Low-order Method Research Articles

Infrared Thermography Measurements over a Tail-Plane Model of a Large Passenger Aircraft

In the frame of the CleanSky 2 MONNALISA project, a wind-tunnel campaign is planned to test a tail-plane model of a large passenger aircraft by using different measurement techniques. Wind-tunnel test results obtained over a systematic series of model geometries will be used to thoroughly validate a low-order numerical method based on physical modelling that will be developed in the project to evaluate the non-linear aerodynamic characteristics of aircraft lifting surfaces. The present paper describes the main results of the infrared thermography measurements performed in the first wind-tunnel entry to be used to tune RANS solvers and for the fine calibration of the low-order numerical method.

A New Approach to Reduce Transverse Shear Locking of Reissner–Mindlin Plate Based on The Cell-Centered Finite Element Method

This study investigates a new approach to solve a shear locking free on a low-order cell-centered finite element method (FECC) of the Reissner–Mindlin plate. The proposed method pursues the Lagrange piecewise of linear polynomial functions to attain the transverse displacement and rotations on the dual sub-mesh derived from the general one. The suggested theory is employed in numerical computation to verify the effectiveness and accurate performance. Detailed studies are carried out on several mesh types to compare with the analytical solutions reported by the low-order mimetic finite difference method [Beir ao Da Veiga, L., Lovadina, C. and Mora, D. [2013] “Numerical results for mimetic discretization of Reissner–Mindlin plate problems,” Calcolo 50, 209–237] and high-order mixed finite element methods [Chinosi, C. and Lovadina, C. [1995] “Numerical analysis of some mixed finite element methods for Reissner–Mindlin plates,” Comput. Mech. 16, 36–44]. The numerical results indicated that the present scheme based on the low-order FECC approximation can effectively predict on even the distorted meshes, satisfy the optimized convergence and be uniformly stable to the thin plate thicknesses.

Speeding Up Explicit Numerical Evaluation Methods for Micromagnetic Simulations Using Demagnetizing Field Polynomial Extrapolation

The performance of numerical micromagnetic models is limited by the demagnetizing field computation, which typically accounts for the majority of the computation time. For magnetization dynamics simulations, explicit evaluation methods are in common use. Higher-order methods call for the evaluation of all effective field terms, including the demagnetizing field, at all sub-steps. Here, a general method of speeding up such explicit evaluation methods is discussed by skipping the demagnetizing field computation at sub-steps, and instead approximating it using polynomial extrapolation based on stored previous exact computations. This approach is tested for a large number of explicit evaluation methods, both adaptive and fixed time-step, ranging from second-order up to fifth-order. The polynomial approximation order should be matched to the evaluation method order. In this case, we show higher-order methods with polynomial extrapolation are more accurate than lower-order methods with a full evaluation of the demagnetizing field. Moreover, for higher-order methods, we show it is possible to achieve a factor of 2 or more computation speedup with no decrease in solution accuracy.

Leading-edge-vortex tailoring on unsteady airfoils using an inverse aerodynamic approach

In this paper, we present an approach to obtain a desired leading-edge vortex (LEV) shedding pattern from unsteady airfoils through the execution of suitable motion kinematics. Previous research revealed that LEV shedding is associated with the leading-edge suction parameter (LESP) exceeding a maximum threshold. A low-order method called LESP-modulated discrete vortex method (LDVM) was also developed to predict the onset and termination of LEV shedding from an airfoil undergoing prescribed motion kinematics. In the current work, we present an inverse-aerodynamic formulation based on the LDVM to generate the appropriate motion kinematics to achieve a prescribed LESP variation, and thus, the desired LEV shedding characteristics from the airfoil. The algorithm identifies the kinematic state of the airfoil required to attain the target LESP value through an iterative procedure performed inside the LDVM simulation at each time step. Several case studies are presented to demonstrate design scenarios such as tailoring the duration and intensity of LEV shedding, inducing LEV shedding from the chosen surface of the airfoil, promoting or suppressing LEV shedding during an unsteady motion on demand, and achieving similar LEV shedding patterns using different maneuvers. The kinematic profiles generated by the low-order formulation are also simulated using a high-fidelity unsteady Reynolds-averaged Navier–Stokes method to confirm the accuracy of the low-order model.

