Method Of Difference Potentials Research Articles

High order numerical simulation of the transmission and scattering of waves using the method of difference potentials

The method of difference potentials generalizes the method of Calderon’s operators from PDEs to arbitrary difference equations and systems. It offers several key advantages, such as the capability of handling boundaries/interfaces that are not aligned with the discretization grid, variable coefficients, and nonstandard boundary conditions. In doing so, the complexity of the algorithm remains comparable to that of an ordinary finite difference scheme on a regular structured grid.Previously, we have applied the method of difference potentials to solving several variable coefficient interior Helmholtz problems with fourth and sixth order accuracy. We have employed compact finite difference schemes as a core discretization methodology. Those schemes enable high order accuracy on narrow stencils and hence require only as many boundary conditions as needed for the underlying differential equation itself. Numerical experiments corroborate the high order accuracy of our method for variable coefficients, regular grids, and non-conforming boundaries.In the current paper, we extend the previously developed methodology to exterior problems. We present a complete theoretical analysis of the algorithm, as well as the results of a series of numerical simulations. Specifically, we study the scattering of time-harmonic waves about smooth shapes, subject to various boundary conditions. We also solve the transmission/scattering problems, in which not only do the waves scatter off a given shape but also propagate through the interface and travel across the heterogeneous medium inside. In all the cases, our methodology guarantees high order accuracy for regular grids and non-conforming boundaries and interfaces.

A High-Order Numerical Method for the Helmholtz Equation with Nonstandard Boundary Conditions

We describe a high-order accurate methodology for the numerical simulation of time-harmonic waves governed by the Helmholtz equation. Our approach combines compact finite difference schemes that provide an inexpensive venue toward high-order accuracy with the method of difference potentials developed by Ryaben'kii. The latter can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. The method of difference potentials can accommodate nonconforming boundaries on regular structured grids with no loss of accuracy due to staircasing. It introduces a universal framework for treating boundary conditions of any type. A significant advantage of this method is that changing the boundary condition within a fairly broad variety does not require any major changes to the algorithm and is computationally inexpensive. In this paper, we address various types of boundary conditions using the method of difference potentials. We demonstrate the resulting numerical capabilities by solving a range of nonstandard boundary value problems for the Helmholtz equation. These include problems with variable coefficient Robin boundary conditions (including discontinuous coefficients) and problems with mixed (Dirichlet/Neumann) boundary conditions. In all our simulations, we use a Cartesian grid and a circular boundary curve. For those test cases where the overall solution was smooth, our methodology has consistently demonstrated the design fourth-order rate of grid convergence, whereas when the regularity of the solution was not sufficient, the convergence slowed down, as expected. We also show that every additional boundary condition requires only an incremental additional expense.

GS41 クロムモリブデン鋼引き抜き丸棒の破壊強度異方性

Rolled or drawing metallic materials have anisotropy of mechanical properties because of the orientation of crystal grains. Fracture toughness values of these materials also have anisotropy due to the reason. The anisotropy of fracture toughness of the rolled metallic materials has been researched experimentally in our laboratory. The fact that the rolled metallic materials have strong anisotropy of fracture toughness became clear. The relations of rolling plane and crack plane have an influence on the fracture toughness values. The shapes or distributions of crystal grains of rolled metallic materials become plane-symmetrical. On the other hand, these of drawing metallic materials become axis-symmetrical. The purpose of this study is to make clear the anisotropy of the fracture toughness of the drawing metallic. Chromium-molybdenum steel (SCM435) round bar was used for this study. The specimens of six kinds of directions were processed in this material. The fracture toughness tests were carried out by the difference potential method. According to the tests, the fracture toughness anisotropy of this material was not remarkable as rolled materials.

Discrete Calderon’s projections on parallelepipeds and their application to computing exterior magnetic fields for FRC plasmas

Confining dense plasma in a field reversed configuration (FRC) is considered a promising approach to fusion. Numerical simulation of this process requires setting artificial boundary conditions (ABCs) for the magnetic field because whereas the plasma itself occupies a bounded region (within the FRC coils), the field extends from this region all the way to infinity. If the plasma is modeled using single fluid magnetohydrodynamics (MHD), then the exterior magnetic field can be considered quasi-static. This field has a scalar potential governed by the Laplace equation. The quasi-static ABC for the magnetic field is obtained using the method of difference potentials, in the form of a discrete Calderon boundary equation with projection on the artificial boundary shaped as a parallelepiped. The Calderon projection itself is computed by convolution with the discrete fundamental solution on the three-dimensional Cartesian grid.

Erratum to: The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

Erratum to: The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

The method of difference potentials was originally proposed by Ryaben'kii and can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the level of a finite-difference scheme on a regular structured grid. The method of difference potentials assembles the overall solution of the original boundary value problem by repeatedly solving an auxiliary problem. This auxiliary problem allows a considerable degree of flexibility in its formulation and can be chosen so that it is very efficient to solve. Compact finite difference schemes enable high order accuracy on small stencils at virtually no extra cost. The scheme attains consistency only on the solutions of the differential equation rather than on a wider class of sufficiently smooth functions. Unlike standard high order schemes, compact approximations require no additional boundary conditions beyond those needed for the differential equation itself. However, they exploit two stencils--one applies to the left-hand side of the equation and the other applies to the right-hand side of the equation. We shall show how to properly define and compute the difference potentials and boundary projections for compact schemes. The combination of the method of difference potentials and compact schemes yields an inexpensive numerical procedure that offers high order accuracy for non-conforming smooth curvilinear boundaries on regular grids. We demonstrate the capabilities of the resulting method by solving the inhomogeneous Helmholtz equation with a variable wavenumber with high order (4 and 6) accuracy on Cartesian grids for non-conforming boundaries such as circles and ellipses.

