The generalized finite element method: an example of its implementation and illustration of its performance

The generalized finite element method (GFEM) was introduced in Reference 1 as a combination of the standard FEM and the partition of unity method. The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation. The essential boundary conditions can be imposed exactly as in the standard FEM. Adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation. This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM. Copyright © 2000 John Wiley & Sons, Ltd.     

Static response of a spindle-shaped tensairity column to axial compression

Tensairity is a new lightweight structural concept consisting of struts and cables stabilized by a textile membrane, which is inflated by low pressurized air. In order to estimate the potential of Tensairity beams towards applications including axial compressive loads, full-scale compression experiments were conducted on a simply-supported spindle shaped Tensairity column. The column was subjected to axial compressive loading for various levels of internal air-pressure in order to quantify its effect on local and global response, and it was found that the axial stiffness of the column increases with air-pressure and eventually reaches a plateau. Displacements were measured in several positions along the span, whereas axial forces were experimentally determined by strain gauges measurements. The experimental results were compared to finite element and analytical predictions, yielding good correlation for low air-pressure levels, whereas for higher ones, local imperfections led to significant deviations. Comparisons of the Tensairity column to similar truss-type structures with comparable stiffness revealed the superiority of the concept in terms of transportation volume and in-situ deployment.     

A posteriori error estimation for finite element solutions of Helmholtz’ equation. part I: the quality of local indicators and estimators

This paper contains a first systematic analysis of a posteriori estimation for finite element solutions of the Helmholtz equation. In this first part, it is shown that the standard a posteriori estimates, based only on local computations, severely underestimate the exact error for the classes of wave numbers and the types of meshes employed in engineering analysis. This underestimation can be explained by observing that the standard error estimators cannot detect one component of the error, the pollution error, which is very significant at high wave numbers. Here, a rigorous analysis is carried out on a one-dimensional model problem. The analytical results for the residual estimator are illustrated and further investigated by numerical evaluation both for a residual estimator and for the ZZ-estimator based on smoothening. In the second part, reliable a posteriori estimators of the pollution error will be constructed. © 1997 by John Wiley & Sons, Ltd.     

A posteriori error estimation for finite element solutions of Helmholtz' equation—Part II: estimation of the pollution error

In part I of this investigation, we proved that the standard a posteriori estimates, based only on local computations, may severely underestimate the exact error for the classes of wave-numbers and the types of meshes employed in engineering analyses. We showed that this is due to the fact that the local estimators do not measure the pollution effect inherent to the FE-solutions of Helmholtz' equation with large wavenumber. Here, we construct a posteriori estimates of the pollution error. We demonstrate that these estimates are reliable and can be used to correct the standard a posteriori error estimates in any patch of elements of interest. © 1997 John Wiley & Sons, Ltd.     

Antagonistic Action of Fe and Mn in Mediterranean-Type Plants Irrigated with Wastewater Effluents Following Biological Treatment

The present study examined the ability of certain Mediterranean plants to absorb Mn ++ when irrigated with wastewater effluents following treatment, the relative absorption of Fe ++ and Mn ++ , as well as the endurance of such plants to high Mn ++ concentrations in the water used for their irrigation. The experimental plants used in the study were Olea europaea , Nerium oleander and Geranium sp . These plants were irrigated with secondary treatment wastewater effluents, with water commonly used for irrigation in the area, and with water enriched with high concentrations of Mn ++ . The results showed that irrigation of the plants with wastewater treatment effluents resulted in slightly elevated concentrations of Mn ++ and Fe ++ in the leaves, the roots, and in the soil. On the contrary, irrigation with water enriched with high concentrations of Mn ++ (as high as 1000 mg/L) resulted in correspondingly high Mn ++ concentrations in the plants under study. The accumulation of Mn ++ was much higher in the roots than in the leaves of these plants. In the case of Mn ++ enriched water irrigation, the corresponding concentrations of Fe ++ in the roots, in the leaves, and in the soil indicated the existence of strong antagonism between these two elements. All plants exhibited tolerance to high concentrations of Mn ++ .     

High‐order layerwise finite element for the damped free‐vibration response of thick composite and sandwich composite plates

A high‐order layerwise finite element methodology is presented, which enables prediction of the damped dynamic characteristics of thick composite and sandwich composite plates. The through‐thickness displacement field in each discrete layer of the laminate includes quadratic and cubic polynomial distributions of the in‐plane displacements, in addition to the linear approximations assumed by linear layerwise theories. Stiffness, mass and damping matrices are formulated from ply to structural level. Interlaminar shear stress compatibility conditions are imposed on the discrete layer matrices, leading to both size reduction and prediction of interlaminar shear stresses at the laminate interfaces. The C¹ continuous finite element implemented yields an element damping matrix in addition to element stiffness and mass matrices. Application cases include thick [0/90/0], [±θ]_S and [±θ] composite plates with interlaminar damping layers and sandwich plates with composite faces and foam core. In the latter case, modal frequencies and damping were also experimentally determined and compared with the finite element predictions. Copyright © 2008 John Wiley & Sons, Ltd.     

Recent experiences with error estimation and adaptivity, Part II: Error estimation for h-adaptive approximations on grids of triangles and quadrilaterals

In Part I, residual and flux projection error estimators for finite element approximations of scalar elliptic problems were reviewed; numerical studies on the performance of these estimators were presented for finite element approximations of the solution of Poisson's equation on uniform grids of hierarchic triangles of order p (1 p 7). Here further numerical experiments are given which also include error estimators for the vector-valued problem of plane elastostatics and implementations for h-adaptive grids of triangles and quadrilaterals which are constructed using an algorithm of equidistribution of error coupled with h-refinement or h-remeshing schemes. A detailed numerical study of several flux-projectors for h-adaptive grids of bilinear and biquadratic quadrilaterals is conducted; a flux equilibration iteration, which may be employed in some cases to improve flux projection estimates, is also included. FAor the case of grids of quadrilaterals, several versions of the element residual estimators, which differ by the approximate flux employed for the calculation of the boundary integral term in the definition of the local problems, are compared. The numerical experiments confirm the good overall performance of residual estimates and indicate that flux projection estimates, which are now operational in several commercial codes, may be divergent when they are employed to estimate the error in even order h-adaptive approximations.     

Investigating the enhancement of three-dimensional diffraction tomography by using multiple illumination planes

The three dimensional (3-D) extension of the two well-known diffraction tomography algorithms, namely, direct Fourier interpolation (DFI) and filtered backpropagation (FBP), are presented and the problem of the data needed for a full 3-D reconstruction is investigated. These algorithms can be used efficiently to solve the inverse scattering problem for weak scatterers in the frequency domain under the first-order Born and Rytov approximations. Previous attempts of 3-D reconstruction with plane-wave illumination have used data obtained with the incident direction restricted at the xy plane. However, we show that this restriction results in the omission of the contribution of certain spatial frequencies near the omegaz axis for the final reconstruction. The effect of this omission is studied by comparing the results of reconstruction with and without data obtained from other incident directions that fill the spatial frequency domain. We conclude that the use of data obtained for incident direction in only the xy plane is sufficient to achieve a satisfactory quality of reconstruction for a class of objects presenting smooth variation along the z axis, while abrupt variations along the z axis cannot be imaged. This result should be taken into account in the process of designing the acquisition geometry of a tomography scanner.