Galerkin Least-squares Method Research Articles

Stabilized equal lower-order finite element methods for simulating Brinkman equations in porous media

This paper demonstrates the mixed formulation of the Brinkman problem using linear equal-order finite element methods in porous media modelling. We introduce Galerkin least-squares (GLS) and least-squares (LS) finite element methods to address the incompatibility of finite element spaces, treating velocity, pressure, and vorticity as independent variables. Theoretical analysis examines coercivity and continuity, providing error estimates. Demonstrating resilience in theoretical findings, these methods achieve optimal convergence rates in the L 2 norm by incorporating stabilization terms with low-order basis functions. Numerical experiments validate theoretical predictions, showing the effectiveness of the GLS method and addressing finite element space incompatibility. Additionally, the GLS method exhibits promising capabilities in handling the Brinkman equation at low permeability compared to the LS method. The study reveals an increase in the average pressure difference in the Brinkman problem compared to the Stokes equations as the inlet velocity rises, providing insights into the behaviour of Brinkman equations.

Isogeometric analysis and Augmented Lagrangian Galerkin Least Squares Methods for residual minimization in dual norm

We explore how recent advances in Isogeometric analysis, Galerkin Least-Squares methods, and Augmented Lagrangian techniques can be applied to solve nonstandard problems, for which there is no classical stability theory, such as that provided by the Lax–Milgram lemma or the Banach-Necas-Babuska theorem. In particular, we consider continuation problems where a second-order partial differential equation with incomplete boundary data is solved given measurements of the solution on a subdomain of the computational domain. The use of higher regularity spline spaces leads to simplified formulations and potentially minimal multiplier space. We show that our formulation is inf-sup stable, and given appropriate a priori assumptions, we establish optimal order convergence.

Numerical solution of magnetohydrodynamics effects on a generalised power law fluid model of blood flow through a bifurcated artery with an overlapping shaped stenosis.

This paper presents a numerical analysis of blood flow in a diseased vessel within the presence of an external magnetic field. The blood flow was considered to be incompressible and fully developed, in that the non-Newtonian nature of the fluid was characterised as a generalised power law model for shear-thinning, Newtonian, and shear-thickening fluids. The impact of a transverse directed external magnetic field on blood flow through a stenosed bifurcated artery was investigated. The arterial geometry was considered as a bifurcated channel with overlapping shaped stenosis. The problem was treated mathematically using the Galerkin Least-Squares (GLS) method. The implementation of this numerical method managed to overcome the numerical instability faced by the classical Galerkin technique when adopted to a highly viscous flow. The benefit of GLS in circumventing the Ladyzhenskaya-Babuška-Brezzi (LBB) condition was utilized by evaluating both the velocity and pressure components at corner nodes of a unstructured triangular element. The non-linearity that emerged from the convective terms was then treated using the Newton-Raphson method, while the numerical integrals were computed using a Gaussian quadrature rule with six quadrature points. The findings obtained from this study were then compared with available results from the literature as well as Comsol multiphysics software to verify the accuracy and validity of the numerical algorithms. It was found that the application of magnetic field was able to overcome flow reversal by 39% for a shear-thinning fluid, 26% for a Newtonian fluid, and 27% for a shear-thickening fluid. The negative pressure and steep wall shear stress which occurs at the extremities of an overlapping stenosis throat were diminished by rise in magnetic intensity. This prevented thrombosis occurrence and produced a uniform calm flow.

Dynamic Response of Heat Transfer in Magnetohydrodynamic Blood Flow Through a Porous Bifurcated Artery with Overlapping Stenosis

The influence of heat transfers on magnetohydrodynamic blood flow treated as a Newtonian rheological model in permeable bifurcated arteries with a magnetic field applied in a transverse direction has been presented in this study. The blood flow is considered steady, incompressible, electrically conducting and porous as it consists of fatty accumulation on blood cells. A mathematical model governing such flow phenomena is developed by the Navier-Stokes and energy equations. The governing equations with appropriate boundary conditions are solved numerically by using a stabilized form of the finite element method, called the Galerkin least-squares method. The implementation of this method managed to overcome two sources of undesirable pathologies suffered by the Classical Galerkin method in solving the incompressible flow. This involves the requirement to satisfy the Babuška-Brezzi stability condition and the inherent instability arising from the approximations of highly advective flows. As a result, stable and converged solutions are generated for the velocity, temperature, pressure, wall shear stress and heat transfer coefficient in the constricted bifurcated artery by employing an equal order (quadratic triangular element) of interpolation function for velocity, pressure and temperature subspaces. The developed algorithm is validated with reported findings from previous literature where a close agreement has been achieved. Effects of the porosity parameter and Hartmann number in the presence of heat on blood flow characteristics at several locations along the branch artery are computed and presented graphically. The finding obtained from this work may be beneficial for cryosurgery and hyperthermia treatment in analysing the temperature distributions inside the vessel.

