Simple Finite Difference Scheme Research Articles

Efficient FEM-Based Modal Circuit Representation of Arbitrarily Shaped Plate Pairs

A novel finite element method-based technique for the calculation of all circuit parameters of an efficient broadband equivalent network representation of parallel-plane pairs with arbitrary geometry is presented. It is based on an eigenvalue analysis in combination with the solution of the quasi-static Poisson equation for the magnetic vector potential. Compared with a recently presented simple finite difference scheme, the suggested FEM approach considerably reduces the modeling and computational effort, enabling an adaptive meshing in the presence of small circular ports. The proposed approach is validated by a practical example, demonstrating the considerable computational savings and the versatility for practical applications.

Enhanced method for the evaluation of the thermal impact of dry machining processes

From today’s point of view the modelling of machining operations is a promising tool to extend the productivity and the precision of future industrial manufacturing. The importance of predictive simulation and compensation of thermally induced workpiece deformation during machining is especially important in dry machining because of the absence of cooling lubricants. Since the simulation results mainly depend on the boundary conditions of the model, a detailed knowledge of them is necessary. In this case the most important boundary condition is the intensity and possibly the distribution of the surface heat flux representing the heat flow into the workpiece resulting from the chip formation. The surface heat flux cannot be measured directly. One possible way to determine surface heat fluxes is to employ a thermal model of the machining process and match simulated and measured time and space dependent temperature fields. This procedure is time-consuming and is in most cases subjective because the congruency of temperature fields is rated manually, e.g. by the position of single isotherms. Therefore an enhanced method for the determination of surface heat fluxes is proposed in this paper. The method is based on nonlinear optimisation techniques and a simple finite difference scheme for numerical solution of the heat equation (WORHP-FDM). The procedure is objective between repeat measurements and works in a fully automated manner. The implementation is validated by the comparison to an analytical solution of the moving heat source based on the model of Carslaw and Jaeger and then applied to measured thermal images from milling experiments.

Efficient Broadband Circuit-Modeling Approach for Parallel-Plane Structures of Arbitrary Shape

A methodology for creating highly efficient and inherently stable equivalent-circuit models for arbitrarily shaped parallel-plane pairs is presented. Based on Green's function with accelerated convergence, a general impedance expression is derived for circular ports, comprising the static capacitance and inductances and a small number of resonant terms. It can be directly cast into an equivalent multiport circuit for arbitrary number of ports, enabling the application in a SPICE circuit-simulation environment for broadband system analysis with active, passive, and nonlinear components. The equivalent-circuit elements are determined by a simple finite-difference scheme in combination with an analytical port model, which allows us to perform the required modal and the quasi-static analysis with a uniform mesh. Validation by numerical full-wave field simulations demonstrates the versatility of the suggested approach and the high computational savings of several orders of magnitude in frequency and time domain.

Stable higher order finite-difference schemes for stellar pulsation calculations

Context: Calculating stellar pulsations requires a sufficient accuracy to match the quality of the observations. Many current pulsation codes apply a second order finite-difference scheme, combined with Richardson extrapolation to reach fourth order accuracy on eigenfunctions. Although this is a simple and robust approach, a number of drawbacks exist thus making fourth order schemes desirable. A robust and simple finite-difference scheme, which can easily be implemented in either 1D or 2D stellar pulsation codes is therefore required. Aims: One of the difficulties in setting up higher order finite-difference schemes for stellar pulsations is the so-called mesh-drift instability. Current ways of dealing with this defect include introducing artificial viscosity or applying a staggered grids approach. However these remedies are not well-suited to eigenvalue problems, especially those involving non-dissipative systems, because they unduly change the spectrum of the operator, introduce supplementary free parameters, or lead to complications when applying boundary conditions. Methods: We propose here a new method, inspired from the staggered grids strategy, which removes this instability while bypassing the above difficulties. Furthermore, this approach lends itself to superconvergence, a process in which the accuracy of the finite differences is boosted by one order. Results: This new approach is shown to be accurate, flexible with respect to the underlying grid, and able to remove mesh-drift.

Seismic waveform simulation with pseudo-orthogonal grids for irregular topographic models

In seismic waveform simulation, an irregular topography such as mountainous areas cannot be simplified to a flat surface. Even for marine seismic, a rough water bottom cannot be treated as a planar interface numerically. A body-fitted grid scheme will accurately present an earth model with an irregular topography. As it is a structured grid, then a simple finite difference scheme can be used as an efficient solver for waveform simulation. The pseudo-orthogonal property of grids is obtained by solving Poisson's equation. Investigation reveals that grids should have the acute angles >67° (90° for completely orthogonal) and the cell-size change rate <5 per cent, so that meshes are in a good orthogonality suitable for finite difference operation in waveform modelling. The acoustic wave equation and the absorbing boundary condition are reformulated from the physical space to the computational space. Waveform simulation and eventually tomographic inversion using a realistically complicated velocity model with a curved surface demonstrate the effectiveness of developed technology that works for irregular topographic models.

