Least-squares finite element approximations to the Timoshenko beam problem

In this paper a least-squares finite element method for the Timoshenko beam problem is proposed and analyzed. The method is shown to be convergent and stable without requiring extra smoothness of the exact solutions. For sufficiently regular exact solutions, the method achieves optimal order of convergence in the H^1-norm for all the unknowns (displacement, rotation, shear, moment), uniformly in the small parameter which is generally proportional to the ratio of thickness to length. Thus the locking phenomenon disappears as the parameter tends to zero. A sharp a posteriori error estimator which is exact in the energy norm and equivalent in the H^1-norm is also briefly discussed.

[1]  Suh-Yuh Yang,et al.  A least-squares finite element method for incompressible flow in stress-velocity-pressure version☆ , 1995 .

[2]  Jinn-Liang Liu,et al.  Exact a posteriori error analysis of the least squares finite element method , 2000, Appl. Math. Comput..

[3]  Suh-Yuh Yang,et al.  Least squares nite element methodsfor the elasticity problem , 2022 .

[4]  Likang Li,et al.  Discretization of the Timoshenko Beam problem by thep and theh-p versions of the finite element method , 1990 .

[5]  T. A. Manteuffel,et al.  First-Order System Least Squares for Velocity-Vorticity-Pressure Form of the Stokes Equations, with Application to Linear Elasticity , 1996 .

[6]  Ivo Babuška,et al.  A Posteriori Error Analysis of Finite Element Solutions for One-Dimensional Problems , 1981 .

[7]  Ching L. Chang,et al.  Finite element method for the solution of Maxwell's equations in multiple media , 1988 .

[8]  Thomas A. Manteuffel,et al.  First-Order System Least Squares (FOSLS) for Convection-Diffusion Problems: Numerical Results , 1998, SIAM J. Sci. Comput..

[9]  Ralf Kornhuber,et al.  A posteriori error estimates for elliptic problems in two and three space dimensions , 1996 .

[10]  Bo-nan Jiang,et al.  The Origin of Spurious Solutions in Computational Electromagnetics , 1996 .

[11]  Thomas A. Manteuffel,et al.  First-Order System Least Squares (FOSLS) for Planar Linear Elasticity: Pure Traction Problem , 1998 .

[12]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[13]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[14]  D. Arnold.  Discretization by finite elements of a model parameter dependent problem , 1981 .

[15]  J. Oden,et al.  A unified approach to a posteriori error estimation using element residual methods , 1993 .

[16]  Ching L. Chang,et al.  Analysis of a two-stage least-squares finite element method for the planar elasticity problem , 1999 .

[17]  Suh-Yuh Yang,et al.  Least-squares finite element methods for the elasticity problem , 1997 .

[18]  M. Gunzburger,et al.  Analysis of least squares finite element methods for the Stokes equations , 1994 .

[19]  T. Hughes,et al.  Petrov-Galerkin formulations of the Timoshenko beam problem , 1987 .

[20]  I. Babuska,et al.  On locking and robustness in the finite element method , 1992 .

[21]  Randolph E. Bank,et al.  A posteriori error estimates based on hierarchical bases , 1993 .

[22]  R. Bank,et al.  Some a posteriori error estimators for elliptic partial differential equations , 1985 .

[23]  Weimin Han,et al.  Finite element methods for Timoshenko beam, circular arch and Reissner-Mindlin plate problems , 1997 .

[24]  Milton E. Rose,et al.  A COMPARATIVE STUDY OF FINITE ELEMENT AND FINITE DIFFERENCE METHODS FOR CAUCHY-RIEMANN TYPE EQUATIONS* , 1985 .

[25]  John J. Nelson,et al.  Least-Squares Finite Element Method for the Stokes Problem with Zero Residual of Mass Conservation , 1997 .

[26]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[27]  T. Manteuffel,et al.  First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity , 1997 .

[28]  D. M. Bedivan,et al.  Least squares methods for optimal shape design problems , 1995 .

[29]  Graham F. Carey,et al.  Adaptive refinement for least‐squares finite elements with element‐by‐element conjugate gradient solution , 1987 .

[30]  Ching L. Chang,et al.  An error analysis of least-squares finite element method of velocity-pressure-vorticity formulation for Stokes problem , 1990 .

[31]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[32]  Thomas J. R. Hughes,et al.  Mixed Petrov-Galerkin methods for the Timoshenko beam problem , 1987 .