Finite Difference Procedure Research Articles

Numerical Prediction of The Gas Flows in a Reciprocating Engine Cylinder during Intake Stoke (1st Report: Inviscid Flow).(Dept.M)

This paper presents a method for predicting the in-cylinder gas flows during the intake stroke of four-stroke engines. A simplified engine model with a deep-bowl piston configuration, and a central intake valve was used. An inviscid two-dimensional axisymmetric flow was investigated. The model used a finite difference procedure with an expanding/contracting grid in axisymmetric representation. The major source of troubles for the convergence of the solution was the unsteady turbulent jet flow issuing from the intake valve which required very small time intervals for the solution compared with that during the compression stroke where there is no jet flow exists. Results show a complicated flow field during the intake stroke in engine like-geometries.

Solving nonlinear differential equations with differentiable quantum circuits

We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations, and compute density, temperature and velocity profiles for the fluid flow in a convergent-divergent nozzle.

A localisation technique based on radial basis function partition of unity for solving Sobolev equation arising in fluid dynamics

This paper develops a numerical approach for finding the approximate solution of the Sobolev model. This model describes many natural processes, such as thermal conduction for different media and fluid evolution in soils and rocks. The proposed method approximates the unknown solution with the help of two main stages. At a first stage, the time discretization is performed by means of a second-order finite difference procedure. At a second stage, the space discretization is accomplished using the local radial basis function partition of unity collocation method based on the finite difference (LRBF-PUM-FD). The major disadvantage of global techniques is the high computational burden of solving large linear systems. The LRBF-PUM-FD significantly sparsifies the linear system and reduces the computational burden, while simultaneously maintaining a high accuracy level. The time-discrete formulation is studied in terms of the stability and convergence analysis via the energy method. Three examples are illustrated to verify the efficiency and accuracy of the method.

Development of Experimental and Numerical Methods for the Analysis of Active Clearance Control Systems1

Abstract The ever increasing performance requirements of modern aeroengines necessitate the development of effective ways to improve efficiency and reduce losses. Casing temperature control is particularly critical from this point of view, since thermal expansion directly affects the blade tip clearance and thus the associated leakages. To limit the turbine tip flows, active clearance control (ACC) systems have been implemented over the last decades. These systems are usually based upon impingement cooling, generated by a series of perforated manifolds enclosing the turbine casing. When dealing with aeroengine low pressure turbines, the current trend in increasing the engine bypass ratio, so as to enhance the system propulsive efficiency, pushes the limits of ACC traditional design performance. The reduction of the pressure head at the ACC system inlet requires lower nozzle-to-target distances as well as denser impingement arrays to compensate the reduction of the jets' Reynolds number. Literature correlations for the impingement heat transfer coefficient estimation are then out of their confidence range and also RANS numerical approaches appear not suitable for future ACC designs. In this work, methodologies for the development of accurate and reliable tools to determine the heat transfer characteristics of low pressure ACC systems are presented. More precisely, this paper describes a custom designed finite difference procedure capable of solving the inverse conduction problem on the target plate of a test sample. The methodology was successfully applied to an experimental setup for the measurement of the thermal loads on a target plate of a representative low pressure ACC impinging system. The experimental outcomes are then used to validate a suitable numerical approach. Results show that RANS model is not able to mimic the experimental trends, while scale-resolving turbulence models provide a good reconstruction of the experimental evidences, thus allowing to obtain a correct interpretation of flow and thermal phenomena for ACC systems.

The application of singular value decomposition to electric field propagation

We demonstrate that the singular value decomposition algorithm can be combined with the fast Fourier transform or finite difference procedures to generate straightforward and accurate one-way Helmholtz equation electric field propagation methods.

Numerical modeling of NPZ and SIR models with and without diffusion

In this paper, the two biological models i.e. Nitrogen, Phytoplankton and Zooplankton (NPZ) and whooping cough SIR models (Charpentier et al., 2010) are being modified and solved numerically by finite difference and meshless methods. Diffusion process has been added to the existing models (Charpentier et al., 2010) so that a unidimensional movement of three species can be incorporated in the models. The effects of diffusion has been studied in both the models. An operator splitting method coupled with the meshless and finite difference procedures, is being considered for numerical solution of the two biological models with and without diffusion. A one step explicit meshless procedure is also applied for the numerical solution of the nonlinear models. The NPZ model contains the concentration of Nitrogen, Phytoplankton and Zooplankton and the whooping cough model contains susceptible, infected, and recovered classes of the population. Equilibrium points of both models have been investigated. Stability of equilibrium points regarding SIR model has been studied. The basic reproduction number of SIR model is also determined. Due to non-availability of the exact solution, the numerical results obtained are mutually compared and their correctness is being verified by the theoretical results as well.

