Original Numerical Method Research Articles

A theoretical study on Krauklis wave characteristics

Abstract The Krauklis wave is a special seismic phenomenon in fluid saturated fracture medium. The wave can prompt a unique resonance effect and enhance the amplitude at special frequencies. These frequencies have a quantitative relationship with the fracture geometry parameters and can be used for quantitative interpretation of geometry parameters. Such frequency information can be transmitted to body waves by the transformation between the Krauklis wave and the body wave. Both P- and S-waves become frequency dependent. In this study, an original numerical method is brought out to solve the equation of the Krauklis wave dispersion relation. The method has fine computational performance, and the frequency band for numerical solution is extended to the megahertz level. The dispersion, resonance, and attenuation of the Krauklis wave can be analyzed within the entire frequency range for Krauklis wave existence. What is more, the formation mechanism, existence, and observability are illuminated. The analysis shows that there are upper limits of frequency and fracture aperture for Krauklis wave existence, but within the frequency band for artificial seismic and micro-seismic exploration, the Krauklis wave exists widely. For experimental research, the frequency and fracture aperture should be well designed to ensure the generation of the Krauklis wave. The attenuation of the Krauklis wave can suppress the resonance effect. The influence of the attenuation should be taken into account, when the wave is used for seismic characterization of fracture reservoirs or micro-seismic monitoring of hydraulic fracturing.

Towards accelerating particle‐resolved direct numerical simulation with neural operators

AbstractWe present our ongoing work aimed at accelerating a particle‐resolved direct numerical simulation model designed to study aerosol–cloud–turbulence interactions. The dynamical model consists of two main components—a set of fluid dynamics equations for air velocity, temperature, and humidity, coupled with a set of equations for particle (i.e., cloud droplet) tracing. Rather than attempting to replace the original numerical solution method in its entirety with a machine learning (ML) method, we consider developing a hybrid approach. We exploit the potential of neural operator learning to yield fast and accurate surrogate models and, in this study, develop such surrogates for the velocity and vorticity fields. We discuss results from numerical experiments designed to assess the performance of ML architectures under consideration as well as their suitability for capturing the behavior of relevant dynamical systems.

A unified two-scale gas–liquid multi-fluid model with capillarity and interface regularization through a mass transfer between scales

In this contribution, we derive a gas–liquid two-scale multi-fluid model with capillarity effects to enable a novel interface regularization approach for multi-fluid models. As this unified modelling is capable of switching from the interface representation of a separated to a disperse regime it lays a new way of modelling regime transitions as it occurs in atomization processes. Above a preset length threshold at large scale, a multi-fluid diffuse interface model resolves the dynamics of the interface while, at small-scale, a set of geometric variables is used to characterize the interface geometry. These variables result from a reduced-order modelling of the small-scale kinetic equation that describes a collection of liquid inclusions. The flow model can be viewed as a two-phase two-scale mixture, and the equations of motion are obtained thanks to the Hamilton’s Stationary Action Principle, which requires to specify the kinetic and potential energies at play. We particularly focus on modelling the effects of capillarity on the mixture’s energy by including dependencies on additional variables accounting for the interface’s geometry at both scales. The regularization of the large-scale interface is then introduced as a local and dissipative process. The local curvature is limited via a relaxation towards a modified Laplace equilibrium such that an inter-scale mass transfer is triggered when the mean curvature is too high. We propose an original numerical method and assess the properties and potential of the modelling strategy on the relevant test-case of a two-dimensional liquid column in a compressible gas flow.

Flow development region in the boundary layer: two-component molecular viscosity and partial slip

