An introduction to Isogeometric Analysis

Transcription

1 An introduction to Isogeometric Analysis A. Buffa 1, G. Sangalli 1,2, R. Vázquez 1 1 Istituto di Matematica Applicata e Tecnologie Informatiche - Pavia Consiglio Nazionale delle Ricerche 2 Dipartimento di Matematica, Università di Pavia Supported by the ERC Starting Grant: GeoPDEs n R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

2 1 Approximation of vector fields and differential forms Construction of the discrete spaces The commuting De Rham diagram Maxwell eigenproblem: B-splines discretization Maxwell eigenproblem: NURBS discretization R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

3 Part I Approximation of vector fields and differential forms R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

4 Setting of the problem Let Ω be a Lipschitz domain described by a NURBS mapping F : Ω Ω, We define the functional spaces Ω being the unit cube. H(d, Ω) = {u L 2 (Ω) : du L 2 (Ω)}, u d,ω = u 0 + du 0 H 1 (Ω)/R grad H(curl, Ω) curl H(div, Ω) div L 2 (Ω). R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

5 Setting of the problem Let Ω be a Lipschitz domain described by a NURBS mapping F : Ω Ω, We define the functional spaces Ω being the unit cube. H(d, Ω) = {u L 2 (Ω) : du L 2 (Ω)}, u d,ω = u 0 + du 0 H 1 (Ω)/R grad H(curl, Ω) curl H(div, Ω) For FEM, we have the following compatible discretization Nodal FE grad Edge FE curl Face FE div L 2 (Ω). div Disc. FE. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

6 Setting of the problem Let Ω be a Lipschitz domain described by a NURBS mapping F : Ω Ω, We define the functional spaces Ω being the unit cube. H(d, Ω) = {u L 2 (Ω) : du L 2 (Ω)}, u d,ω = u 0 + du 0 H 1 (Ω)/R grad H(curl, Ω) curl H(div, Ω) For FEM, we have the following compatible discretization Nodal FE grad Edge FE curl Face FE div L 2 (Ω). div Disc. FE. Approximation of vector fields: we seek for a NURBS or Splines discretization compatible with this structure. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

7 Setting of the problem Let Ω be a Lipschitz domain described by a NURBS mapping F : Ω Ω, We define the functional spaces Ω being the unit cube. H(d, Ω) = {u L 2 (Ω) : du L 2 (Ω)}, u d,ω = u 0 + du 0 H 1 (Ω)/R grad H(curl, Ω) curl H(div, Ω) For FEM, we have the following compatible discretization Nodal FE grad Edge FE curl Face FE div L 2 (Ω). div Disc. FE. Approximation of vector fields: we seek for a NURBS or Splines discretization compatible with this structure. Applications: electromagnetic problems, Darcy s flow, Stokes flow... R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

8 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: univariate B-splines and NURBS S p α/n p α : univariate Splines/NURBS of degree p and regularity α at knots. { } d dx v : v S α p S p 1 α 1 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

9 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: multivariate B-splines and NURBS S p 1,p 2,p 3 α 1,α 2,α 3 = S p 1 α 1 S p 2 α 2 S p 3 α 3, N p 1,p 2,p 3 α 1,α 2,α 3 = N p 1 α 1 N p 2 α 2 N p 3 α 3. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

10 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: multivariate B-splines and NURBS S p 1,p 2,p 3 α 1,α 2,α 3 = S p 1 α 1 S p 2 α 2 S p 3 α 3, N p 1,p 2,p 3 α 1,α 2,α 3 = N p 1 α 1 N p 2 α 2 N p 3 α 3. Let us not keep track of the regularity α in what follows... R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

11 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: multivariate B-splines and NURBS S p 1,p 2,p 3 α 1,α 2,α 3 = S p 1 α 1 S p 2 α 2 S p 3 α 3, N p 1,p 2,p 3 α 1,α 2,α 3 = N p 1 α 1 N p 2 α 2 N p 3 α 3. Let us not keep track of the regularity α in what follows... Differential operators: we look for discrete spaces in Ω such that R Ŝ0 ĝrad Ŝ1 dcurl Ŝ2 cdiv Ŝ3 0 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

