Finite Difference Numerical Method Research Articles

Seal dynamics of supercritical carbon dioxide turbine with the uncertain rotor nonlinear motion

The actual rotor motion and seal dynamics are nonlinear and uncertain. The uncertain effects include the seal flow and the rotor system parameters. While the influence of system parameters makes it more difficult to accurately predict the seal dynamics. A rotor nonlinear whirling model was established to achieve the synchronous solution of rotor motion and seal simulation. The coupling effect between rotor system parameters and seal aerodynamic performance was adopted. The actual seal dynamics for SCO2 turbine were obtained and evaluated by the numerical finite difference method. In the rotor nonlinear whirling model, the rotor has the uncertain continuous free motion. The influence of the system main parameters, such as elastic recovery stiffness and unbalanced mass force, on seal dynamics is revealed. And the dynamic coefficients beyond the artificial specific disturbance frequency are shown. The results show that the rotor nonlinear whirling can better predict the seal dynamics. The rotation speed, elastic recovery stiffness and unbalanced mass eccentricity have different significant effects on seal dynamics.

Numerical Analysis of Low-Enthalpy Deep Geothermal Energy Extraction Using a Novel Gravity Heat Pipe Design

Geothermal energy, derived from the Earth’s internal heat, can be harnessed due to the geothermal gradient between the Earth’s interior and its surface. This heat, sustained by radiogenic decay, varies across regions, and is highest near volcanic areas. In 2020, 108 countries utilised geothermal energy, with an installed capacity of 15,950 MWe for electricity and 107,727 MWt for direct use in 2019. Low-enthalpy sources require binary systems for power production. Open-loop systems face issues like scaling, difficult water treatment, and potential seismicity, while closed-loop systems, using abandoned petroleum or gas wells, reduce costs and environmental impacts greatly. The novel geothermal gravity heat pipe (GGHP) design eliminates parasitic power consumption by using hydrostatic pressure for fluid circulation. Implemented in an abandoned well in north-east (NE) Slovenia, the GGHP uses a numerical finite difference method to model heat flow. The system vaporises the working fluid in the borehole, condenses it at the surface, and uses gravitational flow for circulation, maintaining efficient heat extraction. The model predicts that continuous maximum capacity extraction depletes usable heat rapidly. Future work will explore sustainable heat extraction and potential discontinuous operation for improved efficiency.

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

Highly confined epsilon-near-zero and surface phonon polaritons in SrTiO3 membranes

Recent theoretical studies have suggested that transition metal perovskite oxide membranes can enable surface phonon polaritons in the infrared range with low loss and much stronger subwavelength confinement than bulk crystals. Such modes, however, have not been experimentally observed so far. Here, using a combination of far-field Fourier-transform infrared (FTIR) spectroscopy and near-field synchrotron infrared nanospectroscopy (SINS) imaging, we study the phonon polaritons in a 100 nm thick freestanding crystalline membrane of SrTiO3 transferred on metallic and dielectric substrates. We observe a symmetric-antisymmetric mode splitting giving rise to epsilon-near-zero and Berreman modes as well as highly confined (by a factor of 10) propagating phonon polaritons, both of which result from the deep-subwavelength thickness of the membranes. Theoretical modeling based on the analytical finite-dipole model and numerical finite-difference methods fully corroborate the experimental results. Our work reveals the potential of oxide membranes as a promising platform for infrared photonics and polaritonics.

Topological nano-switches in higher-order topological insulators

We consider multi-terminal transport through a flake of rectangular shape of a two-dimensional topological insulator in the presence of an in-plane magnetic field. This system has been shown to be a second-order topological insulator, thus exhibiting corner states at its boundaries. The position of the corner states and their decay length can be controlled by the direction of the magnetic field. In the leads we assume that the magnetic field is absent and therefore we have helical one-dimensional propagating states characteristic of the spin-Hall effect. Using a low-energy effective Hamiltonian we show analytically that, in a two-terminal setup, transport can be turned on and off by a rotation of the in-plane magnetic field. Similarly, in a three terminal configuration, the in-plane magnetic field can be used to turn on and off the transmission between neighbouring contacts, thus realising a directional switch. Analytical calculations are supplemented by a numerical finite-difference method. For small values of the Fermi energy and field strength, the analytical results agree exceptionally well with the numerics. The effect of disorder is also addressed in the numerical approach. We find that the switching functionality is remarkably robust to the presence of strong disorder stemming from the topological nature of the states contributing to the electron transport.