The subject of this study is the flow development region of laminar incompressible fluid flow in the boundary layer. This flow is an example where a direct application of the Navier-Stokes equations of gradient-free laminar incompressible fluid flow, in which the molecular viscosity is assumed to be a constant value independent of spatial coordinates, leads to a redefinition of the mathematical model. It is about the fluid boundary layer in the region of flow establishment in the motion problem of a semi-infinite plane, where the pressure gradient is zero. There is a situation when the number of equations is equal to three (two equations of momentum conservation and the equation of continuity), and the number of unknowns is equal to two - the number of the speed component. As a logical solution to the obtained inconsistency, it is proposed, as was already done for the problem of stationary motion of a plane and the problem of acceleration of a plane, to depart from the false statement about the constancy of molecular viscosity in the gradient-free boundary layer of an incompressible flow and consider molecular viscosity as a function of spatial coordinates. The need to consider the variable nature of molecular viscosity led to the discovery of another flaw in the Navier-Stokes theory. This non-trivial flaw was discovered during the application of the original numerical analytical method for solving the flow development region problem. The Navier-Stokes equations are supplemented by boundary conditions. The most important condition is the condition of fluid non-slipping on the surface of a solid body, which, by the way, does not follow any physical law. As a result, on the surface of a half-plane (or a moving body), the component of the velocity, which coincides with the direction of motion, has a constant value equal to the velocity of the body. It immediately follows from the continuity equation that the normal derivative of the normal component of the velocity must be equal to zero along the surface of the plane (body), since the longitudinal derivative of the velocity becomes zero. However, it is quite obvious that the velocity component normal to the surface of the plane (body) changes across the boundary layer in the region of current development, which indicates the presence of a normal gradient (both components) of the velocity. The conflict or contradiction is overcome by moving away from the generally accepted condition of non-slipping to the condition of partial non-slipping, or essentially the presence of sliding. Even with the sudden braking of any vehicle, the complete stop does not occur instantly, but after some finite time and distance, so in the case of the motion of a body in a stationary fluid, there is not an instant sticking, but a gradual one - from complete sliding, when a particle of liquid has just met a moving plane (body), to complete non-slipping at the end (and further) of the flow development region. Research methods. This work uses purely theoretical methods based on the use of calculus of variations, laws of physics, and ideas from everyday life. Conclusions. An improved model of a viscous Newtonian fluid in the area of flow development in the boundary layer was derived. On the basis of assumptions about the variable nature of the molecular viscosity, which already has two components, and the departure from the non-slipping condition, analytical solutions for both components of the velocity and both components of the molecular viscosity were obtained. A comparison of the obtained results with the results of other studies is presented.

Parametric Optimization of Thin-Walled 3D Beams with Perforation Based on Homogenization and Soft Computing.

The production of thin-walled beams with various cross-sections is increasingly automated and digitized. This allows producing complicated cross-section shapes with a very high precision. Thus, a new opportunity has appeared to optimize these types of products. The optimized parameters are not only the lengths of the individual sections of the cross section, but also the bending angles and openings along the beam length. The simultaneous maximization of the compressive, bending and shear stiffness as well as the minimization of the production cost or the weight of the element makes the problem a multi-criteria issue. The paper proposes a complete procedure for optimizing various open sections of thin-walled beam with different openings along its length. The procedure is based on the developed algorithms for traditional and soft computing optimization as well as the original numerical homogenization method. Although the work uses the finite element method (FEM), no computational stress analyses are required, i.e., solving the system of equations, except for building a full stiffness matrix of the optimized element. The shell-to-beam homogenization procedure used is based on equivalence strain energy between the full 3D representative volume element (RVE) and its beam representation. The proposed procedure allows for quick optimization of any open sections of thin-walled beams in a few simple steps. The procedure can be easily implemented in any development environment, for instance in MATLAB, as it was done in this paper.

Experimental investigation and modeling of thermal oxidation deposition of RP-3 jet fuel under high Reynolds number

The modeling and numerical prediction of thermal oxidation deposition process under high heat flux and high Reynolds number are challenging. In this paper, experimental investigations on RP-3 jet fuel were carried out under wide range of wall heat flux (0.095–3.6 MW/m2), residence time (1.95–0.06 s), and inlet Reynolds number (522–5220). As turbulence intensified, the enhanced convective mass transfer transferred more precursors into the laminar sublayer to promote the formation of deposits, while the decreased residence time, the decreased laminar sublayer thickness, and the increased wall shear force inhibited the deposition process. These factors together led to the variation that the average deposition rate per unit fuel first decreased, then increased and decreased again in the later stage. Since current numerical methods ignored the effect of turbulence diffusion on mass transfer and the kinetic parameters of global reactions depicted the formation of SMORS and insoluble precursors were based on experimental data under low Reynolds number, it could not precisely simulate the promotion of insoluble precursors formation by enhanced convective mass transfer. Based on the experimental data, two functions were proposed to modify the convective mass transfer process of original numerical method. The error between the experimental data and the original simulated results could reach more than 500%, while the error decreased to less than 15% with the application of modified functions.

Topological reconstruction of a stretched transparent surface relief grating via an optical diffraction pattern.

The optical characterization of transparent and stretchable patterned surfaces replicated from the fabrication of quasicrystal structures on azopolymer thin films is presented. The complexity of the quasicrystal surface fabrication is obtained by superimposed multiple light exposures. Azopolymer surface patterns are used as a replica molding master. The microscopic elongation of nanocavities induced by macroscopic stretchings of the elastomeric quasicrystal replication is characterized via optical diffraction. An original numerical method is presented to reconstruct the structured surface deduced from the optical diffraction measurements. The measurements show that drastic topologic changes, e.g., going from cavities to a canal, happens on the surface. This could be ingeniously used for creating actionable structured surfaces or nanoparticles trapping surfaces.