High order approximation of Hodge Laplace problems with local coderivatives on cubical meshes

In mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex, the exterior derivative operators are computed exactly, so the spatial locality is preserved. However, the numerical approximations of the associated coderivatives are nonlocal and can be regarded as an undesired effect of standard mixed methods. For numerical methods with local coderivatives, a perturbation of low order mixed methods in the sense of variational crimes has been developed for simplicial and cubical meshes. In this paper we extend the low order method to all high orders on cubical meshes using a new family of finite element differential forms on cubical meshes. The key theoretical contribution is a generalization of the linear degree, in the construction of the serendipity family of differential forms, and this generalization is essential in the unisolvency proof of the new family of finite element differential forms.

R-adaptive algorithms for supersonic flows with high-order Flux Reconstruction methods

The present paper addresses the development and implementation of the first r-adaptive mesh refinement (r-AMR) algorithm for a high-order Flux Reconstruction solver. The r-refinement consists on nodal re-positioning while keeping the number of mesh nodes and their connectivity frozen. The developed algorithm is based on physics-driven spring-analogies, where the mesh can be seen as a network of fictitious springs. While AMR increases the local mesh density, the high-order Flux Reconstruction method potentially provides a more accurate detection of complex flow features over relatively coarser mesh, when compared to low-order methods. In this work, a concise overview of the Flux Reconstruction method and spring-based AMR techniques will be given, followed by some promising results of r-AMR applied to benchmark high-order steady-state supersonic flow simulations.

A comparison between experiments and numerical simulations on a scale model of a Horizontal-Axis current turbine

This work refers to towing-tank experiments and numerical simulations of the hydrokinetic turbines for the GEMSTAR system, designed at the University of Naples. The experimental campaign was aimed to validate the quality of the turbine design procedure, utilising low order methods such as Blade Element Momentum theory (BEMT). Discrepancies were found between BEMT results and experiments. Therefore, higher order CFD numerical models were employed to investigate on the rotor hydrodynamic behaviour, simulating the experiments. Blade section pressure distributions, obtained from 3D simulations, were compared with pressure distributions obtained from 2D calculations to explain discrepancies between BEMT and experiments. It was found that pressure distributions from 2D calculations are significantly different from the ones extracted from CFD, especially at inboard sections. As shown previously in literature for wind turbines, the presence of complex 3D phenomena arising from the combination of rotation and pressure distribution on the blades was demonstrated in water turbines. Finally, a so-called 2.5D approach was developed to improve BEMT results, with the sectional forces extracted from 3D simulations. The 2.5D BEMT was validated against the results of a test series with variable collective pitch, showing an acceptable agreement to the observed data for slight pitch angle variations.

Investigation of a four-bladed propeller inflow at yaw

An investigation has been conducted which looks to understand the effect of yaw on the inflow of a 4-bladed propeller. Both experimental and numerical work has been conducted, with the test data used to validate and compare against the numerical methods. The numerical methods involve fully resolved CFD and the low-order Crigler method. Comparing at a yaw angle of 15°, good agreement is found between the numerical and experimental results with adjustments made to account for discrepancies between the setups. More work is still required in the experimental and numerical branches to improve the extraction of data with minimal losses and improve the fidelity of simpler methods without a great expansion of computer cost.

Determination of quadrupole polarizabilities of the excited states of alkali-metal atoms

The scalar and tensor components of the electric quadrupole (E2) polarizabilities of the first two excited states of all the alkali-metal atoms are determined. To validate the calculations, we have evaluated the ground state E2 polarizabilities of these atoms and compared them with the literature values. We could not find the ground state E2 polarizability value for Fr in the literature to compare with our result. The dominant parts of these quantities are estimated by combining the precisely calculated E2 transition matrix elements of many low-lying transitions with the experimental energies, while the other contributions are estimated using lower-order methods. Our estimated values for the ground states of the above atoms are in good agreement with the literature values suggesting that our estimated E2 polarizabilities for the excited states of the alkali atoms, which were not known earlier except for the Li atom, are also quite accurate. These reported E2 polarizabilities could be useful in guiding many precision measurements in the alkali atoms.

