Finite Difference Discretization Method Research Articles

Dynamics and bifurcations of a discrete‐time Lotka–Volterra model using nonstandard finite difference discretization method

A newly disclosed nonstandard finite difference method has been used to discretize a Lotka–Volterra model to investigate the critical normal form coefficients of bifurcations for both one‐parameter and two‐parameter bifurcations. The discrete‐time prey–predator model exhibits a variety of local bifurcations such as period‐doubling, Neimark–Sacker, and strong resonances. Critical normal form coefficients are determined to reveal dynamical scenarios corresponding to each bifurcation point. We also investigate the complex dynamics of the model numerically by Matlab package using MatcotM based on numerical continuation technique. The numerical continuation validates the theoretical analysis, which is discussed from an ecological perspective.

Mathematical and simulation-based analysis of the thermal and electrical performance of a flat plate solar air heater coupled with thermoelectric generators (TEGs)

The thermal performance of a flat plate solar air heater coupled with thermoelectric generators has been studied. A model of the collector was designed and simulations using SCILAB software were carried out to assess its performance. The design was carried out by analyzing the various heat and mass transfer processes which occurred in the components of the air heating system. The analysis resulted in the formulation of mathematical expressions that modeled the conversion of solar energy from the sun into thermal energy absorbed by the air flowing through the collector. The temperatures of the various components of the collector were determined using the finite forward difference discretization method. The cumulative energy generated by the heat source was 87 MJ while the energy gained by the heated exit air amounted to 37 MJ, bringing about a system thermal efficiency of 42%. The electrical efficiency of the model was 8.3%.

Far field calculation of distorted Gaussian beams.

Far field calculations of beams, such as laser beams, are often applied in optical engineering. Current beam propagation methods fail in certain range parameters due to high storage requirements of the algorithms. This paper presents a new beam propagation method for far field calculations of distorted Gaussian beams in a homogeneous medium including optical elements, such as lenses. The method works even in the case of a large distance from the observation plane to the beam waist and can be applied to larger divergence angles. This new simulation technique factors out the phase of the Gaussian TEM00 beam and solves the resulting partial differential equation by suitable finite difference or finite element discretization methods.

Transient behavior of imperfect bi-directional functionally graded sandwich plates under moving loads

An investigation of dynamic behaviors of a sandwich plate containing an imperfect two dimensional functionally graded (2D-FG) core surrounded by two faces on a two-parameter elastic foundation and subjected to a moving load is carried out in this paper. The present sandwich solid is composed of a porous 2D-FG core covered by two homogenous layers. It is assumed that the middle layer has micro voids dispersed uniformly and unevenly through the layer thickness. The fundamental equations are governed within the framework of first-order-shear deformation theory by utilizing Hamilton’s principle, von-Karman geometrical nonlinearity and the principal of mixtures. Newmark direct integration procedure is implemented to transform the dynamic equations into a static form and then the kinetic dynamic relaxation numerical technique in conjunction with the finite difference discretization method are employed to solve the nonlinear partial differential governing equations. Finally, the effects of porosity fraction and scattering patterns, boundary constrains, the variation of materials’ grading indexes and elastic foundation constants on the transient performances of the plate are studied in detail.

Performance Analysis of an Adsorption Refrigeration System Working on Activated Carbon/Methanol Pair Using Finned Tube Type Adsorber Bed

Adsorption refrigeration, being a unique and eco-friendly technology, has gained popularity over conventional refrigeration systems. The present study is aimed at developing an annular finned tube adsorber model which serves as a thermal compressor in adsorption refrigeration systems. The mathematical model is addressed numerically using finite difference discretization method and explicit scheme was used for the solution. The generalized model has been simulated for activated carbon–methanol working pair. The system has an optimum cycle time of 1800s. It was found to have a highest refrigeration capacity of 260.66 kJ/kg at a regeneration temperature of 393 K and evaporator temperature of 283 K. The highest COP (Coefficient of Performance) achieved by the system is 0.3706 at a regeneration temperature of 353 K and evaporator temperature of 283 K. A highest SCP (Specific Cooling Power) of 144.8 W/kg was obtained at an evaporator temperature of 283 K and regeneration temperature of 393 K.

