Finite Difference Method Research Articles

Spontaneous Photonic Jammed Packing of Core-Shell Colloids in Conductive Aqueous Inks for Non-Iridescent Structural Coloration.

Integrating structural colors and conductivity into aqueous inks has the potential to revolutionize wearable electronics, providing flexibility, sustainability, and artistic appeal to electronic components. This study aims to introduce bioinspired color engineering to conductive aqueous inks. Our self-assembly approach involves mixing poly(3,4-ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT:PSS) with sulfonic acid-modified polystyrene (sPS) colloids to generate non-iridescent structural colors in the inks. This spontaneous structural coloration occurs because PEDOT:PSS and sPS colloids can self-assemble into core-shell structures and reversibly cluster into photonic aggregates of maximally random jammed packing within the aqueous environment, as demonstrated by small-angle X-ray scattering. Dissipative particle dynamics simulation confirms that the self-assembly aggregation of PEDOT:PSS chains and sPS colloids can be manipulated by the polymer-colloid interactions. Utilizing the finite-difference time-domain method, we demonstrate that the photonic aggregates of the core-shell colloids achieve close to maximum jammed packing, making them suitable for producing vivid structural colors. These versatile conductive inks offer adjustable color saturation and conductivity, with conductivity levels reaching 36 S cm-1 through the addition of polyethylene glycol oligomer, while enhanced water resistance and mechanical stability are achieved by doping with a cross-linker, poly(ethylene glycol) diglycidyl ether. With these unique features, the inks can create flexible, patterned circuits through processes like coating, writing, and dyeing on large areas, providing eco-friendly, visually appealing colors for customizable, stylish, comfortable, and wearable electronic devices.

Semi-Discretized Approximation of Stability of Sine-Gordon System with Average-Central Finite Difference Scheme

Semi-Discretized Approximation of Stability of Sine-Gordon System with Average-Central Finite Difference Scheme

Solution of the Poisson equation by the boundary integral method

Purpose The boundary integral method (BIM) is very attractive to practicing engineers as it reduces the dimensionality of the problem by one, thereby making the procedure computationally inexpensive compared to its peers. The principal feature of this technique is the limitation of all its computations to only the boundaries of the domain. Although the procedure is well developed for the Laplace equation, the Poisson equation offers some computational challenges. Nevertheless, the literature provides a couple of solution methods. This paper revisits an alternate approach that has not gained much traction within the community. The purpose of this paper is to address the main bottleneck of that approach in an effort to popularize it and critically evaluate the errors introduced into the solution by that method. Design/methodology/approach The primary intent in the paper is to work on the particular solution of the Poisson equation by representing the source term through a Fourier series. The evaluation of the Fourier coefficients requires a rectangular domain even though the original domain can be of any arbitrary shape. The boundary conditions for the homogeneous solution gets modified by the projection of the particular solution on the original boundaries. The paper also develops a new Gauss quadrature procedure to compute the integrals appearing in the Fourier coefficients in case they cannot be analytically evaluated. Findings The current endeavor has developed two different representations of the source terms. A comprehensive set of benchmark exercises has successfully demonstrated the effectiveness of both the methods, especially the second one. A subsequent detailed analysis has identified the errors emanating from an inadequate number of boundary nodes and Fourier modes, a high difference in sizes between the particular solution and the original domains and the used Gauss quadrature integration procedures. Adequate mitigation procedures were successful in suppressing each of the above errors and in improving the solution accuracy to any desired level. A comparative study with the finite difference method revealed that the BIM was as accurate as the FDM but was computationally more efficient for problems of real-life scale. A later exercise minutely analyzed the heat transfer physics for a fin after validating the simulation results with the analytical solution that was separately derived. The final set of simulations demonstrated the applicability of the method to complicated geometries. Originality/value First, the newly developed Gauss quadrature integration procedure can efficiently compute the integrals during evaluation of the Fourier coefficients; the current literature lacks such a tool, thereby deterring researchers to adopt this category of methods. Second, to the best of the author’s knowledge, such a comprehensive error analysis of the solution method within the BIM framework for the Poisson equation does not currently exist in the literature. This particular exercise should go a long way in increasing the confidence of the research community to venture into this category of methods for the solution of the Poisson equation.

