Finite Difference Formulation Research Articles

Inverse-Coefficient Problem of Heat Transfer in Layered Nanostructures

The rapid development of electronics is leading to the creation and use of small electronic components, including nanoelements of a complex layered structure. The search for effective methods for cooling electronic systems dictates the need for the development of methods for the numerical analysis of heat transfer in nanostructures. A characteristic feature of the energy transfer in such systems is the dominant role of contact thermal resistance at interlayer interfaces. Since the contact resistance depends on a number of factors associated with the manufacturing technology of heterostructures, it is of great importance to determine the corresponding coefficients from the results of temperature measurements. The purpose of this paper is to evaluate the possibility of reconstructing the thermal resistance coefficients at the interfaces between layers by solving the inverse problem of heat transfer. The complex of algorithms includes two major blocks: a block for solving the direct heat transfer problem in a layered nanostructure and an optimization block for solving the inverse problem. The direct problem is formulated in an algebraic (finite difference) form under the assumption of a constant temperature within each layer due to their small thickness. The inverse problem is solved in the extreme formulation and optimized using zero-order methods that do not require calculating the derivatives of the optimized function. As a basic optimization algorithm, the Nelder–Mead method is used in combination with random restarts to search for the global minimum. The results of identifying the contact thermal resistance coefficients obtained in the context of a quasi-real experiment are presented. The accuracy of the solution of the identification problem is estimated as a function of the number of layers in the heterostructure and the measurement error. The results are planned to be used in the new technique of multiscale modeling of thermal regimes of the electronic component base of the microwave range, when identifying the thermal conductivity coefficients of the heterostructure.

Applying the Fokker\u2013Planck equation to grating-based x-ray phase and dark-field imaging

X-ray imaging has conventionally relied upon attenuation to provide contrast. In recent years, two complementary modalities have been added; (a) phase contrast, which can capture low-density samples that are difficult to see using attenuation, and (b) dark-field x-ray imaging, which reveals the presence of sub-pixel sample structures. These three modalities can be accessed using a crystal analyser, a grating interferometer or by looking at a directly-resolved grid, grating or speckle pattern. Grating and grid-based methods extract a differential phase signal by measuring how far a feature in the illumination has been shifted transversely due to the presence of a sample. The dark-field signal is extracted by measuring how the visibility of the structured illumination is decreased, typically due to the presence of sub-pixel structures in a sample. The strength of the dark-field signal may depend on the grating period, the pixel size and the set-up distances, and additional dark-field signal contributions may be seen as a result of strong phase effects or other factors. In this paper we show that the finite-difference form of the Fokker–Planck equation can be applied to describe the drift (phase signal) and diffusion (dark-field signal) of the periodic or structured illumination used in phase contrast x-ray imaging with gratings, in order to better understand any cross-talk between attenuation, phase and dark-field x-ray signals. In future work, this mathematical description could be used as a basis for new approaches to the inverse problem of recovering both phase and dark-field information.

Analysis of rigid pavements on ground improved by geosynthetics and stone columns

Stone columns and geosynthetics have been widely used as effective stabilisation techniques to improve the load–settlement response of foundations. In view of this, an analysis of rigid pavements resting on geosynthetic-reinforced granular bed–stone column improved subgrade soil has been carried out. A detailed idealisation of the problem has been done using elements of Kelvin–Voigt body, Pasternak shear layer and Winkler springs, respectively, to simulate the behaviour of subgrade soil, granular fill and stone columns. A non-linear constitutive relationship for these elements has been considered. Governing differential equations for the soil–pavement system have been developed by modelling the geosynthetic as a rough elastic membrane and the pavement as a thin plate. These equations have been written in their finite-difference form and solved using the lower–upper (LU) decomposition method with the help of appropriate boundary conditions. A detailed parametric study to depict the influence of various factors affecting the pavement behaviour has been carried out. Quantitative evaluation of this influence has been done to enable the deflection and bending moment in the pavement and the tension mobilised in the geosynthetic layer to be obtained.

On the discrete Christoffel symbols

The piecewise flat space–time is equipped with a set of edge lengths and vertex coordinates. This defines a piecewise affine coordinate system and a piecewise affine metric in it, the discrete analogue of the unique torsion-free metric-compatible affine connection or of the Levi-Civita connection (or of the standard expression of the Christoffel symbols in terms of metric) mentioned in the literature, and, substituting this into the affine connection form of the Regge action of our previous work, we get a second-order form of the action. This can be expanded over metric variations from simplex to simplex. For a particular periodic simplicial structure and coordinates of the vertices, the leading order over metric variations is found to coincide with a certain finite difference form of the Hilbert–Einstein action.

