Finite Difference Approximation Method Research Articles

Theoretical studies for the detailed electronic structure of organotin(IV) derivatives of 4-fluoroanthranilic acid, X-ray structure of n-dibutyltin bis(4-fluoroanthranilate)

The present study describes the detailed DFT-based electronic structure calculations without any symmetry constraints being performed on di- and triorganotin derivatives of 4-fluoroanthranilic acid (AFA), viz. Me2SnL2 (1), n-Bu2SnL2 (2), Me3SnL (3), and Ph3SnL (4) (where L= monoanion of AFA). The structural and atomic charge analysis have confirmed the previously reported our experimental results, i.e. bicapped tetrahedral and distorted tetrahedral geometry for di- and triorganotin derivatives, respectively, and all the complexes 1–4 contain a positively charged central tin atom surrounded by a negatively charged system. The single crystal X-ray structure of complex 2 (n-Bu2SnL2) also suggests that AFA acts as a monoanion and bidentate ligand resulting a bicapped tetrahedral environment around tin. Various population analysis such as Mulliken (MPA), Hirshfeld (HPA), and natural population analysis (NPA) have been employed to calculate the charges at all the atoms. A finite difference approximation method has been used to explain the charge distribution within the studied complexes 1–4 based on the conceptual global and local reactivity DFT-based descriptions, frontier molecular-orbital analysis (FMOA), and molecular electrostatic potential maps (MEP). Further, the conceptual DFT-based global reactivity descriptors and frontier molecular orbital analysis show that the interactions between CT-DNA and complexes 1–4 may be groove binding or electrostatic. The integral equation formalism-polarizable continuum model (IEF-PCM) has been employed to calculate the UV–visible spectra of 1–4 in the solvent field, whereas the same in the gas phase has been obtained with the help of time-dependent DFT (TD-DFT) at the same level of theory. The intramolecular charge distribution in 1–4 has been investigated with the help of natural bond orbital (NBO) analysis. The topological and energetic parameters at the selected bond critical points in the coordination sphere of complexes 1–4 have been calculated using atoms-in-molecules (AIM) theory. The analysis of different kinds of non-covalent interactions (NCI) in complexes 1–4 has also been investigated.

HNS: An efficient hermite neural solver for solving time-fractional partial differential equations

Neural network solvers represent an innovative and promising approach for tackling time-fractional partial differential equations by utilizing deep learning techniques. L1 interpolation approximation serves as the standard method for addressing time-fractional derivatives within neural network solvers. However, we have discovered that neural network solvers based on L1 interpolation approximation are unable to fully exploit the benefits of neural networks, and the accuracy of these models is constrained to interpolation errors. In this paper, we present the high-precision Hermite Neural Solver (HNS) for solving time-fractional partial differential equations. Specifically, we first construct a high-order explicit approximation scheme for fractional derivatives using Hermite interpolation techniques, and rigorously analyze its approximation accuracy. Afterward, taking into account the infinitely differentiable properties of deep neural networks, we integrate the high-order Hermite interpolation explicit approximation scheme with deep neural networks to propose the HNS. The experimental results show that HNS achieves higher accuracy than methods based on the L1 scheme for both forward and inverse problems, as well as in high-dimensional scenarios. This indicates that HNS has significantly improved accuracy and flexibility compared to existing L1-based methods, and has overcome the limitations of explicit finite difference approximation methods that are often constrained to function value interpolation. As a result, the HNS is not a simple combination of numerical computing methods and neural networks, but rather achieves a complementary and mutually reinforcing advantages of both approaches. The data and code can be found at https://github.com/hsbhc/HNS.

Damage identification in steel plate structures considering bending and torsion moments using estimated seismic responses

This paper proposes a model-free damage identification method and suggests an effective index for damage diagnosis in steel plate-like structures considering both the torsion and bending moments using only acceleration data, with no need for geometric characteristics. The basis for damage detection is the difference between healthy and damaged structural responses to different ground accelerations. In doing so, two Peak Ground Acceleration (PGA) scales, 0.05 (g), and 0.5 (g), labeled for the Loma Prieta ground motion, are considered. Four-node meshes are applied to reflect the torsion moments in the steel plate behavior. A multi-objective function, Fisher Information Matrix, is utilized, and four sensor locations are optimized to identify the locations most sensitive to damage. AutoRegressive Moving Average with exogenous input (ARMAX) system identification method is employed to obtain the structural displacement from the acceleration data and, subsequently, the displacement state equations of the nodes. With the aid of the structural displacement and its derivatives obtained through the central finite difference approximation method, damage to the plate structure is detected at PGA = 0.05 (g) and PGA = 0.50 (g) considering thickness changes based on the peak values manifested in the proposed damage detection index. Furthermore, the recorded structural acceleration responses at the desired scales of PGAs are fed into the 2D wavelet transform to detect damage imposed on the plate structure using the diagonal component of the wavelet transform. The results confirm that the proposed nonmodel-based approach successfully detects the single and multiple damage scenarios and can be used for a quick and judicious assessment of the steel plate behavior immediately after damage.

