Simple Finite Difference Research Articles

FDTD ANALYSIS OF CHIRAL DISCONTINUITIES IN WAVEGUIDES

A simple finite difference time domain (FDTD) scheme is proposed for modeling three-dimensional (3D) nondispersive chiral media. Based on the recently reported new BI-FDTD mesh method and rearranged curl equations, this scheme implements a simple leapfrog algorithm. By adding the mirror layer, the perfect electric conductor (PEC) condition is implemented in the BI-FDTD mesh method of 3D problem. Results of this scheme are presented for the scattering coefficients of discontinuity in waveguides, which are partially filled with chiral or achiral media. The validation is performed by comparing the results with those obtained from the literature and software simulation.

Model-reduced gradient-based history matching

Gradient-based history matching algorithms can be used to adapt the uncertain parameters in a reservoir model using production data. They require, however, the implementation of an adjoint model to compute the gradients, which is usually an enormous programming effort. We propose a new approach to gradient-based history matching which is based on model reduction, where the original (nonlinear and high-order) forward model is replaced by a linear reduced-order forward model and, consequently, the adjoint of the tangent linear approximation of the original forward model is replaced by the adjoint of a linear reduced-order forward model. The reducedorder model is constructed with the aid of the proper orthogonal decomposition method. Due to the linear character of the reduced model, the corresponding adjoint model is easily obtained. The gradient of the objective function is approximated, and the minimization problem is solved in the reduced space; the procedure is iterated with the updated estimate of the parameters if necessary. The proposed approach is adjointfree and can be used with any reservoir simulator. The method was evaluated for a waterflood reservoir with channelized permeability field. A comparison with an adjoint-based history matching procedure shows that the model-reduced approach gives a comparable quality of history matches and predictions. The computational efficiency of the model-reduced approach is lower than of an adjoint-based approach, but higher than of an approach where the gradients are obtained with simple finite differences.

A SPLITTING METHOD FOR THE CAHN–HILLIARD EQUATION WITH INERTIAL TERM

P. Galenko et al. proposed a Cahn–Hilliard model with inertial term in order to model spinodal decomposition caused by deep supercooling in certain glasses. Here we analyze a finite element space semidiscretization of their model, based on a scheme introduced by C. M. Elliott et al. for the Cahn–Hilliard equation. We prove that the semidiscrete solution converges weakly to the continuous solution as the discretization parameter tends to 0. We obtain optimal a priori error estimates in energy norm and related norms, assuming enough regularity on the solution. We also show that the semidiscrete solution converges to an equilibrium as time goes to infinity and we give a simple finite difference version of the scheme.

Simulation of dust loaded V-I characteristics of a commercial thermal power plant precipitator

A new approach based on finite difference method, is proposed for the simulation of electrical conditions in a dc energized wire-duct electrostatic precipitator with and without dust loading. Simulated voltage-current characteristics with and without dust loading were compared with the measured characteristics for analyzing the performance of a precipitator. The simple finite difference method gives sufficiently accurate results with reduced mesh size. The results for dust free simulation were validated with published experimental data. Further measurements were conducted at a thermal power plant in India and the results compares well with the measured ones.

A nonlinear iteration method for solving a two-dimensional nonlinear coupled system of parabolic and hyperbolic equations

A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes.

On the Maxwell−Stefan Approach to Diffusion: A General Resolution in the Transient Regime for One-Dimensional Systems

