Shearlet-Based Energies for Anisotropic Phase-Field Models

该项目分析了基于新型剪切波能量的相场模型。相场模型被用于许多需要解析尖锐界面的应用中。这对于适当地模拟物理过程是必要的，例如，模拟在过饱和的化学溶液或气体中形成的晶体。另一个例子是对断裂的模拟，其中裂缝可以理解为断裂材料位移函数中的一个尖锐界面。相场是通过最小化底层物理模型给出的能量来计算的。事实上，在许多应用中，这种能量必须是各向异性的，以适当地模拟物理现象。例如，在断裂模拟的情况下，各向异性意味着裂纹将根据材料倾向于向某些方向发展。不幸的是，各向异性能量的最小化是一个极具挑战性的数值问题。实际上，最小化需要求解拟线性或非线性偏微分方程。因此，建立的数值方法需要限制允许的各向异性，同时仍然需要相当复杂的数值计算。在这个项目中，我们开发了一种新颖的方法，可以在保持低数值复杂性的同时模拟广泛的各向异性。因此，我们将调用基于所谓shearlet系统的能量。该系统是应用谐波分析的表征系统，具有很强的方向敏感性。实际上，所提出的基于剪切波的能量的最小化只需要求解一个线性算子方程——相对于求解拟线性或非线性微分方程的必要性，这是一个相当大的改进。此外，我们将分析剪切波系统的强方向灵敏度在多大程度上允许对许多类型的各向异性进行建模。我们将特别强调三个最突出的相场模型。它们用于结晶建模，断裂模拟以及信号和图像处理中的应用，例如去噪或图像分离。

The proposed project analyzes phase-field models based on novel shearlet-based energies. Phase-field models are employed in many applications where sharp interfaces need to be resolved. This is necessary to appropriately simulate physical processes such as, for instance, the simulation of a crystal taking shape in an oversaturated chemical solution or gas. Another example is given by the simulation of a fracture, where the crack can be understood as a sharp interface in the displacement function of the fracturing material. The phase-field is computed by minimizing an energy that is given by the underlying physical model. In fact, in many applications, this energy is necessarily anisotropic to model physical phenomena appropriately. For example, in the case of fracture simulation anisotropy means that the crack will prefer to evolve in certain directions depending on the material. Unfortunately, the minimization of anisotropic energies is a highly challenging numerical problem. In fact, the minimization requires solving quasilinear or nonlinear partial differential equations. Consequently, established numerical methods need to restrict the admissible anisotropies while still requiring considerably complex numerical computations. In this project, we develop a novel method which allows modeling a broad range of anisotropies while keeping the numerical complexity low. Therefore, we will invoke energies based on so-called shearlet systems. The systems are representation systems from applied harmonic analysis that exhibit a strong directional sensitivity. Indeed, the minimization of the proposed shearlet-based energies requires only to solve a linear operator equation---a considerable improvement over the necessity to solve quasilinear or nonlinear differential equations. Moreover, we will analyze in how far the strong directional sensitivity of shearlet systems allows the modeling of numerous types of anisotropies. We will place particular emphasis on the three most prominent phase-field models. These are used for the modeling of crystallization, the simulation of fracture, and applications in signal- and image processing such as denoising or image separation.