Isogeometric mortaring

Transcription

1 Isogeometric mortaring E. Brivadis, A. Buffa, B. Wohlmuth, L. Wunderlich IMATI E. Magenes - Pavia Technical University of Munich A. Buffa (IMATI-CNR Italy) IGA mortaring 1 / 29

2 1 Introduction Splines Approximation spaces and properties 2 Non conforming interfaces Mortar method Numerical validation 3 Final remarks A. Buffa (IMATI-CNR Italy) IGA mortaring 2 / 29

3 IGA is based on spline theory B-Splines are defined by the Cox-DeBoor formulae: { 1 if ξi ζ < ξ N i,0 (ζ) = i+1, 0 otherwise, N i,p (ζ) = ζ ξ i N i,p 1 (ζ) + ξ i+p+1 ζ N i+1,p 1 (ζ). ξ i+p ξ i ξ i+p+1 ξ i+1 A. Buffa (IMATI-CNR Italy) IGA mortaring 3 / 29

4 ξ" Quadratic (p=2) basis functions for an non-uniform based onopen, spline theory knot vector:5! 4 IGA is B-Splines are defined by the formulae:" Ξ =Cox-DeBoor {0,0,0,1,2,3,4,4,5,5,5} 1 if ξi ζ < ξi+1, Ni,0 (ζ) = 0 ξ" 0 otherwise, ξ" ξi+p+1 ζ ζ ξi 1 Ni,p (ζ) = Ni,p 1 (ζ) + Ni+1,p 1 (ζ). ξi+p ξi - control pointsξi+p+1 ξi+1 - knots ts F(ξ) = P i Ci Ni,p (ξ) : Quadratic basis h-refined Curve control points A. Buffa (IMATI-CNR Italy) IGA mortaring knots 3 / 29

5 4 4 IGA is based on spline theory B-Splines are defined by the Cox-DeBoor formulae: 6/18/ if ξi ζ < ξi+1, 2 Ni,0 (ζ) = 0 0 otherwise, h-refined Surface ξi+p+1 ζ ζ ξi 1 Ni,p (ζ) = Ni,p 1 (ζ) + Ni+1,p 1 (ζ). ξi+p ξi - control pointsξi+p+1 ξi+1 - knots ts F(ξ) = P i Ci Ni,p (ξ) : Quadratic basis h-refined Surface Control net Mesh Control net A. Buffa (IMATI-CNR Italy) IGA mortaring 3 / 29

6 4 4 IGA is based on spline theory B-Splines are defined by the Cox-DeBoor formulae: 6/18/ if ξi ζ < ξi+1, 2 Ni,0 (ζ) = 0 0 otherwise, h-refined Surface ξi+p+1 ζ ζ ξi 1 Ni,p (ζ) = Ni,p 1 (ζ) + Ni+1,p 1 (ζ). ξi+p ξi - control pointsξi+p+1 ξi+1 - knots ts F(ξ) = P i Ci Ni,p (ξ) : Quadratic basis h-refined Surface Control net NURBS: projection of splines in Rd+1... no need for thismesh talk. Control net A. Buffa (IMATI-CNR Italy) IGA mortaring 3 / 29

7 Construction of approximation spaces F The geometry Ω and its NURBS parametrization F is given by CAD general geometry: unstructured collection of patches. A. Buffa (IMATI-CNR Italy) IGA mortaring 4 / 29

8 Construction of approximation spaces F The geometry Ω and its NURBS parametrization F is given by CAD general geometry: unstructured collection of patches. The discrete space on Ω is the push-forward of Spline/NURBS A. Buffa (IMATI-CNR Italy) IGA mortaring 4 / 29

9 Construction of approximation spaces F The geometry Ω and its NURBS parametrization F is given by CAD general geometry: unstructured collection of patches. The discrete space on Ω is the push-forward of Spline/NURBS Refinement by knot insertion and degree elevation, geometry unchanged. A. Buffa (IMATI-CNR Italy) IGA mortaring 4 / 29

10 Construction of approximation spaces F The geometry Ω and its NURBS parametrization F is given by CAD general geometry: unstructured collection of patches. The discrete space on Ω is the push-forward of Spline/NURBS Refinement by knot insertion and degree elevation, geometry unchanged. A. Buffa (IMATI-CNR Italy) IGA mortaring 4 / 29

11 General geometries are multi-patch Ω (2) Ω (1) Ω (3) (b) T-shaped, but not hanging Globally unstructured Locally structured A. Buffa (IMATI-CNR Italy) IGA mortaring 5 / 29

