Shearlet-Based Energies for Anisotropic Phase-Field Models

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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.