Knut Morten Okstad

Error estimation based on Superconvergent Patch Recovery using statically admissible stress fields

Norges teknisk-naturvitenskapelige universitet REPORT NTNU Institutt for konstruksjonsteknikk Title Date Author Sign. In this paper we investigate an approach for a posteriori error estimation based on recovery of an improved stress eld. The qualitative properties of the recovered stress eld necessary to obtain a conservative error estimator, i.e. an upper bound on the true error, are given. A speciic procedure for recovery of an improved stress eld is then developed. The procedure can be classiied as superconvergent patch recovery (SPR) enhanced with approximate satisfaction of the interior equilibrium and the natural boundary conditions. Herein the interior equilibrium is satissed a priori within each nodal patch. Compared to the original SPR-method, which usually underestimates the true error, the present approach gives a more conservative estimate. The performance of the developed error es-timator is illustrated by investigating two plane strain problems with known closed-form solutions. Abstract In this paper we investigate an approach for a posteriori error estimation based on recovery of an improved stress eld. The qualitative properties of the recovered stress eld necessary to obtain a conservative error estimator, i.e. an upper bound on the true error, are given. A speciic procedure for recovery of an improved stress eld is then developed. The procedure can be classiied as superconvergent patch recovery (SPR) enhanced with approximate satisfaction of the interior equilibrium and the natural boundary conditions. Herein the interior equilibrium is satissed a priori within each nodal patch. Compared to the original SPR-method, which usually underestimates the true error, the present approach gives a more conservative estimate. The performance of the developed error estimator is illustrated by investigating two plane strain problems with known closed-form solutions.

DYNAMIC RESPONSE AND FLUID/STRUCTURE INTERACTION OF SUBMERGED FLOATING TUNNELS

Abstract Alternative approaches to stochastic dynamic response analysis of submerged floating tunnels subjected to wave loading are presented. For the purpose of establishing force, damping and mass coefficients for structural elements with three-dimensional flow conditions, fluid/structure interaction is modeled as finite element implementation of the Navier–Stokes equation. The numerical examples emphasize the effects of wave direction, shortcrestedness, damping, geometrical stiffness and frequency dependence in mass and damping coefficients.

ERROR ESTIMATION AND ADAPTIVITY IN EXPLICIT NONLINEAR FINITE ELEMENT SIMULATION OF QUASI-STATIC PROBLEMS

Abstract A program module for error estimation with application to nonlinear finite element (FE) analysis of shell structures is coupled with the adaptive solution procedure in the explicit FE code LS-DYNA. The error estimation module provides estimates of the local and global errors and element-level refinement indicators. Hence, selective refinement of the mesh in areas where the local error is relatively large compared with a user-defined tolerance is made possible. Furthermore, the relative global error is estimated giving a measure of the overall accuracy of the FE model. Projection-type error estimators based on the L2-norm of the stress vector and the accumulated plastic strain are used to predict the discretization error by comparison of the FE solution with an improved C0-continuous solution obtained by the SPR-method. Three example problems including both material and geometric nonlinearities are provided. The numerical results show that the error estimates capture phenomena such as diffuse necking and local buckling, and give meshes with high resolution in areas with large deformations or high stress gradients.

SUPERCONVERGENT PATCH RECOVERY FOR PLATE PROBLEMS USING STATICALLY ADMISSIBLE STRESS RESULTANT FIELDS

In this paper, we study an approach for recovery of an improved stress resultant field for plate bending problems, which then is used for a posteriori error estimation of the finite element solution. The new recovery procedure can be classified as Superconvergent Patch Recovery (SPR) enhanced with approximate satisfaction of interior equilibrium and natural boundary conditions. The interior equilibrium is satisfied a priori over each nodal patch by selecting polynomial basis functions that fulfil the point-wise equilibrium equations. The natural boundary conditions are accounted for in a discrete least-squares manner. The performance of the developed recovery procedure is illustrated by analysing two plate bending problems with known analytical solutions. Compared to the original SPR-method, which usually underestimates the true error, the present approach gives a more conservative error estimate. Copyright

