Generalization of robustness test procedure for error estimators. Part II: test results for error estimators using SPR and REP

In this part of the paper we shall use the formulation given in the first part to assess the quality of recovery‐based error estimators using two recovery methods, i.e. superconvergent patch recovery (SPR) and recovery by equilibrium in patches (REP). The recovery methods have been shown to be asymptotically robust and superconvergent when applied to two‐dimensional problems. In this study we shall examine the behaviour of the recovery methods on several three‐dimensional mesh patterns for patches located either inside or at boundaries. This is performed by first finding an asymptotic finite element solution, irrespective of boundary conditions at far ends of the domain, and then applying the recovery methods. The test procedure near kinked boundaries is explained in a step‐by‐step manner. The results are given in a series of tables and figures for various cases of three‐dimensional mesh patterns. It has been experienced that the full superconvergent property is generally lost due to presence of boundary layer solution and the definition of the recoveries near boundaries though the results of the robustness test is still within an acceptable range. Copyright © 2005 John Wiley & Sons, Ltd.     

Rayleigh wave correction for the BEM analysis of two‐dimensional elastodynamic problems in a half‐space

A simple, elegant approach is proposed to correct the error introduced by the truncation of the infinite boundary in the BEM modelling of two‐dimensional wave propagation problems in elastic half‐spaces. The proposed method exploits the knowledge of the far‐field asymptotic behaviour of the solution to adequately correct the BEM displacement system matrix for the truncated problem to account for the contribution of the omitted part of the boundary. The reciprocal theorem of elastodynamics is used for a convenient computation of this contribution involving the same boundary integrals that form the original BEM system. The method is formulated for a two‐dimensional homogeneous, isotropic, linearly elastic half‐space and its implementation in a frequency domain boundary element scheme is discussed in some detail. The formulation is then validated for a free Rayleigh pulse travelling on a half‐space and successfully tested for a benchmark problem with a known approximation to the analytical solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Stress recovery and error estimation for the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a stress recovery procedure based on a modal interpretation of the scaled boundary finite‐element method solution process, using the superconvergent patch recovery technique. The recovered stresses are superconvergent, and are used to calculate a recovery‐type error estimator. A key feature of the procedure is the compatibility of the error estimator with the standard recovery‐type finite element estimator, allowing the scaled boundary finite‐element method to be compared directly with the finite element method for the first time. A plane strain problem for which an exact solution is available is presented, both to establish the accuracy of the proposed procedures, and to demonstrate the effectiveness of the scaled boundary finite‐element method. The scaled boundary finite‐element estimator is shown to predict the true error more closely than the equivalent finite element error estimator. Unlike their finite element counterparts, the stress recovery and error estimation techniques work well with unbounded domains and stress singularities. Copyright © 2002 John Wiley & Sons, Ltd.     

Element‐wise a posteriori estimates based on hierarchical bases for non‐linear parabolic problems

We show that the issue of a posteriori estimate the errors in the numerical simulation of non‐linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right‐hand side depending on known data of the problem and the computed numerical solution. A procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right‐hand side the residual of the p‐FEM solution is introduced. The boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs. We prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation. When the error being estimated is that of a elliptic problem constant free local lower bounds are obtained. The local error estimation procedure is applied to non‐linear parabolic differential equations in several space dimensions. Some numerical experiments for both the elliptic and the non‐linear parabolic cases are provided. Copyright © 2005 John Wiley & Sons, Ltd.     

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 © 1999 John Wiley & Sons, Ltd.     

