Finite Difference Formulation Research Articles

LARGE EDDY SIMULATION OF TURBULENT CHANNEL FLOW USING ALGEBRAIC WALL MODEL

A large eddy simulation (LES) of a turbulent channel flow is performed by using the third order low-storage Runge-Kutta method in time and second order finite difference formulation in space with staggered grid at a Reynolds number, <TEX>$Re_{\tau}=590$</TEX> based on the channel half width, <TEX>${\delta}$</TEX> and wall shear velocity, <TEX>$u_{\tau}$</TEX>. To reduce the calculation cost of LES, algebraic wall model (AWM) is applied to approximate the near-wall region. The computation is performed in a domain of <TEX>$2{\pi}{\delta}{\times}2{\delta}{\times}{\pi}{\delta}$</TEX> with <TEX>$32{\times}20{\times}32$</TEX> grid points. Standard Smagorinsky model is used for subgrid-scale (SGS) modeling. Essential turbulence statistics of the flow field are computed and compared with Direct Numerical Simulation (DNS) data and LES data using no wall model. Agreements as well as discrepancies are discussed. The flow structures in the computed flow field have also been discussed and compared with LES data using no wall model.

Graph-theoretic sensitivity analysis of multi-domain dynamic systems: theory and symbolic computer implementation

A graph-theoretic formulation for the sensitivity analysis of multi-domain dynamic systems is presented in this article. In this formulation, a linear graph is used to capture the system topology and a graph-theoretic formulation is used to generate the system equations and the sensitivity equations simultaneously from the linear graph. Symbolic generation ensures automated, accurate, and efficient generation of sensitivity equations. The proposed method of sensitivity analysis avoids differentiation of complicated algebraic expressions, and the corresponding equation swells, by using pre-formed (and stored) expressions that are combined according to the topology captured by the linear graph. To illustrate the application of graph-theoretic sensitivity analysis to multi-domain systems, an electric motor-driven slider-crank mechanism and an automotive torque converter are considered as examples. The generation of the sensitivity equations is demonstrated, and the results are validated using a finite difference formulation. Also, the efficiencies of the generated equations are gauged by measuring the associated computational costs. Finally, some features of multi-domain multibody systems that pose unique challenges to the theory and software implementation are presented, along with a brief description of the symbolic software platform that has been used to implement the algorithm.

Time-domain numerical scheme based on low-order partial-fraction models for the broadband study of frequency-dispersive liquid crystals

A novel finite-difference time-domain formulation for the modeling of electromagnetic wave propagation in frequency-dispersive liquid crystals with random orientation is presented. The dispersion of the complex ordinary and extraordinary permittivities of nematic liquid crystals is described by a generalized model based on the sum of partial fractions. The proposed dispersive model encompasses traditional approaches, such as Drude, Drude–Lorentz, and modified-Lorentz functions; it can also capture arbitrary dispersion properties of experimentally characterized nematic mixtures via the vector fitting technique. The accuracy of the formulation is demonstrated in a series of benchmark examples in optical and terahertz frequencies.

Buckling of Nonlocal Columns with Allowance for Selfweight

AbstractThis paper presented new analytical solutions for the elastic buckling of Eringen’s nonlocal columns with allowance for selfweight. A discrete column model on the basis of the central finite difference formulation and the equivalent Hencky bar-chain model also were presented for this buckling problem. The discrete column model allows one to determine the buckling solutions of columns constructed from repetitive cells. In addition, one may use the discrete buckling solutions to calibrate the small-length scale coefficient e0 in the Eringen’s nonlocal column model. It was found that e0 values varied from 0.289 to 0.373 with respect to increasing selfweight for clamped-free column case, from 0.289 to 0.276 for the pinned-pinned column case, and from 0.289 to 0.281 for the clamped-clamped column case.