Sensitivity Analysis and Joint Estimation of Parameters and States for All-Solid-State Batteries

All-solid-state batteries (ASSBs) are considered to be the next generation of lithium-ion batteries. Physics-based models (PBMs) can effectively simulate the internal electrochemical reactions and provide critical internal states for battery management. In order to promote the onboard applications of PBMs for ASSBs, in this article, the parameter sensitivity of a typical PBM is analyzed, and a joint estimation method for states and parameters based on sigma-point Kalman filtering (SPKF) is proposed. First, to obtain accurate sensitivity analysis results, approaches from different principles, including local sensitivity, elementary effect test, and variance-based methods, are applied. Then, for the battery model based on partial differential equations, a nonlinear state-space model is constructed by using the finite-difference discretization method. Finally, the SPKF algorithm is employed to conduct the joint estimation of model parameters and lithium-ion concentrations. The results from constant current and dynamic cycles show that two parameters, namely maximum lithium-ion concentration and minimum lithium-ion concentration, have the most influence on the model results. The joint estimation method is validated in three different cases, and the mean absolute errors of the estimated voltage and state of charge (SOC) are below 2.1 mV and 1.5%, respectively.

A PROPOSED METHODOLOGY TO EVALUATE SPECIFIC APPLICATIONS OF EXTENDED SURFACES IN INDUSTRIAL EQUIPMENT

The concept of Passive Cooling fits into a number of technological options involving constructive elements, whereby the heat exchange area of a surface being cooled by a surrounding fluid medium occurs. Thus, the increase in area results in an increase in the surface heat dissipation rate for the refrigerant, inferring only a single cost, i.e. the construction cost. Among the available technical possibilities, the use of extended surfaces, known as fins, stands out. The present work focuses on the use of fins on three very common electrical devices in the industrial environment, which present a great need for cooling.They are: three-phase transformer, frequency inverter and induction electric motor. The study is dedicated to a theoretical evaluation of the use of the fin profiles used in these equipments, since the manufacturers do not make their design data available to the public. Thus, a methodology was developed for this evaluation, including comparison of the results between the original profiles of these equipments with an alternative profile of similar design. Within the process of calculating the heat dissipation rates by the fins, the axial temperature distribution profiles were numerically determined using the Finite Difference discretization method, together with the Gauss-Seidel iterative method for the resolution of the discretized equations. The computational code was developed using MatLab® software. The results of the present work attested to the proper use of the original profiles employed by the manufacturers.

Nonlinear bending of functionally graded sandwich plates under mechanical and thermal load

In this paper, the nonlinear bending of circular/annular sandwich plates with functionally graded material face sheets is studied under mechanical and thermal loads. The governing equations are based on first-order shear theory and nonlinear strain- displacement relations of van Karman. Nonlinear equilibrium equations are obtained by combining the numerical method of dynamic relaxation with finite difference discretization method. Also, material properties are studied in two conditions including temperature dependent and temperature independent conditions. In order to verify the accuracy of the research, the results are compared with references as well as Finite element package of Abaqus. Finally, the effects of some parameters such as the thickness ratio of core to face sheet, boundary conditions, temperature and grading index on nonlinear bending are investigated.

A fully implicit mimetic finite difference scheme for general purpose subsurface reservoir simulation with full tensor permeability

In the previous article Abushaikha et al. (2017) [1], we presented a fully-implicit mixed hybrid finite element (MHFE) method for general-purpose compositional reservoir simulation. The present work extends the implementation for mimetic finite difference (MFD) discretization method. The new approach admits fully implicit solution on general polyhedral grids. The scheme couples the momentum and mass balance equations to assure conservation and applies a cubic equation-of-state for the fluid system. The flux conservativity is strongly imposed for the fully implicit approach and the Newton-Raphson method is used to linearize the system. We test the method through extensive numerical examples to demonstrate the convergence and accuracy on various shapes of polyhedral. We also compare the method to other discretization schemes for unstructured meshes and tensor permeability. Finally, we apply the method through applied computational cases to illustrate its robustness for full tensor anisotropic, highly heterogeneous and faulted reservoirs using unstructured grids.

