Mixed Finite Element Approach Research Articles

A mixed finite difference and boundary element approach to one-dimensional Burgers' equation

This paper presents a mixed method for the numerical solution of the one-dimensional Burgers' equation. This method uses mixed boundary elements in association with finite differences. Two standard problems are used to validated the algorithm. Comparisons are made with some of the existing numerical schemes and analytical solutions. The proposed method performs well.

The Andra Couplex 1 Test Case: Comparisons Between Finite-Element, Mixed Hybrid Finite Element and Finite Volume Element Discretizations

We present here results for the Andra Couplex 1 test case, obtained with the code Cast3m. This code is developped at the CEA (Commissariat р l'щnergie atomique) and is used mainly to solve problems of solid mechanics, fluid mechanics and heat transfers. Different types of discretization are available, among them finite element, finite volume and mixed hybrid finite element method. Cast3m is also a componant of the platteform Alliances (co-developped by Andra, CEA), which will be used by Andra for the safety calculation of an underground waste disposal in year 2004. We solve the Darcy equation for the water flow and a convection–diffusion transport equation for the Iodine 129 which escapes from a repository cave into the water. The water flow is calculated with a MHFE discretization. It is shown that this method provides sharp results even on relatively coarse grids. The convection–diffusion transport equation is discretized with FE (Finite Element), MHFE (Mixed Hybrid Finite Element) and FV (Finite Volume) methods. In our comparison, we point out the differences of these methods in term of accuracy, respect of the maximum principle and calculations cost. Neither the finite element nor the mixed hybrid finite element approach respects the maximum principle. This results in the presence of negative concentrations near the repository cave, whereas FV calculations respect the monotonicity. We show that mass lumping techniques suppress this problem but with strong restrictions on the grid. FE and MHFE approaches are more accurate than FV for the diffusion equation, but the overall results are equivalent since the advective terms are dominant in the far field and are discretized with centered schemes. We conclude by studying the influence of the grid: a very fine grid near the repository solves almost all the problems of monotonicity, without employing mass lumping techniques. We also observed a very important increase of the accuracy on a structured grid made up of rectangles.

Simulation of wear in gears with flank interference—a mixed FE and analytical approach

Gears in precision mechanisms, such as industrial robots, are often designed to have no backlash. Deviations from the ideal gear geometries and unfavourable deformations during operation may cause interference between interacting tooth flanks, resulting in high and strongly varying friction. Mild wear of the tooth flanks may improve such conditions. Therefore, this theoretical study has been conducted of wear in spur gears with interference. A mixed finite element (FE) and analytical approach is used. The FE method is used to determine contact loads between the interacting gear teeth. The main drawback with FE analyses of this type of problem is normally the computation time needed. Therefore, a novel FE meshing method is used, giving a dense FE mesh in the contact regions and a coarse mesh in the rest of the teeth. Based on the FE determined loads between the interacting teeth, the contact pressures and the contact widths are then easily determined using analytical expressions based on Hertz theory. The wear of a point on a tooth flank is determined by integrating the product of sliding distance and contact pressure during the time it is in contact with its mating flank. The results show that the wear of the gear tooth flanks may eliminate the interference and consequently the high and strongly varying friction will decrease dramatically. However, the transmission error will change, since the gears will not preserve their accurate involute profiles.

Active control of axisymmetric shells with piezoelectric layers: a mixed laminated theory with a high order displacement field

This paper presents the development of a semi-analytical axisymmetric shell finite element model with embedded and/or surface bonded piezoelectric ring actuators and/or sensors for active damping vibration control of the structure. A mixed finite element approach is used, which combines the equivalent single-layer higher order shear deformation theory, to represent the mechanical behavior with a layerwise discretization in the thickness direction to represent the distribution of the electric potential of each piezoelectric layer of the frusta conical finite element. To achieve a mechanism of active control of the dynamic response of the structure a feedback control algorithm is used coupling the sensor and active piezoelectric layers. Two illustrative examples are presented and discussed.

An expanded mixed finite element approach via a dual–dual formulation and the minimum residual method

We apply an expanded mixed finite element method, which introduces the gradient as a third explicit unknown, to solve a linear second-order elliptic equation in divergence form. Instead of using the standard dual form, we show that the corresponding variational formulation can be written as a dual–dual operator equation. We establish existence and uniqueness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart–Thomas elements. In addition, we show that the corresponding dual–dual linear system can be efficiently solved by a preconditioned minimum residual method. Some numerical results, illustrating this fact and the rate of convergence of the mixed finite element method, are also provided.

A primal mixed formulation for exterior transmission problems in ${\bf R}^2$

We combine a primal mixed finite element approach with a Dirichlet-to-Neumann mapping (arising from the boundary integral equation method) to study the weak solvability and Galerkin approximations of a class of linear exterior transmission problems in potential theory. Our results are mainly based on the Babuska-Brezzi theory for variational problems with constraints. We establish the uniqueness of solution for the continuous and discrete formulations, and show that finite element subspac es of Lagrange type satisfy the discrete compatibility conditions. In addition, we provide the error analysis, including polygonal approximations of the domain, and prove strong convergence of the Galerkin solutions. Moreover, under additional regularity assumptions on the solution of the continuous formulation, we obtain the asymptotic rate of convergence O(h).

