![]() In order to verify our simulation code, we compare our results with those reported for two test models. Each arrow represents a single vector shape function on each edge of the element. Illustration (b) shows a quadratic edge-based element. Each node (solid circles) has three field components. Illustration (a) shows a linear node-based element. Finally, using Galerkin's method, we formulate equation (2) into the linear system of equations which are solved iteratively for A using the conjugate gradient method.Ī schematic of the difference between the node-based and the edge-based elements in a tetrahedral cell. For these reasons, we adopt edge elements for our FEM simulator. Namely, the edge elements guarantee the continuity only along the tangential component. In the edge-based FEM, since the three components at the element's nodes are replaced by a single vector shape function defined on each edge of the elements, the tangential component on an edge is continuous across all the element boundaries. Figure 1b is an example of an edge element. This new FEM approach uses edge elements and assigns the degrees of freedom to the edges rather than to the nodes of the element. Recently, a revolutionary approach has been developed by Nedelec. To avoid this conflict, the node-based FEM requires special care in enforcing the boundary conditions. This usual requirement in the node-based FEM is in conflict with the discontinuity of the normal magnetic potential across the conductivity boundaries. At its nodes, node-based elements impose continuity across element boundaries in all three spatial components. The node-based element ( Figure 1a) is usually used in FEM calculations. Consider now which element is a conformity in the FEM analysis. The complete details of the edge-based FEM is reported in the EM engineering literature. We solve for the magnetic vector potential A in the whole computational domain. The simulator proposed here is based on the edge-based FEM. We develop a new 3-D simulation code using a different method from the solvers reported so far. It is of great significance to test such forward solvers for a 3-D model before extending them to a realistic complex structure. The difficulty in developing a 3-D forward solver, however, is in the evaluation of the solution towards a realistic complex 3-D Earth model. ![]() Various special solutions have also been devised, including quasi-analytical solutions. In order to map such large-scale electrical conductivity heterogeneities, recently several numerical forward solvers for global EM induction in a 3-D sphere have been devised. Joint interpretation of the seismic and electric structures will extend our knowledge of the Earth's interior. In view of the sensitivity of conductivity to temperature, partial fraction of melt, composition and volatiles, determination of the 3-D electrical conductivity image of the Earth's interior is essential. Other geophysical parameters, such as heat flux and lithosphere thickness, also suggest that the presence of such 3-D electrical conductivity is to be expected. Furthermore, Schultz and Larsen makes it evident that there exists lateral conductivity variations at upper and mid-mantle depths. This inconsistency strongly suggests that the data are influenced by lateral heterogeneity in the internal electrical conductivity structure. In global EM induction studies, 1-D electrical conductivity structures are often estimated by various workers, however, the several models do not seem to be consistent with each other. Recent studies using global and regional seismic tomography indicate large-scale lateral heterogeneity in seismic velocities of the Earth's mantle.
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