Stability estimates for radial basis function methods applied to linear scalar conservation laws

We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete ℓ2-norm intrinsic to each of the three methods. The results show that Kansa's method and RBF-PUM can be ℓ2-stable in time under a sufficiently large oversampling of the discretized system of equations. The RBF-FD method in addition requires stabilization of the spurious jump terms due to the discontinuous RBF-FD cardinal basis functions. Numerical experiments show an agreement with our theoretical observations.

Solving a fractional diffusion PDE using some standard and nonstandard finite difference methods with conformable and Caputo operators

IntroductionFractional diffusion equations offer an effective means of describing transport phenomena exhibiting abnormal diffusion pat-terns, often eluding traditional diffusion models.MethodsWe construct four finite difference methods where fractional derivatives are approximated using either conformable or Caputo operators.ResultsStability of the proposed schemes is analyzed using von Neumann stability analysis, and conditions are established to preserve positivity. Consistency analysis is performed for all methods, and numerical results with fractional parameters (α) set to 0.75, 0.90, 0.95, and 1.0 are presented.DiscussionThe rate of convergence in time for the four methods is computed.

Theoretical investigation of the combined effects of solar energy and thermal buoyancy around a laminar jet placed in a porous medium

The current research examines the combined effects of solar energy and thermal buoyancy around a laminar jet placed in a porous medium. The governing boundary layer equations are dimensionalized by using appropriate dimensionless variables. The numerical solution of the dimensionless boundary layer equations is obtained using the finite difference method. The impact of physical parameters, which are Darcy number, dimensionless porous medium inertia coefficient, Prandtl number, radiation parameter, and dimensionless fluid's absorption parameter, on velocity and temperature profile, is shown graphically, while the influence of the above parameters on the heat transfer rate is presented in tabular form. It is keenly observed that for Darcy number velocity profile decreases while reverse behavior is noted for temperature distribution. For the dimensionless radiation parameter, both the velocity and temperature profile decrease. The main novelty of the current work is to improve the thermal performance of natural convection heat transfer system in the presence of thermal radiation placed in porous medium.

Influence of thermal radiation and magnetic field on convective transfer within an inclined square enclosure filled with Cu-water nanofluid

The present work investigates the effects of thermal radiation and magnetic field on the laminar convective transfer process on copper-based nanofluid within an inclined square enclosure. The studied configuration is a rigid-walled cavity subjected to a horizontal temperature gradient where the left wall is maintained at a hot temperature Th and subjected to a magnetic field B0. In contrast, the right wall is kept at a cold temperature Tc. The horizontal walls are adiabatic. The dimensionless governing equations are solved using the finite difference method and the UPWIND scheme to solve the convective terms. Formulated using the stream function, vorticity, and temperature. Effects on mean Nusselt numbers are investigated at different Rayleigh numbers 104,105,106, Hartmann numbers HA=0to100, tilt angles γ=0,π/6,π/4andπ/3, nanofluid volume fractions 0%≤φ≤6% and different radiation parameters ( Rd=0,1,2,3 ). The findings show that the HTR increased 3.6 times by augmenting the Ra from 104 to 106. Also, increasing the Ha number from 0 to 100 causes an 83.16% reduction in the HTR. Considering the radiation mode of heat transfer causes an increase in HTR. The average Nusselt values increase by 59.72% for the concentration of 6% while increasing the radiation parameter from 0 to 3. On the other hand, an increase in volume fraction leads to a deterioration in mean Nusselt numbers (Numean).

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

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

An inverse-free Getz-Marsden dynamic system and its eleven-instant discrete model for time-variant linear equations solving

Time-variant linear equations (TVLEs) are fundamental problems appeared in many practical applications. There is an increasing demand for algorithms that can solve TVLEs with high accuracy. This paper proposes an inverse-free continuous-time Getz–Marsden dynamic system (CTGMDS) for solving TVLEs. Additionally, a general eleven-instant finite-difference method (FDM) is proposed to discretize the CTGMDS, resulting in a general eleven-instant discrete-time GMDS (11-DTGMDS) model. Theoretically, the proposed 11-DTGMDS has truncation error of seven-order precision that delivers higher precision than the existing FDMs. This paper also compares five DTGMDS models using other FDMs to discretize the CTGMDS. The convergence of the CTGMDS and 11-DTGMDS is proven by theoretical analysis. Numerical validations are performed by comparing the 11-DTGMDS and other DTGMDSs, showing that the 11-DTGMDS provides higher solution accuracy. Finally, a path-tracking task is completed by applying the 11-DTGMDS model to a Franka Emika Panda robot, validating the practicality of the 11-DTGMDS.

