Galerkin Least-squares Method Research Articles

Finite element approximations for quasi-Newtonian flows employing a multi-field GLS method

This article concerns stabilized finite element approximations for flow-type sensitive fluid flows. A quasi-Newtonian model, based on a kinematic parameter of flow classification and shear and extensional viscosities, is used to represent the fluid behavior from pure shear up to pure extension. The flow governing equations are approximated by a multi-field Galerkin least-squares (GLS) method, in terms of strain rate, pressure and velocity (D-p-u). This method, which may be viewed as an extension of the formulation for constant viscosity fluids introduced by Behr et al. (Comput Methods Appl Mech 104:31---48, 1993), allows the use of combinations of simple Lagrangian finite element interpolations. Mild Weissenberg flows of quasi-Newtonian fluids--using Carreau viscosities with power-law indexes varying from 0.2 to 2.5--are carried out through a four-to-one planar contraction. The performed physical analysis reveals that the GLS method provides a suitable approximation for the problem and the results are in accordance with the related literature.

A numerical investigation of inertia flows of Bingham-Papanastasiou fluids by an extra stress-pressure-velocity galerkin least-squares method

This article is concerned with finite element approximations for yield stress fluid flows through a sudden planar expansion. The mechanical model is composed by mass and momentum balance equations, coupled with the Bingham viscoplastic model regularized by Papanastasiou (1987) equation. A multi-field Galerkin least-squares method in terms of stress, velocity and pressure is employed to approximate the flows. This method is built to circumvent compatibility conditions involving pressure-velocity and stress-velocity finite element subspaces. In addition, thanks to an appropriate design of its stability parameters, it is able to remain stable and accurate in high Bingham and Reynolds flows. Numerical simulations concerning the flow of a regularized Bingham fluid through a one-to-four sudden planar expansion are performed. For creeping flows, yield stress effects on the fluid dynamics of viscoplastic materials are investigated through the ranging of Bingham number from 0.2 to 100. In the sequence, inertia effects are accounted for ranging the Reynolds number from 0 to 50. The numerical results are able to characterize accurately the morphology of yield surfaces in high Bingham flows subjected to inertia.

BDDC preconditioning for high-order Galerkin Least-Squares methods using inexact solvers

A high-order Galerkin Least-Squares (GLS) finite element discretization is combined with a Balancing Domain Decomposition by Constraints (BDDC) preconditioner and inexact local solvers to provide an efficient solution technique for large-scale, convection-dominated problems. The algorithm is applied to the linear system arising from the discretization of the two-dimensional advection–diffusion equation and Euler equations for compressible, inviscid flow. A Robin–Robin interface condition is extended to the Euler equations using entropy-symmetrized variables. The BDDC method maintains scalability for the high-order discretization of the diffusion-dominated flows, and achieves low iteration count in the advection-dominated regime. The BDDC method based on inexact local solvers with incomplete factorization and p = 1 coarse correction maintains the performance of the exact counterpart for the wide range of the Peclet numbers considered while at significantly reduced memory and computational costs.

Space–time Galerkin least-squares method for the one–dimensional advection–diffusion equation

The advection–diffusion equation has a long history as a benchmark for numerical methods. The standard Galerkin finite-element method is accurate but lacks sufficient stability, whereas the least-squares method is stable but not accurate for small values of diffusion coefficient. One possibility to overcome it is to use a stabilized finite-element method such as Galerkin least-squares and this is the standard approach taken in this paper. The space–time GLS has been constructed by using both linear and quadratic B-spline shape functions. If advection dominates over diffusion the numerical solution is difficult, especially if boundary layers are to be resolved. The design of stability parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show that the behaviour of the method with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution.

A bubble-stabilized least-squares finite element method for steady MHD duct flow problems at high Hartmann numbers

In this paper we devise a stabilized least-squares finite element method using the residual-free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow. We convert the original system of second-order partial differential equations into a first-order system formulation by introducing two additional variables. Then the least-squares finite element method using C 0 linear elements enriched with the residual-free bubble functions for all unknowns is applied to obtain approximations to the first-order system. The most advantageous features of this approach are that the resulting linear system is symmetric and positive definite, and it is capable of resolving high gradients near the layer regions without refining the mesh. Thus, this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the Hartmann number. Before incoorperating the bubble functions into the global problem, we apply the Galerkin least-squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements. Therefore, we indeed introduce a two-level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual-free bubble functions. Numerical results confirming theoretical findings are presented for several examples including the Shercliff problem.

