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

Some topics in: Isogeometric Analysis (IGA) G. Sangalli 1 , 2 in collaboration with: A. Buffa 2 , R. Vázquez 2 1 University of Pavia, 2 IMATI-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 Splines, NURBS, control pts., CAD and d.o.f.s representation 1 Isogeometric De Rham compatible fields 2 Unstructured discretization by T-splines 3 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 { ξ 1 , ..., ξ n + p + 1 } : � 1 if ξ i ≤ ξ < ξ i + 1 B i , 0 ( ξ ) = 0 otherwise. ξ − ξ i ξ i + p + 1 − ξ B i , p ( ξ ) = B i , p − 1 ( ξ ) + B i + 1 , p − 1 ( ξ ) . ξ i + p − ξ i ξ i + p + 1 − ξ i + 1 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 C p − 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 C i = • and B i = gives the parametrization F ( ξ ) = � i C i B i ( ξ ) of a B-spline curve: F [ 0 , 5 ] − − − − − − − − → 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 ( h 2 ) 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 ( h 2 ) 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 R 2 is the projection of a B-spline in R 3 n n C ( ξ ) = [ C w x ( ξ ) , C w � � y ( ξ )] w i B i , p ( ξ ) = C i � n i , p ( ξ ) = C i R i , p ( ξ ) . C w z ( ξ ) i = 1 w b i B b b i = 1 i = 1 G. Sangalli (Univ. of Pavia) Topics in IgA Workshop @ Bicocca 8 / 23

Control polygon convergence Given the knot vector (mesh) M = { ξ i } i = 1 ,..., n + p + 1 = { 0 , 0 , 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 5 } we introduce by knot averaging the Greville mesh M G = { γ i } 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 M G : λ i ( γ j ) = δ i , j ξ i = ◦ , γ i = + , B 1 in blue, λ 1 in black G. Sangalli (Univ. of Pavia) Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence Given the knot vector (mesh) M = { ξ i } i = 1 ,..., n + p + 1 = { 0 , 0 , 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 5 } we introduce by knot averaging the Greville mesh M G = { γ i } 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 M G : λ i ( γ j ) = δ i , j ξ i = ◦ , γ i = + , B 2 in blue, λ 2 in black G. Sangalli (Univ. of Pavia) Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence Given the knot vector (mesh) M = { ξ i } i = 1 ,..., n + p + 1 = { 0 , 0 , 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 5 } we introduce by knot averaging the Greville mesh M G = { γ i } 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 M G : λ i ( γ j ) = δ i , j ξ i = ◦ , γ i = + , B 3 in blue, λ 3 in black G. Sangalli (Univ. of Pavia) Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence Given the knot vector (mesh) M = { ξ i } i = 1 ,..., n + p + 1 = { 0 , 0 , 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 5 } we introduce by knot averaging the Greville mesh M G = { γ i } 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 M G : λ i ( γ j ) = δ i , j ξ i = ◦ , γ i = + , B 4 in blue, λ 4 in black G. Sangalli (Univ. of Pavia) Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence Given the knot vector (mesh) M = { ξ i } i = 1 ,..., n + p + 1 = { 0 , 0 , 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 5 } we introduce by knot averaging the Greville mesh M G = { γ i } 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 M G : λ i ( γ j ) = δ i , j ξ i = ◦ , γ i = + , B 5 in blue, λ 5 in black G. Sangalli (Univ. of Pavia) Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence Given the knot vector (mesh) M = { ξ i } i = 1 ,..., n + p + 1 = { 0 , 0 , 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 5 } we introduce by knot averaging the Greville mesh M G = { γ i } 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 M G : λ i ( γ j ) = δ i , j ξ i = ◦ , γ i = + , B 6 in blue, λ 6 in black G. Sangalli (Univ. of Pavia) Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence Given the knot vector (mesh) M = { ξ i } i = 1 ,..., n + p + 1 = { 0 , 0 , 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 5 } we introduce by knot averaging the Greville mesh M G = { γ i } 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 M G : λ i ( γ j ) = δ i , j ξ i = ◦ , γ i = + , B 7 in blue, λ 7 in black G. Sangalli (Univ. of Pavia) Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence Given the knot vector (mesh) M = { ξ i } i = 1 ,..., n + p + 1 = { 0 , 0 , 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 5 } we introduce by knot averaging the Greville mesh M G = { γ i } 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 M G : λ i ( γ j ) = δ i , j ξ i = ◦ , γ i = + , B 8 in blue, λ 8 in black G. Sangalli (Univ. of Pavia) Topics in IgA Workshop @ Bicocca 9 / 23

Control polygon convergence Given the knot vector (mesh) M = { ξ i } i = 1 ,..., n + p + 1 = { 0 , 0 , 0 , 1 , 2 , 3 , 4 , 4 , 5 , 5 , 5 } we introduce by knot averaging the Greville mesh M G = { γ i } 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 C i B i ( · ) then the control polygon is parametrized by F C ( · ) = � n i = 1 C i λ i ( · ) and it holds � F ( · ) − F C ( · ) � L ∞ ≃ h 2 , 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: � c i B i ◦ F − 1 ( x ) , φ ( x ) = x ∈ spline curve , i = 1 ,..., n the associated control field shares the same d.o.f.s but is push-forward of piecewise linears through F C : � c i λ i ◦ F − 1 φ C ( x ) = C ( x ) , x ∈ control polygon. i = 1 ,..., n 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: � c i B i ◦ F − 1 ( x ) , φ ( x ) = x ∈ spline curve , i = 1 ,..., n the associated control field shares the same d.o.f.s but is push-forward of piecewise linears through F C : � c i λ i ◦ F − 1 φ C ( x ) = C ( x ) , x ∈ control polygon. i = 1 ,..., n distance between the two functions is O ( h 2 ) , if the d.o.f.s are chosen wisely, φ delivers approximation error O ( h p + 1 ) while φ C delivers approximation error O ( h 2 ) . G. Sangalli (Univ. of Pavia) Topics in IgA Workshop @ Bicocca 10 / 23