Subdomain-based error techniques for generalized finite element approximations of problems with singular stress fields

This paper considers four types of error measures, each tailored to the generalized finite element method. Particular attention is given to two-dimensional elasticity problems with singular stress fields. The first error measure is obtained using the equilibrated element residual method. The other three estimators overcome the necessity of equilibrating the residue by employing a subdomain strategy. In this strategy, the partition of unity (PoU) property is used to decompose the error problem into local contributions over each patch of elements. The residual functional of the error problem is the same for the subdomain estimators, but the bi-linear form is different for each one of them. In the second estimator, the bi-linear form is weighted by the PoU functions associated with the patch over which the error problem is stated. No weighting appears in the bi-linear form of the third estimator. The fourth measure is proposed as an alternative strategy, in which the products of the PoU functions and test functions are introduced as weights in the weighted integral statement of the differential equation describing the error problem. The linear form of the local error problem is then identical to that of the other subdomain techniques, while the bi-linear form is stated differently, with the PoU functions directly multiplying the test functions. The goal of this study is to investigate the performance of the four estimators in two-dimensional elasticity problems with geometries that produce singularities in the stress field and concentration of the error in the numerical solution.     

An adaptive finite element approach for frictionless contact problems

The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behaviour and infinitesimal deformation is presented. The approximated solution of this problem is obtained by using the finite element method. A penalization or augmented‐Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh refinement in this problem is proposed, together with its mathematical justification. Up to the present time, this mathematical proof is restricted to the penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators. Copyright © 2001 John Wiley & Sons, Ltd.     

An h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 John Wiley & Sons, Ltd.     

Boundary elements for half‐space problems via fundamental solutions: A three‐dimensional analysis

An efficient solution technique is proposed for the three‐dimensional boundary element modelling of half‐space problems. The proposed technique uses alternative fundamental solutions of the half‐space (Mindlin's solutions for isotropic case) and full‐space (Kelvin's solutions) problems. Three‐dimensional infinite boundary elements are frequently employed when the stresses at the internal points are required to be evaluated. In contrast to the published works, the strongly singular line integrals are avoided in the proposed solution technique, while the discretization of infinite elements is independent of the finite boundary elements. This algorithm also leads to a better numerical accuracy while the computational time is reduced. Illustrative numerical examples for typical isotropic and transversely isotropichalf‐space problems demonstrate the potential applications of the proposed formulations. Incidentally, the results of the illustrative examples also provide a parametric study for the imperfect contact problem. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis of three‐dimensional fracture mechanics problems: A two‐scale approach using coarse‐generalized FEM meshes

This paper presents a generalized finite element method (GFEM) based on the solution of interdependent global (structural) and local (crack)‐scale problems. The local problems focus on the resolution of fine‐scale features of the solution in the vicinity of three‐dimensional cracks, while the global problem addresses the macro‐scale structural behavior. The local solutions are embedded into the solution space for the global problem using the partition of unity method. The local problems are accurately solved using an hp‐GFEM and thus the proposed method does not rely on analytical solutions. The proposed methodology enables accurate modeling of three‐dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts. The boundary conditions for the local problems are provided by the coarse global mesh solution and can be of Dirichlet, Neumann or Cauchy type. The effect of the type of local boundary conditions on the performance of the proposed GFEM is analyzed. Several three‐dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance, both in terms of problem size and CPU time, are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Refined h-adaptive finite element procedure for large deformation geotechnical problems

Adaptive finite element procedures automatically refine, coarsen, or relocate elements in a finite element mesh to obtain a solution with a specified accuracy. Although a significant amount of research has been devoted to adaptive finite element analysis, this method has not been widely applied to nonlinear geotechnical problems due to their complexity. In this paper, the h-adaptive finite element technique is employed to solve some complex geotechnical problems involving material nonlinearity and large deformations. The key components of h-adaptivity including robust mesh generation algorithms, error estimators and remapping procedures are discussed. This paper includes a brief literature review as well as formulation and implementation details of the h-adaptive technique. Finally, the method is used to solve some classical geotechnical problems and results are provided to illustrate the performance of the method.     

Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems

In this paper we present two types of local error estimators for the primal finite‐element‐method (FEM) by duality arguments. They are first derived from the (explicit) residual error estimation method (REM) and then—as a new contribution—from the (implicit) posterior equilibrium method (PEM) using improved boundary tractions, gained by local post‐processing with local Neumann problems, with applications in elastic problems. For the displacements a local error estimator with an upper bound is derived and also a local estimator for stresses. Furthermore—for better numerical efficiency—the residua are projected energy‐invariant onto reference elements, where the local Neumann problems have to be solved. Comparative examples between REM‐ and PEM‐type local estimators show superior effectivity indices for the latter one. Copyright © 2001 John Wiley & Sons, Ltd.     

