SUMMARYPatch recovery based on superconvergent derivatives and equilibrium (SPRE), an enhancement of the Superconvergent Patch Recovery (SPR), is studied for linear elasticity problems. The paper also presents a further improvement for recovery of derivatives near boundaries, SPREB, where either tractions or displacements are prescribed. This is made by inclusion of weighted residual errors at boundary points in the patch recovery. A pronounced improvement in the post processed gradients of the finite element solution is observed by this method.
SUMMARYThis paper studies a time-discontinuous Galerkin finite element method for structural dynamic problems, by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels. A new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations. By using the jumps of the displacements and the velocities in the total energy norm as error indicators, an adaptive time-stepping procedure for selecting the proper time step size is described. Numerical examples including both single-DOF and multi-DOF problems are used to illustrate the performance of these algorithms. Comparisons with the exact results and/or the results by the Newmark integration scheme are given. It is shown that the time-discontinuous Galerkin finite element method discussed in this study possesses good accuracy (third order) and stability properties, its numerical implementation is not difficult, and the higher computational cost needed in each time step is compensated by use of a larger time step size.
SUMMARYIn this paper a postprocessing technique is developed for determining first-order derivatives (fluxes, stresses) at nodal points based on derivatives in superconvergent points. It is an extension of the superconvergent patch recovery technique presented by Zienkiewicz and Zhu. In contrast to that technique all flux or stress components are interpolated at the same time, coupled by equilibrium equations at the superconvergent points. The equilibrium equations and use of one order higher degree of interpolation polynomials of stress give a dramatic decrease in error of recovered derivatives even at boundaries.
SUMMARYIn most plate elements using the Reissner-Mindlin assumptions, the interpolations used for the lateral displacements (w) and the rotation (8) involve the independent representation of each variable by its nodal values, usually with identical interpolations. To ensure a higher order of expansion for displacement MI its representation is linked in the present paper with both sets of nodal variables.Conditions necessary for the use of such expansions are established here and the paper shows the development of a linear quadrilateral element (Q4BL) whose performance and robustness are good (although it possesses one singularity if only three degrees of freedom are prescribed).In Part 11 we apply the identical formulation to develop a triangular element (T3BL) which performs equally well and is fully robust.
SUMMARYThis paper discusses implementation and adaptivity of the Discontinuous Galerkin (DG) "nite element method as applied to linear and non-linear structural dynamic problems. By the DG method, both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels. Both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived. They are third-order accurate and, while the implicit procedure is unconditionally stable, the explicit one is conditionally stable. An h-adaptive procedure based on the Zienkiewicz}Zhu error estimate using the SPR technique is applied. Numerical examples are presented to show the suitability of the DG method for both linear and non-linear structural dynamic analysis.
SUMMARYIn this paper, we present a post-processing technique and an a posteriori error estimate for the Newmark method in structural dynamic analysis. By post-processing the Newmark solutions, we derive a simple formulation for linearly varied third-order derivatives. By comparing the Newmark solutions with the exact solutions expanded in the Taylor series, we achieve the local post-processed solutions which are of fifth-order accuracy for displacements and fourth-order accuracy for velocities in one step. Based on the post-processing technique, a posteriori local error estimates for displacements, velocities and, thus, also the total energy norm error estimate are obtained. If the Newmark solutions are corrected at each step, the post-processed solutions are of third-order accuracy in the global sense, i.e. one-order improvement for the original Newmark solutions is achieved. We also discuss a method for estimating the global time integration error. We find that, when the total energy norm is used, the sum of the local error estimates will give a reasonable estimate for the global error. We present numerical studies on a SDOF and a 2-DOF example in order to demonstrate the performance of the proposed technique.
SUMMARYA simple u posteriori local error estimate for Newmark time integration schemes in dynamic analysis is presented, based on the concept of a so called 'post-processing' technique. In conjunction with the error estimate, an adaptive timestepping algorithm is described, which adjusts the time step size so that the local error of each time step is within a prescribed error tolerance. Numerical examples given in the paper indicate that the error estimate is asymptotically convergent, computationally efficient and convenient, and the adaptive time-stepping scheme can predict a nearly optimal step size from time to time, thus making the numerical solution reliable in an efficient manner.
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