12 General geometries are multi-patch Ω (2) Ω (1) Ω (3) (b) T-shaped, but not hanging Globally unstructured Locally structured A. Buffa (IMATI-CNR Italy) IGA mortaring 5 / 29

13 General geometries are multi-patch Ω (2) Ω (1) Ω (3) (b) T-shaped, but not hanging Globally unstructured Locally structured Question: How to enhance flexibility? Question: How to treat non conforming interfaces? A. Buffa (IMATI-CNR Italy) IGA mortaring 5 / 29

14 Perfect setting for adaptivity if splines can support local refinement T-splines : Pure geometric modeling approach: Sederberg et al 2004, Hughes, Scott, Evans, Li, Zhang,... Pavia team courtesy of M. Scott LR-splines Dokken et al. 2013, Bressan 2013 Definition R n, spline theory of LR-Splines Hierarchical splines Kraft 1998,... the closest to adaptive finite elements on quadrangular meshes A. Buffa (IMATI-CNR Italy) IGA mortaring 6 / 29

15 Hierarchical splines Kraft 1998, Giannelli, Jüttler, Simeon, Speelers, Voung , B.-Giannelli C. Giannelli et al. / Computer Aided Geometric Design 29 (2012) ins (b) hierarchical mesh (a) nested domains (b) hierarchical me Fig. thesplinehierarchyaccordingtorelation(1),i.e.,ω l 2. Anestedsequenceofdomainsfortheconstructionofthesplinehierarchyaccordingtorelation(1),i.e.,Ω Ω l+1 for l = 0,...,2, for the two- l dimensional case. (a) Only dyadic refinement is considered. (b) Ω l+1 is restricted to be the union of supports of B-splines of level l. (c) The boundaries of adjacent domains Ω s of B-splines of level l. l and Ω l+1 must be disjoint. Ω l+1 (d) The modified support definition (2) is not used and H 0 is initialized as the set of B-s must be disjoint. ed and H 0 completely contained is initialized as the set of B-splines in B 0 in Ω 0. whose support is Moreover, the definition in Kraft s thesis (Kraft, 1998) additionally considers the auxiliary su ω l = { x Ω l β B l : x supp β supp β Ω l }, 98) additionally considers the auxiliary subdomain for l = 0,...,N 1, which represents the biggest subset of Ω l such that H l B spans the restr that these subdomains are nested as well, A. Buffa (IMATI-CNR Italy) IGA mortaring 7 / 29 p β Ω l},

16 Hierarchical splines Kraft 1998, Giannelli, Jüttler, Simeon, Speelers, Voung , B.-Giannelli C. Giannelli et al. / Computer Aided Geometric Design 29 (2012) ins (b) hierarchical mesh (a) nested domains (b) hierarchical me Fig. thesplinehierarchyaccordingtorelation(1),i.e.,ω l 2. Anestedsequenceofdomainsfortheconstructionofthesplinehierarchyaccordingtorelation(1),i.e.,Ω Ω l+1 for l = 0,...,2, for the two- l dimensional case. (a) Only dyadic refinement is considered. (b) Ω l+1 is restricted to be the union of supports of B-splines of level l. (c) The boundaries of adjacent domains Ω s of B-splines of level l. l and Ω l+1 must be disjoint. Definition of truncated-basis Ω l+1 (d) The modified ensuring support definition good (2) isspline not used and properties H 0 is initialized as the set of B-s must be disjoint. ed and H 0 completely contained is initialized as the set of B-splines in B 0 in Ω 0. whose support is Giannelli, Juettler, Speelers Moreover, the definition in Kraft s thesis (Kraft, 1998) additionally considers the auxiliary su ω l = { x Ω l β B l : x supp β supp β Ω l }, 98) additionally considers the auxiliary subdomain for l = 0,...,N 1, which represents the biggest subset of Ω l such that H l B spans the restr that these subdomains are nested as well, A. Buffa (IMATI-CNR Italy) IGA mortaring 7 / 29 p β Ω l}, V 0 V 1 V 2... V J Refinement has to contain at least one function : (p + 1) 2 elements

17 Adaptivity with hierarchical splines B. and Giannelli, 2014 Residual based estimator: η Q = h Q Au h f L 2 (Q) (no jumps) Dörfler marking E(u h, M) θe(u h, T), 0 < θ < 1; Suitable local quasi-interpolant and their approximation properties A. Buffa (IMATI-CNR Italy) IGA mortaring 8 / 29

