Nonlinear Dynamics manuscript No. (will be inserted by the editor) Isogeometric collocation for nonlinear dynamic analysis of Cosserat rods with frictional contact Oliver Weeger · Bharath Narayanan · Martin L. Dunn Received: date / Accepted: date Abstract We present a novel isogeometric collocation method for nonlinear dynamic analysis of three-dimen- sional, slender, elastic rods. The approach is based on the geometrically exact Cosserat model for rod dynam- ics. We formulate the governing nonlinear partial dif- ferential equations as a first-order problem in time and develop an isogeometric semi-discretization of position, orientation, velocity and angular velocity of the rod centerline as NURBS curves. Collocation then leads to a nonlinear system of first-order ordinary differen- tial equations, which can be solved using standard time integration methods. Furthermore, our model includes viscoelastic damping and a frictional contact formula- tion. The computational method is validated and its practical applicability shown using several numerical applications of nonlinear rod dynamics. Keywords Isogeometric analysis · Collocation method · Cosserat rod model · Nonlinear dynamics · Frictional contact 1 Introduction Modeling and simulation of thin deformable bodies has wide-spread applications in engineering, sciences and animation, such as vibrations of bridges, cables, drill strings and rigs, and machines [1–3], deformation of woven and knitted textiles [4], additively manufactured O. Weeger · B. Narayanan · M.L. Dunn Singapore University of Technology and Design, SUTD Digital Manufacturing and Design Centre, 8 Somapah Road, Singapore 487372, Singapore, E-mail: oliver [email protected], E-mail: bharath [email protected], E-mail: martin [email protected]structures [5], hair and fiber modeling [6, 7], and bio- dynamic structures such as the double helix of DNA molecules and arterial pathways [8]. In many of these problems, complex dynamic behavior and rod-to-rod contact interactions are essential aspects for the accu- rate modeling of physical effects and thus accurate, ro- bust and efficient computational discretization methods are required for their numerical solution. For the static and dynamic modeling of 3-dimension- al (3D), slender beam structures subject to large defor- mations and rotations, the Cosserat rod model [9–11] has been employed successfully in many of the afore- mentioned problems. It covers nonlinear, geometrically exact deformation behavior, general loading conditions by external forces and moments, and can be extended to include viscous damping [12–14]. It leads to a non- linear partial differential equation (PDE) in space and time, which usually has to be solved by numerical meth- ods due to its complexity, wherefore various kinds of discretization schemes have been proposed. Most com- monly, first a spatial semi-discretization is carried out, either using finite element (FEM) [15,16] or finite differ- ence methods (FDM) [6, 13, 17], and then time integra- tion is performed using standard methods for ordinary differential equations (ODEs). In [18], p-FEM was used in combination with harmonic balance to solve the pe- riodic vibration problem. Dynamics of rod structures, i.e. meshes or nets of interconnected Cosserat rods, were also investigated using these methods [19, 20]. In this work, we apply an isogeometric collocation method for the spatial semi-discretization. Isogeomet- ric analysis (IGA) was first introduced by Hughes et al. in 2005 [21] and has since attracted increasing inter- est in the computational engineering community. This novel concept aims at bridging the gap between the two largely disjunct domains of computer-aided design
14
Embed
Isogeometric collocation for nonlinear dynamic …...applications of nonlinear rod dynamics. Keywords Isogeometric analysis Collocation method Cosserat rod model Nonlinear dynamics
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Nonlinear Dynamics manuscript No.(will be inserted by the editor)
Isogeometric collocation for nonlinear dynamic analysis ofCosserat rods with frictional contact
Oliver Weeger · Bharath Narayanan · Martin L. Dunn
Received: date / Accepted: date
Abstract We present a novel isogeometric collocation
method for nonlinear dynamic analysis of three-dimen-
sional, slender, elastic rods. The approach is based on
the geometrically exact Cosserat model for rod dynam-
ics. We formulate the governing nonlinear partial dif-
ferential equations as a first-order problem in time and
develop an isogeometric semi-discretization of position,
orientation, velocity and angular velocity of the rod
centerline as NURBS curves. Collocation then leads
to a nonlinear system of first-order ordinary differen-
tial equations, which can be solved using standard time
integration methods. Furthermore, our model includes
viscoelastic damping and a frictional contact formula-
tion. The computational method is validated and its
practical applicability shown using several numerical
applications of nonlinear rod dynamics.