Interface-resolved detailed numerical simulation of evaporating two-phase flows with robin boundary conditions on irregular domains

In this paper, our previous finite difference discretization method for Robin boundary conditions is implemented into an interface-resolved detailed numerical simulation framework to solve evaporation problems on an irregular interface. The framework utilizes a conservative level set method for describing the interface, a ghost fluid method for addressing the discontinuities on the interface, and a revised heat flux based model for predicting the evaporation rate to improve the accuracy, stability and applicability of the simulations. Specific care has been devoted to the accurate treatments of the involved complex boundary conditions on the interface. Numerical results are in excellent agreements with the analytical solutions, indicating the success of the methods for the above-mentioned sub-processes coupled with the fluid solver. The numerical framework is robust and achieves first-order overall convergence accuracy. Then, the non-uniform evaporation of single droplets is simulated to demonstrate the fascination of such an interface-resolved framework on in-depth understanding of physical mechanisms and therefore on modification of traditional evaporation models. Finally, rigorous validations are conducted in a three-dimensional configuration of droplet evaporation. In summary, the framework proposed in this paper represents a promising tool for non-uniform evaporation modelling, which will be applied to more practical problems in future work.

A finite difference discretization method for heat and mass transfer with Robin boundary conditions on irregular domains

This paper proposes a finite difference discretization method for simulations of heat and mass transfer with Robin boundary conditions on irregular domains. The level set method is utilized to implicitly capture the irregular evolving interface, and the ghost fluid method to address variable discontinuities on the interface. Special care has been devoted to providing ghost values that are restricted by the Robin boundary conditions. Specifically, it is done in two steps: 1) calculate the normal derivative in cells adjacent to the interface by reconstructing a linear polynomial system; 2) successively extrapolate the normal derivative and the ghost value in the normal direction using a linear partial differential equation approach. This method produces second-order accurate solutions for both the Poisson and heat equations with Robin boundary conditions, and first-order accurate solutions for the Stefan problems. The solution gradients are of first-order accuracy, as expected. It is easy to implement in three-dimensional configurations, and can be straightforwardly generalized into higher-order variants. The method thus represents a promising tool for practical heat and mass transfer problems involving Robin boundary conditions.

Modeling dual reflux-pressure swing adsorption processes: Numerical solution based on the finite volume method

Even if the suitability of Finite Volume discretization methods for Pressure Swing Adsorption modelling is well-established, when simulating the peculiar process called Dual Reflux-Pressure Swing Adsorption, simulations relying only on Finite Difference discretization methods are available in literature. Therefore, a detailed single-bed model of the Dual Reflux-Pressure Swing Adsorption process, the solution of which is based on the Finite Volume discretization method, is proposed in this work and validated against experimental data. Details of the system representation, the numerical schemes and the implementation of the boundary conditions are given. Moreover, a comparison between the performances and limitations of the Finite Volume and Finite Difference discretization methods when simulating a Dual Reflux-Pressure Swing Adsorption process are discussed through two case studies, leading to conclude that Finite Volume Method is the most effective approach for simulating this specific separation process.

Nutrient modeling for a semi-intensive IMC pond: an MS-Excel approach.

