# American Institute of Mathematical Sciences

February  2016, 9(1): 33-42. doi: 10.3934/dcdss.2016.9.33

## Isogeometric collocation mixed methods for rods

 1 Dipartimento di Ingegneria Civile e Architettura, Università di Pavia, Via Ferrata 3, 27100 Pavia, Italy, Italy 2 Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Cozzi 53, 20125 Milano, Italy 3 Institute of Applied Mechanics, TU Braunschweig, Bienroder Weg 87, 38106 Braunschweig, Germany 4 Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy

Received  September 2014 Revised  February 2015 Published  December 2015

Isogeometric collocation mixed methods for spatial rods are presented and studied. A theoretical analysis of stability and convergence is available. The proposed schemes are locking-free, irrespective of the selected approximation spaces.
Citation: Ferdinando Auricchio, Lourenco Beirão da Veiga, Josef Kiendl, Carlo Lovadina, Alessandro Reali. Isogeometric collocation mixed methods for rods. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 33-42. doi: 10.3934/dcdss.2016.9.33
##### References:
 [1] K. Arunakirinathar and B. D. Reddy, Mixed finite element methods for elastic rods of arbitrary geometry, Numerische Mathematik, 64 (1993), 13-43. doi: 10.1007/BF01388679. [2] F. Auricchio, L. Beirão da Veiga, T. J. R. Hughes, A. Reali and G. Sangalli, Isogeometric collocation methods, Mathematical Models and Methods in Applied Sciences, 20 (2010), 2075-2107. doi: 10.1142/S0218202510004878. [3] F. Auricchio, L. Beirão da Veiga, T. J. R. Hughes, A. Reali and G. Sangalli, Isogeometric collocation for elastostatics and explicit dynamics, Computer Methods in Applied Mechanics and Engineering, 249 (2012), 2-14. doi: 10.1016/j.cma.2012.03.026. [4] F. Auricchio, L. Beirão da Veiga, J. Kiendl, C. Lovadina and A. Reali, Locking-free isogeometric collocation methods for spatial Timoshenko rods, Comput. Methods Appl. Mech. Engrg., 263 (2013), 113-126. doi: 10.1016/j.cma.2013.03.009. [5] L. Beirão da Veiga, A. Buffa, J. Rivas and G. Sangalli, Some estimates for $h-p-k-$refinement in isogeometric analysis, Numerische Mathematik, 118 (2011), 271-305. doi: 10.1007/s00211-010-0338-z. [6] L. Beirão da Veiga, C. Lovadina and A. Reali, Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods, Computer Methods in Applied Mechanics and Engineering, 241/244 (2012), 38-51. doi: 10.1016/j.cma.2012.05.020. [7] D. Chapelle, A locking-free approximation of curved rods by straight beam elements, Numerische Mathematik, 77 (1997), 299-322. doi: 10.1007/s002110050288. [8] J. A. Cottrell, T. J. R. Hughes and Y. Bazilevs, Isogeometric Analysis. Towards Integration of CAD and FEA, Wiley, 2009. [9] J. A. Cottrell, A. Reali, Y. Bazilevs and T. J. R. Hughes, Isogeometric analysis of structural vibrations, Computer Methods in Applied Mechanics and Engineering, 195 (2006), 5257-5296. doi: 10.1016/j.cma.2005.09.027. [10] C. de Boor, A Practical Guide to Splines, Springer, 2001. [11] S. Demko, On the existence of interpolation projectors onto spline spaces, Journal of Approximation Theory, 43 (1985), 151-156. doi: 10.1016/0021-9045(85)90123-6. [12] T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194 (2005), 4135-4195. doi: 10.1016/j.cma.2004.10.008. [13] T. J. R. Hughes, A. Reali and G. Sangalli, Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of $p$-method finite elements with $k$-method NURBS, Computer Methods in Applied Mechanics and Engineering, 197 (2008), 4104-4124. doi: 10.1016/j.cma.2008.04.006. [14] R. W. Johnson, A B-spline collocation method for solving the incompressible Navier-Stokes equations using an ad hoc method: The Boundary Residual method, Computers & Fluids, 34 (2005), 121-149. doi: 10.1016/j.compfluid.2004.03.005. [15] A. Reali, An isogeometric analysis approach for the study of structural vibrations, Journal of Earthquake Engineering, 10 (2006), 1-30. [16] D. Schillinger, J. A. Evans, A. Reali, M. A. Scott and T. J. R. Hughes, Isogeometric collocation methods: Cost comparison with Galerkin methods and extension to hierarchical discretizations, Computer Methods in Applied Mechanics and Engineering, 267 (2013), 170-232.

