Accurate Solver Research Articles

Cagniard–DeHoop technique-based computation of retarded zero-thickness partial elements

Integral-equation based methods for the solution of Maxwell’s equations require that the physical system under analysis be meshed into elementary volumes and surfaces. This happens for the method of moments (MoM) and the partial element equivalent circuit (PEEC) method. Then, interaction integrals describing the electric and magnetic field coupling between these elementary regions need to be computed. This is typically done in the frequency domain by resorting to numerical quadrature schemes. In the time domain (TD), brute approximations are typically done leading to simplified schemes which lack the accuracy especially for electrically large problems when propagation delays are important. Such approximations are quite poor also for close elementary regions which are very strong and, thus very important to the overall solution. Hence, in the perspective of developing an accurate time domain solver, it is desirable to have time domain analytical or quasi-analytical forms of the interaction integrals. In this work, we shall derive quasi-closed-form expressions for retarded coefficients as they appear in the partial element equivalent circuit (PEEC) method. To this aim, the Cagniard–DeHoop (CdH) technique exploiting pertinent integration path deformation in the complex-domain leads to semi-analytical forms. The analysis is carried out for a pair of parallel (but non-coplanar) and orthogonal fundamental surface elements as they occur in the modeling of the electric field coupling due to free or bound charges on the surface of conductors and dielectrics. The same results hold also for the magnetic field coupling assuming that currents flow within thin conductors. The accuracy of the proposed approach is tested for representative parallel and orthogonal patches.

Анализ точности и формальное обоснование численного решения нелинейных уравнений методами Ньютона и Хэйли при использовании арифметики с плавающей запятой

The paper describes particularities of using floating-point arithmetics for finding solutions of nonlinear equations by the means of numerical approximation. Analysis of scientific literature shows scarcity of works studying these methods of numerical solvers in presence of limitations and particularities imposed by algebras of floating-point numbers despite well-known significance of these aspects when it comes to overall accuracy, predictability and usability of numerical solvers. Therefore, the paper describes some interesting results of theoretical and experimental study of these wellknown Newton's and Halley's methods from the point of view of their implementability and problems that arise when floating-point arithmetic is used. The analysis is conducted both theoretically and experimentally using floating-point machines. On one hand, the experiments demonstrate correspondence between the predicted efficiency factors and ones that are measured, but on the other hand these measurements contradict intuitively predicted behavior of solvers if no floating-point specifics are taken into account.

An Accurate and Robust Hybrid Hllc Riemann Solver on Triangular Grids

An Accurate and Robust Hybrid Hllc Riemann Solver on Triangular Grids

Fast Solver for Implicit Continuous Set Model Predictive Control of Electric Drives

This paper proposes a fast and accurate solver for implicit Continuous Set Model Predictive Control for the current control loop of synchronous motor drives with input constraints, allowing for reaching the maximum voltage feasible set. The related control problem requires an iterative solver to find the optimal solution. The real-time certification of the algorithm is of paramount importance to move the technology toward industrial-scale applications. A relevant feature of the proposed solver is that the total number of operations can be computed in the worst-case scenario. Thus, the maximum computational time is known a priori. The solver is deeply illustrated, showing its feasibility for real-time applications in the microseconds range by means of experimental tests. The proposed method outperforms general-purpose algorithms in terms of computation time, while keeping the same accuracy.

Methodology for composite materials shrinkage definition for use in shipbuilding and marine technology

Deformations of steel material in shipbuilding and marine technology applications as a result of mechanical or temperature influences are a well-known problem. However, in the modern shipbuilding industry, the application of alternative materials, especially composite materials, in the structure and for the equipment of the ship is increasingly represented. Consequently, there is a need to determine the deformation and change of characteristics of such composite materials as a result of various mechanical, and especially temperature influences that cause the so-called shrinkage. The basic composite production process involves connecting the matrix with a catalyst and accelerators that create temperature, then the material shrinks by cooling when it can change its dimensions and characteristics. Also, in order to achieve the best possible mechanical properties, composite materials are specially heated and then cooled according to strictly defined processes and curves. The ability to predict the characteristics and parameters of such deformations is important in the context of the application of composite materials. To define such deformations, different methods are used within individual numerical solvers, whose results can differ significantly from each other. Therefore, the authors in this paper present an established methodology for predicting mechanical and temperature deformations, and modelling of composite materials, based on the analysis of analytical methods and numerical solvers with the aim of defining the most accurate numerical solver. By applying the presented methodology, it is expected to raise the level of accuracy and quality of composite materials production as well as to raise the quality of design solutions and efficiency of production procedures during shipbuilding in particular, but also within different marine technology applications and during the product’s life cycle.

