Finite element method mesh refinement

Mesh refinement - h- and p-versions of the finite element method... 

The trial functions in the finite element method are not limited to linear ones. Quadratic functions and even higher-order functions are frequently used. The same considerations hold as for boundary value problems The higher-order trial functions converge faster, but require more work. It is possible to refine both the mesh h and the power of polynomial in the trial function p in an hp method. Some problems have constraints on some of the variables. For flow problems, the pressure must usually be approximated by using a trial function that is one order lower than the polynomial used to approximate the velocity. 

The boundary conditions are applied in the finite element method in a different way than in the finite difference method, and then the linear algebra problem is solved to give the approximation of the solution. The solution is known at the grid points, which are the points between elements, and a form of the solution is known in between, either linear or quadratic in position as described here. (FEMLAB has available even higher order approximations.) The result is still an approximation to the solution of the differential equation, and the mesh must be refined and the procedure repeated until no further changes are noted in the approximation. 

Up to this point of their curricula the students have not encountered the finite element method. Sometimes this is a real obstacle for the use of software with such method. A simplistic explanation of such method was done, for example in order to allow the students to understand the need of manipulating the mesh, refining certain areas. 

The comparison between the finite element and analytical solutions for a relatively small value of a - 1 is shown in Figure 2.25. As can be seen the standard Galerkin method has yielded an accurate and stable solution for the differential Equation (2.80). The accuracy of this solution is expected to improve even further with mesh refinement. As Figmre 2.26 shows using a = 10 a stable result can still be obtained, however using the present mesh of 10 elements, for larger values of this coefficient the numerical solution produced by the standard... 

Abstract The aim of this chapter is to introduce special numerical techniques. The first part covers special finite element techniques which reduce the size of the computational models. In the case of the substructuring technique, internal nodes of parts of a finite element mesh can be condensed out so that they do not contribute to the size of the global stiffiiess matrix. A post computational step allows to determine the unknowns of the condensed nodes. In the case of the submodel technique, the results of a finite element computation based on a coarse mesh are used as input, i.e., boundary conditions, for a refined submodel. The second part of this chapters introduces alternative approximation methods to solve the partial differential equations which describe the problem. The boundary element method is characterized by the fact that the problem is shifted to the boundary of the domain and as a result, the dimensionality of the problem is reduced by one. In the case of the finite difference method, the differential equation and the boundary conditions are represented by finite difference equations. Both methods are introduced based on a simple one-dimensional problem in order to demonstrate the major idea of each method. Furthermore, advantages and disadvantages of each alternative approximation methods are given in the light of the classical finite element simulation. Whenever possible, examples of application of the techniques in the context of adhesive joints are given. 

In this work, a numerical solver based on Finite Volume Method (FVM) is developed to solve the governing equations. The solver has been successfully applied in injection molding filling simulation [2]. Numerical experiments confirm the reliability and efficiency of the solver. Currently the proposed solver can handle tetra, hexa, prism, pyramid, and mixing elements. Prism layer element can also be used for analysis to improve thermal boundary resolution while without extensive refining of mesh. This is valuable in mold cooling analysis that may involve millions of elements. 

---