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C1-Continuous stress recovery in finite element analysis
Abstract
A Penalized Discrete-Least-Squares (PDLS) variational principle is employed to recover C1-continuous, smooth stress fields from finite element stresses. The error functional involves discrete least-squares with a penalty constraint to enforce C1 continuity of the recovered stresses. The recovery procedure uses a smoothing element analysis (SEA), with element-based interpolation functions, to minimize the error functional. SEA/PDLS recovers a superconvergent stress field of higher accuracy and continuity than the underlying, ‘consistent’ finite element stress field. General and specialized formulations of the functional are given, and an appropriate discretization strategy for the SEA is discussed. Numerical results for both one- and two-dimensional stress fields are presented and compared with the corresponding solutions for the superconvergent patch recovery (SPR) method of Zienkiewicz and Zhu. The results demonstrate that, in addition to achieving a higher degree of continuity, SEA/PDLS produces generally more accurate stresses than SPR. Although an additional global analysis is required, results indicate that two ‘coarse’ mesh analyses (finite element and smoothing) produce solutions comparable to a single finite element analysis with a much greater degree of refinement and more computational effort. The procedure is unique in that the recovered stress field is essentially C1 continuous. The results are shown to be insensitive to the value of the penalty parameter above a threshold, and hence uncertainty in the choice of an appropriate penalty value is eliminated.
... The main weakness of the iFEM is that for shells in principle it requires triaxial strain measurements for every element of the mesh, which is hardly feasible due to cost and space constraints. To alleviate this problem, weights were natively introduced in the first iFEM formulation [15,16], and strain interpolation/extrapolation techniques, such as the Smoothing Element Analysis (SEA) [19][20][21], have more recently been introduced [22,23]. ...
... To avoid setting the strain to zero in the elements missing experimental measurements, strain interpolation and extrapolation techniques have been developed [19,21] and coupled with the iFEM [23]: they interpolate or extrapolate the measurements between sensors so that measure-less elements are fed by extrapolated/interpolated strain values. It should be noted that in the literature the terms interpolation and extrapolation might be used interchangeably to avoid repetition and to facilitate reading, in accordance with the iFEM literature, even though in most cases interpolation is actually performed [22,23]. ...
... It should be noted that in the literature the terms interpolation and extrapolation might be used interchangeably to avoid repetition and to facilitate reading, in accordance with the iFEM literature, even though in most cases interpolation is actually performed [22,23]. Among the possible strain interpolation and extrapolation techniques, the Smoothing Element Analysis (SEA) [19,21] is one of the most popular choices. In a recent paper, the SEA is compared to polynomial fittings, outlining the sensitivity of the iFEM solution to the SEA parameters [22]. ...
The inverse Finite Element Method (iFEM) employing a network of strain sensors reconstructs the full-field displacement on beam or shell structures, independently of the loading conditions and of the material properties. However, the iFEM in principle requires triaxial strain measurements for each inverse element, which is practically hardly possible due to space and cost constraints. To relieve this issue, some strain values fed as input to the iFEM are typically computed using strain pre-extrapolation/interpolation techniques, and the iFEM solution is computed minimizing a weighted functional: elements missing experimental measurements are assigned low weights, which are generally set to arbitrarily low values taken from the literature. This paper proposes the use of a Gaussian Process as a strain pre-extrapolation and interpolation technique, which natively provides the extrapolation uncertainty, which in turn is used as a metric to assign the functional weights, and it enables the computation of the uncertainty on the reconstructed displacement field. The proposed approach is tested on a virtual and an experimental case study; advantages and limitations of the proposed technique are discussed.
... The transfer strategy proposed in this work is based on a modified smoothing ele-ment analysis (SEA) methodology. SEA is a post-processing recovery procedure that has been developed over the last several years by Tessler, Riggs and coworkers15161718. SEA is based on a penalized discrete-least-square (PDLS) variational principle which combines discrete-least-squares and a penalty constraint in a single variational form. ...
... The two dimensional formulation , which is of interest herein, is reviewed subsequently. More detailed information on SEA can be found in15161718. ...
... To minimize the error functional, the finite element methodology is adopted and therefore, the problem domain Ω is discretized with smoothed field is (nearly) C 1 continuous. Results have shown that the method is remarkably robust with respect to the value of α, and a wide range of values can be used [16]. The third, optional, term is a 'curvature' control term that provides stability should the discrete input data be insufficient, because either there are too few data points or their spatial distribution is poor, to define uniquely the smoothed field for the given smoothing mesh. ...
Mode of access: World Wide Web. Thesis (Ph. D.)--University of Hawaii at Manoa, 2004. Includes bibliographical references (leaves 145-149). Electronic reproduction. Also available by subscription via World Wide Web x, 149 leaves, bound ill. 29 cm
... The Penalized Discrete Least-Squares (PDLS) stress recovery (smoothing) technique developed for twodimensional linear elliptic problems [1][2][3] is adapted here to three-dimensional shell structures. The surfaces are restricted to those which have a 2-D parametric representation, or which can be built-up of such surfaces. ...
