Upwind Petrov-Galerkin Method Research Articles

Corium interface flow dynamics investigation during severe accident in pressurized water reactors using compressive advection interface capturing method

Past nuclear accident occurrences raised strong concerns which led to research on nuclear safety. One of the major causes of nuclear accidents is the impeded circulation of core coolant, leading to decay heat removal cessation and rapid temperature rise. If uncontrolled, this results in critical heat flux, loss of coolant accidents, and core dryout. Detailed melted core relocation (i.e., nuclear fuel, graphite, and zircaloy) needs to be investigated through interface capture and multimaterial flow model coupling, which have not been done in previous studies. This work aims to investigate the impacts of temperature and core material composition on the flow dynamics during core relocation. In this study, mass fraction is discretized using a streamlined upwind Petrov–Galerkin method spatially and a modified Crank–Nicolson method temporally to accurately capture fluid interfaces using a high-order accurate flux-limiter. Two core material composition cases (individual material properties case and bulk material properties case) were considered to assess the impact of temperature and core materials composition on both flow dynamics and computational time. Temperature has a significant impact on core material transport and corium flow dynamics during core relocation. Bulk materials properties case has greater impact of temperature on its corium resulting in faster materials transport, but with higher computation time.

A comparison of variational upwinding schemes for geophysical fluids, and their application to potential enstrophy conserving discretisations

Methods for stabilising the turbulent cascade of potential enstrophy are analysed and compared for a compatible finite element discretisation of the rotating shallow water equations. These different approaches to upwinding the potential vorticity include the well-known anticipated potential vorticity method (APVM), streamwise upwind Petrov-Galerkin (SUPG) method, and a more recent method where the trial functions are evaluated downstream within the reference element. In all cases the upwinding scheme conserves both potential vorticity and energy, since the antisymmetric structure of the equations is preserved. The APVM leads to a symmetric definite correction to the potential enstrophy that is dissipative and inconsistent. The SUPG scheme introduces a consistent correction to the APVM scheme that acts as a backscatter term ensuring a richer depiction of turbulent dynamics. The downwinded trial function formulation results in the advection of downwind corrections, which analysis and numerical experiments show to be quantitatively similar to the SUPG scheme for a turbulent shear flow. The main difference between the SUPG and downwinded trial function schemes is in the energy conservation and residual errors. If just two nonlinear iterations are applied then the energy conservation errors are improved for the downwinded trial function formulation, reflecting a smaller residual error than for the SUPG scheme.We also present new temporal formulations by which potential enstrophy is exactly integrated across each time level. Results using these formulations are observed to be stable in the absence of any dissipation, despite the aliasing of grid scale turbulence. Using such a formulation and the APVM with a coefficient O(100) times smaller that its regular value leads to turbulent spectra that are greatly improved at the grid scale over the SUPG and downwinded trial function formulations with unstable potential enstrophy errors.

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.

Saddle point least squares discretization for convection-diffusion

We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models.

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.

A Variational Multiscale Method for Particle Dispersion Modeling in the Atmosphere

A LES model is proposed to predict the dispersion of particles in the atmosphere in the context of Chemical, Biological, Radiological and Nuclear (CBRN) applications. The code relies on the Finite Element Method (FEM) for both the fluid and the dispersed solid phases. Starting from the Navier-Stokes equations and a general description of the FEM strategy, the Streamline Upwind Petrov-Galerkin (SUPG) method is formulated putting some emphasis on the related assembly matrix and stabilization coefficients. Then, the Variational Multiscale Method (VMS) is presented together with a detailed illustration of its algorithm and hierarchy of computational steps. It is demonstrated that the VMS can be considered as a more general version of the SUPG method. The final part of the work is used to assess the reliability of the implemented predictor/multicorrector solution strategy.

Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements

Within finite element models of fluids, vector-valued fields such as velocity or momentum variables are commonly discretised using the Raviart-Thomas elements. However, when using the lowest-order quadrilateral Raviart-Thomas elements, standard finite element discretisations of the vector transport equation typically have a low order of spatial accuracy. This paper describes two schemes that improve the accuracy of transporting such vector-valued fields on two-dimensional curved manifolds.The first scheme that is presented reconstructs the transported field in a higher-order function space, where the transport equation is then solved. The second scheme applies a mixed finite element formulation to the vector transport equation, simultaneously solving for the transported field and its vorticity. An approach to stabilising this mixed vector-vorticity formulation is presented that uses a Streamline Upwind Petrov-Galerkin (SUPG) method. These schemes are then demonstrated, along with their accuracy properties, through some numerical tests. Two new test cases are used to assess the transport of vector-valued fields on curved manifolds, solving the vector transport equation in isolation. The improvement of the schemes is also shown through two standard test cases for rotating shallow-water models.

On the performance of a Chimera-FEM implementation to treat moving heat sources and moving boundaries in time-dependent problems

Problems with moving sources and moving inner boundaries in transient regime are of high interest in many research fields and engineering applications. One approach to properly tackle such problems is based on the Chimera method for non-matching grids, where each moving object is defined on a fine mesh that moves across the fixed coarse background. In this way, the high gradients around moving sources or boundaries are captured by the fine mesh without need of globally fine fixed meshes or adaptive refinement. In this work, Chimera is implemented in the framework of the finite element method, following an arbitrary Lagrangian–Eulerian formulation for the moving meshes and a standard Eulerian formulation for the fixed background mesh. Further, in case of convection-dominated problems, the scheme is stabilized using the streamline upwind Petrov–Galerkin method. The coupling between the moving fine meshes and the coarse fixed one is achieved via Dirichlet boundary conditions and a high-order interpolation algorithm. The performance of the proposed methodology in terms of accuracy and stability is assessed by means of numerical tests to be compared with equivalent problems solved using fixed meshes. Further tests serve to highlight the good performance of the proposed Chimera-based finite element method to address both convection-dominated problems with multiple moving boundaries and sources, and three-dimensional arc welding processes. • An algebraic-based Chimera-FEM scheme for unsteady problems is proposed. • Remeshing is avoided for large movements of heat sources and boundaries. • Advective-dominant problems can be addressed with Chimera-FEM and SUPG. • Good accuracy is obtained compared to conventional one-domain approaches. • The scheme is suitable to be used on 3D welding process simulations.

Extending FEniCS to work in higher dimensions using tensor product finite elements

We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is structured as a Cartesian product of two lower dimensional subdomains. In this study we consider Dirichlet problems for scalar linear partial differential equations, though the methodology can be extended to non-linear problems. The utilization of tensor product finite elements allows us to construct a global system of linear algebraic equations that only relies on the finite element infrastructure of the lower dimensional subdomains contained in FEniCS. We demonstrate the effectiveness of our methodology in four distinctive test cases. The first test case is a Poisson equation posed in a four dimensional domain which is a Cartesian product of two unit squares solved using the classical Galerkin finite element method. The second test case is the wave equation in space–time, where the computational domain is a Cartesian product of a two dimensional space grid and a one dimensional time interval. In this second case we also employ the Galerkin method. The third test case is an advection dominated advection–diffusion equation where the global domain is a Cartesian product of two one dimensional intervals in which the streamline upwind Petrov–Galerkin method is applied to ensure discrete stability. The final test case uses the Galerkin approach to solve a Poisson problem on a Cartesian product of two intervals with a spatially varying, non-separable diffusivity term In all cases, a p=1 basis is used and optimal L2 convergence rates of order hp+1 of the errors are achieved with respect to h refinement.

A new continuous finite element SN method for solving first-order neutron transport equation

A continuous finite element method (FEM) based on B-spline wavelet on the interval (BSWI) is proposed to solve the first-order neutron transport equation with discrete ordinate (SN) angular discretization. Two technologies are applied in this method, one of technologies is that the streamline upwind Petrov-Galerkin (SUPG) method is utilized to stabilize the first-order neutron transport equation when the FEM is used. Another is that the scaling functions of BSWI are applied to replace the usual Lagrange basis function. Based on the above two technologies, the stabilized weak form of neutron transport equation applying the SUPG technology based on BSWI is derived. Finally, some numerical examples are utilized to verify the proposed method. The results demonstrate that the method can treat the void region problem, and compared the FEM with SUPG technology, it can obtain fast convergence and numerical precision under the condition of much less grid. Overall, this method has good accuracy, high computational efficiency and wide application.

