High order FEM vs. IgA by J.A.Evans, T.J.Hughes and A.Reali, 2014

Transcription

1 High order FEM vs. IgA by J.A.Evans, T.J.Hughes and A.Reali, 2014 Rainer Schneckenleitner JKU Linz Jan 20, 2017 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

2 Overview 1 Introduction 2 The Elliptic eigenvalue problem 3 The Elliptic boundary value problem 4 The Parabolic initial-value problem 5 The Hyperbolic initial-value problem 6 Own results Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

3 Introduction Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

4 Introduction Spectral approximation properties of FE and B-splines Investigation of eigenvalue approximation in former papers Good approximation quality of B-spline FE approximation diverged with p Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

5 Introduction What are the eects of this results to BVP and IVP? This question will be answered throughout this presentation Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

6 Introduction Need approximations from a global perspective Pythagorean eigenvalue error theorem pertains to all modes Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

7 Introduction Only the Laplace operator is considered Domain Ω R d is bounded and connected Ω is Lipschitz H m (Ω) := {f L 2 D α f L 2, α m} V (H m (Ω)) n closed Functions in V satisfy appropriate boundary conditions d, m, n N Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

8 Introduction (, ) and a(, ) are symmetric, bilinear with 1 a(v, w) v E w E 2 w 2 = a(w, w) E 3 (v, w) v w 4 w 2 = (w, w) where E is the energy norm which is equivalent to the (H m (Ω)) n norm on V, is the L 2 (Ω) n norm, v, w V Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

9 The Elliptic eigenvalue problem Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

10 The Elliptic eigenvalue problem Continuous eigenvalue problem formulation Find eigenvalues λ l R + and eigenfunctions u l V for l N, s.t., for all w V λ l (w, u l ) = a(w, u l ) 0 < λ 1 λ 2... and (u k, u l ) = δ kl u l 2 E = a(u l, u l ) = λ l and a(u k, u l ) = 0, l k Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

11 The Elliptic eigenvalue problem Discrete eigenvalue problem formulation Find eigenvalues λ h l R+ and eigenfunctions u h l V h, s.t., for all w h V h λ h l (w h, u h l ) = a(w h, u h l ) 0 < λ h 1 λh 2 λh N where dim(v h ) = N and (u h k, uh l ) = δ kl u h l 2 E = a(uh l, uh l ) = λh l and a(uh k, uh l ) = 0, l k Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

12 The Elliptic eigenvalue problem Comparison of {λ h l, uh l } to {λ l, u l } for l = 1,..., N is important Based on Pythagorean eigenvalue error theorem as introduced by G. Strang and G. Fix λ h l λ l λ l + ul h u l 2 = u h l u l 2 E, l = 1,..., N λ l Figure: Graphical representation of the Pythagorean eigenvalue error theorem Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

13 The Elliptic eigenvalue problem Variational model problem where λ l (w, u l ) = a(w, u l ) a(w, u l ) = (w, u l ) = w x wu l dx u l x dx and with homogeneous Dirichlet boundary conditions and Ω = (0, 1) Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

14 The Elliptic eigenvalue problem For l N, Eigenvalues are λ l = π 2 l 2 Eigenfunctions are u l = 2sin(lπx) Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

15 The Elliptic eigenvalue problem Figure: Pythagorean eigenvalue error theorem budget and L 2 -eigenfunction error for quadratic elements. (a) C 1 -continuous B-splines; (b) C 0 continuous nite elements, N = 99 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

16 The Elliptic eigenvalue problem Figure: Pythagorean eigenvalue error theorem budget and L 2 -eigenfunction error for cubic elements. (a) C 2 -continuous B-splines; (b) C 0 continuous nite elements, N = 99 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

17 The Elliptic eigenvalue problem Figure: Pythagorean eigenvalue error theorem budget for quartic elements. (a) C 3 -continuous B-splines; (b) C 0 continuous nite elements, N = 99 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

18 The Elliptic eigenvalue problem Results of this section An accurate eigenvalue does not imply an accurate eigenfunction The higher the eigenvalue the greater the eigenfunction error is false B-splines yield better approximations outlier modes at the end of the spectrum outlier modes do not spoil the accuracy in the interior Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

19 The Elliptic boundary value problem Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

20 The Elliptic boundary value problem Let f (L 2 (Ω)) n Continuous variational problem Find u V such that for all w V a(w, u) = (w, f ) Discrete variational problem Find u h V h such that for all w h V h a(w h, u h ) = (w h, f ) Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

21 The Elliptic boundary value problem Approximation of the continuous problem by the discrete one Eigenfunction expansion u = d l u l and u h = l=1 N l=1 d h l uh l where d l and dl h denote the Fourier-coecients of the continuous and the discrete solutions, respectively Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