Semi-intensive Indian Major Carp (IMC) culture was practised in polythene lined dugout ponds at the Aquacultural Farm of Indian Institute of Technology, Kharagpur, West Bengal for 3 consecutive years at three different stocking densities (S.D), viz., 20,000, 35,000 and 50,000 numbers of fingerlings per hectare of water spread area. Fingerlings of Catla, Rohu and Mrigal were raised at a stocking ratio of 4:3:3. Total ammonia nitrogen (TAN) value along with other fishpond water quality parameters was monitored at 1 day intervals to ensure a good water ecosystem for a better fish growth. Water exchange was carried out before the TAN reached the critical limit. Field data on TAN obtained from the cultured fishponds stocked with three different stocking densities were used to study the dynamics of TAN. A developed model used to study the nutrient dynamics in shrimp pond was used to validate the observed data in the IMC pond ecosystem. Two years of observed TAN data were used to calibrate the spreadsheet model and the same model was validated using the third year observed data. The manual calibration based on the trial and error process of parameters adjustments was used and several simulations were performed by changing the model parameters. After adjustment of each parameter, the simulated and measured values of the water quality parameters were compared to judge the improvement in the model prediction. Forward finite difference discretization method was used in a MS-Excel spreadsheet to calibrate and validate the model for obtaining the TAN levels during the culture period. Observed data from the cultured fishponds of three different S.D were used to standardize 13 model parameters. The efficiency of the developed spreadsheet model was found to be more than 90% for the TAN estimation in the IMC cultured fishponds.

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection and the square phase field crystal model – are carried out to verify the efficiency of the scheme.

Simulation of DFIG wind turbines for transient studies: An alternative approach based on symbolic–numeric computations

This paper describes an alternative approach based on symbolic and numeric computations to simulate wind turbines equipped with doubly fed induction generator (DFIG). The actuator disk theory is used to represent the aerodynamic part, and the one-mass model simulates the mechanical part. The 5th-order induction generator is selected to model the electric machine, being this approach suitable to estimate the DFIG performance under transient conditions. The corresponding non-linear integro-differential equation system has been reduced to a linear state-space system by using an ad hoc local linearization. This novel symbolic–numeric computation (SNC) method has been implemented by using two different software-packages, with the purpose of solving simultaneously a remarkable number of individual wind turbine models submitted to different wind speed profiles and/or grid voltage waveforms.The obtained results are compared with traditional finite difference discretization (FDD) methods, widely proposed for this type of studies. The results offer a remarkable agreement between the proposed SNC method and the FDD solutions, considering variations of both wind speed profiles and electrical transient events. In this way, real voltage dips collected in Spanish wind farms are used as input voltage waveforms to evaluate the suitability of the proposed solution under transient disturbances when a large amount of individual wind turbines are simultaneously simulated. In this way, a comparison of computational time costs considering different number of wind turbines and integration time-steps is also included in the paper. From this comparison, the proposed SNC method presents clear advantages in terms of computational time cost requirements under both steady-state and transient studies.

Evaluation of discretization and integration methods for the analysis of finite hydrodynamic bearings with surface texturing

Efficient numerical methods are essential in the analysis of finite hydrodynamic bearings with surface texturing. This is especially evident in optimization and parametric studies where the discretization and integration methods are used to solve the governing two-dimensional Reynolds equation multiple times. Performance comparison studies of the methods are thus required to select the method that is most suitable for a particular bearing geometry. In this work, we conduct a comprehensive and systematic comparison of typical implementations of the finite difference, finite volume, finite element and spectral element discretization methods together with the Newton–Cotes formula and Gauss quadratures for hydrodynamic bearings governed by the two-dimensional Reynolds equation. The methods were evaluated by comparing the approximation errors in the calculation of the maximum pressure, load capacity, coefficient of friction, and minimum film thickness for parallel and convergent bearings textured with elliptical grooves or trapezoidal dimples. The number of degrees of freedom required by the methods to achieve the error cut-off values of 5%, 1% and 0.1% were calculated. Our results demonstrate that the spectral element method uses up to 72 times fewer degrees of freedom than the other methods for the same cut-off values. Also, our study revealed that the shape of the groove/dimple and the bearing convergence ratio can have a significant effect on the approximation errors of the numerical methods used. Specifically, for piecewise-linear texture features (e.g. trapezoidal dimples) and positive convergence ratio it is easier, for the methods, to accurately approximate the solution. In such cases, the finite volume and finite element methods are reasonable choices and provide a good trade-off between the ease of implementation and approximation errors. The worse performance was observed for the finite difference and thus this method is not recommended when the computational efficiency and the accuracy of results are of importance.