... Further testing involving more complex, practical structures is necessary. and error estimation for two-dimensional linear elliptic problems can be found [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], while there are relatively few applications to shell structures [17]. ...
... in which w q is a discrete weight for the sample point x q ; is a normalization factor, which would typically be the sum of the discrete weights; is a dimensionless penalty parameter; (x) is a weight density function; and s ; i is the partial derivative of s with respect to x i . The ÿrst term in equation (1) represents the error between the smoothed stress ÿeld and the sample data. The second term represents a penalty functional which, for su ciently large, enforces the derivatives of the smoothed stress ÿeld to be equal to the  s i . ...
The Penalized Discrete Least-Squares (PDLS) stress recovery (smoothing) technique developed for two dimensional linear elliptic problems is adapted here to three-dimensional shell structures. The surfaces are restricted to those which have a 2-D parametric representation, or which can be built-up of such surfaces. The proposed strategy involves mapping the finite element results to the 2-D parametric space which describes the geometry, and smoothing is carried out in the parametric space using the PDLS-based Smoothing Element Analysis (SEA). Numerical results for two well-known shell problems are presented to illustrate the performance of SEA/PDLS for these problems. The recovered stresses are used in the Zienkiewicz-Zhu a posteriori error estimator. The estimated errors are used to demonstrate the performance of SEA-recovered stresses in automated adaptive mesh refinement of shell structures. The numerical results are encouraging. Further testing involving more complex, practical structures is necessary.
... A somewhat di erent recovery strategy has been proposed by Tessler et al. 11,12]. Their method is based on a penalized discrete least-square variational principle which combines discrete least-square and a penalty constraint in a single variational form. ...
... The boundary conditions on @ are u = u on @ u (11) Q s = t on @ t (12) 4 where u = fw; x ; y g T and t = fV z ; M x ; M y g T are prescribed displacement/rotations and prescribed tractions along @ u and @ t , respectively. With s = fm xx ; m yy ; m xy ; q x ; q y g T being the stress resultant vector, the`stress resultant-to-boundary traction' transformation matrix, Q, is given as Q = ...
... Near the boundary @ t we can extend the number of constraints on s by introducing the natural boundary conditions (12). We then get the resulting functional ...
Indexing terms Stikkord Superconvergent Patch Recovery for Plate Problems using Statically Admissible Stress Resultant Fields Report no. R-19-96 1996-09-01 1998-01-08 (revised version) Knut Morten Okstad, Trond Kvamsdal and Kjell Magne Mathisen 31 Open Finite Element Method Stress Recovery Plate Bending Problems Elementmetoden Spenningsberegning Plateproblemer 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-square manner. The perform...
... However, one of the drawbacks of the iFEM is that in principle it requires triaxial strain measurements for each shell element, on both the top and bottom surfaces of the shell. To alleviate this problem, strain interpolation and extrapolation techniques have been developed and compared [48,49]: among them, the Smoothing Element Analysis (SEA) [50][51][52] is one of the most adopted techniques [48,49]. ...
... Its formulation is based on the minimization of a least-square penaltyconstrained functional, similarly to what occurs in the iFEM. The interested reader can refer to its complete formulation in [50][51][52]. In brief, the structure is discretized with a shell triangular mesh where the degrees of freedom (dof) are representative of the interpolated strain field. ...
In the past two decades, the aerospace industry has massively shifted from aluminum-made components to composite materials such as carbon fiber reinforced polymers (CFRP), striving for more fuel efficient and lighter aircrafts. Consequently, traditional joints have been replaced by adhesive bonded interfaces, which are also the most common choice to repair damaged components. Although adhesive bonding is the most efficient choice for permanent connections, it is not free of disadvantages: one of the most common failure modes, the debonding of the two laps, is very problematic to detect and predict in practice. Therefore, frequent inspections must be performed to ensure structural safety, increasing maintenance costs, and lessening the availability of the platforms. The development of innovative sensing technologies has allowed for a close monitoring of structural interfaces, and several structural health monitoring techniques have been proposed to monitor adhesive bonded connections. Sensitivity and correlation between measurements and debonding entity has been demonstrated in the literature: nevertheless, hardly any technique has been proposed and quantitively evaluated to estimate the debonding entity independently of the applied loads, such as misalignment-induced torsion, which is a major confounding influence in the traditional backface strain gauge technique. This paper proposes the inverse finite element method (iFEM) as a load and material independent approach to infer the debonding entity from strain measurements in adhesive-bonded joints. Two approaches to estimate the debonding entity with the iFEM are compared on cracked leap shear specimens representative of CFRP repair patches: one is based on anomaly indexes, the other on performing a model selection with multiple iFEM models including different damages. The latter demonstrates satisfactory performances; thus, it is considered a significant scientific advancement in this field.
... [3][4][5][6], and the references therein. The main focus has been on least squares smoothing, local projection of optimal sampling points, and the super-convergent patch recovery method [7][8][9][10][11][12][13][14][15][16][17], but relatively little attention has been given to the use of the element nodal point forces, see Refs. [18][19][20][21], and the references therein. ...
