Isogeometric Analysis: some approximation estimates for NURBS · Introduction to Isogeometric...

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Introduction to Isogeometric Analysis B-splines and NURBS Approximation theory for NURBS Conclusions

Isogeometric Analysis:some approximation estimates for NURBS

L. Beirao da Veiga, A. Buffa, Judith Rivas, G. Sangalli

Euskadi-Kyushu 2011Workshop 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, Isogeometricanalysis: CAD, finite elements, NURBS, exact geometry andmesh refinement, CMAME (2005).

Improve the connection between numerical simulation ofphysical phenomena and Computer Aided Design (CAD).

Eliminate/reduce the approximation of the computationaldomain and the need of remeshing

Tools:

Use the geometry provided by CAD

Represent unknown fields in the same basis functions usedfor the geometry

Tools:

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 un ∈ Xn such that

B(un, vn) = F(vn), ∀ vn ∈ Yn.

Xn ⊂ X, Yn ⊂ Y subspaces of dimension nFEM: piecewise polynomials

Spectral methods: Orthogonal (global) polynomials

IGA: B-splines, Non-Uniform Rational B-Splines (NURBS)

B(u, v) = F(v), ∀ v ∈ Y.

Xn ⊂ X, Yn ⊂ Y subspaces of dimension n

FEM: piecewise polynomials

B(u, v) = F(v), ∀ v ∈ Y.

Why to use Isogeometric Analysis?

In engineering problems geometry is usually defined by ComputerAided Design (CAD).

CAD and FEM use different geometry descriptions

CAD and IGA use the same geometry description

Why to use Isogeometric Analysis?

In engineering problems geometry is usually defined by ComputerAided Design (CAD).

CAD and FEM use different geometry descriptionsCAD and IGA use the same geometry description

B-splines in one dimension

n ∈ N number of basis functionsp ∈ N0 degree of polynomialsΞ := 0 = ξ1 6 ξ2 6 · · · 6 ξn+p+1 = 1 knot vectorζ1, . . . , ζm mesh in [0, 1]rj number of repetitions of ζj in Ξk = k1, . . . , km, kj = p − rj + 1

Bi,p piecewise polynomial of degree p and continuousderivatives up to the order kj − 1 at knot ζj, with compactsupport in [ξi, ξi+p+1]

Spk(Ξ) = spanB1,p, . . . , Bn,p

n ∈ N number of basis functionsp ∈ N0 degree of polynomialsΞ := 0 = ξ1 6 ξ2 6 · · · 6 ξn+p+1 = 1 knot vectorζ1, . . . , ζm mesh in [0, 1]rj number of repetitions of ζj in Ξk = k1, . . . , km, kj = p − rj + 1Bi,p piecewise polynomial of degree p and continuousderivatives up to the order kj − 1 at knot ζj, with compactsupport in [ξi, ξi+p+1]

B-splines in higher dimensions

B-splines in d dimensions

Bi1...id(x1, . . . , xd) = Bi1,p1(x1) . . . Bid,pd(xd).

Sp1,...,pdk1,...,kd

= Sp1k1⊗ · · · ⊗ S

= spanBi1...idn1,...,ndi=1,...,id=1.

B-spline curves in Rd

F(x) =∑n

i=1 Bi,p(x)Ci, Ci ∈ Rd control points.

B-spline surfaces in Rd

F(x, y) =∑n1,n2

i,j=1 Bij(x, y)Cij, Cij ∈ Rd control points.

Sp1,...,pdk1,...,kd

= Sp1k1⊗ · · · ⊗ S

= spanBi1...idn1,...,ndi=1,...,id=1.

F(x) =∑n

F(x, y) =∑n1,n2

Sp1,...,pdk1,...,kd

= Sp1k1⊗ · · · ⊗ S

= spanBi1...idn1,...,ndi=1,...,id=1.

F(x) =∑n

F(x, y) =∑n1,n2

NURBS in Ω ⊂ Rd

NURBS in Rd are conic projections of B-splines in Rd+1.