Research on the identification method of the pre-tightening state of the matching surface of the aero-engine disk-drum rotor

The aero-engine rotor system is connected by one or more stages of mounting edges through bolts, and the pre-tightening state of the mating surface will directly affect the dynamic performance of the rotor and even the whole machine. However, there is little research on the identification method of the pre-tightening state of the mating surface of the disk-drum rotor with a multi-bolt flange connection. Therefore, based on the vibration theory, this paper analyzes the feasibility of the pre-tightening state identification method of the mating surface. Then, from the perspective of the assembly process and structural characteristics of the rotor system, this paper takes the high-pressure compressor disk-drum rotor, high-pressure turbine disk-drum rotor, and high-pressure rotor system as the research objects, respectively, and uses the finite element method and experimental method to analyze the influence of the pre-tightening state change of the mating surface on the rotor dynamic characteristics. Finally, according to the obtained regularity cognition, three pre-tightening state identification methods of multi-bolt connection mating surface are proposed: low-order vibration phase difference method, amplitude variation method of higher-order frequency response function (FRF), and axial tension-sensitive natural frequency offset method. The proposed methods can effectively identify the preload state of the mating surface of the disk-drum rotor in engineering due to improper assembly or harsh environment and provide a new research idea for the dynamic characteristic detection and assembly quality evaluation of aero-engine rotor.

High-Order Spectral/-Based Solver for Compressible Navier–Stokes Equations

High-order spectral element methods are becoming more established for solving problems in computational fluid dynamics. In this paper, we introduce new stabilization parameters for the solution of compressible Navier–Stokes equations in conjunction with high-order spectral element methods. We previously defined a new set of stabilization parameters for Euler equations in a high-order spectral/ framework. In this paper, we examine the definition of these stabilization parameters in the context of solutions of full Navier–Stokes equations, and we report good agreement with previously published results in the literature. We establish spectral convergence of errors for Kovasznay flow. We then validate the definition of the stabilization parameter for Couette flow, flat plate, and compression corner problems. In a later example, we solve the flow past a cylinder and benchmark results with previously published results obtained with low-order methods and other approaches in compressible flow computations. We further solve hypersonic flow past a wedge and supersonic flow past a Flight Investigation of Re-Entry program (FIRE) entry capsule. We explore the definition of the shock-capturing parameter based on the parameter. The introduced definitions are found to provide excellent results for the full Navier–Stokes equations.

A Cut-Cell Polyhedral Finite Element Model for Coupled Fluid Flow and Mechanics in Fractured Reservoirs

Summary Mechanical deformation induced by injection and withdrawal of fluids from the subsurface can significantly alter the flow paths in naturally fractured reservoirs. Modeling coupled fluid flow and mechanical deformation in fractured reservoirs relies on either sophisticated gridding techniques or enhancing the variables (degrees of freedom) that represent the physics to describe the behavior of fractured formation accurately. The objective of this study is to develop a spatial discretization scheme that cuts the “matrix” grid with fracture planes and utilizes traditional formulations for fluid flow and geomechanics. The flow model uses the standard low-order finite volume method with the compartmental embedded discrete fracture model (cEDFM). Due to the presence of nonstandard polyhedra in the grid after cutting/splitting, we use numerical harmonic shape functions within a polyhedral finite element (PFE) formulation for mechanical deformation. To enforce fracture-contact constraints, we use a penalty approach. We provide a series of comparisons between the approach that uses conforming unstructured grids and an unstructured discrete fracture model (uDFM) with the new cut-cell PFE formulation. The manuscript validates and compares both methods for linear elastic, single-fracture slip, and Mandel’s problems with tetrahedral, Cartesian, and perpendicular-bisectional (PBI) grids. Finally, the paper presents a fully coupled 3D simulation with multiple inclined intersecting faults activated in shear by fluid injection, which caused an increase in effective reservoir permeability. Our approach allows for great reduction in the complexity of the (gridded) model construction while retaining the solution accuracy together with great savings in the computational cost compared with uDFM. The flexibility of our model with respect to the types of grid polyhedra allows us to eliminate mesh artifacts in the solution of the transport equations typically observed when using tetrahedral grids and two-point flux approximation.

