The convergence of a mixed finite element method for steady convective incompressible viscous flow

Some results on the convergence of the assumed deviatoric stress-pressure-velocity mixed finite element method for steady, convective, incompressible, viscous flow are given. An abstract error estimate is proved, which shows that the same LBB conditions for hybrid finite element method for Stokes flow are also applicable to the present method. An unusual term appears in the estimate, the rate of convergence for this term is examined. To make our idea clear, the same finite element method is applied to single elliptic equations first.This work was supported by the Science Foundation of Academia Sinica, No. (84)-103     

A boundary element method for steady incompressible thermoviscous flow

A boundary element formulation is presented for moderate Reynolds number, steady, incompressible, thermoviscous flows. The governing integral equations are written exclusively in terms of velocities and temperatures, thus eliminating the need for the computation of any gradients. Furthermore, with the introduction of reference velocities and temperatures, volume modelling can often be confined to only a small portion of the problem domain, typically near obstacles or walls. The numerical implementation includes higher order elements, adaptive integration and multiregion capability. Both the integral formulation and implementation are discussed in detail. Several examples illustrate the high level of accuracy that is obtainable with the current method.     

A semi-implicit stabilized particle Galerkin method for incompressible free surface flow simulations

A new particle Galerkin method is introduced to solve the Naiver-Stokes equations in a Lagrangian fashion. The present method aims to suppress key numerical instabilities observed in the strong form Lagrangian particle methods such as smoothed particle hydrodynamics (SPH), incompressible SPH, and moving particle semi-implicit for incompressible free surface flow simulations. It is well-known that strong form Lagrangian particle methods usually rely on ad hoc particle stabilization techniques based on particle shifting, artificial viscosity, or density-invariant condition due to some formulation inconsistency issues. In the present method, we introduce a momentum-consistent velocity smoothing algorithm which is used to combine with the second-order rotational incremental pressure-correction scheme to stabilize the pressure field as well as to enforce the consistency of Neumann boundary condition. To further impose slip-free or nonslip boundary conditions for the fluid flow, a penalty method which is free of ghost or dummy particles is developed. Finally, a particle insertion-deletion adaptive scheme is proposed when the violent fluid flow is considered. Four numerical examples are studied to validate the accuracy and stability of the present method.     

On stable equal-order finite element formulations for incompressible flow problems

In this work we discuss stable equal-order finite element formulations for incompressible flow problems based on Petrov–Galerkin methods, constructed by adding to the classical Galerkin formulation least-squares of the governing equations. Continuous and discontinuous pressure interpolations are considered. Numerical results are presented reinforcing the numerical analysis.     

An improved hybrid boundary node method for solving steady fluid flow problems

This paper describes the application of an improved hybrid boundary node method (hybrid BNM) for solving steady fluid flow problems. The hybrid BNM is a boundary type meshless method, which combined the moving least squares (MLS) approximation and the modified variational principle. It only requires nodes constructed on the boundary of the domain, and does not require any ‘mesh’ neither for the interpolation of variables nor for the integration. As the variables inside the domain are interpolated by the fundamental solutions, the accuracy of the hybrid BNM is rather high. However, shape functions for the classical MLS approximation lack the delta function property. Thus in this method, the boundary condition cannot be enforced easily and directly, and its computational cost is high for the inevitable transformation strategy of boundary condition. In the method we proposed, a regularized weight function is adopted, which leads to the MLS shape functions fulfilling the interpolation condition exactly, which enables a direct application of essential boundary conditions without additional numerical effort. The improved hybrid BNM has successfully implemented in solving steady fluid flow problems. The numerical examples show the excellent characteristics of this method, and the computation results obtained by this method are in a well agreement with the analytical solutions, which indicate that the method we introduced in this paper can be implemented to other problems.     