Integrating deep neural networks with full-waveform inversion: Reparameterization, regularization, and uncertainty quantification

Full-waveform inversion (FWI) is an accurate imaging approach for modeling the velocity structure by minimizing the misfit between recorded and predicted seismic waveforms. However, the strong nonlinearity of FWI resulting from fitting oscillatory waveforms can trap the optimization in local minima. We have adopted a neural-network-based full-waveform inversion (NNFWI) method that integrates deep neural networks with FWI by representing the velocity model with a generative neural network. Neural networks can naturally introduce spatial correlations as regularization to the generated velocity model, which suppresses noise in the gradients and mitigates local minima. The velocity model generated by neural networks is input to the same partial differential equation (PDE) solvers used in conventional FWI. The gradients of the neural networks and PDEs are calculated using automatic differentiation, which back propagates gradients through the acoustic PDEs and neural network layers to update the weights of the generative neural network. Experiments on 1D velocity models, the Marmousi model, and the 2004 BP model determine that NNFWI can mitigate local minima, especially for imaging high-contrast features such as salt bodies, and it significantly improves the inversion in the presence of noise. Adding dropout layers to the neural network model also allows analyzing the uncertainty of the inversion results through Monte Carlo dropout. NNFWI opens a new pathway to combine deep learning and FWI for exploiting the characteristics of deep neural networks and the high accuracy of PDE solvers. Because NNFWI does not require extra training data and optimization loops, it provides an attractive and straightforward alternative to conventional FWI.

Development of a WENO-type numerical solver for two-phase two-fluid six-equation model

Advanced discretization methods are pursued to improve accuracy of numerical solver for two-phase two-fluid six-equation model. The Weighted Essentially-Non-Oscillatory (WENO) method was a popular high-order discretization one and was successfully applied to many applications. However, development of the WENO-type numerical solver for two-phase two-fluid six-equation model was limited, which was partly due to the lack of analytical eigenvalues and eigenvectors. In previous work, the author derived an approximate analytical eigenvalues and eigenvectors for the two-phase flow model. The analytical eigenvalues and eigenvectors were formulated in a compact and structured way and were valid for arbitrary form of equation of state. The analytical eigenvalues and eigenvectors enable the development of a new WENO-type numerical solver for the two-phase flow model. In this work, a new WENO-type numerical solver is developed. Numerical tests show that the newly developed WENO-type solver works very well and is capable of capturing all the characteristic waves and sharp discontinuities. The order of accuracy study shows that the accuracy of WENO is at least third-order for a smooth solution.

A consistent treatment of moving boundaries with thermal convection for lattice Boltzmann method

Particulate flows with thermal convection exist widely, and the lattice Boltzmann method is proved to be an accurate and efficient solver. Two issues are crucial, namely, the treatment of the velocity and temperature boundaries at the particle surface and the initialization of the fresh fluid nodes due to the movement of the particle. However, these two issues are treated separately, and the inconsistency decreases the numerical accuracy and stability. Thus, a consistent treatment of these two issues is presented, which includes the prediction and correction steps, and it is validated by four numerical experiments. The present method is of the second-order accuracy, and it provides a consistent treatment for particulate flows with thermal convection.

High-Fidelity Simulation of Turbulent Flow Past Gaussian Bump

A spanwise-periodic computation of a turbulent flow past a Gaussian bump is performed in the form of a hybrid direct numerical simulation and wall-resolved large-eddy simulation. A fourth-order spatially accurate flow solver is employed to perform the simulation, using 10.2 billion grid points for a Reynolds number of 170,000 based on the bump height. The key findings from the simulation are reported in the acceleration and deceleration flow regions associated with the bump shape. Significant anisotropy in the normal Reynolds stresses, along both the wall-normal and streamwise directions, is observed within the acceleration region. The ratio between the Reynolds shear stress and turbulent kinetic energy in that region also experiences significant deviations from the norms of a zero pressure gradient turbulent boundary layer. The chosen Reynolds number generates strong flow separation in the adverse pressure gradient region, which is in contrast with a previous simulation at half the Reynolds number that only indicated incipient separation. An internal layer generated in the acceleration region evolves into a free shear layer that develops in the deceleration region and separates. Proper modeling of this inner layer appears crucial to predict the flow separation. Surface curvature effects on the attached flow development are also discussed.