A 3D fully thermo–hydro–mechanical coupling model for saturated poroelastic medium

Thermo–hydro–mechanical (THM) coupling prevails in all sorts of underground activities, and numerical simulation is an effective tool to understand THM coupling. The essence of performing THM coupling simulation lies in solving a system of partial differential equations (PDEs) with displacements (U), pore pressure (P) and temperature (T) as the primary variables. In previous studies, only the UPT and U-PT schemes are reported to solve the system PDEs. Another promising scheme UP-T was rarely explored. In addition, although multiple solution schemes are theoretically possible, the performance thereof varies. Few studies have ever compared the performance of different solution schemes. Moreover, the rock mass in hydrothermal fields is highly permeable wherein heat advection dominates. Advection-dominated heat transfer may lead to numerical oscillation, but how to stabilize the simulation in the 3D case was inadequately reported. In this study, we developed a new THM coupling simulator HENGYI in the context of the finite element method. The UP-T solution scheme was verified to be robust enough to simulate large engineering problems. The Streamline Upwind Petrov Galerkin method was generalized to 3D and applied to HENGYI to address the advection-dominated heat transfer. The performance of the UPT, UP-T and U-PT schemes was compared and the applicability of different solution schemes was found to be highly associated with the degree of coupling between primary variables. Great details are provided regarding the development of HENGYI. Thus HENGYI not only bridges the research gaps mentioned above but also provides a comprehensive understanding of THM coupling simulation techniques.

Topology optimization of natural convection heat transfer using SEMDOT algorithm based on the reduced-order model

Due to the strong nonlinear nature and the mutual effect between the temperature field and the velocity field, it is very difficult to solve the natural convection heat transfer problem. To this end, the Gauss Seidel iterative algorithm based on the reduced-order model is proposed to decouple temperature and pressure. In this paper, the governing equations are discretized by finite element method using bilinear shape functions. To suppress the oscillation of the solution of the governing equations, Streamline Upwind Petrov-Galerkin method (SUPG) is adopted. The main idea of the Gauss Seidel iterative algorithm is: given an initial temperature, pressure and temperature are obtained successively by solving only two nonlinear equations, which greatly improves the computational efficiency. Compared with full order Navier-Stokes model, the Gauss Seidel iterative algorithm for reduced order model has similar results. Then, in order to generate accurate boundaries, Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT) based on density method (SIMP) is studied. Finally, some numerical examples demonstrate the feasibility and effectiveness of the proposed scheme.

An Experimental Investigation of Viscoelastic Flow in a Contraction Channel.

In order to assess the predictive capability of the S–MDCPP model, which may describe the viscoelastic behavior of the low-density polyethylene melts, a planar contraction flow benchmark problem is calculated in this investigation. A pressure-stabilized iterative fractional step algorithm based on the finite increment calculus (FIC) method is adopted to overcome oscillations of the pressure field due to the incompressibility of fluids. The discrete elastic viscous stress splitting (DEVSS) technique in combination with the streamline upwind Petrov-Galerkin (SUPG) method are employed to calculate the viscoelastic flow. The equal low-order finite elements interpolation approximations for velocity-pressure-stress variables can be applied to calculate the viscoelastic contraction flows for LDPE melts. The predicted velocities agree well with the experimental results of particle imagine velocity (PIV) method, and the pattern of principal stress difference calculated by the S-MDCPP model has good agreement with the results measured by the flow induced birefringence (FIB) device. Numerical and experimental results show that the S-MDCPP model is capable of accurately capturing the rheological behaviors of branched polymers in complex flow.