Ω = (0, 1)d:

w :=∑n1,...,nd

i1=1,...,id=1 wi1...idBi1...id , wi1...id > 1, weighting function

Ri1...id =wi1...idBi1...id

wNURBS basis functions in Ω

Ω ⊂ Rd:

F : Ω→ Ω, F =∑n1,...nd

i1=1,id=1 Ci1...idRi1...id geometrical map

K = F (Q), Q cartesian mesh in Ω

Ni1...id = Ri1...id F−1 NURBS basis functions in Ω

Np1,...,pdk1,...,kd

(K) = spanNi1...id

NURBS in Ω ⊂ Rd

Ω = (0, 1)d:

w :=∑n1,...,nd

Ω ⊂ Rd:

F : Ω→ Ω, F =∑n1,...nd

Np1,...,pdk1,...,kd

(K) = spanNi1...id

NURBS in Ω ⊂ Rd

Ω = (0, 1)d:

w :=∑n1,...,nd

Ω ⊂ Rd:

F : Ω→ Ω, F =∑n1,...nd

Np1,...,pdk1,...,kd

(K) = spanNi1...id

Properties of B-splines and NURBS

They form a partition of unity (for open knot vectors).

They are Ck continuous, with 0 6 k 6 p − 1.

The support of each basis function is compact.

NURBS represent exactly a wide class of curves, e.g. conicsections.Three kinds of refinement can be performed:

h-refinement = mesh refinementp-refinement = degree elevationk-refinement = regularity adjustment

h-, p- and k-refinement can be performed without changingthe geometry

Geometry description and refinement

Coarsest mesh: geometry description

Parametric domain, Ω = (0, 1)2 Physical domain Ω

wijBij

(wijBij

) F −1

The geometrical map F and the weight w are fixed at the coarsestlevel of discretization!

Coarsest mesh: geometry description

Parametric domain, Ω = (0, 1)2 Physical domain Ω

wijBij

(wijBij

) F −1

First refinementParametric domain, Ω = (0, 1)2 Physical domain Ω

wijBij

(wijBij

) F −1

Second refinement... and so onParametric domain, Ω = (0, 1)2 Physical domain Ω

wijBij

(wijBij

) F −1

A priori error estimates

Lemma (Cea)

Suppose Xn ⊂ X, is a family of finite-dimensional subspaces of aHIlbert space X. Suppose B : X × X→ R is a bounded, coercivebilinear form and F : X→ R is a continuous functional. Then theproblem of finding un ∈ Xn such that

B(un, vn) = F(vn), ∀ vn ∈ Xn,

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, un and n such that

||u − un||X 6 C infwn∈Xn

||u − wn||X.

Previous results: hp-estimates for FEM

Theorem

Let Ω ⊂ R2 be a polygon, T a parallelogram mesh in Ω with atmost one hanging node per edge and let h denote its diameter.Then, for any 2 6 s 6 p + 1 and any u ∈ Hs(Ω), there existsΠu ∈ S

p0(T) such that

||u − Πu||H1(Ω) 6 Chs−1p−(s−1)|u|Hs(Ω),

where C is a constant independent of h and p.

C. Schwab, p- and hp- Finite Element Methods. Theory andapplications in Solid and Fluid Mechanics, Oxford UniversityPress, Oxford (1998).

Previous results: h-estimates for NURBS in 2D

Theorem

Let Ω = F (Ω) ⊂ R2, K = F (Q) a mesh in Ω and let h denote itsdiameter. Then, for any 0 6 ` 6 s 6 p + 1 and for any u ∈ Hs(Ω),there exists Πu ∈ N

pk(K) such that

|u − Πu|H`(Ω) 6 C(w, F , p, k)hs−`||u||Hs(Ω),

where C is a constant independent of h, but possibly depending onp 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) ⊂ R2, K = F (Q) a mesh in Ω with diameter h.Then, if 2k 6 p + 1, for any u ∈ Hs(Ω), with 2k 6 s 6 p + 1, andfor any 0 6 ` 6 s 6 p + 1 there exists Πu ∈ N

pk(K) such that

|u − Πu|H`(Ω) 6 C(w, F )hs−`(p − k + 1)−(s−`)‖u‖Hs(Ω),

where C is a constant independent of h, p and k.