Turbulent liquid film evaporation in a partially heated wall along a vertical tube

AbstractIn this numerical study, the evaporative heat and mass transfer of a turbulent falling liquid film in a finite vertical tube are investigated. The liquid film flows in the tube's inner wall, whose outer wall is partially subject to thermal flux. Here, different configurations corresponding to thermal flux imposed on different external surface wall percentages are examined. External face zones where the heat flux is not applied are maintained insulated. The nonlinear set of parabolic mass, momentum, energy, and mass fraction conservation equations combined with boundary and interfacial conditions are treated numerically using implicit finite difference procedure. For falling liquid film analysis, an adapted Van Driest turbulence model is used. For the present work, it is supposed that gas flows in a laminar regime. We examine in this paper the impact of the percentage of heated surface area on flows as well as on heat and mass transfer. Obtained results for a partially heated wall are compared with those produced for an entirely heated wall.

Effect of window-to-wall-area ratio on thermal performance of building wall materials in Elazığ, Turkey.

In this study, the effect of glazing-to-total-wall-area ratio on the thermal performance of different wall materials is numerically investigated in terms of heat transmission load. The investigation was performed for a South-facing wall in Elazığ, Turkey. The heat transmission load through walls and windows are determined separately for summer and winter climate conditions. In this analysis, the frame area of the window is not considered. Therefore, whereas the glazing area on uninsulated and insulated walls is increased from 0% to 100%, the heat gain and losses are calculated separately according to the glazing type. The transmission loads through the wall are determined by an implicit finite difference procedure under steady periodic conditions. Concrete, briquettes, bricks, and autoclaved aerated concrete are selected as structure materials. Results show that in the uninsulated wall, the wall material affected the glazing area, whereas in the insulated wall, the effect of wall material on glazing area is insignificant.

Application of Finite Difference Method in Multifractal Techniques to Identify Pb–Zn Anomalies: a Case Study from Khomein, Iran

Separation of geochemical anomalies from background is an important and essential issue in the exploration of mineral deposits. Statistical methods are used to separation of anomaly from background. Fractal/multifractal and singularity methods are very suitable for the separation of weak anomalies. For interpolation, various methods could be used such as most distance-weighting methods, kriging, spline interpolation, interpolating polynomials, and finite difference methods. This paper addressed application of multifractal techniques together with finite difference procedure to detect the anomalous regions. Fractal models such as concentration–area, spectrum–area, and singularity index model were selected to identify Pb–Zn anomalies. In this research, 170 samples from 33 boreholes were taken in the Haft-Savaran area and analyzed for determination of Pb–Zn anomaly. Singularity method was a suitable method for separating Pb and Zn anomalies. Also, the results showed that the geochemical anomaly has occurred along NE–SW trend and is consistent with the sulfide mineralization process and the faults of the region.

Second-Order Finite Difference/Spectral Element Formulation for Solving the Fractional Advection-Diffusion Equation

The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equation. The time variable has been discretized by a second-order finite difference procedure. The stability and the convergence of the semi-discrete formula have been proven. Then, the spatial variable of the main PDEs is approximated by the spectral element method. The convergence order of the fully discrete scheme is studied. The basis functions of the spectral element method are based upon a class of Legendre polynomials. The numerical experiments confirm the theoretical results.

A finite-difference procedure to solve weakly singular integro partial differential equation with space-time fractional derivatives

The main aim of the current paper is to propose an efficient numerical technique for solving space-time fractional partial weakly singular integro-differential equation. The temporal variable is based on the Riemann–Liouville fractional derivative and the spatial direction is based on the Riesz fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order $${{\mathcal {O}}}(\tau ^{\frac{3}{2}})$$. Also, the space variable is discretized using a finite difference scheme with second-order accuracy. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency, applicability and simplicity of the proposed technique.

