Isogeometric Analysis (IGA) G. Sangalli 1 , 2 in collaboration with: - - PowerPoint PPT Presentation

Some topics in:

Isogeometric Analysis (IGA)

• G. Sangalli1,2

in collaboration with: A. Buffa2, R. Vázquez2

1University of Pavia, 2IMATI-CNR “E. Magenes” (Pavia),

Workshop on: Discretization Methods for Polygonal and Polyhedral Meshes September 17-19, 2012 University of Milano-Bicocca

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 1 / 23

Goal and features of IGA

accurate and less costy mesh construction simplify mesh refinement allow discrete fields with higher global regularity

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 2 / 23

Goal and features of IGA

accurate and less costy mesh construction simplify mesh refinement allow discrete fields with higher global regularity directly based on CAD tools

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 2 / 23

Why IGA?

CAD control points are the natural unknowns in simulation of solid deformation isoparametric paradigm (picture from: [Xu, Mourrain, Duvigneau, Galligod, CAD, 2011] )

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 3 / 23

Outline

1

Splines, NURBS, control pts., CAD and d.o.f.s representation

2

Isogeometric De Rham compatible fields

3

Unstructured discretization by T-splines

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 4 / 23

B-splines 1D

B-splines are (a basis for) piecewise polynomials of degree p defined from a knot vector {ξ1, ..., ξn+p+1}: Bi,0(ξ) =

• 1 if ξi ≤ ξ < ξi+1

0 otherwise. Bi,p(ξ) = ξ − ξi ξi+p − ξi Bi,p−1(ξ) + ξi+p+1 − ξ ξi+p+1 − ξi+1 Bi+1,p−1(ξ).

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 5 / 23

B-splines 1D

Unlike finite element shape functions: number of continuous derivatives is (p−“knot multiplicity”) at the knots: maximum regularity is Cp−1 definition of a B-splines depends on the knots in its support element d.o.f.s cannot be defined (for maximum regularity)

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 5 / 23

B-spline curve

Linear combination of Ci = • and Bi = gives the parametrization F(ξ) =

i CiBi(ξ) of a B-spline curve:

[0, 5]

F

− − − − − − − − →

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 6 / 23

Spline curve

CAD users interact with the control pts to input/modify the curve the control polygon connects the control points the relation between the curve and its control polygon depends on the B-spline basis (partition of unity, ONTP , ... )

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 7 / 23

Spline curve

CAD users interact with the control pts to input/modify the curve the control polygon connects the control points the relation between the curve and its control polygon depends on the B-spline basis (partition of unity, ONTP , ... ) the control polygon converges O(h2) to the curve for h-refinement (knot-insertion) or O(p) for p-refinement (degree-raising)

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 7 / 23

Spline curve

CAD users interact with the control pts to input/modify the curve the control polygon connects the control points the relation between the curve and its control polygon depends on the B-spline basis (partition of unity, ONTP , ... ) the control polygon converges O(h2) to the curve for h-refinement (knot-insertion) or O(p) for p-refinement (degree-raising)

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 7 / 23

NURBS curve

A NURBS curve in R2 is the projection of a B-spline in R3 C(ξ) = [Cw

x (ξ), Cw y (ξ)]

Cw

z (ξ)

=

n

• i=1

Ci wiBi,p(ξ) n

b i=1 wb iBb i,p(ξ) = n

• i=1

CiRi,p(ξ).

