Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

The paper introduces a methodology to compute strict upper and lower bounds for linear‐functional outputs of the exact solutions of the advection–diffusion–reaction equation. The bounds are computed using implicit a posteriori error estimators from stabilized finite element approximations of the exact solution. The new methodology extends the a posteriori error estimates yielding bounds for the standard Galerkin formulation to be able to obtain bounds for stabilized formulations. This methodology is combined with both hybrid‐flux and flux‐free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the flux‐free technique. Copyright © 2012 John Wiley & Sons, Ltd.     

Dispersion analysis of stabilized finite element methods for acoustic fluid interaction with Reissner–Mindlin plates

The application of stabilized finite element methods to model the vibration of elastic plates coupled with an acoustic fluid medium is considered. A complex‐wavenumber dispersion analysis of acoustic fluid interaction with Reissner–Mindlin plates is performed to quantify the accuracy of stabilized finite element methods for fluid‐loaded plates. Results demonstrate the improved accuracy of a recently developed hybrid least‐squares (HLS) plate element based on a modified Hellinger–Reissner functional, consistently combined with residual‐based methods for the acoustic fluid, compared to standard Galerkin and Galerkin gradient least‐squares plate elements. The technique of complex wavenumber dispersion analysis is used to examine the accuracy of the discretized system in the representation of free waves for fluid‐loaded plates. The influence of fluid and coupling matrices resulting from consistent implementation of pressure loading in the residual for the plate equation is examined and clarified for the different finite element approximations. Copyright © 2001 John Wiley & Sons, Ltd.     

A residual‐based finite element method for the Helmholtz equation

A new residual‐based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter‐element boundaries. The inclusion of residuals on inter‐element boundaries distinguishes this method from the well‐known Galerkin least‐squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

On a two‐level finite element method for the incompressible Navier–Stokes equations

We consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions, where the finite‐dimensional space(s) employed consist of piecewise polynomials enriched with residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method (TLFEM) is described and its application to the Navier–Stokes equation is displayed. Numerical solutions employing the TLFEM are presented for three benchmark problems. We compare the numerical solutions using the TLFEM with the numerical solutions using a stabilized method. Copyright © 2001 John Wiley & Sons, Ltd.     

An element‐free method for in‐plane notch problems with anisotropic materials

This study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.     

Inf–sup testing of upwind methods

We propose inf–sup testing for finite element methods with upwinding used to solve convection–diffusion problems. The testing evaluates the stability of a method and compactly displays the numerical behaviour as the convection effects increase. Four discretization schemes are considered: the standard Galerkin procedure, the full upwind method, the Galerkin least‐squares scheme and a high‐order derivative artificial diffusion method. The study shows that, as expected, the standard Galerkin method does not pass the inf–sup tests, whereas the other three methods pass the tests. Of these methods, the high‐order derivative artificial diffusion procedure introduces the least amount of artificial diffusion. Copyright © 2000 John Wiley & Sons, Ltd.     

A stabilized conforming nodal integration for Galerkin mesh‐free methods

Domain integration by Gauss quadrature in the Galerkin mesh‐free methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes. A strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration. For convergence, an integration constraint (IC) is introduced as a necessary condition for a linear exactness in the mesh‐free Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy IC using a divergence theorem. No numerical control parameter is involved in the proposed strain smoothing stabilization. The numerical results show that the accuracy and convergent rates in the mesh‐free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method. It is also demonstrated that the Gauss integration method fails to meet IC in mesh‐free discretization. For this reason the proposed method provides even better accuracy than Gauss integration for Galerkin mesh‐free method as presented in several numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.     

One‐dimensional dispersion analysis for the element‐free Galerkin method for the Helmholtz equation

The standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wave numbers due to the dispersion, unless highly refined meshes are used, leading to unacceptable resolution times. The paper presents an application of the element‐free Galerkin method (EFG) and focuses on the dispersion analysis in one dimension. It shows that, if the basis contains the solution of the homogenized Helmholtz equation, it is possible to eliminate the dispersion in a very natural way while it is not the case for the finite element methods. For the general case, it also shows that it is possible to choose the parameters of the method in order to minimize the dispersion. Finally, theoretical developments are validated by numerical experiments showing that, for the same distribution of nodes, the element‐free Galerkin method solution is much more accurate than the finite element one. Copyright © 2000 John Wiley & Sons, Ltd.     

An assumed‐gradient finite element method for the level set equation

The level set equation is a non‐linear advection equation, and standard finite‐element and finite‐difference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution. We recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front/interface being captured. As with the original level set equation, the use of an extensional velocity helps maintain this signed‐distance function. For some interface‐evolution problems, this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve. Further, we find that sufficient accuracy is available through a standard Galerkin formulation without any stabilization or discontinuity‐capturing terms. Several numerical experiments are conducted to assess the ability of the proposed assumed‐gradient level set method to capture the correct solution, particularly in the presence of discontinuities in the extensional velocity or level‐set gradient. We examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a Hamilton–Jacobi equation with convex/non‐convex Hamiltonian. Importantly, discretizations based on structured and unstructured finite‐element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well. Copyright © 2005 John Wiley & Sons, Ltd.     