22 The Elliptic boundary value problem λ l d l = f l def. = (u l, f ) λ h l d h l = f h l def. = (u h l, f ) u(x) = l=1 f l λ u l l (x) and u h (x) = N l=1 f l h λ h ul h(x) l Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

23 The Elliptic boundary value problem The error in the solution is e(x) = u h (x) u(x) = e(x) + e (x) with e(x) = e (x) = N e l (x) = l=1 N e l (x) = l=1 N l=1 l=n+1 ( f h l λ h l ) ul h (x) f l u l (x) λ l ( f l λ l u l (x) ) Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

24 The Elliptic boundary value problem For a specic l 1,..., N ( ( e l 2 f e l E e l E λ l λ 1/2 2 l λ 1/2 l ( ( e l E 2 f e l E λ l where e l = u h l u l 1 + e l E λ 1/2 l ) ( 1 + e l E λ 1/2 l 1 + e l 2 E λ l ) e l 2 E λ l + 1 e l 2 E 2 λ l )) ) Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

25 The Elliptic boundary value problem Discrete solution can be large in error Elliptic boundary-value problems are usually forgiving Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

26 The Parabolic initial-value problem Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

27 The Parabolic initial-value problem Let f L 2 ((0, T ); (L 2 (Ω)) n ) and U (L 2 (Ω)) n ) and T R + Continuous variational problem Find u V T such that for all w V and almost all t (0, T ) w, u (t) + a(w, u(t)) = (w, f (t)) t (w, u(0)) = (w, U) where V T := {v L 2 ((0, T ); V ) : v t L2 ((0, T ); V )} and, is the duality pairing Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

28 The Parabolic initial-value problem Semi-discrete variational problem Find u h V h T such that for all w h V h and almost all t (0, T ) w h, uh t (t) + a(w h, u h (t)) = (w h, f (t)) (w h, u h (0)) = (w h, U) where V h T := {v L2 ((0, T ); V h ) : v t L2 ((0, T ); V h )} and, is the duality pairing Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

29 The Parabolic initial-value problem yield and u(t) = d l (t)u l and u h (t) = l=1 N l=1 d h l (t)uh l d l (t) + λ ld l (t) = f l (t) def = (u l, f (t)) d h d l (0) = U l def = (u l, U) l (t) + λh l d l h (t) = f l h def (t) = (ul h, f (t)) d h l (0) = U h l def = (u h l, U) Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

30 The Parabolic initial-value problem Solving the ordinary dierential equations yield t d l (t) = U l exp( λ l t) + exp( λ l (t τ))f l (τ)dτ and d h l (t) = U h l exp( λh l t) + t 0 0 exp( λ h h l (t τ))fl (τ)dτ Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

31 The Parabolic initial-value problem From the Fourier-coecients, we obtain u(x, t) = and u h (x, t) = ( U l exp( λ l t) + l=1 t N t (U hl exp( λhl t) + l=1 0 0 ) exp( λ l (t τ))f l (τ)dτ u l (x) ) exp( λ h h l (t τ))fl (τ)dτ ul h (x) The error is e(x, t) = u h (x, t) u(x, t) = e(x, t) + e (x, t) Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

32 The Parabolic initial-value problem e(x, t) is caused by eigenvalue and eigenfunction errors Errors in decay rates are only due to eigenvalue errors Initial error is due to projection error Important are times up to t = O(λ 1 l ) Similar for f e l 0 as λ l t e l E 0 as λ l t Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

33 The Hyperbolic initial-value problem Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

34 The Hyperbolic initial-value problem Let f L 2 ((0, T ); (L 2 (Ω)) n ), U 0 (L 2 (Ω)) n ), U 1 V and T R + Continuous variational problem Find u V T such that for all w V and almost all t (0, T ) where V T := {v L 2 ((0, T ); V ) : v 2 v t 2 w, 2 u (t) + a(w, u(t)) = (w, f (t)) 2 t (w, u(0)) = (w, U 0 ) w, u t (0) = w, U 1 t L2 ( (0, T ); (L 2 (Ω)) n) and L 2 ((0, T ); V )} and, is the duality pairing Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

35 The Hyperbolic initial-value problem Semi-discrete variational problem Find u h V h T such that for all w h V h and almost all t (0, T ) w h, 2 u h t (t) + a(w h, u h (t)) = (w h, f (t)) 2 where V h T := {v L2 ((0, T ); V h ) : v 2 v t 2 (w h, u h (0)) = (w h, U 0 ) w h, uh t (0) = w h, U 1 t L2 ( (0, T ); V h) and L 2 ((0, T ); V h )} and, is the duality pairing Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