Pulsatile flow of blood and Cattaneo Christov heat flux model in diseased arteries with variable properties

ABSTRACT The current effort is executed to interpret the heat transmission of combined stenosis and aneurysm in an artery, taking into account variable properties such as viscous dissipation and thermal radiation. We reduced the two directional energy and momentum equations to unidirectional flow by normalizing them under the mild stenotic assumption. After that, a mathematical framework for the physical problem is developed. An intricate nonlinear set of partial differential equations results from this formulation. A suitable numerical method of finite difference is employed on consequent equations to solve them. Afterward, graphical results predict the region of flow and the influence of involved parameters is shown in pictorial form. Furthermore, these solutions are employed to test how certain parameters, like streamlines, flow rate, and wall shear stress, affect blood flow. Using streamlines, we will also observe the general behavior of blood flow patterns.

Generation Of Scaled Long-Period Ship Waves In A Pump-Driven Flume

Experimental investigations on the generation of long-period, primary ship waves using a pump-driven technique in a closed-circuit flume are presented. Long-period ship waves can have a major impact on river banks and engineering structures. As larger ship dimensions accompanied by a corresponding rise in loads are expected in the future, the characterisation of long-period ship waves and understanding their impact on bank protection and engineering structures is crucial, especially for appropriate design. Our study combines a closed-loop PID controlled wave generation with a numerical finite difference method (FDM) model. This setup offers the advantage of generating repeatable waves at arbitrary positions along the test section of the laboratory flume. The experiments conducted demonstrate a good reproduction of a scaled long-period ship wave signal (1:10) in a distance of approximately 9 m from the flume inlet. The combination of the PID controlled pump-driven technique with a numerical model provides a solid basis for further studies on ship-induced loads, with the aim to predict future wave loads, better understand the wave-structure-interaction and to mitigate potential damage.

Numerical-analytical approach to solving problems of non-stationary thermal conductivity of a non-thin annular plate

This paper considers the first stage of calculating the initial boundary value problem of non-stationary thermal conductivity of cylindrical bodies using a modified method of lines, namely dimension reduction of the original differential equations, initial and boundary conditions. The original equations of thermal conductivity are defined in a cylindrical coordinate system in a spatial setting. An object is a cylindrical body with commensurate dimensions. This area of research is relevant, because when calculating the load bearing elements of structures to thermal effects, the first step is to determine the distribution of temperature fields. Boundary conditions are considered as conditions of convective heat transfer, which by means of boundary transition are transformed into boundary conditions of the first and second types. Dimension reduction with respect to spatial coordinate is performed by the Bubnov-Galorkin-Petrov projection method using local basis functions. These functions are called "cover" functions, which are related to the lines drawn on the domain of the task. Normalized trigonometric series are used to reduce the dimensionality of the equations with respect to circular coordinate. All transformations are performed in index form. In addition to differential equations, the projection method reduces the dimensionality of the initial and boundary conditions. In this paper, the most optimal form of writing reduced equations is determined, which provides the ease of reducing the dimensionality of the original differential equations. Initial and boundary conditions take into account the impact of the environment. All this makes it possible to set a reduced initial boundary value problem for further calculation by numerical finite difference methods.

Bifurcation and asymptotics of cubically nonlinear transverse magnetic surface plasmon polaritons

Linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface of a metal and a dielectric. Metals are typically dispersive, i.e. the dielectric function depends on the frequency. We prove the bifurcation of localized SPPs in dispersive media in the presence of a cubic nonlinearity and provide an asymptotic expansion of the solution and the frequency. The problem is reduced to a system of nonlinear differential equations in one spatial dimension by assuming a plane wave dependence in the direction tangential to the (flat) interfaces. The number of interfaces is arbitrary and the nonlinear system is solved in a subspace of functions with the H1-Sobolev regularity in each material layer. The corresponding linear system is an operator pencil in the frequency parameter due to the material dispersion. The studied bifurcation occurs at a simple isolated eigenvalue of the pencil. For geometries consisting of two or three homogeneous layers we provide explicit conditions on the existence of eigenvalues and on their simpleness and isolatedness. Real frequencies are shown to exist in the nonlinear setting in the case of PT-symmetric materials. We also apply a finite difference numerical method to the nonlinear system and compute bifurcating curves.

