Accurate Discretization Research Articles

Analysis of a meshless generalized finite difference method for the time-fractional diffusion-wave equation

In this paper, a generalized finite difference method (GFDM) is proposed and analyzed for meshless numerical solution of the time-fractional diffusion-wave equation. Two (3−α)-order accurate temporal discretization schemes are presented by using the L1 formula and the original H2N2 or fast H2N2 formulas to discretize the time-fractional derivative of order α∈(1,2). The stability of the temporal discretization schemes is analyzed. Then, the time-fractional diffusion-wave initial-boundary value problem is transformed into a series of time-independent integer-order boundary value problems, and discrete linear algebraic systems are built by the application of the GFDM. Accuracy analysis of the GFDM with both original H2N2 and fast H2N2 formulas is presented in theory, and numerical experimental results are provided to verify the theoretical results and the effectiveness of the proposed meshless method.

Second order accurate particle-in-cell discretization of the Navier-Stokes equations

We propose the use of PolyPIC transfers [10] to construct a second order accurate discretization of the Navier-Stokes equations within a particle-in-cell framework on MAC grids. We investigate the accuracy of both APIC [16,17,8] and quadratic PolyPIC [10] transfers and demonstrate that they are suitable for constructing schemes converging with orders of approximately 1.5 and 2.5 respectively. We combine PolyPIC transfers with BDF-2 time integration and a splitting scheme for pressure and viscosity and demonstrate that the resulting scheme is second order accurate. Prior high order particle-in-cell schemes interpolate accelerations (not velocities) from the grid to particles and rely on moving least squares to transfer particle velocities to the computational grid. The proposed method instead transfers velocities to particles, which avoids the accumulation of noise on particle velocities but requires the polynomial reconstruction to be performed using polynomials that are one degree higher. Since this polynomial reconstruction occurs over the regular grid (rather than irregularly distributed particles), the resulting weighted least squares problem has a fixed sparse structure, can be solved efficiently in closed form, and is independent of particle coverage.

A mixed finite element spatial discretization scheme and a higher‐order accurate temporal discretization scheme for a strongly discontinuous electromagnetic interface condition in ideal sliding electrical contact problems

AbstractSliding electrical contact involves multiple conductors sliding in contact at different speeds, with current flowing through the contact surfaces. The Lagrangian method is commonly used to describe the electromagnetic field in order to overcome the trouble of convective dominance, especially in high‐speed sliding electrical contact problems. However, to maintain correct field continuity, magnetic vector potential and scalar potential taken as variables cannot be continuous simultaneously at the ideal sliding electrical contact interface. This involves a strongly discontinuous condition for variables. Further, commonly used spatial–temporal discretization algorithms are invalid, for example, the classic finite element (CFEM) framework does not allow discontinuous variables, and the backward Euler method in time domain introduces a significant interface error source associated with the velocity of relative motion. To accurately handle strongly discontinuous conditions in numerical calculations, a mixed nodal finite element scheme and a higher‐order accurate temporal discretization scheme are introduced. In this scheme, classical finite element method is performed in each subdomain, and the derivative terms at the boundary are added as new variables. The effectiveness and accuracy of the above methods are verified by comparing them with a standard solution in a two‐dimensional railgun model and analyzing the current density distribution in a three‐dimensional railgun model.

Improved sequential convex programming based on pseudospectral discretization for entry trajectory optimization

Sequential convex programming (SCP) is widely used to solve entry trajectory optimization problems. However, challenges persist in scenarios with strict constraints, such as unsolvable convex subproblems, iterative solution oscillations, and slow convergence rates. This study introduces an enhanced SCP algorithm designed to address these limitations. First, hp Radau pseudospectral discretization is used instead of trapezoidal discretization to improve the efficiency in solving subproblems while maintaining discretization accuracy. Second, the trust region is adaptively updated on the basis of trajectory information during iterations. Additionally, constraint relaxation and virtual control are introduced to facilitate smooth iteration at the initial stage. The regularization technique is also utilized to improve the convergence rates. Finally, the proposed algorithm is validated through two examples: maximum-terminal-velocity entry and maximum-terminal-longitude entry. The results show that these two problems cannot be effectively solved using the basic SCP algorithm. However, the proposed algorithm, along with the trust-region SCP algorithm and GPOPS, can solve them efficiently. With comparable accuracy in the obtained solutions, the algorithm proposed in this paper requires only half the CPU time of the trust-region SCP algorithm and just 5 % of the computing time of the general-purpose open-source solver GPOPS.