A Large Scale Benchmark for the Incompressible Navier-Stokes Equations

We introduce a collection of benchmark problems in 2D and 3D (geometry description and boundary conditions), including simple cases with known analytic solution, classical experimental setups, and complex geometries with fabricated solutions for evaluation of numerical schemes for incompressible Navier-Stokes equations in laminar flow regime. We compare the performance of a representative selection of most broadly used algorithms for Navier-Stokes equations on this set of problems. Where applicable, we compare the most common spatial discretization choices (unstructured triangle/tetrahedral meshes and structured or semi-structured quadrilateral/hexahedral meshes). The study shows that while the type of spatial discretization used has a minor impact on the accuracy of the solutions, the choice of time integration method, spatial discretization order, and the choice of solving the coupled equations or reducing them to simpler subproblems have very different properties. Methods that are directly solving the original equations tend to be more accurate than splitting approaches for the same number of degrees of freedom, but numerical or computational difficulty arise when they are scaled to larger problem sizes. Low-order splitting methods are less accurate, but scale more easily to large problems, while higher-order splitting methods are accurate but require dense time discretizations to be stable. We release the description of the experiments and an implementation of our benchmark, which we believe will enable statistically significant comparisons with the state of the art as new approaches for solving the incompressible Navier-Stokes equations are introduced.

A posteriori Finite-Volume local subcell correction of high-order discontinuous Galerkin schemes for the nonlinear shallow-water equations

We design and analyze a new discretization method for the nonlinear shallow water equations, which is based on an equivalent representation of arbitrary high-order Discontinuous Galerkin (DG) schemes through piecewise constant modes on a sub-grid, together with a selective a posteriori local correction of the sub-interface reconstructed flux. This new approach, based on F. Vilar (2019) [101], allows to combine at the subcell scale the excellent robustness properties of the Finite-Volume (FV) lowest-order method and the high-order accuracy of the DG method. For any order of polynomial approximation, the resulting algorithm is shown to: (i) accurately handle strong shocks with no robustness issues; (ii) ensure the preservation of the water height positivity at the subcell level; (iii) preserve the class of motionless steady states (well-balancing); (iv) retain the highly accurate subcell resolution of DG schemes. These assets are numerically illustrated through an extensive set of test-cases, with a particular emphasize put on the use of very-high order polynomial approximations on coarse grids.

Impact of Angular and Spatial Source Distribution Approximations on Convergence Performance of Nonlinear Acceleration Methods for MOC in Slab Geometry

The convergence performance of nonlinear acceleration methods for the method of characteristics (MOC) with flat source (FS) approximation (FS MOC) or linear source (LS) approximation (LS MOC) is numerically investigated by focusing on the spatial and angular approximations in the acceleration calculations. The convergence of nonlinear acceleration depends on the consistency of the calculation models between the higher-order and lower-order (acceleration) methods. The convergence of four acceleration methods is evaluated to clarify the relationship between model consistency and convergence performance. These methods consist of FS or LS for the spatial source distribution and P1 or discrete angle for the angular distribution, i.e., (1) FS analytic coarse mesh finite difference (ACMFD) acceleration (FS ACMFD), (2) LS ACMFD, (3) FS angular-dependent discontinuity factor MOC (ADMOC) acceleration (FS ADMOC), and (4) LS ADMOC. The ACMFD and ADMOC accelerations are based on P1 and discrete angle approximations, respectively. The FS MOC and LS MOC are considered higher-order methods. The FS MOC and LS MOC with five acceleration methods, i.e., the aforementioned four acceleration methods and the conventional coarse mesh finite difference acceleration method, are used to perform fixed-source calculations in one-group one-dimensional homogeneous slab geometry, and the spectral radii are numerically evaluated. The numerical results indicate that (1) the nonlinear acceleration methods that are unconditionally stable for FS MOC also show similar convergence properties for LS MOC in one-dimensional slab geometry; (2) better convergence is observed when the consistency of higher- and lower-order models is high; and (3) when a coarse mesh is optically thick, the spatial homogenization degrades the convergence performance, even if spatial and angular approximations are consistent between the higher- and lower-order models.

