Three-field Formulation Research Articles

Preconditioned Uzawa Algorithm for the Velocity-Pressure-Stress Formulation of Viscoelastic Flow Problems

This paper is devoted to the development of efficient iterative methods for solving the systems of algebraic equations arising from a spectral element discretization of the equations governing the flow of an Oldroyd B fluid. The governing equations are written in terms of velocity, pressure and extra-stress, giving rise to the so-called three-field formulation. The convection terms are treated using an operator-integration-factor splitting method. The remaining terms in the system of equations constitute a generalized Stokes problem. This problem is formulated in terms of an Uzawa operator applied to the pressure. Efficient preconditioners are developed for inverting this operator that are independent of the numerical and physical parameters of the problem.

Three-field formulation of finite element analysis of compressible viscoplastic powders

Three-field formulation of finite element analysis of compressible viscoplastic powders

Multidisciplinary Simulation of the Maneuvering of an Aircraft

A computational methodology for the simulation of the transient aeroelastic response of an unrestrained and flexible aircraft during high-G maneuvers is presented. The key components of this methodology are: (a) a three-field formulation for coupled fluid/structure interaction problems; (b) a second-order time-accurate and geometrically conserva- tive flow solver for CFD computations on unstructured dynamic meshes; (c) a corotational finite element method for the solution of geometrically nonlinear and unrestrained structural dynamics problems; (d) a robust method for updat- ing an unrestrained and unstructured moving fluid mesh; and (e) a second-order time-accurate staggered algorithm for time-integrating the coupled fluid/structure semi-discrete equations of motion. This computational methodology is illus- trated with the simulation on a parallel processor of several three-dimensional high-G pullup maneuvers of the Langley Fighter in the transonic regime, using a detailed finite element aeroelastic model.

Finite element analysis of compressible viscoplasticity using a three-field formulation

Finite element analysis of compressible viscoplasticity using a three-field formulation

Error estimates for the three-field formulation with bubble stabilization

In this paper we prove convergence and error estimates for the so-called 3-field formulation using piecewise linear finite elements stabilized with boundary bubbles. Optimal error bounds are proved in $L^2$ and in the broken $H^1$ norm for the internal variable $u$, and in suitable weighted $L^2$ norms for the other variables $\lambda$ and $\psi$.

A stabilised large-strain elasto-plastic Q1-P0 method

The mean dilatation method effectively involves bi-linear displacements and a constant pressure and is often known as the Q1-P0 formulation. Its non-linear implementation was originally derived as a three-field formulation which included the volume ratio via the Jacobian, J, of the deformation gradient as an additional separate variable. However, the latter term was not directly required in the numerical implementation once J was assumed constant along with the pressure. This formulation will here be termed the non-linear Q1-P0 method. It is known to give good solutions for many practical large-strain elasto-plastic problems. However, for some problems, it has been shown to be prone to severe ‘hour-glassing’. With a view to remedying this situation, we here re-visit the three-field formulation and derive a modified form, which although variationally valid, is over-stiff in comparison to the original procedure (here simply called the Q1-P0 method). However, the concepts lead to a natural method for stabilising the Q1-P0 technique. The associated tangent stiffness matrix is symmetric. Copyright © 1999 John Wiley & Sons, Ltd.

Finite element analysis of compressible viscoplasticity using a three-field formulation: Application to metal powder hot compaction

In the present study, a finite element model has been formulated to simulate the hot forging stage in powder metallurgy manufacturing route. The compacted material is assumed to obey a purely viscoplastic compressible flow rule. A three-field formulation (velocity, volumetric strain rate and pressure) has been developed. The associated three-dimensional finite element discretization is detailed. In order to take advantage of an automatic remeshing procedure for linear tetrahedra, the compatible P1+/P1/P1 element is used (4-node element plus additional degrees of freedom and bubble interpolation for velocity). The complete model includes thermomechanical coupling and friction. The formulation is validated versus an analytic solution of uniaxial free compaction and applied to the hot forging of an automotive connecting rod preform.

A comparative study of magnetostatics using three-field and conventional vector potential formulations

The three-field finite element formulation for electromagnetics was proposed as an alternative to the existing vector potential formulations. In the formulation, three fields, that is (i) the vector potential, (ii) the divergence of the vector potential and (iii) a scalar variable which is adjoint to (ii), are interpolated independently and the imposition of the gauge condition can be relaxed. A comparative study of magnetostatic problems is presented which includes the calculations by conventional vector potential formulations using second order nodal elements. It is shown that the three-field formulation gives better results than conventional vector potential formulations with and without gauging.

On some mixed finite element methods for incompressible and nearly incompressible finite elasticity

We compare some mixed methods based on different variational formulations, namely a displacement-pressure formulation employed by de Borst and coworkers, the three-field formulation investigated by Simo and Taylor and a two-field formulation which is directly based on an energy functional. It emerges that all these yield the same discrete results if the stored energy function contains a volumetric contribution 1/2k(J−1)2 whereJ is the volume dilatation, i.e., the Jacobian determinant of the deformation, andk is the bulk modulus. The equivalence holds for arbitrary 3D and plane strain elements. In the numerical examples the mixed formulations are discretized by the quadrilateral Q1/P0 and Q2/P1 elements and the triangular Crouzeix-Raviart P2+/P1 element. We also compare with standard displacement elements and the enhanced strain Q1/E4 element.

Collocation in finite element analysis of constrained problems

The concept of three-field displacement finite element formulations of constrained problems is explored in this paper. The approach studied uses collocation within the element domain to enforce the constraints of the problem and resembles mixed formulations because in addition to the approximation of the displacement field, all other auxiliary deformation variables that enter the constraint conditions are interpolated independently. Plate-bending as described by the Reissner–Mindlin theory is used as a model problem for the development of the approach. Two triangular plate elements are formulated based on this methodology, and their performance is investigated using several patch tests. Numerical examples are also considered to study the element behavior under locking conditions (which occur with the classical displacement approach at the thin plate limit). The ability of the elements to overcome locking is established using mathematical arguments and practical examples. The influence of the position of collocation points on the computed results is evaluated through sensitivity studies, with the aim to identify the optimal set. Results are compared with those obtained from the exact solutions and the associated classical displacement model with selective reduced integration of the constrained energy terms. Key words: finite elements, collocation, discrete constraints, mixed methods, thick plates, bending of plates, shear deformations.

Basic features of the time finite element approach for dynamics

Very general weak forms may be developed for dynamic systems, the most general being analogous to a Hu-Washizu three-field formulation, thus paralleling well-established weak methods of solid mechanics. In this work two different formulations are developed: a pure displacement formulation and a two-field mixed formulation. With the objective of developing a thorough understanding of the peculiar features of finite elements in time, the relevant methodologies associated with this approach for dynamics are extensively discussed. After having laid the theoretical bases, the finite element approximation and the linearization of the resulting forms are developed, together with a method for the treatment of holonomic and nonholonomic constraints, thus widening the horizons of applicability over the vast world of multibody system dynamics. With the purpose of enlightening on the peculiar numerical behavior of the different approaches, simple but meaningful examples are illustrated. To this aim, significant parallels with elastostatics are emphasized.