IsoGeometric Analysis: Be zier techniques in Numerical Simulations

Transcription

1 IsoGeometric Analysis: Be zier techniques in Numerical Simulations Ahmed Ratnani IPP, Garching, Germany July 30, / 1 1/1

2 Outline Motivations Computer Aided Design (CAD) Zoology Splines/NURBS Finite Elements Be zier curves Tensor Product surfaces B-Spline curves Be zier triangular surfaces The IsoGeometric Approach Impact of the k-refinement strategy The discrete DeRham diagram Maxwell equations MultiGrid Methods Adaptive meshes Be zier techniques in Computational Plasma Physics 2/ 1 2/1

3 Motivations Engineering Analysis Process Finite Elements Analysis (FEA) models are created from CAD representations Fixing CAD geometry and creating FEA models accounts more than 80% of overall analysis time and is a major engineering bottelneck The geometry is approximated in the FEA mesh 3/ 1 3/1

4 Motivations Engineering Analysis Process Finite Elements Analysis (FEA) models are created from CAD representations Fixing CAD geometry and creating FEA models accounts more than 80% of overall analysis time and is a major engineering bottelneck The geometry is approximated in the FEA mesh 3/ 1 3/1

5 Motivations 4/ 1 4/1

6 Motivations Even if the p-form: C(x(t )) = ni=0 t i Pi, is a natural description for curves, it presents some disadvantages : the curve is not necessary regular everywhere. à non-efficient approximation, the points (Pi )0 i n do not have any geometric interpretation, unstable numerical evaluation. 5/ 1 5/1

7 Motivations Even if the p-form: C(x(t )) = ni=0 t i Pi, is a natural description for curves, it presents some disadvantages : the curve is not necessary regular everywhere. à non-efficient approximation, the points (Pi )0 i n do not have any geometric interpretation, unstable numerical evaluation. Properties à Bezier techniques provide a geometric-based method for describing and manipulating polynomial curves and surfaces. brings sophisticated mathematical concepts into a highly geometric and intuitive form. this form facilitates the creative design process. Bezier techniques are an excellent choice in the context of numerical stability of floating point operations[farouki & Rajan]. à Bezier techniques are at the core of 3D Modeling or Computer Aided Geometric Design (CAGD). 5/ 1 5/1

8 Computer Aided Design Be zier curves Rather than use {1, t,, t n } as a basis of Π<n+1, we can take Bernstein polynomials; this leads to the Be zier-form. Therefore, it is equivalent to the p-form and writes : n C(x(t )) = Bin (t )Pi, 0 t 1 (1) i =0 where Bin denote Bernstein polynomials : Bin (t ) = n i t i (1 t )n i = n! t i (1 t )n i, i! (n i )! 0 t 1 (2) The sequence (Pi )0 i n is called control points. Conics can be descrived exactly using Non-Rational Be zier arcs. 6/ 1 6/1

9 Computer Aided Design Be zier curves 7/ 1 7/1

10 Computer Aided Design Be zier curves 7/ 1 7/1

11 Computer Aided Design Be zier curves 7/ 1 7/1

12 Computer Aided Design Be zier curves 7/ 1 7/1

13 Computer Aided Design Be zier curves 7/ 1 7/1

14 Computer Aided Design Be zier curves Invariance under some transformations : rotation, translation, scaling; it is sufficient to transform the control points, Bin (t ) 0, 0 t 1 partition of unity : ni=0 Bin (t ) = 1, 0 t 1 B0n (0) = Bnn (1) = 1 each Bin has exactly one maximum in [0, 1], at Bn i are symmetric with respect to i n 1 2 recursive property : Bin (t ) = (1 t )Bin 1 (t ) + tbin 11 (t ), and Bin (t ) = 0, if i < 0 or, i > n deriving a curve : C 0 (t ) = n{ ni = 01 Bin 1 (t ) (Pi +1 Pi )}, then : C 0 (0) = n (P1 P0 ) C 00 (0) = n(n 1) (P0 2P1 + P2 ) C 0 ( 1 ) = n ( Pn Pn 1 ) C 00 (1) = n (Pn 2Pn 1 + Pn 2 ) (3) (4) DeCasteljau algorithm: C n (t; P0,, Pn ) = (1 t )C n 1 (t; P0,, Pn 1 ) + t C n 1 (t; P1,, Pn ) (5) 8/ 1 8/1

