High-order Finite Element Method Research Articles

MHD Mixed Convection of Nanofluid Flow Ag- Mgo/water in a Channel Contain a Rotational Cylinder

This investigation explores the mixed convection phenomenon of a hybrid nanofluid flowing within a Channel oriented horizontally featuring a triangular cavity attached to the lower channel wall. The cavity houses a rotating circular cylinder with a radius of R=0.12. The fluid within the enclosure is a water-based nanofluid containing Ag-MgO nanoparticles. Two scenarios of combined natural and forced convection are examined: in Case 1, all channel walls, the left wall of the cavity, and the surface of the rotating cylinder are adiabatic, while the inclined wall of the cavity is isothermal. In Case 2, all channel walls, the ready wall of the cavity, and the surface of the rotating cylinder are adiabatic, except for the left wall of the cavity, which is isothermal. The dimensionless governing equations are solved under steady flow conditions for various physical parameters using a higher-order and stable Galerkin-based finite element method implemented through the software package FlexPDE. Key governing parameters such as Re number (falling within the amplitude of 10 to 100), Ri number (ranging from Ri=0.1 to 30), and the location of cylinder rotation (Xo = 0.65 to 0.9) are simulated. The study presents results in terms of average Nusselt numbers and contour maps of streamlines and isotherms. It is observed that the average Nusselt number increases with higher Reynolds number, angular rotation speed, cylinder location, and Richardson number. Specifically, heating the inclined surface suggests that the optimal location for the rotating cylinder is Xo = 0.9, whereas if the left wall of the cavity is heated, the preferred location for the cylinder is Xo = 0.75. These findings are corroborated by comparing them with previous research, demonstrating significant agreement.

Analysis and Computation of a Weak Galerkin Scheme for Solving the 2D/3D Stationary Stokes Interface Problems with High-Order Elements

Abstract In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝ d (d = 2, 3). This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ≥ 1 for the velocity and polynomials of degree k – 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L 2-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.

Fully higher-order coupling of finite element and level set methods for two-phase flow with a new explicit projection method

This work presents the development of a higher-order finite element method for modeling unsteady, incompressible and inviscid two-phase flows using the level set method. A new explicit projection method is introduced to solve the incompressible Euler equation, where the pressure and the velocity fields are solved separately. To implement high order temporal schemes, the Hodge decomposition in this projection method is not applied to the velocity but to its time derivative. A discontinuous Galerkin discretization is used to solve both the level set and the nonlinear advection terms present in the Euler equations, while a Galerkin discretization is used to solve the pressure Poisson equation. The present method is a fully high order numerical approach, both in space and temporal dimensions. Numerical results provided by several 2D benchmark test cases in naval hydrodynamics (i.e hydrostatic equilibrium, sloshing and dam break) validate the computational code, illustrating the power of the proposed method. Compared to the CFD codes based on low order VOF/finite-volume methods, the present approach demonstrates greater accuracy on coarse meshes, leading to reduced CPU time.

Integrated CFD-FEM approach for external gear pump vibroacoustic field prediction

In this work, a three-dimensional fluid-structural and vibro-acoustics coupled model of a gear pump is presented. Gear pumps represent the majority of the positive displacement machines used for flow generation in fluid power systems. Typically, the dynamics of gear pumps are dependent on the characteristics of the fluid dynamics inside the pump, which translates into vibrations to the housing. These vibrations then propagate to the surrounding medium and emit sound. The purpose of this study is to propose a three-dimensional fully integrated computational model to simulate the complete gear pump behaviour from the fluid dynamics to the structural vibrations, up to the vibroacoustic response. In this hybrid configuration, a transient CFD (Computational Fluid Dynamics) model is first developed to simulate the 3D flow field with a deforming and re-meshing approach to take into account the variation of the volume between the gears. Second, the internal pressure field calculated at each time step is then used as a loading into a structural FEM (Finite Element Method) model of the gear pump to compute the actual stresses and deformations of the housing. Third, the results of the structural response are used as excitation in a vibroacoustic sub-model to simulate the radiated noise using a high-order FEM technique. The comparison between the numerical and experimental flow curves validates the CFD model. The sound power calculations from the vibroacoustic model show good comparison with the sound intensity measurements around the pump casing, confirming the validity of the proposed coupled model. The described CFD-FEM approach proves to be a powerful gear pump design tool: it provides a reliable estimate of gear pump working parameters such as fluid power and vibroacoustic characteristics, starting from the CAD (Computer-Aided Design) geometry of the components. Furthermore, being one of the first multi-domain simulation models of a gear pump, this work can be useful to researchers as a starting point to correlate the noise emitted with the internal flow in full 3D conditions.