Exponential finite elements for diffusion–advection problems

A new finite element method for the solution of the diffusion–advection equation is proposed. The method uses non‐isoparametric exponentially‐varying interpolation functions, based on exact, one‐ and two‐dimensional solutions of the Laplace‐transformed differential equation. Two eight‐noded elements are developed and tested for convergence, stability, Peclet number limit, anisotropy, material heterogeneity, Dirichlet and Neumann boundary conditions and tolerance for mesh distortions. Their performance is compared to that of conventional, eight‐ and 12‐noded polynomial elements. The exponential element based on two‐dimensional analytical solutions fails basic tests of convergence. The one based on one‐dimensional solutions performs particularly well. It reduces by about 75% the number of elements and degrees of freedom required for convergence, yielding an error that is one order of magnitude smaller than that of the eight‐noded polynomial element. The exponential element is stable and robust under relatively high degrees of heterogeneity, anisotropy and mesh distortions. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical accuracy of a Padé‐type non‐reflecting boundary condition for the finite element solution of acoustic scattering problems at high‐frequency

The present text deals with the numerical solution of two‐dimensional high‐frequency acoustic scattering problems using a new high‐order and asymptotic Padé‐type artificial boundary condition. The Padé‐type condition is easy‐to‐implement in a Galerkin least‐squares (iterative) finite element solver for arbitrarily convex‐shaped boundaries. The method accuracy is investigated for different model problems and for the scattering problem by a submarine‐shaped scatterer. As a result, relatively small computational domains, optimized according to the shape of the scatterer, can be considered while yielding accurate computations for high‐frequencies. Copyright © 2005 John Wiley & Sons, Ltd.     

ASYMPTOTIC SOLUTIONS FOR PREDICTED NATURAL FREQUENCIES OF TWO-DIMENSIONAL ELASTIC SOLID VIBRATION PROBLEMS IN FINITE ELEMENT ANALYSIS

In order to assess the discretization error of a finite element solution, asymptotic solutions for predicted natural frequencies of two-dimensional elastic solid vibration problems in the finite element analysis are presented in this paper. Since the asymptotic solution is more accurate than the original finite element solution, it can be viewed as an alternative solution against which the original finite element solution can be compared. Consequently, the discretization error of the finite element solution can be evaluated. Due to the existence of two kinds of two-dimensional problems in engineering practice, both the plane stress problem and the plane strain problem have been considered and the corresponding asymptotic formulae for predicted natural frequencies of two-dimensional solids by the finite element method have been derived from the fact that a discretized finite element system approaches a continuous one if the finite element size approaches zero. It has been demonstrated, from the related numerical results of three examples, that the present asymptotic solution, which can be obtained by simply using the corresponding formula without any further finite element calculation, is indeed more accurate than the original finite element solution so that it can be considered as a kind of corrected solution for the discretization error estimation of a finite element solution.     

A p‐hierarchical adaptive procedure for the scaled boundary finite element method

This paper describes a p‐hierarchical adaptive procedure based on minimizing the classical energy norm for the scaled boundary finite element method. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh element‐wise one order higher, is used to represent the unknown exact solution. The optimum mesh is assumed to be obtained when each element contributes equally to the global error. The refinement criteria and the energy norm‐based error estimator are described and formulated for the scaled boundary finite element method. The effectivity index is derived and used to examine quality of the proposed error estimator. An algorithm for implementing the proposed p‐hierarchical adaptive procedure is developed. Numerical studies are performed on various bounded domain and unbounded domain problems. The results reflect a number of key points. Higher‐order elements are shown to be highly efficient. The effectivity index indicates that the proposed error estimator based on the classical energy norm works effectively and that the reference solution employed is a high‐quality approximation of the exact solution. The proposed p‐hierarchical adaptive strategy works efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

