Generalizing the isogeometric concept: weakening the tight coupling between geometry and simulation in IGA

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Generalizing the isogeometric concept:

weakening the tight coupling between geometry and simulation in IGA

Satyendra Tomar University of Luxembourg

Joint work with

E. Atroshchenko, S. Bordas and G. Xu Financial support from:

FP7-PEOPLE-2011-ITN (289361) F1R-ING-PEU-14RLTC

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Focus on presenting latest results, raising some open questions, and identifying some new challenges

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Outline

1 Motivation

Different degrees for geometry and solution Different basis for geometry and solution

2 Patch tests

Various partitioning of the domain

Various combinations of degrees and knots/weights

3 Some numerical results Patch test results Convergence results

4 Conclusions

Outline 3

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Outline

1 Motivation

2 Patch tests

4 Conclusions

Outline 3

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Outline

1 Motivation

2 Patch tests

4 Conclusions

Outline 3

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Outline

1 Motivation

2 Patch tests

4 Conclusions

Outline 3

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Outline

1 Motivation

2 Patch tests

4 Conclusions

Motivation 4

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Frequently encountered geometries in numerics

Motivation Different degrees for geometry and solution 5

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Frequently encountered geometries in numerics

Straight-sided polygonal domains (including L-shaped)

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Frequently encountered geometries in numerics

Straight-sided polygonal domains (including L-shaped) Curved boundaries, typically, circular and elliptical shapes

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Frequently encountered geometries in numerics

Straight-sided polygonal domains (including L-shaped) Curved boundaries, typically, circular and elliptical shapes Polynomial degree required to represent such a geometry?

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Frequently encountered geometries in numerics

Typically, pg“1,2,3!!

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Frequently encountered geometries in numerics

Typically, pg“1,2, . . .5. . .20!!

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HOFEIM

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HOFEIM

Polynomial degree for the numerical solution?

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HOFEIM

Polynomial degree for the numerical solution?

If the analytical solution is expected to be sufficiently regular, thep- or hp-method can be employed (withpu ąpg) to obtain higher accuracy

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Different basis for geometry and numerical solution?

Motivation Different basis for geometry and solution 7

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Different basis for geometry and numerical solution?

Various splines basis in practice B-Splines, NURBS, T-Splines,

LR-Splines, (truncated)Hierarchical B-Splines,

PHT-Splines, Generalized B-Splines,add your choice

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Different basis for geometry and numerical solution?

PHT-Splines, Generalized B-Splines,add your choice Combining various basis

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Different basis for geometry and numerical solution?

Geo NURBS, T-Splines

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Different basis for geometry and numerical solution?

Solution B-Splines, LR-Splines, (truncated)Hierarchical B-Splines, PHT-Splines, Generalized B-Splines,add your choice

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Different basis for geometry and numerical solution?

Solution B-Splines, LR-Splines, (truncated)Hierarchical B-Splines, PHT-Splines, Generalized B-Splines,add your choice

R. Sevilla, S. Fernandez Mendez, and A. Huerta. NURBS-enhanced finite element method (NEFEM).Int. J. Numer. Meth.

Engrg.,76, 56-83, 2008.

B. Marussig, J. Zechner, G. Beer, T.P. Fries. Fast isogeometric boundary element method based on independent field approximation.Comput. Methods Appl. Mech. Engrg.,284, 458-488, 2015. (ECCOMAS 2014, arxiv/1406.3499) Talk of S. Elgeti on Monday, 2016.05.30 (Spline-based FEM for fluid flow on deforming domains)

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Recall

The main idea of isogeometric analysis (IGA)

J.A. Cottrell, A. Reali, Y. Bazilevs, T.J.R. Hughes. Isogeometric analysis of structural vibrations.Comput. Methods Appl.

Mech. Engrg.,195, 5257-5296, 2006

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Motivation summarized

Standard paradigm of IGA

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Motivation summarized

Geometry and simulation spaces are tightly integrated, i.e. same space for geometry and numerical solution

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Motivation summarized

Situations where this tight integration can berelaxedforimproved solution quality

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Motivation summarized

§ Geometry of the domain is simple enough to be represented by low order NURBS, but the solution is sufficiently regular. Higher order approximation delivers superior results.