A numerical investigation of the solution of a class of fourth–order eigenvalue problems

This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corners on domain boundaries. Recent computational results of Bjorstad & Tjostheim, using a highly accurate spectral Legendre–Galerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with the numerical method. We demonstrate that similar accuracy is also achieved using an unstructured finite–element solver which may be applied to problems on domains with arbitrary geometries. A number of results obtained from this mixed finite–element approach are then presented for a variety of domains. These include a family of circular sector regions, for which the oscillatory behaviour is studied as a function of the internal angle, and another family of (symmetric and non–convex) domains, for which the parity of the least eigenfunction is investigated. The paper not only verifies existing asymptotic theory, but also allows us to make a new conjecture concerning the eigenfunctions of the biharmonic operator.

Behaviour of Lagrangian triangular mixed fluid finite elements

The behaviour of mixed fluid finite elements, formulated based on the Lagrangian frame of reference, is investigated to understand the effects of locking due to incompressibility and irrotational constraints. For this purpose, both linear and quadratic mixed triangular fluid elements are formulated. It is found that there exists a close relationship between the penalty finite element approach that uses reduced/selective numerical integration to alleviate locking, and the mixed finite element approach. That is, performing reduced/ selective integration in the penalty approach amounts to reducing the order of pressure interpolation in the mixed finite element approach for obtaining similar results. A number of numerical experiments are performed to determine the optimum degree of interpolation of both the mean pressure and the rotational pressure in order that the twin constraints are satisfied exactly. For this purpose, the benchmark solution of the rigid rectangular tank is used. It is found that, irrespective of the degree of mean and the rotational pressure interpolation, the linear triangle mesh, with or without central bubble function (incompatible mode), locks when both the constraints are enforced simultaneously. However, for quadratic triangle, linear interpolation of the mean pressure and constant rotational pressure ensures exact satisfaction of the constraints and the mesh does not lock. Based on the results obtained from the numerical experiments, a number of important conclusions are arrived at.

Quantum interface conditions for quasi‐Fermi levels in an abrupt heterojunction

We present a semi‐quantum model including tunneling effects across an abrupt heterojunction. The discontinuity of the effective masses and the energy bands are considered. The quantum transmission conditions for the quasi‐Fermi levels are obtained using WKB approximation. We use mixed finite element approach and a two dimensional mesh which is double‐valued for quasi‐Fermi levels at a heterojunction. A GaAs/GaAIAs heterojunction diode is then simulated using both drift‐diffusion and semi‐quantum model by varying doping density at low temperature.

Development of semianalytical axisymmetric shell models with embedded sensors and actuators

This paper presents the development of two semianalytical axisymmetric shell finite element models, which have the possibility of having embedded and/or surface-bonded piezoelectric ring actuators and/or sensors. A mixed finite element approach is used, which combines the equivalent single-layer higher-order shear deformation theory, to represent the mechanical behavior with a layerwise discretization in the thickness direction to represent the distribution of the electrical potential of each piezoelectric layer of the frusta conical finite element. The electrical potential function is represented through a layerwise discretization in the thickness direction and can be assumed linear or quadratic with two or six electrical potential ring nodes per piezoelectric layer. The displacement field and the electrical potential are expanded by Fourier series in the circumferential direction, considering symmetric and anti-symmetric terms. Several examples are presented and discussed to illustrate the accuracy and capabilities of both models.

Primal mixed formulations for the coupling of FEM and BEM. Part I: linear problems

We apply the boundary integral equation method and a primal mixed finite element approach to study the weak solvability and Galerkin approximations of linear interior transmission problems arising in potential theory and elastostatics. The existence and uniqueness of solution of the resulting weak formulations and of the associated discrete schemes are derived by using the classical theory for variational problems with constraints. Suitable finite element subspaces of Lagrange type satisfying the compatibility conditions are utilized for defining the Galerkin scheme. The error analysis and corresponding rates of convergence are also provided.

Short Communication: A note on optimal hybrid V-cycle multilevel algorithms for mixed finite element systems with penalty term

In this paper, we investigate some cost-effective hybrid V-cycle multilevel algorithms for the discrete systems that arise when a mixed finite element approach is used to solve certain second-order elliptic boundary value problems. By introducing a small penalty parameter, the perturbed indefinite system can be reduced to a symmetric positive definite system involving the unknown flux alone. We study the numerical behaviour of some hybrid V-cycle multilevel algorithms with optimal computational complexity based on the hierarchical spatial decomposition approach proposed by Cai, Goldstein and Pasciaks for the reduced system. © 1997 by John Wiley & Sons, Ltd.

