Galerkin Least-squares 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.

The effect of slip on the development of flow separation due to a bump in a channel based on a two-fluid turbulence model

In the paper, two-fluid turbulence model simulations were used to calculate the flow of a statistically two-dimensional turbulent boundary layer over a hump. In this work, the finite element method is used for the numerical implementation of the turbulence equations. Galerkin least squares (GLS) stabilization was used to stabilize the discretized equations. The results of the two fluid models were compared with experimental measurements by Matai, R., Durbin P. A. and the SST turbulence model which exists in the Comsol Multiphysics software package. The implementation of the Comsol Multiphysics software package showed the stability and good accuracy of the two-fluid turbulence model than the SST turbulence model.

Study of the SST turbulence model for the 2D NASA wall-mounted hump separated FLOW problem with plenum case

The article presents the results of the standard SST turbulence model in the Comsol Multiphysics 6.1 software package for 2D NASA wall-mounted hump separated flow with plenum case problems, which are presented in the NASA database. In this work, the finite element method is used for the numerical implementation of the turbulence equations. To stabilize the discretized equations, Galerkin least squares (GLS) stabilization and crosswind propagation stabilization were used. The results were taken for different computational grids. The results obtained are compared with the results of experimental data.

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.

Turbulent flow out of a convex curve in a channel using the SST turbulence model

The paper presents the results of the SST turbulence model in the Comsol Multiphysics software package for problems of convex boundary layer curvature, which are presented in the NASA database. In this work, the finite element method is used for the numerical implementation of the turbulence equations. To stabilize the discretized equations, Galerkin least squares (GLS) stabilization and crosswind propagation stabilization were used. The results obtained are compared with the results of experimental data.

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.

Down-Hole Electromagnetic Heating of Deep Aquifers for Renewable Energy Storage

Electromagnetic (EM) heating is an emerging method for storing renewable energy, such as photovoltaic solar and wind electric power, into aquifers. We investigate how the captured energy increases the temperature of a prototypical deep aquifer for a six-month period and then to which extent the stored energy can be recovered during the consecutive six months. Water injected at a constant flow rate is simultaneously heated using a high-frequency electromagnetic microwave emitter operating at the water natural resonance frequency of 2.45 GHz. The coupled reservoir flow and EM heating are described using Darcy’s and the energy balance equations. The latter includes a source term accounting for the EM wave propagation and absorption, modeled separately using Maxwell’s equations. The equations are solved numerically by the Galerkin least-squares finite element method. The approach was validated using EM-heating input data obtained from controlled laboratory experiments and then was applied to the aquifer. We found that after six years of alternate storage and recovery, up to 77% of the injected energy is recovered when considering realistic heat losses estimated from field data. Even when heat losses are increased by a factor of two, up to 69% of the injected energy is recovered in this case. This shows that down-hole EM heating is a highly effective method for storing renewable energies, capable of helping to solve their inherent intermittency.

PINNs and GaLS: A Priori Error Estimates for Shallow Physics Informed Neural Networks Applied to Elliptic Problems

Physics Informed Neural Networks (PINNs) have recently gained popularity for solving partial differential equations, given the fact they escape the curse of dimensionality. In this paper, we present Physics Informed Neural Networks as an underdetermined point matching collocation method then expose the connection between Galerkin Least Squares (GALS) and PINNs, to develop an a priori error estimate, in the context of elliptic problems. In particular, techniques that belong to the realm of least squares finite elements and Rademacher complexity analysis are used to obtain the error estimate.

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.

A Stress Mapping Immersed Boundary Method for Viscous Flows

This work introduces an immersed boundary method for two-dimensional simulation of incompressible Navier-Stokes equations. The method uses flow field mapping on the immersed boundary and performs a contour integration to calculate immersed boundary forces. This takes into account the relative location of the immersed boundary inside the background grid elements by using inverse distance weights, and also considers the curvature of the immersed boundary edges. The governing equations of the fluid mechanics are solved using a Galerkin-Least squares finite element formulation. The model is validated against a stationary and a vertically oscillating circular cylinder in a cross flow. The results of the model show acceptable accuracy when compared to experimental and numerical results.

A least-squares Galerkin approach to gradient and Hessian recovery for nondivergence-form elliptic equations

Abstract We propose a least-squares method involving the recovery of the gradient and possibly the Hessian for elliptic equation in nondivergence form. As our approach is based on the Lax–Milgram theorem with the curl-free constraint built into the target (or cost) functional, the discrete spaces require no inf-sup stabilization. We show that standard conforming finite elements can be used yielding a priori and a posteriori convergence results. We illustrate our findings with numerical experiments with uniform or adaptive mesh refinement.

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.

An analysis of classical techniques for consistent stabilisation of the advection–diffusion–reaction equation finite element solution

We analyse and study instability problems related to the solution of the advection–diffusion–reaction equation (ADR) using a standard finite element scheme. With this aim, this work has been carried out in the following way: first, three weak formulations are obtained from the general problem. In specific, we study the existence and uniqueness of the solution for each of the aforementioned formulations. Second, we analyse the general theory of consistent stabilisation techniques for the ADR equation, which includes: streamline upwind/Petrov-Galerkin (SUPG), and Galerkin/least-squares (GLS). Third, we study and develop, for linear triangular elements, two of the most important subgrid-scale techniques, i.e. algebraic subgrid scale (ASGS), and orthogonal subgrid scale (OSS). This includes the study of an expression for a stabilisation parameter based on an ADR equation's Fourier analysis. Finally, as conclusion, all these stabilisation techniques are put in context with the SUPG technique for a better comparison as well as understanding of their underlying features for linear triangular elements.

The variational multiscale formulation for the fully-implicit log-morphology equation as a tensor-based blood damage model.

We derive a variational multiscale (VMS) finite element formulation for a viscoelastic, tensor-based blood damage model. The tensor equation is numerically stabilized by a logarithmic shape tensor description that prevents unphysical, negative eigenvalues. The resulting VMS stabilization terms for this so-called log-morph equation are presented together with their special numerical treatment. Results for a 2D rotating stirrer test case obtained from log-morph simulations with both SUPG and VMS stabilization show significantly improved numerical behavior if compared with Galerkin/least squares (GLS) stabilized untransformed morphology simulation results. The newly proposed method is also successfully applied to a state-of-the-art centrifugal ventricular assist device (VAD), and clear advantages of the VMS stabilization compared with the SUPG-stabilized formulation are presented.

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.

Hartmann and Reynolds Numbers Effects in the Newtonian Blood Flow of a Bifurcated Artery with an Overlapping Stenosis

Abstract Blood flow through a bifurcated artery with the presence of an overlapping stenosis located at parent’s arterial lumen under the action of a uniform external magnetic field is studied in this paper. Blood is treated as an electrically conducting fluid which exhibits the Magnetohydrodynamics principle and it is characterized by a Newtonian fluid model. The governing equations are discretized using a stabilization technique of finite element known as Galerkin least-squares. The maximum velocity and pressure drop evaluated in this present study are compared with the results found in previous literature and COMSOL Multiphysics. The solutions found in a satisfactory agreement, thus verify the source code is working properly. The effects of dimensionless parameters of Hartmann and Reynolds numbers in the fluid’s velocity and pressure are examined in details with further scientific discussions.