Continuity Equation Research Articles

Semi-Lagrangian simulation of particle laden flows using an SPH framework

Particle–laden flows occur in many natural and industrial systems and simulating them can be particularly challenging. The coupling of smoothed particle hydrodynamics (SPH) with the discrete element method (DEM) can effectively simulate particle-laden multiphase fluid dynamical systems, due to the shared Lagrangian nature of both methods. However, this approach has some inherent shortcomings, including a prohibitively small time step when dealing with small particles. An alternative approach is to use an Eulerian-Eulerian reference frame, usually using finite element or finite volume discretisations. In this approach, momentum and continuity equations are solved for each discrete phase as well as the continuous phase, and particles are modelled by means of concentration and velocity fields. This approach can suffer from strong numerical diffusion in the advection of the concentrations when absolute velocities of the phases are high, whilst their relative velocities are small. This numerical diffusion can obscure important aspects of the behaviour as it can smooth out details, especially in the particle concentration fields. In order to mitigate the shortcomings of these existing techniques, we present a new SPH-based semi-Lagrangian framework for solving the momentum and continuity equations for all phases in particle-laden flows. In this framework, the discrete and continuous phases move relative to a reference frame that moves at a momentum-averaged velocity. By focusing on velocities relative to the reference frame our method substantially reduces numerical diffusion compared to traditional Eulerian-Eulerian approaches, at a lower computational cost compared to Lagrangian-Lagrangian approaches. The simulation approach is validated by means of comparisons to both computational and experimental results for a number of relevant systems including particle-liquid separation in an inclined channel, particle sedimentation in a liquid, and gas-particle fluidised beds. This new method is shown to compare very favourably in terms of both the accuracy of the results and the computational cost required to achieve them.

An optimal parameterized Newton-type structure-preserving doubling algorithm for impact angle guidance-based 3D pursuer/target interception engagement

The proposed strategy, finite-time state-dependent Riccati equation (FT-SDRE)-based impact angle guidance, is generally employed to solve the 3D pursuer/target interception model with fixed lateral accelerations. This article expands its application to a general scenario where the lateral acceleration of a target may change. To achieve this, we approximate the accelerations of the azimuth and elevation angles of the target in the inertial frame via second-order finite difference schemes and develop a high-performance FT-SDRE algorithm with structure-preserving doubling algorithms (SDAs). As a result, the update frequency of the controller can be increased, and better guidance of the pursuer can be obtained to address the high maneuverability of the target during the entire interception procedure. At every state of the FT-SDRE, a modified Newton–Lyapunov method is employed to solve the continuous algebraic Riccati equation (CARE), and a new simplified SDA with adaptive optimal parameter selection is proposed for solving the associated Lyapunov equation. Our numerical results demonstrate that the FT-SDRE algorithm accelerated by our proposed methods is approximately three times faster than the FT-SDRE algorithm, in which the MATLAB functions icare and lyap are used to solve the CARE and the Lyapunov equation, respectively, throughout the entire interception procedure. In other words, the control frequency can be increased threefold. In our benchmark cases where the target maneuvers with nonlinear lateral acceleration, the target can be intercepted earlier via the proposed FT-SDRE algorithm.

General Definitions of Information, Intelligence, and Consciousness from the Perspective of Generalized Natural Computing

Atoms themselves have no thoughts and cannot be thinking. Why does the human body, which is composed of atoms, have consciousness? The widely used concepts of information and intelligence in today’s science, which are related to this, do not yet have appropriate general definitions. Answering these interesting questions is a crucial issue for technological development in the historical context of human society entering the era of intelligence. The key lies in how to fully utilize the existing fundamental theories subtly related to information science. Here we attempt to give the definition of general information and general intelligence from the perspective of generalized natural computing, based on the least action principle, Hamilton-Jacobi equation, dynamic programming, reinforcement learning, and point out the relationship between the two. The least action principle for describing conservative systems can be seen as an intelligent manifestation of natural matter, and its equivalent form, the Hamilton-Jacobi equation, can be extended to describe quantum phenomena and is a special case of continuous dynamic programming equations. Dynamic programming is an efficient optimization method under deterministic models, while reinforcement learning, as a manifestation of biological intelligence, is its model-free version. The statement that reinforcement learning is the most promising machine learning method has a profound physical foundation. General information is defined as the degree to which a certain environmental element determines the behavior of the subject. General intelligence is defined as the automatic optimization ability of the action or value function of a system with a certain degree of conservatism. Intelligence is a basic property of material systems, rather than an emergent property that only complex systems possess. Consciousness is an advanced intelligent phenomenon, a reconstruction of quasi conservative systems based on complex systems.