Short Communication: A note on optimal hybridV-cycle multilevel algorithms for mixed finite element systems with penalty term

In this paper, we investigate some cost-effective hybrid V-cycle multilevel algorithms for the discrete systems that arise when a mixed finite element approach is used to solve certain second-order elliptic boundary value problems. By introducing a small penalty parameter, the perturbed indefinite system can be reduced to a symmetric positive definite system involving the unknown flux alone. We study the numerical behaviour of some hybrid V-cycle multilevel algorithms with optimal computational complexity based on the hierarchical spatial decomposition approach proposed by Cai, Goldstein and Pasciaks for the reduced system. © 1997 by John Wiley & Sons, Ltd.

Combination of mixed finite element and Dirichlet-to-Neumann methods in nonlinear plane elasticity

We study the solvability and Galerkin approximation of an exterior hyperelastic interface problem arising in plane elasticity. The weak formulation is obtained from an appropriate combination of a mixed finite element approach with a Dirichlet-to-Neumann method. The derivation of our results is based on some tools from nonlinear functional analysis and the Babuska-Brezzi theory for variational problems with constraints.

Mixed finite element approach for shape optimal design of large displacement contact problems

The paper is concerned with the shape optimal design of a large displacement contact problem. Contact with the prescribed friction is assumed. The optimal design problem consists of finding the domain boundary in the contact zone to minimize the contact stress. The mixed finite element method is used to discretize this optimization problem. The necessary optimality condition is formulated. The augmented Lagrangian approach is used to solve numerically the state system. The conjugate gradient method with projection is used as an optimization method. Numerical examples are provided.

A discrete formulation for elastic solids with damaging interfaces

An elastic solid with embedded interfaces, loci of possible displacement discontinuities, is considered here. Decohesion and quasi-brittle fracture processes are simulated by making use of softening interface laws which relate tractions to displacements jumps. A discrete formulation in terms of interface variables only, is obtained. The space discretization is carried out by means of a mixed finite element approach in which all interface variables are modelled. Study of uniqueness of the rate problem formulated in terms of interface variables and of stability of the equilibrium states is presented. Some examples are shown in order to clarify the formulation.

Arclength continuation methods and applications to 2D drift‐diffusion semiconductor equations

Shows how a strongly non‐linear semiconductor equation can be solved via homotopy deformation methods combined with the arclength continuation procedures. The fundamental goal of these methods is to overcome the instabilities or the failure of the classical Newton‐Raphson’s schemes which appear when the non‐linearity is strong or near limit or bifurcation points. The system, in its artificial transient form, is discretized by the non‐linear implicit scheme with local time steps. Uses the mixed finite element (MFE) approach. Presents numerical results, in two dimension, for a realistic device: an Abrupt Heterojunction Bipolar Transistor working as an amplifier.

Analysis of beam contact problems via optimal control theory

A class of large-deflection contact problems for beams is treated within the framework of optimal control theory, using an automated software system developed for variational optimization (e.g., trajectory optimization). Analysis of both Euler-Bernoulli and Timoshenko-type models is presented. Essentially exact numerical solutions are obtained by shooting, and the accuracy of a very efficient mixed finite element approach on which the automated software is based is demonstrated. Advantages of posing the problem within the optimal control framework are discussed. These include 1) being able to establish from the Euler-Bernoulli equations the existence of a physically inappropriate touch-point solution and 2) the ease by which an integral of the governing equations can be found to independently check the accuracy of a numerical solution

Effect of preform shape on buckling of quasi-isotropic thermoplastic composite laminates during sheet forming

Abstract An important issue in sheet forming of continuous fibre-reinforced thermoplastic composites is the tendency of the laminates to buckle out of plane under rapid forming conditions. This paper outlines a method used to predict the stress patterns responsible for buckling, and presents experimental results that correspond with the predictions. A finite element formulation for ideal fibre-reinforced Newtonian fluids, featuring the twin kinematic constraints of material incompressibility and fibre inextensibility, is used. A mixed penalty finite element approach is adopted, with independent interpolation of tension and velocity solution fields. An analysis model consistent with an assumption of plane stress is used. For multi-ply lay-ups, each ply is analysed individually, and average stress predictions for the laminate are obtained on this basis. A detailed comparison between numerical stress predictions and experimental buckling patterns is presented for central indentation of circular unidirectional, cross-ply and quasi-isotropic preforms. Parameters influencing the magnitude and location of peak tangential stresses include tangential fibre lengths and diaphragm/composite viscosity ratios. The effect of sheet width and shape on the instability patterns is investigated for quasi-isotropic laminates of different shapes, using both numerical and experimental techniques.

Multilevel Iteration for Mixed Finite Element Systems with Penalty

In this paper, the authors consider the solution of the discrete systems that arises when a mixed finite element approach is used to approximate the solution of second-order elliptic boundary value problems. By the introduction of a penalty parameter, these equations can be approximated by the solution of a symmetric and positive definite penalty system on the velocity subspace. Iterative procedures are developed and analyzed for this penalty system based on the hierarchical basis approach as well as on the standard multigrid approach. Finally, numerical experiments are presented that illustrate the convergence behavior suggested by the theory.