Higher-order galerkin finite element method for nonlinear coupled reaction-diffusion models

In this article, generalized nonlinear coupled reaction-diffusion (NCRD) models are analyzed using the higher-order Galerkin finite element method (FEM). The authors present finite element analysis for one- and two-dimensional NCRD models by applying quadratic Lagrange basis functions. Optimal order convergence in space is proved by applying quadratic FEM in spatial discretization and utilizing the Crank-Nicolson (CN) scheme for time discretization. To address the nonlinearity of NCRD models, CN extrapolation and predictorcorrector schemes are employed. The stability analysis for generalized NCRD models and the applied CN scheme are discussed by utilizing the eigenvalues method. Finally, to assess the accuracy and efficiency of the proposed scheme and to capture different patterns of NCRD models, various one- and two-dimensional models are considered. The obtained results are numerically and graphically compared with existing literature. It is observed that quadratic basis functions in FEM improve accuracy and efficiency of the results.

High-order splitting finite element methods for the subdiffusion equation with limited smoothing property

In contrast with the diffusion equation which smoothens the initial data to C ∞ C^\infty for t > 0 t>0 (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac–Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.

NekMesh: An open-source high-order mesh generation framework

High-order spectral element simulations are now becoming increasingly popular within the computational modelling community, as they offer the potential to deliver increased accuracy at reduced cost compared to traditional low-order codes. However, to support accurate, high-fidelity simulations in complex industrial applications, there is a need to generate curvilinear meshes which robustly and accurately conform to geometrical features. This is, at present, a key challenge within the mesh generation community, with only a few open-source tools able to generate curvilinear meshes for complex geometries. We present NekMesh: an open-source mesh generation package which is designed to enable the generation of valid, high-quality curvilinear meshes of complex, three-dimensional geometries for performing high-order simulations. We outline the software architecture adopted in NekMesh, which uses a pipeline of processing modules to provide a flexible, CAD-independent high-order mesh processing tool, capable of both generating meshes for a wide range of use cases, as well as post-processing linear meshes from a range of input formats for use with high-order simulations. A number of examples in various application areas are presented, with a particular emphasis on challenging aeronautical and fluid dynamics test cases. Program summaryProgram title: NekMesh (version 5.4.0)CPC Library link to program files:https://doi.org/10.17632/d82hjm4v6r.1Licensing provisions: MITProgramming language: C++External routines/libraries: Boost, TinyXML, OpenCASCADE, Triangle, TetGen, HDF5Nature of problem: NekMesh is a high-order mesh generation framework with the goal of providing a robust framework to automate the process of generating valid meshes for complex three-dimensional CAD geometries.Solution method: Energy minimisation, solid body models and other techniques based around high-order finite element methods.Additional comments including restrictions and unusual features: The supplied cases require a system with 16384 MB of RAM for the automotive example and 32768 MB of RAM for the variational optimisation example.

High-order finite element method for atomic structure calculations

We introduce featom, an open source code that implements a high-order finite element solver for the radial Schrödinger, Dirac, and Kohn-Sham equations. The formulation accommodates various mesh types, such as uniform or exponential, and the convergence can be systematically controlled by increasing the number and/or polynomial order of the finite element basis functions. The Dirac equation is solved using a squared Hamiltonian approach to eliminate spurious states. To address the slow convergence of the κ=±1 states due to divergent derivatives at the origin, we incorporate known asymptotic forms into the solutions. We achieve a high level of accuracy (10−8 Hartree) for total energies and eigenvalues of heavy atoms such as uranium in both Schrödinger and Dirac Kohn-Sham solutions. We provide detailed convergence studies and computational parameters required to attain commonly required accuracies. Finally, we compare our results with known analytic results as well as the results of other methods. In particular, we calculate benchmark results for atomic numbers (Z) from 1 to 92, verifying current benchmarks. We demonstrate significant speedup compared to the state-of-the-art shooting solver dftatom. An efficient, modular Fortran 2008 implementation, is provided under an open source, permissive license, including examples and tests, wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines. Program summaryProgram Title: featomCPC Library link to program files:https://doi.org/10.17632/962fzmm7f7.1Licensing provisions: MITProgramming language: Fortran with command-line interfacesNature of problem: Solution of the Schrödinger, Dirac, and Kohn–Sham equations of density functional theory for isolated atoms.Solution method: A high-order finite element method is used to assemble a generalized eigenproblem that is then solved for eigenvalues and orbitals. The Poisson equation is solved using the finite element method also. Self-consistent field equations are solved using Pulay mixing.Additional comments including restrictions and unusual features: Our solution methodology for the Dirac equation does not suffer from spurious states. We maintain high accuracy even for κ=±1 states. Unlike other methods, our approach is not limited to Coulombic or self-consistent potentials and can handle non-uniform meshes, including exponential meshes.Restrictions: Spherical symmetry.

