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Introduction to the Finite Element Method in by Anastasis Polycarpou

By Anastasis Polycarpou

This sequence lecture is an advent to the finite point procedure with purposes in electromagnetics. The finite point strategy is a numerical approach that's used to resolve boundary-value difficulties characterised by means of a partial differential equation and a suite of boundary stipulations. The geometrical area of a boundary-value challenge is discretized utilizing sub-domain components, known as the finite components, and the differential equation is utilized to a unmarried aspect after it truly is delivered to a ''weak'' integro-differential shape. a collection of form services is used to symbolize the first unknown variable within the aspect area. a collection of linear equations is acquired for every point within the discretized area. an international matrix method is shaped after the meeting of all parts. This lecture is split into chapters. bankruptcy 1 describes one-dimensional boundary-value issues of functions to electrostatic difficulties defined through the Poisson's equation. The accuracy of the finite point procedure is evaluated for linear and better order components through computing the numerical blunders according to diverse definitions. bankruptcy 2 describes two-dimensional boundary-value difficulties within the components of electrostatics and electrodynamics (time-harmonic problems). For the second one classification, an soaking up boundary situation was once imposed on the external boundary to simulate undisturbed wave propagation towards infinity. Computations of the numerical errors have been played that allows you to review the accuracy and effectiveness of the tactic in fixing electromagnetic difficulties. either chapters are observed via a few Matlab codes which might be utilized by the reader to unravel one- and two-dimensional boundary-value difficulties. those codes could be downloaded from the publisher's URL: This lecture is written essentially for the nonexpert engineer or the undergraduate or graduate scholar who desires to study, for the 1st time, the finite point procedure with functions to electromagnetics. it's also particular for study engineers who've wisdom of alternative numerical options and need to familiarize themselves with the finite aspect technique. The lecture starts off with the fundamentals of the tactic, together with formulating a boundary-value challenge utilizing a weighted-residual process and the Galerkin strategy, and maintains with enforcing all 3 forms of boundary stipulations together with soaking up boundary stipulations. one other very important subject of emphasis is the advance of form services together with these of upper order. In basic phrases, this sequence lecture offers the reader with all info useful for somebody to use effectively the finite aspect technique to one- and two-dimensional boundary-value difficulties in electromagnetics. it truly is compatible for novices within the box of finite parts in electromagnetics.

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Extra resources for Introduction to the Finite Element Method in Electromagnetics

Sample text

9 FINITE ELEMENT SOLUTION OF THE ELECTROSTATIC BOUNDARY-VALUE PROBLEM After going through the major steps of the FEM, we are now in the position to use this powerful numerical method to solve the electrostatic BVP at hand. 6: Discretization of the domain using four linear finite elements compute the electric potential distribution between two parallel plates separated by a distance d and positioned normal to the x-axis. The leftmost plate is maintained at a constant potential V0 whereas the rightmost plate is grounded.

This is demonstrated by plotting the electric field distribution between the two parallel plates for a two-element mesh and a four-element mesh using quadratic shape functions. 17(b), respectively. Once again, the finite element solution, for both the electrostatic potential and the electric field, becomes more accurate as the number of quadratic elements increases. The accuracy of the numerical solution can be evaluated by computing the numerical error, based either on the area bounded between the two curves or the definition of L2 norm, as a function of the number of quadratic elements.

As was indicated in Chapter 1, these interpolation functions must satisfy certain key requirements. First, they must guarantee continuity of the primary unknown quantity across interelement boundaries. Second, they must be at least once differentiable since the governing differential equation at hand is of second order and, third, they must be complete polynomials to provide sufficient representation of the solution’s behavior in the finite element domain. Initially, we will concentrate on the development of interpolation functions based on linear/bilinear elements and, then, move on to higher order elements based on the Lagrange polynomials.

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