Keywords Isogeometric analysis · Collocation
method · Cosserat rod model · Nonlinear dynamics ·Frictional contact
1 Introduction
Modeling and simulation of thin deformable bodies has
wide-spread applications in engineering, sciences and
animation, such as vibrations of bridges, cables, drill
strings and rigs, and machines [1–3], deformation of
woven and knitted textiles [4], additively manufactured
O. Weeger · B. Narayanan · M.L. DunnSingapore University of Technology and Design,SUTD Digital Manufacturing and Design Centre,8 Somapah Road, Singapore 487372, Singapore,E-mail: oliver [email protected],E-mail: bharath [email protected],E-mail: martin [email protected]
structures [5], hair and fiber modeling [6, 7], and bio-
dynamic structures such as the double helix of DNA
molecules and arterial pathways [8]. In many of these
problems, complex dynamic behavior and rod-to-rod
contact interactions are essential aspects for the accu-
rate modeling of physical effects and thus accurate, ro-
bust and efficient computational discretization methods
are required for their numerical solution.
For the static and dynamic modeling of 3-dimension-
al (3D), slender beam structures subject to large defor-
mations and rotations, the Cosserat rod model [9–11]
has been employed successfully in many of the afore-
mentioned problems. It covers nonlinear, geometrically
exact deformation behavior, general loading conditions
by external forces and moments, and can be extended
to include viscous damping [12–14]. It leads to a non-
linear partial differential equation (PDE) in space and
time, which usually has to be solved by numerical meth-
ods due to its complexity, wherefore various kinds of
discretization schemes have been proposed. Most com-
monly, first a spatial semi-discretization is carried out,
either using finite element (FEM) [15,16] or finite differ-
ence methods (FDM) [6,13,17], and then time integra-
tion is performed using standard methods for ordinary
differential equations (ODEs). In [18], p-FEM was used
in combination with harmonic balance to solve the pe-
riodic vibration problem. Dynamics of rod structures,
i.e. meshes or nets of interconnected Cosserat rods, were
also investigated using these methods [19,20].
In this work, we apply an isogeometric collocation
method for the spatial semi-discretization. Isogeomet-
ric analysis (IGA) was first introduced by Hughes et
al. in 2005 [21] and has since attracted increasing inter-
est in the computational engineering community. This
novel concept aims at bridging the gap between the
two largely disjunct domains of computer-aided design
2 O. Weeger et al.
(CAD) and computational analysis by using spline-based
function representations within the numerical discretiza-
and Cd3 = 0.004 Nm2s. Both rods are clamped at one
end and a harmonic load nz = −10 sin 5t N is applied
at the free end of the top rod in z-direction. Each rod
is discretized with p = 4 and ` = 16. We use points at
equal intervals on the rod for our coarse contact search,
and contact parameters kc = 5000 N/m, preg = 10−6
m and µ = 0. The simulation is run for a total time of
5 seconds with a time step size of 0.001 s.
As rod 1 moves up and down, it triggers an oscil-
latory motion in the second rod by coming in and out
of contact with it. Figure 7 shows four snapshots of the
most characteristic states of deformation. From Fig. 8,
which shows the end point z-positions of both rods, we
can clearly see the perturbation of the second rod as
soon as the first comes into contact with it. Once con-
tact is lost and rod 1 continues to move upward, rod 2
keeps oscillating, but due to damping the magnitude of
oscillation diminishes and is virtually zero when rod 1
comes down again. In this way, a stable periodic mo-
tion is established, as can be seen in Fig. 8, altogether
showing that our framework with the contact imple-
mentation works well in the dynamic regime.
6.4 Frictional contact problems
Next, we introduce and investigate two model problems
for dynamic motion of rods with frictional, sliding con-
tact.
6.4.1 Swinging rod in frictional contact
As a first example with frictional contact, we study the
highly nonlinear dynamic motion of a rod under a grav-
ity load and investigate the influence of the friction co-
efficient µ.
The rod is pinned at one end and allowed to swing
downwards under the influence of gravity. It has length
L = 1.0 m, cross-section radius R = 0.005 m, Young’s
modulus E = 50 MPa, and Poisson’s ratio ν = 0.5.