Difference potential based active noise shielding in three dimensional settings

Active noise shielding based on the Difference Potential Method has been developed and validated in previous publications. The key ability of the method, to provide global noise cancellation while automatically preserving wanted sound in multi-connected domains, has been demonstrated clearly in 1D experiments. In this paper, the tests are extended to 3D. The shielded domain is a 1mx1mx1m box with 1-3 sides open. Up to 4 control sources were used on each of the open sides to shield the domain from external noise. Noise attenuation of around 10dB was achieved up to about 350 Hz. The attenuation is lower than the >20dB attenuation achieved in previous 1D experiments. A reason for this is the sensitivity of the cancellation to the positioning of sensors and controls and the physical size of the control sources. Furthermore, up till now the method relies on separate measurements of the total sound field without the control sources. In realistic applications where noise and wanted sound are non-stationary, the total sound field has to be measured in real time with the control sources operating. In such cases a new mathematical formulation is needed, and this paper will discuss how such a solution can be achieved.

Synchronous exploration for the control of real-time external noise suppression in a three-dimensional subdomain

A mathematical (difference) model is proposed for a real-time active shielding device that shields an acoustic field in a given subdomain from the influence of sound sources located in an additional subdomain. An algorithm for computing the current control ensuring a prescribed process is based on information produced by the author’s technique of synchronous weak noise exploration. This information can be measured in real time. Active control problems for nonstationary solutions of linear difference equations in a three-dimensional domain consisting of two subdomains are studied using the difference potential method. The shape of the domain and the boundary conditions may depend on time, while the coefficients may depend on time and spatial coordinates. If the difference problem is a mathematical model of sound propagation, the goal of control is to change the acoustic field in the given subdomains, for example, to shield the acoustic field in one subdomain from the undesirable influence (noise) of sources located in the other subdomain.

Experimental Validation of the Active Noise Control Methodology Based on Difference Potentials

To achieve active noise cancellation over a large area, it is often necessary to get a measure of the physical properties of the noise source to devise a counter measure. This, however, is not practical in many cases. A mathematical approach, the Difference Potential Method, can provide an alternative solution for active shielding over a large area. In this approach, the cancellation of unwanted noise requires only measurements near the boundary surface but not at the source itself, and it does not require any other information. Moreover, the solution based on difference potentials applies to bounded domains in the presence of acoustic sources inside the domain to be shielded. This paper reports on the results of experimental validation. It has been demonstrated that while preserving the wanted sound, the developed approach can cancel out the unwanted noise. The volumetric noise cancellation offered by the proposed methodology along with leaving the wanted sound unchanged is a unique feature compared to other techniques available in the literature. It can be most useful in the context of applications related to civil aviation, in particular, for eliminating the exterior noise inside the passenger compartments of both current and future generation of commercial aircraft.

Generalized Calderon-Ryaben'kii's potentials

Calderón–Ryaben'kii potentials provide the foundation for the difference potential method, which is an efficient way for solving boundary-value problems (BVPs) in arbitrary domains. This method allows us to reduce a uniquely solvable and well-posed BVP to a pseudo-differential boundary equation. The general theory of Calderón–Ryaben'kii potentials is considered via the theory of distributions. The definition of Calderón–Ryaben'kii potentials is based on the notion of a clear trace. The criterion of the clear trace is formulated. Partial differential equations of the first order and the second order are considered as particular examples. On the basis of the Calderón–Ryaben'kii potential theory, a solution of the active sound control problem is obtained in a general formulation. For the first time, the solution of the problem takes into account the feedback of the active shielding sources on the input (measurement) data. The exact transfer of the boundary conditions from the original boundary to an artificial boundary is also considered.

Algorithm composition scheme for problems in composite domains based on the difference potential method

An algorithm composition scheme for the numerical solution of boundary value problems in composite domains is proposed and illustrated using an example. The scheme requires neither difference approximations of the boundary conditions nor matching conditions on the boundary between the subdomains. The scheme is suited for multiprocessor computers.

On the Method of Difference Potentials

We provide a brief general overview of the difference potentials method (DPM) and outline its key features that enable the new computational capabilities. We present three examples of the actual applied problems that have been solved with this method, and also comment on the connections between the new potentials and the Calderon---Seeley potentials.

DIFFERENCE-POTENTIAL TORQUE COMPUTATION WITH MAGNET INTERPOLAR GAPS

ABSTRACT The mean and difference potential method is applied to the problem of calculating cogging torque characteristics of permanent magnet machines in which the pole piece is narrower than the pole pitch. It is shown that altering the magnet radial thickness instead of its demagnetization characteristic yields accurate torques while permitting economic finite element meshes

Nondegeneracy of the equations related to the method of difference potentials

Nondegeneracy of the equations related to the method of difference potentials

The algorithm of the method of difference potentials

The algorithm of the numerical solution of the boundary integral equations, obtained using a reduction of the Calderon-Seeley type, is described. The algorithm is based on using difference potentials to approximate the surface potentials which develop as a result of the reduction. The constructions are made using as an example boundary value problems for the Laplace equation in a plane. The results of calculations are presented.

A Difference Potential Method for Induction Sondes

The response of a general induction sonde in a heterogeneous formation is computed using a difference potential technique. This difference potential method effectively transforms the source coil singularities into a continuum of finite sources distributed throughout the formation. The resulting equations are solved via finite-difference techniques. The apparent conductivity responses are corroborated by the solutions of an integral equation formulation for two invaded thin bed examples and a homogeneous formation example.