Stress Recovery Procedure Research Articles

Development and validation of a simulation methodology to analyse mechanical systems moving on flexible bodies

A new multibody simulation tool, allowing the simulation of moving parts on flexible bodies, is described. This methodology is implemented by an approach using computer-aided design and mode-component synthesis, and illustrated with a test case on a reinforced concrete highway bridge under traffic testing. Dynamic traffic tests and relative multibody simulations are reported and compared in terms of acceleration. Moreover, using a new stress recovery procedure implemented by the authors for a particular multibody simulation and finite element analysis environment, a comparison was realised between stress test measurements and numerical ones. This demonstrates how much this approach allows the definition of load spectra useful in durability analysis.

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part II: Adaptive refinement

In Part II of this study, an automatic adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D linear boundary value problems is suggested. Based in the theoretical development and the numerical experiments done in Part I of this study, the Zienkiewicz and Zhu (Z–Z) error estimation scheme is combined with a new stress recovery procedure for the a posteriori error estimation of the adaptive refinement procedure. By considering the a priori convergence rate of the RKPM and the estimated error norm, an adaptive refinement strategy for the determination of optimal point distribution is proposed. In the suggested adaptive refinement scheme, the local refinement indicators used are computed by considering the partition of unity property of the RKPM shape functions. In addition, a simple but effective variable support size definition scheme is suggested to ensure the robustness of the adaptive RKPM procedure. The performance of the suggested adaptive procedure is tested by using it to solve several benchmark problems. Numerical results indicated that the suggested refinement scheme can lead to the generation of nearly optimal meshes for both smooth and singular problems. The optimal convergence rate of the RKPM is restored and thus the effectivity indices of the Z–Z error estimator are converging to the ideal value of unity as the meshes are refined.

A posteriori error estimation based on the superconvergent Recovery by Compatibility in Patches

The present work deals with an a posteriori error estimator for linear finite element analysis, based on a stress recovery procedure called Recovery by Compatibility in Patches. The key idea of this procedure is to recover improved stresses by minimizing the complementary energy over patches of elements. Displacements computed by the finite element analysis are prescribed on the boundary of the patch. Here, a new form of this recovery procedure is presented. Adopting a different patch configuration, centred upon an element instead of a node, allows to drastically simplify the recovery process, thus improving efficiency and making the implementation in finite element codes much easier. The robustness tests demonstrate that the error estimator associated to the new form of the recovery procedure retains the very good properties of the original one, such as superconvergence. The numerical results on two common benchmark problems confirm the effectiveness of the proposed error estimator, which appears to be competitive with those currently available. Copyright © 2006 John Wiley & Sons, Ltd.

Patch recovery based on complementary energy

AbstractIn this paper a new stress recovery procedure is presented. The formulation is very simple and based on improving stresses by enforcing compatibility over local patches of elements. This is obtained by minimizing the complementary energy, properly defined for the patch thought as a separate system, among an assumed set of equilibrated stress fields. The resultant implementation is simple, cost effective and numerically stable. Several numerical tests evidence an excellent performance which promises a wide applicability of the new procedure. Copyright © 2004 John Wiley & Sons, Ltd.

Local recovery of through-the-thickness stresses in laminated composite plates

In this paper, an effective procedure is presented for the local recovery of through-the-thickness stresses in laminated composite plates, which incorporates the local refinement using mesh superposition technique and a predictor–corrector post processor for the evaluation of stresses. The stress recovery procedure is based on accurate values of discrete superconvergent stresses and nodal displacements of patch of elements. The mesh superposition technique conveniently provides locally refined mesh for accurate and computationally economical recovery of stresses. To demonstrate the effectiveness of the proposed approach, stresses of several composite plates are evaluated. The composite plates are subjected to thermomechanical loading and either simply supported or clamped boundary conditions.

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

AbstractThe 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.