A comparative study between finite difference explicit and implicit method for predicting pressure distribution in a petroleum reservoir

Predicting pressure distribution in a petroleum reservoir is principal to the reservoir’s evaluation and maintenance, as pressure changes with space and time. A convenient approach to effectively achieve this task is to formulate fluid flow equations based on the reservoir characteristics and solve them numerically. Numerical method provides solutions to mathematical fluid flow models developed in a reservoir simulation. This study provides numerical solutions, using both finite difference explicit and implicit method, to a mathematical model by developing MATLAB codes to ascertain the pressure distribution for a singlephase, one-dimensional, slightly compressible fluid flow in a petroleum reservoir. Series of numerical simulations were carried out during the first year of production using timestep sizes of 1, 2 and 3 days, respectively. The explicit method gave poor result (negative values for pressures) for timestep of 1 day, an outcome that is not accurate to describe the problem being solved but gave acceptable pressure results for timestep of 2 and 3 days. This shows that the efficacy of the explicit method is reliant on the chosen timestep and simulation time. In contrast, the implicit method gave a quite satisfactory results for all timesteps, and including less than 1 day, confirming the robustness and unconditionally stable nature of the implicit method. A commercial simulator (CMG software) was also employed to build a one-dimensional black oil model to validate the aforesaid results of which a close match was observed between the simulator results and the numerical solutions. This study provides insights to reservoir’s pressure profile during hydrocarbon recovery beforehand so that efficient pressure maintenance decisions can be made to achieve economic hydrocarbon recovery throughout the life of the reservoir. Keywords: finite difference formulation; explicit and implicit method; petroleum reservoir; pressure distribution; numerical reservoir simulation

Axisymmetric compact finite-difference lattice Boltzmann method for blood flow simulations.

An axisymmetric compact finite-difference lattice Boltzmann method is proposed to simulate both Newtonian and non-Newtonian flow of blood through a lumen. The curvature of the arteries could be accurately resolved using body-fitted mesh owing to the proposed finite-difference formulation. The axisymmetric nature of the flow, as well as the non-Newtonian nature of blood, are incorporated into the lattice Boltzmann equation using separate source terms. Using Chapman-Enskog expansion it is shown that the resulting lattice Boltzmann equation with these additional source terms recovers the macroscopic axisymmetric hydrodynamic equations. The solver is verified for (1) steady inflow of a Newtonian fluid through a stenosed lumen, (2) temporally developing pulsatile flow (Womersley flow) through a straight lumen with Newtonian fluid, and (3) steady inflow of a non-Newtonian fluid through a straight lumen. The solver is then applied to simulate the steady flow of a non-Newtonian fluid through a stenosed lumen, and it was found that a smaller recirculation zone and lower WSS values are obtained when compared with the flow of a Newtonian fluid. The capability of the solver to simulate spatially developing (velocity-driven) pulsatile flow is then demonstrated by simulating physiological pulsatile flow through an axisymmetric abdominal aortic aneurysm. From this simulation, the cycle-averaged wall shear stress is observed to have a steep gradient going from a minimum (negative) to a maximum (positive) value towards the distal end of the aneurysm, which is prone to the risk of rupture. An iterative procedure to select the geometric and flow parameters for unsteady inflow condition in the lattice Boltzmann method framework is demonstrated that accurately resolves all the timescales to achieve incompressibility. Overall, the present solver seems to be promising to simulate axisymmetric flow of blood with steady and pulsatile inflows while considering the blood rheology.

Nonlocality of one-dimensional bilinear hardening–softening elastoplastic axial lattices

The static behaviour of an elastoplastic axial lattice is studied in this paper through both discrete and nonlocal continuum analyses. The elastoplastic lattice system is composed of piecewise linear hardening–softening elastoplastic springs connected between each other via nodes, loaded by concentrated tension forces. This inelastic lattice evolution problem is ruled by some difference equations, which are shown to be equivalent to the finite difference formulation of a continuous elastoplastic bar problem under distributed tension load. Exact solutions of this inelastic discrete problem are obtained from the resolution of this piecewise linear difference equations system. Localization of plastic strain in the first elastoplastic spring, connected to the fixed end, is observed in the softening range. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process, based on a rational asymptotic expansion of the associated pseudo-differential operators. The continualized lattice-based model is equivalent to a distributed nonlocal continuous elastoplastic theory coupled to a cohesive elastoplastic model, which is shown to capture efficiently the scale effects of the reference axial lattice. The hardening–softening localization process of the nonlocal elastoplastic continuous model strongly depends on the lattice spacing, which controls the size of the nonlocal length scales. An analogy with the one-dimensional lattice system in bending is also shown. The considered microstructured elastoplastic beam is a Hencky bar-chain connected by elastoplastic rotational springs. It is shown that the length scale calibration of the nonlocal model strongly depends on the degree of the difference equations of each lattice model (namely axial or bending lattice). These preliminary results valid for one-dimensional systems allow possible future developments of new nonlocal elastoplastic models, including two- or even three-dimensional elastoplastic interactions.

