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Foundations of Analytical and Numerical
Field Computation
Stephan RussenschuckCERN, TE-MCS, 1211 Geneva, Switzerland
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Permanent Magnet Circuits
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Pole shimming
Rogowski profiles
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Different Incarnations of Maxwell’s Equations
Integral form
Local form
Global form
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Directional Derivative
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The Differential Operators
Conclusion: This is horrible, so let’s try the geometrical approach
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Maxwell’s House
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Maxwell’s Equations in Differential Form
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Maxwell’s House
Would be even more symmetric with magnetic monopoles
Outeroriented
Inneroriented
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Maxwell’s Facade
Only forCartesian components
Constant permeablity and no sources
No sources
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Method of Separation
How do you solve differential equations: Look them up in a book
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Solution of Laplace’s Equation
What have we won? If we know the field at a reference radius, we know it everywhere inside
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Multipoles and Scaling Laws
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Ideal Pole Shape of Conventional Magnets
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Numerical Field Computation
Principles of numerical field computation–
Formulation of the Problem–
Weighted residual–
Weak form–
Discretization–
Numerical exampleTotal vector potential formulation–
Weak form in 3-DElement shape functions–
Global shape functions–
Barycentric coordinatesMesh generation
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The Model Problem (1-D)
or
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Shape Functions
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Shape Functions
Cramer’s rule
What have we won? We can express the field in the element as a function of the node potentials using known polynomials in the spatial coordinates
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The Weighted Residual
What have we won? Removal of the second derivative, a way to incorporate Neumann boundary conditions
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Galerkin’s Method
Linear equation system for the nodepotentials
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Numerical Example
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Numerical Example
Essential boundary conditions (Dirichlet)
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Higher order elements
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Two Quadratic Elements
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Curl-Curl Equation
Problem in 3-D: Gauging
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Weak Form in the FEM Problem
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Weak Form in the FEM Problem
Conclusion: 3-D is more complicated than addition just one dimension in space; it’s a different mathematics, and thus often a separate software package
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Weak Form in the FEM Problem
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Meshing the Coil
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Nodal versus Edge-Elements
Notice: Finer discretization does not help! Use edge-elements, or a different formulation (scalar potential, whenever possible. Remember: This problem does not exist in 2-D
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Total Scalar Potential / Reduced Scalar Potential
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Shape Functions
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Barycentric Coordinates
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Barycentric Coordinates
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Higher Order Elements
Higher accuracy of the field solution, but also better modeling of the iron contour
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Mapped Elements
Use of the same shape functions for the transformation of the elements
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Mapped Elements
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Transformation of Differential Operators
Complicated Easy
But how about the Jacobian being singular?
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Collinear Sides yield Singular Jacobi Matrices
Note: Bad meshing is not a trivial offence
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Topology Decomposition
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Paving and Mesh Closing in Simple Domains
The number of nodes is less than 6
The domian does not contain “bottlenecks” , i.e., C2/a approaches 4π
The biggest inner angle is less then π
For triangles: a+b < c
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Examples for FEM Meshes
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Point Based Morphing
Always use morphing (if available) for sensitivity analysis
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Magnet Extremities
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Reduced Vector Potential Formulation
Advantages: No meshing of the coil, no cancellation errors, distinction between source field and iron magnetization
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Source, Reduced, Total Field
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BEM-FEM Coupling (Elementary Model Problem)
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The FEM Part (Vector Laplace Equation)
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FEM Part
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BEM Part
From Green’s second theorem:
Vector Laplace Weighted Residual
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BEM Part
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BEM-FEM Coupling
FEM
BEM
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Always check convergence of your computation
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