Computational modeling of microalgal biofilm growth in heterogeneous rotating algal biofilm reactors (RABRs) for wastewater treatment

Rotating algal biofilm reactors (RABRs) are innovative systems designed to cultivate microalgae biofilms efficiently. In this paper, we have developed a novel mathematical model to accurately capture the growth dynamics of algae biofilms within RABR. By considering the spatial heterogeneity of the RABR, we introduce a PDE-based model that addresses the spatial variations across the substratum, enabling a more accurate simulation of biofilm growth in RABRs. The photosynthesis process is modeled through reactive kinetics, driving the growth of the algae biofilm. To analyze the system's behavior, we employ finite difference numerical methods to solve the complex PDE model. We then conduct extensive numerical simulations to understand algae biofilm growth in the RABR environment under various operational factors and environmental conditions. One primary focus in these simulations is to investigate the impact of various harvesting strategies, harvesting frequencies, light intensity, and light exposure on the overall biomass productivity of the algae biofilm. The numerical results provide valuable insights into optimizing algae biofilm growth and designing harvesting techniques in RABR systems. Our proposed novel mathematical model provides an effective platform for the theoretical investigation and design of RABRs for wastewater treatment.

Fracture prediction and application in carbonate reservoirs based on the rescaled range analysis- finite difference numerical analysis methods

For carbonate reservoirs, the study of fractures is crucial. In this paper, the identification and prediction of fractures in carbonate reservoirs are carried out by applying R/S-FD numerical analysis method by using core, thin section and imaging logging data. The analysis concludes that the AC, CAL, DEN, and GR curves are very sensitive to the response of fracture development segments as calculated by the rescaled range analysis (R/S) analysis method, but some fractures are still misidentified. After finite difference (FD) calculation on the basis of R/S calculation, high-angle structural fractures can be accurately identified, and the R/S-FD fracture identification results of different curves can be verified with the results of core and imaging logging, and the fracture identification accuracy can reach 71.69%. The number of fracture development segments is positively correlated with the cumulative oil production, the number of producing layers and the cumulative thickness of producing layers, and the wells with more fracture development segments have higher cumulative oil production, more producing layers and larger cumulative thickness of producing layers. This method improves the accuracy of fracture identification in carbonate reservoirs, which is of great significance for oil and gas exploration.

Numerical modeling of Carreau–Yasuda fluid with induced magnetic field in a heated curved channel having ciliary walls

Cilia motion is commonly located in mammalian cells of living organisms which continually grow and feature conspicuously. The ciliary apparatus is associated with cell cycle movement and proliferation, and cilia play a dynamic part in human and animal growth and everyday life. The motivation of these applications, this article is to present a metachronal wave analysis for non-Newtonian Carreau–Yasuda fluid inside a curved channel with ciliated walls. Hall effect is encountered in the momentum equations to measure the magnetic field while the viscous dissipation and thermal radiation terms are considered to address the energy analysis. The curvilinear coordinates are used due to the curved nature of flow geometry in the derivation of flow equations. The computations for axial velocity, pressure rise, temperature field, mass concentration, and stream function are carried out under low Reynolds number and long wavelength approximation in the wave frame of reference by utilizing appropriate numerical implicit finite difference method (FDM). The implementation of numerical procedure and graphical representation of the computations are accomplished using MATLAB language. Furthermore, the skin friction and Nusselt number at the channel walls are determined for a variety of critical parameter assessments.

Heat and mass transfer characteristics of vertical falling film cylindrical plastic surfaces under partial wetting conditions for liquid desiccant regeneration

The regenerator plays a dominant role in the overall energy-economic performance of liquid desiccant systems. Nevertheless, very few studies have been reported on falling film liquid desiccant regenerators. Most of the previous studies targeted experimental/numerical modelling needs of regeneration process under design conditions. In the current research workexperimental investigation of regeneration process is carried out to study the coupled heat and mass transfer process for adiabatic regeneration of LiCl liquid desiccant over outer surface of vertical Plain and Modified polypropylene cylindrical surfaces under design (complete wetting) and part load (partial wetting) operating conditions. Modified cylindrical surfaces are developed to compensate the poor wetting of the plastic surfaces. New generalised one-dimensional numerical philosophy is proposed to accommodate the need of part load operating conditions by inclusion of actual wetting of the working surface as well as convective heat transfer happening from dry patches of solid surface to the flowing air. The heat and mass transfer coefficients are determined by solving the governing coupled heat and mass balance non-linear differential equation using the numerical finite difference method. It is found that the Modified surface offered an average improvement of 35.8% in the mass transfer coefficient under partial wetting conditions. New Sh and Nu number correlations are proposed by incorporating heat and mass transfer driving potentials and new parameter wetting factor. The developed correlations predicted the change in humidity and temperature across the regenerator with an average error of 6.2% and 7.5%, whereas from the existing two correlations found in the literature, the first one predicted the same experimental observations with an average error of 54.6% and 36.2%, while second one predicted with an average error of 77.3% and 22.0%, respectively. The results highlighted that the assumption of complete wetting of working surface under part load operating conditions leads to significant error in prediction of air outlet conditions. The proposed correlations will be helpful for numerical modelling and simulation of falling film regenerators under wide range of liquid loading conditions.