Photorealistic visualization and fluid animation: coupling of Maya with a two-phase Navier-Stokes fluid solver

We have coupled the three-dimensional solver for the two-phase incompressible Navier-Stokes equations NaSt3DGPF with Autodesk Maya. Maya is the industry standard software framework for the creation of three-dimensional animations. The parallel level-set-based fluid solver NaSt3DGPF simulates the interaction of two fluids like air and water. It uses high-order finite difference discretization methods that are designed for physics applications. By coupling both applications, we are now able to set up scientific fluid simulations in an easy-to-use user interface. Moreover, the rendering techniques provided by Maya allow us to create photorealistic visualizations for computational fluid dynamics problems and support the creation of highly visually realistic fluid simulations for animation movies. Altogether, we obtain an easy usable and fully coupled fluid animation toolkit for two-phase fluid simulations. These are the first published results of the full integration of a physics-oriented, high-order grid-based parallel two-phase fluid solver in Maya, at least to our knowledge.

Equivalent-Circuit Modeling of Nonradiative Surface Plasmon Energy Transfer Along the Metallic Nanowire

An analytical methodology for establishing an equivalent-circuit network of nonradiative surface plasmon (SP) energy transport along the metallic nanowire (MNW) is presented. To find out the passive elements for MNW, the SP dispersion and damping relation through modified Bessel function electromagnetic (EM) field expansion was derived, thus demonstrating the low-pas transmission-line (TL) model. Specifically, the low-pass TL parameters, such as series impedance (Z) and shunt admittance ( Y) can be calculated based on the lumped-element model and harmonic-voltage (current) distribution. Furthermore, the equivalent-circuit parameters, such as resistance (R), inductance (L), capacitance (C) and conductance (G), are obtained by employing the finite difference (FD) discretization method such as T-cell RLCG networks. These equivalent-circuit elements can be verified by the HSPICE circuit simulation and 3-D scattered finite-difference time-domain (FDTD) method. Finally, the parallel MNWs are modeled as equivalent-circuit networks by using the electrostatic coupling.

3D edge energy transport in stellarator configurations

The finite difference discretization method is used to solve the electron energy transport equation in complex 3D edge geometries using an unstructured grid. This grid is generated by field-line tracing to separate the radial and parallel fluxes and minimize the numerical diffusion connected with the strong anisotropy of the system. The influence of ergodicity on the edge plasma transport in the W7-X stellarator is investigated in this paper. Results show that the combined effect of ergodicity and the radial plasma diffusion leads to the efficient smoothing of the temperature profiles in the finite-β case.

Finite difference scheme for solving general 3D convection–diffusion equation

The effect of ergodisation (either by additional coils like in the TEXTOR-DED tokamak or by intrinsic plasma effects like in the W7-X stellarator) defines the need for transport models being able to describe this properly. A prerequisite for this is the concept of local magnetic coordinates allowing a correct discretization with mimimized numerical errors. In the present paper we employ the finite difference discretization method which allows the numerical simulation of energy transport in complex 3D edge geometries (in particular for W7-X) using a custom-tailored unstructured grid in local magnetic coordinates. This grid is generated by field-line tracing to guarantee an exact discretization of the dominant parallel transport. Therefore the radial and parallel fluxes are almost perfectly separated which minimizes the numerical diffusion connected with strong anisotropy of the system ( D ||/ D ⊥∼10 7). In addition, the parallel and radial directions can be treated independently in the numerical method. Along the magnetic field lines we use the standard Patankar concept to discretize the convection–diffusion equation. In order to solve a quasi-isotropic problem in a plane (toroidal cut), we modified the finite volume method to obtain its finite difference representation. The radial fluxes on plasma cross-sections are interpolated using a constrained Delaunay triangulation (keeping the structural information for magnetic surfaces if they exist). The general concept of an upwind scheme is used to discretize the convective terms. The first tests for W7-X geometry were successfully performed.