15 Computer Aided Design Tensor Product surfaces Bezier patchs of arbitrary degrees A Be zier patch of degrees (p, q ) is defined as p,q x(s, t ) = xij Bi (s )Bj (t ), s, t [0, 1] (6) i,j =0 where (xij )0 i p,0 j q are called control points Properties Endpoint interpolation: The patch passes through the four corner control points {(s, t ) = (0, 0), (0, 1), (1, 0), (1, 1)}, Each boundary corredponds to a Bezier curve Symmetry in the parametric domain Affine invariance (applied to the control points) Convex Hull, C 1 patchs can be easily created by solving (local/global) linear systems, using the endpoints derivatives and moving some specific control points, 9/ 1 9/1

16 Computer Aided Design Tensor Product surfaces Geometric Operations (Exact) Subdivision, (Exact) Degree Elevation, (Inexact) Patchs merge, (Exact if a Spline description is used) (Inexact) Degree Reduction Figure : A mapping as a cubic Bezier patch (left) parametric domain with its domain points, (right) the resulting physical domain 10 / 10/11

17 Computer Aided Design B-Splines curves For a fixed number of control points, we have a fixed number of degrees of freedom to control the Be zier curve (= p + 1). For a better control of the curve, one can subdivise it into a given number of Be zier curves (using a refinement algorithm). How to insure that the local regularity of the curve is preserved, when controling these curves? Need the notion of Macro-Elements or Macro-Patchs, where given regularities are imposed between elements. 11 / 11/11

18 Computer Aided Design B-Splines curves (a) (b) (a) Original curve given as Be zier curves. (a) The quadratic B-spline curve and its control points. The knot vector is T = {000, 41, 12, 43, 111}. 12 / 12/11

19 Computer Aided Design B-Splines curves Figure : (left) A quadratic B-Spline curve and its control points using the knot vector T = { }, (right) the corresponding B-Splines. 13 / 13/11

20 Computer Aided Design B-Splines To create a family of B-splines, we need a non-decreasing sequence of knots T = (ti )16i 6N +k, also called knot vector, with k = p + 1. Each set of knots Tj = {tj,, tj +p } will generate a B-spline Nj. Definition (B-Spline serie) The j-th B-Spline of order k is defined by the recurrence relation: Njk = wjk Njk 1 + (1 wjk+1 )Njk+ 11 where, wjk (x ) = x tj tj +k 1 tj Nj1 (x ) = χ[tj,tj +1 [ (x ) for k 1 and 1 j N. 14 / 14/11

21 Computer Aided Design B-Splines 2 1 N1 N2 N3 N4 N5 N6 N7 N N1 N2 N3 N4 N5 N6 N7 N8 0.8 N1 N2 N3 N4 N5 N6 N7 N Figure : B-splines functions associated to the knot vector T = { }, of order k = 1, 2, 3 Figure : Quadratic B-Splines for T = {000, 111}, T = {000, 21, 111} and T = {000, 21, 3 3 }. 4 4, / 15/11

22 Computer Aided Design Be zier triangular surfaces Barycentric coordinates Let T = {v1, v2, v3 } be a non-degenerate triangle in the plance. Then for all point P in the plane, there exists τ = {τ1, τ2, τ3 } such that P = 3i =1 τi vi. τ is unique if one add the normalization constraint 3i =1 τi = 1 (will be assumed during this talk) P T if and only if τ 0 Affine invariance: if the triangle T together with the point P are transformed by an affine transformation, the transformed point has unchanged barycentric coordinates Bernstein polynomials on triangles Let λ be a multi-index such that λ = n, T a triangle, and x a point in the plane, with τ as barycentric coordinates with respect to T. Bernstein polynomials are defined using the barycentric coordinates. Bλn (τ ) = n! τ1 λ1 τ2 λ2 τ3 λ3 λ1!λ2!λ3! (7) Let ξ ijk = iv1 +jvd2 +kv3. The set Dd,T = {ξ ijk, i + j + k = d } is the set of domain-points. 16 / 16/11

23 Computer Aided Design Be zier triangular surfaces Properties of Bernstein polynomials Partition of Unity: λ =n Bλn = 1, Positivity: Bλn (τ ) 0 if and only if τ 0, Bλn has its maximum at τ = λ n (a domain point) Bernstein triangular patchs As for the rectangular case, the Bernstein patch can be defined as x( τ ) = xλ Bλn (τ ), τ [0, 1] (8) λ =n Properties of Bernstein Series Endpoint interpolation: The patch passes through the three corner control points that are x(1, 0, 0) = v1, x(0, 1, 0) = v2, x(0, 0, 1) = v3 Each boundary corredponds to a Bezier curve Symmetry in the parametric domain, Convex Hull Affine invariance (applied to the control points) C 1 patchs can be easily created by solving (local) linear systems, using the endpoints derivatives and moving some specific control points, 17 / 17/11

