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Finite Element Analysis

Click here to learn about finite integration technique (new for February 2015!)

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This page (and its companions) were contributed by Jackson Jones who is the proprietor of a new EM software tool company, Principles of Prediction, which is just getting started. 

Questions are numbered for reference, please feel free to contact This email address is being protected from spambots. You need JavaScript enabled to view it. if you want more information, or have suggestions for this page!

1. What exactly is a "mesh" and why does it take so much memory?

A mesh boils down to a finite set of points in space. The finite set of points defined by a mesh are the points at which approximate solutions to Maxwell's equations will be obtained. The distance between points in the mesh must be small enough that the approximation of a finite h in the formula for the derivatives is valid. This can take a lot of points and that is why meshes take up so much memory.

2. Should engineers care what it means for h to be small?

Yes. In some circumstances engineers need to be conscientious with the mesh used by a finite element program. In a sense, the value of h used in the formula for the derivatives is related to the density of the mesh. Some finite element software will create a mesh based on a frequency that is input into the program. If a system is designed to operate across a frequency band, or at multiple frequency bands, it can make a difference in both convergence time and accuracy which frequency is chosen for the mesh. Indeed, convergence at one frequency may not mean that all frequencies have converged, and because of the way meshes are created, it can in some instances be faster to mesh at different frequencies for broadband or multi-band systems rather than meshing at one frequency and waiting for all the other frequencies to converge.

3. What are "linear algebraic equations?"

A linear algebraic equation is an equation in which the unknowns are linear. For example x=7 is a linear algebraic equation, but x2=7 is not. Linear Algebra is the study of these types of equations. One can also have a set of linear algebraic equations in multiple unknowns, such as {x+y=2, x-y=0}

4. What is the "matrix form" of linear algebraic equations?

It is a fact of linear algebra that any set of linear algebraic equations can be written using matrices and vectors. Generally, the equations take the form Ax=b, where A is a known matrix, b is a known "solution vector", and x is a vector representing the unknowns to be solved for. For example, the set ofequations {x+y=2,x-y=0} can be written in matrix form as follows:

1 1 x 2

1 -1 y 0

More to come!

 

Author : Jackson Jones

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