensional continua are treated with one set of governing equations. Secondly, the reformulated boundary value problem applies to parametrized and implicit geometries. Therefore, the formulation is more general than classical ones as it does not rely on parametrizations implying curvilinear coordinate systems and the concept of co- and contravariant base vectors. This leads to the third unification: TDC-based models are applicable to two fundamentally different numerical approaches. On the one hand, one may use the classical Surface FEM where the geometry is discretized by curved one-dimensional elements for ropes and two-dimensional surface elements for membranes. On the other hand, it also applies to recent Trace FEM approaches where the geometry is immersed in a higher-dimensional background mesh. Then, the shape functions of the background mesh are evaluated on the trace of the immersed geometry and used for the approximation. As such, the Trace FEM is a fictitious domain method for partial differential equations on manifolds. The numerical results show that the proposed finite strain theory yields higher-order convergence rates independent of the numerical methodology, the dimension of the manifold, and the geometric representation type.
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