Evgrafov, Anton

Gersborg, Allan Roulund

Gravesen, Jens

Nguyen, Dang Manh

Nielsen, Peter Nørtoft

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Parametrisation in Iso geometric AnalysisA first reportNguyen, Dang Manh; Nørtoft, Peter; Gersborg, Allan Roulund; Evgrafov, Anton ; Gravesen, JensPublication date:2009Document VersionEarly version, also known as pre-printLink back to DTU OrbitCitation (APA):Nguyen, D. M., Nielsen, P. N., Gersborg, A. R., Evgrafov, A., & Gravesen, J. (2009). Parametrisation in Isogeometric Analysis: A first report.Parametrisation in Iso Geometric Analysis,a first reportNguyen Dang Manh∗, Peter Nørtoft Nielsen∗†,Anton Evgrafov∗, Allan Roulund Gersborg†, Jens Gravesen∗Technical University of DenmarkOctober 21, 20091 IntroductionThis is a report of our first experiments in the field of iso geometric analysis.The idea was simply to get acquainted with B-splines, finite element analy-sis, and optimisation. It turned out that already a simple one dimensionalexample shows that the accuracy of the method is highly dependent of theparametrisation. So we find it worthwhile to report the results.We consider the simple one dimensional eigenvalue problem of findingthe frequencies of longitudinal vibration of a rod, se Figure 1. The governingx0 1u(x) longitudinal displacementFigure 1: Rod.1Figure 1: We consider longitudinal vibration of a rod.equation is Helmholtz’s equation with Dirichlet boundary conditions4u = d2udx2= −λu, u(0) = u(1) = 0. (1)The solution is of course well known. The eigenvalues and eigenfunctionsareλn = ω2n un(x) = sin(ωnx) ωn = npi. (2)∗Department of Mathematics, {d.m.nguyen,p.n.nielsen,a.evgrafov,j.gravesen}@mat.dtu.dk†Department of Mechanical Engineering, arg@mek.dtu.dk1If we multiply (1) with a test function v and integrate, then we obtainthe weak form of the equations∫ 10λu v dx = −∫ 10d2udx2v dx = −[d2udx2v]10+∫ 10dudxdvdxdx =∫ 10dudxdvdxdx.That is ∫ 10dudxdvdxdx = λ∫ 10u v dx. (3)We now write u as a finite linear combination of basis functions φi, i =1, . . . , N ,u(x) =N∑i=1uiφi(x) (4)and let v run through the set of basis functions. We obtain a system oflinear equationsN∑i=1ui∫ 10φ′i(x)φ′j(x) dx = λN∑i=1ui∫ 10φi(x)φj(x) dx for all j = 1, . . . , N.(5)They can be written in matrix formKu = λMu, (6)whereu =u1...uN , (7)K is the stiffness matrixK = 〈φ′1, φ′1〉 . . . 〈φ′1, φ′N 〉......〈φ′N , φ′1〉 . . . 〈φ′N , φ′N 〉 , (8)M is the mass matrixM = 〈φ1, φ1〉 . . . 〈φ1, φN 〉... ...〈φN , φ1〉 . . . 〈φN , φN 〉 , (9)and the inner product between two functions is〈φ, ψ〉 =∫ 10φ(t)ψ(t) dt. (10)20 0.2 0.4 0.6 0.8 100.20.40.60.810 0.2 0.4 0.6 0.8 100.20.40.60.81Figure 2: The basis functions in traditional finite element analysis. To theleft linear and to the right quadratic.In traditional finite element analysis the basis functions φi are chosenas piecewise polynomials that join together in a piecewise C0 fashion, seeFigure 2. The result of using traditional finite element analysis on a uniformgrid is shown in Figure 3.0 0.2 0.4 0.6 0.8 111.11.21.31.41.5FEM, N=101n/Nωn/(npi)  p=1p=2p=3Figure 3: The normalised spectra resulting from finite element analysis withshape functions of degree 1, 2, and 3.2 Iso Geometric AnalysisIn iso geometric analysis the, normally complicated, physical domain Ω isparametrised by a map F : Ω̂ → Ω, where the parameter domain Ω̂ issupposed to be simple and easy to discretize. In the present case we have3Ω = Ω̂ = [0, 1]. The parametrisation is of the formF (t) =N∑i=1xiNpi (t), (11)where Npi are B-splines of degree p on some knot vector t, see Figure 4. The0 0.2 0.4 0.6 0.8 100.20.40.60.810 0.2 0.4 0.6 0.8 100.20.40.60.810 0.2 0.4 0.6 0.8 100.20.40.60.810 0.2 0.4 0.6 0.8 100.20.40.60.81Figure 4: B-splines of degree 1, 2, 3, and 5 on a uniform knot vector.basis functions are of the formφi = Npi ◦ F−1, i = 1, . . . , N. (12)The left hand side of (3) becomes∫ 10dudxdvdxdx =∫ 10dudtdtdxdvdtdtdxdxdtdt =∫ 101F ′(t)dudtdvdtdt (13)and the right hand side becomesλ∫ 10u v dx = λ∫ 10u(t) v(t)F ′(t) dt (14)whereu(t) =N∑i=1uiNp(t). (15)4Once more the equation can be written matrix form (6), where the entriesof the stiffness and mass matrix now becomesKij =∫ 10Npi′(t)Npj′(t)F ′(t)dt, (16)Mij =∫ 10Npi (t)Npj (t)F′(t) dt. (17)We use Gaussian quadrature to evaluate the integrals. Observe that (17) canbe evaluated exactly if we