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Finite difference technique the finite difference

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Finite Difference Technique The finite difference technique has been used as a general means to solve partial differential equations. The reasons for its widespread application are many. The technique is relatively easy to apply, as well as being general. It is equally applicable to direct current fields, to quasistatic or transient fields and to linear and nonlinear problems. In its simplest form, the formulation of the field equations consists of simply replacing the partial derivatives by appropriate difference formulas. A solution can then be obtained for the dependent variable at discrete points within the solution region either by an iterative process or by the solution of a system of algebraic equations, depending on which finite difference formula is applied. The application of the finite difference technique is complicated by problems of convergence and stability of the solution as well as by restrictions on the discretization process. Although regular sets of discretization points (grids) are easy to handle, irregular grids are not. Discretization of complex geometries into regular grids is not practical and irregular grids may in some cases make the solution nonconvergent. In field problems, the inability to properly discretize small areas (such as air gaps or discontinuities) is detrimental to the finite difference technique. In addition, the technique is a nodal technique and cannot take into account distributed parameters such as current densities, conductivities and permeability. These have to be described as equivalent nodal quantities with all the associated errors. The obtained solution is valid only at the nodal points. Finite Difference Representation If an attempt is made to solve a partial differential equation such as Eq. 100 or 101, it should be possible either to integrate the equation or to represent the partial derivatives in terms of the unknowns themselves at discrete points in space. The finite difference algorithm is an implementation of the second approach. Considering Fig. 20, where a general function is described, the true derivative dy ·( dx ) –1 at a point x i is the tangent to the curve at this point. An approximation to the derivative can be found by taking two 97 Modeling of Electromagnetic Testing F IGURE 20. General function and finite difference approximation to true derivative. y y = f ( x ) Tangent at x i Approximate derivative at x i x i x x i x i + x x
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points, (one point on each side of x i ) and passing a straight line through them. If the two points are chosen to be equally spaced about point x i (as in Fig. 20), the following expression for the slope of the line can be obtained: (106) By denoting in short form y’ ( x i ) as y’ i , y ( x i + dx ) as y ( i +1) and y ( x i dx ) as y ( i –1) , a simpler expression linking the approximation y’ ( x i ) to the function value at x ( i –1) and x ( i +1) can be written: (107) The same result can be obtained formally by using a taylor series expansion. By expanding the function y = f ( x ) about the point x i for x = x i x and x = x i +
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  • Fall '19
  • Wind, The Land, Magnetic Field, Dodd, Modeling of Electromagnetic Testing

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