A novel physics-aware graph network using high-order numerical methods in weather forecasting model

In this paper, we present a novel architecture for accurate weather forecasting within the framework of Physics-aware Graph Networks (PaGN), which aims to improve the prediction of weather time series by integrating both data and physical equations defined on sparsely distributed spatial domains. This approach employs geometric deep learning by accounting for both supervised loss and physics loss. Solely in supervised learning, data-driven training demands a large amount of data, whereas integrating physics information can reduce this requirement, resulting in faster training. The physics awareness also provides a stronger inductive bias and better generalization, directing the machine learning process by clarifying the complexity of weather data. Thus, a more accurate discretization of physics equations can enhance the physics awareness of the complex weather dynamics. Meanwhile, weather data is essentially gathered at observatories that are not located densely, since covering the Earth with numerous observatories is prohibitively expensive. Given its reliance on graph structure, the proposed PaGN is well-suited for handling sparsely distributed data particularly defined on arbitrary geometric domain. One of the main contributions of this study is the use of high-order numerical methods approximating physics equations defined on graph neural networks for achieving accuracy improvements. The fractional graph Laplacian operator is further integrated with the physics equations to account for the non-local characteristics of complex weather data. For computational efficiency, the accuracy improved PaGN architecture is smoothly implemented in the embedding space. A series of numerical tests on weather datasets is performed to verify the improved accuracy, robustness, and applicability of the proposed methodology. We first carry out the numerical study on the synthetic datasets obtained from the physics equations with an extra forcing term. We then extensively investigate the accuracies of weather forecasting for various configurations in terms of the geophysical regions and the accuracy-improved PaGN approaches for diffusion and wave equations. Moreover, we emphasize that the fractional graph Laplacian operator incorporated into our PaGN model can further improve the accuracy for weather forecasting.

A Fast Algebraic Multigrid Solver and Accurate Discretization for Highly Anisotropic Heat Flux I: Open Field Lines

A Fast Algebraic Multigrid Solver and Accurate Discretization for Highly Anisotropic Heat Flux I: Open Field Lines

High-order discretization–based self-adaptive turbulence eddy simulation for supersonic base flow with PHengLEI software

PurposeThe purpose of the present study is to develop a new numerical framework that can predict the supersonic base flow more accurately, including the development of axisymmetrically separated shear layer and recompression shock. To this end, two aspects are improved and combined, i.e. a newly self-adaptive turbulence eddy simulation (SATES) turbulence modeling method and a high-order discretization numerical scheme. Furthermore, the performance of the new numerical framework within a general-purpose PHengLEI software is assessed in detail.Design/methodology/approachSatisfactory prediction of the supersonic separated shear layer with unsteady wake flow is quite challenging. By using a unified turbulence model called SATES combining high-order accurate discretization numerical schemes, the present study first assesses the performance of newly developed SATES for supersonic axisymmetric separation flows. A high-order finite differencing-based compressible computational fluid dynamics (CFD) code called PHengLEI is developed and several different numerical schemes are used to investigate the effects on shock-turbulence interactions, which include the monotonic upstream-centered scheme for conservation laws (MUSCL), weighted compact nonlinear scheme (WCNS) and hybrid cell-edge and cell-node dissipative compact scheme (HDCS).FindingsCompared with the available experimental data and the numerical predictions, the results of SATES by using high-order accurate WCNS or HDCS schemes agree better with the experiments than the results by using the MUSCL scheme. The WCNS and HDCS can also significantly improve the prediction of flow physics in terms of the instability of the annular shear layer and the evolution of the turbulent wake.Research limitations/implicationsThe small deviations in the recirculation region can be found between the present numerical results and experimental data, which could be caused by the inaccurate incoming boundary layer condition and compressible effects. Therefore, a proper incoming boundary layer condition with turbulent fluctuations and compressibility effects need to be considered to further improve the accuracy of simulations.Practical implicationsThe present study evaluates a high-order discretization-based SATES turbulence model for supersonic separation flows, which is quite valuable for improving the calculation accuracy of aeronautics applications, especially in supersonic conditions.Originality/valueFor the first time, the newly developed SATES turbulence modeling method combining the high-order accurate WCNS or HDCS numerical schemes is implemented on the PHengLEI software and successfully applied for the simulations of supersonic separation flows, and satisfactory results are obtained. The unsteady evolutions of the supersonic annular shear layer are analyzed, and the hairpin vortex structures are found in the simulation.