... Therefore, the benefit of imposing the principle in both forms is that differential equilibrium over the element is satisfied more closely than if the principle were only imposed in traction form. Finally, from Eqs. (9) and (13) it is evident that the two principle of virtual work statements are only independent of each other if the assumed space for s ðmÞ contains functions of high enough order. ...
We present in this paper a novel approach to stress calculations in finite element analysis. Rather than using the stress assumption employed in establishing the stiffness matrix, the element nodal point forces are used, in a simple way, to enhance the finite element stress predictions at a low computational cost. While this paper focuses on the improvement of the stress accuracy, the proposed procedure can also be used as a basis for error estimation. Moreover, the procedure is quite general, and has the potential for many applications in finite element analysis.
... To further improve the iFEM results' accuracy, the input strain field can be pre-extrapolated in the element's locations in which physical sensors are not available, obtaining the definition of the input strain field on the whole structure. This can be carried out with different approaches according to the specific problem, such as polynomial fitting and smoothing element analysis [57][58][59][60]. These techniques are purely based on the acquired strain measurements from sensors, thus being defined as data-based approaches, only providing a more continuous and smooth strain field in the considered domain. ...
The inverse finite element method (iFEM) is a model-based technique to compute the displacement (and then the strain) field of a structure from strain measurements and a geometrical discretization of the same. Different literature works exploit the error between the numerically reconstructed strains and the experimental measurements to perform damage identification in a structural health monitoring framework. However, only damage detection and localization are performed, without attempting a proper damage size estimation. The latter could be based on machine learning techniques; however, an a priori definition of the damage conditions would be required. To overcome these limitations, the present work proposes a new approach in which the damage is systematically introduced in the iFEM model to minimize its discrepancy with respect to the physical structure. This is performed with a maximum likelihood estimation framework, where the most accurate damage scenario is selected among a series of different models. The proposed approach was experimentally verified on an aluminum plate subjected to fatigue crack propagation, which enables the creation of a digital twin of the structure itself. The strain field fed to the iFEM routine was experimentally measured with an optical backscatter reflectometry fiber and the methodology was validated with independent observations of lasers and the digital image correlation.
... The first one can be called weighting, it was introduced in the very first iFEM formulation [15]. The second one is the use of strain interpolation or extrapolation techniques, such as the Smoothing Element Analysis (SEA) [30][31][32], which have been introduced only more recently [33,34]. ...
The inverse Finite Element Method (iFEM) has recently gained much popularity within the Structural Health Monitoring (SHM) field since, given sparse strain measurements, it reconstructs the displacement field of any beam or shell structure independently of the external loading conditions and of the material properties. However, in principle, the iFEM requires a triaxial strain measurement for each inverse finite element, which is seldom feasible in practical applications due to both costs and cabling-related limitations. To alleviate this problem several techniques to pre-extrapolate the measured strains have been developed, so that interpolated or extrapolated strain values are inputted to elements without physical sensors: the benefit is that the required number of sensors can be reduced. Nevertheless, whenever the monitored components comprise regions of different thicknesses, each region of constant thickness must be extrapolated separately, due to thickness-induced discontinuities in the strain field. This is the case in many practical applications, especially those concerning fiber-reinforced composite laminates. This paper proposes to extrapolate the measured strain field in a thickness-normalized space, where the thickness-induced trends are removed; this novel method can significantly decrease the number of required sensors, effectively reducing the costs of iFEM-based SHM systems. The method is validated in a simple but informative numerical case study, highlighting the potentialities and benefits of the proposed approach for more complex application scenarios.
... The Smoothing Element Analysis (SEA) is a technique generally applied [16,6,17,5] to smoothen discrete data over a certain domain. Initially, the method was mainly devised in order to recover C 1 continuous stresses from the output of Finite Element simulations so that posteriori error estimates were possible. ...
... However, when dealing with experimental tests, the cost of sensors, acquisition system limitations and physical constraints often limit sensor installation on the whole structure, preventing the full input strain field definition. In this case, the elements free from any sensor can leverage on pre-extrapolated strain measurements, for example, exploiting the Smoothing Element Analysis (SEA) [40,41,[45][46][47][48]. However, since pre-extrapolated strains are reasonably less accurate than sensor's strains, a small weighting coefficient w (·) will be associated to these elements (e.g., 10 −5 ), while unitary value is assumed for elements including physical sensors. ...
The inverse Finite Element Method (iFEM) is receiving more attention for shape sensing due to its independence from the material properties and the external load. However, a proper definition of the model geometry with its boundary conditions is required, together with the acquisition of the structure’s strain field with optimized sensor networks. The iFEM model definition is not trivial in the case of complex structures, in particular, if sensors are not applied on the whole structure allowing just a partial definition of the input strain field. To overcome this issue, this research proposes a simplified iFEM model in which the geometrical complexity is reduced and boundary conditions are tuned with the superimposition of the effects to behave as the real structure. The procedure is assessed for a complex aeronautical structure, where the reference displacement field is first computed in a numerical framework with input strains coming from a direct finite element analysis, confirming the effectiveness of the iFEM based on a simplified geometry. Finally, the model is fed with experimentally acquired strain measurements and the performance of the method is assessed in presence of a high level of uncertainty.