A Galerkin/least‐squares finite element formulation for consolidation

A Galerkin/least‐squares (GLS) finite element formulation for problem of consolidation of fully saturated two‐phase media is presented. The elimination of spurious pressure oscillations appearing at the early stage of consolidation for standard Galerkin finite elements with equal interpolation order for both displacements and pressures is the goal of the approach. It will be shown that the least‐squares term, based exclusively on the residuum of the fluid flow continuity equation, added to the standard Galerkin formulation enhances its stability and can fully eliminate pressure oscillations. A reasonably simple framework designed for derivation of one‐dimensional as well as multi‐dimensional estimates of the stabilization factor is proposed and then verified. The formulation is validated on one‐dimensional and then on two‐dimensional, linear and non‐linear test problems. The effect of the fluid incompressibility as well as compressibility will be taken into account and investigated. Copyright © 2001 John Wiley & Sons Ltd.     

Non‐linear version of stabilized conforming nodal integration for Galerkin mesh‐free methods

A stabilized conforming (SC) nodal integration, which meets the integration constraint in the Galerkin mesh‐free approximation, is generalized for non‐linear problems. Using a Lagrangian discretization, the integration constraints for SC nodal integration are imposed in the undeformed configuration. This is accomplished by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal integration in the undeformed configuration. The proposed method is independent to the path dependency of the materials. An assumed strain method is employed to formulate the discrete equilibrium equations, and the smoothed deformation gradient serves as the stabilization mechanism in the nodally integrated variational equation. Eigenvalue analysis demonstrated that the proposed strain smoothing provides a stabilization to the nodally integrated discrete equations. By employing Lagrangian shape functions, the computation of smoothed gradient matrix for deformation gradient is only necessary in the initial stage, and it can be stored and reused in the subsequent load steps. A significant gain in computational efficiency is achieved, as well as enhanced accuracy, in comparison with the mesh‐free solution using Gauss integration. The performance of the proposed method is shown to be quite robust in dealing with non‐uniform discretization. Copyright © 2002 John Wiley & Sons, Ltd.     

An efficient time‐domain perfectly matched layers formulation for elastodynamics on spherical domains

Many practical applications require the analysis of elastic wave propagation in a homogeneous isotropic media in an unbounded domain. One widely used approach for truncating the infinite domain is the so‐called method of perfectly matched layers (PMLs). Most existing PML formulations are developed for finite difference methods based on the first‐order velocity‐stress form of the elasticity equations, and they are not straight‐forward to implement using standard finite element methods (FEMs) on unstructured meshes. Some of the problems with these formulations include the application of boundary conditions in half‐space problems and in the treatment of edges and/or corners for time‐domain problems. Several PML formulations, which do work with FEMs have been proposed, although most of them still have some of these problems and/or they require a large number of auxiliary nodal history/memory variables. In this work, we develop a new PML formulation for time‐domain elastodynamics on a spherical domain, which reduces to a two‐dimensional formulation under the assumption of axisymmetry. Our formulation is well‐suited for implementation using FEMs, where it requires lower memory than existing formulations, and it allows for natural application of boundary conditions. We solve example problems on two‐dimensional and three‐dimensional domains using a high‐order discontinuous Galerkin (DG) discretization on unstructured meshes and explicit time‐stepping. We also study an approach for stabilization of the discrete equations, and we show several practical applications for quality factor predictions of micromechanical resonators along with verifying the accuracy and versatility of our formulation. Copyright © 2014 John Wiley & Sons, Ltd.     

A non‐oscillatory method for spallation studies

This paper introduces a non‐oscillatory method, the finite element flux‐corrected transport (FE‐FCT) method for spallation studies. This method includes the implementation of a one‐dimensional FCT algorithm into a total Lagrangian finite element method. Consequently, the FE‐FCT method can efficiently eliminate fluctuations behind shock wave fronts without smearing them. In multidimensional simulations, the one‐dimensional FCT algorithm is used on each grid line of the structured meshes to correct the corresponding component of nodal velocities separately. The requirement of structured meshes is satisfied by using an implicit function so that arbitrary boundaries of the simulated object can be described. In this paper, the proposed FE‐FCT method is applied in spallation studies. One‐ and two‐dimensional examples show this non‐oscillatory method could be one of the candidates to accurately predict spallation and the spall thickness. Copyright © 2005 John Wiley & Sons, Ltd.     

