Node-based meshless methods: A way of generalizing numerical modeling techniques

Summary form only given. Science and Engineering Problems are normally quantitatively described with mathematical models. These mathematical models are often represented by mathematic equations which are often difficult to solve due to irregular boundaries and materials. To address the issue, numerical methods have been developed to find the approximate solutions of the equations. Thanks to the tremendous increase in computing power of modern computers, numerical methods have become widely used techniques nowadays and consequently a large number of numerical methods have been developed. On one hand, the numerical methods have led to solutions of many problems that could not be solved before; on the other hand, they have caused confusions and become challenging to understand and choose. In this talk, we will report our recent finding on the essence of the numerical methods for solving Maxwell´s equations: a numerical method is essentially a special case of the method of weighted residuals (minimization). In other words, numerical methods can be unified under the framework of the method of weighted residuals, and a new method can be developed with selection of different basis and testing functions. From there, we will show the recently emerging node-based meshless methods can be used as a way of generalizing or embodying numerical modelling techniques.