12 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: multivariate B-splines and NURBS S p 1,p 2,p 3 α 1,α 2,α 3 = S p 1 α 1 S p 2 α 2 S p 3 α 3, N p 1,p 2,p 3 α 1,α 2,α 3 = N p 1 α 1 N p 2 α 2 N p 3 α 3. Let us not keep track of the regularity α in what follows... Differential operators: we look for discrete spaces in Ω such that R Ŝ0 ĝrad Ŝ1 dcurl Ŝ2 cdiv Ŝ3 0 Ŝ 0 = S p 1,p 2,p 3 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

13 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: multivariate B-splines and NURBS S p 1,p 2,p 3 α 1,α 2,α 3 = S p 1 α 1 S p 2 α 2 S p 3 α 3, N p 1,p 2,p 3 α 1,α 2,α 3 = N p 1 α 1 N p 2 α 2 N p 3 α 3. Let us not keep track of the regularity α in what follows... Differential operators: we look for discrete spaces in Ω such that R Ŝ0 ĝrad Ŝ1 dcurl Ŝ2 cdiv Ŝ3 0 Ŝ 0 = S p 1,p 2,p 3 ĝrad : Ŝ0 S p 1 1,p 2,p 3 S p 1,p 2 1,p 3 S p 1,p 2,p 3 1 Ŝ1 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

14 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: multivariate B-splines and NURBS S p 1,p 2,p 3 α 1,α 2,α 3 = S p 1 α 1 S p 2 α 2 S p 3 α 3, N p 1,p 2,p 3 α 1,α 2,α 3 = N p 1 α 1 N p 2 α 2 N p 3 α 3. Let us not keep track of the regularity α in what follows... Differential operators: we look for discrete spaces in Ω such that R Ŝ0 ĝrad Ŝ1 dcurl Ŝ2 cdiv Ŝ3 0 Ŝ 1 = S p 1 1,p 2,p 3 S p 1,p 2 1,p 3 S p 1,p 2,p 3 1 ĉurl : Ŝ1 S p 1,p 2 1,p 3 1 S p 1 1,p 2,p 3 1 S p 1 1,p 2 1,p 3 Ŝ2 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

15 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: multivariate B-splines and NURBS S p 1,p 2,p 3 α 1,α 2,α 3 = S p 1 α 1 S p 2 α 2 S p 3 α 3, N p 1,p 2,p 3 α 1,α 2,α 3 = N p 1 α 1 N p 2 α 2 N p 3 α 3. Let us not keep track of the regularity α in what follows... Differential operators: we look for discrete spaces in Ω such that R Ŝ0 ĝrad Ŝ1 dcurl Ŝ2 cdiv Ŝ3 0 Ŝ 2 = S p 1,p 2 1,p 3 1 S p 1 1,p 2,p 3 1 S p 1 1,p 2 1,p 3 div : Ŝ2 S p 1 1,p 2 1,p 3 1 Ŝ3 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

16 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: multivariate B-splines and NURBS S p 1,p 2,p 3 α 1,α 2,α 3 = S p 1 α 1 S p 2 α 2 S p 3 α 3, N p 1,p 2,p 3 α 1,α 2,α 3 = N p 1 α 1 N p 2 α 2 N p 3 α 3. Let us not keep track of the regularity α in what follows... Differential operators: we look for discrete spaces in Ω such that R Ŝ0 The sequence is exact. ĝrad Ŝ1 dcurl Ŝ2 cdiv Ŝ3 0 Generalization of nodal, edge and face elements with higher regularity. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

