A locking-free stabilized mixed finite element method for linear elasticity: the high order case

In this paper, we propose a locking-free stabilized mixed finite element method for the linear elasticity problem, which employs a jump penalty term for the displacement approximation. The continuous piecewise k-order polynomial space is used for the stress and the discontinuous piecewise \((k-1)\)-order polynomial space for the displacement, where we require that \(k\ge 3\) in the two dimensions and \(k\ge 4\) in the three dimensions. The method is proved to be stable and k-order convergent for the stress in \(H(\mathrm {div})\)-norm and for the displacement in \(L^2\)-norm. Further, the convergence does not deteriorate in the nearly incompressible or incompressible case. Finally, the numerical results are presented to illustrate the optimal convergence of the stabilized mixed method.     

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 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.     

The analysis of incompressible hyperelastic bodies by the finite element method

A method is developed for the finite element analysis of problems involving incompressible hyperelastic bodies; the constitutive relation is based on a class of strain-energy functions due to Ogden [4], which involve sums of real powers of principal stretches. Incremental equilibrium equations are derived from a rate form of the principle of virtual work and an additional set of equations which express the condition of incompressibility in an average manner, is appended to the equilibrium equations. Examples of solutions are given and compared either with closed-form solutions or with numerical solutions found using conventional approaches.     

Higher order meshless schemes applied to the finite element method in elliptic problems

This paper presents selected approximation techniques, typical for the meshless finite difference method (MFDM), although applied to the finite element method (FEM). Finite elements with standard or hierarchical shape functions are coupled with higher order meshless schemes, based upon the correction terms of a simple difference operator. Those terms consist of higher order derivatives, which are evaluated by means of the appropriate formulas composition as well as a numerical solution, which corresponds to the primary interpolation order, assigned to element shape functions. Correction terms modify the right-hand sides of algebraic FE equations only, yielding an iterative procedure. Therefore, neither re-generation of the stiffness matrix nor introduction of any additional nodes and/or degrees of freedom is required. Such improved FE-MFD solution approach allows for the optimal application of advantages of both methods, for instance, a high accuracy of the nodal FE solution and a derivatives’ super-convergence phenomenon at arbitrary domain points, typical for the meshless FDM. Existing and proposed higher order techniques, applied in the FEM, are compared with each other in terms of the solution accuracy, algorithm efficiency and computational complexity.In order to examine the considered algorithms, numerical results of several two-dimensional benchmark elliptic problems are presented. Both the accuracy of a solution and the solution’s derivatives as well as their convergence rates, evaluated on irregular and structured meshes as well as arbitrarily irregular adaptive clouds of nodes, are taken into account.     

A new local stabilized nonconforming finite element method for the Stokes equations

In this paper, we propose and study a new local stabilized nonconforming finite method based on two local Gauss integrations for the two-dimensional Stokes equations. The nonconforming method uses the lowest equal-order pair of mixed finite elements (i.e., NCP ₁–P ₁). After a stability condition is shown for this stabilized method, its optimal-order error estimates are obtained. In addition, numerical experiments to confirm the theoretical results are presented. Compared with some classical, closely related mixed finite element pairs, the results of the present NCP ₁–P ₁ mixed finite element pair show its better performance than others.     

Higher order dual Lagrange multiplier spaces¶for mortar finite element discretizations

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements. In particular, we focus on dual Lagrange multiplier spaces. These non-standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We construct locally supported and continuous dual basis functions for quadratic finite elements starting from the discontinuous quadratic dual basis functions for the Lagrange multiplier space. In particular, we compare different dual Lagrange multiplier spaces and piecewise linear and quadratic finite elements. The optimality of the associated mortar method is shown. Numerical results illustrate the performance of our approach. Received: July 2002 / Accepted: November 2002     

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 high order isoparametric finite element method for the computation of waveguide modes

We have studied the approximation of optical waveguide eigenvalues by a high order isoparametric vector finite element method. Isoparametric mappings are used for the approximation of domains with curved boundaries or curved material interfaces. Eigenvalue convergence for curved elements is investigated. Numerical results verify the predicted order of convergence and show the remarkable accuracy of the method.     

Direct numerical simulation of turbulent channel flows using a stabilized finite element method

Direct numerical simulations (DNS) of incompressible turbulent channel flows at Re_τ = 180 and 395 (i.e., Reynolds number, based on the friction velocity and channel half-width) were performed using a stabilized finite element method (FEM). These simulations have been motivated by the fact that the use of stabilized finite element methods for DNS and LES is fairly recent and thus the question of how accurately these methods capture the wide range of scales in a turbulent flow remains open. To help address this question, we present converged results of turbulent channel flows under statistical equilibrium in terms of mean velocity, mean shear stresses, root mean square velocity fluctuations, autocorrelation coefficients, one-dimensional energy spectra and balances of the transport equation for turbulent kinetic energy. These results are consistent with previously published DNS results based on a pseudo-spectral method, thereby demonstrating the accuracy of the stabilized FEM for turbulence simulations.     

