An efficient methodology for stress-based finite element approximations in two-dimensional elasticity

We present an efficient approach to compute the matrices used in finite element formulations where self-equilibrated (zero divergence) approximations of the stress field are used. The fundamental aspect of this approach is that it is applicable to polynomial approximations of high degree (it was tested up to degree 10), providing closed-form expressions for the elementary matrices involved. The components of the stress field are defined as a function of the coordinates in the master element, facilitating the postprocessing of the finite element results and allowing for the consideration of geometric parameters in reduced order models.     

Pointwise error estimation and adaptivity for the finite element method using fundamental solutions

In this paper, we present a goal-oriented a posteriori error estimation technique for the pointwise error of finite element approximations using fundamental solutions. The approach is based on an integral representation of the pointwise quantity of interest using the corresponding Green's function, which is decomposed into an unknown regular part and a fundamental solution. Since only the regular part must be approximated with finite elements, very accurate results are obtained. The approach also allows the derivation of error bounds for the pointwise quantity, which are expressed in terms of the primal problem and the regular part problem. The presented technique is applied to linear elastic test problems in two-dimensions, but it can be applied to any linear problem for which fundamental solutions exist.     

Element by element a posteriori error estimation of the finite element analysis for three-dimensional elastic problems

An a posteriori error estimation method for finite element solutions for three-dimensional elastic problems is presented based on the theory developed by the authors for two-dimensional problems.¹ The error is estimated for the finite element solutions obtained using three-dimensional 8-node elements with a linear interpolation function in an arbitrary hexahedron. The method is successfully applied to three-dimensional elastic problems. In order to decrease computing time and memory use, the error is estimated element by element. The major difficulty in the element-wise error estimation technique is satisfying the self-equilibrium condition of applied forces, especially in three-dimensional problems. These forces are mainly due to traction discontinuity on the element boundaries. The difficulty is circumvented by employing an element-wise optimal procedure. It is also shown that a very accurate stress solution can be obtained by adding estimated error to the original finite element solutions.     

A priori error estimation of finite element models from first principles

The quality of finite element computational results can be assessed only by providing rational criteria for evaluating errors. Most exercises in this direction are based ona posteriori error estimates, based primarily on experience and intuition. If finite element analysis has to be considered a rational science, it is imperative that procedures to computea priori error estimates from first principles are made available. This paper captures some efforts in this direction.     

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.     

Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation

When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non-dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one-dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H¹-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h-p-version).     

Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity

An error control technique aimed to assess the quality of smoothed finite element approximations is presented in this paper. Finite element techniques based on strain smoothing appeared in 2007 were shown to provide significant advantages compared to conventional finite element approximations. In particular, a widely cited strength of such methods is improved accuracy for the same computational cost. Yet, few attempts have been made to directly assess the quality of the results obtained during the simulation by evaluating an estimate of the discretization error. Here we propose a recovery type error estimator based on an enhanced recovery technique. The salient features of the recovery are: enforcement of local equilibrium and, for singular problems a “smooth + singular” decomposition of the recovered stress. We evaluate the proposed estimator on a number of test cases from linear elastic structural mechanics and obtain efficient error estimations whose effectivities, both at local and global levels, are improved compared to recovery procedures not implementing these features.     

Computable finite element error bounds for Poisson's equation

Theoretical error bounds of the form are often available for finite element solutions U of elliptic problems. In this form the estimates give the order of convergence of the method but are of little practical value for estimating the size of the error because the magnitudes of the constant K and the theoretical solution u are unknown. An exception occurs in the case of the equation ?²u/?x² + ?²u/?y² + f = 0 in a rectangle where the Ritz–Galerkin finite element solution involves piecewise linears over a regular triangular grid. In this case where α = 1 and Barnhill and Gregory¹ have obtained the theoretical value 0·93√2 for K. In this note calculations are carried out for a variety of problems and the quantity K* = ∥u – U∥_E/h∥f∥_L2 measured and compared with K. The values of K* obtained fit into a well defined pattern from which we conclude that the theoretical constant K is of the correct order of magnitude.     

A-posteriori error estimates for the finite element method

Computable a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h → 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one-dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self-adjoint operator of the second order. Generalizations to more general one-dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a-posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.     

Influence functions and goal‐oriented error estimation for finite element analysis of shell structures

In this paper, we first present a consistent procedure to establish influence functions for the finite element analysis of shell structures, where the influence function can be for any linear quantity of engineering interest. We then design some goal‐oriented error measures that take into account the cancellation effect of errors over the domain to overcome the issue of over‐estimation. These error measures include the error due to the approximation in the geometry of the shell structure. In the calculation of the influence functions we also consider the asymptotic behaviour of shells as the thickness approaches zero. Although our procedures are general and can be applied to any shell formulation, we focus on MITC finite element shell discretizations. In our numerical results, influence functions are shown for some shell test problems, and the proposed goal‐oriented error estimation procedure shows good effectivity indices. Copyright © 2005 John Wiley & Sons, Ltd.     

A semi-analytical finite element for laminated composite plates

This paper presents a semi-analytical finite element solution for laminated composite plates. The method is based on a mixed variational principle that involves both displacements and stresses. Finite element meshes are only used in the plane of plate, while the through thickness distributions of displacements and stresses are obtained using the method of state equations. Numerical results show that the rate of convergence of the new method is fast and the solutions can be very close to corresponding exact three-dimensional ones. The use of a recursive formulation of the state equations leads to an algebra equation system, from which solution are sought, whose dimension is independent of the numbers of layers of the plate considered.     

