# Numerical Modeling: Boundary Element Method vs. Finite Element Method

### Key Takeaways

• The boundary element method is an interesting method to solve engineering problems, especially those involving differential equations.

• BEM requires discretization on the body’s surface, simplifying the data needed to process and solve a problem.

• In numerical modeling, non-linear, nonsymmetric, inhomogeneous problems can be solved faster using BEM vs. FEM.

Numerical modeling and simulation provide an easier, cheaper, and highly efficient way of determining solutions for complex mathematical equations

In engineering, numerical modeling and simulation are widely used to solve complex problems. The development of computational numerical modeling has been advantageous in simplifying the problem-solving procedure. In numerical modeling, various methods such as the discrete element method (DEM), finite difference method (FDM), boundary element method (BEM), and finite element method (FEM) are used to develop models and solve problems. While these methods are mostly analogous, here we will compare the boundary element method vs. the finite element method for numerical modeling.

## The Need for Numerical Modeling

In engineering designs, we come across several mathematical calculations, and the task of solving them involves devising the best possible solution. Conducting experiments is one method to reach a solution, but it requires significant effort, time, and money. Despite being prepared, technical difficulties and troubleshooting may delay a whole project.

Numerical modeling and simulation eliminate the need for experimental setup and provide an easier, cheaper, faster, accurate, and highly efficient way of determining solutions for complex mathematical equations. Numerical models are developed to match the physical system perfectly, and solutions can be analyzed and verified with a real system.

Most engineering systems are governed by differential equations or partial differential equations. With the help of numerical methods such as BEM and FEM, we can streamline approaches to the physical system and solve it in no time.

## The Boundary Element Method

The boundary element method solves engineering problems involving differential equations. BEM is also referred to as the boundary integral equation method (BIEM), as it is efficient in solving boundary value problems. Compared to other numerical methods, BEM is suitable for solving boundary integral equations and other physical problems with complicated boundaries.

BEM can be applied widely to electromagnetic problems associated with electrical machines, actuators, antennas, waveguides, actuators, and others. BEM requires discretization only on a body’s surface, which simplifies data required to process and solve the problem. BEM is a versatile numerical method because it easily generates surface mesh with the aid of discontinuous elements. Additionally, it provides highly accurate solutions obtained with boundary elements.

## Advantages of the Boundary Element Method

The advantages of BEM can be listed as:

• Boundary discretization makes the numerical method simpler.
• Mesh formation is easier in BEM for 3D problems.
• High accuracy is achieved with BEM, as it is a semi-analytical method.
• Suitable for open boundary problems and moving boundary problems.
• BEM can be combined with analytical methods and numerical methods such as FEM.

## The Boundary Element Method vs. Finite Element Method

It is not accurate to say BEM is superior to FEM. Both have advantages and disadvantages depending on the type of physical domain where it is used to solve the problem. The table below compares the boundary element method vs. the finite element method:

 Parameter BEM FEM Type of numerical method Boundary based numerical method Domain-based numerical method Discretization Boundary discretization (Surfaces and curves are discretized in 3D and 2D problems, respectively) Full domain or volume discretization Formulation Based on ordinary or partial differential equations Based on boundary integral equations Ease of use High Less Memory requirements Less due to boundary discretization Comparatively higher

In numerical modeling, non-linear, nonsymmetric, inhomogeneous problems can be solved faster using these methods. For numerical modeling of a physical system using BEM, more knowledge on fundamental solutions is required. The depth of knowledge on fundamental solutions is one criterion for choosing between the boundary element method vs. the finite element method.