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 8 / 23

Control polygon convergence

Given the knot vector (mesh) M = {ξi}i=1,...,n+p+1 = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} we introduce by knot averaging the Greville mesh MG = {γi}i=1,...,n = {0, 0.5, 1.5, 2.5, 3.5, 4, 4.5, 5} consider also the usual piecewise linear Lagrange basis on MG: λi(γj) = δi,j ξi = ◦, γi = +, B1 in blue, λ1 in black

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence

Given the knot vector (mesh) M = {ξi}i=1,...,n+p+1 = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} we introduce by knot averaging the Greville mesh MG = {γi}i=1,...,n = {0, 0.5, 1.5, 2.5, 3.5, 4, 4.5, 5} consider also the usual piecewise linear Lagrange basis on MG: λi(γj) = δi,j ξi = ◦, γi = +, B2 in blue, λ2 in black

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence

Given the knot vector (mesh) M = {ξi}i=1,...,n+p+1 = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} we introduce by knot averaging the Greville mesh MG = {γi}i=1,...,n = {0, 0.5, 1.5, 2.5, 3.5, 4, 4.5, 5} consider also the usual piecewise linear Lagrange basis on MG: λi(γj) = δi,j ξi = ◦, γi = +, B3 in blue, λ3 in black

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence

Given the knot vector (mesh) M = {ξi}i=1,...,n+p+1 = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} we introduce by knot averaging the Greville mesh MG = {γi}i=1,...,n = {0, 0.5, 1.5, 2.5, 3.5, 4, 4.5, 5} consider also the usual piecewise linear Lagrange basis on MG: λi(γj) = δi,j ξi = ◦, γi = +, B4 in blue, λ4 in black

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence

Given the knot vector (mesh) M = {ξi}i=1,...,n+p+1 = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} we introduce by knot averaging the Greville mesh MG = {γi}i=1,...,n = {0, 0.5, 1.5, 2.5, 3.5, 4, 4.5, 5} consider also the usual piecewise linear Lagrange basis on MG: λi(γj) = δi,j ξi = ◦, γi = +, B5 in blue, λ5 in black

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence

Given the knot vector (mesh) M = {ξi}i=1,...,n+p+1 = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} we introduce by knot averaging the Greville mesh MG = {γi}i=1,...,n = {0, 0.5, 1.5, 2.5, 3.5, 4, 4.5, 5} consider also the usual piecewise linear Lagrange basis on MG: λi(γj) = δi,j ξi = ◦, γi = +, B6 in blue, λ6 in black

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence

Given the knot vector (mesh) M = {ξi}i=1,...,n+p+1 = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} we introduce by knot averaging the Greville mesh MG = {γi}i=1,...,n = {0, 0.5, 1.5, 2.5, 3.5, 4, 4.5, 5} consider also the usual piecewise linear Lagrange basis on MG: λi(γj) = δi,j ξi = ◦, γi = +, B7 in blue, λ7 in black

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence

Given the knot vector (mesh) M = {ξi}i=1,...,n+p+1 = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} we introduce by knot averaging the Greville mesh MG = {γi}i=1,...,n = {0, 0.5, 1.5, 2.5, 3.5, 4, 4.5, 5} consider also the usual piecewise linear Lagrange basis on MG: λi(γj) = δi,j ξi = ◦, γi = +, B8 in blue, λ8 in black

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence

Given the knot vector (mesh) M = {ξi}i=1,...,n+p+1 = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} we introduce by knot averaging the Greville mesh MG = {γi}i=1,...,n = {0, 0.5, 1.5, 2.5, 3.5, 4, 4.5, 5}

Theorem ([de Boor, BOOK, 2001] )

If the curve is parametrized by F(·) = n

i=1 CiBi(·) then the control

polygon is parametrized by FC(·) = n

i=1 Ciλi(·) and it holds

F(·) − FC(·)L∞ ≃ h2,

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 9 / 23

Isogeometric fields and control fields

Isogeometric discrete fields are (suitable) push-forward of splines: φ(x) =

• i=1,...,n

ci Bi ◦ F−1(x), x ∈ spline curve, the associated control field shares the same d.o.f.s but is push-forward

• f piecewise linears through FC:

φC(x) =

• i=1,...,n

ci λi ◦ F−1

C (x),

x ∈ control polygon.