The diffusion process in a multicomponent system can be formulated in a general form by the generalized Maxwell-Stefan equations. This formulation is able to describe the diffusion process in different systems, such as, for instance, bulk diffusion (in the gas, liquid, and solid phase) and diffusion in microporous materials (membranes, zeolites, nanotubes, etc.). The Maxwell-Stefan equations can be solved analytically (only in special cases) or by numerical approaches. Different numerical strategies have been previously presented, but the number of diffusing species is normally restricted, with only few exceptions, to three in bulk diffusion and to two in microporous systems, unless simplifications of the Maxwell-Stefan equations are considered. In the literature, a large effort has been devoted to the derivation of the analytic expression of the elements of the Fick-like diffusion matrix and therefore to the symbolic inversion of a square matrix with dimensions n x n (n being the number of independent components). This step, which can be easily performed for n = 2 and remains reasonable for n = 3, becomes rapidly very complex in problems with a large number of components. This paper addresses the problem of the numerical resolution of the Maxwell-Stefan equations in the transient regime for a one-dimensional system with a generic number of components, avoiding the definition of the analytic expression of the elements of the Fick-like diffusion matrix. To this aim, two approaches have been implemented in a computational code; the first is the simple finite difference second-order accurate in time Crank-Nicolson scheme for which the full mathematical derivation and the relevant final equations are reported. The second is based on the more accurate backward differentiation formulas, BDF, or Gear's method (Shampine, L. F. ; Gear, C. W. SIAM Rev. 1979, 21, 1.), as implemented in the Livermore solver for ordinary differential equations, LSODE (Hindmarsh, A. C. Serial Fortran Solvers for ODE Initial Value Problems, Technical Report; https://computation.llnl.gov/casc/odepack/odepack_ home.html (2006).). Both methods have been applied to a series of specific problems, such as bulk diffusion of acetone and methanol through stagnant air, uptake of two components on a microporous material in a model system, and permeation across a microporous membrane in model systems, both with the aim to validate the method and to add new information to the comprehension of the peculiar behavior of these systems. The approach is validated by comparison with different published results and with analytic expressions for the steady-state concentration profiles or fluxes in particular systems. The possibility to treat a generic number of components (the limitation being essentially the computational power) is also tested, and results are reported on the permeation of a five component mixture through a membrane in a model system. It is worth noticing that the algorithm here reported can be applied also to the Fick formulation of the diffusion problem with concentration-dependent diffusion coefficients.

Active contours with selective local or global segmentation: A new formulation and level set method

A novel region-based active contour model (ACM) is proposed in this paper. It is implemented with a special processing named Selective Binary and Gaussian Filtering Regularized Level Set (SBGFRLS) method, which first selectively penalizes the level set function to be binary, and then uses a Gaussian smoothing kernel to regularize it. The advantages of our method are as follows. First, a new region-based signed pressure force (SPF) function is proposed, which can efficiently stop the contours at weak or blurred edges. Second, the exterior and interior boundaries can be automatically detected with the initial contour being anywhere in the image. Third, the proposed ACM with SBGFRLS has the property of selective local or global segmentation. It can segment not only the desired object but also the other objects. Fourth, the level set function can be easily initialized with a binary function, which is more efficient to construct than the widely used signed distance function (SDF). The computational cost for traditional re-initialization can also be reduced. Finally, the proposed algorithm can be efficiently implemented by the simple finite difference scheme. Experiments on synthetic and real images demonstrate the advantages of the proposed method over geodesic active contours (GAC) and Chan–Vese (C–V) active contours in terms of both efficiency and accuracy.

Wavelet–Galerkin solutions of one dimensional elliptic problems

Daubechies wavelet bases are used for numerical solution of partial differential equations of one dimension by Galerkin method. Galerkin bases are constructed from Daubechies functions which are compactly supported and which constitute an orthonormal basis of L 2 ( R ) . Theoretical and numerical results are obtained for elliptic problems of second order with different types of boundary conditions. Optimal error estimates are also obtained. Comparison of solutions with simple finite difference method suggests that for this class of problems, the present method will provide a better alternative to other classical methods. The methodology can be generalized to multidimensional problems by taking care of some technical facts.

An events based algorithm for distributing concurrent tasks on multi-core architectures

In this paper, a programming model is presented which enables scalable parallel performance on multi-core shared memory architectures. The model has been developed for application to a wide range of numerical simulation problems. Such problems involve time stepping or iteration algorithms where synchronization of multiple threads of execution is required. It is shown that traditional approaches to parallelism including message passing and scatter-gather can be improved upon in terms of speed-up and memory management. Using spatial decomposition to create orthogonal computational tasks, a new task management algorithm called H-Dispatch is developed. This algorithm makes efficient use of memory resources by limiting the need for garbage collection and takes optimal advantage of multiple cores by employing a hungry pull strategy. The technique is demonstrated on a simple finite difference solver and results are compared to traditional MPI and scatter-gather approaches. The H-Dispatch approach achieves near linear speed-up with results for efficiency of 85% on a 24-core machine. It is noted that the H-Dispatch algorithm is quite general and can be applied to a wide class of computational tasks on heterogeneous architectures involving multi-core and GPGPU hardware.

