Mixed Hybrid Finite Element Research Articles

Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity?

Selected groundwater flow scenarios are used in a two‐way comparison between the mixed hybrid finite element method and the standard finite element method (also called the conforming finite element method). The simulations presented are performed in the bidimensional case with a triangular space discretization because of its practical interest for hydrogeologists. The basic idea of the mixed procedure is to approximate both the hydraulic potential and the velocity simultaneously and to satisfy an exact water balance for each element. By contrast, the conforming finite element method calculates the potential field everywhere and then calculates the velocity by differentiation of the potential. The conventional approach results in an elementwise constant velocity which can be subject to significant problems because of the normal component discontinuity of the velocity. The mixed hybrid finite element method provides velocities everywhere in the field, as well as potentials at the center of each element and each edge. Moreover, the normal component of the velocity field is continuous between adjacent elements. The results of the simulations are presented in the form of streamlines. To avoid the problem of velocity discontinuity, the method of Cordes and Kinzelbach (1992) is used; it allows the construction of a continuous velocity field from potentials obtained by the conforming finite element method. The comparison studies show that the mixed hybrid finite element is superior to the conforming method in terms of accuracy. It is also superior to the conforming method in terms of computational effort. The potential fields obtained by the mixed hybrid and the conforming finite element methods are the same.

Continuous groundwater velocity fields and path lines in linear, bilinear, and trilinear finite elements

Darcy velocity fields obtained from finite element solutions of heads in groundwater flow exhibit discontinuities at element boundaries, thus giving rise to inaccuracies in path line construction. Several methods have been developed to resolve those inaccuracies including global postprocessing of velocity fields, mixed hybrid finite element formulations, and stream function formulations. All of these techniques either lead to a considerable increase in the computational effort or are not general enough for all purposes. The point of view taken here is that standard finite element flow models yield exact water balances, which in the usual approach to path line computation are not fully used. By starting out from patches with flux‐conserving boundaries, a new postprocessing procedure is derived that allows construction of a continuous flux distribution over the whole model domain. In this procedure only the head gradients of adjacent elements are used, thus avoiding the solution of another global system of equations. Furthermore, application of a stream function interpolation inside each patch allows the fast computation of path lines without any time stepping. The path lines computed with the proposed procedure are compared to path lines computed by usual particle tracking and examples are presented showing that the new method is vastly superior to traditional particle tracking.

Mixed‐hybrid finite elements and streamline computation for the potential flow problem

AbstractAn important class of problems in mathematical physics involves equations of the form −∇ · (A∇ϕ) = f. In a variety of problems it is desirable to obtain an accurate approximation of the flow quantity u = −A∇ϕ. Such an accurate approximation can be determined by the mixed finite element method. In this article the lowest‐order mixed method is discussed in detail. The mixed finite element method results in a large system of linear equations with an indefinite coefficient matrix. This drawback can be circumvented by the hybridization technique, which leads to a symmetric positive‐definite system. This system can be solved efficiently by the preconditioned conjugate gradient method. After approximating u by the lowest‐order mixed finite element method, streamlines and residence times can be determined easily and accurately by computations at the element level.

Finite-element modelling of thermoelectric materials and devices

This paper deals with the modelling of materials and devices in which electric and thermal effects and their cross effects (thermoelectric effects) are important. A phenomenological model based on the thermodynamics of irreversible processes (TIP) and its mixed-hybrid finite-element implementation are presented. It is argued that TIP provides a unified approach to sensor modelling. Mixed-hybrid finite elements are used to resolve the thermoelectric interaction terms accurately. A numerical test example is given to demonstrate the feasibility of this approach.

A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems

The need for more precise simulation of the transport of pollutants by underground water has drawn the attention of hydrologists to a class of approximation techniques known by the generic name of Mixed Finite Element Methods. The basic idea here is to approximate simultaneously the pressure P and its gradient, or more generally a gradient related velocity field q in such a way that both P h and q h can be proved to converge, in adequate norms, to their continuous counterparts, and that the approximated velocity field q h retains one important feature of the exact velocity field q, namely that it is continuous on each element and has a continuous normal component when passing from one element to the other

A mixed hybrid finite element for three-dimensional elastic crack analysis

A new three-dimensional crack tip element is proposed, which is based on a mixed hybrid stress/displacement model. A truncated series expansion of eigenfunctions for the straight semi-infinite crack is deduced and assumed for the internal stress and displacement fields in the element. The basic approach of constructing these hybrid elements is outlined. Their good capability, efficiency and accuracy for analyzing three-dimensional elastic crack problems are demonstrated by first numerical examples.

