Foundations of Analytical and Numerical Field Computation Stephan Russenschuck, CERN-AT-MEL Stephan Russenschuck CERN, TE-MCS, 1211 Geneva, Switzerland 1
Permanent Magnet Circuits 2
Rogowski profiles Pole shimming 3
Different Incarnations of Maxwell s Equations Integral form Global form Stephan Russenschuck, CERN-AT-MEL Local form 4
Directional Derivative 5
The Differential Operators Stephan Russenschuck, CERN-AT-MEL Conclusion: This is horrible, so let s try the geometrical approach 6
Maxwell s House 7
Maxwell s Equations in Differential Form 8
Maxwell s House Inner oriented Outer oriented Stephan Russenschuck, CERN-AT-MEL Would be even more symmetric with magnetic monopoles 9
Maxwell s Facade Constant permeablity and no sources Only for Cartesian components Stephan Russenschuck, CERN-AT-MEL No sources 10
Method of Separation Stephan Russenschuck, CERN-AT-MEL How do you solve differential equations: Look them up in a book 11
Solution of Laplace s Equation Stephan Russenschuck, CERN-AT-MEL What have we won? If we know the field at a reference radius, we know it everywhere inside 12
Multipoles and Scaling Laws 13
Ideal Pole Shape of Conventional Magnets 14
Numerical Field Computation Stephan Russenschuck, CERN-AT-MEL Principles of numerical field computation Formulation of the Problem Weighted residual Weak form Discretization Numerical example Total vector potential formulation Weak form in 3-D Element shape functions Global shape functions Barycentric coordinates Mesh generation 15
The Model Problem (1-D) or 16
Shape Functions 17
Shape Functions Cramer s rule Stephan Russenschuck, CERN-AT-MEL 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 18
The Weighted Residual Stephan Russenschuck, CERN-AT-MEL What have we won? Removal of the second derivative, a way to incorporate Neumann boundary conditions 19
Galerkin s Method Linear equation system for the node potentials 20
Numerical Example 21
Numerical Example Essential boundary conditions (Dirichlet) 22
Higher order elements 23
Two Quadratic Elements 24
Curl-Curl Equation Problem in 3-D: Gauging 25
Weak Form in the FEM Problem 26
Weak Form in the FEM Problem Stephan Russenschuck, CERN-AT-MEL 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 27
Weak Form in the FEM Problem 28
Meshing the Coil 29
Nodal versus Edge-Elements Stephan Russenschuck, CERN-AT-MEL 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 30
Total Scalar Potential / Reduced Scalar Potential 31
Shape Functions 32
Barycentric Coordinates 33
Barycentric Coordinates 34
Higher Order Elements Stephan Russenschuck, CERN-AT-MEL Higher accuracy of the field solution, but also better modeling of the iron contour 35
Mapped Elements Stephan Russenschuck, CERN-AT-MEL Use of the same shape functions for the transformation of the elements 36
Mapped Elements 37
Transformation of Differential Operators Complicated Easy Stephan Russenschuck, CERN-AT-MEL But how about the Jacobian being singular? 38
Collinear Sides yield Singular Jacobi Matrices Stephan Russenschuck, CERN-AT-MEL Note: Bad meshing is not a trivial offence 39
Topology Decomposition 40
Paving and Mesh Closing in Simple Domains The number of nodes is less than 6 The domian does not contain bottlenecks, i.e., C 2 /a approaches 4π The biggest inner angle is less then π For triangles: a+b < c Stephan Russenschuck, CERN-AT-MEL 41
Examples for FEM Meshes 42
Point Based Morphing Stephan Russenschuck, CERN-AT-MEL Always use morphing (if available) for sensitivity analysis 43
Magnet Extremities 44
Reduced Vector Potential Formulation Stephan Russenschuck, CERN-AT-MEL Advantages: No meshing of the coil, no cancellation errors, distinction between source field and iron magnetization 45
Source, Reduced, Total Field 46
BEM-FEM Coupling (Elementary Model Problem) 47
The FEM Part (Vector Laplace Equation) 48
FEM Part 49
BEM Part Vector Laplace Weighted Residual From Green s second theorem: Stephan Russenschuck, CERN-AT-MEL 50
BEM Part 51
BEM-FEM Coupling BEM FEM 52
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Always check convergence of your computation 55