Entropically damped form of artificial compressibility for explicit simulation of incompressible flow

An alternative artificial compressibility (AC) scheme is proposed to allow the explicit simulation of the incompressible Navier-Stokes (INS) equations. Traditional AC schemes rely on an artificial equation of state that gives the pressure as a function of the density, which is known to enforce isentropic behavior. This behavior is nonideal, especially in viscously dominated flows. An alternative, the entropically damped artificial compressibility (EDAC) method, is proposed that employs a thermodynamic constraint to damp the pressure oscillations inherent to AC methods. The EDAC method converges to the INS in the low-Mach limit, and is consistent in both the low- and high-Reynolds-number limits, unlike standard AC schemes. The proposed EDAC method is discretized using a simple finite-difference scheme and is compared with traditional AC schemes as well as the lattice-Boltzmann method for steady lid-driven cavity flow and a transient traveling-wave problem. The EDAC method is shown to be beneficial in damping pressure and velocity-divergence oscillations when performing transient simulations. The EDAC method follows a similar derivation to the kinetically reduced local Navier-Stokes (KRLNS) method [Borok et al., Phys. Rev. E 76, 066704 (2007)]; however, the EDAC method does not rely on the grand potential as the thermodynamic variable, but instead uses the more common pressure-velocity system. Additionally, a term neglected in the KRLNS is identified that is important for accurately approximating the INS equations.

Global Region Based Segmentation of Satellite and Medical Imagery with Active Contours and Level Set Evolution on Noisy Images

In this paper, we proposed a novel global segmentation method for satellite images with active contour model on noisy images with ten percentage of salt and pepper. It was implemented with a special technique selective binary and Gaussian filtering regularized level set evolution. First we selectively penalize the level set function to be binary and then use a Gaussian smoothing kernel to regularize it. The advantages of our method is a new region based signed pressure force(SPF) function is proposed, which can step effectively the contour at weak or blurred edges and automatically detect the interior and exterior boundaries with the initial contour being anywhere in the images effected with noise. The proposed method can implement by the simple finite difference scheme. Experiments on satellite images with noise demonstrate the advantages of the proposed method over the Chan-vase (CV) active contour in terms of the number of Iterations.

A Generalized MAC Scheme on Curvilinear Domains

We propose a simple finite difference scheme for Navier--Stokes equations in primitive formulation on curvilinear domains. With proper boundary treatment and interplay between covariant and contravariant components, the spatial discretization admits exact Hodge decomposition and energy identity. As a result, the pressure can be decoupled from the momentum equation with explicit time stepping. No artificial pressure boundary condition is needed. In addition, it can be shown that this spatially compatible discretization leads to uniform inf-sup condition, which plays a crucial role in the pressure approximation of both dynamic and steady state calculations. Numerical experiments demonstrate the robustness and efficiency of our scheme.

Energy-based method for near-real time modeling of sound field in complex urban environments

Prediction of the sound field in large urban environments has been limited thus far by the heavy computational requirements of conventional numerical methods such as boundary element (BE) or finite-difference time-domain (FDTD) methods. Recently, a considerable amount of work has been devoted to developing energy-based methods for this application, and results have shown the potential to compete with conventional methods. However, these developments have been limited to two-dimensional (2-D) studies (along street axes), and no real description of the phenomena at issue has been exposed. Here the mathematical theory of diffusion is used to predict the sound field in 3-D complex urban environments. A 3-D diffusion equation is implemented by means of a simple finite-difference scheme and applied to two different types of urban configurations. This modeling approach is validated against FDTD and geometrical acoustic (GA) solutions, showing a good overall agreement. The role played by diffraction near buildings edges close to the source is discussed, and suggestions are made on the possibility to predict accurately the sound field in complex urban environments, in near real time simulations.

The wave equation with viscoelastic attenuation and its application in problems of shallow-sea acoustics

A suitable tool for the simulation of low frequency acoustic pulse signals propagating in a shallow sea is the numerical integration of the nonstationary wave equation. The main feature of such simulation problems is that in this case the sound waves propagate in the geoacoustic waveguide formed by the upper layers of the bottom and the water column. By this reason, the correct dependence of the attenuation of sound waves in the bottom on their frequency must be taken into account. In this paper we obtain an integro-differential equation for the sound waves in the viscoelastic fluid, which allows to simulate the arbitrary dependence of acoustic wave attenuation on frequency in the time domain computations. The procedure of numerical solution of this equation based on its approximation by a system of differential equations is then considered and the methods of artificial limitation of computational domain are described. We also construct a simple finite-difference scheme for the proposed equation suitable for the numerical solution of nonstationary problems arising in the shallow-sea acoustics.

Analysis and approximation of a reliable model

In this paper we consider the problem of approximating the dynamical system that models reliability of a system consisting of two machines separated by a finite storage buffer. The system is described as a distributed parameter system defined by a coupled partial and ordinary differential equations and formulated as an abstract Cauchy problem. To derive the dynamical solution and some instantaneous indexes of the model, we present a simple finite difference scheme and establish the convergence of this scheme by employing Trotter–Kato Theorem. Numerical results are given to illustrate the effectiveness of the scheme.