... In turn, this equation necessitates the calculation of the gradient of p p . In the present implementation, this was done by postprocessing the (u p , p p ) primary results by means of a least squares global smoothing procedure [36][37] typically used in structural FE calculation for accurate stress-recovery. Figure 3 shows the comparison between the normal incidence surface impedance of the glass wool layer calculated with the Treasuri2/FE DMAP and the analytical one evaluated by means of the standard transfer-matrix method, on the frequency range [300-1000]Hz. ...
Porous materials are extensively used in the construction of automotive sound package parts, due to their intrinsic capability of dissipating energy through different mechanisms. The issue related to the optimization of sound package parts (in terms of weight, cost, performances) has led to the need of models suitable for the analysis of porous materials' dynamical behavior and for this, along the years, several analytical and numerical models were proposed, all based on the system of equations initially developed by Biot. In particular, since about 10 years, FE implementations of Biot's system of equations have been available in commercial software programs but their application to sound package parts has been limited to a few isolated cases. This is due, partially at least, to the difficulty of smoothly integrating this type of analyses into the virtual NVH vehicle development. This paper presents a tool, called Treasuri2FE, that implements Biot's system of equations and is fully integrated inside Nastran. Treasuri2FE allows including in an efficient and flexible way sound package parts into Nastran vehicle FE models used for NVH analyses. All the steps that are necessary to obtain this, in particular the management of the incompatibility between the FE mesh of the sound package part and the FE mesh of the vehicle, are automatically carried out within Nastran. Furthermore, the tool can be used in conjunction with Nastran ACMS and DMP, to improve computational efficiency. In this way, the evaluation of the effect of a sound package part on the vibro-acoustic response of a vehicle can smoothly be introduced into virtual NVH vehicle development, making this important type of CAE analysis available to decision-makers. The paper presents the tool and its functionalities, together with application examples on full-vehicle models up to 500Hz.
... Usually, the assessment of the discretization errors is performed "a-posteriori", thus, starting with a mesh Mo and constructing successively meshes MI' M2 etc., a number of iterations becomes necessary until the prescribed accuracy is established. Up to now, considerable success has been achieved with error estimators based on smoothing of derivatives (stresses), on the residual of differential equation, on the super-convergent patch recovery technique (SPR) [1][2][3][4][5] etc. ...
An alternative approach to carry out adaptive finite element analysis is presented, with target of reducing the computational effort as well as the number of iterations. The proposed method is based on the use of special large finite elements constructed by the Coons interpolation method. In contradistinction to known adaptive finite element methods which iteratively refine the solution, in the present method the solution is approximated in almost one step by the use of the special finite elements of known performance. Numerical examples dealing with elastostatic stress analysis under plane stress conditions are presented in order to illustrate the proposed methodology and demonstrate its good performance. INTRODUCTION Significant achievements have been made on the theory and computational algorithms for the development of adaptive strategies in recent years. Iterative techniques such as the h-, r-, p-, h-p-, h-r-adaptive finite element methods have been developed in order to compute stress concentrations in problems of mechanics. The h-and r-versions focus, keeping the interpolation order within an element constant, on constructing a mesh that approximates the global solution, the p-scheme focuses, keeping the mesh constant, on improving the interpolation order within the elements, while the h-p-and h-r-methods represent combinations of them. The application of the above methods is based on a reliable error estimator that provides local information for the con~truction of a new mesh for a finite element solution of higher accuracy, since the exact solution is generally unknown. Usually, the assessment of the discretization errors is performed "a-posteriori", thus, starting with a mesh Mo and constructing successively meshes MI' M2 etc., a number of iterations becomes necessary until the prescribed accuracy is established. Up to now, considerable success has been achieved with error estimators based on smoothing of derivatives (stresses), on the residual of differential equation, on the super-convergent patch recovery technique (SPR) [1-5] etc.
... A continuous smooth stress field can also be obtained in a more computationally expensive manner by assembling each H and Σ in (3.3) into corresponding global tensors, in a process analogous to the assembly of the global stiffness. In this case, since C 0 -continuous interpolation functions are normally used for stress recovery, variations in global stress smoothing include the introduction of a penalty term in the error function that enforce C 1 -continuity of the recovered stress field (Riggs et al., 1997). Hinton and Campbell propose using the stress smoothing technique for linear least square fit of reduced integrated elements. ...
... The underlying assumption in these kind of methods is that an average of the consistent stress field is more accurate than the field itself [Hinton and Campbell, 1974]. Since C 0 -continuous interpolation functions are normally used for stress recovery, variations to global stress smoothing include the introduction of a penalty term in the error function that enforces C 1 -continuity of the recovered stress field [Riggs et al., 1997]. ...
A class of mixed finite elements based on the Hu-Washizu functional is introduced as a strategy to reduce the spurious stress phenomena encountered with position-based (standard) formulations for geometrically exact membrane and cable theories. The stress recovery procedure inherent to this mixed formulation is shown to be closely related to, and for some cases is equivalent to, standard a posteriori stress recovery techniques. The conditions for numerical stability and for optimal, superconvergent stress recovery of the mixed formulation are established. Selected examples compare the performance of this class of elements with that of the standard formulation and demonstrate that reduction of spurious stresses is obtained for membranes, and that the stresses are more accurate than those optimally sampled in the standard formulation. It is shown that this mixed formulation is also suitable for near inextensible problems.