A high-order residual-based viscosity finite element method for incompressible variable density flow

In this paper, we introduce a high-order accurate finite element method for incompressible variable density flow. The method uses high-order Taylor-Hood velocity-pressure elements in space and backward differentiation formula (BDF) time stepping in time. This way of discretization leads to two main issues: (i) a saddle point system that needs to be solved at each time step; (ii) a stability issue when the viscosity of the flow goes to zero or if the density profile has a discontinuity. We address the first issue by using Schur complement preconditioning and artificial compressibility approaches. We observed similar performance between these two approaches. To address the second issue, we introduce a modified artificial Guermond-Popov viscous flux where the viscosity coefficients are constructed using a newly developed residual-based shock-capturing method. Numerical validations confirm high-order accuracy for smooth problems and accurately resolved discontinuities for problems in 2D and 3D with varying density ratios.

Finite element-based invariant-domain preserving approximation of hyperbolic systems: Beyond second-order accuracy in space

This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a high-order finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from Abgrall et al. (2017). The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting.

Meshing strategies for 3d geo-electromagnetic modeling in the presence of metallic infrastructure

In 3D geo-electromagnetic modeling, an adequate discretisation of the modeling domain is crucial to obtain accurate forward responses and reliable inversion results while reducing the computational cost. This paper investigates the mesh design for subsurface models, including steel-cased wells, which is relevant for many exploration settings but still remains a numerically challenging task. Applying a goal-oriented mesh refinement technique and subsequent calculations with the high-order edge finite element method, simulations of 3D controlled-source electromagnetic models in the presence of metallic infrastructure are performed. Two test models are considered, each needing a distinct version of approximation methods to incorporate the conductive steel casings of the included wells. The influence of mesh quality, goal-oriented meshing, and high-order approximations on problem sizes, computational cost, and accuracy of electromagnetic responses is investigated. The main insights of our work are: (a) the applied numerical schemes can mitigate the computational burden of geo-electromagnetic modeling in the presence of steel artifacts; (b) investigating the processes driving the meshing of models with embedded metallic infrastructures can lead to adequate strategies to deal with the inversion of such electromagnetic data sets. Based on the modeling results and analyses conducted, general recommendations for modeling strategies are proposed when performing simulations for challenging steel infrastructure scenarios.

Internet Data Bandwidth Optimization and Prediction in Higher Learning Institutions Using Lagrange’s Interpolation: A Case of Lagos State University of Science and Technology

Episodic edifices have a diversity of significant solicitations in contemporary machineries and engineering owing to their exclusive electromagnetic properties. Frequently used episodic edifices comprise; occurrence selective surfaces, visual grilles, phased collection projections, photonic bandgap supplies, and numerous metamaterials. The scrutiny of episodic edifices has all the time been a significant area in computational electromagnetics. This episode, describes a precise and effectual arithmetical study, grounded on a higher-order finite element method (FEM), for depicting the electromagnetic properties of an episodic edifices. Grounded on the Floquet theory, episodic frontier conditions and radioactivity conditions are foremost resultant for the unit cell of an episodic edifice. The FEM is formerly applied to unravel Maxwell’s reckonings in the unit cell. To augment the precision and effectiveness of the FEM, rounded elements are employed to discretize the unit cell and higherorder course basis functions are used to enlarge the electrical arena. The asymptotic waveform evaluation (AWE) is applied to implement wild frequency and rawboned curves. To prove the proficiency of the projected FEM, we apply it to the scrutiny of episodic absorbers, incidence selective edifices, and phased collection aerial. For the aerial analysis, a severe waveguide port condition is industrialized to precisely model the aerial feed edifices. In all the occurrences premeditated, acceptable outcomes are obtained. Key words : Episodic, edifice, absorbers, waveguide, electromagnetic. DOI : 10.7176/CEIS/10-1-04