A multiphase flow model of water droplets dielectrophoretic-induced air dehumidification phenomena

Air humidity in indoor spaces plays a critical role in human comfort and health. Dehumidification systems are used for building humidity controls, but they can take significant energy consumption, especially in geographic locations with high outdoor humidity and warm climates. Consequently, there is a growing demand for innovative dehumidification processes that consume minimal energy. Dielectrophoretic air dehumidification represents one such promising approach. However, it has not garnered significant attention due to the absence of engineering models and simulation tools capable of evaluating its performance and limitations at large-scale airflows. A new numerical multiphase CFD model, which is also experimentally validated, is developed in a customized Reacting Foam solver based on OpenFOAM® version 9. The newly developed model seeks to decrease substantial energy consumption and lower costs by leveraging the dielectrophoretic phenomenon to regulate moisture levels in the air. The solver integrates a hybrid Eulerian-Lagrangian framework to track the droplet's trajectory and growth rate while solving the continuum equations for the moist air. An electrospray produces electrically charged droplets, which grow during their in-flight trajectories as water vapor condenses onto their surfaces. The role of electrostatic forces in promoting vapor condensation within a high-gradient electrical field is investigated, and the dielectrophoretic vapor nucleation process on charged water droplets is discussed. The CFD model was validated against results from the literature and from proof-of-concept experiments conducted by the authors, which showed a 2 % air dehumidification with a single electrospray and airflow rate of 5 cubic feet per minute. The simulation results indicated that augmenting the number of electrically charged spray droplets increased the dehumidification of the air to 25 %. The initial mean droplet diameter, the orientation of the injector and relative humidity significantly influence the assessment of dehumidification. Scaling up this approach to larger airflow volumes is identified as a potential future research direction.

Flow dynamics over cylinder in two‐dimensional benchmark problem: A finite element approximation

This research investigates 2D benchmark flow around a circular cylinder, utilizing the incompressible Navier–Stokes equations alongside the continuity and energy equations. The numerical solution is achieved through finite element discretization for space variable, combined with a second‐order Crank–Nicolson scheme for time integration. The computational results are derived using the FEATFLOW finite element based library package. Our study focuses on the dimensionless form of the flow equations and examines three key dimensionless parameters: drag, lift, and pressure drop. Upon applying finite element method (FEM) discretization, the system is converted into a set of linear or nonlinear ordinary differential equations or algebraic equations for steady‐state scenarios. We then apply the Newton–Raphson method as the outer nonlinear solver, and the multigrid method for efficiently resolving the linear subproblems. To ensure numerical accuracy, we evaluated the and errors, confirming that the experimental order of convergence matches the theoretical predictions. Flow profiles were both graphically represented and tabulated, offering a detailed understanding of the simulation results.

Continuity Equation of Transverse Kähler Metrics on Sasakian Manifolds

We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique η-Einstein metric in the basic Bott–Chern cohomological class of the initial transverse Kähler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the Kähler metrics studied by La Nave and Tian, and also counterparts of the Sasaki–Ricci flow studied by Smoczyk, Wang, and Zhang.

The Continuity Equation is Superior to the Pressure Half-time Method in Calcific Mitral Stenosis: A Comparison with Cardiac Catheterization.

The Continuity Equation is Superior to the Pressure Half-time Method in Calcific Mitral Stenosis: A Comparison with Cardiac Catheterization.

Mean field limit for congestion dynamics

This paper addresses congested transport, which can be described, at macroscopic scales, by a continuity equation with a pressure variable generated from the hard-congestion constraint (maximum value of the density). The main goal of the paper is to show that, in one spatial dimension, this continuum PDE can be derived as the mean-field limit of a system of ordinary differential equations that describes the motion of a large number of particles constrained to stay at some finite distance from each others. To show that these two models describe the same dynamics at different scale, we will rely on both the Eulerian and Lagrangian points of view and use two different approximations for the density and pressure variables in the continuum limit.

