Precorrected Fast Fourier Transform Research Articles

Acceleration of a wave-structure interaction solver by the Parareal method

Potential flow theory-based solvers are commonly used in ocean engineering to investigate the interactions between ocean waves and floating bodies. Depending on assumptions, several methods have been proposed. Among them, the Weak-Scatterer method is an interesting trade-off in the sense that this approach is not limited in theory by the small wave amplitudes and small body motions assumptions of linear methods. Moreover, this approach is in practice more stable than the fully non-linear methods. An implementation of the Weak-Scatterer method is the WS-CN code (Letournel, 2015; Chauvigné, 2016; Wuillaume, 2019).The computational time of the WS-CN code which is considered in the present study is relatively long for engineering purposes. In order to reduce it, the present paper presents an implementation of the Parareal method in the WS-CN code. The Parareal method is an algorithm for parallelizing a simulation in time that can accelerate the complete simulation (Lions, 2001) . This is a key difference in comparison to other acceleration techniques which have been studied in the literature (e.g. the Fast Multipole Method (FMM), the precorrected Fast Fourier Transform (pFFT) method, … ). To the authors’ knowledge, the present study is the first to couple the Parareal method to a potential flow theory-based wave-structure interaction solver. It is shown that the method can significantly reduce the computational time for small wave steepness, but that the performance decreases rapidly with increasing steepness.

Development and Application of a Pre-Corrected Fast Fourier Transform Accelerated Multi-Layer Boundary Element Method for the Simulation of Shallow Water Acoustic Propagation

Because of the complexities associated with the domain geometry and environments, accurate prediction of acoustics propagation and scattering in realistic shallow water environments by direct numerical simulation is challenging. Based on the pre-corrected Fast Fourier Transform (PFFT) method, we accelerated the classical boundary element method (BEM) to predict the acoustic propagation in a multi-layer shallow water environment. The classical boundary element method formulate the acoustics propagation problem as a linear equation system in the form of [A]{x}={b}, where [A] is an N×N dense matrix composed of influence coefficients. Solving such linear equation system requires O(N2/N3) computational cost for iterative/direct methods. The developed method, PFFT-BEM, can effectively reduce the computational efforts for direct numerical simulations from O(N2~3) to O(Nlog N), where N is the total number of boundary unknowns. To numerically simulate the sound propagation in a shallow water environment, we applied the first-order non-reflecting boundary condition in the truncated numerical domain boundary to eliminate the errors due to reflected waves. Multi-layer coupled formulation was used to include the environment inhomogeneity in PFFT-BEM. Through multiple convergence tests on the number of layers and elements, we validated and quantified the accuracy of PFFT-BEM. To demonstrate the usefulness and capability of the developed PFFT-BEM, we simulated three-dimensional (3D) underwater sound propagation through 3D geometries to check the efficacy of the established classical method: the 3D Parabolic equation model. Finally, PFFT-BEM was employed to simulate sound propagation through a complex multi-layer shallow water environment with internal waves. The “3D+T” results obtained by PFFT-BEM compared well with the physical test, thereby proving the capability and correctness of this method.

A fast multi-layer boundary element method for direct numerical simulation of sound propagation in shallow water environments

Direct three-dimensional (3D) numerical simulations of acoustic fields in range-dependent shallow water environments remains a challenge due to environmental complexities and large computational cost. We develop an efficient 3D boundary element method (BEM) to solve the Helmholtz equation for shallow water acoustic propagation, which utilizes a Pre-corrected Fast Fourier Transform (PFFT) approach to reduce the computational effort from O(N2∼3) to O(Nlog⁡N) where N is the total number of boundary unknowns. To account for inhomogeneous media, the method allows for arbitrary number of coupled multi-layer BEM sub-domains. With O(Nlog⁡N) efficiency and the use of massively parallel high-performance computing platforms, we are able to conduct multi-layer 3D direct numerical simulations of low-mid frequency acoustics over kilometer ranges. We perform extensive validations of the method and provide two shallow water waveguide examples benchmarked against theoretical solutions. To illustrate the efficacy and usefulness of the PFFT-BEM method, we perform 3D large-scale direct numerical simulations to assess the performance of two established canonical models: axisymmetric coupled mode model for 3D seamount; and Kirchhoff approximation and perturbation theory for 3D rough surface scattering.