A complete in phase FinitDiff procedure for DiffEquns in chemistry

A new FinitDiff ($$=$$ Finite Difference) procedure is obtained for the efficacious solution of the DiffEquns ($$=$$ Differential Equations) in quantum chemistry.

Improved finite difference analysis of dynamic responses of concrete members reinforced with FRP bars under explosion

An improved finite difference (FD) procedure to predict the dynamic response of concrete members reinforced with fiber reinforced polymer (FRP) bars under explosion is proposed. This method no longer needs the empirical formulae of the moment-curvature relationship, which is only suitable for concrete members reinforced with steel bars but inappropriate for those reinforced with FRP bars. The equilibrium of the horizontal force of the structural element, considering the strain rate effects of the concrete and the reinforcing bars, is established to calculate the depth of the neutral axis based on a numerical solution of nonlinear equations. A layered-section-based analysis model is used to calculate the strain rate effect of the compressive concrete. The Euler-Bernoulli’s beam theory is solved numerically using an explicit FD scheme. The accuracy of the FD model is validated by the experiments of the concrete members reinforced with steel bars and FRP bars, respectively. The FD results are also compared with those obtained by single-degree-of-freedom (SDOF) analysis and finite element (FE) analysis, respectively. The FD procedure is accurate, applicable and convenient to predict the dynamic responses of the flexural concrete members reinforced with FRP bars, as well as those of the members reinforced with steel bars.

Dynamic Sensitivity Analysis Based on Sherman–Morrison–Woodbury Formula

The sensitivity analysis plays a significant role in structural optimization, and repetitive calculations are necessary for the analysis procedure. A reanalysis approach of frequency response functions (FRFs) is proposed based on the Sherman–Morrison–Woodbury (SMW) formula. Multiple perturbations are considered as the change of elements in the dynamic stiffness matrix, and the SMW formula is used as an efficient mathematical tool for solving the problem of repetitive calculations. The perturbed FRFs are directly calculated by the SWM formula, and then the sensitivity of the FRFs is obtained by using the finite difference procedure. The basic theory of the proposed method is illustrated by adopting a mass–spring system, which is extended to truss structures. A five-bay truss and a cantilever plate are adopted to verify the method. The lumped mass, the damping, and the cross-sectional area can be regarded as design variables. Results show that the FRF sensitivity is obtained efficiently by using the initial FRFs and parameter perturbations to avoid repeated finite element analyses, and the FRF sensitivity of simultaneously perturbed multiparameters can be determined by using the proposed method.

On the convergence analysis of a hybrid numerical method for multicomponent transport in porous media

In this article, the convergence of a hybrid numerical method introduced in Daripa & Dutta (2017) [11] has been established. This method integrates a discontinuous finite element method with a modified method of characteristics (MMOC) in combination with finite difference (FD) procedures, and has been successfully applied to solve a coupled system of nonlinear equations that arises in multicomponent two-phase porous media flows. The present convergence analysis is focused on the MMOC-FD procedure for a nonlinear system of transport equations. For this purpose, an analogous single-component system of transport equations has been considered and some key ideas for possible extension to multicomponent systems have been briefly discussed. Error estimates have been obtained and these estimates have also been shown to be consistent with realistic numerical simulations of flows arising in enhanced oil recovery processes.

Green methanol synthesis process from carbon dioxide via reverse water gas shift reaction in a membrane reactor

In the present work, carbon dioxide hydrogenation to methanol via reverse water gas shift reaction (CAMERE) in an industrial scale was modeled and optimized. In this process, syngas is first produced by CO2 hydrogenation through reverse water gas shift (RWGS) reaction over Ni/Al12O19 catalyst and next the syngas is conveyed to a reactor as the feedstock to produce methanol. The inner tubes of methanol synthesis reactor were coated by a water perm-selective membrane for removal of H2O, as the cause of catalyst poisoning. A precise two-dimensional model solved by finite-difference procedure was employed to evaluate both RWGS and methanol synthesis membrane reactors performance. Also, the operating conditions of the RWGS reactor were optimized by differential evolution (DE) technique to gain a maximum methanol production rate. Moreover, the results of the methanol production reactor from the CAMERE process were compared with the conventional route (CR) in which methanol is produced from coal and natural gas. In the case of methanol synthesis membrane reactor, CAMERE process was superior to CR due to achieving 20.8% increase in methanol production rate. The results of this process modeling provide a good initial insight for green methanol production form indirect CO2 conversion.

