Conformal Grids Research Articles

Incorporation of Conductor Loss in the Unconditionally Stable ADI-FDTD Method

A surface-impedance boundary condition (SIBC) is presented for an unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method. The conformal SIBC formulation based on locally conformal grids is capable of modeling the conductor loss of arbitrarily-shaped lossy metal structures. The work is described in the framework of the finite-integration technique (FIT) formulation of the ADI-FDTD. The paper focuses on the proposed ADI-FDTD SIBC formulation and its extensive validation. For this purpose, cylindrical and spherical cavity resonators are used for numerical tests. The quality factor Q is directly proportional to wall loss and is particularly sensitive to the accuracy of loss calculation. The resonator structure consists only of a vacuum-filled metal cavity and is thus free of additional sources of error that are caused by other extensions to the basic ADI-FDTD algorithm. The formulation is validated by comparison with analytic results and numerical data calculated using CST Microwave Studio (MWS). The convergence rate of the results is of second order, i.e., the error reduces to one quarter as the mesh resolution is doubled.

Study on MPI-Based Parallel Modified Conformal FDTD for 3-D Electrically Large Coated Targets by Using Effective Parameters

MPI-based parallel modified conformal finite difference time domain (P-MC-FDTD) method based on effective parameters for solving the scattering of electrically large coated targets is presented in this paper. The thin coating and free-space in a curved cell are replaced by an equivalent medium. The effective permittivity and electric conductivity for electric field samples on the cells are calculated by weighted-length of the medium and the free space. The modified conformal technique is used for magnetic field samples by dealing with stable conformal grids and unstable-prone conformal grids respectively. Combined with the parallel FDTD algorithm, the radar cross section (RCS) of 3-D electrically large coated targets can be calculated accurately.

A Brinkman penalization method for compressible flows in complex geometries

To simulate flows around solid obstacles of complex geometries, various immersed boundary methods had been developed. Their main advantage is the efficient implementation for stationary or moving solid boundaries of arbitrary complexity on fixed non-body conformal Cartesian grids. The Brinkman penalization method was proposed for incompressible viscous flows by penalizing the momentum equations. Its main idea is to model solid obstacles as porous media with porosity, ϕ, and viscous permeability approaching zero. It has the pronounced advantages of mathematical proof of error bound, strong convergence, and ease of numerical implementation with the volume penalization technique. In this paper, it is extended to compressible flows. The straightforward extension of penalizing momentum and energy equations using Brinkman penalization with respective normalized viscous, η, and thermal, η T, permeabilities produces unsatisfactory results, mostly due to nonphysical wave transmissions into obstacles, resulting in considerable energy and mass losses in reflected waves. The objective of this paper is to extend the Brinkman penalization technique to compressible flows based on a physically sound mathematical model for compressible flows through porous media. In addition to penalizing momentum and energy equations, the continuity equation for porous media is considered inside obstacles. In this model, the penalized porous region acts as a high impedance medium, resulting in negligible wave transmissions. The asymptotic analysis reveals that the proposed Brinkman penalization technique results in the amplitude and phase errors of order O(( ηϕ) 1/2) and O(( η/ η T) 1/4 ϕ 3/4), when the boundary layer within the porous media is respectively resolved or unresolved. The proposed method is tested using 1- and 2-D benchmark problems. The results of direct numerical simulation are in excellent agreement with the analytical solutions. The numerical simulations verify the accuracy and convergence rates.

Numerical generation and grid controls of boundary‐fitted conformal grids in multiply connected regions

AbstractBased on the study of grid generation methods in a simply connected region (Int. J. Numer. Meth. Engng 1997; 40:343), with the aid of a new transformation from a multiply connected region to a simply connected domain, the grid generation in a multiply connected domain is carried out in this paper. 2‐D conformal and orthogonal grids are generated in a multiply connected region, mostly involving a multiple‐blocks method developed in the present study. To avoid grid distortion on the degenerated boundary caused by physical singularities, controls of grids through specified points are implemented, without losing the conformal properties. Further applications of this grids‐controlling technique for points inside the region, the nested grids, and the multiple‐blocks method are introduced and discussed comprehensively. Numerical examples are taken to demonstrate the validity and applicability of the theorems. Copyright © 2006 John Wiley & Sons, Ltd.