Numerical Analysis of Blood Flow Behaviour in a Constricted Porous Bifurcated Artery under the Influence of Magnetic Field

A numerical investigation concerning the effect of arterial constriction on the flow behaviour of blood in a porous bifurcated artery under the influence of a transverse magnetic field has been carried out. The equations of motion governing the flow are derived in the Cartesian coordinate system by treating the fluid as an electrically conducting, Newtonian, incompressible and fully developed blood flow. The blood is considered to flow through a constricted bifurcated artery with overlapping stenosis located at the mother artery. This developed model is consistent with the principles of magnetohydrodynamics. A stabilized form of finite element method, known as the Galerkin least-squares method is employed in solving the governing equation with suitably prescribed boundary conditions. Then, the non-linear systems resulting from the developed model are linearized by an iterative technique called the Newton-Raphson method. The effects of various parameters particularly the severity of stenosis, magnetohydrodynamics constant and permeability parameter on blood flow characteristics are analysed graphically for the velocity profile and streamline pattern. The findings obtained show that the presence of a magnetic field caused a significant alteration in the blood flow behaviour. Slight changes occurred in the flow pattern with the porous medium consideration. The application of a magnetic field could be utilized in controlling the hemodynamic flow of blood by decelerating the flow velocity and reducing the vortex sizes with rising in Hartmann number. While the pathological state of arterial disease as a porous structure reveals that a more porous fluid medium could have slightly promoted the growth in vortex size.

GLS methods for Stokes equations under boundary condition of friction type: formulation-analysis-numerical schemes and simulations

In this paper, we present in two and three dimensional space Galerkin least squares (GLS) methods allowing the use of equal order approximation for both the velocity and pressure modeling the Stokes equations under Tresca’s boundary condition. We propose and analyse two finite element formulations in bounded domains. Firstly, we construct the unique weak solution for each problem by using the method of regularization combined with the monotone operators theory and compactness properties. Secondly, we study the convergence of the finite element approximation by deriving a priori error estimate. Thirdly, we formulate three numerical algorithms namely; projection-like algorithm couple with Uzawa’s iteration, the alternative direction method of multiplier and an active set strategy. Finally some numerical experiments are performed to confirm the theoretical findings and the efficiency of the schemes formulated.

A Stabilized Finite Element Formulation of Non-Newtonian Fluid Model of Blood Flow in A Bifurcated Channel with Overlapping Stenosis

A stabilized form of finite element formulation known as the Galerkin least-squares (GLS) method is implemented here for solving the two-dimensional incompressible non-Newtonian fluid model of blood flow in a diseased artery. The modelling for this type of flow is based on the conservation of mass and momentum equations, coupled with the generalised Newtonian liquid (GNL) constitutive equation characterized by the generalised power law (GPL) model. The flow of blood in this present study are assumed as steady, laminar and fully developed. The finite element algorithms considered herein are first solved for the Newtonian fluid in a straight artery with a bell shaped stenosis for validation purposes. As the efficiency and validity of the proposed algorithms are obtained through comparison with the findings from existing literature and COMSOL Multiphysics 5.2 software. Then, the algorithms are being implemented to the generalised power law fluid model of blood flow in a bifurcated artery with overlapping stenosis located at the parent's arterial lumen. The numerical results illustrate the arising of distinct sizes of vortex shedding downstream of the stenotic region for each generalised power law index.

The Galerkin least squares MLPG method for convection-dominated problems

The meshless method has been widely applied in the computational mechanics, materials science, computational heat transfer and fluid flow. However, the development of a high efficient method for convection term in the computational fluid dynamics is difficult. In this paper, a truly meshless method of the meshless local Petrov-Galerkin (MLPG) method was applied to solve convection-diffusion problem. Meanwhile, the Galerkin least squares (GLS) approximation method was proposed to overcome the influence of convection term. The accuracy and efficiency of the present method were validated by some cases with benchmark solutions. The computational results showed that the GLS method could be used in dealing with convection-diffusion problems with high computational precision.

The Galerkin least squares MLPG method for convection-dominated problems

The meshless method has been widely applied in the computational mechanics, materials science, computational heat transfer and fluid flow. However, the development of a high efficient method for convection term in the computational fluid dynamics is difficult. In this paper, a truly meshless method of the meshless local Petrov-Galerkin (MLPG) method was applied to solve convection-diffusion problem. Meanwhile, the Galerkin least squares (GLS) approximation method was proposed to overcome the influence of convection term. The accuracy and efficiency of the present method were validated by some cases with benchmark solutions. The computational results showed that the GLS method could be used in dealing with convection-diffusion problems with high computational precision.