L. Beirão da Veiga, A. Buffa, J. Rivas, G. Sangalli, Someestimates for h − p − k−refinement in Isogeometric Analysis,to appear in Numer. Math.

Construction of the new projection operator

i(!1,1)

tensorproduct

Legendre polynomials

Definition (Legendre polynomial of degree i)

Li(x) =1

i! 2idi

((x2 − 1)i), i = 0, 1, . . .

Definition (L2(−1, 1)-orthogonal projection of order N ∈ N)

πNϕ(x) =N∑

ϕiLi(x), where ϕi =2i + 1

−1ϕ(x)Li(x) dx.

Definition (Primitives of Legendre polynomials)

For n > 0, Ψi,n is the n-th primitive of Li,

Ψi,0(x) = Li(x), Ψi,n(x) =∫ x

−1Ψi,n−1(ξ) dξ, n = 1, 2, . . .

Li(x) =1

i! 2idi

((x2 − 1)i), i = 0, 1, . . .

πNϕ(x) =N∑

−1ϕ(x)Li(x) dx.

Ψi,0(x) = Li(x), Ψi,n(x) =∫ x

−1Ψi,n−1(ξ) dξ, n = 1, 2, . . .

Li(x) =1

i! 2idi

((x2 − 1)i), i = 0, 1, . . .

πNϕ(x) =N∑

−1ϕ(x)Li(x) dx.

Ψi,0(x) = Li(x), Ψi,n(x) =∫ x

−1Ψi,n−1(ξ) dξ, n = 1, 2, . . .

Projection operator in (−1, 1)

Definitionp, k nonnegative integers;

Sp(Λ) set of polynomials of degree 6 p in Λ = (−1, 1);

πp,k : Hk(Λ)→ Sp(Λ) is defined as:

(πp,ku)(k)(x) =πp−ku(k)(x), x ∈ Λ,

(πp,ku)(`)(−1) =u(`)(−1), ` = 0, 1, . . . , k − 1,

If u(k)(x) =∑∞

i=0 αiLi(x), then

πp,ku(x) =p−k∑i=0

αiΨi,k(x) +k−1∑m=0

u(m)(−1)(x + 1)m

If p > 2k − 1, (πp,ku)(`)(1) = u(`)(1), ` = 0, 1, . . . , k − 1.

(πp,ku)(`)(−1) =u(`)(−1), ` = 0, 1, . . . , k − 1,

If u(k)(x) =∑∞

i=0 αiLi(x), then

u(m)(−1)(x + 1)m

If p > 2k − 1, (πp,ku)(`)(1) = u(`)(1), ` = 0, 1, . . . , k − 1.

(πp,ku)(`)(−1) =u(`)(−1), ` = 0, 1, . . . , k − 1,

If u(k)(x) =∑∞

i=0 αiLi(x), then

u(m)(−1)(x + 1)m

If p > 2k − 1, (πp,ku)(`)(1) = u(`)(1), ` = 0, 1, . . . , k − 1.

Spline approximation on the reference domain [0, 1]

Definitionp, k nonnegative integers, with 2k − 1 6 p;

0 = ζ1 < ζ2 < · · · < ζm = 1, Ii = (ζi, ζi+1), 1 6 i 6 m − 1;

Ti : Λ→ Ii linear mapping.

The (local) projection operator πip,k : Hk(Ii)→ Sp is defined as:

πip,ku Ti =

(πp,k(u Ti)

The (global) projection operator πp,k : Hk(0, 1)→ Spk is defined as:

(πp,ku)|Ii = πip,k, i = 0, . . . , m − 1.

Remark

(πp,ku)(`)(ζi) = u(`)(ζi), 1 6 i 6 m, 0 6 ` 6 k − 1.