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Motivation summarized

§ Solution has low regularity (e.g. corner singularity) but the curved boundary can be represented by higher order NURBS.

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Motivation summarized

§ In shape/topology optimization, the constraint of using the same space is particularly undesirable.

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Motivation summarized

§ In shape/topology optimization, the constraint of using the same space is particularly undesirable.

§ Standard tools for the geometry/boundary but different

(spline-)basis for solution (to exploit features like local refinement).

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Outline

1 Motivation

2 Patch tests

4 Conclusions

Patch tests 10

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Some historical background of the patch test

I. Babuska and R. Narasimhan. The Babuska-Brezzi condition and the patch test: an example.Comput. Methods Appl.

Mech. Engrg.,140, 183-199, 1997.

G.P. Bazeley, Y.K. Cheung, B.M. Irons, and O.C. Zienkiewicz. Triangular elements in plate bending - conforming and nonconforming solutions, inProceedings of the Conference on Matrix Methods in Structural Mechanics, Wright Patterson Air Force Base, Dayton, Ohio, 547-576, 1965.

F. Stummel. The generalized patch test.SIAM J. Numer. Anal.,16(3), 449-471, 1979.

M. Wang. On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements.SIAM J.

Numer. Anal.,39(2), 363-384, 2001.

O.C. Zienkiewicz, and R.L. Taylor. The finite element patch test revisited: A computer test for convergence, validation and error estimates.Comput. Methods Appl. Mech. Engrg.,149, 223-254, 1997.

T.J.R. Hughes.The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall Inc., 1987.

Patch tests Various partitioning of the domain 11

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Some historical background of the patch test

I. Babuska and R. Narasimhan. The Babuska-Brezzi condition and the patch test: an example.Comput. Methods Appl.

Mech. Engrg.,140, 183-199, 1997.

G.P. Bazeley, Y.K. Cheung, B.M. Irons, and O.C. Zienkiewicz. Triangular elements in plate bending - conforming and nonconforming solutions, inProceedings of the Conference on Matrix Methods in Structural Mechanics, Wright Patterson Air Force Base, Dayton, Ohio, 547-576, 1965.

F. Stummel. The generalized patch test.SIAM J. Numer. Anal.,16(3), 449-471, 1979.

M. Wang. On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements.SIAM J.

Numer. Anal.,39(2), 363-384, 2001.

O.C. Zienkiewicz, and R.L. Taylor. The finite element patch test revisited: A computer test for convergence, validation and error estimates.Comput. Methods Appl. Mech. Engrg.,149, 223-254, 1997.

T.J.R. Hughes.The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall Inc., 1987.

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Original geometry parametrization of the domain

The geometry is exactly represented by NURBS of degrees1x2 Basic parametrization by one element, defined by 2 knot vectors

Σ“ t0,0,1,1u, Π“ t0,0,0,1,1,1u.

Together with NURBS basis, this is given by the following set of6 control points, where the third value denotes the weight.

Pr0,0s:“ t1,0,1u, Pr1,0s:“ t2,0,1u, Pr0,1s:“ t1,1,1{?

2u, Pr1,1s:“ t2,2,1{? 2u, Pr0,2s:“ t0,1,1u, Pr1,2s:“ t0,2,1u.

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Parametrization of the domain for the patch-test I

θ“π{2

A C

E

G

H

D r“1 I

r“2 F

B θ“0

Quarter annulus region

For patch-test in 2D, one-timeh-refinement in both directions Consider the refined knot vectors

Σ“ t0,0,s,1,1u, Π“ t0,0,0,t,1,1,1u.

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Parametrization of the domain for the patch-test II

Shape A Uniform curvilinear elementss“t “1{2

Shape B For non-uniform curvilinear elements, shift the pointsB, D,F,H andI. Set

t1:“1´t`t{?

2, t2:“t`? 2t1,

Updated set of control points in non-homogenized form t1,0,1u, t1`s,0,1u, t2,0,1u

t1, t

?2t₁,t1u, tp1`sq,p1`sqt

?2t₁ ,t1u, t2,

?2t t1

,t1u t

?2p1´tq t2

,1,t2

2u,t

?2p1`sqp1´tq t2

,p1`sq,t2

2u,t2?