Parameterized reduced order models from a single mesh using hyper-dual numbers

In order to assess the predicted performance of a manufactured system, analysts must consider random variations (both geometric and material) in the development of a model, instead of a single deterministic model of an idealized geometry with idealized material properties. The incorporation of random geometric variations, however, potentially could necessitate the development of thousands of nearly identical solid geometries that must be meshed and separately analyzed, which would require an impractical number of man-hours to complete. This research advances a recent approach to uncertainty quantification by developing parameterized reduced order models. These parameterizations are based upon Taylor series expansions of the system׳s matrices about the ideal geometry, and a component mode synthesis representation for each linear substructure is used to form an efficient basis with which to study the system. The numerical derivatives required for the Taylor series expansions are obtained via hyper-dual numbers, and are compared to parameterized models constructed with finite difference formulations. The advantage of using hyper-dual numbers is two-fold: accuracy of the derivatives to machine precision, and the need to only generate a single mesh of the system of interest. The theory is applied to a stepped beam system in order to demonstrate proof of concept. The results demonstrate that the hyper-dual number multivariate parameterization of geometric variations, which largely are neglected in the literature, are accurate for both sensitivity and optimization studies. As model and mesh generation can constitute the greatest expense of time in analyzing a system, the foundation to create a parameterized reduced order model based off of a single mesh is expected to reduce dramatically the necessary time to analyze multiple realizations of a component׳s possible geometry.

Numerical simulation of premixed H2–air cellular tubular flames

The detailed flame structure of laminar premixed cellular flames in the tubular domain is simulated in 2D using a fully-implicit primitive variable finite difference formulation that includes multicomponent transport and detailed chemical kinetics. Numerical results for H2/air flames are presented and compared against spatially resolved experimental measurements of temperature and chemical species including atomic H and OH. The experimental results compare well for flame structure and cell number, despite the numerical model under-predicting the peak temperature by 200 K. Numerical experiments were performed to assess the ability for cellular tubular flames to impact experimental and numerical investigations of practical flames. The cellular flame structure is found to provide a highly sensitive geometry that is useful for validating diffusive transport modelling approximations. This capability is exemplified through the development of a simple and accurate approximation for thermal diffusion (i.e. the Soret effect) that is suitable for practical combustion codes.

Minimum divergence viscous flow simulation through finite difference and regularization techniques

We develop a new algorithm to simulate single- and two-phase viscous flow through a three-dimensional Cartesian representation of the porous space, such as those available through X-ray microtomography. We use the finite difference method to discretize the governing equations and also propose a new method to enforce the incompressible flow constraint under zero Neumann boundary conditions for the velocity components. Finite difference formulation leads to fast parallel implementation through linear solvers for sparse matrices, allowing relatively fast simulations, while regularization techniques used on solving inverse problems lead to the desired incompressible fluid flow.Tests performed using benchmark samples show good agreement with experimental/theoretical values. Additional tests are run on Bentheimer and Buff Berea sandstone samples with available laboratory measurements. We compare the results from our new method, based on finite differences, with an open source finite volume implementation as well as experimental results, specifically to evaluate the benefits and drawbacks of each method. Finally, we calculate relative permeability by using this modified finite difference technique together with a level set based algorithm for multi-phase fluid distribution in the pore space. To our knowledge this is the first time regularization techniques are used in combination with finite difference fluid flow simulations.

Scale effects in the static response of a one-dimensional quasi-brittle damage lattice

In this paper, we investigate the failure of a discrete elastic-damage axial system using both a discrete and an equivalent continuum approach. The microstructured damage chain consists of a one-dimensional damage lattice with direct nearest–neighbour interactions, which is composed of a series of periodic elastic-damage springs (axial lattice system treated in terms of discrete damage mechanics). We show that the damage lattice equations are equivalent to the centred finite difference formulation of a continuum damage mechanics (CDM) evolution problem. Such a discrete damage system reveals some scale effects on both the structural strength and stiffness. The nonlocal CDM in the hardening branch and cohesive damage model in the softening branch considered here are built using a continualisation procedure applied to the nonlinear difference equations of the lattice system. With this procedure, the difference equations to be solved are approximated by higher order differential equations. Using a rational asymptotic method, the continualised model appears to be equivalent to a nonlocal CDM model in the damage propagation zone. A finite length cohesive model is obtained in the softening range. A comparison of the discrete and the continuous problems for damage chains brings out the effectiveness of the new micromechanics-based nonlocal and cohesive continuum damage model, especially for capturing scale effects.