24 Computer Aided Design Be zier triangular surfaces Geometric Operations (Exact) Subdivision, (Exact) Degree Elevation, (Inexact) Patchs merge, (Exact if a Spline description is used) (Inexact) Degree Reduction c300 ξ 300 c201 ξ 201 ξ 210 x x ξ 120 x ξ 111 x ξ 102 x x ξ 021 x ξ 012 ξ 030 ξ 003 c210 x x c120 x c111 x c102 x x c021 x c012 c030 c003 Figure : (left) Domain points and (right) B-coefficients for a cubic polynomial 18 / 18/11

25 Computer Aided Design Be zier triangular surfaces Figure : Examples of Bezier triangulations (left) quadratic (middle) cubic (right) quadratic, curved triangles 19 / 19/11

26 Splines/NURBS Finite Elements The IsoGeometric Approach F F Q Patch Q K Physical Domain K Patch Physical Domain Grid generation: the use of h/p/k-refinement keeps the mapping F unchanged. Compact support Partition of Unity Affine covariance IsoParametric concept Error estimates in Sobolev norms 20 / 20/11

27 Splines/NURBS Finite Elements The IsoGeometric Approach Refinement strategies Refining the grid can be done in 3 different ways. This is the most interesting aspects of B-splines basis. h-refinement by inserting new knots. It is the equivalent of mesh refinement of the classical finite element method. p-refinement by elevating the B-spline degree. It is the equivalent of using higher finite element order in the classical FEM. k-refinement by increasing / decreasing the regularity of the basis functions (increasing / decreasing multiplicity of inserted knots). the use of k-refinement strategy is more efficient than the classical p-refinement, as it reduces the dimension of the basis. 21 / 21/11

28 Splines/NURBS Finite Elements Impact of the k-refinement strategy In the following table, we show the impact of the k-refinement on the resolution of the Poisson s equation, on a square domain using: B-splines of degree p and minimal regularity (i.e. C 0 ) B-splines of degree p and maximal regularity (i.e. C p 1 ) p=2 p=3 p=5 number C p of d.o.f C number of nnz C p 1 C cpu-superlu C p 1 C cpu-cg C p 1 C Table : Impact of the k-refinement on the resolution of the Poisson equation on a grid for quadratic, cubic and quintic B-splines. 22 / 22/11

29 Splines/NURBS Finite Elements The discrete DeRham diagram Vector fields transformations H 1 (Ω) ı0 H 1 (P ) grad g grad H (curl, Ω) ı1 H (curl, P ) curl g curl H (div, Ω) ı2 H (div, P ) div f div L2 ( Ω ) ı3 L2 (P ) Vector fields approximations H 1 (P ) g grad e 0 Π V H (curl, P ) g curl e1 Π g grad Wcurl H (div, P ) f div L2 (P ) e2 Π g curl Wdiv e3 Π f div X p,p Sα,α 23 / 23/11

30 Splines/NURBS Finite Elements The discrete DeRham diagram Vector fields transformations H 1 (Ω) ı0 H 1 (P ) grad g grad H (curl, Ω) ı1 H (curl, P ) curl g curl H (div, Ω) ı2 H (div, P ) div f div L2 ( Ω ) ı3 L2 (P ) Vector fields approximations H 1 (P ) g grad H (curl, P ) e 0 Π V g curl e1 Π g grad Wcurl H (div, P ) f div L2 (P ) e2 Π g curl Wdiv e3 Π f div X 1,p p,p 1 Sαp 1,α Sα,α 1 23 / 23/11

31 Splines/NURBS Finite Elements The discrete DeRham diagram Vector fields transformations H 1 (Ω) ı0 H 1 (P ) grad g grad H (curl, Ω) ı1 H (curl, P ) curl g curl H (div, Ω) ı2 H (div, P ) div f div L2 ( Ω ) ı3 L2 (P ) Vector fields approximations H 1 (P ) g grad e 0 Π V H (curl, P ) g curl e1 Π g grad Wcurl H (div, P ) f div L2 (P ) e2 Π g curl Wdiv e3 Π f div X p,p 1 p 1,p Sα,α 1 Sα 1,α 23 / 23/11

