Mathematical and computational problems in energy driven pattern formation

能量驱动模式形成中的数学和计算问题

• 批准号: RGPIN-2021-04114
• 负责人: Williams, John
• 金额: $ 1.47万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758601
• 关键词: Mathematical; computational; problems; energy; driven

The configuration of any complex physical systems can be described by an associated energy. The dynamics then follow the gradient flow of this energy as the system dissipates energy. The first natural step in studying these systems is to analyze the critical points of the energy. For the systems of interest in this proposal these critical states and their energies depend on multiple parameters. There is generally some noise in the original physical systems being modeled thus systems generically evolve to global minimizers of the energy.? Most energy based models in the applied sciences involve parameters which often cannot be determined with absolute certainty. It is therefore important to study such models for whole parameter ranges, and in particular, to detect parameter values at which the system behaviour changes qualitatively. A first step towards accomplishing this goal is the understanding of the set of equilibria of the model and their respective energies. The long-term goal of my research program is the development of computational tools which can simulate complex mathematical models with accurate computed error bounds where every computation is an existence proof. Towards this goal I will work with HQP over the next five years on developing methods to address questions related to the Ohta-Kawasaki energy and related models from material science. The project will tie together various research objectives over three interconnected themes: Rigorous computation of phase diagrams; Extensions of the Ohta-Kawasaki energy and Confinement and geometric effects.

任何复杂物理系统的构型都可以用一个相关的能量来描述。然后，当系统耗散能量时，动力学遵循能量的梯度流。研究这些系统的第一个自然步骤是分析能量的临界点。对于本方案中感兴趣的系统，这些临界态及其能量依赖于多个参数。在被建模的原始物理系统中通常存在一些噪声，因此系统通常演化为能量的全局最小化。在应用科学中，大多数基于能量的模型涉及的参数往往不能绝对确定。因此，重要的是研究整个参数范围的这种模型，特别是检测系统行为在其上发生定性变化的参数值。实现这一目标的第一步是理解模型的一组均衡及其各自的能量。我的研究计划的长期目标是开发计算工具，这种工具可以用准确的计算误差范围来模拟复杂的数学模型，其中每一次计算都是存在的证明。为了实现这一目标，我将在未来五年与HQP合作开发方法，以解决与太田-川崎能源相关的问题，以及材料科学的相关模型。该项目将在三个相互关联的主题上将不同的研究目标联系在一起：相图的严格计算；太田-川崎能量和限制以及几何效应的扩展。