Three-dimensional contact formulation for assessment of dynamic interaction of pantograph and overhead conductor rail system

Overhead conductor rail (OCR) is an important power supply device in tunnel railway lines. The penalty method is widely adopted to model the dynamic interaction behaviour of the pantograph-OCR system (POCRs), while it is difficult to determine a proper contact stiffness value. This paper proposes a three-dimensional contact formulation for the POCRs based on the Lagrange multiplier method. Accurate contact forces can be obtained easily by this contact formulation without the trouble of determining the contact stiffness. Contact force is defined in the global coordinate system as a three-dimensional vector. Coulomb law of friction is adopted in this paper to describe the tangential contact force, and the train speed is included in the contact formulation. The central finite difference approximation method is utilised, and an iteration procedure is elaborated to calculate the accurate contact force vector. The proposed contact formulation is verified by comparing simulation results with in-line measured data. The maximum statistical relative error of the contact force is only 2.63%. The advantages of the proposed contact formulation are also demonstrated by comparing it with the penalty method.

A computational point of view on teaching derivatives

The strong connection between mathematics and informatics highlights the important role of scientific computing in many applications. Unfortunately, mathematics is traditionally taught without investigating possible connections between abstract problem solving and the use of algorithms capable of being implemented on a computer. Since mathematical theory and computing practice are taught separately, many students fail to appreciate the utility of mathematics. In this paper, we briefly explain how a typical lecture on obtaining derivatives in differential calculus can benefit from examples implemented in high-level languages like MATLAB, Python or R. Such examples can help to guide the students to a better understanding of the theoretical concepts and limits of finite precision floating-point arithmetic. We argue that, while the historical findings of Leibniz and Newton are good starting points for introducing finite difference approximation methods, this does not preclude new approaches to numerically computing derivatives using Infinite Computer Arithmetic.

Study of cylindrical and spherical dust acoustic solitons and quasiperiodic structures in a quantum dusty plasma

The solitonic and quasiperiodic structures of dust acoustic (DA) waves are investigated in a three components quantum dusty plasma composed of mobile negative dust grains, ions, and inertialess electrons. The reductive perturbation method is employed to derive A deformed Korteweg–de Vries (dKdV) equation in planar and nonplanar geometries, and its numerical solutions are obtained using the two level finite difference approximation method. The influence of geometries on DA solitons is discussed. It is observed that in nonplanar geometries, DA solitons travel at different speeds in comparison to one-dimensional planar ones. Furthermore, in the planar geometry, the bifurcation of DA traveling waves has been analyzed on the framework of the dKdV equation. By adding an external periodic force to the derived dKdV equation, the quasiperiodic behaviors of DA waves are presented.

Insight into the dynamics of micropolar fluid about a vertical cone when nonlinear thermal radiation is significant: The case of triple mixed convection

AbstractThis novel research investigates the nonlinear triple diffusive combined convective micropolar liquid flow past a vertical cone in the presence of nonlinear thermal radiation, cross‐diffusion, and a convective boundary condition. We aim to analyze this present problem using nonsimilar transformations. This report presents the significance of nonlinear mixed convection, energy flux due to the concentration gradient, and mass flux due to the temperature gradient and nonlinear thermal radiation in the dynamics of the fluid subject with micropolar fluid is presented. The differential equations defining the boundary‐layer parameters are then transformed into dimensionless view, taking into account the nonsimilar transformation. Furthermore, the method of quasilinearization and implicit finite difference approximation is used to work out the nondimensional governing equations for the solution. The velocity pattern diminishes, while dimensionless temperature and concentration distributions enhance with growing values of microrotation parameter. Furthermore, species concentrations of the fluid increase with increasing Soret effect values, while opposite results appear for mass transfer rates. Also, drag coefficient enhances for assisting buoyancy flow whilst diminishes for opposing buoyancy flow with increasing values of the microrotation parameter. The microrotation pattern reduces with growing values of the nonsimilarity characteristics. Furthermore, the Prandtl number is displayed on a comparison graph, and the results are very similar.