Learning fluid physics from highly turbulent data using sparse physics-informed discovery of empirical relations (SPIDER)

We show how a complete mathematical model of a physical process can be learned directly from data via a hybrid approach combining three simple and general ingredients: physical assumptions of smoothness, locality and symmetry, a weak formulation of differential equations and sparse regression. To illustrate this, we extract a complete system of governing equations of fluid dynamics – the Navier–Stokes equation, the continuity equation and the boundary conditions – as well as the pressure-Poisson and energy equations, from numerical data describing a highly turbulent channel flow in three dimensions. Whether they represent known or unknown physics, relations discovered using this approach take the familiar form of partial differential equations, which are easily interpretable and readily provide information about the relative importance of different physical effects. The proposed approach offers insight into the quality of the data, serving as a useful diagnostic tool. It is also remarkably robust, yielding accurate results for very high noise levels, and should thus be well suited for analysis of experimental data.

MATHEMATICAL MODEL AND NUMERICAL METHODS OF FILTRATION PROCESSES OF LIQUID SOLUTIONS

Filtration processes of liquid solutions are fundamental in many natural and industrial applications, such as environmental protection, chemical engineering, water purification, and petroleum extraction. This article develops a mathematical model for describing filtration processes and explores various numerical methods for solving the governing equations. The model is based on Darcy’s law, continuity equation, and constitutive relations of liquid solutions in porous media. Numerical methods, including finite difference, finite element, and finite volume approaches, are discussed with applications to various filtration scenarios. We also provide analysis of the stability, convergence, and efficiency of these methods.

A hybrid SPH model of the Navier-Stokes and Boussinesq equations for simulating wave propagation and deformation

A two-way hybrid model based on the smoothed particle hydrodynamics (SPH) method is proposed in this paper to simulate wave propagation and deformation effectively. The hybrid model is composed of two submodels: the first submodel is based on the Boussinesq equations (B-SPH) for simulating wave propagation, and the second submodel is based on the Navier-Stokes (N-S) equations (NS-SPH) for simulating wave deformation caused by interactions with structures. The exchange of information between the two submodels is realized through an open-boundary method: the results derived from the B-SPH submodel are transferred to the open boundaries of the NS-SPH submodel through a relaxation function; the information derived from the NS-SPH submodel is matched to the open boundaries of the B-SPH submodel through the continuity equation. To obtain a stable pressure field in the NS-SPH submodel, the particle shifting technique (PST) is implemented. The subsequent validation of the hybrid model includes four typical numerical examples related to solitary and regular waves. The agreement between the numerical results and the experimental data indicates that the proposed hybrid model is suitable for simulating the propagation and deformation of shallow water waves. Compared with the traditional NS-SPH model, which uses N-S equations to describe the whole fluid domain, the hybrid model has greater computational efficiency.

Statistical-Physics-Informed Neural Networks (Stat-PINNs): A machine learning strategy for coarse-graining dissipative dynamics

Machine learning, with its remarkable ability for retrieving information and identifying patterns from data, has emerged as a powerful tool for discovering governing equations. It has been increasingly informed by physics, and more recently by thermodynamics, to further uncover the thermodynamic structure underlying the evolution equations, i.e., the thermodynamic potentials driving the system and the operators governing the kinetics. However, despite its great success, the inverse problem of thermodynamic model discovery from macroscopic data is in many cases non-unique, meaning that multiple pairs of potentials and operators can give rise to the same macroscopic dynamics, which significantly hinders the physical interpretability of the learned models. In this work, we consider the problem of deriving the macroscopic (continuum) equations from microscopic (particle) data, and encode knowledge from statistical mechanics to resolve this non-uniqueness for the first time. The proposed machine learning framework, named as Statistical-Physics-Informed Neural Networks (Stat-PINNs), is here developed for purely dissipative isothermal systems. Interestingly, it only uses data from short-time particle simulations to learn the thermodynamic structure, which can in turn be used to predict long-time macroscopic evolutions. We demonstrate the approach for particle systems with Arrhenius-type interactions, common to a wide range of phenomena, such as defect diffusion in solids, surface absorption, and chemical reactions. Our results from Stat-PINNs can successfully recover the known analytic solution for the case with long-range interactions and discover the hitherto unknown potential and operator governing the short-range interaction cases. We compare our results with direct particle simulations and an analogous approach that solely excludes statistical mechanics, and observe that, in addition to recovering the unique thermodynamic structure, statistical mechanics relations can increase the robustness and predictive capability of the learning strategy.