18 Adaptivity with hierarchical splines B. and Giannelli, 2014 Residual based estimator: η Q = h Q Au h f L 2 (Q) (no jumps) Dörfler marking E(u h, M) θe(u h, T), 0 < θ < 1; Suitable local quasi-interpolant and their approximation properties The theory of adaptive methods Convergence and optimality! Morin, Nochetto, Siebert Binev, Dahmen, DeVore A. Buffa (IMATI-CNR Italy) IGA mortaring 8 / 29

19 Adaptivity with hierarchical splines B. and Giannelli, 2014 Residual based estimator: η Q = h Q Au h f L 2 (Q) (no jumps) Dörfler marking E(u h, M) θe(u h, T), 0 < θ < 1; Suitable local quasi-interpolant and their approximation properties The theory of adaptive methods Convergence and optimality!... but this is another story... Morin, Nochetto, Siebert Binev, Dahmen, DeVore A. Buffa (IMATI-CNR Italy) IGA mortaring 8 / 29

20 Non conforming interfaces and mortaring A. Buffa (IMATI-CNR Italy) IGA mortaring 9 / 29

21 Non conforming interfaces and mortaring Let Ω be a computational domain in R n, we want to solve the Laplace problem (or linear elasticity with minor changes) div (A u) = f with boundary conditions Ω = Γ D Γ N. u = 0 on Γ D and (A u) n = h on Γ N We suppose that Ω = N i Ω i, Ω i = F i ( Ω), Γ ij = Ω i Ω j, F i are splines (or NURBS) non compatible meshes at the interfaces Γ ij A. Buffa (IMATI-CNR Italy) IGA mortaring 9 / 29

22 About the admissible partition of the domain m(2) 1 s(1) n 2 n 1 s(2) m(1) s(1) n 1 2 s(2) n 2 m(1) m(2) 3 Decomposition can be conforming or non-conforming We can handle the case when Γ ij is a face of either Ω i or Ω j. There is the need for cross-point treatment/reduction A. Buffa (IMATI-CNR Italy) IGA mortaring 10 / 29

23 About the admissible partition of the domain m(2) 1 s(1) n 2 n 1 s(2) m(1) s(1) n 1 2 s(2) n 2 m(1) m(2) 3 Decomposition can be conforming or non-conforming We can handle the case when Γ ij is a face of either Ω i or Ω j. There is the need for cross-point treatment/reduction Non compatible geometries interfaces at the interfaces Γ ij (?) A. Buffa (IMATI-CNR Italy) IGA mortaring 10 / 29

24 Non conforming interfaces and mortaring Let S p ( T j ) be the space of tensor product splines/nurbs of degree p, on the knot mesh T j. in each subdomain Ω j, V j = {v j H 1 (Ω j ) : v F j S p ( T j )} A. Buffa (IMATI-CNR Italy) IGA mortaring 11 / 29

25 Non conforming interfaces and mortaring Let S p ( T j ) be the space of tensor product splines/nurbs of degree p, on the knot mesh T j. in each subdomain Ω j, V j = {v j H 1 (Ω j ) : v F j S p ( T j )} N V = {v L 2 (Ω) : v Ωj V j, v ΓD = 0} v 2 V = v 2 H 1 (Ω j ). i=1 A. Buffa (IMATI-CNR Italy) IGA mortaring 11 / 29

26 Non conforming interfaces and mortaring Let S p ( T j ) be the space of tensor product splines/nurbs of degree p, on the knot mesh T j. in each subdomain Ω j, V j = {v j H 1 (Ω j ) : v F j S p ( T j )} N V = {v L 2 (Ω) : v Ωj V j, v ΓD = 0} v 2 V = v 2 H 1 (Ω j ). Interface numbering and spaces Σ 0 = n I l=1 i=1 Γ l, l (i l, j l ) : Γ l = Ω il Ω jl. Continuity across Σ 0 imposed via Lagrange multipliers: M = {λ L 2 (Σ 0 ) : λ l = λ Γl M l } M l to be chosen properly! A. Buffa (IMATI-CNR Italy) IGA mortaring 11 / 29

27 Variational formulation of the problem Find u h V, λ h M such that a(u h, v h )+b(λ h, v h ) = R(v h ) b(µ h, u h ) = 0 v h V µ h M where a(u, v) = A u v b(λ, v) = λ l [u] [u] = u il u jl Ω l Γ l R(v) is the RHS taking into account also Neumann BC... A. Buffa (IMATI-CNR Italy) IGA mortaring 12 / 29

