Mathematical Aspects of Atomistic and Multiscale Modelling

原子和多尺度建模的数学方面

• 批准号: RGPIN-2021-03489
• 负责人: Ortner, Christoph
• 金额: $ 3.5万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=742579
• 关键词: Mathematical; Aspects; Atomistic; Multiscale; Modelling

The basic building blocks of materials are their constituent atoms which create bonds between one another via suitable arrangements of their associated electrons. Material deformation and failure (e.g., cracking of a turbine blade) affects macro-scopic scales but initiates and propagates at the scale of atoms and electrons, a coupling which is largely ignored in conventional (continuum mechanics) models. The ongoing search for predictive descriptions of material failure and many other macroscopic processes increasingly focuses on the atomic scale and requires computational models that take into account both atomistic and electronic structure as well as macroscopic information such as strain. Quantum mechanical models of electronic structure provide accurate and transferable (i.e., widely applicable) models for forces acting between atoms, but their extreme computational cost makes them unsuitable for simulating complex atomistic processes such as plasticity, cracking or chemical reactions, that span many length- and time-scales. The canonical alternative to study atomistic mechanisms are interatomic potentials (or, empirical force fields), which are empirical models with poor accuracy and extremely limited transferability across applications. Accurate but still computationally efficient models for interatomic forces would immediately have applications across a wide range of disciplines, in particular materials science and bio-chemistry. An exciting development, starting ca 2010, is to build interatomic potentials from universal approximators (machine learning); that is, to treat the construction of interatomic potentials as an approximation problem instead of a modelling problem. This paradigm shift creates an opportunity for a mathematical theory to formalise this extremely rich approximation problem and support and accelerate the next innovation steps. The aim of the proposed program is to explore this problem from the perspective of applied mathematics, in particular applied analysis, approximation theory and numerical analysis. Mathematical techniques will take center-stage but incorporate and synthesize ideas from physics, chemistry and data-science to establish a disciplined approach to support and where appropriate lead the design of the next generation of interatomic potentials. Impact in the sciences and industry will be achieved through close collaboration with science groups and through software co-development. In the long term the research will broaden into similar coarse-graining challenges in other atomistic modelling scenarios. For example, many techniques will be transferrable to developing new tight-binding models, classical DFT models, or coarse-grained dynamical systems. They can also be adapted to analyze databases of atomic structures to accelerate, e.g., automated discovery of structure-property relations with applications in drug and material design.

材料的基本构建块是它们的组成原子，这些原子通过其相关电子的适当排列在彼此之间形成键。材料变形和失效（例如，涡轮机叶片的破裂）影响宏观尺度，但在原子和电子尺度上引发和传播，这是在常规（连续介质力学）模型中很大程度上被忽略的耦合。材料失效和许多其他宏观过程的预测性描述的持续研究越来越多地集中在原子尺度上，并需要考虑原子和电子结构以及宏观信息（如应变）的计算模型。电子结构的量子力学模型提供了精确的和可转移的（即，广泛适用的）原子间作用力的模型，但其极端的计算成本使其不适合模拟复杂的原子过程，如塑性、开裂或化学反应，这些过程跨越许多长度和时间尺度。研究原子机制的经典替代方法是原子间势（或经验力场），这是一种精确度很差的经验模型，并且在应用中的可移植性非常有限。精确但仍然计算效率高的原子间力模型将立即在广泛的学科中得到应用，特别是材料科学和生物化学。从2010年开始，一个令人兴奋的发展是从通用近似器（机器学习）构建原子间势;也就是说，将原子间势的构建视为近似问题而不是建模问题。这种范式转变为数学理论创造了一个机会，使这个极其丰富的近似问题正式化，并支持和加速下一个创新步骤。该计划的目的是从应用数学的角度探索这个问题，特别是应用分析，近似理论和数值分析。数学技术将占据中心地位，但将整合和综合物理学，化学和数据科学的思想，以建立一种纪律严明的方法来支持并在适当的情况下领导下一代原子间势的设计。通过与科学团体的密切合作和软件共同开发，将在科学和工业领域产生影响。从长远来看，这项研究将扩展到其他原子建模场景中类似的粗粒度挑战。例如，许多技术将可转移到开发新的紧束缚模型，经典DFT模型或粗粒度动力学系统。它们还可以适用于分析原子结构的数据库，以加速，例如，自动发现药物和材料设计中的结构-性质关系。