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Isogeometric Analysis: some approximation estimates for NURBS L. Beirao da Veiga, A. Buffa, Judith Rivas, G. Sangalli Euskadi-Kyushu 2011 Workshop on Applied Mathematics BCAM, March t0th, 2011

Outline of the talk 1 Introduction to Isogeometric Analysis 2 B-splines and NURBS 3 Approximation theory for NURBS 4 Conclusions

What is Isogeometric Analysis? Novel technique for the discretization of PDEs T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, CMAME (2005).

What is Isogeometric Analysis? Novel technique for the discretization of PDEs Aim: T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, CMAME (2005). Improve the connection between numerical simulation of physical phenomena and Computer Aided Design (CAD). Eliminate/reduce the approximation of the computational domain and the need of remeshing

What is Isogeometric Analysis? Novel technique for the discretization of PDEs Aim: Tools: T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, CMAME (2005). Improve the connection between numerical simulation of physical phenomena and Computer Aided Design (CAD). Eliminate/reduce the approximation of the computational domain and the need of remeshing Use the geometry provided by CAD Represent unknown fields in the same basis functions used for the geometry

What is Isogeometric Analysis? Variational formulation of a BVP: Find u X such that B(u, v) = F(v), v Y.

What is Isogeometric Analysis? Variational formulation of a BVP: Find u X such that B(u, v) = F(v), v Y. X, Y reflexive Banach spaces B: X Y R continuous bilinear form F: Y R continuous linear functional Galerkin approximation: Find u n X n such that B(u n, v n ) = F(v n ), v n Y n. X n X, Y n Y subspaces of dimension n

Why to use Isogeometric Analysis? In engineering problems geometry is usually defined by Computer Aided Design (CAD). CAD and FEM use different geometry descriptions

Why to use Isogeometric Analysis? In engineering problems geometry is usually defined by Computer Aided Design (CAD). CAD and FEM use different geometry descriptions CAD and IGA use the same geometry description

B-splines in one dimension n N number of basis functions p N 0 degree of polynomials Ξ := {0 = ξ 1 ξ 2 ξ n+p+1 = 1} knot vector {ζ 1,..., ζ m } mesh in [0, 1] r j number of repetitions of ζ j in Ξ k = {k 1,..., k m }, k j = p r j + 1 B i,p piecewise polynomial of degree p and continuous derivatives up to the order k j 1 at knot ζ j, with compact support in [ξ i, ξ i+p+1 ] S p k (Ξ) = span{b 1,p,..., B n,p }

B-splines in higher dimensions B-splines in d dimensions B i1...i d (x 1,..., x d ) = B i1,p 1 (x 1 )... B id,p d (x d ). S p 1,...,p d k 1,...,k d = S p 1 k 1 S p d k d = span{b i1...i d } n 1,...,n d i=1,...,i d =1. B-spline curves in R d F(x) = n i=1 B i,p(x)c i, C i R d control points.

NURBS in Ω R d NURBS in R d are conic projections of B-splines in R d+1.

NURBS in Ω R d NURBS in R d are conic projections of B-splines in R d+1. Ω = (0, 1) d : w := n 1,...,n d i 1 =1,...,i d =1 w i 1...i d B i1...i d, w i1...i d 1, weighting function R i1...i d = w i 1...i d B i1...i d w NURBS basis functions in Ω Ω R d : F : Ω Ω, F = n1,...n d i 1 =1,i d =1 C i 1...i d R i1...i d geometrical map K = F (Q), Q cartesian mesh in Ω N i1...i d = R i1...i d F 1 NURBS basis functions in Ω N p 1,...,p d k 1,...,k d (K) = span{n i1...i d }

Properties of B-splines and NURBS They form a partition of unity (for open knot vectors). They are C k continuous, with 0 k p 1. The support of each basis function is compact. NURBS represent exactly a wide class of curves, e.g. conic sections. Three kinds of refinement can be performed: h-refinement = mesh refinement p-refinement = degree elevation k-refinement = regularity adjustment h-, p- and k-refinement can be performed without changing the geometry

Geometry description and refinement Coarsest mesh: geometry description Parametric domain, Ω = (0, 1) 2 Physical domain Ω F { R ij = w } { ijb ij N ij = w ( ) } wij B ij F 1 w

Geometry description and refinement Coarsest mesh: geometry description Parametric domain, Ω = (0, 1) 2 Physical domain Ω F { R ij = w } { ijb ij N ij = w ( ) } wij B ij F 1 w The geometrical map F and the weight w are fixed at the coarsest level of discretization!

