T. Strouboulis

The design and analysis of the Generalized Finite Element Method

Abstract In this paper, we introduce the Generalized Finite Element Method (GFEM) as a combination of the classical Finite Element Method (FEM) and the Partition of Unity Method (PUM). The standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data; e.g., the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point, etc. The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape-functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space. In this way, the local approximability afforded by the special functions is included in the approximation, while maintaining the existing infrastructure of finite element codes. The major features of the GFEM are: (1) the essential boundary conditions can be imposed exactly as in the standard FEM, unlike other partition of unity based methods where this is a major issue; (2) the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation (this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods); and (3) linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver. The power of the GFEM for solving problems in domains with complex geometry with less error and less computer resources than the standard FEM is illustrated by numerical examples.

The generalized finite element method

This paper describes a pilot design and implementation of the generalized finite element method (GFEM), as a direct extension of the standard finite element method (SFEM, or FEM), which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the FEM. The development of the GFEM is illustrated for the Laplacian in two space dimensions in domains which may include several hundreds of voids, and/or cracks, for which the construction of meshes used by the FEM is practically impossible. The two main capabilities are: (1) It can construct the approximation using meshes which may overlap part, or all, of the domain boundary. (2) It can incorporate into the approximation handbook functions, which are known analytically, or are generated numerically, and approximate well the solution of the boundary value problem in the neighborhood of corner points, voids, cracks, etc. The main tool is a special integration algorithm, which we call the Fast Remeshing approach, which is robust and works for any domain with arbitrary complexity. The incorporation of the handbook functions into the GFEM is done by employing the partition of unity method (PUM). The presented formulations and implementations can be easily extended to the multi-material medium where the voids are replaced by inclusions of various shapes and sizes, and to the case of the elasticity problem. This work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the GFEM, which is needed before analogous implementations are attempted in the three-dmensional and nonlinear cases, which are the cases of main interest for future work.

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

A model study of the quality of a posteriori error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles

Abstract This paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems. In this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles. A completely numerical methodology for the analysis of the quality of estimators is presented. Results from the application of the methodology to the study of the quality of several well-known error estimators are reported. In subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals, non-uniform grids, at boundaries, grid-interfaces and near-singular points.

Generalized finite element method using mesh-based handbooks: application to problems in domains with many voids

This paper describes a new version of the generalized finite element method, originally developed [Int. J. Numer. Methods Engrg. 47 (2000) 1401; Comput. Methods Appl. Mech. Engrg. 181 (2000) 43; The design and implementation of the generalized finite element method, Ph.D. thesis, Texas AM Comput. Methods Appl. Mech. Engrg. 190 (2001) 4081], which is well suited for problems set in domains with a large number of internal features (e.g. voids, inclusions, cracks, etc.). The main idea is to employ handbook functions constructed on subdomains resulting from the mesh-discretization of the problem domain. The proposed new version of the GFEM is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features.

Pollution error in the h-version of the finite element method and the local quality of the recovered derivatives

Abstract In this paper, we address the quality of the solution derivatives which are recovered from finite element solutions by local averaging schemes. As an example, we consider the Zienkiewicz-Zhu superconvergent patch-recovery scheme (the ZZ-SPR scheme), and we study its accuracy in the interior of the mesh for finite element approximations of solutions of Laplaces equation in polygonal domains. We will demonstrate the following: (1) In general, the accuracy of the derivatives recovered by the ZZ-SPR or any other local averaging scheme may not be higher than the accuracy of the derivatives computed directly from the finite element solution. (2) If the mesh is globally adaptive (i.e. it is nearly equilibrated in the energy-norm) then we can, practically always, gain in accuracy by employing the recovered derivatives instead of the derivatives computed directly from the finite element solution. (3) It is possible to guarantee that the recovered solution-derivatives have higher accuracy than the derivatives computed directly from the finite element solution, in any patch of elements of interest, by employing a mesh which is adaptive only with respect to the patch of interest (i.e. it is nearly equilibrated in a weighted energy-norm). (4) In practice, we are often interested in obtaining highly accurate derivatives (or heat-fluxes, stresses, etc.) only in a few critical regions which are identified by a preliminary analysis. A grid which is adaptive only with respect to the critical regions of interest may be much more economical for this purpose because it may achieve the desired accuracy by employing substantially fewer degrees of freedom than a globally adaptive grid which achieves comparable accuracy in the critical regions.

Pollution-error in the h -version of the finite-element method and the local quality of a-posteriori error estimators

Abstract In this paper we study the pollution-error in the h- version of the finite element method and its effect on the local quality of a-posteriori error estimators. We show that the pollution-effect in an interior subdomain depends on the relationship between the mesh inside and outside the subdomain and the smoothness of the exact solution. We also demonstrate that it is possible to guarantee the quality of local error estimators in any mesh-patch in the interior of a finite-element mesh by employing meshes which are sufficiently refined outside the patch.

A posteriori estimation and adaptive control of the error in the quantity of interest. Part I: A posteriori estimation of the error in the von Mises stress and the stress intensity factor

In this paper we address the problem of a posteriori estimation of the error in an engineering quantity of interest which is computed from a finite element solution. As an example we consider the plane elasticity problem with the von Mises stress and the stress intensity factor, as the quantities of interest. The estimates of the error in the von Mises stress at a point are obtained by partitioning the error into two components with respect to the element which includes the point, the local and the pollution errors, and by constructing separate estimates for each component. The estimates of the error in the stress intensity factors are constructed by employing an extraction method. We demonstrate that our approach gives accurate estimates for rather coarse meshes and elements of various degrees. In Part II we will address the problem of the adaptive control of the error in the quantity of interest (the goal of the computation), and the construction of goal-adaptive meshes for one or multiple goals.

Guaranteed computable bounds for the exact error in the finite element solution Part I: One-dimensional model problem

Abstract This paper addresses the computation of guaranteed upper and lower bounds for the energy norm of the exact error in the finite element solution, and the exact error in any bounded linear functional. These bounds are constructed by employing approximate solutions of the element residual problems with equilibrated residual loads. The one-dimensional setting is used for the clarity of the ideas. All the arguments employed can be extended to the higher-dimensional case which will be discussed in Part II of this paper. The main result presented here is that the computed bounds are guaranteed for the exact error and not the error with respect to an enriched finite element solution, like the bounds proposed by other investigators and the bounds are guaranteed for any mesh, however coarse it may be. The quality of the bounds can be controlled by employing an inexpensive iterative scheme.

FINITE ELEMENT SIMULATION OF ROUGH RICE DRYING

ABSTRACT Milled, brown and rough rice samples were dried in the laboratory with heated air and diffusivities of the endosperm, bran and husk were evaluated with a search technique using the finite element method. The endosperm had the highest diffusivity. The husk had a slightly higher diffusivity than the bran because the trapped air inside the husk was considered as an integral part of the husk. The finite element method predicted rough rice drying that was in good agreement with experimental results.