17 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: multivariate B-splines and NURBS S p 1,p 2,p 3 α 1,α 2,α 3 = S p 1 α 1 S p 2 α 2 S p 3 α 3, N p 1,p 2,p 3 α 1,α 2,α 3 = N p 1 α 1 N p 2 α 2 N p 3 α 3. Let us not keep track of the regularity α in what follows... Differential operators: we look for discrete spaces in Ω such that R Ŝ0 The sequence is exact. ĝrad Ŝ1 dcurl Ŝ2 cdiv Ŝ3 0 Generalization of nodal, edge and face elements with higher regularity. But what about NURBS? R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

18 Vector fields - approximation in Ω Buffa, Sangalli, V., 2010 To recall some notation: multivariate B-splines and NURBS S p 1,p 2,p 3 α 1,α 2,α 3 = S p 1 α 1 S p 2 α 2 S p 3 α 3, N p 1,p 2,p 3 α 1,α 2,α 3 = N p 1 α 1 N p 2 α 2 N p 3 α 3. Let us not keep track of the regularity α in what follows... Differential operators: we look for discrete spaces in Ω such that N 0 ĝrad N 1 d curl N 2 The derivative of a NURBS is not a NURBS. The diagram for NURBS does not hold true. c div N 3 D. Arnold (et al.) teach us that N i are then not suitable for vector fields approximations R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

19 Vector fields - SPLINES approximation in Ω Spaces on Ω are defined just by push forward: S i = ι i (Ŝi), N i = ι i ( N i ). R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

20 Vector fields - SPLINES approximation in Ω Spaces on Ω are defined just by push forward: S i = ι i (Ŝi), N i = ι i ( N i ). ι 0 ( φ) F = φ, standard mapping F R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

21 Vector fields - SPLINES approximation in Ω Spaces on Ω are defined just by push forward: S i = ι i (Ŝi), N i = ι i ( N i ). ι 1 (û) F = DF T û, curl conforming mapping F R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

22 Vector fields - SPLINES approximation in Ω Spaces on Ω are defined just by push forward: S i = ι i (Ŝi), N i = ι i ( N i ). ι 2 (û) F = (detdf) 1 DF û, Piola mapping F R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

23 Vector fields - SPLINES approximation in Ω Spaces on Ω are defined just by push forward: S i = ι i (Ŝi), N i = ι i ( N i ). ι 3 (φ) F = (detdf) 1 φ, 3-forms mapping F R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

24 Vector fields - SPLINES approximation in Ω Spaces on Ω are defined just by push forward: S i = ι i (Ŝi), N i = ι i ( N i ). For splines, everything then works thanks to the geometric structure: H 1 (Ω)/R grad H(curl, Ω) curl H(div, Ω) div L 2 (Ω) S 0 /R grad S 1 curl S 2 div S 3 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

25 Vector fields - SPLINES approximation in Ω Spaces on Ω are defined just by push forward: S i = ι i (Ŝi), N i = ι i ( N i ). For splines, everything then works thanks to the geometric structure: grad H 1 (Ω)/R H(curl, Ω) H(div, Ω) L 2 (Ω) Π 0? Π 1? Π 2? Π 3? S 0 /R curl grad S 1 curl S 2 div S 3 How can we define the commuting interpolants Π i? div R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

26 Vector fields - SPLINES approximation in Ω Spaces on Ω are defined just by push forward: S i = ι i (Ŝi), N i = ι i ( N i ). For splines, everything then works thanks to the geometric structure: grad H 1 (Ω)/R H(curl, Ω) H(div, Ω) L 2 (Ω) Π 0? Π 1? Π 2? Π 3? S 0 /R curl grad S 1 curl S 2 div S 3 How can we define the commuting interpolants Π i? Construct (commuting) one-dimensional projectors. Construct the projectors in Ω with tensor product structure. Pull back from Ω to Ω. div R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

27 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. Π p S s = s, s S α, p u H k (I ) C u H k ( e I ), u Hk (0, 1), 0 k p + 1, where I = (ξ j, ξ j+1 ), and Ĩ is a local extension. L.L. Schumaker. Spline functions: basic theory, Y. Bazilevs, L. Beirão da Veiga, J. Cottrell, T.J.R. Hughes, G. Sangalli, R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