Geometry description and refinement First refinement Parametric domain, Ω = (0, 1) 2 Physical domain Ω F { R ij = w } { ijb ij N ij = w ( ) } wij B ij F 1 w The geometrical map F and the weight w are fixed at the coarsest level of discretization!

Geometry description and refinement Second refinement... and so on Parametric domain, Ω = (0, 1) 2 Physical domain Ω F { R ij = w } { ijb ij N ij = w ( ) } wij B ij F 1 w The geometrical map F and the weight w are fixed at the coarsest level of discretization!

A priori error estimates Lemma (Cea) Suppose X n X, is a family of finite-dimensional subspaces of a HIlbert space X. Suppose B: X X R is a bounded, coercive bilinear form and F: X R is a continuous functional. Then the problem of finding u n X n such that B(u n, v n ) = F(v n ), v n X n, has a unique solution. If u X is the solution of B(u, v) = F(v), v X, then there exists a constant C independent of u, u n and n such that u u n X C inf w n X n u w n X.

Previous results: hp-estimates for FEM Theorem Let Ω R 2 be a polygon, T a parallelogram mesh in Ω with at most one hanging node per edge and let h denote its diameter. Then, for any 2 s p + 1 and any u H s (Ω), there exists Πu S p 0 (T) such that u Πu H 1 (Ω) Ch s 1 p (s 1) u H s (Ω), where C is a constant independent of h and p. C. Schwab, p- and hp- Finite Element Methods. Theory and applications in Solid and Fluid Mechanics, Oxford University Press, Oxford (1998).

Previous results: h-estimates for NURBS in 2D Theorem Let Ω = F ( Ω) R 2, K = F (Q) a mesh in Ω and let h denote its diameter. Then, for any 0 l s p + 1 and for any u H s (Ω), there exists Πu N p k (K) such that u Πu H l (Ω) C(w, F, p, k)h s l u H s (Ω), where C is a constant independent of h, but possibly depending on p and k. Y. Bazilevs, L. Beirão da Veiga, J. A. Cottrell, T. J. R. Hughes, and G. Sangalli, Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Math. Models Methods Appl. Sci., 16 (2006), pp. 1031 1090.

Our result: hpk-estimates for NURBS in 2D Theorem Let Ω = F ( Ω) R 2, K = F (Q) a mesh in Ω with diameter h. Then, if 2k p + 1, for any u H s (Ω), with 2k s p + 1, and for any 0 l s p + 1 there exists Πu N p k (K) such that u Πu H l (Ω) C(w, F )h s l (p k + 1) (s l) u H s (Ω), where C is a constant independent of h, p and k. L. Beirão da Veiga, A. Buffa, J. Rivas, G. Sangalli, Some estimates for h p k refinement in Isogeometric Analysis, to appear in Numer. Math.

Construction of the new projection operator (!1,1) I i T i tensor product F

Legendre polynomials Definition (Legendre polynomial of degree i) L i (x) = 1 i! 2 i d i dx i ( (x 2 1) i), i = 0, 1,...

Legendre polynomials Definition (Legendre polynomial of degree i) L i (x) = 1 i! 2 i d i dx i ( (x 2 1) i), i = 0, 1,... Definition (L 2 ( 1, 1)-orthogonal projection of order N N) π N ϕ(x) = N i=0 ˆϕ i L i (x), where ˆϕ i = 2i + 1 2 1 1 ϕ(x)l i (x) dx. Definition (Primitives of Legendre polynomials) For n 0, Ψ i,n is the n-th primitive of L i, Ψ i,0 (x) = L i (x), Ψ i,n (x) = x 1 Ψ i,n 1 (ξ) dξ, n = 1, 2,...

Projection operator in ( 1, 1) Definition p, k nonnegative integers; S p (Λ) set of polynomials of degree p in Λ = ( 1, 1); π p,k : H k (Λ) S p (Λ) is defined as: ( π p,k u) (k) (x) =π p k u (k) (x), x Λ, ( π p,k u) (l) ( 1) =u (l) ( 1), l = 0, 1,..., k 1, If u (k) (x) = i=0 α il i (x), then p k π p,k u(x) = α i Ψ i,k (x) + i=0 k 1 m=0 u (m) ( 1) (x + 1)m m!

Spline approximation on the reference domain [0, 1] Definition p, k nonnegative integers, with 2k 1 p; {0 = ζ 1 < ζ 2 < < ζ m = 1}, I i = (ζ i, ζ i+1 ), 1 i m 1; T i : Λ I i linear mapping. The (local) projection operator π i p,k : Hk (I i ) S p is defined as: π i p,k u T i = ( π p,k (u T i ) ). The (global) projection operator π p,k : H k (0, 1) S p k is defined as: (π p,k u) Ii = π i p,k, i = 0,..., m 1.