2p1´tq t2

,2,t2

2u t0,1,1u, t0,1`s,1u, t0,2,1u

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Parametrization of the domain for the patch-test III

Shape C Add another parameterδ, two interior points changed as tp1`sqt1

t1`δ , p1`sqt

?2pt1`δq,t1`δu,t

?2p1`sqp1´tq

t2`2δ ,p1`sqt2

t2`2δ ,t2

2 `δu

0.0 0.5 1.0 1.5 2.0

Quarter annulus region with non-uniform elements, (s“2{3,t “1{8, δ “1{2)

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Various combinations of degrees and knots/weights

pu “pg, andΣu“Σg (isogeometric case) pu “pg, andΣu‰Σg

pu ăpg, andΣu“Σg

pu ăpg, andΣu‰Σg

pu ąpg, andΣu“Σg

pu ąpg, andΣu‰Σg

Patch tests Various combinations of degrees and knots/weights 16

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Various combinations of degrees and knots/weights

pu “pg, andΣu“Σg (isogeometric case) pu “pg, andΣu‰Σg

pu ăpg, andΣu“Σg

pu ăpg, andΣu‰Σg

pu ąpg, andΣu“Σg

pu ąpg, andΣu‰Σg

Total number of cases

3 choices of element shapes, and 6 choices of degrees/knots, total of 18 cases !!

Patch tests Various combinations of degrees and knots/weights 16

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Outline

1 Motivation

2 Patch tests

4 Conclusions

Some numerical results 17

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Results for various choices of p

u

, p

g

, Σ

u

, Σ

g L²error in the solutionu“1`x `y

pu “pg, i.e.pu “pg “1x2 pu ăpg, i.e.pu “1x2,pg “2x3 pu ąpg, i.e.pu “2x3,pg “1x2

Some numerical results Patch test results 18

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Results for various choices of p

u

, p

g

, Σ

u

, Σ

pandΣ Shape A Shape B Shape C

pu“pg,Σu“Σg 8.2469e-016 8.5695e-016 1.5017e-015

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Results for various choices of p

u

, p

g

, Σ

u

, Σ

pu “pg,Σu“Σg 8.2469e-016 8.5695e-016 1.5017e-015 pu ăpg,Σu“Σg 8.5489e-016 8.8080e-016 1.5172e-015

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Results for various choices of p

u

, p

g

, Σ

u

, Σ

pu“pg,Σu “Σg 8.2469e-016 8.5695e-016 1.5017e-015 puăpg,Σu “Σg 8.5489e-016 8.8080e-016 1.5172e-015 puąpg,Σu “Σg 1.1979e-015 1.8185e-015 3.6019e-015

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Results for various choices of p

u

, p

g

, Σ

u

, Σ

pu“pg,Σu“Σg 8.2469e-016 8.5695e-016 1.5017e-015 puăpg,Σu“Σg 8.5489e-016 8.8080e-016 1.5172e-015 puąpg,Σu“Σg 1.1979e-015 1.8185e-015 3.6019e-015 puăpg,Σu‰Σg 1.1401e-015 1.0907e-015 0.20155

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Results for various choices of p

u

, p

g

, Σ

u

, Σ

pu“pg,Σu“Σg 8.2469e-016 8.5695e-016 1.5017e-015 puăpg,Σu“Σg 8.5489e-016 8.8080e-016 1.5172e-015 puąpg,Σu“Σg 1.1979e-015 1.8185e-015 3.6019e-015 puăpg,Σu‰Σg 1.1401e-015 1.0907e-015 0.20155 puąpg,Σu‰Σg 1.5010e-015 1.4193e-015 0.03019

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Results for various choices of p

u

, p

g

, Σ

u

, Σ

pu“pg,Σu“Σg 8.2469e-016 8.5695e-016 1.5017e-015 puăpg,Σu“Σg 8.5489e-016 8.8080e-016 1.5172e-015 puąpg,Σu“Σg 1.1979e-015 1.8185e-015 3.6019e-015 puăpg,Σu‰Σg 1.1401e-015 1.0907e-015 0.20155 puąpg,Σu‰Σg 1.5010e-015 1.4193e-015 0.03019 pu“pg,Σu‰Σg 7.4477e-016 1.0611e-015 0.20155