A GPU-accelerated Efficient Simulation Tool (EST) for 2D variable-density mud/debris flows over non-uniform erodible beds

Mud/debris flows are among the most challenging gravity-driven flows in environmental and geophysical processes. Natural muddy slurries and debris are solid-laden fluids where the density of the mixture can be more than twice or three times the water density and, hence, the bulk solid phase can represent 40–80% of the flow column volume. Furthermore, these unsteady flows usually occur along steep and irregular terrain which requires a refined non-structured spatial discretization, increasing the computational times of the models. In this work a new upwind Roe-type solver for two-dimensional multi-grain mixture shallow-flow over non-uniform erodible bed is presented. The coupled system of depth-averaged equations is formed by the conservation equations for the mass and momentum of the variable-density mixture, the mass conservation equations for the N different solid phases transported in the flow and the continuity equation for the erodible bed layer, where the different solid phases can be exchanged independently modifying the bed level. The non-Newtonian rheological behavior of the multi-grain mixture is included into the momentum equations using six different basal resistance formulations. An accurate, robust and efficient x-split Augmented Roe (xA-Roe) solver for variable-density flow is derived, which requires a complete reformulation of the averaged-Roe values at the intercell edges and allows the mixture density to participate in the definition of the characteristic wave celerities of the local Riemann problem. The global time step is dynamically controlled by the wave celerities of the coupled system of equations, preserving the scheme stability even for high density gradients. The bed slope and basal resistance source terms are also upwind discretized and included into the intercell numerical fluxes, ensuring a well-balanced flux formulation in steady states and the correct treatment of wet-dry fronts. The proposed model is GPU-accelerated using a CUDA/C++ algorithm and applied to synthetic tests and the USGS debris dambreak experiments over erodible bed, demonstrating its robustness, accuracy and efficiency. Finally, the model is tested against a real-scale and long-term case, the mining-tailing dam failure occured in Brumadinho (Brazil) in 2019. The numerical results show good agreement with the observed field data and the computational cost reduction obtained with the GPU-accelerated algorithm is up to 60 times compared to a CPU-based code.

Stability and accuracy of Engquist–Majda absorbing boundary condition for pseudo spectral time domain method

AbstractLagrange polynomials based Chebyshev Pseudo Spectral Time Domain (L‐CPSTD) method is an accurate time domain solver for electromagnetic problems. It utilizes the Lagrange interpolation polynomials to expand electromagnetic fields. This global interpolation, which uses all field values on all of the grid points to calculate the value at a single point, provides spectral accuracy. However, absorbing boundary conditions (ABCs) must be applied for open space problems. Engquist–Majda ABC is an important one due to its simplicity. Characteristic variables (CVs) can be used to implement the ABCs. In this article, for the first time, stability and accuracy of the Engquist–Majda ABC are proved by using the CVs in the L‐CPSTD solution of Maxwell's equations. The theoretical findings are verified by using the matrix eigenvalue method and the reflection coefficient in one and two dimensional open space examples. The numerical result is also validated by an analytical and an FDTD solution of a parallel‐plate waveguide problem. The efficiency of the proposed ABC is clearly shown.

Multiscale modeling and simulation methods for electromagnetic and multiphysics problems

Multiscale modeling and simulation methods for electromagnetic and multiphysics problems

Immersed boundary finite element method for blood flow simulation

We present an efficient and accurate immersed boundary (IB) finite element (FE) method for numerically solving the incompressible Navier–Stokes equations. Particular emphasis is given to internal flows with complex geometries (blood flow in the vasculature system). IB methods are computationally costly for internal flows, mainly due to the large percentage of grid points that lie outside the flow domain. In this study, we apply a local refinement strategy, along with a domain reduction approach, in order to reduce the grid that covers the flow domain and increase the percentage of grid nodes that fall inside the flow domain. The proposed method utilizes an efficient and accurate FE solver with the incremental pressure correction scheme (IPCS), along with the boundary condition enforced IB method to numerically solve the transient, incompressible Navier–Stokes flow equations. We verify the accuracy of the numerical method using the analytical solution for Poiseuille flow in a cylinder. We further examine the accuracy and applicability of the proposed method by considering flow within complex geometries, such as blood flow in aneurysmal vessels and the aorta, flow configurations which would otherwise be extremely difficult to solve by most IB methods. Our method offers high accuracy, as demonstrated by the verification examples, and high efficiency, as demonstrated through the solution of blood flow within complex geometry on an off-the-shelf laptop computer.