3D electromagnetic simulation of spatial autoresonance acceleration of electron beams

The results of full electromagnetic simulations of the electron beam acceleration by a TE112 linear polarized electromagnetic field through Space Autoresonance Acceleration mechanism are presented. In the simulations, both the self-sustaned electric field and selfsustained magnetic field produced by the beam electrons are included into the elaborated 3D Particle in Cell code. In this system, the space profile of the magnetostatic field maintains the electron beams in the acceleration regime along their trajectories. The beam current density evolution is calculated applying the charge conservation method. The full magnetic field in the superparticle positions is found by employing the trilinear interpolation of the mesh node data. The relativistic Newton-Lorentz equation presented in the centered finite difference form is solved using the Boris algorithm that provides visualization of the beam electrons pathway and energy evolution. A comparison between the data obtained from the full electromagnetic simulations and the results derived from the motion equation depicted in an electrostatic approximation is carried out. It is found that the self-sustained magnetic field is a factor which improves the resonance phase conditions and reduces the beam energy spread.

Well-Balanced, Conservative Finite Difference Algorithm for Atmospheric Flows

The numerical simulation of meso-, convective-, and microscale atmospheric flows requires the solution of the Euler or the Navier–Stokes equations. Nonhydrostatic weather prediction algorithms often solve the equations in terms of derived quantities such as Exner pressure and potential temperature (and are thus not conservative) and/or as perturbations to the hydrostatically balanced equilibrium state. This paper presents a well-balanced, conservative finite difference formulation for the Euler equations with a gravitational source term, where the governing equations are solved as conservation laws for mass, momentum, and energy. Preservation of the hydrostatic balance to machine precision by the discretized equations is essential because atmospheric phenomena are often small perturbations to this balance. The proposed algorithm uses the weighted essentially nonoscillatory and compact-reconstruction weighted essentially nonoscillatory schemes for spatial discretization that yields high-order accurate solutions for smooth flows and is essentially nonoscillatory across strong gradients; however, the well-balanced formulation may be used with other conservative finite difference methods. The performance of the algorithm is demonstrated on test problems as well as benchmark atmospheric flow problems, and the results are verified with those in the literature.

Numerical Analysis of Thermosyphon Solar Water Heaters

This paper presents the modeling and simulation as well as validation of a natural circulation closed thermosyphon glass tube solar collector water heater. Energy conservation equations for the heat transfer fluid flow and the storage tank were written in finite-difference form, integrated and solved to yield the characteristics of the thermosyphon system at different solar insolations and water mass flow rate conditions as well as water temperatures. Comparison between experimental data and numerical prediction of the proposed showed that the model predicted fairly the evacuation of storage tank temperature at various initial temperature of the water at the storage tank.

Non-linear Static Analysis of Offshore Steep Wave Riser

A new solution combining finite difference method and shooting method is developed to analyze the behavior of steep wave riser suffering from current loading. Based on the large deformation beam theory and mechanics equilibrium principle, a set of non-linear ordinary differential equations describing the motion of the steep wave riser are obtained. Then, finite difference method and shooting method are adopted and combined to solve the ordinary differential equations with zero moment boundary conditions at both the seabed end and surface end of the steep wave riser. The resulting non-linear finite difference formulations can be solved effectively by Newton-Raphson method. To improve iterative efficiency, shooting method is also employed to obtain the initial value for Newton-Raphson method. Results are compared with that of FEM by OrcaFlex, to verify the accuracy and reliability of the numerical method.