Immersed boundary technique for turbulent flow simulations

The application of the Immersed Boundary (IB) method to simulate incompressible, turbulent flows around complex configurations is illustrated; the IB is based on the use of non-body conformal grids, and the effect of the presence of a body in the flow is accounted for by modifying the governing equations. Turbulence is modeled using standard Reynolds-Averaged Navier-Stokes models or the more sophisticated Large Eddy Simulation approach. The main features of the IB technique are described with emphasis on the treatment of boundary conditions at an immersed surface. Examples of flows around a cylinder, in a wavy channel, inside a stirred tank and a piston/cylinder assembly, and around a road vehicle are presented. Comparison with experimental data shows the accuracy of the present technique. This review article cites 70 references.

A Parallel Finite-Volume Runge–Kutta Algorithm for Electromagnetic Scattering

A 3D explicit finite volume algorithm has been developed to simulate scattering from complex geometries on parallel computers using structured body conformal curvilinear grids. Most simulations for practical 3D geometries require a large number of grid points for adequate spatial resolution making them suitable to parallel computation. The simulations have been carried out using a multi-block/zonal approach in the message passing paradigm on the SP-2. Each zone is placed on a separate processor and inter-processor communication is carried out using the Message Passing Library/Interface (MPL/MPI). Integration of Maxwell's equations is performed using the four-stage Runge–Kutta time integration method on a dual grid. This method of integrating on a staggered grid gives enhanced dissipative and dispersive characteristics. A scattered field formulation has been used and the Liao boundary condition is used at the outer nonreflecting boundary. The far zone transformation has also been implemented efficiently, using specialized MPL functions to evaluate the far zone scattering results. Results show extremely good comparisons for scattering from the sphere and the ogive with the exact solution and standard FDTD type algorithms. Comparisons for nonaxisymmetric targets like the NASA almond with experimental data has also been found to be extremely good.

The safety and efficacy of chronically implanted subdural electrodes: A prospective study

The popularity of subdural electrodes for the presurgical evaluation of patients with intractable seizures is increasing. However, few reports have prospectively dealt with their efficacy and safety. We conducted a 5-year prospective study of patients evaluated by the California Comprehensive Epilepsy Program, who subsequently underwent subdural electrode implantation at one of two institutions. Efficacy was examined by ultimate outcome with regards to postsurgery resection seizure frequency. Fifty-five patients underwent 58 implant procedures and postresection outcomes were available in 47 patients. Safety was defined by the incidence of expected and unexpected complications, and neuropathologic examination of tissue specimens. The most common expected adverse effects during implant were fever < or = 102 degrees (41%), cerebrospinal fluid leakage (19%), headache (15%), and nausea (4%). There were no infections. Unexpected adverse events included fever > 102 degrees F (5%), migraine (5%), iatrogenic electrode dysfunction (5%), and temporalis muscle fibrosis (5%). The incidence of pathologic findings suggestive of foreign body reaction was 10%. There were no permanent sequelae. Surgical outcomes were excellent in all (> or = 75% seizure reduction) with 50% seizure free regardless of the focus. Subdural electrodes are a safe, easy, and efficacious tool for evaluating seizure foci prior to resective surgery. They should no longer be considered investigational devices.

Analysis of microstrip antennas on a curved surface using the conformal grids FD-TD method

The need for small, portable antennas for mobile communications has spurred the study of microstrip antennas (MSA). MSA are quite flexible and have been used as conformal antennas on arbitrary curved surfaces. The characteristics of conformal MSA can be expected to differ from those of planar models. Dependable numerical analyses will obviate many of the costs and other inconveniences associated with experiments, but as antennas may be mounted on surfaces of arbitrary topological complexity, analysis methods must have a general applicability. The curvilinear finite-difference time-domain (FD-TD) method has shown excellent versatility. This method is applied to analyze microstrip antennas mounted on curved surfaces. The numerical predictions are compared with the experimental values. The results confirm the predictions within acceptable limits and appear to confirm the validity of the method. As a result, it was confirmed that the input impedance and directivity of MSA on curved surfaces are different from the flat MSA. >