0 = ζ1 < ζ2 < · · · < ζm = 1, Ii = (ζi, ζi+1), 1 6 i 6 m − 1;

πip,ku Ti =

(πp,k(u Ti)

(πp,ku)|Ii = πip,k, i = 0, . . . , m − 1.

Remark

(πp,ku)(`)(ζi) = u(`)(ζi), 1 6 i 6 m, 0 6 ` 6 k − 1.

0 = ζ1 < ζ2 < · · · < ζm = 1, Ii = (ζi, ζi+1), 1 6 i 6 m − 1;

πip,ku Ti =

(πp,k(u Ti)

(πp,ku)|Ii = πip,k, i = 0, . . . , m − 1.

Remark

(πp,ku)(`)(ζi) = u(`)(ζi), 1 6 i 6 m, 0 6 ` 6 k − 1.

Error estimate for πp,k

Theoremp, k nonnegative integers, 2k − 1 6 p;

0 = ζ1 < · · · < ζm = 1, Ii = (ζi, ζi+1), hi = ζi+1 − ζi;

u(k) ∈ Hs(Ii) for some 0 6 s 6 κ = p − k + 1.

Then, for ` = 0, . . . , k,

‖u(`) − (πp,ku)(`)‖2L2(Ii)

)2(s+k−`) (κ− s)!(κ+ s)!

(κ− (k − `))!(κ+ (k − `))!

|u(k)|2Hs(Ii).

Consequently, for u ∈ Hσ, k 6 σ 6 p + 1, and ` = 0, . . . , k, thereexists a constant C independent of u, `, σ, , p and k, s.t.

|u − πp,ku|H` 6 Cσ−`(p − k + 1)−(σ−`)|u|Hσ .

Then, for ` = 0, . . . , k,

‖u(`) − (πp,ku)(`)‖2L2(Ii)

)2(s+k−`) (κ− s)!(κ+ s)!

(κ− (k − `))!(κ+ (k − `))!

|u(k)|2Hs(Ii).

Consequently, for u ∈ Hσ(Ii), k 6 σ 6 p + 1, and ` = 0, . . . , k,there exists a constant C independent of u, `, σ, hi, p and k, s.t.

|u − πp,ku|H`(Ii)6 Chi

σ−`(p − k + 1)−(σ−`)|u|Hσ(Ii).

Then, for ` = 0, . . . , k,

‖u(`) − (πp,ku)(`)‖2L2(Ii)

)2(s+k−`) (κ− s)!(κ+ s)!

(κ− (k − `))!(κ+ (k − `))!

|u(k)|2Hs(Ii).

Consequently, for u ∈ Hσ(0, 1), k 6 σ 6 p + 1, and ` = 0, . . . , k,there exists a constant C independent of u, `, σ, h, p and k, s.t.

|u − πp,ku|H`(0,1) 6 Chσ−`(p − k + 1)−(σ−`)|u|Hσ(0,1).

Spline approximation on the reference domain Ω = [0, 1]2

Definition

p = (p1, p2), k = (k1, k2), 2kd − 1 6 pd, d = 1, 2;

Qij = Ii × Jj = (ζi,1, ζi+1,1)× (ζj,2, ζj+1,2) ∈ Qh,

Hk1,k2(Qij) = Hk1(Ii, Hk2(Jj))

Sp1,p2(Qij) = u : Qij → R : u(·, y) ∈ Sp1(Ii), u(x, ·) ∈ Sp2(Jj).

The (local) projection operator Πijp,k : Hk1,k2(Qij)→ Sp1,p2(Qij) is:

Πijp,k = π

ip1,k1⊗ πj

p2,k2.

The (global) projection operator Πp,k : Hk1,k2(Ω)→ Sp1,p2k1,k2

(Qh) is:

(Πp,kv)|Qij = (Πijp,kv),∀Qij ∈ Qh.

Spline approximation on the reference domain Ω = [0, 1]2

Definition

p = (p1, p2), k = (k1, k2), 2kd − 1 6 pd, d = 1, 2;

Qij = Ii × Jj = (ζi,1, ζi+1,1)× (ζj,2, ζj+1,2) ∈ Qh,

Hk1,k2(Qij) = Hk1(Ii, Hk2(Jj))

Sp1,p2(Qij) = u : Qij → R : u(·, y) ∈ Sp1(Ii), u(x, ·) ∈ Sp2(Jj).

