Streamline Upwind Petrov-Galerkin Research Articles

A comparison study of spatial and temporal schemes for flow and transport problems in fractured media with large parameter contrasts on small length scales

AbstractIn this work, various high-accuracy numerical schemes for transport problems in fractured media are further developed and compared. Specifically, to capture sharp gradients and abrupt changes in time, schemes with low order of accuracy are not always sufficient. To this end, discontinuous Galerkin up to order two, Streamline Upwind Petrov-Galerkin, and finite differences, are formulated. The resulting schemes are solved with sparse direct numerical solvers. Moreover, time discontinuous Galerkin methods of order one and two are solved monolithically and in a decoupled fashion, respectively, employing finite elements in space on locally refined meshes. Our algorithmic developments are substantiated with one regular fracture network and several further configurations in fractured media with large parameter contrasts on small length scales. Therein, the evaluation of the numerical schemes and implementations focuses on three key aspects, namely accuracy, monotonicity, and computational costs.

Multilevel Monte Carlo Methods for Stochastic Convection–Diffusion Eigenvalue Problems

We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection–diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element (FE) discretizations of the eigenvalue problem on a hierarchy of increasingly finer meshes. For the discretized, algebraic eigenproblems we use both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA), providing an analysis of the cost in each case. By studying the variance on each level and adapting classical FE error bounds to the stochastic setting, we are able to bound the total error of our MLMC estimator and provide a complexity analysis. As expected, the complexity bound for our MLMC estimator is superior to plain Monte Carlo. To improve the efficiency of the MLMC further, we exploit the hierarchy of meshes and use coarser approximations as starting values for the eigensolvers on finer ones. To improve the stability of the MLMC method for convection-dominated problems, we employ two additional strategies. First, we consider the streamline upwind Petrov–Galerkin formulation of the discrete eigenvalue problem, which allows us to start the MLMC method on coarser meshes than is possible with standard FEs. Second, we apply a homotopy method to add stability to the eigensolver for each sample. Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster convergence of QMC, this improves the overall complexity. We provide detailed numerical results comparing our different strategies to demonstrate the practical feasibility of the MLMC method in different use cases. The results support our complexity analysis and further demonstrate the superiority over plain Monte Carlo in all cases.

SUPG-based stabilized finite element computations of convection-dominated 3D elliptic PDEs using shock-capturing

This study is interested in stabilized finite element computations of advection-dominated convection–diffusion-type partial differential equations. The governing equations are assumed to be defined on the 3D unit cube and stationary. Towards that end, we stabilize the standard (Bubnov–) Galerkin finite element method (GFEM), which typically suffers from yielding numerical approximations polluted with node-to-node nonphysical oscillations in convection dominance, by employing the streamline-upwind/Petrov–Galerkin (SUPG) formulation. In order to achieve better shock representations near sharp layers, the SUPG-stabilized formulation is also enhanced with a simple and residual-based shock-capturing mechanism, the so-called YZβ technique. Several test computations are performed to assess the efficiency and robustness of the proposed formulation. The test problems are examined under more challenging conditions than those previously studied in the literature, i.e., for flows substantially dominated by convection phenomena. The numerical results and comparisons demonstrate that the SUPG-YZβ combination significantly eliminates both globally spread and localized numerical instabilities. Beyond that, the proposed formulation accomplishes that by using only linear elements on relatively coarser meshes and without making use of any adaptive mesh strategies.

A Streamline Upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations Under Optimal Control

Abstract In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.