Dynamics of the black soliton in a regularized nonlinear Schrödinger equation

We consider a family of regularized defocusing nonlinear Schrödinger (NLS) equations proposed in the context of the cubic NLS equation with a bounded dispersion relation. The time evolution is well-posed if the black soliton is perturbed by a small perturbation in the Sobolev space H s ( R ) H^s(\mathbb {R}) with s > 1 2 s > \frac {1}{2} . We prove that the black soliton is spectrally stable (unstable) if the regularization parameter is below (above) some explicitly specified threshold. We illustrate the stable and unstable dynamics of the perturbed black solitons by using the numerical finite-difference method. The question of orbital stability of the black soliton is left open due to the mismatch of the function spaces for the energy and momentum conservation.

COMPLEX RESISTANCE OF A COMPRESSED-BENT ROD TAKINGINTO ACCOUNT ELASTIC COMPLIANCE OF ITS SUPPORT

This work deals with the urgent problem in the mechanics of a deformable solid: studying factors of the stress-strain state of a single-span statically indeterminate beam with complex boundary conditions that is under conditions of complex resistance (axial compression with plane transverse bending). To solve the problem, both analytical methods (the method of forces in the form of “five” support moments, the method of initial parameters) and the numerical finite difference method with a “linear” grid with density n=8 were used. The necessary resolving equations and matrices are given to take into account changes in the rigidity parameter of the right hinged support and variations in concentrated and uniformly distributed loads, both along the axis of the beam and across it. In the final form, diagrams of deflections, bending moments and shear forces were constructed for specific values of bending rigidity and the degree of elastic compliance of the right hinge-yielding support. Reliability of the theoretical principles and applied results obtained by the authors is confirmed on the basis of the given alternative calculation methods.

Impacts of Temperature Dependent Thermal Conductivity and Viscosity on Slipped Flow of Maxwell Nanofluid

The mathematical model to inspect the effects of changeable thermo-physical properties such as thermal conduction, slip effects and viscosity on Maxwellian nanofluid is proposed. The thermal conductivity increases rapidly due to presence of nanoparticles such as metals, carbides, oxides etc. in base fluid. The flow occurs from the stagnated point pass a stretched sheet with slipped conditions. The characteristics of the Brownian motion as well as the thermophoresis processes are also taken into consideration. By means of similarity transformations, the ODEs are reduced from the equations influencing the fluid flow. A built-in solver of MATLAB namely bvp4c which is a collocation formula implementing the Lobatto IIIa finite differences numerical method is applied to solve these transformed equations numerically. The graphs of the numerical outcomes representing impacts of variations in different parameters on the fluid movement, transfer of heat along with mass are analyzed. This investigation leads to an important aspect that as the thermal conductivity in the flow is intensified, the temperature of the fluid reduces with high aggregation of the nanoparticles near the sheet’s surface. Also, the rates of heat and mass transferral depletes due to the relaxation of Maxwellian fluid. Furthermore, the effectiveness of the present numerical computations is determined by carrying out comparisons of heat and mass transferred rates against the previous analytical results for several values of thermophoresis and Prandtl parameters. The effectiveness of its outcomes can be applied in nanoscience technology and polymeric industries for their developments.