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Results for various choices of p

u

, p

g

, Σ

u

, Σ

pu“pg,Σu“Σg 8.2469e-016 8.5695e-016 1.5017e-015 puăpg,Σu“Σg 8.5489e-016 8.8080e-016 1.5172e-015 puąpg,Σu“Σg 1.1979e-015 1.8185e-015 3.6019e-015 puăpg,Σu‰Σg 1.1401e-015 1.0907e-015 0.20155 puąpg,Σu‰Σg 1.5010e-015 1.4193e-015 0.03019 pu“pg,Σu‰Σg 7.4477e-016 1.0611e-015 0.20155

What is expected foru“1`x²`y² ?

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Numerical setup

Analytic solution with Dirichlet BC u “logpa

ppx ´0.1q ˚ px´0.1q ` py ´0.1q ˚ py´0.1qqq

Example A Same knot vectors, same starting approximation for solution as geo (using NURBS)

Example B Same geo, but different knot vectors for geo and solution (still using NURBS), which also represents geo

Example C Same knot vectors for geo and field, geo using NURBS, and solution using B-Splines

Some numerical results Convergence results 19

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Convergence

10^-1 10^-4

10^-3 10^-2 10^-1

↑ slope = 1.82

↑ slope = 3.51

log10(h) log 10(|| u - u h ||)

Convergence for Example A

p_u = p_g pu < p

g pu > p

g

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Convergence

10^-1 10^-5

10^-4 10^-3 10^-2

↑ slope = 1.90

↑ slope = 3.29

log10(h) log 10(|| u - u h ||)

Convergence for Example B

p_u = p_g pu < p

g pu > p

g

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Convergence

10^-1 10^-4

10^-3 10^-2 10^-1

↑ slope = 1.82

↑ slope = 3.51

log10(h) log 10(|| u - u h ||)

Convergence for Example C

p_u = p_g pu < p

g pu > p

g

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Outline

1 Motivation

2 Patch tests

4 Conclusions

Conclusions 21

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Approaches of IGA and GIA

Conclusions 22

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Approaches of IGA and GIA

The main idea of Geometry-Independent Field approximaTion (GIFT)

G. Beer, B. Marussig, J. Zechner, C. D ¨unser, T.P. Fries. Boundary Element Analysis with trimmed NURBS and a generalized IGA approach.http://arxiv.org/abs/1406.3499, 2014.

B. Marussig, J. Zechner, G. Beer, T.P. Fries. Fast isogeometric boundary element method based on independent field approximation.Comput. Methods Appl. Mech. Engrg.,284, 458-488, 2015.

Conclusions 22

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Approaches of IGA and GIA

Conclusions 22

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Approaches of IGA and GIA

J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs.Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, 2009.

Conclusions 22

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Approaches of IGA and GIA

The main idea of geometry-induced analysis (GIA)

Conclusions 22

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Naming convention

O.C. Zienkiewicz, R.L. Taylor, and J.Z. Zhu.The Finite Element Method: Its Basis and Fundamentals. Elsevier, 2013.

Conclusions 23

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Naming convention

pu“pg puăpg puąpg

Iso-parametric Super-parametric Sub-parametric

Conclusions 23

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Naming convention

pu“pg puăpg puąpg

Iso-parametric Super-parametric Sub-parametric Iso-geometric Sub-geometric Super-geometric

Conclusions 23

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Concluding remarks

When knot data are same (same representation/basis)

Conclusions 24

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Concluding remarks

§ various combinations of polynomial degrees pass the test for all kind of parametric elements

Conclusions 24

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Concluding remarks

When knot data are different (allowing different basis)

Conclusions 24

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Concluding remarks

§ various combinations of polynomial degrees pass the teston rectangular elements in parametric domain

Conclusions 24

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Concluding remarks

§ various combinations of polynomial degrees pass the teston rectangular elements in parametric domain

One message

Without any fancy/weird parametric elements,various

combinations of different basis and polynomial degrees pass the test, and can be used in practice

Conclusions 24