Electromagnetic Modeling Of Large Structures Using Integral Equations

Application of electromagnetic techniques to high resolution geotechnical problems will require<br>effective means of calculating the response of complex models having many cells of possibly<br>differing conductivity. To address this need, we have developed an electromagnetic modeling<br>algorithm for large conductivity structures based on the method of system iteration and spatial<br>symmetry reduction using integral equations.<br>Electromagnetic scattering of conductivity structures using the integral equation method has<br>been widely used in geophysical applications [l]-[8]. Recently, [9] introduced the method of<br>system iteration, where a scatterer is divided into substructures and the direct matrix inversion<br>is applied to each substructure only, while the mutual interactions among the substructures<br>introduce equivalent sources. This technique greatly reduces the matrix factorization time and<br>storage requirements. Using the lateral homogeneity of a layered earth, [lo] developed a spatial<br>symmetry reduction scheme which drastically reduces the computation time for forming the<br>scattering impedance matrix by identifying and reducing the redundancy of calculating the<br>scattering matrix elements. This scheme permits enhanced use of the method of system iteration.<br>The method of system iteration is applicable to arbitrary scatterers with arbitrary discretizations.<br>It divides a structure into many substructures and solves the resulting matrix equation<br>using a block iterative method. The block-submatrices usually need to be stored on disk in<br>order to save computer core memory. However, this requires a large disk for large structures. If<br>the body is discretized into equal size cells, it is possible to use the spatial symmetry relations<br>of the Green’s functions to re-generate the scattering matrix in each iteration, thus avoiding<br>the expensive disk storage. This will allow us to calculate the responese of models comprised<br>of tens of thousands of cells on workstation type of computers. Numerical tests show that the<br>algorithm effectively reduces the solution of the modeling problem to an order of O(N2), instead<br>of O(N³), as with direct solvers.