Modeling of bidirectional chloride convection-diffusion for corrosion initiation life prediction of RC square piles under drying-wetting cycle

Modeling chloride penetration mechanism in unsaturated concrete is essential for corrosion initiation life prediction of marine reinforced concrete (RC) structures. This paper presents a bidirectional chloride convection-diffusion model for the chloride concentration distribution of RC square piles in the tidal and splash zone. The finite difference numerical method is used to solve the convection-diffusion governing equations with the drying-wetting cycle. The numerical model is validated by two experimental results and the analytical solution with constant surface chloride conditions. The peak value and convection depth in the chloride distribution curve of RC square piles are investigated. The results show that the periodic drying-wetting chloride environment is the dominant factor of the convection effect formed in the surface layer of the square piles. Moreover, a coupling effect is observed in the diagonal areas of square piles. The corrosion initiation life under a periodic drying-wetting chloride environment is shorter than that with constant surface chloride concentration. Furthermore, the sensitivity analysis results show that the diffusion coefficient, the concrete cover depth, and the variable rate of drying-wetting cycles are the critical parameters for predicting the corrosion initiation life of RC square piles under the drying-wetting cycle.

Nonlinear vibrational analysis and linearization error investigation in size-dependent resonant pressure microsensors

Dynamic and vibrational behavior of nonlinear resonant pressure micro sensors is studied in this paper using a proposed dynamic model. The basic part of the device is a microplate deflects due to external pressure in presence of electrostatic actuation. The frequency changes because of the microplate deformation is used to estimate the external pressure. For modeling of the micro sensor, the nonlinear governing equation of size dependent microplate in polar coordinate system and based on the modified couple stress theory is presented. Assuming specified number of Taylor expansion terms for electrostatic force, the linear and nonlinear ODE is obtained and nonlinear equation is solved both with multiple time scales method (MTS) and finite difference numerical method. Results indicate that considering finite terms of Taylor expansion (1 term for linear and three terms for MTS method) is the source of error in estimation of pressure or system stability. Using sufficient terms of Taylor Expansion, the error in linear and MTS solutions is vanished. Therefore, the limited values of microplate vibrational amplitude is extracted to increase the accuracy of the sensor in linear action. The effect of size dependency, applied voltage on nonlinear behavior and linearization error is also studied.

Study on the Optimization of Pile Length of Micropiles in Soil Landslides

This study summarizes the engineering design and calculation methods of micropiles and proposes a pile length optimization model based on numerical simulation software. Based on the proposed micropile calculation method and optimization method, a specific analysis of a project example was carried out, and a series of calculations, such as micropile design calculation and pile length optimization for the project, was completed. The results show that the miniature pile length optimization model based on numerical simulation finite difference method improves the previous method by automating the optimization process through fast modeling, automatic creation of optimization commands, and output and analysis of optimization results, and realizes the optimization of pile length using numerical simulation, which improves the efficiency of the optimization of the pile length under the premise of guaranteeing accuracy, and achieves the unity of both efficiency and accuracy. The feasibility of this optimization process is proved by engineering examples. The engineering practicability of the micropile design calculation method and optimization method proposed in this study is proved through practice. It provides a reference value for the initial fast and flexible management of small landslides.

Physical nonlinearity in porous functionally graded kirchhoff nano-plates: Modeling and numerical experiment

In this paper, mathematical models of porous functionally graded inhomogeneous plates were constructed. The following hypotheses were used as the foundation: the kinematic Kirchhoff model was used; nano effects were considered by the modified couple stress theory; material properties depend on coordinates (x,y,z), and for metals, they also hinge on the stress-strain state at a point in the plate volume, i.e. there is a dependence σi(ei). This dependence can be any analytical function derived from experimentation, which means that it is possible to take into account elastic-plastic deformations according to the deformation theory of plasticity. The ceramic material is resilient. The required variational and differential equations and the boundary conditions were derived from the Lagrange functional. The plate material's porosity was characterised by six types. For numerical solutions, the methods of reduction of the partial differential equation to the ordinary differential equation were applied, as well as iterative methods nested one into another, based on the following techniques: variational iterations method (VIM), Agranovskii-Baglai-Smirnov method (ABSM) and Birger's method of variable parameters of elasticity. For each of these methods, there are theorems proving their convergence. Each problem that describes the stress-strain state of Kirchhoff nano-plates was solved using the following analytical methods: Exact Navier's solution (ENS), Bubnov-Galerkin method (BGM), Levy solution (LS), numerically analytical Kantorovich-Vlasov (KVM) methods, variational iterations method (VIM), Vaindiner method (VaM) and numerical finite difference method (FDM). The convergence of these methods was investigated and the optimal method for calculating such structures, both in terms of accuracy and speed, was determined for the elliptic 2D PDEs of arbitrary area Ω. Numerical results for both elastic nano-plates and elastoplastic nano-plates have been presented.