Warm Absorber Diagnostics of AGN Dynamics

Abstract Warm absorbers and related phenomena are some of the observable manifestations of outflows or winds from active galactic nuclei (AGNs). Warm absorbers are common in low-luminosity AGNs. They have been extensively studied observationally and are well described by simple phenomenological models. However, major open questions remain. What is the driving mechanism? What is the density and geometrical distribution? How much associated fully ionized gas is there? What is the relation to the quasi-relativistic “ultrafast outflows”? In this paper we present synthetic spectra for the observable properties of warm absorber flows and associated quantities. We use ab initio dynamical models, i.e., solutions of the equations of motion for gas in finite difference form. The models employ various plausible assumptions for the origin of the warm absorber gas and the physical mechanisms affecting its motion. The synthetic spectra are presented as an observational test of these models. In this way we explore various scenarios for warm absorber dynamics. We show that observed spectra place certain requirements on the geometrical distribution of the warm absorber gas, and that not all dynamical scenarios are equally successful at producing spectra similar to what is observed.

Modal Analysis of Circular Symmetrical Plates by Means of Generalized Finite Difference Method

In this paper, a simplified modal analysis procedure of circular plates procedures (on polar domains) through generalized (modernized) finite difference method (abbreviated next as – FDM) is developed.Generally, circular plates are widely used for a plenty of modern civilian and industrial utilities, machine design and many other purposes. They form a spectrum of elements starting with trains’ bogies along with engine pistons, dampers and up to slabs and roofs over circular-shaped buildings, train stationsand other transportation facilities. Nowadays, FDM predominates the numerical solutions of partial differential equations (abbreviated next as – PDE) not less than the method of finite elements (abbreviated next as – FEM). This is wide-famous mathematical-discretization method that is economic to compute and simple to code, less regarding to computation tools in hands and how powerful/less powerful they are, since it bases on replacing each derivative by a difference algebraic quotient in a classical formulation. In a sense, a finite difference formulation offers a more direct approach to the numerical solution of the PDE especially in polar coordinates domain problems considering curvilinear dimensions that even FEM does not.The generalized approach of FDM considers many parameters less regarded by the classical one. Consequently, the use of classical approach negatively affects the accuracy of calculation (convergence to the exact solution values) and the tendency of results, the thing been healed by the generalized approach.

Semi-implicit methods for the dynamics of elastic sheets

Recent applications (e.g. active gels and self-assembly of elastic sheets) motivate the need to efficiently simulate the dynamics of thin elastic sheets. We present semi-implicit time stepping algorithms to improve the time step constraints that arise in explicit methods while avoiding much of the complexity of fully-implicit approaches. For a triangular lattice discretization with stretching and bending springs, our semi-implicit approach involves discrete Laplacian and biharmonic operators, and is stable for all time steps in the case of overdamped dynamics. For a more general finite-difference formulation that can allow for general elastic constants, we use the analogous approach on a square grid, and find that the largest stable time step is two to three orders of magnitude greater than for an explicit scheme. For a model problem with a radial traveling wave form of the reference metric, we find transitions from quasi-periodic to chaotic dynamics as the sheet thickness is reduced, wave amplitude is increased, and damping constant is reduced.

Fast 2D full-waveform modeling and inversion using the Schur complement approach

In most full-waveform inversion (FWI) problems, sufficient prior information is available to constrain the velocity of certain parts of the model, e.g., the water column or, in some cases, near-surface velocities. We take advantage of this situation and develop a fast Schur-complement-based forward modeling and inversion approach by partitioning the velocity model into two parts. The first part consists of the constrained zone that does not change during the inversion, whereas the second part is the anomalous zone to be updated during the inversion. For this decomposition, we partially factorize the governing system of linear equations by computing a Schur complement for the anomalous zone. The Schur complement system is then solved to compute the fields in the anomalous zone, which are then back substituted to compute the fields in the constrained region. For each successive modeling steps with new anomalous zone velocities, the corresponding Schur complement is easily computed using simple algebra. Because the anomalous part of the model is comparatively smaller than the whole model, considerable computational savings can be achieved using our Schur approach. Additionally, we showed that the Schur complement method maintains the accuracy of standard frequency-domain finite difference formulations, but this comes at a slightly higher peak memory requirement. Our FWI workflow shows reduced runtime by 15%–57% depending upon the depth of the water column without losing any accuracy compared to the standard method.