28 Variational formulation of the problem Find u h V, λ h M such that a(u h, v h )+b(λ h, v h ) = R(v h ) b(µ h, u h ) = 0 v h V µ h M where a(u, v) = A u v b(λ, v) = λ l [u] [u] = u il u jl Ω l Γ l R(v) is the RHS taking into account also Neumann BC... Wellposedness and approximation depends only upon the choice of Lagrange multipliers which should guarantee stability! A. Buffa (IMATI-CNR Italy) IGA mortaring 12 / 29

29 Variational formulation of the problem Find u h V, λ h M such that a(u h, v h )+b(λ h, v h ) = R(v h ) b(µ h, u h ) = 0 v h V µ h M where a(u, v) = A u v b(λ, v) = λ l [u] [u] = u il u jl Ω l Γ l R(v) is the RHS taking into account also Neumann BC... Wellposedness and approximation depends only upon the choice of Lagrange multipliers which should guarantee stability! M = {λ L 2 (Σ 0 ) : λ l = λ Γl M l } n I λ 2 M = λ l 2 (H 1/2 00 ) l=1 A. Buffa (IMATI-CNR Italy) IGA mortaring 12 / 29

30 Choice of the Langrange multiplier space Topology for M l is H 1/2 00 (Γ l) I want to have the largest possible set of multipliers such that the form b(λ, v) = Γ l λ l [u] remains uniformly stable Favorite choice: if i l is the slave side, we want M l V il Γl! It contraints all slave functions. But it is known that stability fails with this choice, and there is a need for cross point degree reduction.. A. Buffa (IMATI-CNR Italy) IGA mortaring 13 / 29

31 Choice of the Langrange multiplier space Topology for M l is H 1/2 00 (Γ l) I want to have the largest possible set of multipliers such that the form b(λ, v) = Γ l λ l [u] remains uniformly stable Favorite choice: if i l is the slave side, we want M l V il Γl! It contraints all slave functions. But it is known that stability fails with this choice, and there is a need for cross point degree reduction.. dim(m l ) dim{v V il Γl : v Γl = 0} A. Buffa (IMATI-CNR Italy) IGA mortaring 13 / 29

32 Choice of the Langrange multiplier space Each Γ l is a face of a subdomain Ω i (the slave side) Γ l inherits a spline mapping F l : (0, 1) d 1 Γ l and a parametric mesh on Γ = (0, 1) d 1 denoted as T l. A. Buffa (IMATI-CNR Italy) IGA mortaring 14 / 29

33 Choice of the Langrange multiplier space Each Γ l is a face of a subdomain Ω i (the slave side) Γ l inherits a spline mapping F l : (0, 1) d 1 Γ l and a parametric mesh on Γ = (0, 1) d 1 denoted as T l. Let us start with choices in the parametric space, and then we will map! A. Buffa (IMATI-CNR Italy) IGA mortaring 14 / 29

34 Choice of the Langrange multiplier space Each Γ l is a face of a subdomain Ω i (the slave side) Γ l inherits a spline mapping F l : (0, 1) d 1 Γ l and a parametric mesh on Γ = (0, 1) d 1 denoted as T l. Let us start with choices in the parametric space, and then we will map! Choice 1: same degree, cross point reduction M l 1 = S p ( T l ) 2 1 i h 2h 3h ζ A. Buffa (IMATI-CNR Italy) IGA mortaring 14 / 29

35 Choice of the Langrange multiplier space Each Γ l is a face of a subdomain Ω i (the slave side) Γ l inherits a spline mapping F l : (0, 1) d 1 Γ l and a parametric mesh on Γ = (0, 1) d 1 denoted as T l. Let us start with choices in the parametric space, and then we will map! Choice 2: degree reduction M l 2 = S p 2( T l ) Indeed, it is true that dim( M l 2 ) = { v S p( T il ) Γl : v Γl = 0} No need for degree reduction or other manipulation If stable, it will deliver a slightly suboptimal order : 1/2 suboptimal A. Buffa (IMATI-CNR Italy) IGA mortaring 14 / 29

36 Stability: numerical checks Chapelle-Bathe test Estimate on the inf-sup constant: A. Buffa (IMATI-CNR Italy) IGA mortaring 15 / 29