An original numerical calculation method for impacting contact model of spur gear under lightly loaded condition

In this study, an impacting contact model of spur gear under lightly loaded condition is constructed by considering the pinion and the wheel as rigid body, in which the lubricant is modelled by a linear spring damper combination. The entire motion of the gear system is fell into three phases, i.e.,: meshing state, transient state and non-meshing state. An original numerical calculation method is proposed for this model, and its validation is confirmed by using the Co-simulation of Adams-Matlab/Simulink. Instead of using the Jacobian matrix, the proposed numerical method could effectively avoid the ill condition in the iteration on data resulting from the nonlinearity of gear backlash. The results in this study verified that the algorithm is useful not only can increase the accuracy, but also significantly decrease the calculation time of the entire system. And the original numerical calculation method is employed to illustrate and quantify the influences of the excitation frequency on the impacting contact model of spur gears.

Propagation of Flood Wave in River and Construction Riverbanks – Example of River Velika Morava

In hydraulics area there is an open space for apply of numerical methods, instead of frequently used empirical formulas. At first place numerical methods may be used in river flow hydraulics and ground water flow. Application of numerical methods is given enough correct results and prediction for all parameters in river flow, that is presented in this paper. Natural phenomenon of flood wave is researched and original numerical method is shown, having calculation of water depth as primary goal. There are different amount of flooding in river in correspondence with event frequency which are monitored through numerical calculation and measuring. Numerical calculation of water depth in huge amount of flooding is important for defined defense areas. In this paper is researched flood wave in river Velika Morava, propagated from city Naissus to measuring station Bagrdan, continuing to city Smederevo where is merged with Danube. Flooding is unsteady flow with dominant part of kinetic energy. Propagation of flood wave is analyzed as plane problem and through numerical procedure are calculated values of water depth. Calculation results of propagated flood wave is treated with measured data water depth H and discharge Q. Presented calculated and measured data are compared and they have small differences, therefor results are acceptable as correct. Whole volume of flooded water in river is analyzed for calculation results correction and average cross section of flow is got.

The Non-Smooth Problem of Wheel Motion

Planar motion of a non-deformable wheel under the action of non-ideal unilateral constraints is considered. The mathematical description of this phenomenon has a form of a non-smooth initial value problem. The non-smoothness of this problem means that its solution is determined by an absolutely continuous function having a discontinuous first derivative. For this reason, a collision problem describing abrupt changes of velocity has been formulated next to the equations of motion specifying the acceleration. The non-idealness of constraints means that the constraint reaction force includes also a component resulting from the friction between the wheel and the constraints. Differential equations specifying acceleration of the wheel making contact with the constraints and algebraic equations for determining the changes in the wheel’s velocity at the moment of collision have been formulated in the paper. The principal task in these formulations is to determine the reaction forces of the considered constraints. This task is specified by the relationships between acceleration and the constraint reaction force components. In the description of the collision, these relations refer to the post-collision velocities and reaction force impulses. For determining an approximate solution of the formulated wheel motion problem, an original numerical method and a computer program for wheel motion simulation have been developed. Selected results illustrating the changes in displacements and velocity have been presented.

A new method to integrate newtonian N-body dynamics

In this note we approach the classical, Newtonian, gravitational N-body problem by mean of a new, original numerical integration method. After a short summary of the fundamental characteristics of the problem, including a sketch of some of its mathematical and numerical issues, we present the new algorithm, which is applied to a set of sample cases of initial conditions in the ‘intermediate’ N regime (N = 100). This choice of N is not due to algorithm limitation but just for computational convenience, in what this preliminary work aims mainly to the presentation of the new method and so we wanted just to provide an acceptable although statistical significant comparison with other integration schemes. The proposed algorithm seems to be fast and precise at the same time and so promising for further, more realistic, tests and scientific applications.