A new particle method for simulation of incompressible free surface flow problems

A new Lagrangian particle method called the consistent particle method (CPM), which solves the Navier–Stokes equations in a semi‐implicit time stepping scheme, is proposed in this paper. Instead of using kernel function as in some particle methods, partial differential operators are approximated in a way consistent with Taylor series expansion. A boundary particle recognition method is applied to help define the changing liquid domain. The incompressibility condition of free surface particles is enforced by an adjustment scheme. With these improvements, the CPM is shown to be robust and accurate in long time simulation of free surface flow particularly for smooth pressure solution. Two types of free surface flow problems are presented to verify the CPM, that is, two‐dimensional dam break and liquid sloshing in a rectangular tank. In the dam break example, the CPM solutions of pressure and wave elevation are in good agreement with published experimental results. In addition, an experimental study of water sloshing in tank on a shake table was conducted to verify the CPM solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

A spectral stochastic element free Galerkin method for the problems with random material parameter

This paper presents a spectral stochastic element free Galerkin method (SSEFGM) for the problems involving a random material property. The random material property and resulting system response quantity are represented by a probabilistic spectral expansion techniques (Karhunen–Loeve expansion and Polynomical Chaos series, respectively) and implemented into the element free Galerkin (EFG) analysis. Numerical solutions in 1D linear elastic problem with random elastic modulus are introduced, and compared with those of Monte Carlo simulation (MCS) so as to provide the validation of the proposed approach. The present SSEFGM approach can produce a probabilistic density distribution as well as a first‐ and second‐order statistical moments (mean and variance) of response quantity by a single calculation, which is distinguished from an iterative MCS. Moreover, the method is based on an element free analysis so that there is no need of nodal connectivities, which usually require more time and labourious task than main calculations. Thus the proposed SSEFGM approach can provide an alternative analysis tool for the problems contains a stochastic material property, and demands complex mesh structures. Copyright © 2004 John Wiley & Sons, Ltd.     

Locking in the incompressible limit for the element‐free Galerkin method

Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element‐free Galerkin method. The modal analysis developed here shows that the number of non‐physical locking modes is independent of the dilation parameter (support of the interpolation functions). Thus increasing the dilation parameter does not suppress locking. Nevertheless, an increase in the dilation parameter does reduce the energy associated with the non‐physical locking modes; thus, in part, it alleviates the locking phenomena. This is shown for linear and quadratic orders of consistency. Moreover, the biquadratic order of consistency, as in finite elements, improves the locking behaviour. Although more locking modes are present in the element‐free Galerkin method with quadratic consistency than with standard biquadratic finite elements. Finally, numerical examples are shown to validate the modal analysis. In particular, the conclusions of the modal analysis are also confirmed in an elastoplastic example. Copyright © 2001 John Wiley & Sons, Ltd.     

Volumetric locking in the element free Galerkin method

A new formulation of the Element Free Galerkin (EFG) method is developed for the modelling of incompressible materials. Beginning with a mixed variational principle, a selective reduced integration procedure is developed by implementing nodal quadrature. Numerical examples are provided which compare the performance of the proposed technique to the standard EFG approximation. These studies illustrate the capability of the new formulation to eliminate volumetric locking. For the standard method, however, the degree of volumetric locking is shown to be a function of the local support sizes (domains of influence) of the EFG shape functions. Copyright © 1999 John Wiley & Sons, Ltd.     

A boundary element method without internal cells for solving viscous flow problems

In this paper, a new boundary element method without internal cells is presented for solving viscous flow problems, based on the radial integration method (RIM) which can transform any domain integrals into boundary integrals. Due to the presence of body forces, pressure term and the non-linearity of the convective terms in Navier–Stokes equations, some domain integrals appear in the derived velocity and pressure boundary-domain integral equations. The body forces induced domain integrals are directly transformed into equivalent boundary integrals using RIM. For other domain integrals including unknown quantities (velocity product and pressure), the transformation to the boundary is accomplished by approximating the unknown quantities with the compactly supported fourth-order spline radial basis functions combined with polynomials in global coordinates. Two numerical examples are given to demonstrate the validity and effectiveness of the proposed method.     

eXtended hybridizable discontinuous Galerkin for incompressible flow problems with unfitted meshes and interfaces

The eXtended hybridizable discontinuous Galerkin (X-HDG) method is developed for the solution of Stokes problems with void or material interfaces. X-HDG is a novel method that combines the hybridizable discontinuous Galerkin (HDG) method with an eXtended finite element strategy, resulting in a high-order, unfitted, superconvergent method, with an explicit definition of the interface geometry by means of a level-set function. For elements not cut by the interface, the standard HDG formulation is applied, whereas a modified weak form for the local problem is proposed for cut elements. Heaviside enrichment is considered on cut faces and in cut elements in the case of bimaterial problems. Two-dimensional numerical examples demonstrate that the applicability, accuracy, and superconvergence properties of HDG are inherited in X-HDG, with the freedom of computational meshes that do not fit the interfaces     

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. This work was supported by the German Research Foundation (DFG) under Grant No. Ste 238/35-1.     