Galerkin Least-Squares Multifield Approximations for Flows of Inelastic Non-Newtonian Fluids

The aim of this work is to investigate a Galerkin least-squares (GLS) multifield formulation for inelastic non-Newtonian fluid flows. We present the mechanical modeling of isochoric flows combining mass and momentum balance laws in continuum mechanics with an inelastic constitutive equation for the stress tensor. For the latter, we use the generalized Newtonian liquid model, which may predict either shear-thinning or shear-thickening. We employ a finite element formulation stabilized via a GLS scheme in three primal variables: extra stress, velocity, and pressure. This formulation keeps the inertial terms and has the capability of predicting viscosity dependency on the strain rate. The GLS method circumvents the compatibility conditions that arise in mixed formulations between the approximation functions of pressure and velocity and, in the multifield case, of extra stress and velocity. The GLS terms are added elementwise, as functions of the grid Reynolds number, so as to add artificial diffusivity selectively to diffusion and advection dominant flow regions—an important feature in the case of variable viscosity fluids. We present numerical results for the lid-driven cavity flow of shear-thinning and shear-thickening fluids, using the power-law viscosity function for Reynolds numbers between 50 and 500 and power-law exponents from 0.25 to 1.5. We also present results concerning flows of shear-thinning Carreau fluids through abrupt planar and axisymmetric contractions. We study ranges of Carreau numbers from 1 to 100, Reynolds numbers from 1 to 100, and power-law exponents equal to 0.1 and 0.5. Besides accounting for inertia effects in the flow, the GLS method captures some interesting features of shear-thinning flows, such as the reduction of the fluid stresses, the flattening of the velocity profile in the contraction plane, and the separation of the boundary layer downstream the contraction.

Galerkin least-squares solutions for purely viscous flows of shear-thinning fluids and regularized yield stress fluids

This paper aims to present Galerkin Least-Squares approximations for flows of Bingham plastic fluids. These fluids are modeled using the Generalized Newtonian Liquid (GNL) constitutive equation. Their viscoplastic behavior is predicted by the viscosity function, which employs the Papanastasiou's regularization in order to predict a highly viscous behavior when the applied stress lies under the material's yield stress. The mechanical modeling for this type of flow is based on the conservation equations of mass and momentum, coupled to the GNL constitutive equation for the extra-stress tensor. The finite element methodology concerned herein, the well-known Galerkin Least-Squares (GLS) method, overcomes the two greatest Galerkin shortcomings for mixed problems. There is no need to satisfy Babuška-Brezzi condition for velocity and pressure subspaces, and spurious numerical oscillations, due to the asymmetric nature of advective operator, are eliminated. Some numerical simulations are presented: first, the lid-driven cavity flow of shear-thinning and shear-thickening fluids, for the purpose of code validation; second, the flow of shear-thinning fluids with no yield stress limit, and finally, Bingham plastic creeping flows through 2:1 planar and axisymmetric expansions, for Bingham numbers between 0.2 and 133. The numerical results illustrate the arising of two distinct unyielded regions: one near the expansion corner and another along the flow centerline. For those regions, velocity and pressure fields are investigated for the various Bingham numbers tested.

An algebraic subgrid scale finite element method for the convected Helmholtz equation in two dimensions with applications in aeroacoustics

An algebraic subgrid scale finite element method formally equivalent to the Galerkin Least-Squares method is presented to improve the accuracy of the Galerkin finite element solution to the two-dimensional convected Helmholtz equation. A stabilizing term has been added to the discrete weak formulation containing a stabilization parameter whose value turns to be the key for the good performance of the method. An appropriate value for this parameter has been obtained by means of a dispersion analysis. As an application, we have considered the case of aerodynamic sound radiated by incompressible flow past a two-dimensional cylinder. Following Lighthill’s acoustic analogy, we have used the time Fourier transform of the double divergence of the Reynolds stress tensor as a source term for the Helmholtz and convected Helmholtz equations and showed the benefits of using the subgrid scale stabilization.

CONTINUOUS Q1–Q1 STOKES ELEMENTS STABILIZED WITH NON-CONFORMING NULL EDGE AVERAGE VELOCITY FUNCTIONS

We present a stabilized finite element method for Stokes equations with piecewise continuous bilinear approximations for both velocity and pressure variables. The velocity field is enriched with piecewise polynomial bubble functions with null average at element edges. These functions are statically condensed at the element level and therefore they can be viewed as a continuous Q1–Q1 stabilized finite element method. The enriched velocity-pressure pair satisfies optimal inf–sup conditions and approximation properties. Numerical experiments show that the proposed discretization outperforms the Galerkin least-squares method.