28 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. First idea: tensor products of Π p S, choosing the right degree p. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

29 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. First idea: tensor products of Π p S, choosing the right degree p. The diagram is not commutative, because even the 1D diagram is not: Π p 1 S d dx u d dx Π p S u. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

30 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. Second idea: define Π p 1 A such that the 1D diagram commutes. d dx R H 1 (0, 1) L 2 (0, 1) 0 Π bp 1 R bπ p S S p α d dx A S p 1 α 1 0 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

31 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. Second idea: define So, we define it as Π p 1 A such that the 1D diagram commutes. d dx R H 1 (0, 1) L 2 (0, 1) 0 Π bp 1 R bπ p S S p α Π p 1 A v := d dx Π p S x 0 d dx A S p 1 α 1 0 v(ξ)dξ, v L 2 (0, 1). R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

32 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. Second idea: define So, we define it as Π p 1 A such that the 1D diagram commutes. d dx R H 1 (0, 1) L 2 (0, 1) 0 Π bp 1 R bπ p S S p α Π p 1 A v := d dx Π p S x It is a stable and local projector: 0 d dx Π p 1 p 1 A s = s, s Sα 1, A S p 1 α 1 0 v(ξ)dξ, v L 2 (0, 1). u H k (I ) C u H k ( e I ), u Hk (0, 1), 0 k p. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

33 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. Second idea: define Π p 1 A Next step: in Ω, take tensor products of Π p S For instance, for we define such that the 1D diagram commutes. and Π p 1 A. Ŝ 1 = S p 1 1,p 2,p 3 S p 1,p 2 1,p 3 S p 1,p 2,p 3 1 Π 1 := ( Π p 1 1 A Π p 2 S Π p 3 S ) ( Π p 1 S Π p 2 1 A Π p 3 S ) ( Π p 1 S Π p 2 S Π p 3 1 A ). R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

34 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. Second idea: define Π p 1 A Next step: in Ω, take tensor products of Π p S such that the 1D diagram commutes. and Π p 1 A. With this choice, the diagram in Ω is commutative. ĝrad H 1 ( Ω)/R H(curl, Ω) H(div, Ω) L 2 ( Ω) bπ 0 Π b1 Π b2 Π b3 dcurl cdiv Ŝ 0 /R ĝrad Ŝ 1 d curl Ŝ 2 c div Ŝ3 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

35 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. Second idea: define Π p 1 A Next step: in Ω, take tensor products of Π p S such that the 1D diagram commutes. and Π p 1 A. Last step: define the interpolants in Ω by push forward techniques. ι i (Π i φ) = Π i (ι i (φ)), φ H(d, Ω), i = 0,..., 3 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

36 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. Second idea: define Π p 1 A Next step: in Ω, take tensor products of Π p S such that the 1D diagram commutes. and Π p 1 A. Last step: define the interpolants in Ω by push forward techniques. With the definition of these interpolants, the diagram is commutative. grad H 1 curl div (Ω)/R H(curl, Ω) H(div, Ω) L 2 (Ω) Π 0 Π 1 Π 2 Π 3 S 0 /R grad S 1 curl S 2 div S 3 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

37 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. Second idea: define Π p 1 A Next step: in Ω, take tensor products of Π p S such that the 1D diagram commutes. and Π p 1 A. Last step: define the interpolants in Ω by push forward techniques. With the definition of these interpolants, the diagram is commutative. grad H 1 curl div (Ω)/R H(curl, Ω) H(div, Ω) L 2 (Ω) Π 0 Π 1 Π 2 Π 3 S 0 /R grad S 1 curl S 2 div S 3 The projectors Π i are local and L 2 (Ω)-stable. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

