Error indicators and accuracy improvement of finite element solutions

Error indicators are introduced as part of a simple computational procedure for improving the accuracy of the finite element solutions for plate and shell problems. The procedure is based on using an initial (coarse) grid and a refined (enriched) grid, and approximating the solution for the refined grid by a linear combination of a few global approximation vectors (or modes) which are generated by solving two uncoupled sets of equations in the coarse grid unknowns and the additional degrees of freedom of the refined grid. The global approximation vectors serve as error indicators since they provide quantitative pointwise information about the sensitivity of the different response quantities to the approximation used. The three key elements of the computational procedure are: (a) use of mixed finite element models with discontinuous stress resultants at the element interfaces; (b) operator splitting, or additive decomposition of the finite element arrays for the refined grid into the sum of the coarse grid arrays and correction terms (representing the refined grid contributions); and (c) application of a reduction method through successive use of the finite element method and the classical Bubnov—Galerkin technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Bubnov—Galerkin technique. The similarities between the proposed computational procedure and a preconditioned conjugate gradient (PCG) technique are identified and are exploited to generate from the PCG technique pointwise error indicators. The effectiveness of the proposed procedure is demonstrated by means of two numerical examples of an isotropic toroidal shell and a laminated anisotropic cylindrical panel.