A discontinuous Galerkin least-squares method for div–curl systems

A discontinuous Galerkin (DG) least-squares finite element method is developed for solving a div–curl problem. This finite element method leads to a simple symmetric positive definite system and can work on a general polytopal mesh. Optimal convergence rate in the energy norm is established. The numerical results confirm the proposed theory.

An a posteriori error estimator for the weak Galerkin least-squares finite-element method

Abstract In this paper, we derive an a posteriori error estimator for the weak Galerkin least-squares (WG-LS) method applied to the reaction–diffusion equation. We show that this estimator is both reliable and efficient, allowing it to be used for adaptive refinement. Due to the flexibility of the WG-LS discretization, we are able to design a simple and straightforward refinement scheme that is applicable to any shape regular polygonal mesh. Finally, we present numerical experiments that confirm the effectiveness of the estimator, and demonstrate the robustness and efficiency of the proposed adaptive WG-LS approach.

A stabilised finite element method for the convection–diffusion–reaction equation in mixed form

This paper is devoted to the approximation of the convection–diffusion–reaction equation using a mixed, first-order, formulation. We propose, and analyse, a stabilised finite element method that allows equal order interpolations for the primal and dual variables. This formulation, reminiscent of the Galerkin least-squares method, is proven stable and convergent. In addition, a numerical assessment of the numerical performance of different stabilised finite element methods for the mixed formulation is carried out, and the different methods are compared in terms of accuracy, stability, and sharpness of the layers for two different classical test problems.

A weak Galerkin least-squares finite element method for div–curl systems

In this paper, we introduce a weak Galerkin least-squares method for solving div–curl problem. This finite element method leads to a symmetric positive definite system and has the flexibility to work with general meshes such as hybrid mesh, polytopal mesh and mesh with hanging nodes. Error estimates of the finite element solution are derived. The numerical examples demonstrate the robustness and flexibility of the proposed method.

A Study on the Galerkin Least-Squares Method for the Oldroyd-B Model

Abstract We consider a reduced Galerkin least-squares finite element method for the Oldroyd-B model of viscoelastic fluid flows. Model problems considered are the flow past a planar channel and a 4-to-1 contraction problems. An a priori error estimate for the reduced Galerkin least-squares method is derived and numerical results supporting the estimate are presented.

Finite element investigation of inertia and viscous effects on regularized Herschel–Bulkley flows

Finite element simulations of regularized viscoplastic flows through a planar expansion is performed. The mechanical model is made up of the conservation equations of mass and momentum, coupled with a regularized form of the Herschel–Bulkley viscosity function, based on the Papanastasiou equation. This model is approximated by a three-field Galerkin least-squares method, in terms of the extra-stress tensor, the velocity vector and the pressure field. A sensitivity analysis is performed, aiming at an investigation of the influence of yield stress, shear-thinning and inertia effects on the flow pattern and pressure drop. The results show a strong dependence of the position and shape of the yielded and unyielded regions on Herschel–Bulkley and Reynolds numbers.

Buffeting Analysis of a Cable-Stayed Bridge Using Three-Dimensional Computational Fluid Dynamics

In recent years, a number of studies have been conducted to analyze wind-induced vibrations of a bridge using computational fluid dynamics (CFD). Previous studies have been limited to analyzing two-dimensional sections only, and three-dimensional (3D) analyses of entire bridges have not yet been performed. In this study, a CFD program with fluid-structure interaction is developed to perform buffeting analysis of a 3D cable-stayed bridge. The FEM formulated by the Galerkin least-squares method and arbitrary Lagrangian-Eulerian method are used to perform fluid-structure interaction CFD. The spectral representation method is used to generate the time series turbulence, which is applied to the inlet boundary condition. A supercomputer is used to reduce the large computing time. The analysis model of an existing cable-stayed bridge, which contains an ambient atmospheric region, has been used to calculate the dynamic responses of bridges and the detailed flow of wind. The results of the 3D CFD analysis of the bridge are compared with the results of conventional frequency domain buffeting analysis. The results are essentially in good agreement with those of the conventional analysis. The numerical method developed in this study will be an efficient alternative to wind tunnel tests for verifying wind flow, wind loading, and structural vibrations.

