Novel higher order mass matrices for isogeometric structural vibration analysis

Abstract A set of novel higher order mass matrices are presented for isogeometric analysis of structural vibrations using NURBS. The proposed method for the construction of higher order mass matrices contains two essential steps. Firstly based upon the standard consistent mass matrix a special reduced bandwidth mass matrix is designed. This reduced bandwidth mass matrix meets the requirement of mass conservation while simultaneously preserves the same order of frequency accuracy as the corresponding consistent mass matrix. Subsequently a mixed mass matrix is formulated through a linear combination of the reduced bandwidth mass matrix and the consistent mass matrix. The desired higher order mass matrix is then deduced from the mixed mass matrix by optimizing the linear combination parameter to achieve the most favorable order of accuracy. Both quadratic and cubic formulations are discussed in detail and it is shown that with regard to the vibration frequency, the proposed higher order mass matrices have 6th and 8th orders of accuracy in contrast to the 4th and 6th orders of accuracy associated with the quadratic and cubic consistent mass matrices. A generalization to two dimensional higher order mass matrix is further realized by the tensor product operation on the one dimensional reduced bandwidth and consistent mass matrices. A series of benchmark examples congruously demonstrate that the proposed higher order mass matrices are capable of achieving the theoretically derived optimal accuracy orders for structural vibration analysis.

[1]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[2]  Jia Lu,et al.  Isogeometric contact analysis: Geometric basis and formulation for frictionless contact , 2011 .

[3]  T. Hughes,et al.  Isogeometric Fluid–structure Interaction Analysis with Applications to Arterial Blood Flow , 2006 .

[4]  Paul Steinmann,et al.  Isogeometric analysis of 2D gradient elasticity , 2011 .

[5]  I. Akkerman,et al.  Large eddy simulation of turbulent Taylor-Couette flow using isogeometric analysis and the residual-based variational multiscale method , 2010, J. Comput. Phys..

[6]  Hung Nguyen-Xuan,et al.  Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS‐based isogeometric approach , 2012 .

[7]  Antonio Huerta,et al.  3D NURBS‐enhanced finite element method (NEFEM) , 2008 .

[8]  Roland Wüchner,et al.  Isogeometric shell analysis with Kirchhoff–Love elements , 2009 .

[9]  D. F. Rogers,et al.  An Introduction to NURBS: With Historical Perspective , 2011 .

[10]  T. Hughes,et al.  Isogeometric analysis of the Cahn–Hilliard phase-field model , 2008 .

[11]  T. Hughes,et al.  Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows , 2007 .

[12]  C. Stavrinidis,et al.  New concepts for finite-element mass matrix formulations , 1989 .

[13]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[14]  John A. Evans,et al.  Isogeometric analysis using T-splines , 2010 .

[15]  R. Echter,et al.  A hierarchic family of isogeometric shell finite elements , 2013 .

[16]  Alessandro Reali,et al.  AN ISO GEOMETRIC ANALYSIS APPROACH FOR THE STUDY OF STRUCTURAL VIBRATIONS , 2006 .

[17]  H. Nguyen-Xuan,et al.  Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids , 2011 .

[18]  Xiaoping Qian,et al.  Full analytical sensitivities in NURBS based isogeometric shape optimization , 2010 .

[19]  T. Hughes,et al.  Isogeometric fluid-structure interaction: theory, algorithms, and computations , 2008 .

[20]  Seonho Cho,et al.  Isogeometric shape design sensitivity analysis using transformed basis functions for Kronecker delta property , 2013 .

[21]  T. Q. Bui,et al.  Free vibration and buckling analysis of laminated composite plates using the NURBS-based isogeometric finite element method , 2012 .

[22]  T. Hughes,et al.  ISOGEOMETRIC COLLOCATION METHODS , 2010 .

[23]  Ki-Ook Kim,et al.  A REVIEW OF MASS MATRICES FOR EIGENPROBLEMS , 1993 .

[24]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[25]  Thomas J. R. Hughes,et al.  An isogeometric analysis approach to gradient damage models , 2011 .

[26]  T. Hughes,et al.  Isogeometric collocation for elastostatics and explicit dynamics , 2012 .

[27]  Alessandro Reali,et al.  Duality and unified analysis of discrete approximations in structural dynamics and wave propagation : Comparison of p-method finite elements with k-method NURBS , 2008 .

[28]  Antonio Huerta,et al.  NURBS-enhanced finite element method , 2006 .

[29]  B. Simeon,et al.  Adaptive isogeometric analysis by local h-refinement with T-splines , 2010 .

[30]  T. Rabczuk,et al.  A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis , 2012 .

[31]  Victor M. Calo,et al.  Isogeometric Variational Multiscale Large-Eddy Simulation of Fully-developed Turbulent Flow over a Wavy Wall , 2012 .

[32]  N. Asmar,et al.  Partial Differential Equations with Fourier Series and Boundary Value Problems , 2004 .

[33]  Isaac Fried,et al.  Superaccurate finite element eigenvalue computation , 2004 .

[34]  Thomas J. R. Hughes,et al.  A large deformation, rotation-free, isogeometric shell , 2011 .

[35]  Anh-Vu Vuong,et al.  ISOGAT: A 2D tutorial MATLAB code for Isogeometric Analysis , 2010, Comput. Aided Geom. Des..

[36]  Debasish Roy,et al.  NURBS-based parametric mesh-free methods , 2008 .

[37]  Thomas J. R. Hughes,et al.  NURBS-based isogeometric analysis for the computation of flows about rotating components , 2008 .

[38]  W. Wall,et al.  Isogeometric structural shape optimization , 2008 .

[39]  John A. Evans,et al.  Isogeometric boundary element analysis using unstructured T-splines , 2013 .

[40]  Peter Wriggers,et al.  Contact treatment in isogeometric analysis with NURBS , 2011 .

[41]  Ping Wang,et al.  Adaptive isogeometric analysis using rational PHT-splines , 2011, Comput. Aided Des..

[42]  T. Belytschko,et al.  A generalized finite element formulation for arbitrary basis functions: From isogeometric analysis to XFEM , 2010 .

[43]  Thomas J. R. Hughes,et al.  Isogeometric shell analysis: The Reissner-Mindlin shell , 2010 .

[44]  Yuri Bazilevs,et al.  Rotation free isogeometric thin shell analysis using PHT-splines , 2011 .

[45]  Alessandro Reali,et al.  Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods , 2012 .

[46]  Alessandro Reali,et al.  Studies of Refinement and Continuity in Isogeometric Structural Analysis (Preprint) , 2007 .

[47]  Thomas J. R. Hughes,et al.  Patient-Specific Vascular NURBS Modeling for Isogeometric Analysis of Blood Flow , 2007, IMR.

[48]  Dongdong Wang,et al.  An improved NURBS-based isogeometric analysis with enhanced treatment of essential boundary conditions , 2010 .

[49]  N. Valizadeh,et al.  Extended isogeometric analysis for simulation of stationary and propagating cracks , 2012 .

[50]  Saeed Shojaee,et al.  Free vibration analysis of thin plates by using a NURBS-based isogeometric approach , 2012 .