A posteriori error estimates and adaptivity for finite element solutions in finite elasticity

Methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems. In contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context. In the paper a brief outline of continuum theory of finite elasticity is first given. Using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally. The proposed error estimator is physically interpreted in the energy sense. We then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences.     

Optimization under adaptive error control for finite element based simulations

Optimization problems constrained by complex dynamics can lead to computationally challenging problems especially when high accuracy and efficiency are required. We present an approach to adaptively control numerical errors in optimization problems approximated using the finite element method. The discrete adjoint equation serves as a key tool to efficiently compute both parameter sensitivities and goal-oriented error estimates at the same discretized levels. By using a recovery method for the error estimators, we avoid expensive higher order adjoint calculations. We nest the adaptivity of the mesh within the optimization algorithm, which is responsible for converging both the state and optimization algorithms and thereby allowing the reuse of state, parameters, and reduced Hessian in subsequent optimization iterations. Our approach is demonstrated on a parameter estimation problem for contamination transport in a contact tank reactor. Significant efficiency and accuracy improvements are realized in comparison to uniform grid refinement strategies and black-box optimization methods. A flexible and maintainable software interface was developed to provide access between the underlying linear algebra of a production simulator and advanced numerical algorithms such as optimization and error estimation.     

A micromechanical composite approach for finite element crashworthiness simulation

The present article deals with micromechanical composite modeling. Both analytical and computational micromechanics approaches are described as well as micromechanical modeling of damage. Based on micromechanics of failure theory, a user subroutine including a progressive damage algorithm is programmed for finite element analysis. Three theory-experiment correlations of tubes under a three-point bending test have been carried out using the bi-phase material model developed along with this project. These studies include three-ply schedules.     

A beam finite element for magneto-electro-elastic multilayered composite structures

A new finite element based upon an elastic equivalent single-layer model for shear deformable and straight magneto-electro-elastic generally laminated beam is presented. The element has six degrees of freedom represented by the displacement components and the cross-section rotation of its two nodes. The magneto-electric boundary conditions enter the discrete problem as work-equivalent forces and moments while the electro-magnetic state characterization constitutes a post-processing step. The element possesses the superconvergence property for the static problem of beams with uniform cross-section and homogenous material properties along the beam axis direction. Moreover, it is free from the shear locking phenomenon. The developed finite element is validated through comparison with plane-stress results for piezoelectric and magneto-electro-elastic structures. Last, applications for more complex magneto-electro-elastic systems are described.     

Effective mechanical properties of EM composite conductors: an analytical and finite element modeling approach

An analytical model and numerical approach to predict the effective mechanical properties of a composite conductor consisting of metallic core and insulation layers are presented in this paper. The analytical model was developed based on a two-step homogenizations and mechanics analysis for composite unit cell. The Step 1 homogenization derives the effective properties of the out-wrapped composite insulation layers. The Step 2 homogenization further smears the metallic core and the effective composite insulation layers to develop homogenized mechanical properties for composite conductor according to appropriate homogenization sequences. The procedure of using numerical approach and finite element method to determine the unit cell effective constants were also described and the results of the FEA prediction were presented. The analytical predictions were compared well to the numerical results for the nine material constants that characterize the effective mechanical properties of the composite conductor.     

A three-dimensional multilayer composite finite element for stress analysis of composite laminates

A three-dimensional multilayer composite finite element method has been developed based on a composite variational functional which takes three in-plane strains ε_x, ε_x, ε_xy and three transverse stresses σ_z, σ_yz, σ_xz as the basic variables. The continuity of the transverse stresses σ_z, σ_yz, σ_xz across the laminate thickness is assured a priori by introducing a partial stress field parameter α which is associated with the lower and upper surfaces of a lamina in a laminate. A method has been developed to form the partial stress field based on the assumed displacement field. With this method, a three dimensional (3-D) multilayer composite finite element is formulated for stress analysis of composite laminates. A numerical example is given, which shows some advantages of this composite element.     

Statically admissible finite element solution of the thin plate bending problem with a posteriori error estimation

Two triangular elements of class C⁰ developed on the basis of the principle of complementary work are applied in the static analysis of a thin plate. Some techniques to widen the versatility of the equilibrium approach for the finite element method are presented. Plates of various shapes subjected to diverse types of loading are considered. The results are compared with outcomes obtained by use of the displacement-based finite element method. By use of these two dual types of solutions, the error of the approximate solution is calculated. The lower and upper bounds for the strain energy are found.     

A finite element model for contact analysis of multiple Cosserat bodies

The objective of this paper is to develop a finite element model for multi-body contact analysis of Cosserat materials. Based on the parametric virtual work principle, a quadratic programming method is developed for finite element analysis of contact problems. The contact problem with friction between two Cosserat bodies is treated in the same way as in plastic analysis. The penalty factors, that are normally introduced into the algorithm for contact analysis, have a direct influence on accuracy of solution. There is no available rule for choosing a reasonable value of these factors for simulation of contact problems of Cosserat materials, and they are therefore cancelled through a special technique so that the numerical results can be of high accuracy. Compared with the conventional work on Cosserat elasticity, the newly developed model is on the contact analysis of the Cosserat materials and is seldom found in the existing literatures. Four examples are computed to illustrate the validity and importance of the model developed.