... A number of nodal stress recovery (smoothing) procedures have been proposed in the literature [21,22]. Evaluating directly stresses at the nodes of each element, or using the stress at the Gauss points, in any case requires then some kind of averaging since stresses are in general different for elements sharing the same node. ...
This paper describes a new technique for the determination of the inter-element forces and tractions, as well as stress state at nodes, as a post-processing step after the solution of standard FE-displacement calculation. The work is motivated in the context of a broader development of a procedure to simulate fracture processes using a discrete approach without the need of double-noded interface elements. The technique, easily implementable, is based on the double minimization of an objective function, representing the error between the inter-element stress tractions and the projection of the best-fit stress tensor T along the planes of the interfaces converging at an element corner node. The formulation is illustrated with some basic examples in which the resulting stress tensors and inter-element forces are compared to theoretical solutions and to the results obtained by using a traditional stress average smoothing method. Copyright © 2005 John Wiley & Sons, Ltd.
... The interpolation functions for the MIN3 element have also been used to develop a 'smoothing element analysis' for improved stress recovery in finite element analysis456. ...
Thesis (Ph. D.)--University of Hawaii at Manoa, 2002. Includes bibliographical references (leaves 123-126).
The inverse problem of reconstructing structural deformations based on measured strain data, known as shape sensing, is addressed in this numerical study for a cantilevered plate undergoing complex bending and twisting deformations. The inverse Finite Element Method (iFEM) is the main computational tool used for solving the inverse problem, and the Smoothing Element Analysis (SEA) is used as a pre-processing step seeding additional strain data into a high-fidelity iFEM discretization. This novel SEA-iFEM approach demonstrates that even with a small number of strain sensors and a judicial choice of strain-sensor layouts, accurate shape-sensing reconstructions for plate structures can be obtained.
The inverse Finite Element Method (iFEM) is a model-based technique for the structure’s displacement field computation, based on an optimized strain sensor network. In real applications, sensors cannot be applied to the whole structure due to practical constraints and limitations. In this context, strain pre-extrapolation techniques can predict the strain value where physical sensors are not available, but the sensor network must well describe the strain pattern on the whole structure. However, if sensors are located far from local discontinuities, the strain field will be no more correctly pre-extrapolated, resulting in a wrong displacement field reconstruction. The present work presents a physics-based strain pre-extrapolation, where the physical knowledge of the discontinuity, together with its analytical stress formulation, supports the pre-extrapolation technique and thus the iFEM field reconstruction. The particular cases of a hole in a plate and a hole in a strip are considered in four numerical case studies with increasing complexity, always providing significative result’s improvement with respect to the previous state-of-the-art pre-extrapolation technique.
The inverse Finite Element Method (iFEM) is a model-based technique for structural shape sensing based on a pattern of input strain data experimentally acquired on the structure. It is independent of the external load and material properties, making it a valid approach for composite materials monitoring. In this framework, the iFEM shape sensing capability is experimentally investigated on a carbon fiber reinforced polymer plate subjected to a compressive buckling condition. As logistic constraints impede strain sensor installation in all the desired locations, Smoothing Element Analysis (SEA) and polynomial fittings are used to pre-extrapolate the whole plate's strain field. Finally, after a sensitivity analysis on the input strain pre-extrapolation, the iFEM displacement reconstruction is validated with three lasers' independent measurements and direct FEM results, investigating the local and global shape sensing capability.
This paper refers to an adaptive FE-method based on the use of “pre-constructed” large finite
elements, with the target of reducing computational effort as well as number of iterations. In contradistinction
to known adaptive finite element techniques (h-, p-, r-versions), which iteratively refine the solution, the
present method proposes an one-step refinement procedure based on the use of “large” pre-constructed
elements of known performance. Numerical results for the plane elasticity case demonstrate the good
performance of the proposed method.
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.
Response surface techniques have originally been developed for the construction of approximations on the basis of function values only. In many cases however, derivative information is available inexpensively, and can be used to improve the accuracy and reduce the cost of the response surface. This is, for example, the case when efficient procedures for design sensitivity analysis are available. In the present paper, a framework is given for the construction of response surfaces (RS) using both function values and derivatives. The basis is a weighted least squares formulation, which also includes derivatives. In addition, several scaling schemes are summarized that can be used to balance the relative importance of functions and derivatives. Use of derivatives is potentially hindered by a number of complications, namely (i) functions and their derivatives are noisy, (ii) although function may be continuous the corresponding derivatives may not, and (iii) some derivatives may be expensive to calculate. In order to assess whether these complications obstruct using derivatives in RS building, a series of simple numerical examples is studied. The basic conclusion that can be drawn from the present work is, that inexpensive derivative data can be employed efficiently in RS building. Even when derivatives are noisy and discontinuous, the advantages in terms of accuracy and efficiency can be significant. To take full advantage of information on derivatives, higher order RS might be needed. © 2000 by Van Keulen, Delft University of Technology and Liu and Haftka, University of Florida.