Analysis of multi-frequency ultrasound effect on acoustic microcavitations in [formula omitted]-dimensional fluid

Acoustic microcavitation and associated mechanical effects, such as stress and strain, are significant in decreasing cavitation thresholds and improving cavitation effects at multiple (i.e., single, dual, and triple) frequencies. Ultrasonic excitation and the introduction of microbubbles as cavitation nuclei are both extremely successful techniques. Besides, the enhancing effects brought on by a dual and triple frequency approach, the cavitation dynamics of microbubbles in viscoelastic tissues under the influence of dual and triple frequency excitation are poorly known. This work deals with the modelling and numerical investigation of acoustic cavitation in N-dimensional fluid based on the multifrequency of ultrasound fields. The mathematical model of ultrasonic microcavitation bubble dynamics is formulated by continuity and Navier-Stokes equations in N-dimensions and the equation of pressure due to acoustic multifield. The proposed model is solved and investigated numerically based on the hybrid-B-Spline method. The numerical results illustrate that the behaviour of the microcavitation bubble dynamics increases with a decrease of the value of the N-dimensional fluid. Moreover, the results of the numerical simulation show the importance of the effect of different multifrequency on the behaviour of the microbubble dynamics, as the results obtained showed that in the case of a single frequency, the radius of the microbubbles is higher than in the cases of dual and triple frequency. Additionally, at different initial radii values for cavitation microbubbles, the growth and departure rates for microcavitation dynamics were examined. Finally, in the proposed model, numerical solutions are studied, their stability is verified, and they are compared with previous theoretical works.

Physics-informed neural networks coupled with flamelet/progress variable model for solving combustion physics considering detailed reaction mechanism

In recent years, physics-informed neural networks (PINNs) have shown potential as a method for solving combustion physics. However, current efforts using PINNs for the direct predictions of multi-dimensional flames only use global reaction mechanisms. Considering detailed chemistry is crucial for understanding detailed combustion physics, and how to accurately and efficiently consider detailed mechanisms under the framework of PINNs has not been explored yet and is still an open question. To this end, this paper proposes a PINN/flamelet/progress variable (FPV) approach to accurately and efficiently solve combustion physics, considering detailed chemistry. Specifically, the combustion thermophysical properties are tabulated using several control variables, with the FPV model considering detailed chemistry. Then, PINNs are used to solve the governing equations of continuity, momentum, and control variables with the thermophysical properties extracted from the FPV library. The performance of the proposed PINN/FPV approach is assessed for diffusion flames in a two-dimensional laminar mixing layer by comparing it with the computational fluid dynamics (CFD) results. It has been found that the PINN/FPV model can accurately reproduce the flow and combustion fields, regardless of the presence or absence of observation points. The quantitative statistics demonstrated that the mean relative error was less than 10%, and R2 values were all higher than 0.94. The applicability and stability of this model were further verified on other unseen cases with variable parameters. This study provides an efficient and accurate method to consider detailed reaction mechanisms in solving combustion physics using PINNs.

DualSPHysics+: An enhanced DualSPHysics with improvements in accuracy, energy conservation and resolution of the continuity equation