Study on Water Entry of a 3D Torpedo Based on the Improved Smoothed Particle Hydrodynamics Method

The water entry of a torpedo is a complex nonlinear problem, involving transient impact, free surface deformation, droplet splashing, and fluid–structure coupling, which poses severe challenges to traditional mesh methods. The meshless smoothed particle hydrodynamics (SPH) method shows unique advantages in capturing the complex features of the water entry of the torpedo at different entry angles. However, it still suffers from some inherent shortcomings, such as low surface discretization accuracy, poor discretization flexibility, and low calculation efficiency. In this study, an improved adaptive SPH algorithm is proposed to investigate the water entry of the torpedo accurately and efficiently. This method integrates meshless point generation and adaptive techniques simultaneously. The numerical results demonstrate that when the torpedo vertically enters the water at different velocities, the induced impact loads acting on the head of the torpedo fluctuate significantly with two peak values in the initial stage and thereafter stabilize in a later stage. The impact load acting on the torpedo, the entry depth of the torpedo, the splash height of the droplets, and the size of the cavity generated around the torpedo increase with the increment in the entry velocity. When the torpedo enters the water at different entry angles under the same initial entry velocity, both the vertical and the horizontal movements of the torpedo are observed, which results in more complex variations in parameters along the x- and y-axes. The findings and the corresponding numerical method in this study can provide a certain basis for the future designs of the entry trajectory and the structural bearing capacity of torpedoes.

A parallel multi-resolution Smoothed Particle Hydrodynamics model with local time stepping

Smoothed Particle Hydrodynamics (SPH) model has exhibited remarkable efficacy in addressing strongly nonlinear large deformation problems. However, the expensive computational cost could limit its application. To address this, we propose a novel parallel multi-resolution SPH model with a local time step strategy. This model integrates the multi-resolution model with a Message Passing Interface-based parallelization framework, wherein subdomains discretize the computational domain at varying resolutions. Meanwhile, a local fourth-order Runge-Kutta time integration scheme is introduced, enabling different subdomains to use different time steps. Subdomains of varying resolutions are devoid of overlapping regions, and changes in particle resolution at interfaces are achieved through a dynamic strategy involving particle merging and splitting. Besides that, we conducted a comparative analysis of different SPH discretization formats concerning their impact on the accuracy of multi-resolution particle discretization. Finally, several numerical tests were conducted to validate the accuracy and efficiency of the present multi-resolution SPH model.

A three-stage sequential convex programming approach for trajectory optimization

Recently, sequential convex programming (SCP) has become a potential approach in trajectory optimization because of its high efficiency. To improve stability and discretization accuracy, a three-stage SCP approach based on the hp-adaptive Radau pseudospectral discretization is proposed in this paper. In most instances, the initial subproblem may risk infeasibility due to the undesignated initial guess. Therefore, we design a constraint relaxation stage for the SCP to enhance the feasibility of the subproblem as much as possible. Once the subproblem can be directly solved, the iteration enters the second stage, during which a mesh refinement algorithm based on discretization error analysis is utilized to decrease the discretization error to the tolerance. In the final stage, the damping term is introduced into the objective of the subproblem to suppress the oscillation of the solution and accelerate the convergence. A dual-channel control reentry trajectory optimization and an ascent trajectory optimization are taken as examples, and the simulation results show that the proposed approach outperforms conventional SCP approaches in terms of accuracy and efficiency.

On a Stability of Non-Stationary Discrete Schemes with Respect to Interpolation Errors

The aim of this article is to analyze the efficiency and accuracy of finite-difference and finite-element Galerkin schemes for non-stationary hyperbolic and parabolic problems. The main problem solved in this article deals with the construction of accurate and efficient discrete schemes on nonuniform and dynamic grids in time and space. The presented stability and convergence analysis enables improving the existing accuracy estimates. The obtained stability results show explicitly the rate of accumulation of interpolation and projection errors that arise due to the movement of grid points. It is shown that the cases when the time grid steps are doubled or halved have different stability properties. As an additional technique to improve the accuracy of discretizations on non-stationary space grids, it is recommended to use projection operators instead of interpolation operators. This technique is used to solve a test parabolic problem. The results of specially selected computational experiments are also presented, and they confirm the accuracy of all theoretical error estimates obtained in this article.