Nonlinear, time-domain hydroelastic analysis of flexible offshore structures requires that the structural motion be transferred to the fluid model and the resulting fluid pressure at the fluid-structure interface be transferred from the fluid model to the structure. When the structural mesh and the fluid mesh describe two distinct three-dimensional surfaces, the transfer of displacement and pressure is both difficult and non-unique. In this paper, a new transfer strategy based on the variational-based smoothing element analysis (SEA) technique is presented. The displacement transfer uses the original formulation of the SEA method, although the application of the procedure to displacement transfer is new. For energy conservation during the reverse pressure transfer, the original functional in the SEA method is enhanced with a new term that attempts to conserve the work done by the hydrodynamic forces when obtaining the global structural nodal forces. To evaluate the transfer methodology, the hydrodynamic response of three rigid bodies are considered. Pressure contours, hydrodynamic coefficients, and motions that are calculated based on the data transferred with the proposed method are compared with the results that are obtained from standard rigid-body hydrodynamics theory that does not include a structural finite element model. The method is shown to work very well. In addition, it has general applicability and it can deal with relatively large geometric differences in the meshes.
In recent years the pure displacement formulation for plate elements has not been as popular as other formulations. We revisit the pure displacement formulation for shear-deformable plate elements and propose a family of N-node, displacement-compatible, fully-integrated, pure-displacement, triangular, Mindlin plate elements, MIN-N. The development has been motivated by the relative simplicity of the pure displacement formulation and by the success of the existing 3-node plate element, MIN3. The formulation of MIN3 is generalized to obtain the MIN-N family, which possesses complete, fully compatible kinematic fields, in which the interpolation functions for transverse displacement are one degree higher than those for rotations. General element-level formulas for the thin-limit Kirchhoff constraints are developed. The 6-node, 18 degree-of-freedom element MIN6, with cubic displacement and quadratic rotations, is implemented and tested extensively. Numerical results show that MIN6 exhibits good performance for both static and dynamic analyses in the linear, elastic regime. The results illustrate that the fully-integrated MIN6 element has excellent performance in the thin limit, even for coarse meshes, and that it does not require shear relaxation.
The smoothing element analysis for stress recovery and error estimation is applied to facilitate adaptive finite element solutions of adhesively bonded structures. The formulation is based on the minimization of a penalized discrete least-squares variational principle leading up to the recovery of C 1 -continuous stress fields from discrete, Gauss-point finite element stresses. The smoothed distributions are then used as reference solutions in a posteriors error estimators. Adaptive mesh refinements are performed to predict the linearly elastic response of uniformed and tapered double splice adhesively bonded joints. Key aspects pertaining to specific smoothing strategies, adaptive refinement solutions, and detailed stress distributions are discussed. Consistent comparisons are also presented with Oplinger's one-dimensional adhesive lap joint analysis.
Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990-2000.
A four-node, quadrilateral smoothing element is developed based upon a penalized-discrete-least-squares variational formulation. The smoothing methodology recovers C1-continuous stresses, thus enabling effective a posteriori error estimation and automatic adaptive mesh refinement. The element formulation is originated with a five-node macro-element configuration consisting of four triangular anisoparametric smoothing elements in a cross-diagonal pattern. This element pattern enables a convenient closed-form solution for the degrees of freedom of the interior node, resulting from enforcing explicitly a set of natural edge-wise penalty constraints. The degree-of-freedom reduction scheme leads to a very efficient formulation of a four-node quadrilateral smoothing element without any compromise in robustness and accuracy of the smoothing analysis. The application examples include stress recovery and error estimation in adaptive mesh refinement solutions for an elasticity problem and an aerospace structural component. Copyright © 1999 John Wiley & Sons, Ltd.
A four-node, quadrilateral smoothing element is developed based upon a penalized-discrete-least-squares variational formulation. The smoothing methodology recovers C1-continuous stresses, thus enabling effective a posteriori error estimation and automatic adaptive mesh refinement. The element formulation is originated with a five-node macro-element configuration consisting of four triangular anisoparametric smoothing elements in a cross-diagonal pattern. This element pattern enables a convenient closed-form solution for the degrees of freedom of the interior node, resulting from enforcing explicitly a set of natural edge-wise penalty constraints. The degree-of-freedom reduction scheme leads to a very efficient formulation of a four-node quadrilateral smoothing element without any compromise in robustness and accuracy of the smoothing analysis. The application examples include stress recovery and error estimation in adaptive mesh refinement solutions for an elasticity problem and an aerospace structural component.
Displacement and pressure transfer between fluid and structure meshes in fluid-structure interaction analysis is complicated substantially when the two meshes on the domain interface are different and therefore form two distinct surfaces in three-dimensional space. A new, approximately energy-conserving interfacing strategy with two distinct components - mapping and interpolation - is proposed. The mapping between meshes is based on the assumption of a common parametric-based description of the wetted surface. The interpolation strategy is based on the smoothing element analysis method developed to recover stresses in unite element analysis, with an additional term to impose an energy-conserving constraint. The method is evaluated numerically with several examples from linear hydrodynamics/hydroelasticity. The advantage of these examples is that the components can be evaluated separately, and the performance of the method can be examined more clearly. Numerical results show the method is quite promising.