Vectorial finite element method for neutron transport solving with preconditioning GMRES acceleration

This work proposed a vectorial finite element method (VFEM) for solving neutron transport equation with high accuracy and convergence. The Streamline Upwind Petrov–Galerkin method is used to establish the VFEM for accuracy, and the traditional discrete-ordinate neutron transport equation is organized in vectorial form for high stability and convergence. To further improve the convergence efficiency of VFEM, the preconditioning GMRES is introduced for solving the VFEM linear system, and different preconditioners are imposed, including the Jacobian and Block-Jacobian. Numerical results showed that the proposed VFEM has high accuracy and adaptability for multi-group and multi-dimensional neutron transport simulation. Further performance analyses showed that the proposed VFEM has higher efficiency than the traditional finite element method, while the application of preconditioning GMRESs can further improve the convergence efficiency. This work can provide an innovative and effective unstructured-mesh neutron transport solution format, and provide a valuable reference for further improvement of other numerical methods.

A cell-based smoothed finite element model for the analysis of turbulent flow using realizable k-ε model and mixed meshes

Smoothed finite element method (S-FEM) has been widely employed in computational mechanics, particularly in computational solid mechanics (CSM) and, to a lesser extent, in computational fluid dynamics (CFD). The present work focuses on the development of S-FEM for solving three-dimensional incompressible turbulent flow problems using the realizable k-ε model. The Streamline Upwind Petrov-Galerkin approach combined with Stabilized Pressure Gradient Projection (SUPG/SPGP) was used to restrain/control the spatial oscillation and instability in conjunction with mixed meshes employed to discretize complex geometries. The presented turbulent cell-based S-FEM (CS-FEM) was validated under several common flow types, including mesh turbulence, turbulent channel flow, turbulent flow past a circular cylinder, flow separation around an obstacle, flow in the dimpled channel, and flow in the upper human airway. In particular, the presented work exhibited the latent capability of the CS-FEM in solving turbulent flows near boundary regions and strong convection regions. Meanwhile, the results of the CS-FEM were compared to those of the lowest equal-order finite element method (i.e. P1–P1 or Q1–Q1 elements) in simulating turbulent flows, and it was found that the CS-FEM had better agreement with other published works.

Performance Analysis of Thermal and Surface Roughness Effect of Slider Bearings with Unsteady Fluid Film Lubricant Using Finite Element Method

The streamline upwind Petrov-Galerkin (SUPG) finite element method was used in this study to investigate the thermal and surface roughness effects on an inclined slider bearing with an unsteady fluid film. One-dimensional transverse and longitudinal surface roughness models were considered with the supposition that roughness is stochastic and has a Gaussian random distribution. For simplicity of numerical computation, the irregularity caused by the texture of the surface is transformed into a regular domain. The bearing performance of the combined effect is lower than the thermal and surface roughness effects of the one-dimensional longitudinal surface roughness for all modified Reynolds numbers of nonparallel slider bearings; this means that for nonparallel (w=0.4) between the surface roughness effect and the combined effect condition, there is a decrease of 13% in load-carrying capacity performance and a minimal change in friction force, respectively. However, in the case of nonparallel one-dimensional transverse type slider bearings, the bearing performance of the thermal effect is lower than the combined and surface roughness effects for all modified Reynolds numbers, where between the combined effect and the thermal effect condition, there is a reduction of 19% in load-carrying capacity performance and 2% in friction force practically for all changed Reynolds values, respectively. Furthermore, the combined effects at various temperatures have been investigated. As a result, in both longitudinal and transverse models, in the case of the pad temperature being lower than the slider, the load-carrying capacity performance is higher than in other cases for nonparallel slider bearings, whereas when the slider temperature is lower than the pad temperature, the drag frictional force is the leading one in both models. In general, considering surface texture and inertial effects will increase the performance of a slider. The results obtained are displayed using figures and tables.