38 Construction of the interpolants for Spline spaces Buffa, Rivas, Sangalli, V., 2010 Starting point: we know the local and stable projector Π p S. Second idea: define Π p 1 A Next step: in Ω, take tensor products of Π p S such that the 1D diagram commutes. and Π p 1 A. Last step: define the interpolants in Ω by push forward techniques. With the definition of these interpolants, the diagram is commutative. grad H 1 curl div (Ω)/R H(curl, Ω) H(div, Ω) L 2 (Ω) Π 0 Π 1 Π 2 Π 3 S 0 /R grad S 1 curl S 2 div S 3 The projectors Π i are local and L 2 (Ω)-stable. The same idea can be used in spaces with boundary conditions. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

39 Approximation estimates Buffa, Rivas, Sangalli, V., Assume F : Ω Ω belongs to N 0 and F 1 is piecewise regular. Let K be an element of the mesh on the physical domain: 0 l p u Π i u H(d i,k) Ch l u H l (d i, K) e, i = 0,..., 3, R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

40 Approximation estimates Buffa, Rivas, Sangalli, V., Assume F : Ω Ω belongs to N 0 and F 1 is piecewise regular. Let K be an element of the mesh on the physical domain: 0 l p As a consequence u Π i u H(d i,k) Ch l u H l (d i, K) e, i = 0,..., 3, u Π 1 u H(curl,Ω) Ch l u H l (curl,ω). R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

41 Approximation estimates Buffa, Rivas, Sangalli, V., Assume F : Ω Ω belongs to N 0 and F 1 is piecewise regular. Let K be an element of the mesh on the physical domain: 0 l p As a consequence u Π i u H(d i,k) Ch l u H l (d i, K) e, i = 0,..., 3, u Π 1 u H(curl,Ω) Ch l u H l (curl,ω). Reducing the continuity we can obtain estimates in terms of p. If α = max{α i } p 1 2 then u Π i u H(d i,k) C(h/p) l u H l (d i, K) e, i = 0,..., 3. Beirão da Veiga, Buffa, Rivas, Sangalli (2009). R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

42 Maxwell eigenproblem: B-splines discretization Maxwell eigenproblem Find u H 0 (curl, Ω), u 0 and λ R, λ 0 such that curl u curl v = λ u v v H 0 (curl, Ω) Ω Ω R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

43 Maxwell eigenproblem: B-splines discretization Maxwell eigenproblem Find u H 0 (curl, Ω), u 0 and λ R, λ 0 such that curl u curl v = λ u v v H 0 (curl, Ω) Ω Ω Theorem If there is a commuting diagram with local and L 2 (Ω)-stable projectors, the Galerkin approximation is spurious-free and optimal. D. Arnold, R. Falk, R. Winther, Bulletin A.M.S. (2010) R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

44 Maxwell eigenproblem: B-splines discretization Maxwell eigenproblem Find u H 0 (curl, Ω), u 0 and λ R, λ 0 such that curl u curl v = λ u v v H 0 (curl, Ω) Ω Ω Theorem If there is a commuting diagram with local and L 2 (Ω)-stable projectors, the Galerkin approximation is spurious-free and optimal. D. Arnold, R. Falk, R. Winther, Bulletin A.M.S. (2010) The B-spline discretization fulfills the theorem: it is spectrally correct. Exact CAD description of the geometry. Higher regularity than standard edge elements. Singular functions are approximated correctly. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

45 Maxwell eigenproblem in the patch Ω We solve the eigenvalue problem: Find u 0 and ω 0 such that curl curl u = ω 2 u u n = 0 in Ω, on Ω. We solve with a Galerkin projection on Ŝ1,0, and different degrees p. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

46 Maxwell eigenproblem in the patch Ω We solve the eigenvalue problem: Find u 0 and ω 0 such that curl curl u = ω 2 u u n = 0 in Ω, on Ω. We solve with a Galerkin projection on Ŝ1,0, and different degrees p. In terms of d.o.f., we obtain better convergence rate than edge elements Relative error p = 2 p = 3 p = 4 y = Ch 4 y = Ch 6 y = Ch 8 Relative error IGA, p = 2 IGA, p = 3 IGA, p = 4 FEM, p = 2 FEM, p = 3 FEM, p = Mesh size Degrees of freedom R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