Custom Transient Finite Element for the Modelling of Ultrasonic Control Methods for Downhole Well Integrity

The transient spectral element method can be understood as a specific high-order finite element method that is particularly accurate and fast with a low memory footprint, hence enabling 3D ultrasonic testing simulations on a standard PC. However, particular care must be taken in its settings when seeking both practicality and performance. In comparison, fast semi-analytical simulation using ray tracing is easily accessible to nondestructive testing experts, through the CIVA simulation software. Although this approximate model effectively deals with a wide range of industrial applications, it reaches its limits in certain cases, particularly at relatively low frequencies involving resonance phenomena. To deal with these challenging inspection configurations, CEA-List is working on dedicated finite element models in the CIVA framework to meet specific needs in a compromise between efficiency and accuracy. The case study considered in partnership with SLB perfectly illustrates the benefits of developing such a modelling approach. SLB provides cased-well imaging services using ultrasonic resonance methods to evaluate oil and gas well integrity. Defective cased wells have irregularities, such as corroded casings with pitting, localized cement debonding and channelings. Simulated ultrasonic signals of the actual wells are useful to understand the measurements and are available by finite element modelling with sufficient accuracy. However, the associated computational costs for numerous model cases limit the building of the model database as the digital twin of the actual well logging. The development strategy of dedicated hybrid finite element applications adopted by CEA-List is presented in broad outline. Cross-model comparisons on a reference configuration of pristine cases are presented, highlighting the effective limits of standard CIVA ray models. The contribution offered by the new hybrid finite element approach is then discussed. We discuss the evolution of model functionalities to handle geometries that differ from the canonical case (interface defects, loss of thickness in arbitrary geometries, etc.) from the perspective of a digital twin of this control.

Custom Transient Finite Element Method and Ray Tracing Hybridization Strategies for Ultrasonic Testing Modelling

The transient spectral element method (SEM) is a specific high order finite element method that is particularly accurate and fast with a low memory footprint, hence enabling 3D ultrasonic testing simulations on a standard PC. However, particular care in its settings and hybridization with a semi-analytical solution remains a major challenge when seeking for both practicality and performance. A natural hybrid strategy consists of coupling a field calculated by ray tracing in the defect-free object to a SEM subdomain wherein the perturbation is enclosed, then synthesizing the signal with a reciprocity argument. The difficulty is to define the suited coupling strategy when the flaw response interacts with the back wall. In a highly heterogeneous medium, it may even become necessary to widen the numerical domain to the entire thickness of the inspected object and, therefore, to ensure the performance of the SEM through a custom discretization strategy. Here we present different customized uses of the SEM for practical ultrasonic inspection applications and discuss how they complement the ray tracing solution.

High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems

We design and compute first-order implicit-in-time variational schemes with high-order spatial discretization for initial value gradient flows in generalized optimal transport metric spaces. We first review some examples of gradient flows in generalized optimal transport spaces from the Onsager principle. We then use a one-step time relaxation optimization problem for time-implicit schemes, namely generalized Jordan-Kinderlehrer-Otto schemes. Their minimizing systems satisfy implicit-in-time schemes for initial value gradient flows with first-order time accuracy. We adopt the first-order optimization scheme ALG2 (Augmented Lagrangian method) and high-order finite element methods in spatial discretization to compute the one-step optimization problem. This allows us to derive the implicit-in-time update of initial value gradient flows iteratively. We remark that the iteration in ALG2 has a simple-to-implement point-wise update based on optimal transport and Onsager's activation functions. The proposed method is unconditionally stable for convex cases. Numerical examples are presented to demonstrate the effectiveness of the methods in two-dimensional PDEs, including Wasserstein gradient flows, Fisher–Kolmogorov-Petrovskii-Piskunov equation, and two and four species reversible reaction-diffusion systems.

