A finite element formulation for transient incompressible viscous flows stabilized by local time-steps

Stabilized finite element formulations are well suited for convection dominated flows and for the solution of the incompressible Navier–Stokes equations in primitive variables. In this paper, we present a method where the structure of stabilization terms appear naturally from a least-squares minimization of the time-discretized momentum balance. Local time-steps, chosen according to the time-scales of convection–diffusion of momentum, play the role of stabilization parameters. Numerical solutions of incompressible viscous flows demonstrate the usefulness of the proposed stabilized formulation.     

Higher order stabilized finite element method for hyperelastic finite deformation

This paper presents a higher order stabilized finite element formulation for hyperelastic large deformation problems involving incompressible or nearly incompressible materials. A Lagrangian finite element formulation is presented where mesh dependent terms are added element-wise to enhance the stability of the mixed finite element formulation. A reconstruction method based on local projections is used to compute the higher order derivatives that arise in the stabilization terms, specifically derivatives of the stress tensor. Linearization of the weak form is derived to enable a Newton–Raphson solution procedure of the resulting non-linear equations. Numerical experiments using the stabilization method with equal order shape functions for the displacement and pressure fields in hyperelastic problems show that the stabilized method is effective for some non-linear finite deformation problems. Finally, conclusions are inferred and extensions of this work are discussed.     

Least-squares finite element formulation for shear-deformable shells

We present a least-squares based finite element formulation for the numerical analysis of shear-deformable shell structures. The variational problem is obtained by minimizing the least-squares functional, defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of Hilbert spaces. The use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise, i.e. stability requirements such as inf–sup conditions never arise. The proposed formulation retains the generalized displacements and stress resultants as independent variables and, in view of the nature of the variational setting upon which the finite element model is built, allows for equal-order interpolation. A p-type hierarchical basis is used to construct the discrete finite element model based on the least-squares formulation. Exponentially fast decay of the least-squares functional is verified for increasing order of the modal expansions. Several well established benchmark problems are solved to demonstrate the predictive capability of the least-squares based shell elements. Shell elements based on this formulation are shown to be effective in both membrane- and bending-dominated states.     

Stabilized finite element formulation for elastic–plastic finite deformations

This paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic–plastic solids, such as metals. An updated Lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation. This formulation circumvents the restriction on the displacement and pressure fields due to the Babuška–Brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit, typical in metal forming applications. Moreover, it facilitates the use of low order simplex elements (i.e. P1/P1), reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary. Linearization of the weak form is derived for implementation into a finite element code. Numerical experiments with P1/P1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis.     

A variational Germano approach for stabilized finite element methods

In this paper the recently introduced Variational Germano procedure is revisited. The procedure is explained using commutativity diagrams. A general Germano identity for all types of discretizations is derived. This relation is similar to the Variational Germano identity, but is not restricted to variational numerical methods. Based on the general Germano identity an alternative algorithm, in the context of stabilized methods, is proposed. This partitioned algorithm consists of distinct building blocks. Several options for these building blocks are presented and analyzed and their performance is tested using a stabilized finite element formulation for the convection–diffusion equation. Non-homogenous boundary conditions are shown to pose a serious problem for the dissipation method. This is not the case for the least-squares method although here the issue of basis dependence occurs. The latter can be circumvented by minimizing a dual-norm of the weak relation instead of the Euclidean norm of the discrete residual.     

A stabilized mixed finite element method for Darcy flow

We develop new stabilized mixed finite element methods for Darcy flow. Stability and an a priori error estimate in the “stability norm” are established. A wide variety of convergent finite elements present themselves, unlike the classical Galerkin formulation which requires highly specialized elements. An interesting feature of the formulation is that there are no mesh-dependent parameters. Numerical tests confirm the theoretical results.     

A unified approach for displacement,equilibrium and hybrid finite element models in elasto-plasticity

Finite element models for elasto-plastic incremental analysis are derived from a three-field variational principle. The Newton-Raphson method is applied to solve the nonlinear system of equations which is obtained from the stationarity condition of this principle. The iterative schemes are discussed in detail for pure displacement and for pure equilibrium models from which iterative schemes for hybrid models follow directly. In the displacement model, the compatibility of the strains and the plasticity criterium are satisfied during the whole iterative process, while the equilibrium of the stresses is restored only in the mean after convergence. In the equilibrium model, the plasticity criterium and the compatibility of the strains are verified in the mean during the iterative process; when convergence is achieved, the stresses are locally in equilibrium with the applied external loads. In both cases, a tangential stiffness matrix can be constructed, even for perfectly plastic materials and it allows one to obtain always very good convergence properties. Examples are shown for plane stress and axisymmetric cases.     

Stability of plates using a mixed finite element formulation

A rectangular plate finite element is developed according to a variational principle due to Prager The element is applied to plate stability analysis. The results obtained compare very favorably with results based on previous formulations.     

On one type of finite element simulation in dynamic elasticity

The aim of the present paper is to generalize the finite element approximation for numerical simulation of special types of problems in the dynamic theory of elasticity, described by the two-dimensional mixed boundary value problems, and to demonstrate the property of their numerical solution. Physically the problems describe propagation of elastic waves, generated by a harmonic line source, in a nonhomogeneous anisotropic media. The existence and unicity of the weak solution as well as of the finite element approximation is proved. Convergence of the method is proved for any regular family of triangulations.     