A Simple Full-Vectorial Finite-Difference Equation for Arbitrarily Positioned Dielectric Interfaces

A new, simple, full-vectorial finite-difference equation is derived that allows the interfaces on rectangular dielectric waveguides to be positioned arbitrarily between mesh points.

Thermal loading of explosives—Finite difference method with time step reduction

The method of simulation of heat transfer during thermal loading of energetic materials is introduced. The combination of the simple finite difference method with a time step reduction enables the quick and precise calculation of ignition times. The results are in accordance with published model data. The method was also used to simulate heat transfer during the slow cookoff test and the result agrees very well with experimental data.

A New Variational Level Set Evolving Algorithm for Image Segmentation

Level set methods are the numerical techniques for tracking interfaces and shapes. They have been successfully used in image segmentation. A new variational level set evolving algorithm without re-initialization is presented in this paper. It consists of an internal energy term that penalizes deviations of the level set function from a signed distance function, and an external energy term that drives the motion of the zero level set toward the desired image feature. This algorithm can be easily implemented using a simple finite difference scheme. Meanwhile, not only can the initial contour can be shown anywhere in the image, but the interior contours can also be automatically detected.

Numerical modeling of multidimensional diffusion in the radiation belts using layer methods

A new code using layer methods is presented to solve radiation belt diffusion equations and is used to explore effects of cross diffusion on electron fluxes. Previous results indicate that numerical problems arise when solving diffusion equations with cross diffusion when using simple finite difference methods. We show that layer methods, which are based on stochastic differential equations, are capable of solving diffusion equations with cross diffusion and are also generalizable to three dimensions. We run our layer code using two chorus wave models and a combined magnetosonic wave and hiss wave model (MH wave model). Both chorus and magnetosonic waves are capable of accelerating electrons to MeV levels in about a day. However, for the chorus wave models, omitting cross diffusion overestimates fluxes at high energies and small pitch angles, while for the MH wave model, ignoring cross diffusion overestimates fluxes at high energies and large pitch angles. These results show that cross diffusion is not ignorable and should be included when calculating radiation belt electron fluxes.

Determining moisture content in concrete under simulated precipitation using ultrasonic propagation time measurements

The use of ultrasound is recognised as a means to obtain information about concrete with non-destructive testing. Various aspects of the material can be characterised. However, the scientific literature in this area is quite sparse. The present paper therefore addresses the issue of moisture content, and related compressive strength, in a systematic experimental manner. The moisture content profiles of 24 concrete cubes under constant simulated precipitation over 24 h were investigated using ultrasonic methods. Different values of water/cement ratio, duration of pre-curing vibration and curing regime were used. The propagation time of ultrasonic compression wave pulses gave a good correlation with gravimetric moisture content and it was found that the type of curing and the amount of vibration had the greatest impact on the propagation time of the ultrasonic wave. The moisture diffusion coefficient was estimated using a simple finite difference model and good agreement was achieved with experimental measurements.

On the Thermal Elevation of a 60-Electrode Epiretinal Prosthesis for the Blind

In this paper, the thermal elevation in the human body due to the operation of a dual-unit epiretinal prosthesis to restore partial vision to the blind affected by irreversible retinal degeneration is presented. An accurate computational model of a 60-electrode device dissipating 97 mW power, currently under clinical trials is developed and positioned in a 0.25 mm resolution, heterogeneous model of the human head to resemble actual conditions of operation of the prosthesis. A novel simple finite difference scheme combining the explicit and the alternating-direction implicit (ADI) method has been developed and validated with existing methods. Simulation speed improvement up to 11 times was obtained for the the head model considered in this work with very good accuracy. Using this method, solutions of the bioheat equation were obtained for different placements of the implant. Comparison with in-vivo experimental measurements showed good agreement.