Vortex breakdown mechanics of a laminar confined swirling flow with large expansion ratio using a non-uniform finite difference approximation

Abrupt expansions are a very frequent geometry in mechanical engineering systems, i.e. in combustion chambers, valves, heat exchangers or impinging cooling devices. However, despite the large number of devices that use this geometry, the expanded flow behaviour still needs further research to understand and predict the full system performance. This paper presents the application of the non-uniform finite difference approximation method developed in Sanmiguel et al. for the numerical characterisation of a confined swirling laminar jet discharging with a large expansion ratio. This investigation can be considered an extension of previous work by Revuelta, but now a swirling flow is generated by a rotating pipe upstream the expansion. The structures found when a fully-developed rotating Hagen-Poiseuille flow discharges into a much larger pipe section are summarised in a bifurcation diagram, whose coordinates are the Reynolds number of the jet ( Rej) and the swirl parameter ( L), for which the time-dependent, axisymmetric and incompressible Navier-Stokes equations are integrated numerically. For values of the jet Reynolds number below 200, there is a critical value of the swirl parameter above which stable vortex breakdown appears. For values of the Reynolds number above 200, three different behaviours are observed, and each performance appears for a critical value of the swirl parameter. When increasing the swirl parameter from zero, the flow becomes axisymmetrically unstable, showing an oscillatory behaviour. If further increasing the swirl intensity, the oscillatory flow coexists with a vortex breakdown bubble and, finally, a steady vortex breakdown is reached. The expansion ratio ε considered in all the simulations is 1[Formula: see text]. In previous literature, the exactness of the limiting critical Rej and L values that define these behaviours has been found to be influenced by the variability in the inlet profile conditions, which affects the expanded flow. This enhances the importance in the present investigation to accurately simulate the discharge pipe inlet profiles.

Explicit Finite Difference Approximation Method for Solving the Riesz Space Fractional Percolation Equation

ABSTRACT This paper presents an efficient approximation method for solving the Riesz space fractional percolation equation mixed fractional derivative (RSFPEMFD) with the initial condition (IC) and derivative boundary conditions (DBC) using explicit finite difference method (EFDM). The shifted Grunwald estimate was used for analysis of the mixed fractional derivatives. In addition, the analysis method of consistency, stability, and convergence was used for analyzing the EFDM. To demonstrate the efficiency and validity of the proposed algorithm, some illustrative examples are given and comparing results with the exact solution. The results are displayed when necessary in tables using the MathCAD and MATLAB software package. The EFDM for solving RSFPEMFD has been demonstrated to be effective and reliable.

Development and validation of mathematical model of hydrotropic-reactive extraction of lignin

Abstract Hydrotropes have been largely explored as reactive extraction agent for lignin separation. In this paper, a mathematical model of hydrotropic-reactive extraction of sugarcane bagasse lignin was proposed and validated by experimental data from literature. The mathematical model was developed by assuming the particle is in slab shape, and by considering simultaneous processes of hydrotrope intra particle diffusion, second order reaction of lignin-hydrotrope, and intra-particle soluble delignification product diffusion. The proposed model results in a set of partial differential equations which were then solved by explicit finite difference approximation method. The mathematical model parameters were determined by fitting the model to the hydrotropic reactive extraction experimental data reported by Ansari and Gaikar (2014). Simulations show that the mathematical model of the hydrotropic-reactive extraction were well fitted to the experimental data with the obtained hydrotrope effective diffusivity (D eA ) of 5.0 × 10−11 m2/s, effective diffusivity of soluble lignin product (D eC ) of 9.0 × 10−12 m2/s and reaction rate constant (k r ) of 1.78 × 10−10 m3/(g.s). It was also observed that the reaction was first order to the hydrotrope (n = 1), and one half order to the lignin (m = 0.5). Meanwhile the pseudo-stoichiometric mass ratio of hydrotrope to lignin was 6.4 g hydrotrope/g lignin.

Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients

Space non-integer order convection–diffusion descriptions are generalized form of integer order convection–diffusion problems expressing super diffusive and convective transport processes. In this article, we propose finite difference approximation for space fractional convection–diffusion model having space variable coefficients on the given bounded domain over time and space. It is shown that the Crank–Nicolson difference scheme based on the right shifted Grünwald–Letnikov difference formula is unconditionally stable and it is also of second order consistency both in temporal and spatial terms with extrapolation to the limit approach. Numerical experiments are tested to verify the efficiency of our theoretical analysis and confirm order of convergence.