Simultaneous impacts of MHD and variable wall temperature on transient mixed Casson nanofluid flow in the stagnation point of rotating sphere

A numerical analysis is provided to scrutinize time-dependent magnetohydrodynamics (MHD) free and forced convection of an electrically conducting non-Newtonian Casson nanofluid flow in the forward stagnation point region of an impulsively rotating sphere with variable wall temperature. A single-phase flow of nanofluid model is reflected with a number of experimental formulae for both effective viscosity and thermal conductivity of nanofluid. Exceedingly nonlinear governing partial differential equations (PDEs) subject to their compatible boundary conditions are mutated into a system of nonlinear ordinary differential equations (ODEs). The derived nonlinear system is solved numerically with implementation of an implicit finite difference procedure merging with a technique of quasi-linearization. The controlled parameter impacts are clarified by a parametric study of the entire flow regime. It is depicted that from all the exhibited nanoparticles, Cu possesses the best convection. The surface heat transfer and surface shear stresses in the x - and z -directions are boosted with maximizing the values of nanoparticle solid volume fraction φ and rotation λ . Besides, as both the surface temperature exponent n and the Casson parameter γ upgrade, an enhancement of the Nusselt number is given.

Response sensitivity analysis of laminated composite shells based on higher-order shear deformation theory

Laminated composite shells are widely used as structural components in important aerospace, marine, automobile engineering structures. Thus, appropriate evaluation of sensitivities of responses like deflection, frequency, buckling etc. due to changes in design variables is of great importance for efficient and safe design of such structures. The present paper deals with a comprehensive sensitivity analysis of laminated composite shells using $${C}^{0 }$$ finite element with more accurate theoretical model based on higher-order shear deformation theory (HSDT). The sensitivity analysis of deflection and natural frequency with respect to important design parameters such as material parameters, angle of fiber orientation, radius of curvature, density of materials and external load is presented. Furthermore, sensitivity-based importance factor for each parameter is obtained so that the most important parameters affecting the shell responses can be readily identified. The response sensitivities obtained by the proposed formulation are compared with those obtained by the finite difference procedure. An extensive parametric study has been carried out considering different variables to understand the performance of laminated shell.

Inhomogeneous dielectrics: conformal mapping and finite-element models

Abstract Field singularities in electrostatic and magnetostatic fields require special attention in field calculations, and today finite element methods are normally used, both in homogeneous and in inhomogeneous dielectric cases. Conformal mappings are a traditional tool in the homogeneous case, but two-stage Schwarz-Christoffel + Finite Difference procedures have been proposed for a long time to solve problems in case of inhomogeneous dielectric materials too. This allowed to overcome accuracy problems caused by convex corners in the domain boundary and relevant field singularities, and to easily apply finite difference (FD) solvers in rectangular domains. In this paper, compound procedures Schwarz-Christoffel + Finite Elements Method procedures are suggested, to improve both the accuracy and the speed of second stage calculations. The results are compared to Schwarz-Christoffel + Finite Difference and to direct finite-element calculations, and the small differences analyzed considering a well know case study geometry, i.e., a shielded dielectric-supported stripline geometry.

The Informational Content of Infinite Maturity Bonds

This paper investigates how the market implicitly prices very long-dated cash-flows. We do this empirically, using daily market data on the 3½% War Loan, a UK infinite maturity coupon bond. We price the War Loan as an American option with 100 years of maturity. To this end, we perform daily calibrations to observed swaption prices using the Hull-White (1990) one-factor term structure model. The mean-reversion and volatility parameters resulting from the calibrations are then used in a finite difference method, which is equivalent to the Hull-White (1994 and 1996) trinomial tree procedure. We calculate the constant long-term yield beyond the last observed yield, which equates the results of the finite difference procedure with the observed prices of the War Loan. We find that prior to the financial crisis the constant long-term yield was situated above the last observed yield, whereas after the crisis it was situated below the last observed yield.