Download A Projection Transformation Method for Nearly Singular by Ken Hayami PDF

By Ken Hayami

In 3 dimensional boundary point research, computation of integrals is a crucial point because it governs the accuracy of the research and likewise since it frequently takes the foremost a part of the CPU time. The integrals which ascertain the impression matrices, the interior box and its gradients comprise (nearly) singular kernels of order lIr a (0:= 1,2,3,4,.··) the place r is the gap among the resource element and the mixing aspect at the boundary point. For planar parts, analytical integration can be attainable 1,2,6. even though, it really is changing into more and more very important in useful boundary point codes to take advantage of curved components, similar to the isoparametric components, to version normal curved surfaces. given that analytical integration isn't really attainable for normal isoparametric curved components, one has to depend upon numerical integration. whilst the space d among the resource aspect and the point over which the combination is played is adequately huge in comparison to the aspect measurement (d> 1), the normal Gauss-Legendre quadrature formulation 1,3 works successfully. even though, whilst the resource is really at the point (d=O), the kernel 1I~ turns into singular and the easy program of the Gauss-Legendre quadrature formulation breaks down. those integrals could be referred to as singular integrals. Singular integrals ensue while calculating the diagonals of the impression matrices.

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2- (r '11 + 2 A r '12 + A r '22 ) . D. 6) are only weakly singular. They can be calculated efficiently using polar coordinates on the plane tangent at Xs, as will be shown by numerical experiments in Chapter 7. For the case when e' = e but k:;t: l, it is shown in the following that the numerators of the integral kernels also become zero at the source point xe' , so that the integrals gee kl and hee kl have a even weaker singularity compare to gee" and hee". 42) and let the source xek be x(O, 0).

43) 0. represents the Dirichlet boundary condition and q the Neumann boundary condition. 44) . 44) is solved for the boundary values u and q, the potential u(xs) at an internal point Xs can be calculated by U (xs ) n =i i. (1J 1 ,1J 2) x! 55) dTJ 1 dTJ 2 ,and n is the unit outward normal vector of the boundary element Se at x E Se. 4. 13) when Xs is a node shared by two or more elements as shown in Fig. 8 involves the calculation of the solid angle w subtended by the region V at Xs on S. In order to avoid this, one may use the row sum elimination method in many cases.

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