Unlike in Section 6.1, however, the swinging motion is
hindered by the presence of a second, stiffer rod placed
perpendicular to and directly beneath the first rod. The
stiffer rod is clamped at both ends and has a radius of
0.01 m and Young’s modulus 5.0 GPa. Both rods have
density ρ = 1100 kg/m3. The spatial discretization is
done using p = 4 and ` = 16 and the contact param-
eters kc and preg are set to 500 N/m and 5 · 10−6 m
Isogeometric collocation for nonlinear dynamic analysis of Cosserat rods with frictional contact 11
(a) t = 0.12 s (b) t = 0.33 s
(c) t = 0.63 s (d) t = 0.95 s
Fig. 7: Contacting rods under harmonic loading. The snapshots are taken at the following points: (a) rod 1 first
contacts rod 2; (b) rod 1 and rod 2 are at their bottom most positions; (c) both rods return to their original
unperturbed positions; (d) rod 1 attains its highest amplitude in the upward direction while rod 2 remains at its
original position
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−0.06
−0.04
−0.02
0.00
0.02
0.04
0.06
0.08
time in s
endpointz-positions
rod 1rod 2
Fig. 8: Contacting rods under harmonic loading. Cen-
terline end point z-positions of two perpendicular rods
r1z(1, t) and r2z(1, t) in time interval 0 to 5 s
respectively. The Crank–Nicolson integration scheme is
used with a time step size of 0.001 s.
We vary the friction coefficient and observe its im-
pact on the final position of the pinned rod after t = 1.0
s, see Figure 9. Due to an increase in µ more energy is
dissipated and the ability of the first rod to slide past
the second rod is restrained. For µ = 0 and µ = 0.1, the
rod still manages to finally slip past the stiff rod, but
for higher values of µ the rod gets “stuck” in different
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x [m]
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
y [m
]
µ = 0.0µ = 0.1µ = 0.2µ = 0.3µ = 0.4stiffer rod
Fig. 9: Swinging rod in frictional contact. All snapshots
are taken at t = 1 s and highlight the influence of the
coefficient of friction µ
positions. At the highest value of µ, the rod is closest to
its horizontal position, which is what one would expect.
6.4.2 Three rods in sliding frictional contact
With the second application including frictional con-
tact, we want to validate the effect of the choice of fric-
12 O. Weeger et al.
Fig. 10: Three rods in sliding frictional contact. Dis-
placed rods after t = 5 s for µ = 0 (red), µ = 0.2
(green), and µ = 0.4 (blue)
tion coefficient more quantitatively and also show that
our formulation can handle multiple contacts correctly.
The use case involves three rods of length L = 1.0
m, with the first rod perpendicular to and underneath
the other two parallel rods. All rods have cross-section
radius R = 0.01 m, Young’s modulus E = 1.0 MPa,
Poisson’s ratio ν = 0.5, density ρ = 1100 kg/m3, and no
damping is applied. An isogeometric discretization with
p = 4 and ` = 10 is used, as well as time integration
using the backward Euler method with a time step size
of 0.02 s over atotal integration time of 5.0 s. Each rod
is clamped at one end and free at the other, and an
upward-directed point load of 0.005 N is applied to the
first rod, pushing it onto the other two. For the contact
treatment, parameters kc and preg are set to 500 N/m
and 0.0001 m respectively.
The friction coefficient µ is now varied for each simu-
lation and Fig. 10 demonstrates this visually by showing
three end configurations at t = 5 s for increasing values
of µ. In Figure 11 the end point position of the first
rod is plotted over the friction coefficient µ. In all three
principal directions, the magnitude of displacement in
the case of frictionless contact is the highest. As µ is in-
creased, the displacement decreases in a (roughly) lin-
ear fashion, which is due to the frictional force acting
against the forced motion of the first rod.
Altogether, both applications with frictional contact
provide a good qualitative and quantitative validation
of our methods and implementation and show that also
complex scenarios, where the system behavior depends
on the friction coefficient in a highly nonlinear way, or
with multiple contacts can be handled in a robust and
accurate manner.
7 Summary and conclusions
We have presented an isogeometric collocation method
for the nonlinear dynamic analysis of spatial rods, in-
cluding frictional contact. Our approach is based on the
dynamic model of mechanical deformation of geometri-
cally exact Cosserat rods, for which we have introduced
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
−0.15
−0.10
−0.05
0.00
0.05
friction coefficient µ
end
poi
nt
dis
pla
cem
entu
uxuyuz
Fig. 11: Three rods in sliding frictional contact. End
point displacement of first rod at t = 5 s for varying
coefficient of friction µ
a spatial semi-discretization based on the concept of iso-
geometric collocation. The balance equations of linear
and angular momentum were re-formulated as a first-
order system in time, and the additional kinematic vari-
ables, centerline velocities and angular velocities, along
with the primary unknowns, centerline positions and
rotations, were discretized as spline curves. Collocation
of the equilibrium equations at Greville abscissae leads
to a stiff, nonlinear ODE system, which was then inte-
grated using standard implicit numerical time integra-
tion schemes.