Exact eigensystems for some matrices arising from discretizations

It is known, for example, that the eigenvalues of the N×N matrix A, arising in the discretization of the wave equation, whose only nonzero entries are Akk+1=Ak+1k=-1,k=1,…,N-1, and Akk=2,k=1,…,N, are 2{1-cos[pπ/(N+1)]} with corresponding eigenvectors v(p) given by vk(p)=sin[pkπ/(N+1)],p,k=1,…,N. We show by considering a simple finite difference approximation to the second derivative and using the summation formulae for sines and cosines that these and other similar formulae arise in a simple and unified way.

Non-Newtonian Fluid Flow on a Flat Plate Part 1:Boundary Layer

A modified power-law viscosity for non-Newtonian fluids based on actual measurements is proposed. This realistic model allows removal of the singularities at the leading edge of a flat-plate boundary layer for either shear-thinning or shear-thickening fluids. Under this condition, the boundary-layer equations can be solved numerically by simple finite difference methods that march downstream from the leading edge, as is usually done for Newtonian fluids. Numerical results are presented for the case of a shear-thinning fluid; applying the model to a shear-thickening fluid is straightforward. The effects of this new variable viscosity are explicitly demonstrated by comparing plots of isolines of viscosity and shear rate, the velocity distribution, and the wall shear stress for non-Newtonian and Newtonian fluids.

Pacific trench motions controlled by the asymmetric plate configuration

We present a novel explanation for absolute trench‐normal motions of slabs surrounding the Pacific. Rapid subduction‐zone retreat on the eastern side of the Pacific and slow advance in the west can result from the large‐scale asymmetric plate configuration. We use simple fluid dynamics to explain the mechanical background of this hypothesis, and we use the results of a simple finite difference scheme to estimate the effect on trench motion velocities. The hypothesis is based on two key assumptions. First, we follow the concept of plate‐scale horizontal counterflow in the asthenosphere driven by accretion of asthenosphere into lithosphere and by plate motion. Second, we assume that horizontally wide slabs without large slab windows drift passively in the mantle flow field and do not retreat as a result of flow around the slab. If the asthenosphere transfers flow‐related horizontal shear stress into deeper levels of the mantle, an asymmetry in the plate configuration leads to different net pressure forces on the two slabs and thus affects the retreat behavior. In an ocean with an asymmetric ridge position, the slab of the smaller plate should retreat faster than the slab of the large plate, which may even advance. Also, the domain of a slower moving plate should collapse faster than the domain of the faster plate. Our model explains the counterintuitive negative correlation between slab age and retreat velocity observed in the Pacific. It also accords with the topographic asymmetry of the ridge flanks along the Pacific rise.

Partial proxy simulation schemes for generic and robust Monte Carlo Greeks

We consider a generic framework which allows to calculate robust Monte-Carlo sensitivities seamlessly through simple finite difference approximation. The method proposed is a generalization and improvement of the proxy simulation scheme method (Fries and Kampen, 2005). As a benchmark we apply the method to the pricing of digital caplets and target redemption notes using LIBOR and CMS indices under a LIBOR Market Model. The framework is generic in the sense that it is model and almost product independent. The only product dependent part is the specification of the proxy constraint. This allows for an elegant implementation, where new products may be included at small additional costs.

On the relation between order of accuracy, convergence rate and spectral slope for linear numerical methods applied to multiscale problems

AbstractThe relation between order of accuracy and convergence rate for simple linear finite difference schemes for differentiation and advection is examined theoretically and empirically. For sufficiently smooth functions, i.e. those with sufficiently steep spectral slope, the convergence rate is given by the order of accuracy. For less smooth functions, with shallower spectral slope, differentiation and advection behave slightly differently: the convergence rate of a finite difference derivative is determined entirely by the spectral slope, while the convergence rate of a finite difference advection scheme is determined by an interaction between the spectral slope and the order of accuracy. © Crown copyright 2007. Reproduced with the permission of Her Majesty's Stationery Office. Published by John Wiley & Sons, Ltd.