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##### References:
 [1] K. Arunakirinathar and B. D. Reddy, Mixed finite element methods for elastic rods of arbitrary geometry, Numerische Mathematik, 64 (1993), 13-43. doi: 10.1007/BF01388679. [2] F. Auricchio, L. Beirão da Veiga, T. J. R. Hughes, A. Reali and G. Sangalli, Isogeometric collocation methods, Mathematical Models and Methods in Applied Sciences, 20 (2010), 2075-2107. doi: 10.1142/S0218202510004878. [3] F. Auricchio, L. Beirão da Veiga, T. J. R. Hughes, A. Reali and G. Sangalli, Isogeometric collocation for elastostatics and explicit dynamics, Computer Methods in Applied Mechanics and Engineering, 249 (2012), 2-14. doi: 10.1016/j.cma.2012.03.026. [4] F. Auricchio, L. Beirão da Veiga, J. Kiendl, C. Lovadina and A. Reali, Locking-free isogeometric collocation methods for spatial Timoshenko rods, Comput. Methods Appl. Mech. Engrg., 263 (2013), 113-126. doi: 10.1016/j.cma.2013.03.009. [5] L. Beirão da Veiga, A. Buffa, J. Rivas and G. Sangalli, Some estimates for $h-p-k-$refinement in isogeometric analysis, Numerische Mathematik, 118 (2011), 271-305. doi: 10.1007/s00211-010-0338-z. [6] L. Beirão da Veiga, C. Lovadina and A. Reali, Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods, Computer Methods in Applied Mechanics and Engineering, 241/244 (2012), 38-51. doi: 10.1016/j.cma.2012.05.020. [7] D. Chapelle, A locking-free approximation of curved rods by straight beam elements, Numerische Mathematik, 77 (1997), 299-322. doi: 10.1007/s002110050288. [8] J. A. Cottrell, T. J. R. Hughes and Y. Bazilevs, Isogeometric Analysis. Towards Integration of CAD and FEA, Wiley, 2009. [9] J. A. Cottrell, A. Reali, Y. Bazilevs and T. J. R. Hughes, Isogeometric analysis of structural vibrations, Computer Methods in Applied Mechanics and Engineering, 195 (2006), 5257-5296. doi: 10.1016/j.cma.2005.09.027. [10] C. de Boor, A Practical Guide to Splines, Springer, 2001. [11] S. Demko, On the existence of interpolation projectors onto spline spaces, Journal of Approximation Theory, 43 (1985), 151-156. doi: 10.1016/0021-9045(85)90123-6. [12] T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194 (2005), 4135-4195. doi: 10.1016/j.cma.2004.10.008. [13] T. J. R. Hughes, A. Reali and G. Sangalli, Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of $p$-method finite elements with $k$-method NURBS, Computer Methods in Applied Mechanics and Engineering, 197 (2008), 4104-4124. doi: 10.1016/j.cma.2008.04.006. [14] R. W. Johnson, A B-spline collocation method for solving the incompressible Navier-Stokes equations using an ad hoc method: The Boundary Residual method, Computers & Fluids, 34 (2005), 121-149. doi: 10.1016/j.compfluid.2004.03.005. [15] A. Reali, An isogeometric analysis approach for the study of structural vibrations, Journal of Earthquake Engineering, 10 (2006), 1-30. [16] D. Schillinger, J. A. Evans, A. Reali, M. A. Scott and T. J. R. Hughes, Isogeometric collocation methods: Cost comparison with Galerkin methods and extension to hierarchical discretizations, Computer Methods in Applied Mechanics and Engineering, 267 (2013), 170-232.
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