The method was validated succesfully through sev-
eral numerical studies and computational applications.
Particularily important for many practical applications
is the ability to resolve rod-to-rod contact, which was
succesfully included into our approach by a frictional
contact formulation. Overall, the presented computa-
tional method shows good potential for efficient and ac-
curate nonlinear dynamic rod modeling of challenging
applications, including dynamic contacts with friction
and highly nonlinear system behavior.
Acknowledgements The authors were supported by theSUTD Digital Manufacturing and Design (DManD) Centre,supported by the Singapore National Research Foundation.
References
1. D.Q. Cao, R.W. Tucker, and C. Wang. Cosserat dy-namics and cable-stayed bridge vibrations. In L. Cheng,K.M. Li, and R.M.C. So, editors, Proceedings of theeighth international congress on sound and vibration: 2-6July 2001, the Hong Kong Polytechnic University, HongKong, China, pages 1139–1146. Hong Kong PolytechnicUniversity, Department of Mechanical Engineering, 2001.
Isogeometric collocation for nonlinear dynamic analysis of Cosserat rods with frictional contact 13
2. R.W. Tucker and C. Wang. Torsional vibration controland Cosserat dynamics of a drill-rig assembly. Meccanica,38(1):145–161, 2003.
3. F. Maurin, L. Dede, and A. Spadoni. Isogeometricrotation-free analysis of planar extensible-elastica forstatic and dynamic applications. Nonlinear Dynamics,81:77–96, 2015.
4. D. Durville. Simulation of the mechanical behaviour ofwoven fabrics at the scale of fibers. International Journalof Material Forming, 3:1241–1251, 2010.
5. O. Weeger, Y.S.B. Kang, S.-K. Yeung, and M.L. Dunn.Optimal design and manufacture of active rod structureswith spatially variable materials. 3D Printing and Addi-tive Manufacturing, 3(4):204–215, 2016.
6. F. Bertails, B. Audoly, B. Querleux, F. Leroy, J.-L. Lev-eque, and M.-P. Cani. Predicting natural hair shapes bysolving the statics of flexible rods. In J. Dingliana andF. Ganovelli, editors, Eurographics Short Papers. Euro-graphics, August 2005.
7. F. Bertails-Descoubes, F. Cadoux, G. Daviet, andV. Acary. A nonsmooth newton solver for capturing exactcoulomb friction in fiber assemblies. ACM Transactionson Graphics (TOG), 30(1), 2011.
8. M.B. Rubin. Cosserat Theories: Shells, Rods and Points,volume 79 of Solid Mechanics and Its Applications, chap-ter Cosserat Rods, pages 191–310. Springer Netherlands,2000.
9. K.-J. Bathe and S. Bolourchi. Large displacement anal-ysis of three-dimensional beam structures. Interna-tional Journal for Numerical Methods in Engineering,14(7):961–986, 1979.
10. J.C. Simo. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Computer Meth-ods in Applied Mechanics and Engineering, 49(1):55–70,1985.
11. S.S. Antman. Nonlinear Problems of Elasticity, volume107 of Applied Mathematical Sciences. Springer NewYork, 2005.
12. S.S. Antman. Dynamical problems for geometrically ex-act theories of nonlinearly viscoelastic rods. Journal ofNonlinear Science, 6(1):1–18, 1996.
13. H. Lang, J. Linn, and M. Arnold. Multi-body dynamicssimulation of geometrically exact Cosserat rods. Multi-body System Dynamics, 25(3):285–312, 2011.
14. J. Linn, H. Lang, and A. Tuganov. Geometrically exactCosserat rods with Kelvin–Voigt type viscous damping.Mechnical Sciences, 4:79–96, 2013.
15. D.Q. Cao, D. Liu, and C.H.-T. Wang. Three-dimensionalnonlinear dynamics of slender structures: Cosserat rodelement approach. International Journal of Solids andStructures, 43:760–783, 2006.
16. D.Q. Cao and R.W. Tucker. Nonlinear dynamics of elas-tic rods using the Cosserat theory: Modelling and sim-ulation. International Journal of Solids and Structures,45:460–477, 2008.
