(written by: Willy Yanto Wijaya)
Nowadays, in engineering designing purposes, the role of numerical modeling/ simulation is becoming more and more important. While experiments could cause higher cost or technical difficulties to do, numerical simulation could offer a high efficiency, faster and cheaper way for the designing cycles and development. However, in order to be reliable, simulation results must give relatively accurate results when compared with the actual physical system. Therefore, it is very important to have appropriate physical models of the system to be analyzed; then from these physical models, we derive the mathematical models as well as the numerical models of the system. The solutions of these mathematical models will then be refined and interpreted, then be compared with the actual physical mechanism/ phenomena for verification.
Many of physical phenomena in the nature could be described in differential equations. Often, a physical system could include a group of complex governing partial differential equations, which is difficult to be solved analytically. Therefore, we utilize the numerical methods such as FEM (Finite Element Method) and BEM (Boundary Element Method), which are essentially the numerical approaches to solve the Partial Differential Equations (PDE) of the physical system.
Basically, FEM is a method of dividing a physical system (with PDE characteristics) to be analyzed into smaller pieces (discrete elements). Each of these smaller pieces will have a simpler approximation of the solution. These local approximate solutions will then be put together to obtain a global approximate solution. FEM will include the determination of the discretized elements (cells/ mesh/ grids) as well as the basis function approximated in each cell. In FEM, the physical system (domain) will be thoroughly discretized, which will become one point of its differences with BEM. In elastic analysis, FEM has the root analogous with analytical Rayleigh-Ritz method which is derived from the energy principle.
On the other hand, in BEM, the discretization is restricted only to the system boundary. This BEM is derived through the discretization of an integral equation which is equivalent mathematically to the original PDE. This integral equation will then be defined at the boundary of the domain and it relates the boundary solutions to the solutions at points inside the domain.
Then, what are the advantages of BEM over FEM? Since in BEM, the discretization is done only at the boundary, it will result in more efficient computation and easier to be used compared with FEM. Besides, regarding its characteristics, BEM is also very suitable for modeling symmetrical problems as well as problems which involve infinite domains (such as the case of potential flow past an obstacle). BEM is also especially popular to solve Laplace and Helmholtz problems.
However, regarding its characteristics, BEM also poses several disadvantages compared with FEM. Since the formulation of integral equation can’t be done for all types of PDE, BEM is not so widely applicable compared with FEM. Besides, in the case of inhomogenous and non-symmetric or non-linier problems, fully populated system of equations will often occur in BEM thus making the storage requirement and computational time become increasing significantly. Further, BEM also requires more knowledge about the suitable fundamental solutions compared than if we use FEM for the simulation of the physical system.
Therefore, depending upon the type of physical systems we’d like to analyze, we could then decide which method is more favorable to be used.