Extension of the frequency-domain pFFT method for wave structure interaction in finite depth

To analyze wave interaction with a large scale body in the frequency domain, a precorrected Fast Fourier Transform (pFFT) method has been proposed for infinite depth problems with the deep water Green function, as it can form a matrix with Toeplitz and Hankel properties. In this paper, a method is proposed to decompose the finite depth Green function into two terms, which can form matrices with the Toeplitz and a Hankel properties respectively. Then, a pFFT method for finite depth problems is developed. Based on the pFFT method, a numerical code pFFT-HOBEM is developed with the discretization of high order elements. The model is validated, and examinations on the computing efficiency and memory requirement of the new method have also been carried out. It shows that the new method has the same advantages as that for infinite depth.

An explicit formula for the coherent SH waves’ attenuation coefficient in random porous materials with low porosities

In this paper, the attenuation coefficient of coherent SH waves in random porous material with uniformly randomly distributed elliptical cavities of different aspect ratios is studied. Based on an analysis of the mechanism for attenuation, a simple macro model for the attenuation coefficient is proposed. The macro model says that the attenuation coefficient can be expressed as a function of the mean scattering cross section and the number density of cavities at low porosities. Then, large-scale numerical simulations using the pre-corrected Fast Fourier Transform (pFFT) algorithm accelerated Boundary Element Method (BEM) are conducted to specify this macro model. Finally, this macro model is compared with four theoretical models derived for composite/porous materials with circular inclusions at the porosity p=3.17% and 5%. Results show this macro model agree well with three of them. Compared to the existing theoretical models, the form of this macro model is simple and has a clear physical meaning. In addition, it is applicable to cases with relatively complex cavities.

A precorrected-FFT higher-order boundary element method for wave–body problems

The boundary element method (BEM) has been widely applied in the field of wave interaction with offshore structures, but it is still not easy to use in resolving large-scale problems because of computing costs and computer storage being increased by O( N 2) for the traditional BEM. In this paper a precorrected Fast Fourier Transform (pFFT) higher-order boundary element method (HOBEM) is proposed for reducing the computational time and computer memory by O( N). By using a free-surface Green function for infinite water-depth, the disadvantage of the Fast Multipole Boundary Element Method (FMBEM)—i.e. unable to solve infinite deep-water wave problems—can be overcome. Numerical results from the problems of wave interaction with single- and multi-bodies show that the present method evidently has more advantages in saving memory and computing time, especially for large-scale problems, than the traditional HOBEM. In addition, the optimal variable of pFFT mesh is recommended to minimize time cost.

An efficient high-order boundary element method for nonlinear wave–wave and wave-body interactions

A highly efficient high-order boundary element method is developed for the numerical simulation of nonlinear wave–wave and wave-body interactions in the context of potential flow. The method is based on the framework of the quadratic boundary element method (QBEM) for the boundary integral equation and uses the pre-corrected fast Fourier transform (PFFT) algorithm to accelerate the evaluation of far-field influences of source and/or normal dipole distributions on boundary elements. The resulting PFFT–QBEM reduces the computational effort of solving the associated boundary-value problem from O( N 2∼3) (with the traditional QBEM) to O( N ln N) where N represents the total number of boundary unknowns. Significantly, it allows for reliable computations of nonlinear hydrodynamics useful in ship design and marine applications, which are forbidden with the traditional methods on the presently available computing platforms. The formulation and numerical issues in the development and implementation of the PFFT–QBEM are described in detail. The characteristics of accuracy and efficiency of the PFFT–QBEM for various boundary-value problems are studied and compared to those of the existing accelerated (lower- and higher-order) boundary element methods. To illustrate the usefulness of the PFFT–QBEM, it is applied to solve the initial boundary-value problem in the generation of three-dimensional nonlinear waves by a moving ship hull. The predicted wave profile and resistance on the ship are compared to available experimental measurements with satisfactory agreements.

A Three-Dimensional Precorrected FFT Algorithm for Fast Method of Moments Solutions of the Mixed-Potential Integral Equation in Layered Media

A three-dimensional pre-corrected fast Fourier transform (PFFT) algorithm for the rapid solution of the full-dyadic Michalski-Zheng's mixed potential integral equation is presented. The integral equation is discretized with the Rao-Wilton-Glisson (RWG) method of moments. Handling the method of moments interactions with the dyadic kernel is simplified via representation of the RWG functions in terms of barycentric shape functions. The proposed three-dimensional precorrected FFT method distributes two-dimensional FFT grids nonuniformly along the direction of stratification according to conductor locations within the layers. For P two-dimensional FFT grids each with an average of Np associated triangular elements the method exhibits O(P 2 Np logNp) computational complexity and O(P Np) memory usage. The low-frequency breakdown of the integral equation is eliminated via loop-tree decomposition. A unique combination of O(N logN) computational complexity, fully three-dimensional boundary-element modeling in layered substrates, and full-wave modeling from dc to multi-gigahertz frequencies makes the algorithm particularly useful for characterizing large interconnect networks embedded in multilayered substrates. The method is implemented as the electromagnetic solver in Cadence's Virtuoso RF Designer software.