37 Stability: numerical checks Chapelle-Bathe test Estimate on the inf-sup constant: Testing M l against S p ( T il ) Γl without boundary conditions Stability constant Inverse of the number of elements P5 P5 P5 P4 P5 P3 P5 P2 P5 P1 P5 P0 O(E 1 ) Stability constant Pp Pp Pp P(p 2) Pp P(p 4) p p/p 2k couples are all stable! the pairings are also stable varying p! A. Buffa (IMATI-CNR Italy) IGA mortaring 15 / 29

38 Stability: numerical checks Chapelle-Bathe test Estimate on the inf-sup constant: Testing M l against S p ( T il ) Γl with boundary conditions... the right thing! 1 Stability constant P5 P5 P5 P3 P5 P Inverse of the number of elements Stability constant Pp Pp Pp P(p 2) Pp P(p 4) p p/p 2k couples are all stable! exponential behavior in p! A. Buffa (IMATI-CNR Italy) IGA mortaring 15 / 29

39 Stability: Proof of the inf-sup condition the p/p 2 case We consider M l 2 and can prove the following: µ v Γ inf sup α 0 v L 2 µ L 2 + quasi uniform meshes : inf µ S p 2 v S p H0 1 sup µ S p 2 v S p H0 1 µ v Γ α 0 v 1/2 H µ 1/2 00 (H 00 ) A. Buffa (IMATI-CNR Italy) IGA mortaring 16 / 29

40 Stability: Proof of the inf-sup condition the p/p 2 case We consider M l 2 and can prove the following: µ v Γ inf sup α 0 v L 2 µ L 2 + quasi uniform meshes : Proof In 2D: S p H0 1 x inf µ S p 2 v S p H0 1 sup µ S p 2 v S p H0 1 x µ v Γ α 0 v 1/2 H µ 1/2 00 (H 00 ) Sp 1 L 2 0 Sp 2 is exact choose v S p H0 1 such that 2 xx v = µ and the work with Sobolev norms. In 3D, basically the same applies... A. Buffa (IMATI-CNR Italy) IGA mortaring 16 / 29

41 Stability: Proof of the inf-sup condition the p/p 2 case We consider M l 2 and can prove the following: µ v Γ inf sup α 0 v L 2 µ L 2 + quasi uniform meshes : Proof In 2D: S p H0 1 x inf µ S p 2 v S p H0 1 sup µ S p 2 v S p H0 1 x µ v Γ α 0 v 1/2 H µ 1/2 00 (H 00 ) Sp 1 L 2 0 Sp 2 is exact choose v S p H0 1 such that 2 xx v = µ and the work with Sobolev norms. In 3D, basically the same applies... It is stable!... we need now to go to physical space A. Buffa (IMATI-CNR Italy) IGA mortaring 16 / 29

42 Stability in the physical space the p/p 2 case inf sup µ S p 2 v S p H0 1 Γ µ v v L 2 µ L 2 α 0 inf sup µ M l v V il :v H0 1(Γ l) Γ l µ v v L 2 µ L 2 α 0 A. Buffa (IMATI-CNR Italy) IGA mortaring 17 / 29

43 Stability in the physical space the p/p 2 case inf sup µ S p 2 v S p H0 1 Γ µ v v L 2 µ L 2 α 0 inf sup µ M l v V il :v H0 1(Γ l) Γ l µ v v L 2 µ L 2 α 0 Γ l µ v = Γ ρ µ v ρ = weight, area change.. A. Buffa (IMATI-CNR Italy) IGA mortaring 17 / 29

44 Stability in the physical space the p/p 2 case inf sup µ S p 2 v S p H0 1 Γ µ v v L 2 µ L 2 α 0 inf sup µ M l v V il :v H0 1(Γ l) Γ l µ v v L 2 µ L 2 Γ l µ v = Γ ρ µ v ρ = weight, area change.. and by super-convergence results à la Wahlbin: α 0 Π : L 2 ( Γ) M 2 l ρ µ Π(ρ µ) L 2 ( Γ) Ch µ L 2 ( Γ) A. Buffa (IMATI-CNR Italy) IGA mortaring 17 / 29

45 Stability in the physical space the p/p 2 case inf sup µ S p 2 v S p H0 1 Γ µ v v L 2 µ L 2 α 0 inf sup µ M l v V il :v H0 1(Γ l) Γ l µ v v L 2 µ L 2 Γ l µ v = Γ ρ µ v ρ = weight, area change.. and by super-convergence results à la Wahlbin: α 0 Π : L 2 ( Γ) M 2 l ρ µ Π(ρ µ) L 2 ( Γ) Ch µ L 2 ( Γ) For h small enough the stability holds in physical space! A. Buffa (IMATI-CNR Italy) IGA mortaring 17 / 29