Error estimate for π p,k Theorem p, k nonnegative integers, 2k 1 p; {0 = ζ 1 < < ζ m = 1}, I i = (ζ i, ζ i+1 ), h i = ζ i+1 ζ i ; u (k) H s (I i ) for some 0 s κ = p k + 1. Then, for l = 0,..., k, u (l) (π p,k u) (l) 2 L 2 (I i ) ( hi ) 2(s+k l) (κ s)! (κ (k l))! 2 (κ + s)! (κ + (k l))! u(k) 2 H s (I i ). Consequently, for u H σ (I i ), k σ p + 1, and l = 0,..., k, there exists a constant C independent of u, l, σ, h i, p and k, s.t. u π p,k u H l (I i ) Ch i σ l (p k + 1) (σ l) u H σ (I i ).

Error estimate for π p,k Theorem p, k nonnegative integers, 2k 1 p; {0 = ζ 1 < < ζ m = 1}, I i = (ζ i, ζ i+1 ), h i = ζ i+1 ζ i ; u (k) H s (I i ) for some 0 s κ = p k + 1. Then, for l = 0,..., k, u (l) (π p,k u) (l) 2 L 2 (I i ) ( hi ) 2(s+k l) (κ s)! (κ (k l))! 2 (κ + s)! (κ + (k l))! u(k) 2 H s (I i ). Consequently, for u H σ (0, 1), k σ p + 1, and l = 0,..., k, there exists a constant C independent of u, l, σ, h, p and k, s.t. u π p,k u H l (0,1) Ch σ l (p k + 1) (σ l) u H σ (0,1).

Spline approximation on the reference domain Ω = [0, 1] 2 Definition p = (p 1, p 2 ), k = (k 1, k 2 ), 2k d 1 p d, d = 1, 2; Q ij = I i J j = (ζ i,1, ζ i+1,1 ) (ζ j,2, ζ j+1,2 ) Q h, H k 1,k 2 (Q ij ) = H k 1(I i, H k 2(J j )) S p 1,p 2 (Q ij ) = {u: Q ij R : u(, y) S p 1(I i ), u(x, ) S p 2(J j )}. The (local) projection operator Π ij p,k : Hk 1,k 2 (Q ij ) S p 1,p 2 (Q ij ) is: Π ij p,k = πi p 1,k 1 π j p 2,k 2.

Error estimate for Π p,k Theorem p 1 = p 2 = p; k 1, k 2 be nonnegative integers, 2k d 1 p for d = 1, 2 k = min{k 1, k 2 }, k = max{k 1, k 2 }; Q ij = (ζ i,1, ζ i+1,1 ) (ζ j,2, ζ j+1,2 ), h ij = max{ζ i+1,1 ζ i,1, ζ j+1,2 ζ j,2 },, h = max h ij ; v H σ (Q ij ) with k 1 + k 2 σ p + 1. Then, for all integers 0 l k, there exists a positive constant C, independent of v, σ, l, h, p, k 1 and k 2, such that, v Π p,k v H l ( Ω) Chσ l (p k + 1) (σ l) v H σ ( Ω).

NURBS approximation in the physical domain Ω R 2 Definition p = (p 1, p 2 ), k = (k 1, k 2 ), 2k d 1 p d, d = 1, 2; K ij = F (Q ij ), Q ij Q h, The (local) projection operator for functions defined on K ij is: Π ij N : Hσ (K ij ) N p 1,p 2 k 1,k 2, σ k 1 + k 2 Π ij Πij p,k (w (v F )) N (v) F =. w The (global) projection operator for functions defined on Ω is Π N : H σ (Ω) N p 1,p 2 k 1,k 2, σ k 1 + k 2, Π N (v) Kij = Π ij N (v K ij ) K ij K h.

Error estimate for Π N Theorem p 1 = p 2 = p; k 1, k 2 nonnegative integers, 2k d 1 p for d = 1, 2 k = min{k 1, k 2 }, k = max{k 1, k 2 }; w and F fixed at the coarsest level of discretization; K K h, h K = diam K, h = max h K ; v H σ (K) with k 1 + k 2 σ p + 1; Then for l = 0,..., k, there exists a constant C = C(w, F ), independent of v, σ, l, h, p, k 1 and k 2 such that v Π N (v) H l (Ω) Ch σ l (p k + 1) (σ l) v H σ (Ω).

Concluding remarks We have constructed a new projection operator onto NURBS spaces in 2 dimensions and given error estimates in Sobolev norms which are explicit in the three discretization parameters: degree p, regularity k and mesh size h. A restriction on the regularity must be imposed, namely 2k 1 p.