An iterative factored topography-dependent eikonal solver for anisotropic media

Accurate and efficient eikonal solvers for heterogeneous media play an important role in many areas of seismology, such as seismic tomography, migration, and earthquake localization. Incorporating seismic anisotropy and complex topography remain a computational challenge for finite-difference eikonal solvers. In recent years, the topography-dependent eikonal equation (TDEE) has been proposed as an effective way to calculate seismic traveltimes for isotropic and anisotropic media with irregular topography. However, the Lax-Friedrichs sweeping method used in previous studies to approximate the viscosity solution of TDEE for anisotropic media is more dissipative and needs a much higher number of iterations to converge. In addition, the TDEE solution for the initial point source has an upwind source singularity, which makes all TDEE solvers, even the high-order ones, exhibit polluted convergence and relatively large errors that propagate from the point source to the entire computational domain. To solve these problems, we have formulated the factored topography-dependent anisotropic eikonal (FTDAE) equation in tilted transversely isotropic (TTI) media using the factorization principle. Then, the resulting quartic equation can be numerically solved by using a fixed-point iteration technique based on the simpler elliptical anisotropic eikonal (EAE) equation with a high-order source term due to anelliptical anisotropy introduced by TTI media. At each iteration, the unknown traveltime in the EAE equation is factored into two functions: One of the functions is specified analytically to capture the source singularity, such that the unknown factor is differentiable in the source neighborhood and could be solved by the fast sweeping method. Numerical examples indicate that our FTDAE solver can treat source singularity successfully and achieve high accuracy after just a few iterations, independently of the mesh size, which could provide a more efficient and robust tool for traveltime calculation in the presence of seismic anisotropy and complex surfaces.

A hyperbolic Poisson solver for wall distance computation on irregular triangular grids

This paper presents an accurate and efficient Poisson solver for wall distance computations on irregular triangular grids. It is discussed that a typical formula of computing the wall distance from a numerical solution of a Poisson equation is not very accurate when solution contours are curved and the accuracy can be improved by a second-order formula derived by Taubin. However, the improved formula, as it requires second derivatives of the numerical solution, presents a challenge for a second-order method leading to zeroth-order accurate second-order derivatives on irregular grids. To overcome the difficulty, we develop a hyperbolic Poisson solver with gradients included as additional unknowns. In this method, the solution gradient can be obtained with second-order accuracy by a second-order discretization method, and therefore the second-derivatives can be obtained with first-order accuracy. Moreover, the hyperbolic method eliminates the numerical stiffness arising from the discretization of the Laplacian operator and achieves the ever-growing O(1/h) speed-up in iterative convergence, where h is a typical mesh spacing. Superior accuracy and efficiency of the developed method are demonstrated for irregular triangular grids.

Efficient computation of coordinate-free models of flame fronts

We present an efficient, accurate computational method for a coordinate-free model of flame front propagation of Frankel and Sivashinsky. This model allows for overturned flames fronts, in contrast to weakly nonlinear models such as the Kuramoto–Sivashinsky equation. The numerical procedure adapts the method of Hou, Lowengrub and Shelley, derived for vortex sheets, to this model. The result is a nonstiff, highly accurate solver which can handle fully nonlinear, overturned interfaces, with similar computational expense to methods for weakly nonlinear models. We apply this solver both to simulate overturned flame fronts and to compare the accuracy of Kuramoto–Sivashinsky and coordinate-free models in the appropriate limit. doi:10.1017/S1446181121000079

Numerical study of shock-disturbances interaction in hypersonic inviscid flows with real gas effects using high-order WENO scheme