47 b Maxwell eigenproblem in the patch Ω We solve the eigenvalue problem: Find u 6= 0 and ω 6= 0 such that b curl curl u = ω 2 u in Ω, b u n = 0 on Ω. Fill-in of the matrix with respect to FEM. Sparsity pattern is similar nz = R. Va zquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis nz = Santiago de Compostela, / 17

48 Maxwell eigenproblem in the patch Ω We solve the eigenvalue problem: Find u 0 and ω 0 such that curl curl u = ω 2 u in Ω, u n = 0 on Ω. We solve with continuous fields: the divergence is well defined. It is an oscillating field, and converges to zero with order h p divuh L p = 2 p = 3 p = 4 y = Ch y = Ch 2 y = Ch Mesh size 10 0 R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

49 Maxwell eigenproblem: Fichera s corner We solve the eigenvalue problem: Find u 0 and ω 0 such that curl curl u = ω 2 u in Ω, u n = 0 on Ω. Interaction between edge and corner singularities. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

50 Maxwell eigenproblem: Fichera s corner We solve the eigenvalue problem: Find u 0 and ω 0 such that curl curl u = ω 2 u in Ω, u n = 0 on Ω. Interaction between edge and corner singularities. We get a good approximation of the singular functions. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

51 Maxwell eigenproblem: Fichera s corner We solve the eigenvalue problem: Find u 0 and ω 0 such that curl curl u = ω 2 u in Ω, u n = 0 on Ω. Interaction between edge and corner singularities. We get a good approximation of the singular functions. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

52 Maxwell eigenproblem: Fichera s corner We solve the eigenvalue problem: Find u 0 and ω 0 such that curl curl u = ω 2 u in Ω, u n = 0 on Ω. Interaction between edge and corner singularities. We get a good approximation of the singular functions. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

53 Maxwell eigenproblem: Fichera s corner We solve the eigenvalue problem: Find u 0 and ω 0 such that curl curl u = ω 2 u in Ω, u n = 0 on Ω. Interaction between edge and corner singularities. We get a good approximation of the singular functions. Eigenvalues computation CODE by S. Zaglmayr M. Duruflé IGA, p = 3 d.o.f Eig Eig Eig Eig Eig Eig Eig Eig R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

54 Maxwell source problem: non-convex domain We solve the source problem, with mixed boundary conditions. curl curl u + u = f. The geometry is described exactly with only three elements. Domain with a reentrant edge (as the L-shaped domain). R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

55 Maxwell source problem: non-convex domain We solve the source problem, with mixed boundary conditions. curl curl u + u = f. The geometry is described exactly with only three elements. Domain with a reentrant edge (as the L-shaped domain). The solution is u = grad(r 2/3 sin(2θ/3)), and u H 2/3 ε (curl, Ω). The convergence rate in energy norm is h 2/3, as expected p = 1 p = 2 p = 3 Relative error 10 1 p = 4 y = Ch 2/ Mesh size R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

56 Maxwell eigenproblem: NURBS discretization If we try to discretize the problem with NURBS, in the form: Find u h N 1, u h 0 and λ R, λ 0 such that curl u h curl v h = λ u h v h v h N 1, Ω Ω R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

57 Maxwell eigenproblem: NURBS discretization If we try to discretize the problem with NURBS, in the form: Find u h N 1, u h 0 and λ R, λ 0 such that curl u h curl v h = λ u h v h v h N 1, Ω the diagram does not hold: spurious eigenvalues appear. Ω R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

58 Maxwell eigenproblem: NURBS discretization If we try to discretize the problem with NURBS, in the form: Find u h N 1, u h 0 and λ R, λ 0 such that curl u h curl v h = λ u h v h v h N 1, Ω the diagram does not hold: spurious eigenvalues appear. In the continuous case, we have the equivalent mixed formulation: Find (u, p) H 0 (curl, Ω) H0 1 (Ω), λ R such that curl u curl v + p v = λ u v v H 0 (curl, Ω), Ω Ω Ω q u = 0 q H0 1 (Ω). Ω The two discrete formulations are equivalent for B-splines. Ω R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

