Some Research Of The Fundamental Solutions Method In Plate And Shell Vibration Models And Its Application

In the research area of plate and shell vibration model, there are two major prob-lems, the model construction and the model computation. This is a doctoral thesis which focus on the latter area. The computation of plate and shell model is a very pop-ular area in the last decade. A lot of numerical methods are introduced. In this article, we mainly use the Fundamental Solutions Method (FSM) to solve the plate and shell vibration problems. Two types of questions are mainly considered, the initial boundary value problem, and the inverse source problem. The article points out that FSM is one of the most powerful methods to solve inverse source problems.This thesis consists of six chapters.The first chapter is introduction. We introduce the history and recent research status of the elastic plate and shell models. We also describe several popular numerical methods, particularly the use of FSM in some other models.Chapter2study the static plate, which can be described by biharmonic equation. Firstly we construct the elastic plate model, and give the uniqueness and stability anal-ysis to the biharmonic equation. And then we derive the fundamental solution of the biharmonic equation. In the process of solving the forward problem, we decompose the problem into two easy solving problems, one is to find a particular solution to the biharmonic equation without boundary restrictions, and another is a boundary value problem. This idea can be easily extended to the solution of the inverse internal bound-ary value problem. The inverse source problem to the biharmonic equation is instable, as can be reflected in some numerical examples.In chapter3, we consider the plate vibration model. Similarly, we get the unique-ness and stability property of the governing equation, and derive the fundamental so-lution as well as the basis solution of the equation when there’s no source terms. In order to solve its forward problem, we use some similar method as is used in bihar-monic equation, that is to decompose the hard solving problem into two easy solving problems. In case that the source term can be written as spatial and temporal separation format, we use a general definition of the inverse source problem to plate equation, that is given the vibration mode, so as to determine how strong the source is. After doing some discussion of the stability of the inverse solution, we use FSM to success-fully solve the problem in case that the vibration mode is arbitrary given. Numerical experiment shows that this method works well.In chapter4, we briefly introduce the finite difference method (FDM), the Mesh-less Local Petrov-Galerkin method (MLPG) and the analytical solution to a special boundary value problem. We compare the FDM and MLPG with FSM, and find that they both have their advantages and disadvantages when solving forward problems. But to deal with inverse source problem, FSM is the most powerful.In chapter5, we apply the FSM in the shell vibration model. The fundamental solution could not be explicitly expressed. Luckily the shell equation can be divided into the principle part and the secondary part. We prove that when the curvature radius of the shell is large, we can solve the shell equation by solving a series of plate equation using iterative method. We give some numerical example based on cylinder shell. The idea of iterative method can also be promote to other kinds of shell problems. For solving the inverse source problem of shell equation, we introduce an approximation method, and successfully apply to the main part dominant equation.The last chapter is conclusion. We give some prospects of the use of FSM in the field of plate and shell problem, and present some future research directions.