C. S. Upadhyay

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.

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.

Electro-thermo-elastic formulation for the analysis of smart structures

Coupled electro-thermo-elastic equations applicable for the analysis of smart structures with piezoelectric patches/layers have been derived from the fundamental principles of mass, linear momentum, angular momentum, energy and charge conservation. The relevant constitutive equations have been obtained by using the second law of thermodynamics. The interaction of the electric field and polarization introduces distributed non-linear body force in the piezo material, and in addition renders the stress tensor non-symmetric due to distributed couple. Using the linear equations, and applying a layer-by-layer finite element model, the induced electric potential and mechanical deformations in the piezo and non-piezo core material have been obtained for various cases of actuation and sensing of a smart beam under external mechanical, electrical and thermal loadings. The mathematical formulation and the solution technique have been validated by comparing the results of the present study with those available in the literature. It is also shown that piezo patches can be effectively used for shape control.

A model study of element residual estimators for linear elliptic problems: The quality of the estimators in the interior of meshes of triangles and quadrilaterals

In Babuska et al. (Comput. Meth. appl. Mech. Engng, 114, 307–378 (1994); Int. J. numer. Meth. Engng 37, 1073–1123 (1994)) we presented a model study of a posteriori error estimators in the interior of finite element meshes using a computer-based methodology. In this paper we investigate further the quality of element-residual error estimators. We analyzed several versions of the element-residual estimator and based on this study we propose recipes for robust estimators.

Linear and Nonlinear Analysis of a Smart Beam Using General Electrothermoelastic Formulation

Coupled electrothermoelastic equations applicable to the analysis of smart structures have been derived from first principles. Using the equations and applying a layer-by-layer finite element model, the induced potential and mechanical deformations in the piezo and nonpiezo core material have been obtained for various cases of actuation and sensing of a smart beam under external mechanical load and actuation potential. The present study clearly brings out the essential difference between sensing and actuation. It is also brought out that the interaction between polarization and electric field in the piezo continuum leads to nonlinear distributed body force and nonsymmetric stress tensor. These nonlinear effects are found to have significant influence on the deformation of a smart beam under actuation. Shape control studies of multipatch smart beams have also been investigated.

Validation of recipes for the recovery of stresses and derivatives by a computer-based approach

In this paper, we present a methodology for checking the local quality of recipes for the recovery of stresses or derivatives from finite element solutions of linear elliptic problems. The methodology accounts precisely for the factors which affect the local quality of the recovered quantities, namely, the geometry of the grid, the polynomial degree and the type of the elements, the coefficients of the differential equation and the class of solutions of interest. We give examples of how the methodology can be used to obtain precise conclusions about the quality of a class of recipes, based on least-squares patch-recovery, in the interior of complex grids, like the ones employed in engineering computations. By using this approach, we were able to discover recipes which are much more robust than the ones which are currently in use in the various finite element codes.

e%-superconvergence in the interior of locally refined meshes of quadrilaterals: superconvergence of the gradient in finite element solutions of Laplace's and Poisson's equations

Abstract This paper is the third in a series in which we study the superconvergence of finite element solutions by a computer-based approach. In [1] we studied classical superconvergence and in [2] we introduced the new concept of η%-superconvergence and showed that it can be employed to determine regions of least error for the derivatives of the finite element solution in the interior of any grid of triangular elements. Here we use the same ideas to study the superconvergence of the derivatives of the finite element solution in the interior of complex grids of quadrilaterals of the type used in practical computations.

η%-Superconvergence of finite element approximations in the interior of general meshes of triangles

Abstract In this paper we introduce a new definition of superconvergence — tne η%-superconvergence, which generalizes the classical idea of superconvergence to general meshes. We show that this new definition can be employed to determine the regions of least-error in any element in the interior of any grid by using a computer-based approach. We present numerical results for the standard displacement finite element method for the scalar equation of orthotropic heat-conduction, for meshes of conforming triangles of degree p, 1 ⩽ p ⩽ 5, and elements in the interior of the mesh. The results demonstrate that, unlike classical superconvergence, η%-superconvergence is applicable to the complex grids which are employed in practical engineering computations.

Local quality of smoothening based a-posteriori error estimators for laminated plates under transverse loading

Abstract The local and global quality of various smoothening based a-posteriori error estimators is tested in this paper, for symmetric laminated composite plates subjected to transverse loads. Smoothening based on strain recovery and displacement-field recovery is studied here. Effect of ply orientation, laminate thickness, boundary conditions, mesh topology, and plate model is studied for a rectangular plate. It is observed that for interior patches of elements, both the estimators based on strain or displacement smoothening are reliable. For element patches at the boundary of the domain, all estimators tend to be unreliable (especially for angle-ply laminates). However, the strain recovery based estimator is clearly more robust for element patches at the boundary, as compared to displacement-recovery based error estimators. Globally, all the estimators tested here were found to be very robust.