On solving large strain hyperelastic problems with the natural element method

In this paper, an extension of the natural element method (NEM) is presented to solve finite deformation problems. Since NEM is a meshless method, its implementation does not require an explicit connectivity definition. Consequently, it is quite adequate to simulate large strain problems with important mesh distortions, reducing the need for remeshing and projection of results (extremely important in three‐dimensional problems). NEM has important advantages over other meshless methods, such as the interpolant character of its shape functions and the ability of exactly reproducing essential boundary conditions along convex boundaries. The α‐NEM extension generalizes this behaviour to non‐convex boundaries. A total Lagrangian formulation has been employed to solve different problems with large strains, considering hyperelastic behaviour. Several examples are presented in two and three dimensions, comparing the results with the ones of the finite element method. NEM performs better showing its important capabilities in this kind of applications. Copyright © 2004 John Wiley & Sons, Ltd.     

The numerical solution of two-dimensional,steady flow problems by the finite element method

Finite element equivalents of the equations governing shearing and buoyancy driven flows are derived, and reduced to upwind forms suitable for the solution of problems in which the Reynolds and Rayleigh numbers are large. A modification to the central difference iterative method is studied which increases the Reynolds and Rayleigh numbers for which a central difference form may be used. A comparison is made between the results obtained using the central and upwind forms of the finite element method and those predicted by finite difference methods in the case of flow in a cavity. A mesh refinement study is made. The upwind forms of the finite element equations are applied to the solution of a complex flow problem involving the flow of glass in a throated furnace in the case of constant- and temperature- dependent viscosity and conductivity.     

Periodic Galerkin finite element method of tidal flow

This paper presents the finite element method of the analysis of tidal flow. Assuming that tidal flow is periodic, the Galerkin approach is employed as the numerical integration procedure in time using a trigonometric function as the interpolation function. The present method has shown to be suitable for computation especially from the point of computing time and numerical stability.     

Orthotropic enriched element free Galerkin method for fracture analysis of composites

A new approach for modeling discrete cracks in two-dimensional orthotropic media by the element free Galerkin method is described. For increasing the solution accuracy, recently developed orthotropic enrichment functions used in the extended finite element method are adopted along with a sub-triangle technique for enhancing the Gauss quadrature accuracy near the crack. An appropriate scheme for selecting the support domains near a crack is employed to reduce the computational cost. In this study, mixed-mode stress intensity factors are obtained by means of the interaction integral to determine the fracture properties. Several problems are solved to illustrate the effectiveness of the proposed method and the results are compared with available results of other numerical or (semi-) analytical methods.     

Continuum damage growth analysis using element free Galerkin method

This paper presents an elasto-plastic element free Galerkin formulation based on Newton-Raphson algorithm for damage growth analysis. Isotropic ductile damage evolution law is used. A study has been carried out in this paper using the proposed element free Galerkin method to understand the effect of initial damage and its growth on structural response of single and bi-material problems. Asimple method is adopted for enforcing EBCs by scaling the function approximation using a scaling matrix, when non-singular weight functions are used over the entire domain of the problem definition. Numerical examples comprising of oneand two-dimensional problems are presented to illustrate the effectiveness of the proposed method in analysis of uniform and non-uniform damage evolution problems. Effect of material discontinuity on damage growth analysis is also presented.     

On a mixed finite element method for clamped anisotropic plate bending problems

The paper deals with a new mixed finite element method of solution of the bending problem of clamped anisotropic/orthotropic/isotropic plates with variable/constant thickness. This new mixed method gives simultaneous approximations to displacement u and bending and twisting moment tensor (ψ_ij)_{1 ≤ i, j ≤ 2}. Computer implementation procedures for this mixed method are given along with results of numerical experiments on a good number of interesting problems.