This paper presents an enhanced version of the well-known SPH (Smoothed Particle Hydrodynamics) open-source code DualSPHysics for the simulation of free-surface fluid flows, leading to the DualSPHysics+ code. The enhancements are made through incorporation of several schemes with respect to stability, accuracy and energy/volume conservation issues in simulating incompressible free-surface fluid flows within the weakly compressible SPH formalism. The Optimized Particle Shifting (OPS) scheme is implemented to improve the accuracy of particle shifting vectors in the free-surface region. To mitigate energy dissipation and maintain consistency, the artificial viscosity in δ-SPH is substituted with a Riemann stabilization term, leading to the δR-SPH. The Velocity divergence Error Mitigating (VEM) and Volume Conservation Shifting (VCS) schemes are adopted in DualSPHysics+ to mitigate the velocity divergence error and improve the volume conservation, and hence to enhance the resolution of the continuity equation. To further reduce both the instantaneous and accumulated errors in velocity divergence, a Hyperbolic/Parabolic Divergence Cleaning (HPDC) scheme is incorporated in addition to the VEM scheme. The implementations of the introduced schemes on both CPU and GPU-based versions of the DualSPHysics+ code along with details on the compilation, running and computational performance are presented. Validations in terms of accuracy, energy conservation and convergence of DualSPHysics+ are shown via several relevant benchmarks. It is demonstrated that a better velocity divergence error cleaning in both instantaneous and accumulated errors can be achieved by the combination of VEM and HPDC. Meanwhile, the excessive energy dissipation by the artificial viscosity is shown to be suppressed by adopting the Riemann stabilization term. Enhanced resolution of the continuity equation along with improved energy conservation of DualSPHysics+ advance the SPH-based simulation of incompressible free-surface fluid flows.

Different built-in electric fields for transmission-mode GaAs photocathodes through doping engineering: Design and modeling

In order to enhance the emission performance of transmission-mode GaAs photocathodes used in optoelectronic field, different exponential doping structures are designed for the GaAs emission layer to generate different types of built-in electric fields. These built-in electric fields include the constant, increasing and decreasing types, which can help photoelectron transport toward the emission surface. By solving the one-dimensional continuity equation using the finite difference method, the electron concentration distribution and quantum efficiency of the three types of exponential doping GaAs photocathodes are derived. Meanwhile, the light absorption distributions in the emission layer are simulated by the finite difference time domain method. By comparing the optical absorption distribution and the electron concentration distribution, it is concluded that, among the three types of exponential doping structures, the exponential doping structure generating the increasing built-in electric field has the most sufficient long-wave light absorption capacity and the strongest photoelectron transport ability, thereby achieving the highest quantum efficiency. This theoretical work can help understand the mechanism of improving the photoemission performance of GaAs photocathodes through doping engineering.

MODELING THE ENTRY OF AIR CONTAMINANTS INTO A ROOM

A mathematical model of air contaminant (products of human activity) inflow into the isolated air space has been developed. On the basis of the formula modified by us the simulation of human respiration with carbon dioxide, water vapor and heat emission is implemented. The model also takes into account the heat input from the human body through clothing. Applying numerical modelling ANSYS CFD (Computational Fluid Dynamics) on the basis of continuity equations and Reynolds-Averaged Navier-Stokes equations "RANS" (Reynolds-Averaged Navier-Stokes) the following results on air medium state change in the isolated space were obtained: - the human respiratory cycle is modelled at simultaneous heat transfer from the body surface through clothes into the studied air space; - the exponential equation of the trend line of concentration to observation time was obtained; - monitoring and rendering (visualization) of changes in concentration, temperature and relative humidity in the space under study by time along the room height was performed. These results and regularities served as initial data for solving a number of model non-stationary problems on aerodynamics and heat and mass transfer in the room. The inverse problem of general exchange ventilation was to be solved. Changes in the state of the air environment initially contaminated with carbon dioxide, heat and water vapors were studied when people were in the studied space and the supply and exhaust ventilation was operating. Of the four air change schemes planned for the study, the results for one schemes are presented in this publication. The dynamics of assimilation of excess heat, humidity and carbon dioxide made it possible to assess the efficiency of ventilation systems and to predict improvements in their energy efficiency when air parameters are brought up to standard values.

Research on hydrodynamics of the thermal agent flow for the beet pulp filtration drying

