Boundary Element FormulationDr Hatem R. WasmiA Prof in Applied Mechanics

Types of boundary element formulationsTypes of BE Formulation

Solution of the boundary integral equationAfter forming the integral equations, a numerical scheme has to be set up to solve such equations.In order to do so, the boundary of the problem is discretized into elements, where the unknown source functions (potentials or generalized displacements) are assumed to vary using polynomials (constant, linear, quadratic, etc).It can be seen that such boundary elements represent virtual discretisation (not a physical one as that used in the finite element method).Inside these elements, some points called nodes are chosen to approximate the source functions in terms of the nodal values (via the used polynomials).

IntroductionIf you hold a ruler to the end points of a linear function you can draw the function with a pencil, see Fig. 1This is the simplest application of the principle behind the boundary element method: the boundary element values of a function determine a function uniquely.In the language of the mechanics this principle can be expressed as:The displacement and the forces on the surface of a body and the exterior load determine the displacements and the stresses within the body uniquely.

Example: A rodIn case of a rod, as shown in Fig. 2, the corresponding boundary terms are the displacements and the end actions , u(0) = u1 u(l) = u2 , N(0)=-f1 , N(l)= f2 And the influence function, which connect the displacement u(x) in the interior with the boundary values

The boundary elements targetThe function 1 does not only describe the displacement of the rod in fig 3 but the displacement of any rod be in a trussBoundary elements are just a tool to construct these piecewise linear functions. They represent a piece of the boundary along which we model the displacement and the force distribution by simple piecewise polynomials.

In one dimension the boundary consists, of the endpoints of the interval[0,l] so we need no boundary elements. But we face the same problem as in higher dimensions: we must find the unknown boundary values. We utilized a coupling condition between the boundary terms ui and fi

The Matrix Equation of a Rod

In the case of a rod this coupling conditions are:

Where the matrix is the stiffness matrix of the rod and the components pi and the end fixing forces. The end fixing forces are the support reactions fi .when both ends of the rod are fixed (ui = 0 implies fi =pi )When both ends of the rod are fixed (ui = 0 implies fi =pi ). This coupling condition is the key to our problem: of the four

boundary terms u1 , u2 and f1 , f2 in equ. 2 two are prescribed while the conjugated terms are unknownu1 = 0 , u2 =? f1 =?, f2 = 0

Numerical Example• Suppose l=4 m and P0 =10KN/m and EA/L =250 in

previous example.

Solution• Substitute the given data in equation 2 to get on

• This equations can be solved for the two unknowns to get u2 = 0.8 m, f1 = - 400 KN

If we substitute these values and the given boundary data u1 = 0, f2 = 0 into the influence equation (1), we obtain the displacement of the rod:

Additional Example for Comparison between BVM & FEM

• Discretization of gear into 291 finite elements (b) Discretization of gear into 33 boundary elements

• Figure 2. Comparisons of FEM and BEM meshes.

FEM & BEM Mathematical Dependability

The structure of a BEM applicationA typical application of BEM consists of the

following parts:

• Mathematical model

• Representation formula

• Boundary integral equation

• Boundary elements

• Discrete equations

• Solution of the linear system

• Interpretation

The Method of Weighted Residuals

Collocation Method

▪ Method of Moments

▪ Galerkin’s Method

▪ Collocation by Subregions

▪ Least Squares