Isogeometric schemes in rarefied gas dynamics context

Geometrical complexity is typical of missile and reentry systems, small-scale devices with intrinsic milli-to-micro scale features, or in general topology-rich systems. Additionally, at high altitudes or at microscales, when the flow becomes rarefied, the conventional continuum/macroscopic equations produce unsatisfactory results. A microscopic/kinetic description becomes necessary. In this work, we introduce isogeometric (IG) kinetic schemes (KS) for the full Boltzmann equation and related kinetic models (Bhatnagar–Gross–Krook (BGK), Ellipsoidal statistical BGK, and Shakov). These schemes are specifically aimed for modeling flows in topologically complex “multi-patch” geometries such as those constructed using computer-aided geometric design tools; high order accurate; unconditionally time-stable (as per our tests) following generalized-α method; non-linearly stable following streamlined upwind Petrov–Galerkin method applied at every patch level with patch-to-patch coupling enforced either strongly by constraining the interface using single degree of freedom, or weakly using numerical fluxes as in discontinuous Galerkin (useful because the underlying system is first order in space); amenable to nearly-linear parallel efficiency owing to the six-dimensional nature of Boltzmann equation, wherein the convection and collision can be, respectively, performed simultaneously over velocity and physical spaces. In particular, since the surface representation is exact in IG, when surface properties are to be computed, which is often the case in aerodynamic applications, these schemes perform well. A series of verification tests for 1D/2D/3D-3V initial–boundary value and boundary-value flow problems, involving momentum, energy, and diffusive transport, including problems posed on non-conforming domains/patches, is performed to illustrate the stability and accuracy of the proposed method.

Feasibility of streamline upwind Petrov-Galerkin angular stabilization of the linear Boltzmann transport equation with magnetic fields

To accurately model dose in a magnetic field, the Lorentz force must be included in the traditional linear Boltzmann transport equation (LBTE). Both angular and spatial stabilization are required to deterministically solve this equation. In this work, a streamline upwind Petrov-Galerkin (SUPG) method is applied to achieve angular stabilization of the LBTE with magnetic fields. The spectral radius of the angular SUPG method is evaluated using a Fourier analysis method to characterize the convergence properties. Simulations are then performed on homogeneous phantoms and two heterogeneous slab geometry phantoms containing water, bone, lung/air and water for 0.5 T parallel and 1.5 T perpendicular magnetic field configurations. Fourier analysis determined that the spectral radius of the SUPG scheme is unaffected by magnetic field strength and the SUPG free parameter, indicating that the Gauss-Seidel source iteration method is unconditionally stable and the convergence rate is not degraded with increasing magnetic field strength. 100% of simulation points passed a 3D gamma analysis at a 2%/2 mm (3%/3 mm) gamma criterion for both magnetic field configurations in the homogeneous phantom study, with the exception of the 1.5 T perpendicular magnetic field in the pure lung phantom where a 77.4% (87.0%) pass rate was achieved. Simulations in the lung slab geometry phantom resulted in 100% of points passing a 2%/2 mm gamma analysis in a 0.5 T parallel magnetic field, and 97.7% (98.8%) of points passing a 2%/2 mm (3%/3 mm) gamma criterion in a 1.5 T perpendicular magnetic field. For the air slab geometry phantom, 72.1% (79.2%) of points passed a 2%/2 mm gamma criterion in a 0.5 T parallel magnetic field and 90.3% (92.8%) passed the same gamma criterion in a 1.5 T perpendicular magnetic field. While the novel SUPG angular stabilization method shows feasibility in some cases, it was found that the accuracy of this method was degraded for very low density media such as air.

Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model

The aim of the present work is to develop a model able to represent the propagation and transformation of waves in nearshore areas. The focus is on the phenomena of wave breaking, shoaling and run-up. These different phenomena are represented through a hybrid approach obtained by the coupling of non-linear Shallow Water equations with the extended Boussinesq equations of Madsen and Sorensen. The novelty is the switch tool between the two modelling equations: a critical free surface Froude criterion. This is based on a physically meaningful new approach to detect wave breaking, which corresponds to the steepening of the wave's crest which turns into a roller. To allow for an appropriate discretization of both types of equations, we consider a finite element Upwind Petrov Galerkin method with a novel limiting strategy, that guarantees the preservation of smooth waves as well as the monotonicity of the results in presence of discontinuities. We provide a detailed discussion of the implementation of the newly proposed detection method, as well as of two other well known criteria which are used for comparison. An extensive benchmarking on several problems involving different wave phenomena and breaking conditions allows to show the robustness of the numerical method proposed, as well as to assess the advantages and limitations of the different detection methods.