On the Adequacy of Some Low-Order Moments Method to Simulate Certain Particle Removal Processes

The population balance is an indispensable tool for studying colloidal, aerosol, and, in general, particulate systems. The need to incorporate spatial variation (imposed by flow) to it invokes the reduction of its complexity and degrees of freedom. It has been shown in the past that the method of moments and, in particular, the log-normal approximation can serve this purpose for certain phenomena and mechanisms. However, it is not adequate to treat gravitational deposition. In the present work, the ability of the particular method to treat diffusional and convective diffusional depositions relevant to colloid systems is studied in detail.

High-order numerical solutions to the shallow-water equations on the rotated cubed-sphere grid

A novel numerical approach to solving the shallow-water equations on the sphere using high-order numerical discretizations in both space and time is proposed. A space-time tensor formalism is used to express the equations of motion covariantly and to describe the geometry of the rotated cubed-sphere grid. The spatial discretization is done with the direct flux reconstruction method, which is an alternative formulation to the discontinuous Galerkin approach. The equations of motion are solved in differential form and the resulting discretization is free from quadrature rules. It is well known that the time step of traditional explicit methods is limited by the phase velocity of the fastest waves. Exponential integration is employed to enable integrations with significantly larger time step sizes and improve the efficiency of the overall time integration. New multistep-type exponential propagation iterative methods of orders 4, 5 and 6 are constructed and applied to integrate the shallow-water equations in time. These new schemes enable time integration with high-order accuracy but without significant increases in computational time compared to low-order methods. The exponential matrix functions-vector products used in the exponential schemes are approximated using the complex-step approximation of the Jacobian in the Krylov-based KIOPS (Krylov with incomplete orthogonalization procedure solver) algorithm. Performance of the new numerical methods is evaluated using a set of standard benchmark tests.

The H2020 McSAFER Project: Main Goals, Technical Work Program, and Status

This paper describes the main objectives, technical content, and status of the H2020 project entitled “High-performance advanced methods and experimental investigations for the safety evaluation of generic Small Modular Reactors (McSAFER)”. The main pillars of this project are the combination of safety-relevant thermal hydraulic experiments and numerical simulations of different approaches for safety evaluations of light water-cooled Small Modular Reactors (SMR). It describes the goals, the consortium, and the involved thermal hydraulic test facilities, e.g., the COSMOS-H (KIT), HWAT (KTH), and MOTEL (LUT), including the experimental programs. It also outlines the different safety assessment methodologies applied to four different SMR-designs, namely the CAREM (CNEA), SMART (KAERI), F-SMR (CEA), and NuScale. These methodologies are multiscale thermal hydraulics, conventional, low order, and high fidelity neutron physical methods used to demonstrate the inherent safety features of SMR-core designs under postulated design-basis-accident conditions. Finally, the status of the investigations is shortly discussed followed by the dissemination activities and an outlook.

Discretization error estimates for discontinuous Galerkin isogeometric analysis

Isogeometric analysis is a spline-based discretization method to partial differential equations which show the approximation power of a high-order method. The number of degrees of freedom, however, is as small as the number of degrees of freedom of a low-order method. This does not come for free as the original formulation of isogeometric analysis requires a global geometry function. Since this is too restrictive for many kinds of applications, the domain is usually decomposed into patches, where each patch is parameterized with its own geometry function. In simpler cases, the patches can be combined in a conforming way. However, for non-matching discretizations or for varying coefficients, a non-conforming discretization is desired. An symmetric interior penalty discontinuous Galerkin method for isogeometric analysis has been previously introduced. In the present paper, we give error estimates that are explicit in the spline degree. This opens the door towards the construction and the analysis of fast linear solvers, particularly multigrid solvers for non-conforming multipatch isogeometric analysis.

Development of an Unsteady Vortex Lattice Method to Model Propellers at Incidence

The complex aerodynamics of a propeller at incidence involve multiple asymmetric flow conditions around the disk, with each originating from different physical phenomena. These asymmetries give rise to in-plane forces and out-of-plane moments that are difficult to model with traditional low-order computational aerodynamics methods. This paper presents a three-dimensional unsteady vortex lattice method with a free vortex ring wake formulation that has been developed to model these effects for propellers at incidence. Convergence quality is investigated with respect to simulation time, grid resolution, and time-step resolution. Solution accuracy is evaluated by comparing to computational fluid dynamics (CFD) analyses. Good agreement was found with CFD results. The paper concludes with a discussion on the challenges of modeling propellers at incidence with vortex lattice methods.