Simulation of mixed lubrication of rigid plain journal bearing by finite difference method with a skewed discretisation mesh

Numerical simulation of lubrication of journal bearing often uses a rectangular discretisation mesh. Steady-state mixed hydro-dynamic lubrication of rigid plain journal bearing is investigated by using finite difference form of Patir-Cheng average Reynolds equation under Reynolds boundary condition. Two kinds of non-orthogonal discretisation meshes, i.e., helix mesh and symmetric herringbone mesh are considered respectively. Three difference schemes, i.e.: 1) central finite difference; 2) second order upwinding; 3) a new central finite difference considering the film thickness at the projected positions from the nodes of neighbouring rows; are used to difference the shear induced flow term (i.e., Couette term). The effectiveness of the three schemes is discussed using the skewed discretisation mesh by comparing theoretical predictions of load with that found in the literature for smooth lubrication using the rectangular discretisation mesh. The effect of the skewed angle of mesh on the predicted parameters such as load, friction coefficient, attitude angle and maximum pressure is investigated for both smooth and rough surfaces. It is found that generally these schemes may provide a reasonable satisfactory solution. However, when symmetric herringbone mesh is used, a larger relative deviation is found as the skewed angle is larger and the eccentricity ratio is smaller.

Investigation of Thermostressed State of Coating Formation at Electric Contact Surfacing of “Shaft” Type Parts

The forming of coating at electric contact surfacing is considered. The mathematical model of the coating formation is developed. The method of numerical recurrent solution of the finite-difference form of static equilibrium conditions of the selected elementary volume of coating is used. This model considers distribution of thermal properties and geometric parameters along the thermal deformation zone during the process of electric contact surfacing by compact material. It is found that the change of value of speed asymmetry factor leads to increasing of the friction coefficient in zone of surfacing. This provides the forming of the coating of higher quality. The limitation of the technological capabilities of equipment for electric contact surfacing is related to the size of recoverable parts and application of high electromechanical powers. The regulation of the speed asymmetry factor allows for expanding the technological capabilities of equipment for electric contact surfacing. The nomograms for determination of the stress on the roller electrode and the finite thickness of the coating as the function of the initial thickness of the compact material and the deformation degree are shown.

Nonlocal or gradient elasticity macroscopic models: A question of concentrated or distributed microstructure

One dimensional discrete systems, such as axial lattices, may be investigated by using some enriched continuum models. In this paper, strain gradient models (also called gradient elasticity) and stress gradient models (also called nonlocal elasticity) are both shown to be supported by some microstructured physical configurations. Starting from the difference equations associated with each discrete system, a continualization approach is applied to the governing difference equations. Alternatively, one may use energy considerations to derive these higher-order continua. Stress gradient models are built from concentrated microstructure (with direct neighboring interaction) whereas strain gradient models are associated to some distributed microstructure (also with direct neighboring interaction). Each model leads to opposite effect, namely the softening effect of the small scale terms for the stress gradient model (built from concentrated microstructure), and the stiffening effect of the small scale terms for the strain gradient model (built from distributed microstructure) with respect to the asymptotic local model. We also discuss the link between lattice equations, finite difference formulation or finite element formulation of the continuous local problem. The paper concludes that the local neighboring interaction at the discrete scale may transmit some higher-order effects at the macroscopic scale. Hence, the higher-order nature of the macroscopic constitutive laws may not necessarily be seen as the consequence of nonlocal interaction at the lattice scale.

From Ziegler to Beck’s column: a nonlocal approach

This paper is concerned with the dynamic stability of a microstructured elastic column loaded by circulatory forces. This nonconservative lattice (or discrete) problem is shown to be equivalent to the finite difference formulation of Beck’s problem (cantilever column loaded by follower axial force). The lattice problem can be exactly solved from the resolution of a linear difference eigenvalue problem. The first part of the paper deals with the theoretical and numerical analyses of this discrete Beck’s problem, with a particular emphasis on the flutter load sensitivity with respect to the discretization parameters, such as the number of links of the lattice. The second part of the paper is devoted to the elaboration of a nonlocal equivalent continuum that possesses similar mathematical or physical properties as compared to the original lattice model. A continualized nonlocal model is introduced first by expanding the difference operators present in the lattice equations in terms of differential operators. The length scale of the continualized nonlocal model is size independent. Next, Eringen’s nonlocal phenomenological stress gradient is considered and applied at the beam scale in allowance for scale effects of the microstructured Beck column. The nonlocal Euler–Bernoulli beam model is able to capture the softening scale effect of the lattice model, even if the length scale of Eringen’s model appears to be size dependent in this case. The continualized nonlocal continuum slightly differs from the Eringen’s one, in the sense that the length scale affecting the static and the inertia terms differs in the deflection equation. A general parametric study illustrates the capability of each nonlocal model, the phenomenological and the continualized one, with respect to the reference lattice model. Nonlocal Beck’s column is shown to be a transient medium from Ziegler’s column (two-degree-of-freedom system) to the local continuous Beck’s column (with an infinite degree of freedom).