SUPG-stabilized stabilization-free VEM: a numerical investigation

<abstract><p>We numerically investigate the possibility of defining Stabilization-Free Virtual Element discretizations–i.e., Virtual Element Method discretizations without an additional non-polynomial non-operator-preserving stabilization term–of advection-diffusion problems in the advection-dominated regime, considering a Streamline Upwind Petrov-Galerkin stabilized formulation of the scheme. We present numerical tests that assess the robustness of the proposed scheme and compare it with a standard Virtual Element Method.</p></abstract>

WAVEx: Stabilized finite elements for spectral wind wave models using FEniCSx

The prediction of the wind wave spectrum of the ocean using numerical models are an important tool for researchers, engineers, and communities living in coastal areas. The governing equation of the wind wave models, the Wave Action Balance Equation, presents unique challenges for implementing reliable numerical models because it is highly advective, highly nonlinear and high dimensional. Historically, most operational models have utilized finite difference methods, others have used finite volume methods but relatively few attempts at using finite element methods. In this work, we seek to fill this gap by investigating several different finite element discretizations of the Wave Action Balance Equation. The methods, which include streamline upwind Petrov–Galerkin (SUPG), least squares, and discontinuous Galerkin, are implemented and convergence properties are examined for some simplified 2-D test cases. Then, a new spectral wind wave model, WAVEx, is formulated and implemented for the full problem setting. WAVEx uses continuous finite elements along with SUPG stabilization in geographic/spectral space that allows for fully unstructured triangular meshes in both geographic and spectral space. For propagation in time, a second order fully implicit finite difference method is used. When source terms are active, a second order operator splitting scheme is used to linearize the problem. In the splitting scheme, propagation is solved using the implicit method and the nonlinear source terms are treated explicitly. Several test cases, including analytic tests and laboratory experiments, are demonstrated and results are compared to analytic solutions, observations, as well as output from another model that is used operationally.

A time-consistent stabilized finite element method for fluids with applications to hemodynamics

Several finite element methods for simulating incompressible flows rely on the streamline upwind Petrov–Galerkin stabilization (SUPG) term, which is weighted by tau _{text{SUPG}}. The conventional formulation of tau _{text{SUPG}} includes a constant that depends on the time step size, producing an overall method that becomes exceedingly less accurate as the time step size approaches zero. In practice, such method inconsistency introduces significant error in the solution, especially in cardiovascular simulations, where small time step sizes may be required to resolve multiple scales of the blood flow. To overcome this issue, we propose a consistent method that is based on a new definition of tau _{text{SUPG}}. This method, which can be easily implemented on top of an existing streamline upwind Petrov–Galerkin and pressure stabilizing Petrov–Galerkin method, involves the replacement of the time step size in tau _{text{SUPG}} with a physical time scale. This time scale is calculated in a simple operation once every time step for the entire computational domain from the ratio of the L2-norm of the acceleration and the velocity. The proposed method is compared against the conventional method using four cases: a steady pipe flow, a blood flow through vascular anatomy, an external flow over a square obstacle, and a fluid–structure interaction case involving an oscillatory flexible beam. These numerical experiments, which are performed using linear interpolation functions, show that the proposed formulation eliminates the inconsistency issue associated with the conventional formulation in all cases. While the proposed method is slightly more costly than the conventional method, it significantly reduces the error, particularly at small time step sizes. For the pipe flow where an exact solution is available, we show the conventional method can over-predict the pressure drop by a factor of three. This large error is almost completely eliminated by the proposed formulation, dropping to approximately 1% for all time step sizes and Reynolds numbers considered.

High-order moving immersed boundary and its application to a resolved CFD-DEM model

The sharp interface immersed boundary method (IBM) is a powerful tool to simulate the flow around moving objects. This paper extends this method to simulate the motion of particles immersed in a fluid. We introduce a new implicit coupling scheme between the solid and fluid phases, verify the high–order of convergence of the scheme, and validate it using four cases. The fluid dynamics is simulated using a streamline upwind Petrov–Galerkin and pressure-stabilizing Petrov–Galerkin (SUPG/PSPG) stabilizations. The particles’ dynamics and contact are solved using Newton’s second law of motion and a soft sphere contact model derived from the discrete element method. The first validation case is the sedimentation of a lone particle with a Reynolds number smaller than 32. The second validation case is the rise of a positively buoyant particle at a high Reynolds number (2500). The third validation case demonstrates the stability of the scheme when particles are in contact and exhibit the classical drafting, kissing, and tumbling (DKT) behavior. Finally, we use the scheme to study the Boycott effect with a larger number of particles (64) to show the stability of the scheme in cases with numerous particle–particle and particle–wall contacts.

