EN224: Linear Elasticity
Division of Engineering
3.3 The Boundary Element Method
There is an important application of the singular solutions developed in Section 3.2.
It is generally very difficult to find exact solutions to boundary value problems in linear elasticity, and we often resort to numerical techniques to obtain approximate solutions. The boundary element method is a particularly efficient numerical method, particularly for problems where no body forces act on the solid.
There are two common varieties of the boundary element method: these are known as the so called `direct and `indirect formulations.
The Indirect Boundary Element Method
Suppose that we seek
Now, suppose we were to regard our region as a subregion of an infinite solid. Perhaps we could find a distribution of body forces in an infinite solid that would somehow satisfy all our boundary conditions? Surprisingly, this approach turns out to work, and is the basis for the indirect boundary element method.
In fact, it turns out that one only needs to distribute body forces inside and on its boundary. Following standard notation, we will let denote this unknown body force distribution.
We must clearly setwithin the solid. There is no simple way to determine on the boundary, but we note that we can express the displacement and stress fields in the solid as
denotes the normalized Kelvin state.
Now, we could set up a pair of integral equations for the unknown body force distribution on the boundary:
We have shown explicitly that one evaluate the limit as x approaches the boundary, since if one sets blindly, the integrals are strongly singular and are not defined. (The boundary integrals in the expression for displacements contain singularities of order 1/r, while those in the expressions for stress contain singularities of order ).
The integral equations are usually solved using a boundary collocation method. We will describe this procedure briefly, assuming for simplicity that no body forces act within the region of interest.
The boundary is divided into a series of M segments. A nodal point with coordinates separates pairs of boundary segments. A `collocation point with coordinates lies at the center of each boundary segment (collocation points are also sometimes taken to coincide with the nodes).
Now, we assume that the unknown distribution of sources may be interpolated between values at the nodes. Thus
where are the unknown values of at the nodes, and are a set of interpolation functions. It is common to use a piecewise linear interpolation.
We then choose the nodal values so as to satisfy the governing integral equations exactly at the M collocation points. We obtain a set of 3M simultaneous equations for the 3M unknowns:
Once have been determined, stresses and displacements at interior points may be computed by evaluating the appropriate integrals:
The merits and disadvantages of this approach compared with finite element methods should be apparent. The advantages are:
(1) Smaller numbers of DOF for comparable accuracy of solution (particularly in 3D)
(2) Very good accuracy, especially away from the boundaries (because of Saint
(3) Mesh generation is easy
(4) One can compute fields at arbitrary interior points with equal accuracy. Recall
that in FEM one should compute stresses only at integration points for optimal
accuracy. A cumbersome variational recovery procedure is required to interpolate
(1) Matrix to be solved is fully populated and unsymmetric; its conditioning is not
always great either. Solution can be a pain.
(2) Evaluating the singular integrals on the boundaries is a pain.
(3) Computing fields at large numbers interior points can be expensive: the integrals
have to be computed from scratch each time. One can speed things up by using
small numbers of quadrature points when x is far from an element, but if one wants
to plot stress contours over the entire region, for example, its not clear that BEM
offers significant advantages over FEM.
(4) It is virtually impossible to generalize this procedure to nonlinear solids. Even
anisotropic solids are something of a problem since no closed form expressions are
known for the singular solutions (there are some efficient integral representations).
The Direct Boundary Element Method
A very similar procedure may be devised using the results we obtained at the end of the discussion of `Further Singular Solutions. Recall that we devised a method for computing displacements at the interior of an arbitrary region given the tractions and displacements on its boundary:
Let us suppose that body forces vanish, for simplicity. Using the given boundary
conditions, we may write
We now let approach the boundary and regard this as an integral equation for the unknown tractions on and displacements on . As before, we must be careful to compute the integrals using an appropriate limiting procedure, since some of them are singular.
The integral equation is solved using a very similar collocation procedure. It is usual to take the collocation points to coincide with the nodes in this case, so as to make the partitioning of the boundary unambiguous.
There is not much difference between the direct and indirect formulations. Here are some comparisons:
(1) The integrals in the direct formulation are (marginally) less singular
(2) Equation assembly takes slightly more work in the direct formulation; this is
offset partly by the fact that the solution is useful (we actually learn something
about u and t on the boundary, rather than the fictitious body force distribution!)
(3) More work is required in the direct BEM to compute fields at interior points.
Convergence and accuracy appear to be similar in both methods.