The reuse and recycling of secondary raw materials is essential to ensure sustainable development and the rational use of natural resources. One of the most important types of secondary raw materials is plant-based food waste, which is generated in large quantities at food processing plants, including sugar beet pulp, which is a valuable resource for further use. One of the main problems with using beet pulp is its high moisture content, which is ≈ 80 % wt. It is proposed to dry beet pulp by filtration drying, which is highly efficient. The calculation of industrial equipment for filtration drying should take into account the costs of overcoming the hydraulic resistance of a stationary layer of material. The aim of the article was to study the hydrodynamics of the flow of a thermal agent through a stationary layer of beet pulp of different heights and to perform computer modelling of this process. The obtained results will allow predicting the hydraulic resistance of a material layer of different heights and the required pressure drop to ensure the course of the drying process under various possible technological modes. For computer modelling of the flow of a thermal agent through a stationary layer of beet pulp, the porous media method was used in the ANSYS Fluent 2022 R2 software package. The process was simulated on the basis of the Navier-Stokes system of differential equations and the flow continuity equation with the additional use of the Darcy equation to determine the value of the hydraulic resistance of the layer ΔP. Computer simulations of the hydrodynamics of the thermal agent flow through a stationary layer of beet pulp were performed for the range of stationary layer heights H = 90÷110 mm with an interval of 5 mm. The obtained data showed a relative average deviation of 2.19 % from the experimental data, which indicates a high accuracy of calculations in this range. The study for the extended range of layer heights H = 80÷120 mm with an interval of 10 mm showed an increase in the relative average deviation to 4.09 % from the experimental data, but the results are still sufficiently accurate for practical use in the calculation of drying equipment. The obtained dependencies may be used to determine the value of the hydraulic resistance of a stationary layer of material using the filtration drying method and for practical calculations of drying equipment. Together with the results of the kinetic regularities of filtration drying, the obtained experimental data and computer modelling data on the hydrodynamics of the flow of a heat transfer agent through a material layer will allow us to calculate the optimal parameters of filtration drying for beet pulp and to assess the efficiency of filtration drying for drying beet pulp in comparison with other methods of dehydration.

Thermal-fluid behavior analysis of skin cancer photothermal therapy with variation in intratumoral single blood vessel diameter and velocity

Recently, usage of nanoparticles, nanometer sized materials, in several therapeutic techniques including photothermal therapy, which uses elevated temperatures to kill tumor tissue, is on the rise. In this study, implementation of photothermal therapy for blood vessels of various diameters, present in the skin layer of squamous cell carcinoma, is simulated using numerical analysis. The optical properties of the nanoparticles were calculated using the discrete dipole approximation method, and the optical properties of the medium were calculated using the optical coefficients calculation method of a composite medium. The calculated optical properties revealed the temperature distribution in the medium using the energy equation, and the flow rate of blood through various vessel diameters using the continuity and momentum equations. Finally, the calculated temperature distribution was used to quantitatively confirm the therapeutic effect of different photothermal treatment conditions through the apoptotic variable. This is expected to serve as a standard for selecting optimal treatment conditions when performing actual photothermal therapy in the future.

Numerical Simulation and Neural Network Model for Hydromagnetic Nanofluid Convection in a Porous Wavy Channel with Thermal Non-Equilibrium Model

Abstract The present study aims to analyze the nonfluid MHD convective heat transfer in a porous wavy channel with a local thermal non-equilibrium (LTNE) model. Such a model finds applications related to enhancement in thermal performance, increasing the heat transfer coefficient in the compact design of heat exchangers for the aerospace and automotive industries and elevation in the efficiency of the solar collector. A sinusoidal porous wavy LTNE channel containing nanofluid and subjected to the induced and applied magnetic fields is considered. A uniform magnetic field is applied orthogonal to the channel and the induced magnetic field effects are considered due to the large magnetic Reynolds number. The momentum, continuity, energy, and nanoparticle volume fraction equations constitute the coupled nonlinear system of differential equations and are solved using the Galerkin finite element method. The reliability of the technique is assessed by comparing the proposed procedure with the results from earlier sources. A detailed analysis is presented to determine the effects of different physical parameters arising in the system on temperature, nanoparticle concentration, and velocity profiles. As an illustration, the findings exhibit that increasing the modified diffusivity ratio increases the values of the nanoparticle volume fraction whereas, reducing the modified diffusivity ratio enhances the temperature distribution. A higher value of thermal Rayleigh number presents a significant involvement of buoyancy forces, potentially resulting in the development of convective currents. A higher Nield number indicates more effective heat transport from the solid surface to the nanofluid. Consequently, there is a minimal thermal difference between the solid surface and the bulk nanofluid. Effective heat transmission enhances the nanofluid ability to absorb heat and generates a more consistent dispersion of temperature inside the fluid. The performance of the designed algorithms of the artificial neural network, namely, the Levenberg – Marquardt algorithm, in the problem under consideration is evaluated and the methodology is found reasonably precise with the matching of order around 6 to 7 decimal places of accuracy.