Development of Matched Interface and Boundary Algorithm and Applications to Analysis of Thin Plates with Interfaces

Abstract

Plate is an important structural component, which has been widely used in many engineering fields. Apart from uniform bare plates, there are many other kinds of plates such as stepped plates, multi-span plates and plates resting on non-homogeneous elastic foundations. Many factors such as discontinuities of thicknesses, internal supports and non-homogeneous foundations cause abrupt changes of relevant physical quantities and generate interfaces, which may cause some difficulties in solving plate problems.

The method of matched interface and boundary (MIB) is an effective numerical approach to deal with interfaces caused by abrupt changes of physical quantities. MIB and the interpolation formulation of MIB (IMIB) algorithms have been successfully used to solve various interface problems. However, in structural analysis, only stepped rectangular plates have been analyzed and the computational accuracy was not very high. Interfaces lying between grid nodes have been considered in many cases in the existing literature, but interfaces located at grid nodes have not been dealt with. To this end, this thesis summarizes the existing MIB and IMIB algorithms in structural analysis. In view of the features of thin plates, new MIB and IMIB algorithms for dealing with various interfaces in thin plates are developed to improve the adaptability of algorithms and the computational accuracy in this thesis. In the light of different relationships between interfaces and grid nodes as well as different strategies for handling compatibility conditions of interfaces, these MIB and IMIB algorithms can be re-divided into MIB-Ⅰ, MIB-Ⅱ, MIB-Ⅲ and IMIB-Ⅰ, IMIB-Ⅱ, IMIB-Ⅲ, respectively. Advantages and shortcomings of these algorithms are also discussed in details. In addition, for bidirectional cross-interfaces in thin plates, multi-domain MIB (MD-MIB) and multi-domain IMIB (MD-IMIB) algorithms are proposed by further modifying the existing algorithms, which overcomes the limitation that the existing algorithms cannot be used to solve bidirectional cross-interface problems. Through the above development and modification, a MIB based computing framework is preliminarily established to solve problems of thin plates with unidirectional and bidirectional interfaces.

In the analysis of thin plates with unidirectional interfaces, different types of MIB and IMIB algorithms are used to solve free vibration, bending and buckling problems of thin plates. Numerical studies show that MIB and IMIB algorithms have the strong capabilities to solve various interface problems in thin plates. However, there are some differences among them. Additionally, it can also be found from the comparisons of different examples that on the whole, MIB and IMIB algorithms are considered to be equivalent for small half computational bandwidths W such as W≤4, while, there are some obvious differences between them for very large half computational bandwidths W such as W≥5. Especially for IMIB, the use of higher-order interpolation polynomial may decrease the accuracy sharply, which has not been reported in the existing literature.

In the analysis of thin plates with bidirectional cross-interfaces caused by discontinuous loads and non-homogeneous foundations, various types of MD-MIB and MD-IMIB algorithms are utilized to solve bending and free vibration problems of thin plates. Numerical studies indicate that MD-MIB and MD-IMIB algorithms perform very well in dealing with various cross-interfaces. The introduction of MD-MIB and MD-IMIB overcomes the shortcoming of the existing MIB method and is a new development of MIB method. This study is the first attempt to solve bidirectional cross-interface problems in thin plates by using strong form numerical method, i.e., high order central finite difference method. This study further extends the application fields of MIB method from unidirectional interfaces to cross-interfaces in thin plates, which lays a solid foundation for mechanical analysis of complex plated structures.