High-order variational Lagrangian schemes for compressible fluids

We present high-order variational Lagrangian finite element methods for compressible fluids using a discrete energetic variational approach. Our spatial discretization is mass/momentum/energy conserving and entropy stable. Fully implicit time stepping is used for the temporal discretization, which allows for a much larger time step size for stability compared to explicit methods, especially for low-Mach number flows and/or on highly distorted meshes. Ample numerical results are presented to showcase the good performance of our proposed scheme.

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation

We consider the reliable implementation of high-order unfitted finite element methods on Cartesian meshes with hanging nodes for elliptic interface problems. We construct a reliable algorithm to merge small interface elements with their surrounding elements to automatically generate the finite element mesh whose elements are large with respect to both domains. We propose new basis functions for the interface elements to control the growth of the condition number of the stiffness matrix in terms of the finite element approximation order, the number of elements of the mesh, and the interface deviation which quantifies the mesh resolution of the geometry of the interface. Numerical examples are presented to illustrate the competitive performance of the method.

Analysis of effects of moving wall direction in a one sided semi-circular porous cavity on mixed magneto-bioconvection in the presence of hybrid nanofluid with oxytactic bacteria

A computational analysis on the effects of direction of wall movement in a one sided semi-circular porous cavity on mixed magneto-bioconvection of hybrid nanofluid in the presence of oxytactic bacteria has been performed. The working fluid is the hybrid nanofluid. The enclosure is cooled from semi-circle side and heated from moving lid under the effect of uniform magnetic field in the horizontal direction. The numerical solution of governing equations is obtained by using the higher order finite element method (FEM). The reliability of designed solver is tested through the experimental and numerical data available in the literature. Two cases of the boundary conditions are considered related to the tendency of right vertical lid of the cavity. Other parameters are Richardson number, Darcy number, Hartmann number, Peclet number and porosity. It is found that moving lid direction is the most effective parameter for the flow field, isotherms, oxygen and microorganism distributions. Moreover, higher heat transfer rate is obtained in Case-I for different values of Darcy number. Around 10% increment is occurred on mass transfer with different Pe number in case of downward moving lid due to increment of convection mode of heat transfer inside the cavity.

Effects of Internal Boundary Layers and Sensitivity on Frequency Response of Shells of Revolution

New applications introduced capsule designs with features that have not been fully analysed in the literature. In this study, thin shells of revolution are used to model drug delivery capsules both with closed and open designs including perforations. The effects of internal boundary layers and sensitivity on frequency response are discussed in the special case with symmetric concentrated load. The simulations are carried out using high-order finite element method and the frequency response is computed with a very accurate low-rank approximation. Due to the propagation of the singularities induced by the concentrated loads, the most energetic responses do not necessarily include a pinch-through at the point of action. In sensitive configurations, the presence of regions with elliptic curvature leads to strong oscillations at lower frequencies. The amplitudes of these oscillations decay as the frequencies increase. For efficient and reliable analysis of such structures, it is necessary to understand the intricate interplay of loading types and geometry, including the effects of the chosen shell models.

Symmetric and asymmetric Gauss and Gauss–Lobatto quadrature rules for triangles and their applications to high-order finite element analyses

Although symmetric Gauss quadrature rules have been studied extensively, the investigations of asymmetric Gauss quadrature rules and symmetric Gauss–Lobatto (GL) quadrature rules are limited. Moreover, GL quadrature rules are essential for high-order finite element methods like the quadrature element method (QEM) and the spectral element method (SEM) in strong and weak form, and asymmetric Gauss quadrature rules are even more efficient than the symmetric ones. This work first presents new optimization methods for solving symmetric and asymmetric Gauss quadrature rules and then generalizes the techniques to solve GL and hierarchical Gauss–Lobatto (HGL) quadrature rules. Most of the asymmetric Gauss quadrature rules with precision degree range from 1 to 50, and symmetric GL and HGL quadrature rules presented in this work are new. They are attached to this article as MATLAB m-files for future usage. The GL and HGL quadrature rules are (1) used for constructing weak form QEM and HQEM with diagonal-mass-matrix (DMM) that is promising in dynamic analyses, and (2) used as interpolation nodes in high-order finite element analyses like Fekete points. Applying the present asymmetric Gauss quadrature rules and symmetric GL and HGL quadrature rules to in-plane vibration shows that both cases of usage of the GL and HGL quadrature rules can provide results with high accuracy and convergence rate.