On using degenerated solid shell elements in adaptive refinement analysis

Three different degenerated shell elements are studied in an adaptive refinement procedure for the solution of shell problems. The stress recovery procedure expressed in a convective patch co-ordinate system is used for the construction of continuous smoothed stress fields for the a posteriori error estimation. The performance of the stress recovery procedure, the error estimator and the adaptive refinement strategy are tested by solving three benchmark shell problems. It is found that when adaptive refinement is used, the adverse effects of boundary layers and stress singularities are eliminated and all the elements tested are able to achieve their optimal convergence rates. It is also found that the accuracy of the shell elements increases with the number of polynomial terms included in the stress and strain approximations. In addition, if complete Lagrangian polynomial terms are used, the element will be less sensitive to shape distortion than the one in which only complete polynomial terms are employed. Copyright © 1999 John Wiley & Sons, Ltd.

On using degenerated solid shell elements in adaptive refinement analysis

Three different degenerated shell elements are studied in an adaptive refinement procedure for the solution of shell problems. The stress recovery procedure expressed in a convective patch co-ordinate system is used for the construction of continuous smoothed stress fields for the a posteriori error estimation. The performance of the stress recovery procedure, the error estimator and the adaptive refinement strategy are tested by solving three benchmark shell problems. It is found that when adaptive refinement is used, the adverse effects of boundary layers and stress singularities are eliminated and all the elements tested are able to achieve their optimal convergence rates. It is also found that the accuracy of the shell elements increases with the number of polynomial terms included in the stress and strain approximations. In addition, if complete Lagrangian polynomial terms are used, the element will be less sensitive to shape distortion than the one in which only complete polynomial terms are employed. Copyright © 1999 John Wiley & Sons, Ltd.

On using different recovery procedures for the construction of smoothed stress in finite element method

The performance of three different stress recovery procedures, namely, the superconvergent patch recovery technique (SPR), the recovery by equilibrium in patches (REP) and a combined method known as the LP procedure is reviewed. Different order of polynomials and various patch formation strategies have been employed in the numerical studies for the construction of smoothed stress fields. Two 2-D elastostatic problems of different characteristics are used to assess the behaviour of the stress recovery procedures. The numerical results obtained indicate that when the order of polynomial used in the recovery procedure is equal to that of the finite element analysis, the behaviours of all three recovery procedures are very similar and all of them are adequate to provide a reliable recovered stress field for error estimation. In case that the order of polynomial of the recovered stress is increased, the LP procedure seems to give a more stable recovery matrix and a more reliable recovered stress field than the REP procedure. © 1998 John Wiley & Sons, Ltd.

A Three-dimensional Stress Recovery Procedure for Composite Materials

A three-dimensional stress recovery procedure is described in this paper. As usual, in the finite element method, we suppose to know a good approximation of the nodal values of the displacements; starting from the nodal displacements, we evaluate the strain components at Gauss points of a 27-node prismatic element, then the stress components at the same points. Stresses are finally projected to corner nodes with a smoothing procedure. The repeated application of such a procedure to a set of adjacent elements allows the construction of the stress field in a finite region of any deformed body. The proposed method finds its most natural application in the field of composite materials as shown in the numerical applications which reveal a good concordance of results with an equivalent three-dimensional finite element analysis.

Automatic adaptive refinement for shell analysis using nine-node assumed strain element

An automatic adaptive refinement procedure for the analysis of shell structures using the nine-node degenerated solid shell element is suggested. The basic adaptive refinement principle and the effects of singularities and boundary layers on the convergence rate of the nine-node element used are discussed. A new stress recovery procedure based on the patch convective co-ordinate system concept is developed for the construction of a continuous smoothed stress field over the shell domains. The stress recovery procedure is easy to implement, requires a modest computational effort and needs only local patch information. It can be applied to shells with non-uniform thickness as well as to multi-layered shell structures. The smoothed recovered stress obtained is then used with the Zienkiewicz and Zhu error estimator for a posteriori error estimation during the adaptive refinement analysis. Numerical results which are in good agreement with theoretical predictions are obtained and they indicate that the current adaptive refinement procedure can eliminate the effect of singularities inside the problem domains so that a near-optimal convergence rate is achieved in all the numerical examples. This also indicates that the stress recovery procedure can produce an accurate stress field and as a result the error estimator can reflect the error distribution of the finite element solution. Even though in the current study only one type of element is used in the analysis, the whole adaptive refinement scheme can be readily applied to any other types of degenerated solid element. © 1997 John Wiley & Sons, Ltd.