The (local) projection operator Πijp,k : Hk1,k2(Qij)→ Sp1,p2(Qij) is:

Πijp,k = π

ip1,k1⊗ πj

p2,k2.

The (global) projection operator Πp,k : Hk1,k2(Ω)→ Sp1,p2k1,k2

(Qh) is:

(Πp,kv)|Qij = (Πijp,kv),∀Qij ∈ Qh.

Error estimate for Πp,k

Theoremp1 = p2 = p;

k1, k2 be nonnegative integers, 2kd − 1 6 p for d = 1, 2k∗ = mink1, k2, k∗ = maxk1, k2;

Qij = (ζi,1, ζi+1,1)× (ζj,2, ζj+1,2),hij = maxζi+1,1 − ζi,1, ζj+1,2 − ζj,2, , h = max hij;

v ∈ Hσ(Qij) with k1 + k2 6 σ 6 p + 1.

Then, for all integers 0 6 ` 6 k∗, there exists a positive constant C,independent of v, σ, `, h, p, k1 and k2, such that,

|v − Πp,kv|H`(Ω)6 Chσ−`(p − k∗ + 1)−(σ−`)|v|Hσ(Ω)

NURBS approximation in the physical domain Ω ⊂ R2

Definition

p = (p1, p2), k = (k1, k2), 2kd − 1 6 pd, d = 1, 2;

Kij = F (Qij), Qij ∈ Qh,

The (local) projection operator for functions defined on Kij is:

ΠijN : Hσ(Kij)→ N

p1,p2k1,k2

, σ > k1 + k2

ΠijN(v) F =

Πijp,k(w (v F ))

The (global) projection operator for functions defined on Ω isΠN : Hσ(Ω)→ N

p1,p2k1,k2

, σ > k1 + k2,

ΠN(v)|Kij = ΠijN(v|Kij) ∀Kij ∈ Kh.

NURBS approximation in the physical domain Ω ⊂ R2

Definition

p = (p1, p2), k = (k1, k2), 2kd − 1 6 pd, d = 1, 2;

Kij = F (Qij), Qij ∈ Qh,

The (local) projection operator for functions defined on Kij is:

ΠijN : Hσ(Kij)→ N

p1,p2k1,k2

, σ > k1 + k2

ΠijN(v) F =

Πijp,k(w (v F ))

The (global) projection operator for functions defined on Ω isΠN : Hσ(Ω)→ N

p1,p2k1,k2

, σ > k1 + k2,

ΠN(v)|Kij = ΠijN(v|Kij) ∀Kij ∈ Kh.

Error estimate for ΠN

Theoremp1 = p2 = p;

k1, k2 nonnegative integers, 2kd − 1 6 p for d = 1, 2k∗ = mink1, k2, k∗ = maxk1, k2;

w and F fixed at the coarsest level of discretization;

K ∈ Kh, hK = diam K, h = max hK;

v ∈ Hσ(K) with k1 + k2 6 σ 6 p + 1;

Then for ` = 0, . . . , k∗, there exists a constant C = C(w, F ),independent of v, σ, `, h, p, k1 and k2 such that

|v − ΠN(v)|H`(Ω) 6 Chσ−`(p − k∗ + 1)−(σ−`)‖v‖Hσ(Ω).

Concluding remarks

We have constructed a new projection operator onto NURBSspaces in 2 dimensions and given error estimates in Sobolevnorms which are explicit in the three discretizationparameters: degree p, regularity k and mesh size h.

A restriction on the regularity must be imposed, namely2k − 1 6 p.

The case of higher regularity, up to k = p remains open.

Thank you

Concluding remarks

Thank you

Concluding remarks

Thank you

Concluding remarks

Thank you

Isogeometric Analysis: some approximation estimates for NURBS · Introduction to Isogeometric...

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