Anisotropic boundary layer mesh generation for reliable 3D unsteady RANS simulations

This paper proposes a Computational Fluid Dynamics (CFD) framework with the aim of combining consistency and efficiency for the numerical simulation of high Reynolds number flows encountered in engineering applications for aerodynamics. The novelty of the framework is the combination of a Reynolds-Averaged Navier–Stokes (RANS) model with an anisotropic mesh adaptation strategy handling arbitrary immersed geometries by building the corresponding boundary layer meshes. The numerical algorithm consists of robust and accurate solution of the unsteady incompressible Navier–Stokes equations supplemented with a Spalart–Allmaras turbulence model and boundary layer remeshing relying on a specifically designed metric. The flow solver is formulated as a Variational Multiscale (VMS) finite element method for the momentum balance and the incompressibility constraint, and as an upwind Petrov–Galerkin method for the nonlinear turbulent equation. The boundary layer remeshing strategy is flexible as it allows the adaptation of arbitrary coarse meshes by modifying the size and the orientation of elements along the immersed boundary to ensure a smooth gradation along the curvature of the body's geometry. The solver is capable of handling highly stretched anisotropic elements and is shown to successfully predict both mean and fluctuating drag/lift coefficients. Laminar and turbulent test cases in 2D and 3D are presented to assess the performance of this framework against experimental results relevant to external aerodynamics, including an airship and a flying drone.

Optimal adaptivity for the SUPG finite element method

For convection dominated problems, the streamline upwind Petrov–Galerkin method (SUPG), also named streamline diffusion finite element method (SDFEM), ensures a stable finite element solution even on coarse meshes, i.e., in the preasymptotic range. Based on some a posteriori error estimators from the literature, we formulate an adaptive mesh-refining algorithm for SUPG. We prove that the adaptively generated SUPG solutions converge at asymptotically optimal rates towards the exact solution. The main focus thus is on the mathematical proof that the SUPG stabilization does not spoil the asymptotic convergence behavior of the adaptive scheme.

Fluid-structure interaction simulations of venous valves: A monolithic ALE method for large structural displacements.

Venous valves are bicuspidal valves that ensure that blood in veins only flows back to the heart. To prevent retrograde blood flow, the two intraluminal leaflets meet in the center of the vein and occlude the vessel. In fluid-structure interaction (FSI) simulations of venous valves, the large structural displacements may lead to mesh deteriorations and entanglements, causing instabilities of the solver and, consequently, the numerical solution to diverge. In this paper, we propose an arbitrary Lagrangian-Eulerian (ALE) scheme for FSI simulations designed to solve these instabilities. A monolithic formulation for the FSI problem is considered, and due to the complexity of the operators, the exact Jacobian matrix is evaluated using automatic differentiation. The method relies on the introduction of a staggered in time velocity to improve stability, and on fictitious springs to model the contact force of the valve leaflets. Because the large structural displacements may compromise the quality of the fluid mesh as well, a smoother fluid displacement, obtained with the introduction of a scaling factor that measures the distance of a fluid element from the valve leaflet tip, guarantees that there are no mesh entanglements in the fluid domain. To further improve stability, a streamline upwind Petrov-Galerkin (SUPG) method is employed. The proposed ALE scheme is applied to a two-dimensional (2D) model of a venous valve. The presented simulations show that the proposed method deals well with the large structural displacements of the problem, allowing a reconstruction of the valve behavior in both the opening and closing phase.

The Influence of Velocity Field Approximations in Tracer Injection Processes

Although the concentration is the most important variable in tracer injection processes, an efficient and accurate velocity field approximation is crucial to obtain a good physical behaviour for the problem. In this paper we analyse a Stabilized Dual Hybrid Mixed (SDHM) method to solve the Darcy's system in the velocity and pressure variables that involves the conservation of mass and Darcy's law. This approach is locally conservative, free of compromise between the finite element approximation spaces and capable of dealing with heterogeneous media with discontinuous properties. The tracer concentration is solved via a combination of the Streamline Upwind Petrov-Galerkin (SUPG) method in space with an implicit finite difference scheme in time. We also employ a semi-analytical approach (Abbaszadeh-Dehghani analytical solution) to integrate the transport equation. A numerical comparative study using the SDHM formulation, the Galerkin method and a post-processing technique to calculate the velocity field in combination with those concentration approximation methodologies are presented. In all comparisons, the SDHM formulation appears as the most efficient, accurate and almost free of spurious oscillations.