17. J. Spillmann and M. Teschner. CoRdE: Cosserat rod el-ements for the dynamic simulation of one-dimensionalelastic objects. In Dimitris Metaxas and JovanPopovic, editors, Proceedings of the 2007 ACM SIG-GRAPH/Eurographics Symposium on Computer Anima-tion, SCA ’07, pages 63–72, Aire-la-Ville, Switzerland,Switzerland, 2007. Eurographics Association.
18. S. Stoykov and P. Ribeiro. Stability of nonlinear periodicvibrations of 3D beams. Nonlinear Dynamics, 66:335–353, 2011.
19. J. Gratus and R.W. Tucker. The dynamics of Cosseratnets. Journal of Applied Mathematics, 2003(4):187–226,2003.
20. J. Spillmann and M. Teschner. Cosserat nets. Visual-ization and Computer Graphics, IEEE Transactions on,15(2):325–338, March 2009.
21. T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Iso-geometric analysis: CAD, finite elements, NURBS, ex-act geometry and mesh refinement. Computer Methodsin Applied Mechanics and Engineering, 194:4135–4195,2005.
22. R. Bouclier, T. Elguedj, and A. Combescure. Locking freeisogeometric formulations of curved thick beams. Com-puter Methods in Applied Mechanics and Engineering,245–246:144–162, 2012.
23. S.B. Raknes, X. Deng, Y. Bazilevs, D.J. Benson, K.M.Mathisen, and T. Kvamsdal. Isogeometric rotation-free bending-stabilized cables: Statics, dynamics, bend-ing strips and coupling with shells. Computer Methods inApplied Mechanics and Engineering, 263:127–143, 2013.
24. L. Greco and M. Cuomo. B-spline interpolation ofKirchhoff-Love space rods. Computer Methods in Ap-plied Mechanics and Engineering, 256:251–269, 2013.
25. J. Kiendl, K.-U. Bletzinger, J. Linhard, and R. Wuchner.Isogeometric shell analysis with Kirchhoff-Love elements.Computer Methods in Applied Mechanics and Engineer-ing, 198:3902–3914, 2009.
26. W. Dornisch, S. Klinkel, and B. Simeon. IsogeometricReissner-Mindlin shell analysis with exactly calculateddirector vectors. Computer Methods in Applied Mechan-ics and Engineering, 253:491–504, 2013.
27. N. Nguyen-Thanh, J. Kiendl, H. Nguyen-Xuan,R. Wuchner, K.-U. Bletzinger, Y. Bazilevs, andT. Rabczuk. Rotation free isogeometric thin shell anal-ysis using PHT-splines. Computer Methods in AppliedMechanics and Engineering, 200:3410–3424, November2011.
28. J. A. Cottrell, A. Reali, Y. Bazilevs, and T. J. R. Hughes.Isogeometric analysis of structural vibrations. Com-puter Methods in Applied Mechanics and Engineering,195:5257–5296, 2006.
29. O. Weeger, U. Wever, and B. Simeon. Isogeometric anal-ysis of nonlinear Euler-Bernoulli beam vibrations. Non-linear Dynamics, 72(4):813–835, 2013.
30. T. J. R. Hughes, J. A. Evans, and A. Reali. Finite elementand NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Computer Methods inApplied Mechanics and Engineering, 272:290–320, 2014.
31. F. Auricchio, L. Beirao da Veiga, T. J. R. Hughes, A. Re-ali, and G. Sangalli. Isogeometric collocation methods.Mathematical Models and Methods in Applied Sciences,20(11):2075–2107, 2010.
32. A. Reali and T.J.R. Hughes. An introduction to isogeo-metric collocation methods. In G. Beer and S. Bordas,editors, Isogeometric Methods for Numerical Simulation,volume 561 of CISM International Centre for MechanicalSciences, pages 173–204. Springer, 2015.
33. D. Schillinger, J. A. Evans, A. Reali, M. A. Scott, andT. J. R. Hughes. Isogeometric collocation: Cost compari-son with Galerkin methods and extension to adaptive hi-erarchical NURBS discretizations. Computer Methods inApplied Mechanics and Engineering, 267:170–232, 2013.
34. F. Auricchio, L. Beirao da Veiga, T.J.R. Hughes, A. Re-ali, and G. Sangalli. Isogeometric collocation for elas-tostatics and explicit dynamics. Computer Methods inApplied Mechanics and Engineering, 249-252:2–14, 2012.Higher Order Finite Element and Isogeometric Methods.