Ab initio electronic structure calculations using a real-space Chebyshev-filtered subspace iteration method

Ab initio electronic structure calculations within Kohn–Sham density functional theory requires a solution for the Kohn–Sham equation. However, the traditional self-consistent field (SCF) approach of solving the equation using iterative diagonalization exhibits an inherent cubic scaling behavior and becomes prohibitive for large systems. The Chebyshev-filtered subspace iteration (CheFSI) method holds considerable promise for large-system calculations by substantially accelerating the SCF procedure. Here, we employed a combination of the real space finite-difference formulation and CheFSI to solve the Kohn–Sham equation, and implemented this approach in ab initio Real-space Electronic Structure (ARES) software in a multi-processor, parallel environment. An improved scheme was proposed to generate the initial subspace of Chebyshev filtering in ARES efficiently, making it suitable for large-scale simulations. The accuracy, stability, and efficiency of the ARES software were illustrated by simulations of large-scale crystalline systems containing thousands of atoms.

Forced convective heat transfer between assemblages of spherical particles and power‐law non‐Newtonian liquids with velocity slip at the interface

AbstractHeat transfer from spheres can be influenced by a varying degree of slip at the fluid‐particle interface along with the rheology of the surrounding continuous liquid and adjacent spheres. Thus in this study, the effects of dimensionless velocity slip parameter (λ) along with power‐law fluid rheology and other pertinent kinematic flow and heat transfer parameters on isotherm contours, local and average Nusselt numbers of assemblages of spherical slip particles are presented. This is done by adopting a segregated approach where dimensionless momentum and energy equations are solved by SMAC algorithm formulated in spherical coordinates within the finite difference formulation. Before obtaining new results, grid independence studies for either extreme values of power‐law consistency index of non‐Newtonian fluids are carried out. Finally, the major contribution of this study is the development of a correlative equation for the average Nusselt number of assemblages of spherical slip particles in power‐law fluids based on the present results (5880 data points) as a function of pertinent dimensionless parameters.

A two-layer numerical model for simulating the frontal plowing phenomenon of flow-like landslides

Many flow-like landslides entrain material from their paths during motion. At the flow front and along the lateral margins the sliding mass can plow into the path material pushing or entraining the existing soil. Although plowing can be a dominant mechanism for landslide mobility, little attention has been paid to this phenomenon in comparison with other entraining mechanism such as basal scouring. Therefore, establishing a suitable mathematical description is still a challenge. In this paper, a two-layer finite difference model is proposed to simulate frontal plowing. The frontal erodible mass and the sliding mass are simplified as two separate layers based on the assumption that they are immiscible in their propagation processes. The interaction (i.e., thrusting and shear) between the two layers is simulated by the normal force and shear force acting on the two-layer interface. The governing equations for the two-layer model are deduced from the mass and momentum conservations of a soil column and transformed into a finite difference form for numerical solving. Then the proposed model is tested in the back analysis of the Ximiaodian landslide which is a typical loess flow-like landslide located at the south bank of the Jing River, China. The modeling results show that frontal plowing has significant influence on the propagation of this landslide, especially on the final topography of the deposit. Without considering this effect, the thickness of the final deposit tends to be underestimated, while the propagation duration, area and distance are likely to be overestimated. The proposed model can provide more accurate and reliable simulations for rapid flow-like landslides with frontal plowing phenomenon.

Numerical solutions of the singular vortex problem

This study develops a finite-difference numerical formulation to describe the motion of a singular monopole in a quasigeostrophic β-channel model with scale-selective frictional damping, using parameter values typical for the middle-latitude atmosphere and a wide range of viscosities. In this model, the “theoretical” singular vortex is replaced by the equivalent nonsingular vortex of a finite amplitude, consistent with the finite spatial resolution of the model. Numerical experiments demonstrate that at initial stages of the singular-vortex (SV) evolution, this model accurately reproduces the behavior expected from the theoretical considerations of the inviscid case. The numerical model also approximately conserves several invariants of motion derived from the continuous equations and accurately represents their modifications in the presence of friction. The evolution of a singular cyclone in the Northern Hemisphere starts with the development of the dipolar β gyres in the regular component of the flow; these gyres induce initial northward displacement and subsequent westward bending of the cyclone trajectory. At larger times, the β gyres gradually disintegrate, and the singular cyclone in the Northern Hemisphere continues to move northwestward by forming a dipolelike system with the localized secondary regular-field anticyclone northeast of the singular-cyclone center resulting eventually in a friction-assisted steady-state regime. The SV trajectories tend to become more zonally elongated for large vortices and small values of viscosity. These results lay a foundation for numerical consideration of systems of multiple singular vortices, which could provide further insights into our fundamental understanding of the processes underlying the multiscale atmospheric and oceanic variability.

