THE VOLUME INTEGRAL METHOD

The fundamental problem in eddy-current NDE is to calculate the scattered field from a flaw within a workpiece. VIC-3D® uses the methods of modern computational electromagnetics to efficiently compute the current in the workpiece by using volume-integral equations. In a volume-integral equation, we determine the field at a point by summing the effects of the sources at all points, weighting them with a Green's function. When the effects are summed over a given region, or volume, of space, we have a volume-integral equation. The same approach is used in order to compute the magnetization of a ferrite core probe.

The advantage of this method, over that of the finite-element method, is the following. In the finite-element method, it is necessary to construct a grid over the flaw as well as the entire region surrounding the flaw, and to solve for the fields at all points on the grid. In contrast, in the volume-integral method, it is only necessary to construct a grid over the flaw and solve for the currents in the flaw; the Green's function takes care of all regions outside the flaw. This removes the complicated gridding requirements of the finite-element method, and reduces the size of the problem tremendously. That is why VIC-3D® can obtain much more accurate probe responses, and in much less time than a finite-element code, while running on a small personal computer or workstation. And without the complicated gridding, problems can be set up much more quickly and easily.

THE CONJUGATE GRADIENT ALGORITHM AND FAST FOURIER TRANSFORMS

VIC-3D® uses a regular grid to subdivide the flaw into cells. Then the problem of computing the probe response is reduced to solving a matrix equation for the electric and magnetic currents in the cells. When discretized on a regular grid, the volume-integral method has the very nice feature that the resulting matrices are Toeplitz or Hankel (the ij'th element is a function only of i-j or i+j).

Because very little information about the matrix has to be stored, iterative methods, such as the conjugate-gradient algorithm, can be efficiently used to solve very large problems using the small memory resources available on small computers like the PC.

In addition to the iterative solver just discussed, VIC-3D® has a direct solver that can be used for problems that will fit into memory. This solver provides additional speed because it can very efficiently process multiple probe positions. The user can choose the solver, giving him the flexibility of obtaining the optimum solution for both large and small problems.