Numerical simulations of uncertainty quantification and multiscale computation

点击翻译按钮获取中文摘要

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.