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 10 / 23

Isogeometric fields and control fields

Isogeometric discrete fields are (suitable) push-forward of splines: φ(x) =

• i=1,...,n

ci Bi ◦ F−1(x), x ∈ spline curve, the associated control field shares the same d.o.f.s but is push-forward

• f piecewise linears through FC:

φC(x) =

• i=1,...,n

ci λi ◦ F−1

C (x),

x ∈ control polygon. distance between the two functions is O(h2), if the d.o.f.s are chosen wisely, φ delivers approximation error O(hp+1) while φC delivers approximation error O(h2).

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 10 / 23

Isogeometric fields and control fields

Spline curve in blue, control polygon in black

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 11 / 23

Isogeometric fields and control fields

IGA scalar field in blue, control field in black

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 11 / 23

3D extension

A spline space on a tensor product mesh M = M1 ⊗ M2 ⊗ M3 is constructed by tensor product of univariate spaces: Sp1,p2,p3(M) = Sp1(M1) ⊗ Sp2(M2) ⊗ Sp3(M3) A mono-patch spline geometry is parametrized by F ∈ (Sp1,p2,p3(M))3:

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 12 / 23

3D extension

A spline space on a tensor product mesh M = M1 ⊗ M2 ⊗ M3 is constructed by tensor product of univariate spaces: Sp1,p2,p3(M) = Sp1(M1) ⊗ Sp2(M2) ⊗ Sp3(M3) A mono-patch spline geometry is parametrized by F ∈ (Sp1,p2,p3(M))3: “parametric cube”

F

− − − − − − →

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 12 / 23

3D extension

A spline space on a tensor product mesh M = M1 ⊗ M2 ⊗ M3 is constructed by tensor product of univariate spaces: Sp1,p2,p3(M) = Sp1(M1) ⊗ Sp2(M2) ⊗ Sp3(M3) A mono-patch spline geometry is parametrized by F ∈ (Sp1,p2,p3(M))3:

• Ω = “parametric cube”

F

− − − − − − → = Ω

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 12 / 23

3D extension

A spline space on a tensor product mesh M = M1 ⊗ M2 ⊗ M3 is constructed by tensor product of univariate spaces: Sp1,p2,p3(M) = Sp1(M1) ⊗ Sp2(M2) ⊗ Sp3(M3) A mono-patch spline geometry is parametrized by F ∈ (Sp1,p2,p3(M))3:

• Ω = “parametric cube”

F

− − − − − − → = Ω

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 12 / 23

single patch vs. multipatch geometry

single patch:

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 13 / 23

single patch vs. multipatch geometry

single patch:

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 13 / 23

single patch vs. multipatch geometry

single patch: multi-patch:

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 13 / 23

single patch vs. multipatch geometry

single patch: multi-patch:

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 13 / 23

single patch vs. multipatch geometry

single patch: multi-patch:

[Courtesy by T.J.R Hughes et al.]

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 13 / 23

Univariate De Rham complex

Sp(M)

d dζ

− − − − → Sp−1(M), Derivative of p-degree B-splines are linear combination of (p − 1)-degree B-splines: d dζ Bi,p(·) = p |supp(Bi−1,p−1)|Bi−1,p−1(·) − p |supp(Bi,p−1)|Bi,p−1(·),

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 14 / 23

Univariate De Rham complex

Sp(M)

d dζ

− − − − → Sp−1(M), Derivative of p-degree B-splines are linear combination of (p − 1)-degree B-splines: d dζ Bi,p(·) = p |supp(Bi−1,p−1)|Bi−1,p−1(·) − p |supp(Bi,p−1)|Bi,p−1(·), the regularity of Bi,p−1 is one lower than the regularity of Bi,p−1 at the knots,

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 14 / 23

Univariate De Rham complex

Sp(M)

d dζ

− − − − → Sp−1(M), Derivative of p-degree B-splines are linear combination of (p − 1)-degree B-splines: d dζ Bi,p(·) = p |supp(Bi−1,p−1)|Bi−1,p−1(·) − p |supp(Bi,p−1)|Bi,p−1(·), the regularity of Bi,p−1 is one lower than the regularity of Bi,p−1 at the knots, if the Bi,p are associated to the piecewise-linear on the Greville mesh MG, then Bi,p−1 can be associated to piecewise constant functions on the elements of the same Greville mesh MG,

