Automatic mesh generators for a variety of element

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Automatic mesh generators for a variety of element shapes are available and offer a simple and efficient way to define the necessary input data. 166,167 The details of the discretization process are discussed in the literature. 157 The following steps and assumptions are general and applicable to the discretization of a region into finite elements. 1. The solution region is subdivided into finite elements. The number and shape of the elements are not restricted in any way. The element density must be chosen for the geometry of the region and expected gradients in the solution. Small, dense elements must be used in regions of high curvature or high gradients. 2. Material interfaces within the solution region and on the boundaries must coincide with element boundaries. An element cannot cover more than one material. 3. The current density and each conductivity and permeability component are assumed to be constant within the element. Calculated quantities are either nodal values, as in the case for the magnetic vector potential, or quantities associated with the element such as flux densities or energy. In this case, the calculated value is associated with the volume (energy) or with the centroid of the element (flux density). 4. At the outer boundaries of the solution region, the magnetic vector potential is either zero (by ensuring that the discretized region extends far enough to have negligible flux density on the boundary) or is otherwise prescribed from known or calculated conditions. 5. The discretizations for two-dimensional and axisymmetric geometries are identical. The difference between the two formulations manifests itself in the integration over the element volume and in the form of the functional itself. Finite Element Formulation The discretization of the solution region is a geometrical procedure and by itself is not sufficient to ensure that the chosen elements can be used for finite element solution. A set N i of special functions, called interpolating or shape functions, must be chosen for the element: 143 (122) where A i is the nodal vector potential (weber per meter) and n is the number of nodes in the element. These must meet two conditions to ensure convergence of the solution as the size of the elements decreases: (1) at element interfaces C r , continuity must be maintained (compatibility requirement); and (2) within an element C r +1 , continuity must be met (completeness requirement). In these conditions, r is the necessary continuity order of the function at the element boundaries and C r continuity means that the function and its first r derivatives are continuous. For field problems formulated in terms of the magnetic vector potential, only C 0 continuity is necessary, meaning that only the function is continuous. Its first derivatives define the flux density components, which are not necessarily continuous.
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  • Fall '19
  • Wind, The Land, Magnetic Field, Dodd, Modeling of Electromagnetic Testing

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