Analysis of microstrip antennas on a curved surface using the conformal grids fd‐td method

AbstractWith the development of mobile communication, active research is being done on the small‐sized antennas such as microstrip antennas (MSAs). Moreover, MSAs normally are placed on spherical or cylindrical surfaces as a conformal antenna.Conformal MSAs are expected to have different characteristics compared to planar structure. In this case, numerical analysis is needed before the actual experiment.The method of analysis which has wide applications may be considered as efficient for this purpose. One such method is the conformal grids finite‐different time‐domain method (FD‐TD method).In this paper, an MSA on a curved surface is analyzed using the conformal grids FD‐TD method. Moreover, comparison with the experimental values also is performed. The experimental results and the analytical results are in close agreement which shows the validity of the method of analysis. As a result, it is confirmed that in the range of this paper, the input impedance and radiation pattern varies from their values for the planar patch when the patch surface is curved.

Conformal finite-different time-domain (FDTD) with overlapping grids

It is shown how to use many overlapping conformal grids in the neighborhood of the scatterer while using a rectangular grid away from the scatterer for solving Maxwell's equations. The locally conformal grids allow better approximation of the boundary condition on the scattering object while the rectangular grid preserves the simplicity and accuracy of the regular finite-difference time-domain (FDTD) algorithm. These grids overlap typically about three zones. It is shown how to connect together the calculations done in different overlapping grids. >

Missing Link between Virasoro and $\hat{sl(2)}$ Kac-Moody Algebras

We discuss the representations of the fractional level sl(2) affine Kac-Moody algebra and their relation to the Virasoro representations. A practical way of calculation of the free field null vectors and explicit relation between those for both algebras are given with many examples. For the complete degenerate representations some subtle points about the representations on and beyond the boundaries of the conformal grids are clarified

Conformal mappings onto multiply connected regions with specified boundary shapes: A preliminary discussion of computer implementation

Abstract : The author describes a method of calculating conformal mappings of any given finitely connected region onto a region with arbitrarily specified boundary shapes. If the specified shapes are rectangles, than this method can be used to generate conformal grids which should be useful for numerical solution of many partial differential equations, for example in calculating the airflow past an airflow with flaps or the flow of cooling water past fuel pins in a nuclear reactor. He has proved that there exists a conformal mapping of any given finitely connected region onto a region with arbitrarily specified boundary shapes. The construction in the proof has been adapted for computer implementation. Some examples have been worked to determine the region bounded by circles which is the image of a given region in the extended complex plane under a conformal mapping taking infinity to infinity. A region whose outer boundary is a rectangle is mapped conformally to a region with all rectangular boundaries, and the vertices of the outer boundaries correspond. Mappings onto regions bounded by rectangles should be of considerable use in grid generation for numerical solution of partial differential equations. Grids have been calculated for one simple example. (Author)

Solution of boundary-value problems for the Laplace equation by the method of conformal grids

RECENTLY the straight-line method has been attracting more and more attention both among theoreticians and among those using it for practical purposes (see [1]). For one particular class of boundary value problems it possesses obvious advantages over the method of nets with respect to the accuracy of the results and the number of arithmetical operations. Retaining the derivative with respect to one of the variables not only permits us to solve the problem, but also to investigate it. Moreover this method may also be used for the effective solution of non-linear boundary-value problems. To solve equations of the elliptic type in a rectangular region of Integration, the method of straight lines was first used In [2]. Later, many authors tried to use this method of solving problems in trapezoidal regions too. However, as was shown in [3], the method of straight lines was not completely correct for these regions. Very simple regions exist in which the boundary value problem for the set of differential equations of the method of straight lines is insoluble for certain n. In the present paper a solution is obtained, for regions with curvilinear boundaries, of linear and non-linear boundary-value problems for the two-dimensional Laplace equation for a one-dimensional family of curves.