The moving finite element method with streamline-upwind Petrov–Galerkin for magnetohydrodynamic flows problems at high Hartmann numbers

It is widely acknowledged that both the streamline-upwind Petrov–Galerkin (SUPG) method and moving mesh method can enhance the stability of finite element method in solving convection-dominated convection–diffusion problems. However, in the case of strong convection-dominated conditions, they still exhibit numerical oscillations and the solutions will not even converge. In order to address similar phenomena observed in solving magnetohydrodynamic (MHD) flow problems at high Hartmann numbers with strong convective dominance, the MHD equation was first decoupled into two convection–diffusion equations. Subsequently, a new approach called the moving mesh streamline-upwind Petrov–Galerkin method (MM-SUPG) was proposed by coupling the moving mesh technique with SUPG method. The MM-SUPG method contains two independent part: the finite element method with SUPG for solving the decoupled MHD equation and a mesh-redistribution algorithm. To verify its effectiveness for solving the MHD equation at high Hartmann number, a classical MHD fluid flow problem was numerically tested using both the MM-SUPG method and the moving finite element method (MFEM). Compared to the MFEM, the proposed method not only yields stable numerical solutions under strong convective dominance while reducing unnecessary false oscillations, but also improves calculation accuracy to some extent using the same mesh.

Image-guided subject-specific modeling of glymphatic transport and amyloid deposition

The glymphatic system is a brain-wide system of perivascular networks that facilitate exchange of cerebrospinal fluid (CSF) and interstitial fluid (ISF) to remove waste products from the brain. A greater understanding of the mechanisms for glymphatic transport may provide insight into how amyloid beta (Aβ) and tau agglomerates, key biomarkers for Alzheimer’s disease and other neurodegenerative diseases, accumulate and drive disease progression. In this study, we develop an image-guided computational model to describe glymphatic transport and Aβ deposition throughout the brain. Aβ transport and deposition are modeled using an advection–diffusion equation coupled with an irreversible amyloid accumulation (damage) model. We use immersed isogeometric analysis, stabilized using the streamline upwind Petrov–Galerkin (SUPG) method, where the transport model is constructed using parameters inferred from brain imaging data resulting in a subject-specific model that accounts for anatomical geometry and heterogeneous material properties. Both short-term (30-min) and long-term (12-month) 3D simulations of soluble amyloid transport within a mouse brain model were constructed from diffusion weighted magnetic resonance imaging (DW-MRI) data. In addition to matching short-term patterns of tracer deposition, we found that transport parameters such as CSF flow velocity play a large role in amyloid plaque deposition. The computational tools developed in this work will facilitate investigation of various hypotheses related to glymphatic transport and fundamentally advance our understanding of its role in neurodegeneration, which is crucial for the development of preventive and therapeutic interventions.

A framework to model the hydraulic fracturing with thermo-hydro-mechanical coupling based on the variational phase-field approach

