Dr Alain Kassab (University of Central Florida, Orlando, Florida)
This course presents the Finite Element Method (FEM), Boundary Element Method (BEM) and Meshless Methods (MEM) within a unified framework of the method of weighted residuals (MWR). The course begins by introducing the MWR and one dimensional applications of all three methods as well as finite difference and finite volume methods. The basic fundamentals of the finite element method are then developed using simple examples in heat transfer. Particular attention is given to the development of the discrete set of algebraic equations, beginning with simple one-dimensional problems, illustrating the principles of automated generation of finite elements via Gauss quadratures and body-fitted coordinates, the loading matrix, and continuing to two- and three-dimensional elements. Once these principles are established, the concept of boundary element methods are then introduced. The relation of the BEM to the Green’s function approach for the analytical solution of partial differential equations is presented. The advantage of the BEM in reducing the dimensionality of a problem is demonstrated along with applications to problems with infinite domain boundaries. The boundary element technique is a natural extension of the finite element method, and this becomes greatly appreciated by users. The BEM is developed to 2 and 3D problems, symmetric and non-symmetric solvers iterative solvers are presented along with applications in heat transfer and inverse problems. Finally, meshless methods are developed. Here, the focus is on radial basis function (RBF) based meshless formulation in strong form. The method is simple to grasp, and simple to implement. The power of the method is becoming more appreciated with time. The meshless method has been shown to yield solutions with accuracies comparable to finite element methods employing an extensive number of elements, yet requiring no mesh (or connectivity of nodes). We have used it for structural analysis, fluid flow, heat transfer, environmental transport, and various biomedical applications. Ref: Darrell W. Pepper (University of Nevada Las Vegas), Alain J. Kassab (University of Central Florida), Eduardo A. Divo (Embry-Riddle Aeronautical University), An introduction to finite element, boundary element, and meshless methods with applications to heat transfer and fluid, American Society of Mechanical Engineers (ASME), New York, New York, USA, 2014.