Stability analysis of underground space with discontinuous planes using a four-node iso-parametric element numerical manifold method with rigid body rotation

In a large-scale rock structure with discontinuous planes in a rock mass, the pattern of the discontinuous planes strongly influences the deformation and stress distribution behaviors along the structure. A jointed rock mass, however, is generally assumed to be partially and imperfectly connected rather than completely discontinuous. Earlier the authors presented an elastic-plastic analysis of a discontinuous model of a jointed rock mass using a four-node iso-parametric manifold method (Sasaki and Ohnishi, 2001). The authors adapted Drucker-Prager’s elastic-plastic yield criterion, the Jaumann co-rotational derivative to determine the objectivity of stresses for rotation, and a rock-bolting element.In discontinuous models such as the original numerical manifold method (NMM), however, rigid rotation is not introduced in the degrees of freedom in the manner seen in the conventional finite element method (FEM). This paper describes the co-rotational formulation theory (Crisfield and Moita, 1996) introduced for a four-node iso-parametric NMM, along with examples of a model of rockfall activity in underground space. The results compare very well with the values obtained without a rigid body rotation analysis, especially those obtained for block shapes with large rotations.

Numerical Model of Time-Dependent Gas Flows Through Bed of Granular Phase Change Material

A novel mathematical model and original numerical method for investigating time-dependent gas flows through a bed of granular phase change material (PCM) are proposed and described in detail. Such material is modeled as a porous medium, and continua mechanics method are used for constructing the mathematical model. The numerical method is based on a combination of explicit and implicit finite-difference schemes. Comparison of calculation results with known experimental data demonstrates a very good coincidence. The results of the study can be applied in modeling the thermal energy storage with granular PCM in advanced adiabatic compressed air energy storage and other heat storage devices.

Теоретический подход и численный метод поиска квазиоптимального решения нелинейной дискретной задачи большой размерности

At present, the class of NP-complete problems is represented not only by the «classical» issue of the travelling salesman problem and many tasks in the theory of graphs and networks, resolved into it, but also by the problems of nonlinear and stochastic optimization, including, in the discrete setting. The peculiarity of these problems is the lack of constructive algorithms forfinding the optimal solution in polynomial time from the dimension of the problem, which «dooms» the researcher to use non-constructive methods in solving these problems, for example, the exhaustive algorithm. However, these algorithms, being applied to discrete optimization problems, do not allow to solve the problem in a complex way: for example, such methods do not let us obtain dual constraints estimation that are important for the subsequent analysis of the optimal solution. In the article, for a wide class of problems of production and financial planning, which, in the production plan, are reduced to the problems of discrete nonlinear convex optimization of large dimension, the author offers an original numerical method for finding a quasi-optimal solution with a high (predetermined) accuracy of approximation to the optimum. For the specified class of problems, the proposed method is universal, i.e. it can be applied without additional adaptation of the numerical procedure.

Long range numerical simulation of acoustical shock waves in a 3D moving heterogeneous and absorbing medium

Acoustical shock waves can be generated by numerous atmospheric sources, either natural – like thunder and volcanoes – or anthropic – like explosions, sonic boom or buzz saw noise. The prediction of their long-range propagation remains a numerical challenge at 3D because of the large propagation distance to wavelength ratio, and of the high frequency / small wavelength content associated to shocks. In this paper, an original numerical method for propagating acoustical shock waves in three-dimensional heterogeneous media is proposed. Heterogeneities can result from temperature or density gradients and also from atmospheric shear and turbulent flows. The method called FLHOWARD (for FLow and Heterogeneities in a One-Way Approximation of the nonlineaR wave equation in 3D) is based on a one-way solution of a generalized nonlinear wave equation. Even though backscattering is neglected, it does not suffer from the limitations of classical ray theory nor from the angular limitations of the popular parabolic methods. The numerical approach is based on a split-step method, which has the advantage of splitting the original equation into simpler ones associated with specific physical mechanisms: diffraction, flows, heterogeneities, nonlinearities, absorption and relaxation. The method has been developed on parallel architecture for very high demanding 3D configurations using the Single Method Multiple Data paradigm. The method is validated through several test cases. A study of the lateral cut-off of the sonic boom finally illustrates the potentialities of the method for realistic cases.

Numerical model of two-dimensional heterogeneous combustion in porous media under natural convection or forced filtration

A novel mathematical model and original numerical method for investigating the two-dimensional waves of heterogeneous combustion in porous media are proposed and described in detail. The mathematical model is constructed within the framework of the model of interacting interpenetrating continua and includes equations of state, continuity, momentum conservation and energy for solid and gas phases. Combustion, considered in the paper, is due to the exothermic reaction between fuel in the porous solid medium and oxidiser contained in the gas flowing through the porous object. The original numerical method is based on a combination of explicit and implicit finite-difference schemes. A distinctive feature of the proposed model is that the gas velocity at the open boundaries (inlet and outlet) of the porous object is unknown and has to be found from the solution of the problem, i.e. the flow rate of the gas regulates itself. This approach allows processes to be modelled not only under forced filtration, but also under free convection, when there is no forced gas input in porous objects, which is typical for many natural or anthropogenic disasters (burning of peatlands, coal dumps, landfills, grain elevators). Some two-dimensional time-dependent problems of heterogeneous combustion in porous objects have been solved using the proposed numerical method. It is shown that two-dimensional waves of heterogeneous combustion in porous media can propagate in two modes with different characteristics, as in the case of one-dimensional combustion, but the combustion front can move in a complex manner, and gas dynamics within the porous objects can be complicated. When natural convection takes place, self-sustaining combustion waves can go through the all parts of the object regardless of where an ignition zone was located, so the all combustible material in each part of the object is burned out, in contrast to forced filtration.

