The use of exact values at quadrature points in the boundary element method N.D. Stringfellow, R.N.L. Smith Applied Mathematics & OR Group, Cranfield University, Shrivenham, Swindon, U.K., SN6 SLA E-mail: N. D. String fellow@rmcs. cranfield. ac. uk Abstract This paper describes the use of exact values in the solution of boundary element problems and compares the accuracy of the corresponding numerical solutions with those obtained using the traditional interpolation method. The new scheme improves accuracy significantly, perhaps by several orders of magnitude, at little computational cost. 1 Introduction The traditional approach to solving a boundary element problem over a domain il with boundary conditions $ defined over a boundary F has been to take the geometry of F, discretise this geometry and use a set of polynomial interpolation functions to represent each section of the discretised shape. The assembly of entries in the coefficient matrices then consists of using these interpolants to calculate the x and y coordinates (in 2 dimensions) of an approximation to the boundary F together with information related to the boundary conditions and the particular type of problem (potential, elasticity, etc.) which is to be solved. A similar interpolation scheme is used to approximate the values of the boundary conditions along the discretised region.
240 Boundary Elements Figure 1: Example domain for interior problem For many real problems, the geometry may be described exactly in terms of simple shapes such as straight lines, circular arcs and splines. The boundary conditions also frequently consist of a combination of simple functions such as trigonometric functions, polynomials and logarithmic or exponential functions. Using these functions it is possible to calculate the exact values of the x and y coordinates of the geometry and the exact values of the boundary conditions at each quadrature point along an element, and to use these exact values in the calculation of the coefficient matrices. The use of these values immediately remove much of the approximation which is usually carried out before the problem is solved numerically. We first consider the collocation formulation for the 2-dimensional potential boundary element method using polynomial interpolation functions to represent the geometry and boundary conditions. We show how removing interpolation functions from the formulation removes inaccuracies and present numerical results to demonstrate improved convergence. 2 Polynomial Collocation Method Here we consider the traditional collocation approach to the boundary element method for potential problems over a domain fi. For interior problems an example domain is shown in figure 1. The general form of the boundary integral equation for potential problems is:- cu+ / up*dr= I u*pdt (1) Jr Jr
Boundary Elements 241 Figure 2: Discretised boundary where:- c a constant dependent upon the boundary shape u the function values p the derivative values n* and p* fundamental solutions of the PDE F the boundary The boundary is chopped up into "elements" as shown in figure 2 to give the new equation c%+ ]T / up*df= ^2 I u*pdr (2) At this point we would normally assume that u and p may be represented by a polynomial of order n varying over the element F^ and that the geometrical shape of the element can be represented by a polynomial of order ra varying over the element, with ra n for isoparametric elements. Introducing a parameterisation of each element F^ using a parameter t and representing the boundary condition shape functions as </>(t) and the geometry shape functions as ip(t) we obtain a new equation with p* and u* being functions of i >i(t) and J(t) the Jacobian of :,he transformation. Note that equation 3 is not the same as equation 2 but is an approximation to it.
242 Boundary Elements For potential problems we have the kernel functions p* and u* given by p*, u* = log r (4) where r is the vector from the collocation point at which u is the potential, to the point on the boundary of I\, and n is the normal vector to the boundary on IV However both r and n are now only approximated in the terms J?* and u* because the geometry is approximated at these points using the geometry shape functions if)i(t) and the Jacobian J(t) evaluated using the i/ji(t) gives an approximation J(t)dt to dt. Our equation may now be written in which we still have to evaluate the integrals. In order to evaluate the integrals we use quadrature schemes - Gauss quadrature for ordinary non-singular integrals and schemes such as those devised by Smith [1] for the singular u* integrals - and the row-sum property ^y r*dr = o (6) for the p* integrals. After evaluating the integrals we are left with a system of equations Hu = Gt. (7) Collecting the known values on the right hand side and multiplying through we are left with a linear system of equations Ax = b (8) from which we may retrieve x using a linear solver such as Gaussian elimination. 3 Exact Geometry For most practical problems the geometrical composition of the boundary F is defined in terms of simple line sections such as straight lines, circular and elliptical sections. If the boundary is composed of straight lines any interpolant involving at least two points can produce any point x(to) along an element for a given to. A polynomial interpolant of a circular section x(to) will not lie on the circle except at the sampling points. This inaccuracy gives rise to numerical errors in the integrations. As an example, consider a quadratic element passing through the points ( 1,2 v^) (0,0) (1,2 - A/3) which can be used to interpolate a circle
Boundary Elements 243 Figure 3: Interpolant and circle centre (0,2) with radius 2 apparently quite well as shown in figure 3. Closer inspection of the region x G [-0.6, 0.4] as shown in figure 4 reveals that the interpolant is some distance from the "true" boundary. The actual values of the x and y coordinates used in the numerical integrations of the numerical scheme already described are interpolated values at the quadrature points using the polynomial interpolation functions ^ (t). For simple geometries (e.g. circular sections) we know the true x and y coordinates at the quadrature points, and by using these values we remove the interpolation error from the kernel or Green's functions and from the Jacobian. Equation 5 may now be represented as cu + E E (Y ^ re r*=i ^Te = E E (9) where we now use p* and u* instead of p* and u* since we are modelling the exact geometry up to numerical accuracy. Note also that the Jacobian J is not now a function of interpolation functions. 4 Exact Boundary Conditions The function values used in the numerical scheme are also interpolated values at the quadrature points using the shape functions (fri(t). However, for most problems of interest we have analytical expressions for the boundary conditions which can be written as some function of the coordinates of the boundary points. We may therefore use the exact values of the boundary conditions at the quadrature points in order to form the 6 vector in the linear system which we wish to solve. Note that the exact values for the geometry
244 Boundary Elements -0.6-0.58-0.56-0.54-0.52-0.5-0.48-0.46-0.44-0.42-0.4 Figure 4: Close up of interpolant and circle and boundary condition functions may be given at the quadrature points used to evaluate the u* singularities as well as for the Gauss quadrature points. Our code allows the boundary conditions to be denned as symbolic expressions consisting of common functions such as sin(x) or 32 on each section of the boundary. 5 Results In order to analyse the accuracy of the numerical schemes we use two exterior Neumann potential problems. The use of exterior problems ensures that imposing pure Neumann boundary conditions does not result in an underdetermined system. The use of Neumann boundary conditions avoids the necessity to use the row-sum singularity in the right hand side vector b where boundary condition information along the element is reduced to a single boundary condition at the singular node. The particular problems to be considered are those analysed by Kirkup and Henwood [2] which provide varying boundary conditions or geometries that use functions other than polynomials, enabling an effective comparison of the methods. For the two potential problems for which we have analytical expressions for the function values, we use as the error estimator the mean residual error (MRE), defined by 1 " %^# (10) for TV collocation points, with HJ the computed value, and Uj the exact value at the jth collocation point.