Object-Oriented Programming in Field Recovery and Error Estimation

Abstract.This paper considers an object-oriented implementation of a generic program module for field recovery and recovery-based error stimation. The field recovery is based on the superconvergent patch recovery technique by Zienkiewicz and Zhu. The current implementation is problem independent, and is organized as a set of C++ classes based on the software library Diffpack. The program may be run stand-alone as a post-processor, reading finite element data and results from ASCII files. Alternatively, it may be linked into existing simulation codes through a Fortran interface, thereby enabling error estimation within the time-step loop of a transient simulation.The use of the field recovery and error estimation module is demonstrated on an isotropic linear elastic problem with known analytical solution, such that also the exact error, and not only the estimated error, may be computed. The computational efficiency of the object-oriented module is assessed by comparing the time consumption with a similar program implemented in Fortran.

Interactive-adaptive geometrically nonlinear analysis of shell structures

This paper presents an investigation of interactive-adaptive techniques for nonlinear finite element structural analysis. In particular, effective methods leading to reliable automated, finite element solutions of nonlinear shell problems are of primary interest here. This includes automated adaptive nonlinear solution procedures based on error estimation and adaptive step length control, reliable finite elements that account for finite deformations and finite rotations, three-dimensional finite element modeling, and an easy-to-use, easy-to-learn graphical user interface with three-dimensional graphics. A computational environment, which interactively couples a comprehensive geometric modeler, an automatic three-dimensional mesh generator and an advanced nonlinear finite element analysis program with real-time computer graphics and animation tools, is presented. Three examples illustrate the merit and potential of the approaches adopted here and confirm the feasibility of developing fully automated computer aided engineering environments.

Parallel Methods for Fluid-Structure Interaction

A parallel CFD code capable of simulating flow within moving boundaries has been coupled to a beam element structural dynamics code. The coupled codes are used to simulate fluid- structure interaction for a class of applications involving long and slender structures, e.g. suspension bridges and offshore risers. Due to the difference in size and dimensionality of the 3D CFD problem on one side, and the essentially 1D structure problem on the other side, the bulk of the computations are carried out in the CFD code. The parallel efficiency of the coupled codes thus rest on the paralll performance of the CFD code, and on minimizing the amount of communication between the two codes. The CFD code uses implicit time stepping, and is parallelized by a multiblock technique based on a block-Jacobi iteration together with coarse grid correction. To reduce the amount of communication between the CFD code and the structure code, the mesh movement algorithm is split into two parts, where the most computationally intensive part is carried out in parallel within the CFD code. The resulting coupled system has a high parallel efficiency even if the structure code runs on a workstation and the CFD code runs on a parallel supercomputer provided that the size of the CFD problem is sufficiently large.

The use of sparse matrix methods in finite-element codes for structural mechanics applications

Abstract In this study, sparse matrix methods are considered as linear solvers in finite-element programs for structural mechanics applications. The factorization schemes used, both for the standard global solution technique and for the partial factorization in the superelement technique, are based on sparse matrix technology [George and Liu, Computer Solution of Large Sparse Positive Definite Systems , Prentice-Hall, 1981; Duff and Reid, ACM Trans. Math Software 9, 302–325 (1983); Ashcraft et al., Int. J. Supercomputer Appl. 1, 10–30 (1987); Ashcraft, “A vector implementation of the multifrontal method for large sparse symmetric positive definite linear systems,” ETA-TR-51, Boeing Computer Services, Seattle, 1987], and tailored for use with the finite-element method for structural mechanics problems as described in [Damhaug, “Sparse solution of linear finite-element equations,” Dr Ing. Thesis 1992: 76, The Norwegian Institute of Technology, 1992].

Object-Oriented Field Recovery and Error Estimation in Finite Element Methods

In this chapter we study an object-oriented implementation of procedures for field recovery and recovery-based error estimation. The field recovery is based on the superconvergent patch recovery technique by Zienkiewicz and Zhu. The core of the current implementation is problem independent, and is organized as a set of C++ classes based on the software library Diffpack. The use of the developed program module is demonstrated on an isotropic linear elasticity problem and on a stationary NavieriaStokes problem. For both example problems, analytical solutions are available. The exact error may therefore be computed in addition to the estimated error, enabling us to study the effectivity of the estimator. The computational efficiency of the object-oriented program module is assessed by comparing the time consumption with a similar program implemented in FORTRAN.