In this research, a numerical framework for hydraulic fracturing has been formulated, incorporating thermo-hydro-mechanical (THM) coupled effects. While previous studies have reported various hydraulic fracturing models based on the phase-field method, a THM coupling scheme grounded in the variational phase-field approach remains unexplored. The THM coupling is of paramount importance for understanding underground fracture propagation. This study’s principal innovation lies in its extension of the variational phase field model, introducing it to a thermoporoelastic medium, thereby integrating fluid flow and heat transfer elements in fractured materials. In our advanced model, the fluid flow and heat transfer in the matrix and fracture are modeled independently. Subsequently, employing a set of reasonable assumptions, we introduce unified flow and heat transfer equations derived from phase-field calculus. A pivotal contribution of our work is the development of an iterative solution algorithm based on the fixed stress method, pivotal for tackling the THM coupled system. The proposed equations are discretized employing the FEM. Additionally, to address the numerical oscillations in advection-dominant issues that manifest in the heat transfer model, we have applied a generalized Streamline Upwind Petrov Galerkin method for 8-node elements. The accuracy of fracture propagation was assessed by comparing the numerical results of the evolution of fracture length, fracture width at the injection point, and the injection pressure under the K regime with the corresponding analytical solution. Furthermore, the process of heat transfer within a fractured domain, often referred to as the Barends problem, has been studied by proposed model. Additionally, some numerical experiments were conducted to gain insights into the intricate interplay between the thermal–hydraulic–mechanical system during hydraulic fracturing, thereby further emphasizing the significance of our advanced model.

Stabilized isogeometric formulation of the multi-network poroelasticity and transport model (MPET[formula omitted]) for subcutaneous injection of monoclonal antibodies

The MPET2 model couples the multi-network poroelastic theory (MPET) with solute transport equations and provides predictions of the material deformation, fluid dynamics, and solute transport in different compartments of a deformable multiple-porosity medium. MPET2 offers a comprehensive framework for understanding complex porous media across multiple disciplines. Examples of its applications include studying rock formations, soil mechanics and subsurface reservoirs, investigating biological tissues, modeling groundwater flow and contaminant transport, and optimizing the design of porous materials. Despite the wide range of applications of the model, its numerical discretization has received little attention. Here we propose a stabilized formulation of the MPET2 model. To address the unique challenges posed by the discretization of the MPET2 model, we use multiple techniques including the Fluid Pressure Laplacian stabilization, Streamline Upwind Petrov–Galerkin stabilization, and discontinuity capturing. Our spatial discretization is based on Isogeometric Analysis with higher-order continuity basis functions. The fully discretized governing equations are solved simultaneously with a monolithic algorithm. We perform a convergence study of the proposed formulation. Then, we conduct a series of simulations of subcutaneous injection of monoclonal antibodies under different injection conditions. Our simulations show that the stabilized MPET2 formulation can provide oscillation-free solutions for tissue deformation, fluid flow in the interstitial tissue, blood vessels, and lymphatic vessels, drug absorption in blood vessels and lymphatic vessels, as well as drug transport in each compartment. We also study the effects of different injection conditions on drug absorption, showing the potential of the proposed model and algorithm in the future optimization of injection strategy.

Simulation of Drag Crisis in Flow Past a Circular Cylinder using 2D Computations

The unsteady incompressible 2D Navier-Stokes equations are solved using a finite element method with Streamline-Upwind/Petrov-Galerkin (SUPG) and Pressure-Stabilizing/Petrov- Galerkin (PSPG) stabilization terms for high Reynolds number (105 to I06) flow past a cylinder to stimulate drag-crisis. The mean drag coefficient for various Re are compared with experimental data. It is observed that the low at Re =106 separates much later, from the cylinder surface as compared to that at Re =105. The wake at Re =106 is narrower than that at Re = 105. The time-averaged flow field shows a standing bubble just downstream of the flow separation point for the Re = 105 flow. The bubble is not observed for the flow at Re = 106. The energy spectrum of the flow exhibits the characteristics of 2D isotropic turbulence.

Comparing different stabilization strategies for reduced order modeling of viscoelastic fluid flow problems

This paper presents a computational framework for model order reduction of viscoelastic fluid flows, with a particular focus on the stabilization of the reduced order model (ROM) and the efficient approximation of non-linear terms appearing in the governing equations. We compare three different stabilization approaches, two of which are based on offline stabilization only and one that uses both offline and online stabilization. Two main techniques for the treatment of non-linear terms are examined and compared: the discrete empirical interpolation method, and a reduced quadrature method based on a sparse recovery technique. Numerical experiments are conducted on two benchmark flows such as the flow past a sphere and the 4:1 contraction flow problem. We show that significant speedups can be achieved using offline streamline upwind Petrov–Galerkin stabilization together with hyper-reduction. We also discuss the potential accuracy losses of the ROM that could result from mesh-related issues of the full-order model (FOM) and propose possible remedies.