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 14 / 23

Univariate De Rham complex

Sp(M)

d dζ

− − − − → Sp−1(M), Derivative of p-degree B-splines are linear combination of (p − 1)-degree B-splines: d dζ Bi,p(·) = p |supp(Bi−1,p−1)|Bi−1,p−1(·) − p |supp(Bi,p−1)|Bi,p−1(·), the regularity of Bi,p−1 is one lower than the regularity of Bi,p−1 at the knots, if the Bi,p are associated to the piecewise-linear on the Greville mesh MG, then Bi,p−1 can be associated to piecewise constant functions on the elements of the same Greville mesh MG, is we scale the basis, d

dζ is the incidence matrix of MG.

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 14 / 23

Multivariate (3D) spline De Rham complex on Ω

Theorem ([Buffa, Rivas, GS, Vázquez, SINUM, 2011] and [Buffa, GS, Vázquez, 2012, submitted to JCP] )

The following complex is exact R − − − − → X 0

h b ∇

− − − − → X 1

h

• curl

− − − − → X 2

h d div

− − − − → X 3

h −

− − − → 0. where

• X 0

h := Sp1,p2,p3,

• X 1

h := Sp1−1,p2,p3 × Sp1,p2−1,p3 × Sp1,p2,p3−1,

• X 2

h := Sp1,p2−1,p3−1 × Sp1−1,p2,p3−1 × Sp1−1,p2−1,p3,

• X 3

h := Sp1−1,p2−1,p3−1.

Moreover there are commuting L2-stable quasi-local projectors.

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 15 / 23

Multivariate (3D) spline De Rham complex on Ω

By push forward we move the previous complex on the domain of interest: R − − − − → X 0

h b ∇

− − − − → X 1

h

• curl

− − − − → X 2

h d div

− − − − → X 3

h −

− − − → 0

ι0

• 



ι1

• 



ι2

• 



ι3

• 

 R − − − − → X 0

h ∇

− − − − → X 1

h curl

− − − − → X 2

h div

− − − − → X 3

h −

− − − → 0, This is a conforming discretization of the De Rham complex H1(Ω)

∇

− − − − → H(curl; Ω)

curl

− − − − → H(div; Ω)

div

− − − − → L2(Ω) − − − − → 0. and, assuming regularity as needed, of ( H1(Ω) )3

div

− − − − → L2(Ω) − − − − → 0.

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 16 / 23

Control field and control complex

R − − − − → X 0

h ∇

− − − − → X 1

h curl

− − − − → X 2

h div

− − − − → X 3

h −

− − − → 0

I0

h

 

• I1

h

 

• I2

h

 

• I3

h

 

• R −

− − − → Z 0

h ∇

− − − − → Z 1

h curl

− − − − → Z 2

h div

− − − − → Z 3

h −

− − − → 0.

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 17 / 23

Control field and control complex

R − − − − → X 0

h ∇

− − − − → X 1

h curl

− − − − → X 2

h div

− − − − → X 3

h −

− − − → 0

I0

h

 

• I1

h

 

• I2

h

 

• I3

h

 

• R −

− − − → Z 0

h ∇

− − − − → Z 1

h curl

− − − − → Z 2

h div

− − − − → Z 3

h −

− − − → 0. where Z 0

h := Lagrange trilinear FE on the control mesh

Z 1

h := low-order Nédélec hexahedral FE on the edge control mesh

Z 2

h := low-order Nédélec hexahedral FE on the face control mesh

Z 3

h := piecewise constant FE on the control mesh

Ii

h := "isogeometric field" to "control field" operators

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 17 / 23

Control field and control complex

R − − − − → X 0

h ∇

− − − − → X 1

h curl

− − − − → X 2

h div

− − − − → X 3

h −

− − − → 0

I0

h

 

• I1

h

 

• I2

h

 

• I3

h

 

• R −

− − − → Z 0

h ∇

− − − − → Z 1

h curl

− − − − → Z 2

h div

− − − − → Z 3

h −

− − − → 0.