46 Back to our variational problem Find u h V, λ h M such that a(u h, v h )+b(λ h, v) = R(v h ) b(µ h, u h ) = 0 v h V µ h M A. Buffa (IMATI-CNR Italy) IGA mortaring 18 / 29

47 Back to our variational problem Find u h V, λ h M such that a(u h, v h )+b(λ h, v) = R(v h ) b(µ h, u h ) = 0 v h V µ h M It is well-posed and verifies the following error estimate: if u H r (Ω): u u h V C inf v h V u v h V + inf µ h M λ µ h M (1) Ch t + Ch s t = min{p, r 1} (2) s = min{p + 1/2, r 1} for Choice 1: same degree, s = min{p 1/2, r 1} for Choice 2: degree reduction Or, indeed: u u h V C inf u v h V C... v h V Ker(B) A. Buffa (IMATI-CNR Italy) IGA mortaring 18 / 29

48 Numerical validation: problem Ω 1 γ γ 3 γ 2 Ω 3 1 Ω A. Buffa (IMATI-CNR Italy) IGA mortaring 19 / 29

49 Numerical validation: problem u uh V Mortar P4 P4 Mortar P4 P2 Mortar P3 P3 Mortar P3 P1 O(h 2/3 ) u n λ h L 2 (γ 2) Mortar P4 P4 Mortar P4 P2 Mortar P3 P3 Mortar P3 P1 O(h 1/6 ) primal dof number primal dof number 1/6 + 1/2 = 2/3 A. Buffa (IMATI-CNR Italy) IGA mortaring 20 / 29

50 Numerical validation: problem A. Buffa (IMATI-CNR Italy) IGA mortaring 21 / 29

51 Numerical validation: problem 2 u uh L 2 (Ω ) Mortar P4 P4 Mortar P4 P2 Conform P4 O(h 5 ) u uh L 2 (Ω ) Mortar P3 P3 Mortar P3 P1 Conform P3 O(h 4 ) primal dof number primal dof number Multipliers degree does not affect the order for the primal unknown! A. Buffa (IMATI-CNR Italy) IGA mortaring 22 / 29

52 Numerical validation: problem u n λ h L 2 (Γ) Mortar P4 P4 Mortar P4 P2 Mortar P3 P3 Mortar P3 P1 O(h 2 ), O(h 3 ), O(h 4 ),O(h 5 ) primal dof number but it affects the convergence of the multiplier! A. Buffa (IMATI-CNR Italy) IGA mortaring 23 / 29

53 Numerical validation: elasticity A. Buffa (IMATI-CNR Italy) IGA mortaring 24 / 29

54 Numerical validation: elasticity u uh V Mortar P4 P4 Mortar P4 P2 Mortar P4 P0 O(h 2 ), O(h 4 ) u uh V Mortar P4 P4 Mortar P4 P2 Mortar P4 P0 O(h 2 ), O(h 4 ) primal dof number primal dof number A. Buffa (IMATI-CNR Italy) IGA mortaring 25 / 29

55 Numerical validation: elasticity σ n λ h L 2 (Γ ) Mortar P4 P4 Mortar P4 P2 Mortar P4 P0 O(h), O(h 3 ), O(h 4 ) σ n λ h L 2 (Γ ) Mortar P4 P4 Mortar P4 P2 Mortar P4 P0 O(h), O(h 3 ), O(h 4 ) primal dof number primal dof number A. Buffa (IMATI-CNR Italy) IGA mortaring 26 / 29

56 Final remarks This approach can be used to treat contact in a variationally consistent way For patch gluing: the geometric non-matching patch cases should be studied in details The question of exact / inexact integration for interface integrals remains open (robustness of splines thanks to regularity) A. Buffa (IMATI-CNR Italy) IGA mortaring 27 / 29

57 Final remarks Surveys and Codes New Acta Numerica survey paper with several math results: L. Beirão Da Veiga, A. Buffa, G. Sangalli, R. Vázquez, Mathematical analysis of variational isogeometric methods We have two codes available to public : GeoPDEs library is a GNU licensed software available here: IGATools is a C++ dimension independent library igatools A. Buffa (IMATI-CNR Italy) IGA mortaring 28 / 29

58 THANKS! A. Buffa (IMATI-CNR Italy) IGA mortaring 29 / 29