Computational Modeling and Sub-Grid Scale Stabilization of Incompressibility and Convection in the Numerical Simulation of Friction Stir Welding Processes

This paper deals with the computational modeling and sub-grid scale stabilization of incompressibility and convection in the numerical simulation of the material flow around the probe tool in a friction stir welding (FSW) process. Within the paradigmatic framework of the multiscale stabilization methods, suitable pressure and convective derivative of the temperature sub-grid scale stabilized coupled thermomechanical formulations have been developed using an Eulerian description. Norton-Hoff and Sheppard-Wright thermo-rigid-viscoplastic constitutive material models have been considered. Constitutive equations for the sub-grid scale models have been proposed and an approximation of the sub-grid scale variables has been given. In particular, algebraic sub-grid scale (ASGS) and orthogonal sub-grid scale (OSGS) methods for mixed velocity, pressure and temperature P1/P1/P1 linear elements have been considered. Furthermore, it has been shown that well known classical stabilized formulations, such as the Galerkin least-squares (GLS) for incompressible (or quasi-incompressible) problems or the Streamline Upwind/Petrov-Galerkin (SUPG) method for convection dominant problems, can be recovered as particular cases of the multiscale stabilization framework considered. Using a product formula algorithm for the solution of the coupled thermomechanical problem, the resulting algebraic system of equations has been solved using a staggered procedure in which a mechanical problem, defined by the linear momentum balance equation, under quasi-static conditions, and the incompressibility equation, is solved first at constant temperature. Then a thermal problem, defined by the energy balance equation, is solved keeping constant the mechanical variables, i.e. velocity and pressure. The computational model has been implemented in an enhanced version of the finite element software COMET, developed by the authors at the International Center for Numerical Methods in Engineering (CIMNE). Two numerical examples have been considered. The first one deals with the numerical simulation of a coupled thermomechanical flow in a 2D rectangular domain. Steady-state and transient conditions have been considered. The goal of this numerical example has been the comparison between different sub-grid scale stabilization methods for the velocity and temperature equations. In particular, using a GLS stabilization method for the pressure equation, a comparison between SUPG and OSGS convective stabilization methods has been performed. Additionally, using a SUPG stabilization method for the temperature equation, a comparison between GLS and OSGS pressure stabilization methods has been done. The second example deals with the 3D numerical simulation of a representative FSW process. Numerical results obtained have been compared with experimental results available in the literature. A good agreement on the temperature distribution has been obtained and predicted peak temperatures compare well, both in value and position, with the experimental results available.

MULTI-FIELD STABILIZED FINITE ELEMENT APPROXIMATIONS FOR OLDROYD-B FLUID FLOWS

This work concerns with numerical simulations of creeping and inertial flows of viscoelastic fluids. The mechanical model consists of mass and momentum balance equations, coupled with the Oldroyd-B fluid. The model is approximated by a multi-field Galerkin least-squares (GLS) methodology in terms of extra-stress, velocity and pressure. The GLS method, introduced by Hughes et al. (1986) in the context of the Stokes problem for Newtonian fluidflows, allows the use of combinations of equal-order finite element interpolations and remains stable even for elastic- and inertiadominated fluid flows. Some steady simulations of Oldroyd-B fluids, flowing over a slot, are herein carried out. The influence of inertia and fluid viscoelasticity is taken into account ranging the Reynolds and Weissenberg numbers for relevant values of this flow. The results are in accordance to the viscoelastic literature and reassure the fine stability features of the GLS formulation.

A new absolutely stable simplified Galerkin Least-Squares finite element method using nonconforming element for the Stokes problem

We propose a new absolutely stable simplified Galerkin Least-Squares (SGLS) finite element method using equal order linear nonconforming element for the Stokes problem. We prove that our method is absolutely stable and has optimal error estimates. Numerical tests confirm and illustrate the theoretical analysis. Compare with the conforming Galerkin Least-Squares method and conforming SGLS method, our nonconforming method is more accuracy.

Numerical approximations for flow of viscoplastic fluids in a lid-driven cavity

The effect of inertia and rheology parameters on the flow of viscoplastic fluids inside a lid-driven cavity is investigated using a stabilized finite element approximation. The viscoplastic material behavior is described by the model introduced by de Souza Mendes and Dutra [30] – herein called SMD fluid – which is essentially a regularized viscosity function that involves only rheological properties of the material. The incompressible balance equations are coupled with the non-linear SMD model and are approximated by a multi-field Galerkin least-squares method in terms of extra-stress, pressure and velocity. The results obtained confirm the stability features of the multi-field formulation and the appropriateness of the rheological stress regularization introduced by the SMD fluid. The influence of inertia and rheological parameters on the morphology of the material yield surfaces is analyzed and discussed.