Binarization method based on evolution equation for document images produced by cameras

We present an evolution equation-based binarization method for document images produced by cameras. Unlike the existing thresholding techniques, the idea behind our method is that a family of gradually binarized images is obtained by the solution of an evolution partial differential equation, starting with an original image. In our formulation, the evolution is controlled by a global force and a local force, both of which have opposite sign inside and outside the object of interests in the original image. A simple finite difference scheme with a significantly larger time step is used to solve the evolution equation numerically; the desired binarization is typically obtained after only one or two iterations. Experimental results on 122 camera document images show that our method yields good visual quality and OCR performance.

Fast and Regularization less Active Contour

The application of the level set method in image segmentation has been very popular due to its capability of automatically handling changes in topology. However, a re-initialization procedure, which leads to expensive computation, is required in the traditional level set method to keep the level set function as a signed distance function to its interface. A method based on Gaussian filtering and binary level set is proposed for the level set function of region based active contour model (ACM). The proposed level set method is integrated with the global region based Chan-Vese (C-V) ACM for image segmentation. The proposed method can, not only ensure the smoothness of the level set function by Gaussian filtering, but also eliminate the requirement of re-initialization, which is very computationally expensive task. The level set function can also be easily initialized as a binary function, which is more efficient to construct practically than the widely used signed distance function (SDF). Moreover, as the proposed scheme allows using larger time step than what can be used with the standard C-V model, it is tremendously faster than standard C-V model. Finally, the proposed algorithm can be efficiently implemented by the simple finite difference scheme. Experimental results on synthetic and real images shows that the proposed method is more efficient in terms of computational time and accuracy than global region based C-V active contour model.

Adaptive level set evolution starting with a constant function

In this paper, we propose a novel level set evolution model in a partial differential equation (PDE) formulation. According to the governing PDE, the evolution of level set function is controlled by two forces, an adaptive driving force and a total variation (TV)-based regularizing force that smoothes the level set function. Due to the adaptive driving force, the evolving level set function can adaptively move up or down in accordance with image information as the evolution proceeds forward in time. As a result, the level set function can be simply initialized to a constant function rather than the widely-used signed distance function or piecewise constant function in existing level set evolution models. Our model completely eliminates the needs of initial contours as well as re-initialization, and so avoids the problems resulted from contours initialization and re-initialization. In addition, the evolution PDE can be solved numerically via a simple explicit finite difference scheme with a significantly larger time step. The proposed model is fast enough for near real-time segmentation applications while still retaining enough accuracy; in general, only a few iterations are needed to obtain segmentation results accurately.

Energy-based method for near real-time modeling of sound field in complex urban environments

Prediction of the sound field in large urban environments has been limited thus far by the computational heaviness of standard numerical methods such as boundary element (BE) or finite-difference time-domain methods. Recently, a considerable amount of work has been devoted to developing energy-based methods for this application, and results have shown the potential to compete with conventional methods. However, these developments have been limited to 2-D models, and no real description of the phenomena at issue has been exposed. Here, the mathematical theory of diffusion is used to predict the sound field in complete 3-D complex urban environments. A 3-D diffusion equation is implemented by means of a simple finite-difference scheme and applied to two different types of urban configurations. This modeling approach is validated against BE and geometrical acoustic solutions, showing a good overall agreement. The role played by diffraction near the source is discussed, and suggestions are made on the possibility to accurately predict the sound field in complex urban environments in near real time simulations.

A Boussinesq-type model incorporating random wave-breaking

A recent Boussinesq-type model is herein modified to account for breaking waves in shallow water. The model is based on a system of equations in terms of surface elevation and depth-averaged horizontal velocities, in two horizontal dimensions for fully dispersive and weakly nonlinear random waves over any finite water depth. The formulation involves five terms in each momentum equation, including the classical shallow-water equation terms, and only one frequency dispersion term. This work extends the model by including depth-induced wave-breaking in one horizontal dimension, based on the eddy viscosity and surface roller criteria. The modified model was applied to simulate the propagation and wave-breaking of regular and random waves using a simple explicit finite difference scheme. The simulation results were compared with experimental data and with results from one of the most widespread commercial Boussinesq wave models, indicating good agreement in most cases.

Quasinormal modes and Regge poles of the canonical acoustic hole

We compute the quasinormal mode frequencies and Regge poles of the canonical acoustic hole (a black hole analogue), using three methods. First, we show how damped oscillations arise by evolving generic perturbations in the time domain using a simple finite-difference scheme. We use our results to estimate the fundamental QN frequencies of the low multipolar modes $l=1, 2, \ldots$. Next, we apply an asymptotic method to obtain an expansion for the frequency in inverse powers of $l+1/2$ for low overtones. We test the expansion by comparing against our time-domain results, and (existing) WKB results. The expansion method is then extended to locate the Regge poles. Finally, to check the expansion of Regge poles we compute the spectrum numerically by direct integration in the frequency domain. We give a geometric interpretation of our results and comment on experimental verification.

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.

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.

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.