The relationship between a mixed finite element formulation based on the Hu–Washizu (HW) functional and stress recovery techniques is elucidated. Although mixed formulations are primarily motivated by avoidance of locking phenomena in finite element solutions, it is shown that a mixed formulation based on the HW functional, with displacements, stresses and strains as fields, is so intimately related to stress recovery that the formulation itself can also be viewed as a stress recovery method. Copyright © 2000 John Wiley & Sons, Ltd.
A new variational formulation is presented which serves as a foundation for an improved finite element stress recovery and a posteriori error estimation. In the case of stress predictions, interelement discontinuous stress fields from finite element solutions are transformed into a C1-continuous stress field with C0-continuous stress gradients. These enhanced results are ideally suited for error estimation since the stress gradients can be used to assess equilibrium satisfaction. The approach is employed as a post-processing step in finite element analysis. The variational statement used herein combines discrete-least squares, penalty-constraint, and curvature-control functionals, thus enabling automated recovery of smooth stresses and stress gradients. The paper describes the mathematical foundation of the method and presents numerical examples including stress recovery in two-dimensional structures and built-up aircraft components, and error estimation for adaptive mesh refinement procedures.
A variational method for obtaining smoothed stresses from a finite clement derived non-smooth stress field is presented. The method is based on minimizing a functional involving discrete least-squares error plus a penalty constraint that ensures smoothness or the stress field. An equivalent accuracy criterion is developed for the smoothing analysis which results in a C'-continuous smoothed stress field possessing the same order or accuracy as that found at the supcrconvergent optimal stress points of the original finite clement analysis. Application of the smoothing analysis to residual error estimation is also demonstrated.
The existence of optimal points for calculating accurate stresses within finite element models is discussed. A method for locating such points is proposed and applied to several popular finite elements.
A new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes. The estimator allows the global energy norm error to be well estimated and also gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials. When combined with an automatic mesh generator a very efficient guidance process to analysis is available. Estimates other than the energy norm have successfully been applied giving, for instance, a predetermined accuracy of stresses.
An essential phase in moire techniques is the generation of strain fields from optical fringe patterns. In this effort we explore a finite element method based upon a new least-squares penalty-constraint variational principle, which will be shown to be extremely effective in producing accurate, two-dimensional displacement and strain fields over the full-fringe domain. The approach encompasses the full-field analyses of optical fringes, which include: (1) an effective and theoretically sound noise filtering/smoothing and differentiating of the optical intensity fringes, from which displacement fringes are accurately determined, (2) subsequent smoothing/differentiating of the displacement fringes, producing continuous displacement and strain fields of high quality, (3) computational efficiency, (4) the ability to routinely process distorted mismatch fields which occur, for example, with large fields of view, avoiding the need for any special care to eliminate such distortion, (5) ease of implementation within a conventional finite element program. Several examples are carried out which demonstrate the attractiveness of the methodology.
The Superconvergent Patch Recovery (SPR) technique and its enhancements has shown to be an accurate and efficient method to obtain an improved solution. It gives a more accurate solution, used as such, and it can be utilized for an error estimation of the finite element solution. The improved solution has often at least one order higher convergence rate and is much more accurate than the finite element solution. The paper gives a review of different aspects and enhancements of SPR-technique and proposes an additional global iteration procedure, which gives a still higher accuracy. These changes make it possible to give an error estimation of the improved solution. The different error estimations are used in an adaptive process, showing that improved error estimation gives improved adaptivity performance.
An a priori, explicit algebraic procedure for identifying shear locking and excessive solution stiffening in thin shear-deformable plates is explored. Only element-level solutions of element Kirchhoff modes are required to establish the nodal degree-of-freedom constraints. These constraints clearly identify whatever kinematic stiffening might exist. The methodology is demonstrated by the use of a conforming, three-node Mindlin element. Several discretizations of square plates are examined. The results are compared, and fully confirmed, with the corresponding numerical solutions. Examples of alternate discretizations, which alleviate the locking effect, are also presented.
The error estimation processes which have been developed by the authors depend on the accuracy with which stresses (gradients) can be recovered. The recovery methods available currently, such as the L2 projection and averaging, are inaccurate and an improved process is needed. This paper presents a very much improved recovery process yielding superconvergent values throughout the domain. Details of this method, named the superconvergent patch recovery, are presented with test results showing its efficiency. Examples of adaptive, h-based refinement are given, showing high local and overall efficiency indices for error estimation.
L2 projective techniques have proven very useful for postprocessing of derivatives, such as strains, in finite element solutions of elliptic problems. However, these projections often exhibit substantial errors near boundaries. It is shown here that adding the square of the residuals of selected governing equations to the least square form enhances the accuracy. Both global and local projection schemes are considered. Results are presented which show that these enhancements significantly increase the accuracy and convergence of recovered derivatives.