Higher‐order generalized‐α methods for parabolic problems

AbstractWe propose a new class of high‐order time‐marching schemes with dissipation control and unconditional stability for parabolic equations. High‐order time integrators can deliver the optimal performance of highly accurate and robust spatial discretizations such as isogeometric analysis. The generalized‐ method delivers unconditional stability and second‐order accuracy in time and controls the numerical dissipation in the discrete spectrum's high‐frequency region. We extend the generalized‐ methodology to obtain high‐order time marching methods with high accuracy and dissipation control in the discrete high‐frequency range. Furthermore, we maintain the original stability region of the second‐order generalized‐ method in the new higher‐order methods; we increase the accuracy of the generalized‐ method while keeping the unconditional stability and user‐control features on the high‐frequency numerical dissipation. The methodology solves matrix problems and updates the system unknowns, which correspond to higher‐order terms in Taylor expansions to obtain ‐order method for even and ‐order for odd . A single parameter controls the high‐frequency dissipation, while the update procedure follows the formulation of the original second‐order method. Additionally, we show that our method is A‐stable, and for we obtain an L‐stable method. Furthermore, we extend this strategy to analyze the accuracy order of a generic method. Lastly, we provide numerical examples that validate our analysis of the method and demonstrate its performance. First, we simulate heat propagation; then, we analyze nonlinear problems, such as the Swift–Hohenberg and Cahn–Hilliard phase‐field models. To conclude, we compare the method to Runge–Kutta techniques in simulating the Lorenz system.

Forecasting model for hypoid gear elastohydrodynamic lubrication considering entrainment effect

In full consideration of time-varying meshing performance and oil film characteristics, entrainment effect on elastohydrodynamic lubrication (EHL) is developed for establishing accurate forecasting model of the complex hypoid gears. Firstly, dynamic loaded tooth contact analysis (DLTCA) is employed to determine the meshing interface kinematics, load distribution and entrainment speed under dynamic load condition. Then, entrainment speed is introduced into the EHL governing equation and accurate forecasting model is established. With accurate discretization of EHL governing equation, discrete convolution and fast Fourier transformation (DC-FFT) method is employed to compute the flank elastic deformation. Finally, a hypoid gear set in automotive industrial application is exercised to verify entrainment effect on EHL assessment in the proposed model.

Accurate Time-Efficient Thermal Modeling and Discretization Methodology Applied to Electric Machines

The increasing power density of electric machines for electric vehicles necessitates accurate thermal analysis in design and real-time condition monitoring through simulations with computation times not exceeding one minute. Existing thermal models are either accurate and complex while requiring large computation times or they are fast but lack accuracy. To address these challenges, a novel hybrid modeling methodology is proposed, combining 3D lumped parameter thermal networks with thermal resistances extracted from 2D finite element models (FEM) to reduce computation demands while preserving accuracy. Further, a methodology is introduced to define the thermal network discretization for a desired threshold accuracy while minimizing computation time. Analysis of this discretization methodology demonstrates that the desired accuracy can be reliably achieved with limited computational effort. Verification of the modeling methodology against 3D FEM in OpenFOAM shows key component temperature deviations of less than 0.67 °C in steady-state simulations and 0.75 °C in transient simulations, comparable to the 3D FEM uncertainty of 0.39 °C. The computation times for steady-state and transient simulations are 18.3 s and 43.5 s respectively compared to 44 h and 175 h for 3D FEM. This highlights the time efficiency of the proposed methodology while maintaining accuracy for key component temperatures.

Multiharmonic multiscale modelling in 3-D nonlinear magnetoquasistatics: Composite material made of insulated particles

The use of the classical finite element method (FEM) to solve problems with magnetic composites leads to huge linear systems that are impossible to solve. Instead, homogenization and multiscale methods are often used with the composite material replaced by a homogeneous material with the homogenized constitutive law obtained by solving cell-problems representing the mesoscale material structure. For non-linear time-dependent problems, FEM is often used with a time-transient method (TTM) and the solution is obtained one time-step at a time. However, in cases where a steady-state solution is of interest, the multiharmonic method can be faster and more cost effective for the same accuracy of the time discretization. In addition, when solving magnetoquasistatic multiscale problems with TTM, the dynamic hysteresis in the homogenized fields can slow down or even impede the convergence of the macro-scale problem due to the possibly non-continuously differentiable homogenized material laws. This work presents a novel robust modelling approach for non-linear magnetoquasistatic problems combining multiharmonic method with the multiscale method.