On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity

In the case of linear elasticity, a direct connection between the concept of reduced integration with hourglass stabilization and a mixed method can usually be established. In the non-linear case, this is in general not possible. To overcome this difficulty we suggest in this paper a new concept based on a Taylor expansion of the constitutively dependent quantities with respect to the centre of the element. The push-forward of the second (linear) term of the Taylor series for the first Piola–Kirchhoff stress tensor to the current configuration determines the so-called hourglass stabilization part of the residual force vector. Due to the fact that the element uses only one Gauss point and the hourglass stabilization part is computed by means of a simple functional evaluation, the present element technology is very efficient from the computational point of view.In contrast to the 2D case the computation of the Jacobi determinant only in the centre of the 3D element does not yield the correct volume, if the element shape deviates from being a parallelipiped. It is shown in the paper that the error becomes negligibly small for a relatively coarse discretization. The formulation is free of volumetric locking and can compete with shell formulations up to an aspect ratio of about hundred. For bending-dominated problems, at least two elements over the thickness are needed in order to compute the onset of plastification correctly. The element behaves very robustly in finite elasticity and inelasticity, also when large element distortions occur.     

Computation of incompressible bubble dynamics with a stabilized finite element level set method

This paper presents a stabilized finite element method for the three dimensional computation of incompressible bubble dynamics using a level set method. The interface between the two phases is resolved using the level set approach developed by Sethian [Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999], Sussman et al. [J. Comput. Phys. 114 (1994) 146], and Sussman et al. [J. Comput. Phys. 148 (1999) 81–124]. In this approach the interface is represented as a zero level set of a smooth function. The streamline-upwind/Petrov–Galerkin method was used to discretize the governing flow and level set equations. The continuum surface force (CSF) model proposed by Brackbill et al. [J. Comput. Phys. 100 (1992) 335–354] was applied in order to account for surface tension effects. To restrict the interface from moving while re-distancing, an improved re-distancing scheme proposed in the finite difference context [J. Comput. Phys. 148 (1999) 81–124] is adapted for finite element discretization. This enables us to accurately compute the flows with large density and viscosity differences, as well as surface tension. The capability of the resultant algorithm is demonstrated with two and three dimensional numerical examples of a single bubble rising through a quiescent liquid, and two bubble coalescence.     

不可压缩流两重稳定有限体积算法应用研究

通过将局部高斯积分稳定化方法和两重网格算法思想紧密结合,提出了粘性不可压缩流体的两重稳定有限体积算法。将该算法的三种迭代格式进行了效率的分析比较。理论分析和数值实验发现：当粗、细网格尺度比例选择适当时,两重算法与传统算法具有相同精度解的同时,效率大大提高;对不同格式的两重有限体积算法进行比较分析发现：Simple格式计算效率最高,Picard格式次之,Newton格式较低。     

A finite element method for the free vibration of plates allowing for transverse shear deformation

An isoparametric quadrilateral plate bending element is introduced and its use for the free vibration analysis of both thick and thin plates is examined. Plates of rectangular planform and of orthotropic materials are analysed and excellent results are obtained. The element performance is assessed by comparison with well established analytical and numerical solutions based on Mindlin's thick plate theory, three dimensional elasticity solutions and solutions based on thin plate theory. The ease with which the element may be implemented is stressed. The use of an eigenvalue economiser which produces considerable economy in the computer solution is demonstrated. Various mass lumping schemes and numerical integration rules used in the construction of the element mass matrix are also examined.     

Pressure projection stabilized finite element method for Navier–Stokes equations with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Navier–Stokes equation with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. The H ¹ and L ² error estimates for the velocity and the L ² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

OptimalL ∞-estimates for a mixed finite element method for second order elliptic and parabolic problems

Summary A mixed finite element method for second order problems is considered. OptimalL _∞-error estimates for the elliptic as well as for the corresponding parabolic problem are derived.     

Regularity results for discrete solutions of second order elliptic problems in the finite element method

We discuss a discrete version of the De Giorgi-Nash-Moser regularity theory for solutions of elliptic second-oder equations. Working under a fellowship of Consejo Nacional de Investigationes Cientificas y Tecnicas, Argentina Partially supported by National Science Foundation Grant MCS7915171     

Interior and superconvergence estimates for a primal hybrid finite element method for second order elliptic problems

Interior and superconvergence estimates are inevestigated for the primal hybrid-finite-element method proposed by Raviart and Thomas for the Robin problem. Dedicated to Professor S. Faedo on his 70th Birthday