use at least p+1+dp/2e points in the quadrature.In our example we use a uniform knot vector except for the boundaryknots which have full multiplicity. We try four different choice of parametri-sation (11) which of course is given by the control points x = (xi)i=1...N .1. The control points are chosen such that F is the identity map, i.e.,x = F (t) = t. The control points are the Greville abscissas, xi =(ti+1 + · · ·+ ti+p)/p.2. The control points are chosen uniformly, i.e., xi = i−1N−1 .3. The control points are chosen such that the error of a single eigenvalueλk is minimal, i.e. we solve the following optimisation problemminimise(ωnnpi− 1)2. (18)The eigenvalue λn = ω2n is chosen as the one with the maximal errorin 2.4. The control points are chosen such that the maximal relative error ofthe eigenvalues is minimised. We formulate it as the following con-strained optimisation problem.minimiseSsuch that(ωnnpi− 1)2< S for all n = 1, . . . , N .(19)All calculations are done in Matlab with the optimisation toolbox. Theresults can be seen in Figures 5–8.We have not spend much time on the optimisation and it is very likelythat better parametrisations can be obtained by changing the optimisationalgorithm. This, however, is not the main point in this report. We simplywant to point out that the result of iso geometric analysis can be verysensitive to the parametrisation.50 0.2 0.4 0.6 0.8 111.21.41.61.82IGA, N=101, p=1n/Nωn/(npi)  identityuniformoptimise oneoptimise all0 0.2 0.4 0.6 0.8 111.051.11.151.2IGA, N=101, p=1n/Nωn/(npi)  identityuniformoptimise oneoptimise all0 0.2 0.4 0.6 0.8 1−0.015−0.01−0.00500.0050.010.015IGA, N=101, p=1tF(t)−tFigure 5: The results of iso geometric analysis using B-splines of degree 1and four different parametrisations. On the top the normalised spectra, inthe middle a zoom near the line ωnnpi = 1, and below the differences betweenthe parametrisations F and the identity.60 0.2 0.4 0.6 0.8 111.11.21.31.4IGA, N=101, p=2n/Nωn/(npi)  identityuniformoptimise oneoptimise all0 0.2 0.4 0.6 0.8 111.021.041.061.08IGA, N=101, p=2n/Nωn/(npi)  identityuniformoptimise oneoptimise all0 0.2 0.4 0.6 0.8 1−0.015−0.01−0.00500.0050.010.015IGA, N=101, p=2tF(t)−tFigure 6: The results of iso geometric analysis using B-splines of degree 2and four different parametrisations. On the top the normalised spectra, inthe middle a zoom near the line ωnnpi = 1, and below the differences betweenthe parametrisations F and the identity.70 0.2 0.4 0.6 0.8 111.051.11.151.21.25IGA, N=101, p=3n/Nωn/(npi)  identityuniformoptimise oneoptimise all0 0.2 0.4 0.6 0.8 111.011.021.031.041.05IGA, N=101, p=3n/Nωn/(npi)  identityuniformoptimise oneoptimise all0 0.2 0.4 0.6 0.8 1−0.015−0.01−0.00500.0050.010.015IGA, N=101, p=3tF(t)−tFigure 7: The results of iso geometric analysis using B-splines of degree 3and four different parametrisations. On the top the normalised spectra, inthe middle a zoom near the line ωnnpi = 1, and below the differences betweenthe parametrisations F and the identity.80 0.2 0.4 0.6 0.8 111.21.41.61.82IGA, N=101, p=5n/Nωn/(npi)  identityuniformoptimise oneoptimise all0 0.2 0.4 0.6 0.8 111.011.021.031.04IGA, N=101, p=5n/Nωn/(npi)  identityuniformoptimise oneoptimise all0 0.2 0.4 0.6 0.8 1−0.02−0.0100.010.02IGA, N=101, p=5tF(t)−tFigure 8: The results of iso geometric analysis using B-splines of degree 5and four different parametrisations. On the top the normalised spectra, inthe middle a zoom near the line ωnnpi = 1, and below the differences betweenthe parametrisations F and the identity.93 ConclusionIn [1] it was found that using B-splines as basis functions gave considerablebetter accuracy and robustness than traditional finite elements. We haveconfirmed these findings, but have also found that in an iso geometric settingthe performance is very much dependent on the parametrisation.References[1] T.J.R. Hughes, A. Reali, G. Sangalli, Duality and unified analysis ofdiscrete approximations in structural dynamics and wave propagation:Comparison of p-method finite elements with k-method NURBS, Com-put. Methods Appl. Mech. Engrg. 197, 4104–4124 (2008).10

Parametrisation in Iso geometric Analysis:A first report

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Parametrisation in Iso geometric Analysis:A first report

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