A reduced basis approach for rapid and reliable computation of structural linear elasticity problems using mixed interpolation of tensorial components element

As the reduced basis method is well used in some other fields to rapidly solve engineer systems, it is adapted to structural computation here. In this article, a reduced basis approach of computing structural linear elasticity problems is proposed to obtain rapid and reliable outputs using mixed interpolation of tensorial components elements. The procedure is as follows. First, structural computation problems based on finite element formulation should be parametrized. As an example, finite element formulation based on a type of element is analyzed and parameters in the formulation are extracted explicitly. Parametrized structural problem is built by assembling finite element formulation of components with parameters. Then, an approximate subspace of lower dimension based on structural outputs within whole parameter domain is constructed by an adaptive procedure. By projecting the parametrized structural problem onto that subspace, it is reduced to a parametrized lower dimension problem, which can be solved rapidly with random given parameters. The procedure is divided into two phases, naming offline and online procedure. Structures in vehicle design problems are employed to verify the feasibility of the procedure and deviation of the accuracy. The results showed that reduced basis approach of structural computation is applicable and efficient.

[1]  G. Rozza,et al.  On the stability of the reduced basis method for Stokes equations in parametrized domains , 2007 .

[2]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1980 .

[3]  T. A. Porsching,et al.  Estimation of the error in the reduced basis method solution of nonlinear equations , 1985 .

[4]  D. Rovas,et al.  A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations , 2003 .

[5]  Ngoc Cuong Nguyen,et al.  A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations , 2007, J. Comput. Phys..

[6]  Anthony T. Patera,et al.  A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations , 2002, J. Sci. Comput..

[7]  Phill-Seung Lee,et al.  The quadratic MITC plate and MITC shell elements in plate bending , 2010, Adv. Eng. Softw..

[8]  Anthony T. Patera,et al.  Reduced basis approximation and a posteriori error estimation for a Boltzmann model , 2007 .

[9]  K. Bathe.  Finite Element Procedures , 1995 .

[10]  Gui-Rong Liu,et al.  Rapid inverse parameter estimation using reduced-basis approximation with asymptotic error estimation , 2008 .

[11]  G. Liu,et al.  Rapid identification of elastic modulus of the interface tissue on dental implants surfaces using reduced-basis method and a neural network. , 2009, Journal of biomechanics.

[12]  Nguyen Ngoc Cuong,et al.  Certified Real-Time Solution of Parametrized Partial Differential Equations , 2005 .

[13]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[14]  Anthony T. Patera,et al.  Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations , 2002 .

[15]  A. Patera,et al.  A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations , 2005 .

[16]  K. Bathe,et al.  Development of MITC isotropic triangular shell finite elements , 2004 .

[17]  Alexander G Iosilevich,et al.  An evaluation of the MITC shell elements , 2000 .

[18]  S. Ravindran,et al.  A Reduced-Order Method for Simulation and Control of Fluid Flows , 1998 .

[19]  Anthony T. Patera,et al.  Inverse identification of thermal parameters using reduced-basis method , 2005 .

[20]  Anthony T. Patera,et al.  "Natural norm" a posteriori error estimators for reduced basis approximations , 2006, J. Comput. Phys..

[21]  K. ITOy.  REDUCED BASIS METHOD FOR OPTIMAL CONTROL OF UNSTEADY VISCOUS FLOWS , 2006 .

[22]  K. Bathe,et al.  Fundamental considerations for the finite element analysis of shell structures , 1998 .

[23]  Gianluigi Rozza,et al.  Reduced basis method for linear elasticity problems with many parameters , 2008 .

[24]  Gui-Rong Liu,et al.  A novel reduced-basis method with upper and lower bounds for real-time computation of linear elasticity problems , 2008 .

[25]  A. Patera,et al.  Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds , 2005 .