Finite difference modeling of sinking stage curved beam based on revised Vlasov equations

For the static analysis of the sinking stage curved beam, a finite difference model was presented based on the proposed revised Vlasov equations. First, revised Vlasov equations for thin-walled curved beams with closed sections were deduced considering the shear strain on the mid-surface of the cross-section. Then, the finite difference formulation of revised Vlasov equations was implemented with the parabolic interpolation based on Taylor series. At last, the finite difference model was built by substituting geometry and boundary conditions of the sinking stage curved beam into the finite difference formulation. The validity of present work is confirmed by the published literature and ANSYS simulation results. It can be concluded that revised Vlasov equations are more accurate than the original one in the analysis of thin-walled beams with closed sections, and that present finite difference model is applicable in the evaluation of the sinking stage curved beam.

A COMBINED SENSITIVITY ANALYSIS OF SEVEN POTENTIAL EVAPOTRANSPIRATION MODELS

Graphical and partial derivatives approaches were used to analyse the sensitivity of variables for the seven potential evapotranspiration models (PET). The models, which have different data requirements and structures, are Hamon, Hargreaves-Samani, Jensen-Haise, Makkink, Turc, Priestley-Taylor, and Penman. Julian date based mean imputation was used to fill the missing data. Tukey's outlier detection method was employed before estimating the PET. Partial derivative approach was conducted by combining the absolute values of the error term through a root mean square and changing to the finite difference form. According to partial derivatives analysis, Hamon is the most sensitive model followed by Penman, Priestley-Taylor, Hargreaves-Samani, Jensen-Haise, Turc, and Makkink models. Temperature is more sensitive meteorological input in Jensen-Haise and Makkink models while solar radiation is more sensitive ones in Turc and Priestley-Taylor models. Wind speed and relative humidity are the most and less sensitive ones in Penman model. Graphical analysis showed that Hamon was the most sensitive PET model with respect to the temperature while Priestley-Taylor was the one with respect to the solar radiation. Turc is the less sensitive PET model with respect to temperature and solar radiation. Overall, graphical method gives clearly comparison for sensitivity of PET. However, it does not indicate its sensitivity values compared to partial derivative approach.

Numerical analysis of FGM plates with variable thickness subjected to thermal buckling

A numerical solution using finite difference method to evaluate the thermal buckling of simply supported FGM plate with variable thickness is presented in this research. First, the governing differential equation of thermal stability under uniform temperature through the plate thickness is derived. Then, the governing equation has been solved using finite difference method. After validating the presented numerical method with the analytical solution, the finite difference formulation has been extended in order to include variable thickness. The accuracy of the finite difference method for variable thickness plate has been also compared with the literature where a good agreement has been found. Furthermore, a parametric study has been conducted to analyze the effect of material and geometric parameters on the thermal buckling resistance of the FGM plates. It was found that the thickness variation affects isotropic plates a bit more than FGM plates.

An ADE-FDTD Formulation for the Study of Liquid-Crystal Components in the Terahertz Spectrum

An anisotropic auxiliary differential equation finite-difference time-domain formulation is presented in detail for the time-domain study of nematic liquid crystal devices in the terahertz spectrum. The termination of the computation domain is achieved by employing a properly designed convolution perfectly matched layer. The material dispersion and dichroism of the LC complex permittivities is modeled via a modified Lorentzian function that is demonstrated to provide an accurate description for a series of state-of-the-art materials used in LC-THz technology.