Numerical simulations of uncertainty quantification and multiscale computation

数学模型和计算技术在解决科学和工程中的复杂问题中是非常重要的。对于大多数应用，数值模拟通常基于确定性公式。然而，在现实中，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的量化，以考虑不确定性的影响。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，开发一种高效的计算方法是一项具有挑战性的任务。近年来，我们发展了随机配点法和随机范式来研究非线性气动弹性模型。该模型在结构项和初始条件的非线性系数中引入了不确定性。特别注意在存在不确定性的情况下对Hopf分岔和二次分岔的影响。我们目前的目标是开发随机哈密顿系统和随机偏微分方程的有效辛格式。**本课题研究的另一个课题是三维复合材料的多尺度建模与计算。近年来，复合材料研究取得了巨大的进展。复合材料的使用提供了许多新的和令人兴奋的发展的可能性。例如，在航空航天工程中，环保和高效的飞机可以使用更坚固和更轻的材料设计；使用耐热材料的更坚固的结构可以提高航天器的安全性。然而，应该指出的是，传统的计算方法可能无法处理由复合材料引起的问题，因为复合材料通常包含许多不同的空间尺度。即使使用超级计算机，也不可能直接进行数值模拟来解决所有精细尺度特征。因此，一种替代方法，如多尺度模拟就出现了。该项目的目标是为航空航天工程应用中的3D复合材料开发可靠且稳健的多尺度模拟。该数学公式基于一个耦合热弹性系统，该系统能够研究复合材料结构中的材料变形和温度变化。值得注意的是，复合材料通常被建模为具有几种材料的特定夹杂物或孔的矩阵，并且这些夹杂物/孔可以具有六面体网格或随机分布在三维单位体积上。对于长期目标，我们计划考虑几何模型中随机微观结构的不确定性效应。因此，发展起来的复合材料数值模拟包括多尺度建模和不确定性量化。**本提案中提出的项目是一项具有挑战性的研究工作，但它将使许多应用数学和工程界的研究人员和工程师感兴趣并受益。该研究将为知识和技术从学术界向工业界的转移做出重要贡献。

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.