High‐order finite volume schemes for the advection–diffusion equation

A high‐order finite volume method based on piecewise interpolant polynomials is proposed to discretize spatially the one‐dimensional and two‐dimensional advection–diffusion equation. Evolution equations for the mean values of each control volume are integrated in time by a classical fourth‐order Runge–Kutta. Since our work focuses on the behaviour of the spatial discretization, the time step is chosen small enough to neglect the time integration error. Two‐dimensional interpolants are built by means of one‐dimensional interpolants. It is shown that when the degree of the one‐dimensional interpolant q is odd, the proper selection of a fixed stencil gives rise to centred schemes of order q+1. In order not to lose precision due to the change of stencil near boundaries, the degree of the interpolants close to boundaries is raised to q+1. Four test cases with small values of diffusion are integrated with high‐order methods. It is shown that the spatial discretization of the advection–diffusion equation with periodic boundary conditions leads to normal discretization matrices, and asymptotic stability must be assured to bound the spatial discretization error. Once the asymptotic stability is assured by means of the spectra of the discretization matrix, the spatial error is of the order of the truncation error. However, it is shown that the discretization of the advection–diffusion equation with arbitrary boundary conditions gives rise to non‐normal matrices. If asymptotic stability is assured, the spatial order of steady solutions is of the order of the truncation error. But, for transient processes, the order of the spatial error is determined by both the truncation error and the norm of the exponential matrix of the spatial discretization. The use of the pseudospectra of the discretization matrix is proposed as a valuable tool to analyse the transient error of the numerical solution. Copyright © 2001 John Wiley & Sons, Ltd.     

Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

This paper presents the development and application of the finite node displacement (FiND) method to the incompressible Navier–Stokes equations. The method computes high‐accuracy nodal derivatives of the finite element solutions. The approach imposes a small displacement to individual mesh nodes and solves a very small problem on the patch of elements surrounding the node. The only unknown is the value of the solution ( u , p) at the displaced node. A finite difference between the original and the perturbed values provides the directional derivative. Verification by grid refinement studies is shown for two‐dimensional problems possessing a closed‐form solution: a Poiseuille flow and a flow mimicking a boundary layer. For internal nodes, the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. The local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. Computations show that the resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique. Copyright © 2008 John Wiley & Sons, Ltd.     

Computation of accurate nodal derivatives of finite element solutions: the finite node displacement method

This paper presents a new method for extracting high‐accuracy nodal derivatives from finite element solutions. The approach involves imposing a finite displacement to individual mesh nodes, and solving a very small problem on the patch of surrounding elements, whose only unknown is the value of the solution at the displaced node. A finite difference between the original and perturbed values provides the directional derivative. Verification is shown for a one‐dimensional diffusion problem with exact nodal solution and for two‐dimensional scalar advective–diffusive problems. For internal nodes the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. In this case, the local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. We show that the new method gives normal derivatives at boundary points that are consistent with the so‐called ‘auxiliary fluxes’. The resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique. Copyright © 2007 Crown in the right of Canada. Published by John Wiley & Sons, Ltd.     

Error estimates for mixed finite element approximations of nonlinear problems

The finite element methods have proved a very effective tool for the numerical solutions of nonlinear problems arising in elasticity and other related engineering sciences. Relative to linear elliptic theory, little is known about the accuracy and convergence properties of mixed finite element approximation of nonlinear elliptic boundary value problems. The nonlinear problems are much more complicated, since each problem has to be treated individually. This is one of the reasons that there is no unified and general theory for the nonlinear problems. In this paper, the application of the mixed finite element method to a highly nonlinear Dirichlet problem, which arises in the field of oceanography and elasticity is studied and new results involving the error estimates are derived. In fact, some of the results and methods to be described in this paper may be extended to more complicated problems or problems with other boundary conditions. As a special case, we obtain the well known error estimates for the corresponding linear and mildly nonlinear elliptic boundary value problems.