35. L. De Lorenzis, J.A. Evans, T.J.R. Hughes, and A. Reali.Isogeometric collocation: Neumann boundary conditions
14 O. Weeger et al.
and contact. Computer Methods in Applied Mechanicsand Engineering, 284:21–54, 2015.
36. R. Kruse, N. Nguyen-Thanh, L. De Lorenzis, and T.J.R.Hughes. Isogeometric collocation for large deformationelasticity and frictional contact problems. ComputerMethods in Applied Mechanics and Engineering, 296:73–112, 2015.
37. L. Beirao da Veiga, C. Lovadina, and A. Reali. Avoid-ing shear locking for the Timoshenko beam problem viaisogeometric collocation methods. Computer Methodsin Applied Mechanics and Engineering, 241-244:38–51,2012.
38. F. Auricchio, L. Beirao da Veiga, J. Kiendl, C. Lovad-ina, and A. Reali. Locking-free isogeometric collocationmethods for spatial Timoshenko rods. Computer Meth-ods in Applied Mechanics and Engineering, 263:113–126,2013.
39. J. Kiendl, F. Auricchio, T.J.R. Hughes, and A. Reali.Single-variable formulations and isogeometric discretiza-tions for shear deformable beams. Computer Methods inApplied Mechanics and Engineering, 284:988–1004, 2015.Isogeometric Analysis Special Issue.
40. A. Reali and H. Gomez. An isogeometric collocation ap-proach for Bernoulli-Euler beams and Kirchhoff plates.Computer Methods in Applied Mechanics and Engineer-ing, 284:623–636, 2015. Isogeometric Analysis SpecialIssue.
41. O. Weeger, S.-K. Yeung, and M.L. Dunn. Isogeometriccollocation methods for Cosserat rods and rod structures.Computer Methods in Applied Mechanics and Engineer-ing, 316:100–122, 2017. Special Issue on IsogeometricAnalysis: Progress and Challenges.
42. D. Durville. Contact-friction modeling within elasticbeam assemblies: an application to knot tightening. Com-putational Mechanics, 49(6):687–707, 2012.
43. C. Meier, A. Popp, and W. A. Wall. A finite elementapproach for the line-to-line contact interaction of thinbeams with arbitrary orientation. Computer Methods inApplied Mechanics and Engineering, 308:377–413, 2016.
44. C. Meier, W.A. Wall, and A. Popp. A unified approachfor beam-to-beam contact. Computer Methods in AppliedMechanics and Engineering, 315:972–1010, 2017.
45. C. Meier, M.J. Grill, W. A. Wall, and A. Popp. Geomet-rically exact beam elements and smooth contact schemesfor the modeling of fiber-based materials and structures.submitted to International Journal of Solids and Struc-tures, 2016.
46. O. Weeger, B. Narayanan, L. De Lorenzis, J. Kiendl, andM.L. Dunn. An isogeometric collocation method for fric-tionless contact of cosserat rods. submitted to ComputerMethods in Applied Mechanics and Engineering, 2016.
47. M. Bırsan, H. Altenbach, T. Sadowski, V.A. Eremeyev,and D. Pietras. Deformation analysis of functionallygraded beams by the direct approach. Composites PartB: Engineering, 43(3):1315–1328, 2012.
48. Z. Ding, O. Weeger, H.J. Qi, and M.L. Dunn. 4D rod:Rapid 3D self-assembly from 1D soft active composites.submitted to Advanced Functional Materials, 2017.
49. L. A. Piegl and W. Tiller. The NURBS Book. Mono-graphs in Visual Communication. Springer, 1997.
50. H. Gomez and L. De Lorenzis. The variational collocationmethod. Computer Methods in Applied Mechanics andEngineering, 309:152–181, 2016.
51. M. Montardini, G. Sangalli, and L. Tamellini. Optimal-order isogeometric collocation at Galerkin superconver-gent points. arXiv.org pre-print, 2016.
52. B. Simeon. Computational Flexible Multibody Dynam-ics – A Differential-Algebraic Approach. Differential-Algebraic Equations Forum. Springer Berlin Heidelberg,2013.
53. H. Lang and M. Arnold. Numerical aspects in the dy-namic simulation of geometrically exact rods. AppliedNumerical Mathematics, 62(10):1411–1427, 2012.
54. J.T. Oden and J.A.C. Martins. Models and computa-tional methods for dynamic friction phenomena. Com-puter Methods in Applied Mechanics and Engineering,52(1):527–634, 1985.