A superconvergent recovery (smoothing) procedure for finite element stresses is presented and applied to two-dimensional stress fields. The procedure is based on a variational principle involving discrete least-squares with a penalty constraint. A 'continuous' formulation and a 'wireframe' formulation of the principle are discussed and numerical results are compared. Both formulations produce superconvergent stress fields of higher accuracy than the underlying finite element stress field. Although a 'smoothing' analysis is required, results indicate that two 'coarse' mesh analyses (finite element and smoothing) produce the same accuracy as a single, fine mesh finite element analysis. The procedure is unique in that the smoothed stress field is (optionally) C1 continuous.
In this second part of the paper, the issue of a posteriori error estimation is discussed. In particular, we derive a theorem showing the dependence of the effectivity index for the Zienkiewicz–Zhu error estimator on the convergence rate of the recovered solution. This shows that with superconvergent recovery the effectivity index tends asymptotically to unity. The superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the Zienkiewicz–Zhu error estimator to demonstrate accurate estimation of the exact error attainable. Numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient (stress) error estimation. Several examples of the performance of the error estimator in adaptive mesh refinement are also presented.
A technique is described wherein improved stresses can be computed in finite element models based on displacement approximations. The method is based on the idea of consistent stress approximations and it approximates such stresses using the notion of a domain of influence of the stress intensity at a nodal point. Considerable improvement in accuracy of the stresses is obtained with little difficulty.
The theory of conjugate approximations1 is used to obtain consistent approximations of stress fields in finite element approximations based on displacement assumptions. These consistent stresses are continuous across interelement boundaries and involve less mean error than those computed by the conventional approach.
In 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.
The concepts and potential advantages of local and global least squares smoothing of discontinuous finite element functions are introduced.
The relationship between local smoothing and the ‘reduced’ integration' technique is established.
Examples are presented to illustrate the application of the two smoothing techniques to the finite element stresses from several structural analysis problems.
The paper concludes with some practical recommendations for discontinuous finite element function smoothing.
A new superconvergence recovery technique for finite element solutions is presented and discussed for one dimensional problems. By using the recovery technique a posteriori error estimators in both energy norm and maximum norm are presented for finite elements of any order. The relation between the postprocessing and residual types of energy norm error estimators has also been demonstrated.
The existence of optimal points for calculating accurate stresses within finite element models is discussed. A method for locating such points is proposed and applied to several popular finite elements.
A new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes.
The estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials.
When combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable.
Estimates other than the energy norm have successfully been applied giving, for instance, a predetermined accuracy of stresses.
This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.
An analysis, based on the use of Taylor series expansions, is developed to determine accuracy estimates for derivatives in one and two dimensions computed by differentiation of a finite-element interpolant or approximation. The analysis clarifies some issues concerning special points at which the derivatives are believed to be exceptionally accurate with higher convergence rates (superconvergence). Moreover, it leads directly to a class of post-processing strategies for the derivatives and offers a more direct constructive approach to the subject.
A brief survey with a bibliography of superconvergence phenomena in finding a numerical solution of differential and integral equations is presented. A particular emphasis is laid on superconvergent schemes for elliptic problems in the plane employing the finite element method.
Finite element procedures are applied to the modeling, analysis and visualization of experimental moiré data. Smoothing elements
are introduced and evaluated with respect to data sparseness and error. A one-dimensional smoothing element is uniquely coupled
with the method of principal curves to extract moiré fringe centers. A two-dimensional smoothing element is then used to produce
a full-field representation given the fringe locations. The moiré technique is applied to the four-point bend experiment,
and the surface-modeling technique is used to obtain displacement and gradient (strain) information.
A displacement methodology for Mindlin elements, recently employed in the development of an efficient, four-node quadrilateral (MIN4), is the basis for a three-node, explicitly integrated triangular element (MIN3). The approach, herein referred to as anisoparametric, is founded upon two kinematic requirements that are auxiliary to the standard convergence criteria. The initial, complete quadratic deflection field is constrained by ‘continuous’ shear edge constraints to achieve an isoparametric-like, three-node configuration. Much success of the element is due to an innovative shear correction factor concept. Several numerical experiments are performed in the linear elasto-static regime to ascertain the element performance.
Construction of an asymptotically exact recovery based finite element method error estimator
- Pomeranz
S. Pomeranz and B.A. Conder, Construction of an asymptotically exact recovery based finite element method error
estimator, Comput. Methods Appl. Mech. Engrg. (1995), under review.
An asymptotically exact finite element error estimator for planar linear elasticity problems
- Pomeranz
S. Pomeranz and AW. Kirk, An asymptotically exact finite element error estimator for planar linear elasticity problems,
Comput. Methods Appl. Mech. Engrg. (1995), under review.
Application of a variational method for computing smooth stresses, stress gradients, and error estimation in finite element analysis
- A Tessler
- H R Riggs
- S C Macy
A. Tessler, H.R. Riggs and S.C. Macy, Application of a variational method for computing smooth stresses, stress gradients,
and error estimation in finite element analysis, in: J.R. Whiteman, ed., The Mathematics of Finite Elements and
Applications (John Wiley & Sons, Ltd., 1994) 189-198.
The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique
- Zienkiewicz