This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method (FEM) by recursive application of the one-dimensional (1D) element energy projection (EEP) technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems, based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element (FE) model is reached. This conceptual dimension-bydimension (D-by-D) discretization procedure is entirely equivalent to a full FE discretization. As a reverse D-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional (2D) and three-dimensional (3D) problems can be obtained over the domain. Numerical examples of 3D Poisson's equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.
Si YUAN, Yue WU, Qinyan XING
. Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique[J]. Applied Mathematics and Mechanics, 2018
, 39(7)
: 1031
-1044
.
DOI: 10.1007/s10483-018-2345-7
[1] BABUSKA, I. and RHEINBOLDT, W. C. A posteriori error estimates for the finite element method. International Journal for Numerical Methods in Engineering, 12(10), 1597-1615(1978)
[2] ZIENKIEWICZ, O. C. and ZHU, J. Z. The superconvergence patch recovery (SPR) and a posteriori error estimates, part 1:the recovery technique. International Journal for Numerical Methods in Engineering, 33(7), 1331-1364(1992)
[3] ZIENKIEWICZ, O. C. and ZHU, J. Z. The superconvergence patch recovery (SPR) and a posteriori error estimates, part 2:error estimates and adaptivity. International Journal for Numerical Methods in Engineering, 33(7), 1365-1382(1992)
[4] WIBERG, N. E. and LI, X. D. Superconvergent patch recovery of finite-element solution and a posteriori L2 norm error estimate. Communications in Numerical Methods in Engineering, 10(4), 313-320(1994)
[5] BOROOMAND, B. and ZIENKIEWICZ, O. C. Recovery by equilibrium in patches (REP). International Journal for Numerical Methods in Engineering, 40(1), 137-164(1997)
[6] BENEDETTI, A., MIRANDA, S. D., and UBERTINI, F. A posteriori error estimation based on the superconvergent recovery by compatibility in patches. International Journal for Numerical Methods in Engineering, 67(1), 108-131(2006)
[7] YUAN, S. and WANG, M. An element-energy-projection method for post-computation of superconvergent solutions in one-dimensional FEM (in Chinese). Engineering Mechanics, 21(2), 1-9(2004)
[8] STRANG, G. and FIX, G. J. An Analysis of the Finite Element Method, Prentice-Hall, London (1973)
[9] WANG, M. and YUAN, S. Computation of super-convergent nodal stresses of Timoshenko beam elements by EEP method. Applied Mathematics and Mechanics (English Edition), 25(11), 1228-1240(2004) https://doi.org/10.1007/BF02438278
[10] YUAN, S., DU, Y., XING, Q. Y., and YE, K. S. Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method. Applied Mathematics and Mechanics (English Edition), 35(10), 1223-1232(2014) https://doi.org/10.1007/s10483-014-1869-9
[11] YUAN, S. The Finite Element Method of Lines:Theory and Applications, Science Press, BeijingNew York (1993)
[12] PANG, Z. The error estimate of finite element method of lines (in Chinese). Mathematica Numerica Sinica, 15(1), 110-120(1993)
[13] YUAN, S., WANG, M., and WANG, X. An element-energy-projection method for superconvergent solutions in two-dimensional finite element method of lines (in Chinese). Engineering Mechanics, 24(1), 1-10(2007)
[14] YUAN, S., WU, Y., XU, J. J., and XING, Q. Y. Exploration on super-convergent solutions of 3D FEM based on EEP method (in Chinese). Engineering Mechanics, 33(9), 15-20(2016)
Email Alert
RSS