Boundary Elements 245 Figure 5: Square Example Multiple runs were performed, using isoparametric and subparametric elements, with the geometry varying from linear to cubic in the interpolant, and the function values ranging from constant to the same order as the geometry shape. Thefirstexample is a potential problem with Neumann boundary condition *± = * (11) dm?r(d2+4) ^ / imposed, where d is the distance from the centre of the side as shown in figure 5. The second example is a potential problem with Neumann boundary condition du 4 - cos(0) (12) dn 27r(17-8cos(<9)) imposed, where 9 is as shown infigure6. In order to minimise quadrature errors the number of Gauss points used was increased until the solution appeared to be stable. Sixteen points were used for the solutions given here. Although results were produced for different configurations of boundary shape functions, geometry shape functions and numbers of nodes, the general trend can be identified by consideration of just one combination of geometry and boundary condition shape functions. In order to provide a good comparison with standard methods, quadratic isoparametric elements were chosen as the polynomial interpolant elements. Since the mean relative error diminishes rapidly logarithmic scales (base 10) were used when plotting. For thefirstexample shown infigure5 the quadratic polynomial is able to interpolate the linear geometry exactly so that the polynomial interpolant and exact geometry methods are the same. As can be seen infigure7 the
246 Boundary Elements Figure 6: Circle Example 0.1 0.01 interpolant/exact geom i exact geom & b.c.s x- 0.001 0.0001 1e-05 1e-06 1e-07 20 40 60 80 Number of Nodes 100 120 140 Figure 7: Square convergence (log^q scale)
Boundary Elements 247 0.01 0.0001 interpolant Hexact geom x- exact geom & b.c.s - -* 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 20 40 60 80 Number of Nodes 100 120 140 Figure 8: Circle convergence scale) case using the exact boundary condition values at the quadrature points has a consistently lower mean relative error than the other two cases with the same accuracy being achieved with just over half the number of elements used in the polynomial interpolant case. For the second example shown in figure 6 neither the geometry or boundary condition function can be exactly represented using polynomial interpolants and therefore all three methods produce different results. Figure 8 shows that using the exact geometry in this case produces better results than using a polynomial interpolant as shown by Kirkup & Henwood and our results for the use of exact geometry agree closely with theirs. However, the best results are produced by using the exact values of geometry and boundary condition function at the quadrature points. Here the mean relative error (and other error measures used) soon approach machine accuracy, in this case a double precision implementation with 16 digits of accuracy, a dramatic improvement in accuracy over the conventional BEM implementation. 6 Conclusions Good convergence can normally be achieved using polynomial interpolants for the geometry of the boundary and the boundary condition function. Using exact values of x and y coordinates at the quadrature points produces increased accuracy when the boundary cannot be modelled exactly using a
24 8 Boundary Elements polynomial interpolant. Where the boundary condition is a known function which cannot be modelled accurately using a polynomial interpolant, the use of exact boundary condition values at the quadrature points produces smaller errors and perhaps dramatically smaller errors. The use of exact values of x and y coordinates together with exact values of the boundary condition function at quadrature points can lead to a reduction in the number of elements required for a given error tolerance. As the matrix assembly process is O(n^) and the linear solver is O(n?), using exact values at quadrature points can lead to a large reduction in solution time for little extra initial computation. There would seem to be a clear need for more research in this area and we intend to consider a wider range of examples if different boundary conditions and also to consider the effect of exact boundary conditions on vector problems. References [1] Smith R.N.L. Specialised Quadrature Schemes for Boundary Integral Methods. First UK Conference on Boundary Integral Methods, pages 266-275, 1997. [2] Kirkup S.M., Henwood D.J. An Empirical Error Analysis of the Boundary Element Method Applied to Laplace's Equation. Applied Mathematical Modelling, 18:62-38, 1994.