59 Maxwell eigenproblem: NURBS discretization If we try to discretize the problem with NURBS, in the form: Find u h N 1, u h 0 and λ R, λ 0 such that curl u h curl v h = λ u h v h v h N 1, Ω the diagram does not hold: spurious eigenvalues appear. For a NURBS discretization, the two formulations are not equivalent. Find (u h, p h ) N 1 N 0, λ R such that curl u h curl v h + p h v h = λ u h v h v h N 1, Ω Ω Ω q h u h = 0 q h N 0. Ω Ω R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

60 NURBS discretization: numerical results Buffa, Sangalli, V. In preparation The domain Ω is described with a NURBS mapping. Solution of the mixed formulation with a NURBS discretization NURBS map of fine mesh Relative error rd eigenvalue 8 th eigenvalue th eigenvalue y=ch Mesh size The results are spurious-free, with optimal convergence rate. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

61 NURBS discretization: numerical results Buffa, Sangalli, V. In preparation The domain Ω is described with a NURBS mapping. Solution of the mixed formulation with a NURBS discretization Relative error st eigenvalue 2 nd eigenvalue 3 rd eigenvalue 4 th eigenvalue 5 th eigenvalue y=ch 4/3 y=ch NURBS map of the coarsest mesh Mesh size The results are spurious-free, with optimal convergence rate. Also the singular functions are approximated correctly. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

62 NURBS discretization: numerical results Buffa, Sangalli, V. In preparation The domain Ω is described with a NURBS mapping. Solution of the mixed formulation with a NURBS discretization Relative error st eigenvalue 2 nd eigenvalue 3 rd eigenvalue 4 th eigenvalue 5 th eigenvalue y=ch 4/3 y=ch NURBS map of the coarsest mesh Mesh size The results are spurious-free, with optimal convergence rate. Also the singular functions are approximated correctly. The spectral correctness can also be proved for (mixed) NURBS. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

63 Conclusions Isogeometric Analysis has been extended to the approximation of vector fields. Generalization of edge and face elements Higher continuity than standard finite elements. Exact geometry description. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

64 Conclusions Isogeometric Analysis has been extended to the approximation of vector fields. Generalization of edge and face elements Higher continuity than standard finite elements. Exact geometry description. Complete theory for Maxwell, and promising numerical results. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

65 Conclusions Isogeometric Analysis has been extended to the approximation of vector fields. Generalization of edge and face elements Higher continuity than standard finite elements. Exact geometry description. Complete theory for Maxwell, and promising numerical results. Further comparisons with FEM will be done in the future. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

66 Conclusions Isogeometric Analysis has been extended to the approximation of vector fields. Generalization of edge and face elements Higher continuity than standard finite elements. Exact geometry description. Complete theory for Maxwell, and promising numerical results. Further comparisons with FEM will be done in the future. COMING SOON The GeoPDEs code: a research tool for Isogeometric Analysis of PDEs. Open Octave (compatible with Matlab) implementation of the method. Joint work with C. de Falco and A. Reali. R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

67 Acknowledgments Funded by ERC Starting Grant n : GeoPDEs Team Annalisa Buffa (IMATI-CNR) Lourenço Beirão da Veiga (University of Milano) Alessandro Reali (University of Pavia) Giancarlo Sangalli (University of Pavia) Andrea Bressan Durkbin Cho Carlo De Falco Mukesh Kumar Massimiliano Martinelli Judith Rivas Rafael Vázquez R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17

68 Acknowledgments Funded by ERC Starting Grant n : GeoPDEs Team Annalisa Buffa (IMATI-CNR) Lourenço Beirão da Veiga (University of Milano) Alessandro Reali (University of Pavia) Giancarlo Sangalli (University of Pavia) Andrea Bressan Durkbin Cho Carlo De Falco Mukesh Kumar Massimiliano Martinelli Judith Rivas Rafael Vázquez Thanks for your attention! R. Vázquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, / 17