Mathematical Modeling of Reactive Extraction of Solute from Slab Solid Material

Reactive extraction is gaining higher attention due its wide application in various solute separation processes. Here, a mathematical model of reactive extraction in slab has been proposed. The model was developed by considering simultaneous processes of active compound intra particle diffusion, second order elemental reaction of solute-active compound, and intra-particle product diffusion. The obtained partial differential equations (PDEs) were solved using Finite Difference Approximation (FDA) method by using realistic parameters. Concentration profile as well as product yield were evaluated as a function of time. As a result, the model proposed here may serve as a basis design for reactive extraction unit. Sensitivity analyses was conducted to inspect the influence of slab thickness, diffusivity and reaction rate constant to the product yield. Eventually, model validation was conducted by comparing the simulation results with analytical solutions for special cases. Validation results showed that the model gave good agreement with the analytical solution.

Study on thermal parameters of asphalt concrete for countermeasures against high surface temperature of pavement in tunnel

It is reported that the surface temperature of asphalt pavement in a road tunnel in an urban area in summer days gets hot especially when the tunnel is long and the traffic volume is heavy. Since the asphalt pavement absorbs a large amount of heat generated by vehicles and keeps it for a long time, a high-temperature environment in the tunnel also continues for a long time. This phenomenon brings uncomfortableness for users, especially for motor-bikers and maintenance workers. On this background, this study aims to estimate the thermal parameters of the various types of asphalt concrete and investigate their effects on a high-temperature pavement surface. Six types of asphalt concrete specimens were tested in a wind-tunnel where a hot temperature air inside was generated by a forced circulation boiler system. Thermal parameters: specific heat capacity, heat conductivity, heat diffusivity, and heat transfer coefficients were analysed using a one-dimensional partial differential heat transfer equation and a finite difference approximation method. Additionally, the effects of different basement boundary conditions and water spray on the surface temperature were investigated to find effective countermeasures to decrease the surface temperature of asphalt concrete in the tunnel.

Electronic Structure Explanation for the Structure and Reactivity of di-n-Butyltin(IV) Derivative of Glycylphenylalanine

The density functional theory (DFT)-based quantum-chemical calculations have been performed on di-n-butyltin(IV) derivative of glycylphenylalanine (H2L) using the Gaussian 09 software package. The molecular geometry of n-Bu2SnL was optimized at B3LYP/6-31G(d,p)/LANL2DZ(Sn) level of theory without any symmetry constraint. The harmonic vibrational frequencies were computed at the same level of theory to find the true potential energy surface (PES) minima. The various geometrical and thermochemical parameters for the studied complex are obtained in the gas phase. The atomic charges at all the atoms were calculated using Mulliken Population Analysis, Hirshfeld Population Analysis and Natural Population Analysis. The charge distribution within the studied complex is explained on the basis of molecular electrostatic potential maps, the frontier molecular orbital analysis and conceptual-DFT-based reactivity (global as well as local) descriptors, using the finite difference approximation method. The nature of O–Sn, N–Sn, N → Sn and C–Sn bonds is discussed in terms of the conceptual-DFT-based reactivity descriptors. The structural analysis of the studied complex has been carried out in terms of the selected bond lengths and bond angles. The structural and atomic charge analysis suggests a distorted trigonal bipyramidal arrangement consisting of negatively charged centres around the positively charged central Sn atom.

High-order solvers for space-fractional differential equations with Riesz derivative

This paper proposes the computational approach for fractional-in-space reaction-diffusion equation, which is obtained by replacing the space second-order derivative in classical reaction-diffusion equation with the Riesz fractional derivative of order \begin{document}$ α $\end{document} in \begin{document}$ (0, 2] $\end{document} . The proposed numerical scheme for space fractional reaction-diffusion equations is based on the finite difference and Fourier spectral approximation methods. The paper utilizes a range of higher-order time stepping solvers which exhibit third-order accuracy in the time domain and spectral accuracy in the spatial domain to solve some fractional-in-space reaction-diffusion equations. The numerical experiment shows that the third-order ETD3RK scheme outshines its third-order counterparts, taking into account the computational time and accuracy. Applicability of the proposed methods is further tested with a higher dimensional system. Numerical simulation results show that pattern formation process in the classical sense is the same as in fractional scenarios.