32 Splines/NURBS Finite Elements The discrete DeRham diagram Vector fields transformations H 1 (Ω) ı0 H 1 (P ) grad g grad H (curl, Ω) ı1 H (curl, P ) curl g curl H (div, Ω) ı2 H (div, P ) div f div L2 ( Ω ) ı3 L2 (P ) Vector fields approximations H 1 (P ) g grad e 0 Π V H (curl, P ) g curl e1 Π g grad Wcurl H (div, P ) f div L2 (P ) e2 Π g curl Wdiv e3 Π f div X 1,p 1 Sαp 1,α 1 23 / 23/11

33 Splines/NURBS Finite Elements Maxwell Equations We shall only consider in the sequel the TE mode which reads E rot H = J, t H + rot E = 0, t div E = ρ. (9) 1st variational formulation Find (E, H ) H0 (curl, Ω) L2 (Ω) such that d dt Z Ω d dt E ψ dx Z Ω Z H (rot ψ) dx = Ω H ϕ dx + Z Ω Z (rot E) ϕ dx = 0 Ω J ψ dx ψ H0 (curl, Ω), ϕ L2 ( Ω ). (10) (11) 2nd variational formulation Find (E, H ) H (div, Ω) H 1 (Ω) such that d dt Z d dt Ω E ψ dx Z Ω H ϕ dx + Z Ω (rot H ) ψ dx = Z Ω E (rot ϕ) dx = 0 Z Ω J ψ dx ψ H (div, Ω), ϕ H 1 ( Ω ). (12) (13) 24 / 24/11

34 Splines/NURBS Finite Elements Maxwell Equations In the case of the 2nd formulation, we look for Eh Wdiv and Hh V. the linear system writes: Without DeRham sequence MW e = Kh MV h = K T e Using DeRham sequence e = Rh MV h = K T e Time discreatization using Leap-Frog scheme 2nd or 4th, Solving only one linear system at each time step, 25 / 25/11

35 Splines/NURBS Finite Elements Maxwell Equations In the case of the 2nd formulation, we look for Eh Wdiv and Hh V. the linear system writes: Without DeRham sequence MW e = Kh MV h = K T e Using DeRham sequence e = Rh MV h = K T e Time discreatization using Leap-Frog scheme 2nd or 4th, Solving only one linear system at each time step, the use of regular elements leads to better CFL numbers. 25 / 25/11

36 Splines/NURBS Finite Elements Maxwell Equations Test on a square domain, e-05 1e quadratic Ch3 cubic Ch4 quartic Ch5 quintic Ch e-05 1e-06 error (L2 norm) error (L2 norm) 1e-07 quadratic Ch2 cubic Ch3 quartic Ch4 quintic Ch5 1e-08 1e-09 1e-07 1e-08 1e-10 1e-09 1e-11 1e-10 1e-12 1e-11 1e h 0.1 1e h 0.1 Figure : Square test: the L2 norm error for (left) the magnetic field, (right) the electric field 26 / 26/11

37 Splines/NURBS Finite Elements Maxwell Equations Test on a square domain, dim W, m=1 dim W, m=2 dim V, m=1 dim V, m= m=1 m=2 1e-05 error (L2 norm) e-06 1e e h 0.1 Figure : Square test: (left) the dimension of the discrete spaces Wh and Vh, (right) the L2 norm error for the electric field, where the vector knots are multiplicity m = 1, 2 for quadratic B-splines 26 / 26/11

38 Splines/NURBS Finite Elements Maxwell Equations p p p p = = = = LF 2Th LF 2num LF 4Th LF 4num Table : Test case 1: CFL numbers (theoretical and numerical values), for splines of degree p = 2,, 5 p p p p = = = = m= m= Table : Test case 1: CFL, using LF4, for splines of degee p = 2,, 5 for singular knots (m = 1), and doubled knots (m = 2) 27 / 27/11

39 Splines/NURBS Finite Elements MultiGrids Methods One can insert a new knot t, where tj 6 t < tj +1. e = N + 1, e = {t1,.., tj, t, tj +1,.., tn +k } Let N ke = k, T and i 6 j k +1 t ti j k +2 6 i 6 j αi = t t i i + k 1 0 j +1 6 i If Q are the new control points after the insertion of the knot t, then Qi = αi Pi + (1 αi )Pi 1 or equivalently Q = AP The basis transformation A is called the knot insertion matrix of degree k 1 from T to e. T Now let us consider a nested sequence of knot vectors T0 T1... Tn, where #(Ti +1 Ti ) = 1. The knot insertion matrix from Ti to Ti +1 is denoted by Aii +1. It is easy to see that the insertion matrix from T0 to Tn is simply: A := An0 = A10 A21... Ann 1 In the case of 2D, the interpolation operator can be constructed using the Kronecker product, as follows Aη Aξ 28 / 28/11