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 17 / 23

Multi-patch De Rham complex

Multi-patch structure: unstructured hexahedral meshby conformity + gluing of control fields as for FE!

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 18 / 23

The Spline Complex: numerical testing

curl curl u = k2u

Eigenvalues computation CODE by Schoberl Dauge et al. IGA, p = 3 IGA, p = 6 Reliable digits d.o.f. 53982 41691 8421 5436

• Eig. 1.

3.2199939 3.3138052 3.2194306 3.2111746 3.2???e+00

• Eig. 2.

5.8804425 5.8863499 5.8804604 5.8809472 5.88??e+00

• Eig. 3.

5.8804553 5.8863499 5.8804604 5.8809472 5.88??e+00

• Eig. 4.

10.6856632 10.6945143 10.6866214 10.6938099 1.0694e+01

• Eig. 5.

10.6936955 10.6945143 10.6949643 10.7069155 1.0694e+01

• Eig. 6.

10.6937289 10.7005804 10.6949643 10.7069155 1.07??e+01

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 19 / 23

Breaking the tensor product structure: T-splines

[Sederberg, Zheng, Bakenov, and Nasri, 2003] [Sederberg, Cardon, Finnigan, North, Zheng, Lyche, 2004]

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 20 / 23

Breaking the tensor product structure: T-splines

[Sederberg, Zheng, Bakenov, and Nasri, 2003] [Sederberg, Cardon, Finnigan, North, Zheng, Lyche, 2004]

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 20 / 23

Breaking the tensor product structure: T-splines

[Sederberg, Zheng, Bakenov, and Nasri, 2003] [Sederberg, Cardon, Finnigan, North, Zheng, Lyche, 2004]

B(3,5)(s, t) = B[s1, s2, s3, s5, s6](s)B[t2, t4, t5, t6, t7](t) = B[0, 0, 1/3, 1, 1](s)B[0, 1/2, 3/4, 1, 1](t).

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 20 / 23

Linear dependence of blending function on patological T-meshes: B1(s, t) = B[0, 0, 1/2, 1/2, 1](s)B[0, 1/2, 1/2, 1, 1](t), B22(s, t) = B[0, 0, 1/2, 1/2, 2/3](s)B[0, 1/2, 1/2, 1, 1](t), B33(s, t) = B[0, 1/2, 1/2, 2/3, 1](s)B[0, 1/2, 1/2, 1, 1](t),

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 21 / 23

Linear dependence of blending function on patological T-meshes: We have: B1 = B2 + 1

3B3 !!!

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 21 / 23

T-splines (& Co.) present and future

Known results: Analysis-suitable subclass [Li, Zheng, Sederberg, Scott, Hughes, CAGD, 2012] - [Beirão da

Veiga,Buffa, GS, Vázquez, 2012]

De Rham compatible T-splines [Buffa, GS, Vázquez, 2012, submitted to JCP] promising numerical benchmarks and integration with CAD alternative definitions such as LR − splines. . .

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 22 / 23

T-splines (& Co.) present and future

Known results: Analysis-suitable subclass [Li, Zheng, Sederberg, Scott, Hughes, CAGD, 2012] - [Beirão da

Veiga,Buffa, GS, Vázquez, 2012]

De Rham compatible T-splines [Buffa, GS, Vázquez, 2012, submitted to JCP] promising numerical benchmarks and integration with CAD alternative definitions such as LR − splines. . . Future challenges: extraordinary points 3D Analysis suitable T-splines

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 22 / 23

T-splines numerical testing of a wave guide

6 elements on the length (cm 24), dof 7950, cubic splines. |T| = 0.9998, |R| = 0.0025

• G. Sangalli (Univ. of Pavia)

Topics in IgA Workshop @ Bicocca 23 / 23