Non-polynomial quintic spline for numerical solution of fourth-order time fractional partial differential equations

This paper presents a novel approach to numerical solution of a class of fourth-order time fractional partial differential equations (PDEs). The finite difference formulation has been used for temporal discretization, whereas the space discretization is achieved by means of non-polynomial quintic spline method. The proposed algorithm is proved to be stable and convergent. In order to corroborate this work, some test problems have been considered, and the computational outcomes are compared with those found in the exiting literature. It is revealed that the presented scheme is more accurate as compared to current variants on the topic.

In-Plane and Out-of-Plane Elastodynamics of Thin Rings and Seals

Thin curved rings used mostly as seals, including in internal combustion engines undergo complex elastodynamic behavior when subjected to a combination of normal radial loading and tangential shear with friction. In turn, their complex modal behavior often results in loss of sealing, increased friction, and power loss. This paper presents a new finite difference approach to determine the response of thin incomplete circular rings. Two interchangeable approaches are presented; one embedding mass and stiffness components in a unified frequency-dependent matrix, and the other making use of equivalent mass and stiffness matrices for the ring structure. The versatility of the developed finite difference formulation can also allow for efficient modification to account for multiple dynamically changing ring support locations around its structure. Very good agreement is observed between the numerical predictions and experimental measurements, particularly with new precision noncontact measurements using laser Doppler vibrometry. The influence of geometric parameters on the frequency response of a high performance motorsport engine's piston compression ring demonstrates the degree of importance of various geometrical parameters on ring dynamic response.

An efficient computational technique based on cubic trigonometric B-splines for time fractional Burgers' equation

ABSTRACTThis paper presents a computational technique based on cubic trigonometric B-splines for the time fractional Burgers' equation. The nonlinear advection term is approximated by a new linearization technique which is very efficient and significantly reduces the computational cost. The usual finite difference formulation is used to approximate the Caputo time fractional derivative while the derivative in space is discretized using cubic trigonometric B-spline functions. The proposed technique is proved to be globally unconditionally stable. A convergence analysis is discussed to measure the accuracy of the solution. Computational experiments are performed to further confirm the accuracy and stability of the method. Numerical results are compared with those obtained by a scheme based on parametric spline functions. The comparison reveals that the proposed scheme is quite accurate and effective.

A multispecies, multifluid model for laser–induced counterstreaming plasma simulations

The interpenetration of counterstreaming plasmas is an important phenomenon in several application areas, such as astrophysical flows, design of controlled fusion devices, and laser–induced plasma experiments. Multispecies “single-fluid” codes are unable to model this phenomenon due to the single velocity representation for all the species/fluids. Kinetic codes, though capable of modeling interpenetration, are computationally prohibitive for at-scale simulations. In this paper, we propose a multifluid model that solves the fluid equations for each ion fluid or stream. This allows distinct flows that interact with each other through electrostatic and collisional forces. We introduce and describe our code, EUCLID, that uses a conservative finite-difference formulation to discretize the governing equations in space. The 5th-order Monotonicity-Preserving WENO scheme is used for the upwind approximation of the hyperbolic flux, and the explicit 4th-order Runge–Kutta scheme is used for time integration. The code is verified for several benchmark cases and manufactured solutions. We simulate one- and two-dimensional interactions of counterstreaming plasmas in vacuum as well as in the presence of gas fill, where the setups are representative of laser-induced plasma experiments.

A modal ACMFD formulation of the HEXNEM3 method for solving the time-dependent neutron diffusion equation

An analytical coarse-mesh finite-difference (ACMFD) formulation of the HEXNEM3 nodal flux expansion method for solving the two-group neutron diffusion equation in hexagonal-z geometry is developed. In the time-dependent case, where an implicit differencing scheme in time is required for stability, nodal flux expansion necessitates iterating on the group sources. In order to avoid convergence issues, modal decomposition through matrix diagonalisation in the energy domain is first performed, and then the HEXNEM3 nodal expansion model is applied to the modes instead of fluxes, with boundary and continuity conditions on the scalar flux and the net current. The ACMFD formulation of HEXNEM3 has the advantage to produce an explicit non-homogeneous linear algebraic system for either the node-averaged fluxes or modes in all groups and nodes of a three-dimensional problem, with a free choice of any appropriate solution method.