Polarization-independent tetramer metastructure with multi-Fano resonances governed by quasi-bound states in the continuum

We propose what we believe to be a novel metastructure consisting of silicon nanoblock tetramer clusters and investigated theoretically and experimentally how to enhance surface sensing capabilities with polarization-independent properties through bound states in the continuum (BICs). By introducing square defects, three quasi-BIC modes are excited at the wavelengths of 1013.29 nm, 1109.57 nm, and 1310.73 nm with a modulation depth up to 100% and a Q-factor of 16618, while achieving a sensitivity of 256 nm/RIU and a figure of merit of 2519.7 RIU−1. Utilizing the finite-difference method in the time domain (FDTD), we have analyzed the electromagnetic near-field distributions, polarization properties, and robustness corresponding to resonance dips at different wavelengths. Our results reveal that the distinct electromagnetic near-field distributions excited by the quasi-BIC modes are characterized by the strong field confinement within the silica nanoblock tetramer clusters. The modes are primarily governed by magnetic quadrupole (MQ) resonance and toroidal dipole (TD) moment. Additionally, the polarization-independent properties of the metastructure are demonstrated to ensure that they can effectively respond to the incident light regardless of its polarization state, while the robustness is investigated to show the stability and reliability by shifting square defects. Moreover, the transmission spectra of the metastructure is experimentally verified by immersing the sample into CaCl2 solutions in different concentrations. This work plays a pivotal role in creating further novel and high-performance optical metastructures while demonstrating the potential applications of quasi-BIC modes in precision sensing in the future.

Numerical simulation of hybrid nanofluid flow with homogeneous and heterogeneous chemical reaction across an inclined permeable cylinder/plate

The current study investigates the steady two-dimensional (2D) hybrid nanofluid (Hnf) flow over an inclined permeable plate/cylinder. The Hnf flow has been examined in the context of mixed convection, heterogeneous/homogenous chemical reaction, and permeable medium. The Hnf is prepared by dispersing silver (Ag), and iron ferrite (Fe3O4) nanoparticles (NPs) in water. The current research is motivated by the increasing demand for highly efficient cooling devices in a variety of industries and energy-associated operations. The energy transmission and fluid flow are mathematically specified by using a coupled nonlinear system of partial differential equations (PDEs). The system of PDEs is simplified into a dimensionless form of ODEs, which are then further numerically treated with the MATLAB package based on the finite difference method (bvp4c). It has been noticed that the permeability component develops the heat transfer curve while decreasing the flow rate of the fluid. The impact of heat source/sink increases the energy profile. Moreover, the plate surface demonstrates the dominant behavior of energy transportation than a cylinder with the variance of Ag-Fe3O4-NPs.

Impact of north-west Bay of Bengal bathymetry along the off-coast of Digha on the propagation of shallow water waves using finite difference method

Numerical simulation of shallow water equations has been carried out using the Crank-Nicolson scheme of Finite Difference Method (FDM) to study the interaction of waves with different forms of bathymetry especially that of the off-coast of Digha, a tourist spot situated in the north-west coast of Bay of Bengal. MATLAB code has been developed for the simulation. One-dimensional (1-D) linear shallow water model has been considered here. Three different bathymetry profile are chosen for the investigation: (i) flat bathymetry (with or without seamount/basin); (ii) piecewise linear approximation to the real bathymetry of the off-coast of Digha; (iii) real bathymetry of the off-coast of Digha. Real bathymetry data has been obtained from the gridded data set of GEBCO (General Bathymetric Chart of the Oceans) published in 2022. Validation of the present numerical scheme has been performed by considering test problems with results existing in the literature. Comparative study of the effect of the bathymetries, on the propagation of shallow water waves, has been performed and demonstrated with figures and tables. Results of wave-bathymetry interaction obtained here are believed to be new with respect to the present study area and will definitely give some insight into the phenomena in this region.