An element‐wise,locally conservative Galerkin (LCG) method for solving diffusion and convection–diffusion problems

An element‐wise locally conservative Galerkin (LCG) method is employed to solve the conservation equations of diffusion and convection–diffusion. This approach allows the system of simultaneous equations to be solved over each element. Thus, the traditional assembly of elemental contributions into a global matrix system is avoided. This simplifies the calculation procedure over the standard global (continuous) Galerkin method, in addition to explicitly establishing element‐wise flux conservation. In the LCG method, elements are treated as sub‐domains with weakly imposed Neumann boundary conditions. The LCG method obtains a continuous and unique nodal solution from the surrounding element contributions via averaging. It is also shown in this paper that the proposed LCG method is identical to the standard global Galerkin (GG) method, at both steady and unsteady states, for an inside node. Thus, the method, has all the advantages of the standard GG method while explicitly conserving fluxes over each element. Several problems of diffusion and convection–diffusion are solved on both structured and unstructured grids to demonstrate the accuracy and robustness of the LCG method. Both linear and quadratic elements are used in the calculations. For convection‐dominated problems, Petrov–Galerkin weighting and high‐order characteristic‐based temporal schemes have been implemented into the LCG formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

Exponential finite elements for diffusion–advection problems

A new finite element method for the solution of the diffusion–advection equation is proposed. The method uses non‐isoparametric exponentially‐varying interpolation functions, based on exact, one‐ and two‐dimensional solutions of the Laplace‐transformed differential equation. Two eight‐noded elements are developed and tested for convergence, stability, Peclet number limit, anisotropy, material heterogeneity, Dirichlet and Neumann boundary conditions and tolerance for mesh distortions. Their performance is compared to that of conventional, eight‐ and 12‐noded polynomial elements. The exponential element based on two‐dimensional analytical solutions fails basic tests of convergence. The one based on one‐dimensional solutions performs particularly well. It reduces by about 75% the number of elements and degrees of freedom required for convergence, yielding an error that is one order of magnitude smaller than that of the eight‐noded polynomial element. The exponential element is stable and robust under relatively high degrees of heterogeneity, anisotropy and mesh distortions. Copyright © 2005 John Wiley & Sons, Ltd.     

A NURBS‐based interface‐enriched generalized finite element method for problems with complex discontinuous gradient fields

A non‐uniform rational B‐splines (NURBS)‐based interface‐enriched generalized finite element method is introduced to solve problems with complex discontinuous gradient fields observed in the structural and thermal analysis of the heterogeneous materials. The presented method utilizes generalized degrees of freedom and enrichment functions based on NURBS to capture the solution with non‐conforming meshes. A consistent method for the generation and application of the NURBS‐based enrichment functions is introduced. These enrichment functions offer various advantages including simplicity of the integration, possibility of different modes of local solution refinement, and ease of implementation. In addition, we show that these functions well capture weak discontinuities associated with highly curved material interfaces. The convergence, accuracy, and stability of the method in the solution of two‐dimensional elasto‐static problems are compared with the standard finite element scheme, showing improved accuracy. Finally, the performance of the method for solving problems with complex internal geometry is highlighted through a numerical example. Copyright © 2015 John Wiley & Sons, Ltd.     

A finite element method based on C0‐continuous assumed gradients

This article presents an alternative approach to assumed gradient methods in FEM applied to three‐dimensional elasticity. Starting from nodal integration (NI), a general C0‐continuous assumed interpolation of the deformation gradient is formulated. The assumed gradient is incorporated using the principle of Hu‐Washizu. By dual Lagrange multiplier spaces, the functional is reduced to the displacements as the only unknowns. An integration scheme is proposed where the integration points coincide with the support points of the interpolation. Requirements for regular finite element meshes are explained. Using this interpretation of NI, instabilities (appearance of spurious modes) can be explained. The article discusses and classifies available strategies to stabilize NI such as penalty methods, SCNI, α‐FEM. Related approaches, such as the smoothed finite element method, are presented and discussed. New stabilization techniques for NI are presented being entirely based on the choice of the assumed gradient interpolation, i.e. nodal‐bubble support, edge‐based support and support using tensor‐product interpolations. A strategy is presented on how the interpolation functions can be derived for various element types. Interpolation functions for the first‐order hexahedral element, the first‐order and the second‐order tetrahedral elements are given. Numerous examples illustrate the strengths and limitations of the new schemes. Copyright © 2010 John Wiley & Sons, Ltd.     

Hat interpolation wavelet‐based multi‐scale Galerkin method for thin‐walled box beam analysis

The objective of the present work is to propose a new adaptive wavelet‐Galerkin method based on the lowest‐order hat interpolation wavelets. The specific application of the present method is made on the one‐dimensional analysis of thin‐walled box beam problems exhibiting rapidly varying local end effects. Higher‐order interpolation wavelets have been used in the wavelet‐collocation setting, but the lowest‐order hat interpolation is applied here first and a hat interpolation wavelet‐based Galerkin method is newly formulated. Unlike existing orthogonal or biorthogonal wavelet‐based Galerkin methods, the present method does not require special treatment in dealing with general boundary conditions. Furthermore, the present method directly works with nodal values and does not require special formula for the evaluation of system matrices. Though interpolation wavelets do not have any vanishing moment, an adaptive scheme based on multi‐resolution approximations is possible and a preconditioned conjugate gradient method can be used to enhance numerical efficiency. Copyright © 2001 John Wiley & Sons, Ltd.