23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds)
⃝KTH, Stockholm, 2010c
LINEAR ISOGEOMETRIC SHELL ANALYSIS IN MARINE
Geir Skeie, Susanne Støle-Hentschel and Torgeir Rusten
𝜃1 𝜃1 𝜃1
𝜃2 𝜃2 𝜃2
𝜃3 𝜃3 𝑫 𝜃3
𝑎
𝑿𝑗
𝑿𝑘
𝑿𝑎
Figure 1: The degenerated solid approach illustrated in three steps: parent element, linear constraint in normal direction and midplane projection.
implementation found in SESTRA, [6], the linear finite element solver within the SESAM system.
The principle approach is depicted in Figure 1.
The displacement field may be expressed using parameters defined on the shell mid sur-face where the through thickness in-plane displacements are expressed using midplane rotation components and the through thickness out-of-plan displacement component is constant, thus
𝒖(𝜃1, 𝜃2, 𝜃3) = 𝑁𝑎(𝜃1, 𝜃2) (
𝒗𝑎+ 𝜃3𝑡𝑎𝒗𝑑𝑎 )
(1) 𝒗𝑎 denotes the midplane control point translation vector while 𝒗𝑑𝑎 denotes the out-of-midplane displacement corrections. 𝑡𝑎 is the shell thickness at the control point and 𝜃𝑖 are the curvilinear coordinates (parametric coordinates) and the Einstein’s summation convention is used. 𝑁𝑎 denotes the spline (in our setting NURBS) shape functions. 𝒗𝑑𝑎 may be expressed using either two or three rotations where the latter include the drilling rotation;
𝒗𝑑𝑎= −𝑫𝑎× 𝝎 or 𝒗𝑑𝑎= (−𝑨𝑎2Θ𝑎1+ 𝑨𝑎1Θ𝑎2) (2) Θ𝑎𝑖denotes the midplane rotation components and 𝑨𝑎𝛼 is an orthogonal local coordinate system that span the shell tangent plane. 𝝎 denotes the three components of the global rotation vector.
The linearized strains may be expressed in the global coordinate system according to the familiar expression from linearized elasticity. The zero transverse normal stress condition 𝜎33= 0 has to be enforced in the local shell coordinate frame.
Since the formulation may include an additional unknown the formulation needs to be stabi-lized. One criteria for the stabilization is that is should not destroy the sparsity pattern of the patch stiffness matrix. A penalty formulation, suggested by Fox and Simo [4], is used with this in mind. The augmented potential energy functional may be expressed as
Π𝛾(𝒖, 𝝎) = Π(𝒖, 𝝎) + 𝛾 2
∫
Ω
(
𝑨1⋅ ∂𝒖
∂𝜃2 − ∂𝒖
∂𝜃1 ⋅ 𝑨2− 2𝑨1× 𝑨2⋅ 𝝎 )2
(3) where Π(⋅) denotes the potential energy for the shell formulation and 𝛾 is the penalty parameter.
The equations above use a local coordinate triad to express the shell rotations and the local material law. A number of methods to establish the local triad at control points has been suggested in Benson et al. [2].
Geir Skeie, Susanne Støle-Hentschel and Torgeir Rusten
æ æ
à à à à
à
ì ì ì
ì
ì
ò ò
ò
ò ò
ô ô
ô
ô ô
ç ç
ç ç ç
5 10 15 20
0.0 0.2 0.4 0.6 0.8 1.0
ç 2nd order SESTRA
ô 5th order
ò 4th order
ì 3rd order
à 2nd order
æ Analytic
Figure 2: Pinched cylinder with diaphragms; convergence of the displacement under the load.
3 NUMERICAL EXAMPLE
3.1 Pinched cylinder
The shell obstacle course suggested by Belytschko et al. [1] is used to check robustness and accuracy for shell implementation in complex strain states. The problem set includes three problems, namely, the Scordelis-Lo roof, the pinched cylinder diaphragm and finally the pinched hemispherical shell. The cylinder is fully restrained by rigid diagraph at the ends and is subjected to two radial point loads alternating at 90∘. The displacement in the direction of the point load is compared to the reference solution given as ∣𝑢𝑧∣ = 1.8248 × 10−5.
The convergence curves for displacement under the load as a function of the mesh density for the spline and SESTRA formulation is shown in Figure 2.
3.2 Tubular joint
Pipe intersections occur frequently in off-shore applications. A simple two pipe model is shown in Figure 3 where the pipes meet at a right angle. The diameters are different for the two intersecting pipes, 𝐷1 = 2 and 𝐷2 = 1. The pipe segements are 𝐿1 = 10 and 𝐿2 = 5 respectively. The number of patches in the pipe model is 12.
A stress analysis is performed. The two ends of the larger pipe are fixed in all translations and rotations. The free end of the smaller pipe is subject to a line-load with a load intensity of unity in the global z-direction, that is in the longitudinal direction of the larger pipe. The geometry is not exact in this example, that is, the vertical cylinder is not a perfect cylinder in the neighborhood of the intersection.
The convergence of the displacement under the load is shown in Figure 3.
4 CONCLUSIONS
The results are comparable to the current second order shell finite element. However, the SESTRA second order shell element performs in general better than the plain Isogeometric shell implementation. The current Isogeometric shell implementation is based on a pure displacement formulation with no remedies to handle out of plane shear locking or membrane locking. This is due to the fact that higher order approximations are believed to be less susceptible to locking.
IGA is in its infancy and more research is needed to tune the performance in commercial shell type applications.
Geir Skeie, Susanne Støle-Hentschel and Torgeir Rusten
æ æ
æ
æ
æ
æ æ
à à
à
à
ì ì
ì
ì ò ò
ò
ô ô
ô
ô
0 500 1000 1500 2000 2500
0 2.´10-7 4.´10-7 6.´10-7 8.´10-7
ô SESTRA, second order
ò 5th order
ìàæ 4th order3rd order2nd order
Figure 3: Two intersecting pipes; convergence of the displacement under the load. 𝐸 = 2.1×1011, 𝜈 = 0.3, 𝑡 = 0.02 and 𝑓𝑧 = 1.
REFERENCES
[1] Ted Belytschko, Henryk Stolarski, Wing Kam Liu, Nicholas Carpenter, and Jame S.J. Ong.
Stress projection for membrane and shear locking in shell finite elements. Computer Methods in Applied Mechanics and Engineering, 51(1-3):221 – 258, 1985.
[2] D.J. Benson, Y. Bazilevs, M.-C. Hsu, and T.J.R. Hughes. A Large Deformation, Rotation-Free, Isogeometric Shell. Technical Report 09-37, Institute for Computational Engineering and Sciences, The University ofTexas at Austin, 201 East 24th Street, 1 University Station C0200, Austin, TX 78712, USA, 2009.
[3] J. A. Cottrell, T. J .R. Hughes, and A. Reali. Studies of Refinement and Continuity in Isogeometric Structural Analysis. Technical Report 07-05, Institute for Computational En-gineering and Sciences, The University ofTexas at Austin, 201 East 24th Street, 1 University Station C0200, Austin, TX 78712, USA, 2007.
[4] D.D. Fox and J.C. Simo. A drill rotation formulation for geometrically exact shells. Computer Methods in Applied Mechanics and Engineering, 98(3):329 – 343, 1992.
[5] T.J.R. Hughes, J. A. Cottrel, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135—4195, 2005.
[6] SESTRA. SESTRA – Super Element Structural Analysis, User’s Manual. DNV Software, Høvik, Norway, 2008.
23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds) KTH, Stockholm, 2010c