Combined effect of track-bridge interaction on the dynamic response of a simply supported railway bridge

In this work, the objective is to understand the effect of the nonlinear behaviour of the ballasted track, or the mechanism of vertical and horizontal track-bridge interaction behaviour, on the dynamic response of a simply supported railway bridge, carrying a single ballasted track characterized by a fractional order viscoelastic nonlinear Winkler foundation type, and acted by a moving train load with constant speed. The main feature of the model is that the horizontal and vertical interactions at the interface between the bridge deck and the ballasted track, composed by rails, sleepers and ballast, is introduced through a nonlinear cubic force-displacement relation, schematizing the nonlinear stiffness, plus fractional derivative term for simulating the rail bed’s damping behaviour and a system of horizontally orientated spring-damper elements. By applying the two numerical methods, fourth-order Runge-Kutta method and Finite Difference Method, the validation of the dynamic behaviour of the rail-bridge system is performed, and numerical simulations demonstrate the accuracy of the proposed theoretical model for predicting the real behaviour of the studied bridge taking into account the track-bridge interaction effect. The influence of the fractional-order derivative and the distance between sleepers on the critical velocities, associated with resonance effects, is also addressed.

Hydrodynamic response on trajectory tracking of a tethered underwater robot system under hybrid control algorithms of umbilical cable and propellers

A hydrodynamic control model is established in attention to coupling relationships among cable, propellers and robot body of a tethered underwater robot system. In this model the governing equations of umbilical cable and the robot are firstly introduced, then supplementary conditions are coupled into the equations to forming the dynamic mathematical model; finally a hybrid control strategy based on feed-forward negative feedback method for the cable and PID rule for the propellers are integrated in the mathematical model for composing the whole hydrodynamic control model. Both the mathematical model and the control algorithms are proved to be effective and reliable through comparing simulation with the experimental data in existed references. Based on the numerical model constructed in this paper, trajectory tracking of a tethered underwater robot system in different motion combinations are numerically simulated through computational fluid dynamics method. In the numerical simulations, finite difference method is used to solving the kinematic parameters of the mathematical model, while finite volume method is applied on calculating the hydrodynamic forces under a hybrid control manipulations. The robot motion in vertical direction is determined primary by feed-forward negative feedback strategy of adjusting the cable length, while the horizontal movement of the robot is controlled mainly through PID algorithm; The hydrodynamic loading on the robot body are influenced by the flow fields around the robot.

Nonlinear flow phenomenon of a power-law non-Newtonian fluid falling down a cylinder surface

In this paper, we present a comprehensive study of the fingering phenomenon of a power-law non-Newtonian fluid falling down a cylinder surface. A theoretical analysis is firstly carried out and the governing equation describing the film thickness is established for the non-Newtonian fluid denoted by a power-law index n. Using the lubrication theory with dimensionless variables, the partial differential equation for the film thickness is derived, and both two-dimensional flow and three-dimensional flow are investigated. The traveling wave characteristic of the two-dimensional flow is displayed by using the finite difference scheme associated with a Newton iteration technique. The effect of changing the radius of the cylinder, precursor-layer thickness and power-law index is then considered through examining the variation of the capillary waves. Furthermore in three-dimensional complex flow, the fingering patterns for different parameters are simulated. The linear stability analysis based on the traveling wave solutions is given to elucidate the influence of different physical parameters, explaining the physical mechanism of nonlinear flow behaviors in two dimension and three dimension. Results from linear stability analysis show that the power-law index reinforces the fingering instability whether the fluid is shear-thinning or shear-thickening. Moreover, the numerical results illustrate that increasing the radii of the cylinder expands the unstable region. The flow profiles for different power-law coefficients are consistent with the result of the linear stability analysis, proving the unstable role of the power-law coefficient.

Retrieving the time-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer

PurposeIn order to include the non-negligible lag relaxation time feature that is characteristic of heat transfer in biological bodies, the classical Fourier's law of heat conduction has to be generalized as the Maxwell–Cattaneo law resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the unknown time-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from (non-intrusive) measurements of the temperature on an accessible sub-portion of the boundary that may be taken with an infrared scanner.Design/methodology/approachThe nonlinear and ill-posed problem is reformulated as a nonlinear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite-difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Numerical results for a benchmark test example are presented and thoroughly discussed, shedding light on the performance and effectiveness of the proposed methodology.FindingsThe inverse problem of obtaining the time-dependent blood perfusion coefficient and the temperature in the thermal-wave model of bio-heat transfer from extra boundary temperature measurement has been solved. In particular, the uniqueness of the solution to this inverse problem has been established. Furthermore, our proposed computational method demonstrated successful attainment of the perfusion coefficient and temperature, even when dealing with noisy data.Originality/valueThe originalities of the present paper are to account for such a more representative thermal-wave model of heat transfer in biological bodies and to investigate the possibility of determining its time-dependent blood perfusion coefficient from non-intrusive boundary temperature measurements.