Fast Analysis of RCS Over a Frequency Band Using Pre-Corrected FFT/AIM and Asymptotic Waveform Evaluation Technique

The pre-corrected fast Fourier transform (PFFT)/adaptive integral method (AIM) is combined with the asymptotic waveform evaluation (AWE) technique to present fast RCS calculation for arbitrarily shaped three-dimensional PEC objects over a frequency band. The electric field integral equation (EFIE) is used to formulate the problem and the method of moments (MoM) is employed to solve the integral equation. By using the AWE method, the unknown equivalent current is expanded into a Taylor series around a frequency in the desired frequency band. Then, instead of solving the equivalent current at each frequency point, it is only necessary to solve for the coefficients of the Taylor series (called ldquomomentsrdquo) at each expansion point. Since the number of the expansion points is usually much smaller than that of the frequency points, the AWE can achieve fast frequency sweeping. To facilitate the analysis of large problems, in this paper, all the full matrices are stored in a sparse form and the PFFT/AIM method is employed to accelerate all the matrix-vector products on both sides of the matrix equation for the moments. Further, the incomplete LU preconditioner is used at each expansion point to improve the convergence behaviour of the matrix equation for the moments. The present method can deal with much larger problems than the conventional MoM-AWE method since the PFFT/AIM achieves considerable reduction in memory requirement and computation time. Numerical results will be presented to show the efficiency and capability of the method.

A Fast Combined Field Volume Integral Equation Solution to EM Scattering by 3-D Dielectric Objects of Arbitrary Permittivity and Permeability

A fast solution to the combined field volume integral equation (CFVIE) for electromagnetic scattering by large three-dimensional dielectric bodies of arbitrary permittivity and permeability is presented. The CFVIE is formulated in the region of the scatterers by expressing the total fields as the sum of the incident wave and the radiated wave due to both the electric and magnetic polarization currents. The resultant integral equation is solved using the method of moments (MoM). Then the precorrected fast Fourier transform (P-FFT) method is applied to reduce the memory requirement and accelerate the matrix-vector multiplication in the MoM solution. In the implementation of the P-FFT method, two sets of projection operators are constructed respectively for the projections of the electric sources and magnetic sources. In addition, two sets of interpolation operators are also applied respectively for the computation of the vector/scalar potentials and the curl of the vector potentials in the support of the testing functions. The resultant method has a memory requirement of O(N) and a computational complexity of O(NlogN) respectively, where N denotes the number of unknowns.

Algorithms in FastStokes and Its Application to Micromachined Device Simulation

For a wide variety of micromachined devices, designers need accurate analysis of fluid drag forces for complicated three-dimensional (3-D) problems. This paper describes FastStokes, a recently developed 3-D fluid analysis program. FastStokes rapidly computes drag forces on complicated structures by solving an integral formulation of the Stokes equation using a precorrected fast Fourier transform (PFFT)-accelerated boundary element method (BEM). The specializations of the PFFT algorithm to the Stokes flow problem are described, and computational results are presented. Timing results are used to demonstrate that FastStokes scales almost linearly with problem complexity, can easily analyze structures as complicated as an entire comb drive in under an hour, and can produce results that accurately match measured data.

Analysis of scattering by a large array of waveguide-fed wide-slot millimeter wave antennas using precorrected-FFT algorithm

An effective numerical technique employing the precorrected-fast Fourier transform (P-FFT) technique based on the method-of-moments (MoM) is presented to study the scattering from wide-slot antennas fed by waveguides with arbitrary terminations. As the Rao-Wilton-Glisson (RWG) functions are used as the basis and testing functions, the magnetic currents with arbitrary slot directions are considered. In addition, the P-FFT technique is employed to accelerate the MoM procedure for a large slot array. The numerical results are obtained and compared with both the traditional MoM results obtained using the entire-domain basis functions and the experiments results, to demonstrate better effectiveness and accuracy of this technique for millimeter wave scattering from arbitrary wide-slot antennas.

