Foundations of Analytical and Numerical Field Computation

Transcription

1 Foundations of Analytical and Numerical Field Computation Stephan Russenschuck, CERN-AT-MEL Stephan Russenschuck CERN, TE-MCS, 1211 Geneva, Switzerland 1

2 Permanent Magnet Circuits 2

3 Rogowski profiles Pole shimming 3

4 Different Incarnations of Maxwell s Equations Integral form Global form Stephan Russenschuck, CERN-AT-MEL Local form 4

5 Directional Derivative 5

6 The Differential Operators Stephan Russenschuck, CERN-AT-MEL Conclusion: This is horrible, so let s try the geometrical approach 6

7 Maxwell s House 7

8 Maxwell s Equations in Differential Form 8

9 Maxwell s House Inner oriented Outer oriented Stephan Russenschuck, CERN-AT-MEL Would be even more symmetric with magnetic monopoles 9

10 Maxwell s Facade Constant permeablity and no sources Only for Cartesian components Stephan Russenschuck, CERN-AT-MEL No sources 10

11 Method of Separation Stephan Russenschuck, CERN-AT-MEL How do you solve differential equations: Look them up in a book 11

12 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

13 Multipoles and Scaling Laws 13

14 Ideal Pole Shape of Conventional Magnets 14

15 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

16 The Model Problem (1-D) or 16

17 Shape Functions 17

18 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

19 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

20 Galerkin s Method Linear equation system for the node potentials 20

21 Numerical Example 21

22 Numerical Example Essential boundary conditions (Dirichlet) 22

23 Higher order elements 23

24 Two Quadratic Elements 24

25 Curl-Curl Equation Problem in 3-D: Gauging 25

26 Weak Form in the FEM Problem 26

27 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

28 Weak Form in the FEM Problem 28

29 Meshing the Coil 29

30 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

31 Total Scalar Potential / Reduced Scalar Potential 31

32 Shape Functions 32

33 Barycentric Coordinates 33

34 Barycentric Coordinates 34

35 Higher Order Elements Stephan Russenschuck, CERN-AT-MEL Higher accuracy of the field solution, but also better modeling of the iron contour 35

36 Mapped Elements Stephan Russenschuck, CERN-AT-MEL Use of the same shape functions for the transformation of the elements 36

37 Mapped Elements 37

38 Transformation of Differential Operators Complicated Easy Stephan Russenschuck, CERN-AT-MEL But how about the Jacobian being singular? 38

39 Collinear Sides yield Singular Jacobi Matrices Stephan Russenschuck, CERN-AT-MEL Note: Bad meshing is not a trivial offence 39

40 Topology Decomposition 40

41 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

42 Examples for FEM Meshes 42

43 Point Based Morphing Stephan Russenschuck, CERN-AT-MEL Always use morphing (if available) for sensitivity analysis 43

44 Magnet Extremities 44

45 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

46 Source, Reduced, Total Field 46

47 BEM-FEM Coupling (Elementary Model Problem) 47

48 The FEM Part (Vector Laplace Equation) 48

49 FEM Part 49

50 BEM Part Vector Laplace Weighted Residual From Green s second theorem: Stephan Russenschuck, CERN-AT-MEL 50

51 BEM Part 51

52 BEM-FEM Coupling BEM FEM 52

53 53

54 54

55 Always check convergence of your computation 55