On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients

We investigate in a 2D setting the scattering of time-harmonic electromagnetic waves by a plasmonic device, represented as a non-dissipative bounded and penetrable obstacle with a negative permittivity. Using the T-coercivity approach, we first prove that the problem is well-posed in the classical framework Hloc1 if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we propose an original numerical method based on the use of Perfectly Matched Layers at the corners. We emphasize that it is necessary to design an ad hoc technique because the field is too singular to be captured with standard finite element methods.

Numerical modeling of 1D heterogeneous combustion in porous media under free convection taking into account dependence of permeability on porosity

Using numerical experiment the one-dimensional unsteady process of heterogeneous combustion in porous object under free convection is considered when the dependence of permeability on porosity is taken into account. The combustion is due to exothermic reaction between the fuel in the solid porous medium and oxidizer contained in the gas flowing through the porous object. In the present work the process is considered under natural convection, i.e. when the flow rate and velocity of the gas at the inlet to the porous objects are unknown, but the gas pressure at object boundaries is known. The influence of changing of permeability due to the changing of porosity on the solution is investigated using original numerical method, which is based on a combination of explicit and implicit finite-difference schemes. It was shown that taking into account the dependence of permeability on porosity, which is described by some known equations, can significantly change the solution in one-dimensional case. The changing of permeability due to the changing of porosity leads to the speed increasing of both cocurrent and the countercurrent combustion waves, and to the temperature increasing in the combustion zone of countercurrent combustion wave.

New Rock Bolt Model and Numerical Implementation in Numerical Manifold Method

A beam-element-based rock bolt model is proposed in this paper. The proposed model can not only simulate the compression and tension deformations of bolts in the rock block but also capture the interaction between the bolt and the joint. The antitension, antishear, and antirotation of bolts near the joint and the hardening characteristics of bolts after the plastic yielding can be comprehensively reflected using the proposed model. The rock bolt model has been incorporated into the original numerical manifold method (NMM) procedure developed for realistic applications. In the improved NMM procedure, the bolt can be input as the physical mesh, the layout of which is not correlated with the mathematical cover. Hence, the preprocess is simplified, and numerical simulation of a large number of bolts can be easily carried out. The results of this improved procedure for bolt modeling were verified with a direct shear test of an anchored joint. Finally, the procedure was applied to evaluate the reinforcement effect of bolts in the underground powerhouse of the Shuibuya hydropower station. The results show that the deformation behavior of the anchored rock mass and the reinforcement effect of bolts can be well captured by the bolt model.

Dynamic Model Adaptation for Multiscale Simulation of Hyperbolic Systems with Relaxation

In numerous industrial CFD applications, it is usual to use two (or more) different codes to solve a physical phenomenon: where the flow is a priori assumed to have a simple behavior, a code based on a coarse model is applied, while a code based on a fine model is used elsewhere. This leads to a complex coupling problem with fixed interfaces. The aim of the present work is to provide a numerical indicator to optimize to position of these coupling interfaces. In other words, thanks to this numerical indicator, one could verify if the use of the coarser model and of the resulting coupling does not introduce spurious effects. In order to validate this indicator, we use it in a dynamical multiscale method with moving coupling interfaces. The principle of this method is to use as much as possible a coarse model instead of the fine model in the computational domain, in order to obtain an accuracy which is comparable with the one provided by the fine model. We focus here on general hyperbolic systems with stiff relaxation source terms together with the corresponding hyperbolic equilibrium systems. Using a numerical Chapman---Enskog expansion and the distance to the equilibrium manifold, we construct the numerical indicator. Based on several works on the coupling of different hyperbolic models, an original numerical method of dynamic model adaptation is proposed. We prove that this multiscale method preserves invariant domains and that the entropy of the numerical solution decreases with respect to time. The reliability of the adaptation procedure is assessed on various 1D and 2D test cases coming from two-phase flow modeling.