36 The Hyperbolic initial-value problem Proceeding as in the previous chapter yield and d l (t) + λ ld l (t) = f l (t) def = (u l, f (t)) d h l d l (0) = U 0l d l (0) = U 1l def = (u l, U 0 ) def = u l, U 1 (t) + λ h l d l h (t) = f l h def (t) = (ul h, f (t)) d h l (0) = U h 0l def = (u h l, U 0) d l h (0) = U 1l h def = ul h, U 1 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

37 The Hyperbolic initial-value problem The solutions of these ordinary dierential equations are d l (t) = U 0lcos(ω l t) + U 1l sin(ω l t) + 1 t sin(ω l (t τ))f l (τ)dτ ω l ω l 0 and dl h (t) = U 0l h cos(ωh l t) + U 1l h sin(ω h ωl h l t) + 1 ωl h t 0 sin(ωl h h (t τ))fl (τ)dτ with ω l = (λ l ) 1/2 and ω h l = (λh l )1/2 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

38 The Hyperbolic initial-value problem Plug the Fourier-coecients into eigenfunction expansion e(x, t) = u h (x, t) u(x, t) = e(x, t) + e (x, t) Modal error can be bounded by eigenfunctions and eigenvalue errors Modal errors oscillate in time Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

39 The Hyperbolic initial-value problem Numerical investigation of hyperbolic approximations For U 0 = sin(51πx), U 1 = 0 and f = 0 the exact solution to the hyperbolic initial-value problem is u(x, t) = sin(51πx)cos(51πt) with solution coecients { 1/ 2 if l = 51 U 0l = 0 otherwise, U 1l = 0 and f l = 0 on Ω = (0, 1) Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

40 The Hyperbolic initial-value problem Approximation is given by u h (x, t) = N {U0l h cos(ωh l t) + U 1l h sin(ωl h t) l=1 + 1 ω h l ωl h t 0 sin(ωl h h (t τ))fl (τ)dτ}uh l (x) C 3 -continuous quartic B-splines and C 0 -continuous quartic nite elements Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

41 The Hyperbolic initial-value problem Figure: Initial condition coecients U h 0l for C 3 -cont. quartic B-splines(left) and C 0 -cont. quartic FE-solutions(right), N = 99 Single wave and composition of two dierent waves Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

42 The Hyperbolic initial-value problem Figure: Exact and numerical solutions for t = 0, 0.25/51, 0.5/51, 0.75/51, 1/51, N = 99 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

43 The Hyperbolic initial-value problem Figure: Exact and numerical solutions for t = 10, 10.25/51, 10.5/51, 10.75/51, 11/51, N = 99 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

44 Own results Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

45 Own results Own tests with Laplace-eigenvalue problem Figure: Eigenvalue error for polynomial degree 2 and with C 1 -continuity, N = 100 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

46 Own results Figure: Eigenvalue error for polynomial degree 3 and with C 1 and C 2 -continuity, respectively, N = 100 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

47 Own results Figure: Eigenvalue error for polynomial degree 3 and with C 1 and C 2 -continuity, respectively, N = 100 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

48 Own results Figure: Eigenvalue error for polynomial degree 4 and with C 1 and C 3 -continuity, respectively, N = 100 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

49 Own results Figure: Eigenvalue error for polynomial degree 4 and with C 1 and C 3 -continuity, respectively, N = 100 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

50 Own results Figure: Eigenvalue error for polynomial degree 5 and with C 1 and C 4 -continuity, respectively N = 50, 100 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

51 Own results Figure: Eigenvalue error for polynomial degree 5 and with C 1 and C 4 -continuity, respectively, N = 50, 100 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

52 Own results Figure: Eigenvalue error for polynomial degree 6 and with C 3 and C 5 -continuity, respectively N = 50, 100 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

53 Own results Figure: Eigenvalue error for polynomial degree 6 and with C 3 and C 5 -continuity, respectively, N = 50, 100 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

54 Own results Figure: Eigenvalue error for polynomial degree 5 and 6 with C 3 and C 4 -continuity, respectively, N = 50 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

55 Own results Figure: Eigenvalue error for polynomial degree 5 and 6 with C 3 and C 4 -continuity, respectively, N = 75 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

56 Own results Figure: Eigenvalue error for polynomial degree 5 and 6 with C 3 and C 4 -continuity, respectively, N = 100 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

57 Summary First considered the elliptic eigenvalue problem Then looked at corresponding BVP, parabolic IVP and hyperbolic IVP Solution errors can be expressed in therms of eigenvalue and eigenfunction errors B-spline eigenvalue spectrum does not have optical branches B-spline eigenfunction error is indistinguishable in L 2 -norm B-spline approximations are much more accurate Own results for eigenvalue error for dierent p and r Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58

58 References [1] J. A. Evans, T. J. Hughes, and A. Reali. Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems, pages Elsevier, [2] G. Strang and G. Fix. An Analysis of the Finite Element Method. Wellesley-Cambridge Press, second edition, Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20, / 58