New variational-asymptotic formulations for 3-D stress analyses of laminated composite shells with a circular hole

A new approach for three-dimensional stress analyses in composite cylindrical shells is presented. The method of composite expansions along with Hellinger-Reissner variational formulation is employed to derive the interior and edge layer problems for high order approximations. Classical assumptions have been justified and new approximations have been established. These formulations are directed especially towards, new high integrity mixed-hybrid finite element schemes. The expository examples chosen are of cross-ply and angle-ply laminated shells. The circumferential location of the delamination failure initiation, for angle-ply laminates containing a circular hole, is within a sector located symmetrically around the perpendicular direction to the applied load.

Free-form shell analysis by a mixed-hybrid finite element approach

A novel finite element scheme is proposed for the linear elasto-static analysis of free-form thinwalled shells. On the basis of a modified version of the variational theorem due to Hellinger and Reissner, a doubly curved triangular element is developed with six degrees-of-freedom (three translations plus three rotations) per nodal point. For each element, separate trial functions are introduced in order to approximate the displacements and the stress resultants. It is shown that by properly choosing these interpolants, an element is obtained which (i) is capable of exactly representing the rigid body and constant strain modes; (ii) does not possess any spurious kinematic modes; (iii) fulfils C1 interelement continuity and (iv) satisfies exactly the interior domain static equilibrium conditions. Two representative examples have been solved in order to illustrate the advantages of the proposed scheme over existing ones.

A pseudo-complementary finite element approach for the vibration analysis of thin plates in bending

The present paper is concerned with the numerical analysis of the steady-state and transient response of thin elastic plates. Based on a modification of the variational principle due to Hamilton wherein in contrast to the classical formulation not only displacements but also stress resultants represent independent (primal) variables, a new mixed hybrid finite element model is proposed. Introducing separate approximations for the displacement and stress field, stiffness and consistent mass matrix of a triangular plate element with three kinematic degrees of freedom per nodal point are obtained. The performance of the new element scheme is evaluated on the basis of several test examples representing a broad range of circumstances encountered in linear elastokinetic thin plate analysis. The obtained numerical results demonstrate that in terms of efficiency, reliability and accuracy the new element scheme competes most favorably with a series of well-established plate elements.

On the existence and stability conditions for mixed-hybrid finite element solutions based on Reissner’s variational principle

The extensions of Reissner’s two-field (stress and displacement) principle to the cases wherein the displacement field is discontinuous and/or the stress field results in unreciprocated tractions, at a finite number of surfaces (“interelement boundaries”) in a domain (as, for instance, when the domain is discretized into finite elements), is considered. The conditions for the existence, uniqueness, and stability of mixed-hybrid finite element solutions based on such discontinuous fields, are summarized. The reduction of these global conditions to local (“element”) level, and the attendant conditions on the ranks of element matrices, are discussed. Two examples of stable, invariant, least-order elements—a.four-node square planar element and an eight-node cubic element—are discussed in detail.

Finite element analysis of plate systems: 1. Hybrid—mixed finite elements

The analysis of plate systems (Figure 1) with mixed finite elements is limited to geometric smooth partial systems (Figure 1a) with continuous strains and stresses. This is due to the independent approximation of the parameters of the local generalized displacements (displacements, slopes) and generalized stresses (stress resultants at the cross‐section) in the nodes of the partial systems (Figure 1c). Owing to the simultaneous approximation, an additional static constraint of the generalized stresses, apart from the kinematic constraint of the displacements which is usual in the finite element methods, is needed at the assembly, so that discontinuities of the stresses resulting from the equilibrium conditions cannot be determined.

Stability analysis of structures via a new complementary energy method

A new procedure for the analyses of finite deformations and stability of structures, based on a complementary energy principle and an associated hybrid-mixed finite element method, is presented. In this procedure, the description of kinematics is based on the polar-decomposition of the displacement gradient into pure stretch and rigid rotation. The details of the procedure are illustrated through the problems of (i) post-buckling of a column, (ii) the elastica, and (iii) finite-displacements of a transversely loaded beam.

Mixed-hybrid finite element approximations of second-order elliptic boundary value problems: Part 2 — weak-hybrid methods

In this paper we present a new class of hybrid finite element methods for the numerical analysis of second-order elliptic boundary value problems. The methods are characterized by the use of particular solutions of the differential equation being solved, in contrast to conventional hybrid methods in which polynomial approximations are used. As a model problem the Dirichlet problem for Laplace's equation is studied. A priori error estimates are derived and the results of numerical experiments are presented.

Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems

Abstract Mixed-hybrid finite element approximations are described for second-order elliptic boundary-value problems in which independent approximations are used for the solution and its gradient on the interior of an element and the trace of the gradients on the boundary of the element. A priori error estimates are derived with conditions for convergence. Several other finite element models are also obtained as special cases. Some numerical examples are included.