QPDE: Quantum Neural Network Based Stabilization Parameter Prediction for Numerical Solvers for Partial Differential Equations

We propose a Quantum Neural Network (QNN) for predicting stabilization parameter for solving Singularly Perturbed Partial Differential Equations (SPDE) using the Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. SPDE-Q-Net, a QNN, is proposed for approximating an optimal value of the stabilization parameter for SUPG for 2-dimensional convection-diffusion problems. Our motivation for this work stems from the recent progress made in quantum computing and the striking similarities observed between neural networks and quantum circuits. Just like how weight parameters are adjusted in traditional neural networks, the parameters of the quantum circuit, specifically the qubits’ degrees of freedom, can be fine-tuned to learn a nonlinear function. The performance of SPDE-Q-Net is found to be at par with SPDE-Net, a traditional neural network-based technique for stabilization parameter prediction in terms of the numerical error in the solution. Also, SPDE-Q-Net is found to be faster than SPDE-Net, which projects the future benefits which can be earned from the speed-up capabilities of quantum computing.

Space–time computational flow analysis: Unconventional methods and first-ever solutions

The Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method was introduced in 1990 as a moving-mesh method for computational analysis of flows with moving boundaries and interfaces (MBI), which is a wide class of problems that includes fluid–particle and fluid–structure interactions and free-surface and multi-fluid flows. The method was inspired by Thomas Hughes’s 1987 work on space–time finite element methods for elastodynamics. The original DSD/SST method is now called “ST-SUPS”, reflecting its stabilization components, which are the Streamline-Upwind/Petrov–Galerkin (SUPG) method, pioneered by Hughes, and the Pressure-Stabilizing/Petrov–Galerkin (PSPG) method, inspired by Hughes’s work on a Stokes-flow Petrov–Galerkin formulation allowing equal-order interpolations for velocity and pressure. Hughes’s work on the residual-based variational multiscale (RBVMS) method inspired the ST-VMS method, which is the VMS version of the DSD/SST. A number of special methods were introduced in connection with the core methods ST-SUPS and ST-VMS. Hughes’s work on isogeometric analysis (IGA) inspired one of those special methods, the “ST-IGA”, with IGA basis functions not only in space but also in time. The core and special ST methods enabled first-ever solutions in some of the most challenging classes of MBI problems, including particle-laden flows, spacecraft parachute fluid–structure interactions, and car and tire aerodynamics. We provide an overview of the ST methods inspired by Hughes’s work and highlight some of the first-ever solutions in these three classes of MBI problems.

Stabilization in 3‐D FEM and solution of the MHD equations

In this study, we consider the numerical solution of convection‐diffusion typed equations defined in 3‐D domain using the finite element method (FEM) with the stabilized version in order to fill the gap in literature on this area in the sense of different applications. The formulation processed is performed step by step. For this purpose, we started with FEM formulation of the 3‐D Laplace equation. Then, the obtained formulation is extended to the convection‐diffusion equation with standard Galerkin FEM. In order to solve the stability problems for the convection dominated case, the most popular stabilized formulation known as the streamline upwind Petrov‐Galerkin (SUPG) type stabilization method is considered. As a further step, obtained stabilized formulation is used in the solutions of the MHD equations by transforming the coupled system of differential equations to the decoupled convection‐diffusion type equations defined on different 3‐D domains. For the comparison purpose, MHD equations are also formulated with finite volume method (FVM). Obtained numerical results are displayed in terms of a table and figures as the 2‐D slices of the 3‐D plots in order to visualize the effect and accuracy of the proposed numerical scheme over the different test problems.