Solution of Nonlinear Incompressible Hyperelastic Problems by Isogeometric Analysis Method

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Abstract

This article is devoted to the derivation of formulation and isogeometric solution of nonlinear incompressible elastic problems, known as incompressible hyperelasticity. After problem definition, the governing equations are linearized for employing the Newton-Raphson iteration method. Then, the problem is discretized by using concepts of isogeometric analysis method and its solution algorithm is devised. To demonstrate the performance of the proposed approach, the obtained results are compared with finite elements. Due to large deformations in this kind of problems, the finite element method requires a relatively large number of elements, as well as the need for remeshings in some problems, that results in a large system of equations with a high computational cost. In the isogeometric analysis method, using B-Spline and NURBS (Non-Uniform Rational B-Spline) basis functions provides us with a good flexibility in modeling of geometry without any need for further remeshings. The examples studied in this article indicate that by using the isogeometric approach good quality results are obtained with a smaller system of equations and less computational cost. Also, influence of Gauss integration points for the incompressible materials are investigated.

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[1] Tuner MJ, Drill EH, Martin HC, Melosh RJ (1960) Large deflection of structures subject to heating and external load. J Aero Sci 27: 97–106.
[2] Kapur WW, Hartz BJ (1966) Stability of plates using the finite element method. J Engng Mech Die, ASCE 92, 177–195.
[3] Gallagher RJ, Padlog J (1963) Discrete element approach to structural stability. Am Inst Aero & Astro J 1(6): 1437–1439.
[4] Gallagher RJ, Gellatly RA, Padlog J, Mallet RH (1967) A discrete element procedure for thin shell instability analysis. Am Inst Aero & Astro J 5(1): 138–145.
[5] Holand I, Moan T (1969) The finite element in plate buckling, Finite Element Meth. in Stress Analysis. 1st edn. Tapir.
[6] Argyris JH (1964) Recent Advance in matrix method of structure analysis. Pergamon Press.
[7] Argyris JH (1965) Countinua and discountinua. Proc. conf. Matrix Methods in Struct. mech. Air ForceInst. of Tech. Wright Patterson Air Force Base, Ohio.
[8] Oden JT (1967) Numerical formulation of non-linear elasticity problems. Proc. ASCE J. Struct. Dir. 93, ST3 52–90.
[9] Mallet RH, Marcal PV (1968) Finite element analysis of non-linear structures. Proc. ASCE. J. of Struct. Dir. 94, ST9 2081–2105.
[10] Oden JT (1969) Finite element application in non-linear structural analysis. Proc. Conf. on Finite elemnt Meth.Vanderbilt University Tennessee.
[11] Haisler WE, Stricklin JE, Stebbins FJ (1971) Development and evaluation of  solution procedures for geometrically non-linear structural analysis by the discrete stiffnes method. AIAA/ASME, 12th structure, Structural Dynamics & Materials Conf, Anaheim, Californa.
[12] Zinckiewicz OC (1971) The finite element in engeneering science. Mc Graw-Hill, London.
[13] Brebbia C, Connor J (1969) Geometrically non-linear finite element analysis. Proc. ASCE J. Eng. Mech. Dir. Proc. Paper 6516.
[14] Crisfield MA (1991) Nonlinear finite element analysis of solids and structures. vol I & vol II, John Wiley & Sons.
[15] Belytschko T, Liu WK, Moran B (2000) Nonlinera finite element for countinua and structures. John Wiley & Sons.
[16] Zinkiewicz OC, Taylor RL (2000) The finite element method. 5nd edn.  McGraw Hill.
[17] Bonet J, Wood RD (2008) Nonlinear continuum mechanics for finite element analysis. 2nd edn. Cambridge University Press.
[18] Wriggers P (2008) Nonlinear finite element methods. Springer.
[19] Piegel L, Wayne T (1996) The nurbs book, 2nd edn. Springer.
[20] Rogers DF (2001) An introduction to nurbs with historical perspective. Morgan Kaufmann Publishers.
[21] Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: Cad, finite elements, NURBS, exact geometry and mesh refinement, Comput. Meth Appl Mech Engrg 194(39–41): 4135–4195.
Journal of Solid and Fluid Mechanics
Volume 5, Issue 2 - Serial Number 2
July and August 2015
Pages 29-41

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