40 Splines/NURBS Finite Elements MultiGrids Methods: Numerical results 29 / 29/11

41 Splines/NURBS Finite Elements MultiGrids Methods: Numerical results 30 / 30/11

42 Splines/NURBS Finite Elements MultiGrids Methods: Numerical results 31 / 31/11

43 Adaptive meshes The r-refinement strategy Basic ideas Idea: Move the control points in order to have a better resolution in high density regions. Iteratively, by minimizing an estimation of the numerical error (see B. Mourrain papers). To ensure an equi-distribution property: needs a monitor function (works also with a posteriori-estimates). Depending on the method, specific conditions (on the boundary) must be hold in order to keep the exact geometry. à Injectivity property: the geometric transformation F must be a one-to-one map : Example of flux aligned mesh, IGA Figure used CEMRACS in MHD / 32/11

44 Adaptive meshes Optimal transport Definition (L2 Monge-Kantorovich problem) Let ρ0 and ρ1 be two given densities of equal masses, defined in Ω Rd. Find a mapping x0 = Ψ(x), x, x0 Ω, that transfers the density ρ0 to ρ1 and minimizes the transport cost J [Ψ] = Z Ω Ψ(x) x 2 ρ0 (x) dx (14) The density transfert means that, ω Ω, Z Ψ 1 ( ω ) ρ0 (x) dx = Z ρ1 (x0 ) dx0 (15) ω If the mapping Ψ is a smooth one-to-one map, then (Eq.??) writes Ψ ρ1 (Ψ(x)) det = ρ0 (x) (16) x 33 / 33/11

45 Adaptive meshes Optimal transport and MA eq. Existence theorem [Brenier 91] There exists a unique optimal mapping that satisfies the equidistibution principle. This mapping can be written as the gradient of a convex function φ φ is the solution of the Monge-Ampe re eq. det H (φ) = σ Ωc ρ( ξ φ) (17) for the boundary conditions, we ensure that x(ξ ) maps Ωc to Ω: ξ φ( Ωc ) = Ω (18) 34 / 34/11

46 Adaptive meshes Monge-Ampe re equation using Benamou-Froese-Oberman Method (BFO) [Benamou2010] Let us define the operator T : H 2 (Ω) H 2 (Ω) by q 2 1 T [u ] = ( 2 u )2 + 2(f det H (u )) (19) when u H 2 (Ω) is a solution of the Monge-Ampe re equation, then it is a fixed point of the operator T. Algorithm Given an initial value u 0, +1 as the solution of Compute u np 2 n + 1 u = ( 2 u n )2 + 2(f det H (u n )) 35 / 35/11

47 Adaptive meshes Anisotropic case, on a square domain 36 / 36/11

48 Adaptive meshes Anisotropic case, on a square domain 37 / 37/11

49 Adaptive meshes Anisotropic case, on a square domain Number of cells Grid [Delzanno08] Error CPU-time p=2 Error CPU [Sulman11] Error CPU-time p=3 Error CPU p=5 Error CPU Table : Example 1: CPU-time needed to solve the Monge-Ampe re equation and the adaptive Error Eadp using the two grids Picard algorithm using 38 / quadratic, cubica. and quintic B-splines. Ratnani 38/11

50 Adaptive meshes Anisotropic case, on a square domain 39 / 39/11

51 Adaptive meshes Anisotropic case, on a square domain Number of cells Grid [Delzanno08] Error CPU-time p=2 Error CPU p=3 Error [Sulman11] Error CPU-time CPU p=5 Error CPU Table : Example 2: CPU-time needed to solve the Monge-Ampe re equation and the adaptive Error Eadp using the Picard algorithm with the two grids 40 / method. 40/11

52 Adaptive meshes Anisotropic case, on a square domain 41 / 41/11

53 Adaptive meshes Anisotropic case, on a square domain 42 / 42/11

54 Adaptive meshes Anisotropic case, on a square domain 43 / 43/11

55 Adaptive meshes Anisotropic case, on a square domain 44 / 44/11

56 Adaptive meshes Anisotropic case, on a square domain 45 / 45/11

57 Adaptive meshes Example on annulus domain The annulus domain is contructed using NURBS. (NURBS curves model all conics) 46 / 46/11

58 Be zier techniques in Computational Plasma Physics Applications Mesh Generation Kinetic models MagnetoHydrodynamics (MHD) Particle In Cell Semi-Lagrangian schemes Equilibrium Non-linear Reduced MHD Full MHD Tomography 47 / 47/11