Accuracy and stability analysis of horizontal discretizations used in unstructured grid ocean models

One important tool at our disposal to evaluate the robustness of Global Circulation Models (GCMs) is to understand the horizontal discretization of the dynamical core under a shallow water approximation. Here, we evaluate the accuracy and stability of different methods used in, or adequate for, unstructured ocean models considering shallow water models. Our results show that the schemes have different accuracy capabilities, with the A- (NICAM) and B-grid (FeSOM 2.0) schemes providing at least 1st order accuracy in most operators and time integrated variables, while the two C-grid (ICON and MPAS) schemes display more difficulty in adequately approximating the horizontal dynamics. Moreover, the theory of the inertia-gravity wave representation on regular grids can be extended for our unstructured based schemes, where from least to most accurate we have: A-, B, and C-grid, respectively. Considering only C-grid schemes, the MPAS scheme has shown a more accurate representation of inertia-gravity waves than ICON. In terms of stability, we see that both A- and C-grid MPAS scheme display the best stability properties, but the A-grid scheme relies on artificial diffusion, while the C-grid scheme does not. Alongside, the B-grid and C-grid ICON schemes are within the least stable. Finally, in an effort to understand the effects of potential instabilities in ICON, we note that the full 3D model without a filtering term does not destabilize as it is integrated in time. However, spurious oscillations are responsible for decreasing the kinetic energy of the oceanic currents. Furthermore, an additional decrease of the currents’ turbulent kinetic energy is also observed, creating a spurious mixing, which also plays a role in the strength decrease of these oceanic currents.

A convergence‐enhanced Timoshenko beam element for the stochastic nonlinear analysis of reinforced concrete components

AbstractAn effective and precise physical model is generally required for the performance assessment of reinforced concrete (RC) components. In this study, a convergence‐enhanced Timoshenko beam element considering finite deformation is proposed for the stochastic nonlinear analysis of RC components. The unified framework of the rank 2 correction matrix is obtained by approximating the iterative matrix through the linearly independent basis vectors, and two types of Broyden–Fletcher–Goldfarb–Shanno (BFGS) operators are constructed. Then, a consistent solution scheme with superior numerical stability and efficiency is proposed based on the second kind of BFGS operator and improved quasi‐Newton method. Accurate discretization of the displacement fields and description of nonlinear sectional behavior are achieved by the high‐order Lagrangian interpolation functions, concrete damage‐plasticity model and J2 plasticity model. In an updated Lagrangian formulation, these shape functions and constitutive relationships are used to develop a complete secant stiffness with higher‐order terms in the strain tensor and residual force vector. To validate the proposed Timoshenko beam element, numerical applications involving material nonlinearity and geometrical nonlinearity are conducted. The numerical stability and efficiency of the solution scheme, the shear‐locking free technique, and the oscillation of axial force are thoroughly examined and discussed by solution of RC columns, cantilever beam and progressive collapse of planar RC frame. Finally, the parametric study considering the geometric and material randomness indicates that the RC column designed for flexural failure mode suffered considerable degradation of loading capacity and a larger scatter of residual load.

The Impact of Frame Transformations on Power System EMT Simulation

This article investigates the impact of frame transformations on the accuracy of numerical discretization in power system transient and stability studies. As analysed, frame transformations influence the convergence of the numerical discretization. Specifically, for an explicit discretization method (e.g., forward Euler method), the stability of the original system is best preserved in the frame where the system eigenvalue is closer to the origin of the complex plane, e.g., in the stationary frame for inductors and capacitors, and in the synchronous frame for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq-</i> frame controllers of inverters. Simulation results are given to validate the theoretical analysis.

An improved integrated framework based nodal density variable and Voronoi polygon for FE-based topology optimization

In this paper, an improved integrated framework is developed to deal with the numerical instability problems in FE-based topology optimization design methods. The proposed method mainly consists of two coupled computational layers: the upper layer for material geometric description, and the lower layer for structural analysis and sensitivity calculation. In the upper layer, the design domain is discretized by Delaunay triangulation, and a smooth and continuous material density description is obtained by interpolating the densities at nodes of Delaunay triangulation with Shepard interpolants. In the lower layer, the dual Voronoi polygon for Delaunay triangulation is used to generate the finite element mesh, and the virtual element method is applied to handle Voronoi polygon for structural response. Following this framework, the separate description of density distribution avoids the problem of geometric description relying on FE mesh in some existing work. Meanwhile, the use of Voronoi polygon provides more accurate finite element discretization of constantly changing material distributions. Typical numerical examples for the minimum compliance optimization problem reveal that the proposed method can eliminate checkerboards and mesh-dependencies effectively, and get approximate discrete 0/1 solutions without any filter. The simulation results demonstrate the efficiency and robustness of the proposed formulation and numerical techniques.

Multiresolution approximation for shallow water equations using summation-by-parts finite differences

Abstract We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.