ET04.pdf

Algebraic equations in n unknown values of the

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algebraic equations in N unknown values of the magnetic vector potential for the entire solution region. In this way, a repeatable process is performed on each element, a process well adapted for automatic assembly on a computer. The size of the elemental matrix is 4 × 4 (for a four-node element) and the individual contributions to the elemental matrix are summarized below: (135) (136) r N N dv i j i j v , = ωσ s N x N x N y N y dv i j i j v i j , = + 1 µ ( ) = F A A i 0 2 1 1 1 1 π ξ η ξ η r f d d + + ∫ ∫ ( ) , + + ∫ ∫ ( ) 1 1 1 1 f d d ξ η ξ η , dv r drdz J r d d = = [ ] 2 2 π π ξ η dv dx dy J d d = = [ ] ξ η = [ ] N x N y J N N i i i i 1 ξ η = + N N x x N y y i i i η η η = + N N x x N y y i i i ξ ξ ξ = = A y N y A i i i 1 4 = = A x N x A i i i 1 4 A N A i i i ξ η ξ η , , ( ) = ( ) = 1 4 105 Modeling of Electromagnetic Testing
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(137) where the indices i,j vary from 1 to 4. Each coefficient is numerically integrated over the volume of the element using gaussian quadrature 157 and then summed into a complex elemental matrix of the form: (138) where { a } is the 4 × 1 vector of unknowns, { q } is the source vector, [ r ] is the imaginary part and [ s ] is the real part of the elemental matrix. The elemental matrices of all the elements in the solution region are summed into a global system of the form: (139) where there are a total of N equations in N unknowns, N being the total number of nodes in the solution region. This matrix is symmetric and banded. The bandwidth depends on the number of elements, the number of nodes per element and especially on the way the nodes are numbered. The system in Eq. 139 can be solved by any standard solution technique (such as gauss elimination or the conjugate gradient technique) to yield the nodal values of the magnetic vector potential. Boundary Conditions The field equations (formulated in terms of finite elements in Eq. 139 for eddy current problems) can only be solved provided a correct set of boundary conditions is specified. Either dirichlet boundary conditions (for which the function A is known on the boundary) or neumann boundary conditions (for which the first derivative of A is known) can be specified. In the finite element analysis of magnetic field problems, it is more convenient to specify dirichlet boundary conditions because the global matrix in Eq. 139 can accommodate the function value A but not its derivative. Moreover, the neumann boundary conditions are implicit in the formulation in Eq. 139 and need not be specified.
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  • Fall '19
  • Wind, The Land, Magnetic Field, Dodd, Modeling of Electromagnetic Testing

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