In the present study, the shock-disturbances interaction in hypersonic inviscid flows with real gas effects is studied by applying a high-order accurate numerical method with the shock capturing technique. To consider real gas effects, the equilibrium air model is utilized here. The strong conservative form of the unsteady compressible Euler equations in the 2D generalized curvilinear coordinates is formulated and the resulting system of equations for the equilibrium air model is discretized by using the fifth-order finite-difference WENO scheme in space and the explicit third-order TVD Runge–Kutta scheme in time to provide a highly accurate and robust equilibrium airflow solver. The solution methodology adopted based on the high-order WENO scheme with the shock capturing technique can reasonably simulate the strong discontinuities in the solution domain without needing any numerical artifact and this feature makes the solution method suitable for the shock-disturbances interaction in hypersonic flows in which real gas effects are to be considered. To study the physical phenomenon of the shock-disturbances interaction problem in high-speed inviscid equilibrium airflows, two test cases are simulated by applying the solution methodology adopted. In the first test case, the interaction between the incoming fast acoustic wave with the bow shock formed around a cylinder with the nose radius of RN=0.0381 m at M∞=8.03 and T∞=182.333 K is simulated. In the second test case, the cylinder radius is taken RN=2.54×10−2 m and the upstream velocity and temperature are u∞=5590m/s and T∞=1833 K, respectively, and all kinds of free stream disturbances, including the entropy wave and the fast and slow acoustic waves, for different wave numbers are considered. The computations are carried out for both the perfect gas and equilibrium air models and the effects of real gas on both the mean and disturbances fields are also studied. The present study introduces the first-known numerical investigation of the shock-disturbances interaction in hypersonic flow over blunt noses with real gas effects in which the numerical results obtained are thoroughly validated by the analytical/theoretical ones and good agreement is found.

Cattaneo-christov heat flux model of 3D hall current involving biconvection nanofluidic flow with Darcy-Forchheimer law effect: Backpropagation neural networks approach

Recently, the applications of artificial intelligence through soft computing and machine learning algorithms have become the focal point of researcher's consideration on account of their convenience for accurate modelling, ease in simulation and effective assessment. This article endorses soft computing based backpropagated neural networks (BNNs) with Levenberg Marquardt technique (LMT), i.e., BNN-LMT, over a novel mathematical model based on biconvection, second grade combine convection nanofluid (BSCCN) flow associated with Cattaneo-Christove (CC) heat flux model for thermal transportation and viscous dissipation, Darct-Forhheimer (DF) law for permeable medium and Hall (H) current for high intensity electric conductive on flow motion model, i.e., BSCCN-CCDFH flow model. Self-similar transformations are used to reduce the multivariable function model to mathematical system of a single variable. The assessment of thermal buoyancy parameter, Hall parameter, porosity parameter, thermophoresis factor, Lewis number and Peclet number over the flow rate dynamics, energy, nanofluid concentration and microorganism concentration profiles is made through dataset based on Adam numerical solver for different physical quantity based scenarios. The results of exhaustive numerical simulation studies show that the proposed technique BNN-LMT is an efficient, reliable, accurate and rapid convergent stochastic numerical solver exploited viably for the BSCCN-CCDFH flow model having number of physical variations.

High-fidelity kinetic modeling of instabilities and gyromotion physics in nonuniform low-beta plasmas

A fourth-order accurate continuum kinetic Vlasov solver and a systematic method for constructing customizable kinetic equilibria are demonstrated to be powerful tools for the study of nonuniform collisionless low-beta plasmas. The noise-free methodology is applied to investigate two gradient-driven instabilities in 4D (x,y,vx,vy) phase space: the Kelvin–Helmholtz instability and the lower hybrid drift instability. Nonuniform two-species configurations where ion gyroradii are comparable to gradient scale lengths are explored. The approach sheds light on the evolution of the pressure tensor in Kelvin–Helmholtz instabilities and demonstrates that the associated stress tensor deviates significantly from the gyroviscous stress tensor. Even at high magnetization, first-order approximations to finite-gyromotion physics are shown to be inadequate for the Kelvin–Helmholtz instability, as shear scales evolve to become on par with gyromotion scales. The methodology facilitates exploring transport and energy partitioning properties associated with lower hybrid drift instabilities in low-beta plasma configurations. Distribution function features are captured in detail, including the formation of local extrema in the vicinity of particle-wave resonances. The approach enables detailed targeted investigations and advances kinetic simulation capability for plasmas in which gyromotion plays an important role.

Large dust fractions can prevent the propagation of soundwaves

ABSTRACT Dust plays a central role in several astrophysical processes. Hence, the need of dust/gas numerical solutions, and analytical problems to benchmark them. In the seminal dustywave problem, we discover a regime where sound waves cannot propagate through the mixture above a large critical dust fraction. We characterize this regime analytically, making it of use for testing accuracy of numerical solvers at large dust fractions.