Isogeometric modeling of Lorentz detuning in linear particle accelerator cavities

Transcription

1 Isogeometric modeling of Lorentz detuning in linear particle accelerator cavities Mauro Bonafini 1,2, Marcella Bonazzoli 3, Elena Gaburro 1,2, Chiara Venturini 1,3, instructor: Carlo de Falco 4 1 Department of Computer Science, Università degli Studi di Verona, Italy 2 Department of Mathematics, Università degli Studi di Trento, Italy 3 Laboratoire J.A. Dieudonné, Université Nice Sophia Antipolis, France 4 MOX, Politecnico di Milano, Italy PhD Modelling Week September 10, 2016

2 Outline 1 Multiphysics problem 2 Mathematical formulation The electromagnetic problem The elasticity problem 3 The discretization IGA 4 Conclusions

3 Outline 1 Multiphysics problem 2 Mathematical formulation 3 The discretization 4 Conclusions

4 Linear Particle Accelerator (Linac) New International Linear Collider (ILC) project (planning stage) Structure design: two linacs throw particles toward each other at nearly the speed of light. Copyright c ILC, all rights reserved. Rendered and authored by Rey.Hori

5 Linear Particle Accelerator (Linac) The accelerator is a sequence of several TESLA cavities. Copyright c ILC, all rights reserved. Rendered and authored by Rey.Hori

6 Linear Particle Accelerator (Linac) The TESLA cavities are composed of 9 niobium cells. Copyright c ILC, all rights reserved. Rendered and authored by Rey.Hori

7 Electric field of the TM 010 π-mode Like a vibrating string the cavity can operate at different frequencies and the field will have different shapes (modes) By selecting the frequency one selects the operating mode To achieve acceleration the fundamental Transverse Magnetic (TM) mode is used

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45 Multiphysics model Relevant phenomenon: Lorentz detuning The electromagnetic field creates a pressure on the cavity wall They are slightly ( 10nm) deformed Shift of the frequency Electromagnetic problem in Ω C Linear elasticity problem in Ω W Γ CW Ω W Γ ext Ω W p Γ ext Γ CW Γ W Γ W Γ W Γ W undeformed shape Γ C Ω C Γ C Γ C Γ C Γ W Γ W Γ W Γ W deformed shape Γ CW Γ ext Γ ext

46 Outline 1 Multiphysics problem 2 Mathematical formulation The electromagnetic problem The elasticity problem 3 The discretization 4 Conclusions

47 Electromagnetic problem Γ W Γ C Γ W Γ CW Ω W Ω C Γ ext Γ W Γ C Γ W Γ CW Γ ext Maxwell s eigenvalue problem ( ) 1 E = ω 2 ε 0 E in Ω C µ 0 E ( n c = 0 ) on Γ CW 1 E n c = 0 on Γ C µ 0 electric field E eigenvector, angular frequency ω eigenvalue metallic boundary conditions on Γ CW zero tangential component homogeneous Neumann boundary conditions on Γ C due to the symmetry ε 0 electric permittivity, µ 0 magnetic permeability of vacuum Our application: mode E 0 with the smallest frequency ω 0

48 The second order time-harmonic Maxwell s equation The equation can be derived from the classical system of Maxwell s equations: ε E t H + σe = 0 µ H t + E = 0 (εe) = ρ (µh) = 0 (Ampère-Maxwell theorem) (Faraday s law) (Gauss theorem) (Gauss theorem for magnetism) assuming that the initial conditions already satisfy the two Gauss theorems they are automatically satisfied for all time, using the first two equations we obtain ( ) 1 µ E + ε 2 E t 2 + σ E t = 0, by assuming E(x, t) = Re(Ẽ(x)e iωt ), σ = 0, ε = ε 0, µ = µ 0, we get ( ) 1 Ẽ ω 2 ε 0 Ẽ = 0. µ 0