Construction of stabilization operators for Galerkin least-squares discretizations of compressible and incompressible flows

The design and analysis of a class of stabilization operators suitable for space–time Galerkin least-squares finite element discretizations of the symmetrized compressible Navier–Stokes equations is discussed. The obtained stabilization matrix is well defined in the incompressible limit and reduces to the matrix described in [M. Polner, J.J.W. van der Vegt, R.M.J. van Damme, Analysis of stabilization operators for Galerkin least-squares discretizations of the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 195 (2006) 982–1006]. The stabilization matrix is investigated to give necessary and sufficient conditions for its positive definiteness. Under these conditions on the stabilization matrix, the Galerkin least-squares method for the symmetrized compressible Navier–Stokes equations satisfies the entropy condition, which ensures global nonlinear stability of the discretization.

Four-field Galerkin/least-squares formulation for viscoelastic fluids

A new Galerkin/Least-Squares (GLS) stabilized finite element method is presented for computing viscoelastic flows of complex fluids described by the conformation tensor; it extends the well-established GLS method for computing flows of incompressible Newtonian fluids. GLS methods are attractive for large-scale computations because they yield linear systems that can be solved easily with iterative solvers (e.g., the Generalized Minimum Residual method) and because they allow simple combinations of interpolation functions that can be conveniently and efficiently implemented on modern distributed-memory cache-based clusters. Like other state-of-the-art methods for computing viscoelastic flows (e.g., DEVSS-TG/SUPG), the new GLS method introduces a separate variable to represent the velocity gradient; with the aid of this variable, the conservation equations of mass, momentum, conformation, and the definition of velocity gradient are converted into a set of first-order partial differential equations in four unknown fields—pressure, velocity, conformation, and velocity gradient. The unknown fields are represented by low-order (continuous piecewise linear or bilinear) finite element basis functions. The method is applied to the Oldroyd-B constitutive equation and is tested in two benchmark problems—flow in a planar channel and flow past a cylinder in a channel. Results show that (1) the mesh-convergence rate of GLS is comparable to the DEVSS-TG/SUPG method; (2) the LS stabilization permits using equal-order basis functions for all fields; (3) GLS handles effectively the advective terms in the evolution equation of the conformation tensor; and (4) GLS yields accurate results at lower computational costs than DEVSS-type methods.

Meshless Galerkin least-squares method

Collocation method and Galerkin method have been dominant in the existing meshless methods. Galerkin-based meshless methods are computational intensive, whereas collocation-based meshless methods suffer from instability. A new efficient meshless method, meshless Galerkin lest-squares method (MGLS), is proposed in this paper to combine the advantages of Galerkin method and collocation method. The problem domain is divided into two subdomains, the interior domain and boundary domain. Galerkin method is applied in the boundary domain, whereas the least-squares method is applied in the interior domain.The proposed scheme elliminates the posibilities of spurious solutions as that in the least-square method if an incorrect boundary conditions are used. To investigate the accuracy and efficiency of the proposed method, a cantilevered beam and an infinite plate with a central circular hole are analyzed in detail and numerical results are compared with those obtained by Galerkin-based meshless method (GBMM), collocation-based meshless method (CBMM) and meshless weighted least squares method (MWLS). Numerical studies show that the accuracy of the proposed MGLS is much higher than that of CBMM and is close to, even better than, that of GBMM, while the computational cost is much less than that of GBMM.

A note on a recent study of stabilized finite element computations for heat conduction

In a recent paper studying finite element computation of heat transfer processes with dominant sources, for which the classical Galerkin method proves unstable, the authors conclude that Galerkin/least-squares (GLS) stabilization is insufficient while Galerkin-gradient/least-squares (GGLS) stabilization provides good results. It is the intention of this manuscript to correct these conclusions, that are based on a GLS method with a suboptimal parameter and on mislabelling a combined stabilized method as GGLS.

CBS versus GLS stabilization of the incompressible Navier–Stokes equations and the role of the time step as stabilization parameter

AbstractIn this work we compare two apparently different stabilization procedures for the finite element approximation of the incompressible Navier–Stokes equations. The first is the characteristic‐based split (CBS). It combines the characteristic Galerkin method to deal with convection dominated flows with a classical splitting technique, which in some cases allows us to use equal velocity–pressure interpolations. The second approach is the Galerkin‐least‐squares (GLS) method, in which a least‐squares form of the element residual is added to the basic Galerkin equations. It is shown that both formulations display similar stabilization mechanisms, provided the stabilization parameter of the GLS method is identified with the time step of the CBS approach. This identification can be understood from a formal Fourier analysis of the linearized problem. Copyright © 2001 John Wiley & Sons, Ltd.

GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows

In this paper, order one finite elements together with Galerkin least-squares (GLS) methods are used for solving a three-field Stokes problem arising from the numerical study of viscoelastic flows. Stability and convergence results are established, even when the solvent viscosity is small compared to the viscosity due to the polymer chains. An iterative algorithm decoupling velocity–pressure and stress calculations is proposed. The link with the modified elastic viscous split stress (EVSS) method studied in (M. Fortin, R. Guénette, R. Pierre, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79–95; R. Guénette, M. Fortin, J. Non-Newtonian Fluid Mech. 60 (1995) 27–52) is presented. Numerical results are in agreement with theoretical predictions, and with those presented in (M. Fortin, R. Guénette, R. Pierre, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79–95).

On stabilized finite element formulations for incompressible advective–diffusive transport and fluid flow problems

A new approach is presented to obtain stabilized finite element formulations such as streamline-upwind/Petrov–Galerkin (SUPG) and Galerkin-least-squares (GLS). The procedure consists in modifying the equations to be solved and then obtaining the variational equations by the standard Galerkin method. The new formulation generates additional terms involving boundary integrals to standard stabilization techniques. These terms compensate for the lack of consistency of the traditional SUPG and GLS methods for which stabilization terms are added only on the element interiors, while jumps of the residual across element faces are neglected. A physical interpretation is provided of how the modified equations are obtained. It is shown how stabilized formulations such as streamline-upwind (SU) and SUPG are recovered as special cases. Stabilization terms defined on the element interiors are always accompanied by additional boundary integrals. The presence of the boundary integrals is shown to improve the numerical prediction for various viscous and nearly inviscid flows.

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.

A multiscale finite element method for the Helmholtz equation

It is well known that when the standard Galerkin method is applied to the Helmholtz equation it exhibits an error in the wavenumber and the solution does not, therefore, preserve the phase characteristics of the exact solution. Improvements on the Galerkin method, including Galerkin least-squares (GLS) methods, have been proposed. However, these approaches rely on a dispersion analysis of the underlying difference stencils in order to reduce error in the solution. In this paper we propose a multiscale finite element for the Helmholtz equation. The method employs a multiscale variational formulation which leads to a subgrid model in which subgrid scales are incorporated analytically through appropriate Green's functions. It is shown that entirely new and accurate methods emerge and that GLS methods can be obtained as special cases of the more general subgrid model.

H-Adaptive finite element computation of time-harmonic exterior acoustics problems in two dimensions

In this paper we carry out adaptive finite element computations of the Helmholtz equation in two dimensions, in the context of time-harmonic exterior acoustics. The purpose is to demonstrate potentially significant cost savings engendered through adaptivity for propagating solutions at moderate wave numbers ( ka = 2 π, 4 π). The computations are performed on meshes of linear triangles, and are adapted to the solution by locally changing element sizes ( h-refinement). The adaptive procedure involves applying an explicit, residual-based a posteriori error estimator and an h-adaptive strategy, which were derived in a previous paper. The adaptive meshes are then obtained through global mesh regeneration using an advancing front mesh generator. Two different finite element formulations are considered: The Galerkin and Galerkin Least-Squares (GLS) methods. The infinite physical domain is truncated by an artificial exterior boundary, on which a fully coupled Dirichlet-to-Neumann (DtN) boundary condition is applied. Two different problems are computed: Plane wave scattering by a rigid infinite surface (circular and square cross sections), and nonuniform radiation from an infinite circular cylinder. Detailed cost studies with respect to an active column direct solver are performed. For the nonuniform radiation problem, the adaptive mesh is twenty times more cost effective than a uniform mesh (for Galerkin). When coupled with the GLS formulation, the adaptive mesh is forty times more efficient than Galerkin computations on a uniform mesh.

A Galerkin least‐squares finite element method for the two‐dimensional Helmholtz equation

AbstractIn this paper a Galerkin least‐squares (GLS) finite element method, in which residuals in least‐squares form are added to the standard Galerkin variational equation, is developed to solve the Helmholtz equation in two dimensions. An important feature of GLS methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes. Previous work has accomplished this for the one‐dimensional Helmholtz equation using dispersion analysis. In this paper, the selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy. For any given direction of wave propagation, an optimal GLS mesh parameter is determined using two‐dimensional Fourier analysis. In general problems, the direction of wave propagation will not be known a priori. In this case, an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements. The optimal GLS parameters are derived for both consistent and lumped mass approximations. Several numerical examples are given and the results compared with those obtained from the Galerkin method. The extension of GLS to higher‐order quadratic interpolations is also presented.