Boundary Element Methods - Introduction

Figure 1: Simulation of the sound field by a vibrating wheel, calculated by Sysnoise

Boundary Element Methods

Numerical Solutions for Engineering Problems

In the design of engineering structures, numerical simulations play an increasingly important role. This can be attributed to the high costs or practical difficulties related to experiments, which have to confront rapid advances in the computational power and the resulting decrease in the costs for computer simulations. However, in order to supplement or even replace experiments, simulation approaches have to fulfill strong requirements. An essential demand is that the simulations are efficient and lead to accurate and reliable results (Figure 2). This in turn will depend upon the mathematical model of the physical world which the engineer has to choose, and which he or she tries to solve with a specific simulation tool- e.g., the Boundary Element method – by applying assumptions about the loading situation, initial and boundary conditions.

Figure 2: Physical and mathematical modelling

The differential equations that describe the physical phenomena (elastomechanics, acoustics, heat conduction etc.)  can only be solved analytically for a very limited class of problems and even there only for simple geometries. More complex tasks require numerical approaches. Among these approaches, we should mention the Finite Difference Method (FDM), the Finite Volume Method (FVM), the Finite Element Method (FEM), and – more recently – the Boundary element Method (BEM), which have obtained a certain degree of universality, each of these methods with some specific advantages and disadvantages. However, the influence of FDM and FVM in solid physics is rather limited today, so that BEM mainly competes with FEM in a common field, where both of these numerical methods have specific advantages.

The most noticeable difference between FEM and BEM – and one of the important advantages of the latter – concerns the discretisation. While in FEM the complete domain has to be discretised, the BEM discretisation is restricted to the boundary, as depicted in Figure 3. Depending on the complexity of geometry and load case, this can lead to important time saving in the creation and modification of the mesh. Apart from this, Boundary Element Methods usually possess advantages when dealing with stress concentration problems or with problems involving infinite or semi-infinite domains, e.g., acoustics, soil-structure interaction etc.

Figure 3: Discretisation of a FEM model (left) and of a BEM model (right) 

A basic feature of all Boundary Element Methods is their use of fundamental solutions, which are analytically free space solutions of the governing differential equation under the action of point source. The fact that they are exact solutions accounts for some of the advantages of the BEM, viz., the improved accuracy in the calculation of stresses and exterior problems.

In this Boundary Element course, the theory of BEM is introduced at the examples of a bar, a bending beam and heat conduction (Laplace’s equation) in a plate. Important aspects are the design of the boundary integral equation, the fundamental solution, the collocation method, the generation of the system matrices and the application of boundary conditions. Besides, discretisation and interpolation techniques are covered as well as numerical integration methods. A homework project is given in MATLAB in order to underline the crucial points of the boundary element method. As a result, a complete BEM problem is covered. Laplace’s equation is extended by source terms (Poisson’s equation). Substructure techniques are discussed as well.

Besides, the advantage of the BEM with respect to FEM is demonstrated in acoustics, where a vibrating body generates an unbounded sound field in an infinite space. Industrial applications are involved by introducing the commercial program code “Sysnoise”, see Figure 1.

Table 1: Comparison BEM - FEM