RCS computation for a large array of waveguide slots with finite wall thickness using the MoM accelerated by P-FFT algorithm

This correspondence presents an accurate and efficient method of moments (MoM) for analyzing electromagnetic scattering by a large finite array of waveguide slots with finite thickness. In this method, the mixed potential integral equation (MPIE) is utilized onto both upper and lower surfaces of the slots, and the subdomain MoM procedure is implemented to obtain the equivalent magnetic current distributions. The precorrected fast Fourier transform (P-FFT) method is employed to accelerate the entire computational process to reduce significantly the memory requirements for large arrays. During the MoM procedure, the Rao-Wilton-Glisson (RWG) functions are used as the basis and testing functions, with both z- and x-directional magnetic currents considered. This approach extends applicability of the present method to solve the MPIE for characterizing waveguide slots of arbitrary shape and distribution and the capability of the integral equation based fast algorithm. Numerical results are obtained and compared with the experimental results, to verify the accuracy and efficiency of this technique.

Studies on Fast Algorithm for the Scattering of a Large Array of Waveguide-Fed Wide-Slot Antennas in Millimeter Wave Region

We present an effective numerical technique to characterize the scattering of wide-slot antennas fed by waveguides with arbitrary terminations in terms of the method of moment (MoM) and the mixed potential integral equation (MPIE). In particular, the precorrected-fast Fourier transform (P-FFT) eliminates the need to generate and store the usual square impedance matrix andthus leads to speed up the matrix-vector multiplication in the resultant system. This property makes the Rao-Wilton-Glisson (RWG) functions to be useful in simulating electrically large-scale problems. In addition, the scattering from the finite ground surfaces is accounted for in the total scattered field by using the method of equivalent edge currents. The numerical results are presented and compared with both the traditional method of moment results obtained using the entire-domain basis functions and the experimental results, to demonstrate the proposed method to be a good candidate for study on the scattering of arbitrary wide-slot large array.

Large-scale broad-band parasitic extraction for fast layout verification of 3-D RF and mixed-signal on-chip structures

A methodology for efficient parasitic extraction and verification flow for RF and mixed-signal integrated-circuit designs is presented. The implementation of a multiplane precorrected fast Fourier transform (PFFT) computational engine enables the full-wave electromagnetic (EM) simulation of interconnects and passive components. The PFFT algorithm is implemented on a set of two-dimensional fast Fourier transform grids associated with the current sheets corresponding to the conductor loss models. This leads to the full-wave modeling of silicon embedded three-dimensional circuits within the two-and-one-half-dimensional computational framework yielding the O(NlogN) computational complexity and O(N) memory requirements of the algorithm. The broad-band capability of the EM solver is provided through the loop-tree/charge implementation of the PFFT algorithm allowing for robust full-wave modeling from dc to microwaves. The EM verification flow is integrated seamlessly within the Cadence environment allowing for the nonlinear circuit simulation of the entire device. The capability and accuracy of the proposed methodology is demonstrated through EM simulation results for an individual on-chip spiral inductor, as well as a low-noise amplifier.

A fast analysis of electromagnetic scattering by arbitrarily shaped homogeneous dielectric objects

AbstractIn this paper, we present a fast analysis of electromagnetic scattering by arbitrarily shaped 3D homogeneous dielectric objects. The solution is based on the PMCHW integral equation formulation. The method of moments (MoM) is used to solve the integral equations and the precorrected fast Fourier transform (P‐FFT) method is used to eliminate the need to generate and store the square impedance matrix and to speed up the matrix–vector product. Several numerical examples are included, which illustrate the accuracy and capability of the present method. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 38: 30–35, 2003

Boundary-Element Methods In Offshore Structure Analysis

Abstract Boundary-element methods, also known as panel methods, have been widely used for computations of wave loads and other hydrodynamic characteristics associated with the interactions of offshore structures with waves. In the conventional approach, based on the low-order panel method, the submerged surface of the structure is represented by a large number of small quadrilateral plane elements, and the solution for the velocity potential or source strength is approximated by a constant value on each element. In this paper, we describe two recent developments of the panel method. One is a higher-order method where the submerged surface can be represented exactly, or approximated to a high degree of accuracy by B-splines, and the velocity potential is also approximated by B-splines. This technique, which was first used in the research code HIPAN, has now been extended and implemented in WAMIT. In many cases of practical importance, it is now possible to represent the geometry exactly to avoid the extra work required previously to develop panel input files for each structure. It is also possible to combine the same or different structures which are represented in this manner, to analyze multiple-body hydrodynamic interactions. Also described is the pre-corrected Fast Fourier Transform method (pFFT) which can reduce the computational time and required memory of the low-order method by an order of magnitude. In addition to descriptions of the two methods, several different applications are presented.