Shearlet-Based Energies for Anisotropic Phase-Field Models
各向异性相场模型的基于剪切波的能量
基本信息
- 批准号:397355080
- 负责人:
- 金额:--
- 依托单位:
- 依托单位国家:德国
- 项目类别:Research Fellowships
- 财政年份:2018
- 资助国家:德国
- 起止时间:2017-12-31 至 2018-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
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.
拟议的项目分析相场模型的基础上,新的基于剪切波的能量。相场模型在许多应用中,尖锐的界面需要解决。这对于适当地模拟物理过程是必要的,例如,模拟在过饱和化学溶液或气体中形成的晶体。另一个例子是通过裂缝的模拟给出的,其中裂缝可以被理解为压裂材料的位移函数中的尖锐界面。通过最小化由底层物理模型给出的能量来计算相场。事实上,在许多应用中,这种能量必然是各向异性的,以便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倾向于根据材料在某些方向上发展。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,我们开发了一种新的方法,允许建模的各向异性范围广泛,同时保持数值复杂性低。因此,我们将调用基于所谓的剪切波系统的能量。该系统是从应用谐波分析,表现出很强的方向灵敏度的表示系统。事实上,最小化的建议剪切波为基础的能量只需要解决一个线性算子方程-一个相当大的改进,需要解决准线性或非线性微分方程。此外,我们将分析剪切波系统的强方向敏感性在多大程度上允许建模的各种类型的各向异性。我们将特别强调三个最突出的相场模型。这些用于结晶的建模,断裂的模拟,以及信号和图像处理中的应用,如去噪或图像分离。
项目成果
期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces
Sobolev空间混合剪切波-小波框架的逼近性质
- DOI:10.1007/s10444-019-09679-9
- 发表时间:2019
- 期刊:
- 影响因子:1.7
- 作者:Philipp Petersen;Mones Raslan
- 通讯作者:Mones Raslan
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Professor Dr. Philipp Petersen的其他文献
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