Homogenization Theory And 3-D Numerical Simulations For Chemical Vapor Infiltration

This thesis is divided into three parts.Chemical vapor infiltration (CVI) process is used to produce the the Silicon Car-bide Composite Ceramic Materials reinforced by carbon fibers. Multiscale modeling for CVI process was presented by Bai, Yue and Zeng. They set up micro and macro models respectively, and then combine porosity evolution equation to simulate the entire CVI process. The process can be divided into two stages, the stage of small pores dominating densification process and the stage of SiC deposition on surface of fiber bundles only. In the first part, the homogenization theory, which played a fundamental role in the multiscale algorithm, will be rigorously established. The governing system, which is a multi-scale reaction-diffusion equation, is different in the two stages of CVI process, so we will consider the homogenization for the two stages respectively. One of the main features is that the reaction only occurs on the surface of fiber, so it behaves as a singular surface source. The other feature is that in the second stage of the process when the micro pores inside the fiber bundles are all closed, the diffusion only occurs in the macro pores between fiber bundles and we face up with a problem in a locally periodic perforated domain.In the second part, we focus on the numerical simulation to the 3D CVI pro-cess. The goal is to obtain the residual pore distribution in the material since the residual determine the properties of the material. However, the residual porosity is micro information, simulations on the whole domain in micro structure are beyond our computation ability. Based on multiscale modeling, a model for non-periodic structure is proposed to probe into the residual pore distribution in the statisti-cal sense. In the macro scale, under the framework of Heterogeneous Multiscale Method(HMM), we upscale the micro concentration diffusion-reaction equation, es- timate porosity and effective reaction area etc. In the micro scale, we reconstruct the micro diffusion-reaction concentration equation. By the relationship between micro concentration and velocity in normal direction on fibre fronts, and then combine level set equation, we can capture evolution of fronts.In the third part, a fast Poisson solver for three dimensional interface problems with piecewise constant coefficient, which was presented by Shaozhong Deng, Kazu-fumi Ito and Zhilin Li, will be introduced. We recall the methods and present a new scheme for selecting grid points to compute the normal derivatives of solutions at projections for the fast Poisson solver. Three dimensional numerical examples and an application are provided and analyzed.