Numerical simulations of uncertainty quantification and multiscale computation

不确定性量化和多尺度计算的数值模拟

• 批准号: RGPIN-2014-05664
• 负责人: Wong, YauShu
• 金额: $ 1.31万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=668846
• 关键词: Numerical; simulations; uncertainty; quantification; multiscale

Mathematical models and computational techniques are very essential in solving complex problems in science and engineering. For most applications, numerical simulations are usually based on a deterministic formulation. However, in reality, uncertainties are present in one form or the other. Hence, to assess the reliability and accuracy of the numerical simulations, it is important to consider the uncertainty quantification so that the effect of uncertainty is taken into account. To incorporate uncertainty into a mathematical model, random variables or stochastic process must be considered. Hence, it is a challenging task to develop an efficient computational methodology. Recently, we have developed the stochastic collocation method and the stochastic normal form to study the nonlinear aeroelastic model. In this model uncertainties are introduced in the nonlinear coefficients in the structural term and also in the initial conditions. Particular attention is paid on the effect on the Hopf and secondary bifurcations in the presence of uncertainties. Our current objective is to develop efficient symplectic schemes for stochastic Hamiltonian systems and for stochastic partial differential equations. **Another topic to be studied in this proposal is the multiscale modeling and computation for three-dimensional composite materials. In recent years, tremendous progress has been reported in composite materials. The use of the composite materials offers the possibility of many new and exciting developments. For example, in aerospace engineering, an environmentally friendly and efficient aircraft can be designed using tougher and lighter materials; and a stronger structure with heat resistance materials could improve the safety of a spacecraft. However, it should be noted that conventional computational methodology may not be capable of dealing with problems arising from composite materials, which usually contain many different spatial scales. A direct numerical simulation to resolve all the fine scale features is impossible even by make use of supercomputers. Thus, an alternative approach such as the multiscale simulation comes in. The goal of the project is to develop reliable and robust mutliscale simulations for 3D composite materials in aerospace engineering applications. The mathematical formulation is based on a coupled thermoelastic system, which is capable of investigating the material deformations and the temperature variations in a composite structure. It is noted that a composite material is usually modeled as a matrix with specific inclusions or holes of several materials, and the inclusion/hole may have a hexahedral mesh or randomly distributed over a 3D unit volume. For the long-term goal, we plan to consider the uncertainty effect due to the random microstructure in the geometric model. Hence, the developed numerical simulation for composite materials include multiscale modelling and uncertainty quantification.**The project presented in this proposal is a challenging research work, but it will be of interest and benefit to many researchers and engineers in applied mathematics and engineering communities. The research will make an important contribution of the knowledge and technology transfer from academia to industry.

数学模型和计算技术对于解决科学和工程中的复杂问题至关重要。对于大多数应用，数值模拟通常基于确定性公式。但是，实际上，不确定性以一种或另一种形式存在。因此，为了评估数值模拟的可靠性和准确性，重要的是考虑不确定性量化，以便考虑到不确定性的效果。为了将不确定性纳入数学模型，必须考虑随机变量或随机过程。因此，开发有效的计算方法是一项具有挑战性的任务。最近，我们开发了随机搭配方法和随机正常形式，以研究非线性航空弹性模型。在此模型中，在结构项和初始条件下的非线性系数中引入了不确定性。在不确定性的情况下，特别关注对HOPF和次要分叉的影响。我们目前的目标是为随机的哈密顿系统和随机部分微分方程开发有效的征服性方案。 **该提案中要研究的另一个主题是三维复合材料的多尺度建模和计算。近年来，复合材料中已经报道了巨大的进步。复合材料的使用提供了许多新的令人兴奋的发展的可能性。例如，在航空航天工程中，可以使用更强硬，更轻的材料设计环保和高效的飞机。耐热材料具有更强的结构可以提高航天器的安全性。但是，应该指出的是，常规计算方法可能无法处理由复合材料引起的问题，这些材料通常包含许多不同的空间尺度。即使使用超级计算机，也不可能进行直接的数值模拟来解决所有精细的比例功能。因此，诸如多尺度仿真之类的替代方法都出现了。该项目的目的是针对航空航天工程应用中的3D复合材料开发可靠且可靠的mutliscale模拟。数学公式基于耦合的热弹性系统，该系统能够研究复合结构中的材料变形和温度变化。据注意，复合材料通常被建模为具有特定夹杂物或几种材料孔的基质，并且包含/孔可能具有六面体网状网格或在3D单位体积上随机分布。对于长期目标，我们计划考虑由于几何模型中的随机微观结构而引起的不确定性效应。因此，开发的复合材料的数值模拟包括多尺度建模和不确定性量化。这项研究将为知识和技术转移从学术界转移到行业做出重要贡献。