49 Electromagnetic problem Variational formulation Dropping the symbol ( ) 1 E ω 2 ε 0 E = 0 µ 0 we multiply each term by a (vector) test function v V = {v H(curl, Ω C ), v n = 0 on Γ CW } (H(curl, Ω C ) is the space of square integrable functions whose curl is also square integrable) we integrate using for the second term the integration by parts formula ( u) v dx = ( v) u dx (u n) v dσ. Ω C Ω C Γ We obtain [( 1 ) ] ( 1 ) µ E ( v) ω 2 ε 0 E v dx Γ µ E n v dσ = 0. Ω C

50 Electromagnetic problem Variational formulation Dropping the symbol ( ) 1 E ω 2 ε 0 E = 0 µ 0 we multiply each term by a (vector) test function v V = {v H(curl, Ω C ), v n = 0 on Γ CW } (H(curl, Ω C ) is the space of square integrable functions whose curl is also square integrable) we integrate using for the second term the integration by parts formula ( u) v dx = ( v) u dx (u n) v dσ. Ω C Ω C Γ We obtain Ω C [( 1 ] E) ( v) ω 2 ε 0 E v dx = 0. µ 0

51 Electromagnetic problem elasticity problem Radiation pressure on Γ CW produced by the electromagnetic field: p = 1 4 ε 0 (E 0 n c ) (E 0 n c ) µ 0 (H 0 n c ) (H 0 n c ) It depends on the normal component of the electric field, the tangential component of the magnetic field. It is a time-constant value.

52 Linear elasticity problem Ω W p Γ CW Γ ext (2η (S) u + λi u ) = 0 u = 0 in Ω W on Γ W Γ W Γ W ( 2η (S) u + λi u ) n w = p n w on Γ ( CW 2η (S) u + λi u ) n w = 0 on Γ ext Γ C Γ C Γ W Γ W u displacement of each point η, λ Lamé parameters of the material Γ ext (S) u = 1 2 ( u + ut ) symmetric gradient

53 Deformed cavity undeformed shape Compute the deformation deformed shape Ω W = {x + u (x), x Γ W } Γ CW = {x + u Γ CW (x), x Γ CW } Γ C Γ CW Γ CW Ω C Γ C Solve again the Maxwell s eigenvalue problem but in the deformed cell Ω C Compute the shift of the frequency (Lorentz detuning) ω 0 = ω 0 ω 0

54 Outline 1 Multiphysics problem 2 Mathematical formulation 3 The discretization IGA 4 Conclusions

55 Numerical methods Strategy Geometry definition: Choose a representation of the domain Galerkin approach: Choose a discrete representation of the functional spaces where we look for our unknowns (standard example: Finite elements) In particular: Take into account the physical properties of the fields (example: tangential continuity of E ) Γ CW = {x + u Γ CW (x), x Γ CW } elasticity problem: structural problem, the unknown u is the displacement of the geometry

56 Isogeometric analysis (IGA) The deformed domain is obtained by adding the displacement to the original domain Ω W = Ω W + u Approximated geometry + Finite elements CAD exact geometry + Finite elements + interpolation CAD exact geometry + NURBS (IGA) FEM geometry CAD geometry Idea: use the CAD basis functions (B-Splines, NURBS) not only for the geometry but also for the discretization space!

57 Isogeometric analysis (IGA) B-splines Given a non-uniform knot vector {ξ 1,..., ξ n+p+1}, in the parametric domain, B-spline basis functions are: { 1 if ξi ξ < ξ i+1 B i,0 (ξ) = 0 otherwise. B i,p (ξ) = ξ ξ i B i,p 1 (ξ) + ξ i+p+1 ξ B i+1,p 1 (ξ). ξ i+p ξ i ξ i+p+1 ξ i+1 1 B 1,0 1 B 2,0 1 B 3, B 1,1 1 B 2,1 1 B 3, B 1,2 1 B 2,2 1 B 3,

58 Isogeometric analysis (IGA) B-splines B i,p (ξ) = ξ ξ i B i,p 1 (ξ) + ξ i+p+1 ξ B i+1,p 1 (ξ). ξ i+p ξ i ξ i+p+1 ξ i+1 1 B 1,2 1 B 2,2 1 B 3, The support of B i,p is (ξ i, ξ i+p+1 ) The functions B i,p are non-negative and enjoy the partition of unity property The derivative of a B-spline is still a B-spline