Effect of continental and nearshore slopes on tsunami height

In the present study, an attempt is made to understand the effect of continental and nearshore slopes on the transmission, propagation and run-up of tsunami. A numerical model study is carried out using shallow water equations that are solved using Crank-Nicolson finite difference approximation method on a staggered grid. A rectangular solitary wave is considered to propagate over typical continental slopes and nearshore profiles available along the Indian coast. The amplification or attenuation of tsunami characteristics over these cross-sections was studied. It was observed that the tsunami run-up is altered with varying continental slope, water depth on continental shelf and nearshore slope. The study suggests that tsunami height on the continental shelf increases for shallow continental shelf depths. The percentage increase in tsunami height with decrease in continental shelf depth from 200 m to 50 m is found to be 31%. Higher tsunami run-up was observed for deeper continental shelf and steeper continental slope, whereas lower tsunami run-up for deeper continental shelf and flatter continental slope. It was observed that for a steep continental slope (1:0.1), a reduction in nearshore slope (steep to flat) increases tsunami run-up up to a slope of 1:632 and then decreases.

Augmented upwind numerical schemes for the groundwater transport advection‐dispersion equation with local operators

SummaryWhen solute transport is advection‐dominated, the advection‐dispersion equation approximates to a hyperbolic‐type partial differential equation, and finite difference and finite element numerical approximation methods become prone to artificial oscillations. The upwind scheme serves to correct these responses to produce a more realistic solution. The upwind scheme is reviewed and then applied to the advection‐dispersion equation with local operators for the first‐order upwinding numerical approximation scheme. The traditional explicit and implicit schemes, as well as the Crank‐Nicolson scheme, are developed and analyzed for numerical stability to form a comparison base. Two new numerical approximation schemes are then proposed, namely, upwind–Crank‐Nicolson scheme, where only for the advection term is applied, and weighted upwind‐downwind scheme. These newly developed schemes are analyzed for numerical stability and compared to the traditional schemes. It was found that an upwind–Crank‐Nicolson scheme is appropriate if the Crank‐Nicolson scheme is only applied to the advection term of the advection‐dispersion equation. Furthermore, the proposed explicit weighted upwind‐downwind finite difference numerical scheme is an improvement on the traditional explicit first‐order upwind scheme, whereas the implicit weighted first‐order upwind‐downwind finite difference numerical scheme is stable under all assumptions when the appropriate weighting factor (θ) is assigned.

NUMERICAL SIMULATION OF ULTRASONIC DETECTION FOR CONCRETE STRUCTURE BASED ON EQUIVALENT OFFSET MIGRATION

Ultrasonic wave testing is a classic Non-destructive testing (NDT) method to detect, locate and monitor the crack/fracture in construction materials. However, it is still hard to examine those small abnormal bodies since effective reflected signal from abnormity is usually rather weak. In this paper, a new ultrasound imaging technique, equivalent offset migration (EOM), is studied to demonstrate the feasibility and applicability for detecting concrete cracks. Thus, a complex numerical model along with six small scale flaws was built, and then the ultrasonic wave propagation in concrete was modeled by high order finite difference approximation method. Numerical simulation indicates that 1) there exists a strong scattering phenomenon while ultrasound propagates in concrete with multiple small scatter flaws, and 2) EOM is capable of imaging small flaws in concrete with high resolution and accuracy.

Qualitative analysis of MTEM response using instantaneous attributes

This paper introduces new technique for qualitative analysis of multi-transient electromagnetic (MTEM) earth impulse response over complex geological structures. Instantaneous phase and frequency attributes were used in place of the conventional common offset section for improved qualitative interpretation of MTEM data by obtaining more detailed information from the earth impulse response. The instantaneous attributes were used to describe the lateral variation in subsurface resistivity and the visible geological structure with respect to given offsets. Instantaneous phase attribute was obtained by converting the impulse response into a complex form using the Hilbert transform. Conversely, the polynomial phase difference (PPD) estimator was favored over the center finite difference (CFD) approximation method in calculating the instantaneous frequency attribute because it is computationally efficient and has the ability to give a smooth variation of the instantaneous frequency over a common offset section. The observed results from the instantaneous attributes were in good agreement with both the subsurface model used and the apparent resistivity section obtained from the MTEM earth impulse response. Hence, this study confirms the capability of both instantaneous phase and frequency attributes as highly effective tools for MTEM qualitative analysis.

Simultaneous identification of diffusion coefficient, spacewise dependent source and initial value for one‐dimensional heat equation

This paper deals with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and the initial value simultaneously for a one‐dimensional heat equation based on the boundary control, boundary measurement, and temperature distribution at a given single instant in time. By a Dirichlet series representation for the boundary observation, the identification of the diffusion coefficient and initial value can be transformed into a spectral estimation problem of an exponential series with measurement error, which is solved by the matrix pencil method. For the identification of the source term, a finite difference approximation method in conjunction with the truncated singular value decomposition is adopted, where the regularization parameter is determined by the generalized cross‐validation criterion. Numerical simulations are performed to verify the result of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.