Finite Element Method for Boundary Element Method in Deformation Mechanic
The Boundary Element Method (BEM) has become an important and powerful technique for solving field problems of deformation mechanics, such as linear and nonlinear elasticity problems. After more than 50 years of development, it has become an effective tool for the analysis of a wide range of engineering problems. Compared to traditional finite element methods (FEMs), BEMs have the advantages of a reduced computational cost and superior convergence behavior. Furthermore, compared to the traditional method of characteristics (MOC) and method of fundamental solutions (MFS), BEMs demonstrate faster and more accurate solutions.
However, while BEMs are more efficient when it comes to speed and accuracy, they do have some drawbacks, most notably the difficulty associated with formulating and solving numerical boundary integral equations. Generally speaking, solving these equations requires a considerable amount of time and effort on the part of the analyst. In order to overcome this limitation, a technique known as the Finite Element Method (FEM) has been developed. FEMs are based on the concept of approximating the boundary integral equation by subdividing the boundary into small finite element blocks and replacing the boundary integral equation by a set of finite element equations.
The Finite Element Method (FEM) for boundary element solutions is based on the idea that the boundary integral equations can be replaced by a set of finite element equations. The finite element equations are derived from the boundary integral equations by integrating the latter over small finite elements, which are defined over appropriate parts of the boundary (i.e., the elements or nodal points). These elements are thus used to approximate the boundary, and the values at the nodes of the elements are used to solve the boundaries integral equation. Since the integration is not exact but only approximate, some errors may occur in the solution. This can be minimized by using a finer mesh or increasing the number of elements.
Here we give a general description of the FEM for solving BEM problems. First, the physical problem is discretized using the finite element method. This means that the problem is replaced by a series of subdomains, which are discretized into small finite elements (or nodal points). Each element is defined by its nodes and their associated values, which can then be used in the numerical solution of the boundary integral equations. Second, the boundary conditions are imposed on the nodes of the elements. This can be done either by setting these conditions as boundary element boundary conditions or by the use of boundary condition enforcement methods, such as weighting factors or source functions. Third, the boundary element boundary integral equation is then solved by solving a system of linear equations, where the unknown coefficients or displacement components are determined. This can be done either directly or indirectly. Finally, the solution is computed by interpolating the nodal values of the unknown coefficients or displacement components. Thus, the numerical solution of the boundary integral equation is obtained in the form of a set of linear algebraic equations.
Finally, the FEM for boundary element solutions can be used to efficiently solve a variety of problems, including nonlinear elasticity problems, dynamic problems, and coupled problems. Since the FEM is used for numerical calculations, the boundary element boundary integral equations can be handled in exactly the same way as for the finite element equation. Therefore, the same numerical techniques can be applied for the FEM for boundary element solutions as for the finite element solutions.
In conclusion, the Finite Element Method (FEM) for boundary element solutions has become an important and powerful technique for solving field problems of deformation mechanics. It is based on the concept of approximating the boundary integral equations by subdividing the boundary into small finite element blocks and replacing the boundary integral equations by a set of finite element equations. The FEM for boundary element solutions can be used to efficiently solve a variety of problems, including nonlinear elasticity problems, dynamic problems, and coupled problems. Furthermore, compared to traditional finite element methods and the method of characteristics, BEMs demonstrate faster and more accurate solutions.