The stress recovery procedure for the finite element method

The idea of the proposed stress recovery procedure has been explained, peculiarities of its realization have been demonstrated on the example of the plane problem with small deformations, and generalization of this procedure to problems with finite deformations has been carried out. The proposed procedure allows us to construct the stress field for the problems with small and finite deformations without numerical differentiation of the displacement field obtained by the finite element method. Firstly, in accordance with this procedure the distributed forces are defined on three internal surfaces (in general case) within the body. This problem is reduced to solving the systems of linear algebraic equations of low-dimensions with symmetrical and positive definite band matrices. It is one of the advantages of the proposed procedure in contrast with other methods of the stress field construction. Then, using the Cauchy relation, the components of the stress tensor or displacement vector increment derivative are computed for the nodes of the finite element grid.

EQUIVALENCE BETWEEN ENHANCED ASSUMED STRAIN METHOD AND ASSUMED STRESS HYBRID METHOD BASED ON THE HELLINGER-REISSNER PRINCIPLE

An equivalence between the enhanced assumed strain (EAS) method based on the Hu–Washizu principle, recently proposed by Simo and Rifai, and assumed stress hybrid (hybrid) method based on the Hellinger–Reissner principle is investigated. It is proved that not only the displacements but also the stresses of the EAS-elements calculated from the strains are identical to those of the corresponding hybrid-elements at least at the Gauss integration points provided the spaces of the trial functions for enhanced assumed strains and for assumed stresses satisfy the orthogonality and the inclusion or the invertibility condition. By virtue of this equivalence, a stress recovery procedure of the EAS-elements is devised. This procedure is variationally consistent and more efficient than those proposed by Simo and Rifai and Andelfinger and Ramm. Since the classical method of incompatible displacement modes is a special case of the EAS-method, this procedure also can be used to evaluate variationally consistent stresses for the non-conforming elements.

The effectivity of the Zienkiewicz–Zhu error estimate and two 2D adaptive mesh generators

AbstractThe paper reports some numerical experiments on the effectivity of the Zienkiewicz–Zhu error estimate (1987) and comparisons of two adaptive mesh generators: ADMESH, developed by Jin and Wiberg (1990) and MAD2D by Peraire et al. (1987) and Zhu et al. (1990–91). From the experiments we observe that: (1) without any empirical correction, the effectivity of the error estimate for linear triangular elements is good if post‐processed continuous stresses are obtained by adequately using an iterative stress recovery procedure (Zienkiewicz et al., 1985), and is also acceptable by using a simple ‘averaging’ procedure; (2) these two mesh generators are basically comparable.

ON RECOVERY OF STRESSES FOR A MULTI‐LAYERED BEAM ELEMENT

The stress recovery procedures discussed in the present paper refer to a multi‐layered element of assembled Timoshenko beam elements. Directly calculated stresses for a multi‐layered beam model strongly depend on properties of the approximation functions, and are unrealistic. Thus, an enhanced procedure which circumvents the limitations of the interface variables model, and hierarchical model is proposed. Each material layer of the beam element is covered by one quadrilateral 9‐node element, providing a parabolic approximation of displacements. The stresses are evaluated using 2 × 2 Gauss points, projected to corner nodes, and smoothed within material layers. Numerical calculations show very good accordance of stresses yielded by this procedure with 2D results.

On the Variational Foundations of Assumed Strain Methods

So-called assumed strain methods are based on the a-priori assumption of an interpolation for the discrete gradient operator, not necessarily derivable from the displacement interpolation. It is shown that this class of methods falls within the class of variational methods based on the Hu-Washizu principle. The essential point of this equivalence lies in the statement of the appropriate stress recovery procedure compatible with this variational structure. It is noted that most currently existing assumed strain methods fail to perform the stress recovery in a manner consistent with the variational structure discussed herein. Application is made to recently proposed methods such as mode decomposition techniques.

Iterative solution of mixed problems and the stress recovery procedures

AbstractThis note introduces certain stress smoothing procedures and the so called ‘Loubignac‐Cantin’ iteration for restoration of momentum balance in the smoothed stress fields. It is shown that this iteration corresponds precisely to the solution of mixed formulations in which the stresses (or strains) and the displacements are used as primary variables. The procedure has a very wide field of application and promises to add considerable accuracy to f.e.m. results by a small additional expenditure.