A stabilized finite element method for modeling of gas discharges

An efficient stabilized finite element method for modeling of gas discharge plasmas is represented which provides wiggle-free solutions without introducing much artificial diffusion. The stabilization is achieved by modifying the standard Galerkin test functions by means of a weighted quadratic term that results in a consistent Petrov-Galerkin formulation of the charge carriers in the plasma. Using the example of a glow discharge plasma in argon, it is shown that this efficient method provides more accurate results on the same spatial grid than the widely used finite difference approach proposed by Scharfetter-Gummel if the weighting factor is determined in dependence on the local Péclet number and the modified test functions are consistently applied to all terms of the governing equations.     

A finite element formulation for multilayered and thick plates

A bilinear isoparametric finite element concept is used for the numerical analysis of multilayered plates. The underlying theory used allows for transverse shear and normal strains in each layer, thus extending the analysis to very thick plates and laminates. To illustrate the versatility of the multilayered element, three examples are presented and the results are compared with available exact solutions.     

A Dorodnitsyn finite element formulation for turbulent boundary layers

The Dorodnitsyn boundary layer formulation is combined with a modified Galerkin finite element formulation and an implicit, non-iterative marching scheme to generate a computational algorithm that is both accurate and very economical. For four representative pressure gradient cases taken from the 1968 Stanford Turbulent Boundary Layer Conference the Dorodnitsyn finite element formulation is compared with a Dorodnitsyn spectral formulation and a representative finite difference package. All methods produce solutions of high accuracy but the Dorodnitsyn finite element formulation is about ten times more economical than the other methods.     

A finite element formulation for elastoplasticity based on a three-field variational equation

A three-field variational equation, which expresses the momentum balance equation, the plastic consistency condition, and the dilatational constitutive equation in a weak form, is proposed as a basis for finite element computations in hardening elastoplasticity. The finite element formulation includes algorithms for the integration of the elastoplastic rate constitutive equations which are similar to members of the “return mapping” family of algorithms employed in displacement formulations, except that the proposed algorithms are not required to explicitly satisfy the plastic consistency condition at the end of each time step. This condition is imposed globally by the inclusion of a variational equation that suitably constrains the solution. The plastic incompressibility constraint is also treated in an appropriate variational sense. Solution of the nonlinear finite element equations is obtained by use of Newton's method and details of the linearization of the variational equation are given. The formulation is developed for an associative von Mises plasticity model with general nonlinear isotropic and kinematic strain hardening. A number of numerical test examples are provided.     

On the role of mass operators in the group finite element formulation

The influence of the mass operators on the accuracy, economy and computational efficiency of the time-split group finite element formulation is investigated for the viscous flow over a backward-facing step. On a coarse grid it is found that mass operators must be retained adjacent to the computational boundaries to obtain the correct steady-state solution. The present time-split (or approximate factorisation) group finite element formulation is only 18% less economical than a fully lumped (equivalent to a finite difference) formulation. If mass lumping is introduced in the interior only, the solution is almost as accurate as if the full mass operators are retained, but there is little gain in economy in two dimensions. An operation-count estimate indicates that interior mass lumping would be about 40% more economical than retaining the mass operators in three dimensions.     

A finite element quasi-Eulerian method for three-dimensional fluid-structure interactions

A quasi-Eulerian approach is used in the development of a three-dimensional hydrodynamic finite element. With this approach the motion of the computing grid may be different from the motion of the material. Within each element the field variables are represented by trilinear interpolation functions and the pressure field is assumed to be constant. This leads to a set of simple relations for internal nodal forces that are easily coded and computationally efficient. Because the formulation is based upon a rate approach it is applicable to problems involving large displacements. The technique of degeneration is applied to the hexahedron to generate a pentahedral element. The use of the hour-glass dissipating nodal-forces for mesh stabilization is discussed. The procedure used to couple the fluid and structural domains of a problem is presented. The above method is applied to a three-dimensional, fluid-structure interaction problem in the area of reactor safety.     

On the application of a least squares residual-fitting finite element formulation to fluid flow problems

The connection between the least squares residual fitting finite element formulation and reduced integration is considered for two model problems, one of which requires an isoparametric mapping. For both problems a significant improvement in accuracy is achieved when compared with a conventional Galerkin finite element formulation. It is found that the two methods are equivalent for rectangular elements as long as an isoparametric mapping is not required. For triangular elements there is no direct link between the two methods and neither method produces any significant improvement over the use of exact numerical integration.     

A finite element formulation for beams with thin walled cross-sections

The paper presents a finite element model for calculation of stresses and deformations of beams with thin walled cross-sections. The beam model takes into account deformations due to shear. Warping is accounted for by a modified sector coordinate formulation. As interpolation functions between the seven degrees of freedom at each node are used the analytical solutions for the special case of a double symmetric cross-section. Therefore, depending on the external loading, each prismatic beam can in most cases be treated as a single element. The assembly of the beam elements to the global model is performed by use of transition matrices which assures compatibility between the elements in the sence of least squares.     

A finite element formulation for shells of negative Gaussian curvature

An expression for the strain energy of a shell of negative Gaussian curvature, including thickness shear deformations and without neglecting z/R in comparison with unity, is derived. Then a curved trapezoidal finite element formulation based on the principle of minimum potential energy is obtained. The shell element has eight nodes with 40 degrees of freedom and at each node there are three displacements and two rotations. The formulation is applicable for both thin and moderately thick shell analysis. The performance of this finite element is verified by applying it to some problems existing in the literature.