59 Isogeometric analysis (IGA) NURBS - Non-Uniform Rational B-splines A NURBS curve in R2 is the projection of a B-spline in R3 C (ξ) = n X i=1 Bonafini, Bonazzoli, Gaburro, Venturini n X wi Bi,p (ξ) Ci Pn = Ci Ni,p (ξ) ˆ (ξ) ˆ wiˆbi,p i=1 i=1 Particle accelerator cavities 10/09/2016

60 Isogeometric analysis (IGA) TESLA geometry definition - computed via the NURBS package To construct the TESLA cavity geometry: NURBS toolbox. It is a collection of routines for Octave and Matlab for the creation and manipulation of NURBS, developed by M. Spink in From 2009, extended and maintained by Carlo de Falco and Rafael Vázquez.

61 Isogeometric analysis (IGA) The Electric field - computed via the GeoPDEs package To solve the equations: GeoPDEs Free package for Octave and Matlab for solving PDEs using the IGA. Originally (2009) developed in Pavia, by C. de Falco, A. Reali, and R. Vázquez. We employed this package to reproduce some of the results obtained by J. Corno. The field lines for the electric (left) and magnetic (right) field. The solutions have been rescaled so that the total energy is 1J.

62 Isogeometric analysis (IGA) The deformation Deformation of the cavity walls (rescaling factor ).

63 Outline 1 Multiphysics problem 2 Mathematical formulation 3 The discretization 4 Conclusions

64 Summary structure and physics of a Linear Collider, TESLA cavity and the multiphysics problem, mathematical formulation through a coupled electromagnetic and elastic problem Maxwell s equations double curl time-harmonic PDE for (E 0, ω 0 ) elasticity deformation due to radiation pressure displacement u Lorentz detuning initial cavity corresponding ω 0 radiation pressure elastic deformation of the walls new deformed cavity new ω 0 detuning effect ω 0 ω 0 isogeometric analysis: same framework used for both geometry and numerical analysis, GeoPDEs and NURBS Octave packages allow an easy implementation.

65 Outlook First step: solve the eigenvalue problem in the initial cavity (E 0, ω 0 ) extract E b = E ΓC use that as BC to excite the mode ω 0 Second step: 1 set Ω 0 C = Ω C 2 find E V 0 = {v H(curl, Ω 0 C ), v n = 0 on Γ CW } s.t. ( ) 1 (E + E 0 b Ω 0 µ ) ( v) dx ω0ε 2 0 (E + E 0 b ) v dx = 0 C 0 Ω 0 C for all v V 0, with E 0 b an extension of the boundary condition over Ω0 C 3 determine the deformed cavity Ω 1 C (note that Γ C won t move because u = 0 on Γ C anyway) 4 repeat from 2 with the new cavity using the same ω 0

66 Outlook sequence of domains Ω 0 C, Ω1 C, Ω2 C, Ω3 C... natural question: does this sequence converge? if yes, what about Ω inf? What about the resulting field E inf? is this convergence fast? Thank you for your attention!

67 Outlook sequence of domains Ω 0 C, Ω1 C, Ω2 C, Ω3 C... natural question: does this sequence converge? if yes, what about Ω inf? What about the resulting field E inf? is this convergence fast? Thank you for your attention!

68 References J. Cottrell, T. Hughes, Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, C. de Falco, A. Reali, R. Vázquez. GeoPDEs: a research tool for Isogeometric Analysis of PDEs. Adv. Engrg. Softw., 42(12): , R. Vázquez. A new design for the implementation of isogeometric analysis in Octave and Matlab: GeoPDEs 3.0. Comput. Math. Appl., 72(3): , J. Corno, C. de Falco, H. De Gersem, S. Schöps. Isogeometric simulation of Lorentz detuning in superconducting accelerator cavities. Computer Physics Communications, 201, 1-7, 2016.