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.

Singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme

In this paper, we proposed an accurate ϵ-uniformly convergent numerical method to solve singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme. The time-fractional derivative is defined in the sense of Caputo with order η∈(0,1). The time-fractional derivative is discretized by employing the Crank–Nicolson method on a uniform mesh, and an exponential fitted operator scheme along with the standard upwind method is used to mesh-grid the space domain. The truncation error and uniform stability of the discretized problems are examined in order to prove the parameter uniform convergence of the proposed scheme. It is demonstrated that the scheme is ϵ-uniformly convergent of order O((Δt)2−η+Δx), where Δt and Δx represent the step sizes of the time and space domains, respectively. Two numerical examples are provided in order to assess the accuracy of the suggested scheme and validate the theoretical concepts discussed. To demonstrate the efficiency of the numerical scheme presented, comparisons have been made with the numerical solution obtained by the finite difference method that exists in the literature. Consequently, it is observed that the results obtained by the present scheme are more accurate and have a better convergence rate.

Global Stability and Forced Response Analysis of Swirling Flows In Aviation Combustors

Abstract Swirling jets are canonical flow fields used to stabilize flames in many gas turbine combustion systems. These flows amplify background acoustic disturbances and perturb the flame, producing combustion noise and potentially combustion instabilities. Hydrodynamic flow response to acoustic excitations in reacting flows is an important modeling task for understanding combustion noise and combustion instabilities. This study utilizes a global hydrodynamic stability analysis in a Bi-Global and Tri-Global framework to model the vortical hydrodynamic modes of a reacting, swirling LES mean flow based on a commercial nozzle. After conducting an unforced, natural Bi-Global stability analysis to study the unstable eigenmodes of the flow, linearized equations of motion are used to study flow response of the flow to forcing. An empirical velocity transfer function is constructed by determining the flow response to a harmonically varying 150-1500 Hz u' velocity disturbance at the inflow boundary. These disturbances are strongly amplified, with amplification factors ranging from about 20 to 50 and peaking at about 1050 Hz, or st = 0.375. A comparison study with a cartesian, three-dimensional reacting base flow is also performed where the base-flow varies spatially in three-dimensions. The study utilizes a stable, high accuracy, but computationally effective centered finite difference scheme based on a three-dimensional structured mesh. Illustrative results present the qualitative modes and quantitative transfer functions for Bi-Global and Tri-Global stability analyses.

Dynamic Rupture and Strong Ground-Motion Simulations of the 8 January 2022 Ms 6.9 Qinghai Menyuan Earthquake

Abstract The 2022 Ms 6.9 Qinghai Menyuan, China, earthquake is the most destructive earthquake to have occurred near the Lenglongling fault at the western segment of the Qilian–Haiyuan fault since 2016 Ms 6.4 Menyuan earthquake. The 2022 earthquake generated surface rupture measuring about 30 km with an unexpected maximum offset larger than 2.6 m in the epicentral area, and severely damaged the local infrastructure and transportation. To analyze the possible causes of the large surface slip and to reveal the rupture process, we modeled the dynamic rupture and strong ground motion of the 2022 Menyuan earthquake using the curved-grid finite-difference method. In the simulation, the geometry of the fault is constructed based on the observed trace of the surface ruptures. The background tectonic stress field is assumed to be uniform, and the slip-weakening law with the constant friction coefficients is adopted. Our modeling results showed that the rupture model with a focal depth of 6 km and a rupture width of 10 km provides a good fit to the observed surface slips and the field records. We also investigated the effects of the focal depth and the rupture size on the surface slips. It is found that under the same conditions, the dynamic rupture models with a larger rupture size generated greater coseismic slips at the surface. However, only the model with a relatively smaller rupture width produced an Mw∼6.7 event similar to the Menyuan earthquake. In contrast, the influence of the focal depth is less significant. The decrease of the focal depth only leads to a slight increase in surface slip. Our results illustrated that a surface-breaking rupture with a relatively narrow width may physically control the general characteristics of the earthquake. This study provides a new insight into the rupture dynamics of the 2022 Menyuan earthquake.