Temporal Basis Functions Research Articles

Design of Temporal Basis Functions for Time Domain Integral Equation Methods With Predefined Accuracy and Smoothness

A key parameter in the design of integral equation methods for transient electromagnetic scattering is the definition of temporal basis functions. The choice of temporal basis functions has a profound impact on the efficiency and accuracy of the numerical scheme. This paper presents a framework for the design of temporal basis functions with predefined accuracy and varying smoothness properties. The well-known shifted Lagrange basis functions naturally fit in this framework. New spline basis functions will be derived that have the same interpolation accuracy as shifted Lagrange basis functions and with the added advantage of being smooth. Numerical experiments show the positive influence of smoothness on the quadrature error in the numerical integration procedure. The global accuracy in time of the numerical scheme based on shifted Lagrange and spline basis functions has been experimentally analyzed. For a given interpolation error the experiments confirm the expected accuracy for the shifted Lagrange basis functions, but remarkably show a higher order of accuracy for the spline basis functions.

Computer modeling techniques for flapping-wing aerodynamics of a locust

We present an overview of the special computer modeling techniques we have developed recently for flapping-wing aerodynamics of a locust. The wing motion and deformation data is from an actual locust, extracted from high-speed, multi-camera video recordings of the locust in a wind tunnel. The special techniques have been developed around our core computational technique, which is the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation. Here we use the version of the DSD/SST formulation derived in conjunction with the variational multiscale (VMS) method, and this version is called “DSD/SST-VMST.” The special techniques are based on using, in the space–time flow computations, NURBS basis functions for the temporal representation of the motion and deformation of the locust wings. Temporal NURBS basis functions are used also in representation of the motion of the volume meshes computed and in remeshing. In this special-issue paper, we present a condensed version of the material from [1], concentrating on the flapping-motion modeling and computations, and also a temporal-order study from [2].

Space–time techniques for computational aerodynamics modeling of flapping wings of an actual locust

We present the special space---time computational techniques we have introduced recently for computational aerodynamics modeling of flapping wings of an actual locust. These techniques have been designed to be used with the deforming-spatial-domain/stabilized space---time (DSD/SST) formulation, which is the core computational technique. The DSD/SST formulation was developed for flow problems with moving interfaces and was elevated to newer versions over the years, including the space---time version of the residual-based variational multiscale (VMS) method, which is called "DSD/SST-VMST" and used in the computations reported here. The special space---time techniques are based on using, in the space---time flow computations, NURBS basis functions for the temporal representation of the motion and deformation of the locust wings. The motion and deformation data is extracted from the high-speed, multi-camera video recordings of a locust in a wind tunnel. In addition, temporal NURBS basis functions are used in representation of the motion and deformation of the volume meshes computed and also in remeshing. These ingredients provide an accurate and efficient way of dealing with the wind tunnel data and the mesh. The computations demonstrate the effectiveness of the core and special space---time techniques in modeling the aerodynamics of flapping wings, with the wing motion and deformation coming from an actual locust.

A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions

We consider retarded boundary integral formulations of the three-dimensional wave equation in unbounded domains. Our goal is to apply a Galerkin method in space and time in order to solve these problems numerically. In this approach the computation of the system matrix entries is the major bottleneck. We will propose new types of finite-dimensional spaces for the time discretization. They allow variable time-stepping, variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially. The reason is that the basis functions of these spaces are globally smooth and compactly supported. In order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere. In this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiments.

Galerkin reduced-order modeling scheme for time-dependent randomly parametrized linear partial differential equations

SUMMARY In this paper, we consider the problem of constructing reduced-order models of a class of time-dependent randomly parametrized linear partial differential equations. Our objective is to efficiently construct a reduced basis approximation of the solution as a function of the spatial coordinates, parameter space, and time. The proposed approach involves decomposing the solution in terms of undetermined spatial and parametrized temporal basis functions. The unknown basis functions in the decomposition are estimated using an alternating iterative Galerkin projection scheme. Numerical studies on the time-dependent randomly parametrized diffusion equation are presented to demonstrate that the proposed approach provides good accuracy at significantly lower computational cost compared with polynomial chaos-based Galerkin projection schemes. Comparison studies are also made against Nouy's generalized spectral decomposition scheme to demonstrate that the proposed approach provides a number of computational advantages. Copyright © 2012 John Wiley & Sons, Ltd.

Highly undersampled phase‐contrast flow measurements using compartment‐based k–t principal component analysis

The applicability of cine blood flow measurements in a clinical setting is often compromised by the long scan times associated with phase-contrast imaging. In this work, we propose an extension to the k-t principal component analysis method and demonstrate that by definition of spatial compartment-dependent temporal basis functions, significant improvements in reconstruction accuracy can be achieved relative to the original k-t principal component analysis and k-t SENSE formulations. Using this method, it is shown that prospective nominal undersampling of up to 16 corresponding to a net acceleration factor of 8 including training data acquisition can be realized while keeping the error in stroke volume below 5%. As a practical application, the acquisition of cine flow data in the aorta is demonstrated permitting assessment of two-dimensional velocity images and pulse wave velocities at 100 frames per second in a single breathhold per slice.

Time-domain MoM for the analysis of thin-wire structures above half-space media using complex-time Green's functions and band-limited quadratic B-spline temporal basis functions

Abstract The transient response of a thin wire in the presence of a half-space is calculated with a new formulation of the Method of Moments (MoM) in time domain, based on a novel Time-Domain Mixed Potential Integral Equation (TD-MPIE), using complex-time Green's functions. The excitation is a Gaussian voltage source and the solution is obtained by using a Marching-On-in-Time (MOT) procedure. Band-Limited Quadratic B-spline (BL-QB) functions are used as Temporal Basis Functions (TBFs). They are compared with Band-Limited Linear B-spline (BL-LB) interpolation functions. Numerical results show that the solution using BL-QB TBFs is stable and accurate, without late-time instabilities, and efficient in terms of memory usage and computation time.

Electromagnetic Scattering From General Bi-Isotropic Objects Using Time-Domain Integral Equations Combined With PMCHWT Formulations

Electromagnetic scattering by general bi-isotropic objects is calculated by using the time-domain integral equations which are incorporated with the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulations. By introducing a pair of equivalent electric and magnetic sources, electromagnetic fields inside a homogeneous bi-isotropic region can be represented by these sources over its boundary after applying fields splitting method. A series of coupled surface integral equations are obtained after imposing boundary conditions. These equations are then solved numerically using the method of moment (MoM) which involves separate spatial and temporal testing procedures. The Rao-Wilton-Glisson (RWG) functions are used as the spatial expansion and testing functions, and the weighted Laguerre functions are derived as the temporal basis and testing functions. Numerical results such as transient currents, far scattered fields, and normalized radar cross sections are presented and compared with analytical results as well as MoM-based frequency-domain analysis, and good agreements are observed.

A TIME DOMAIN INTEGRAL EQUATION SOLVER FOR SCATTERING FROM GENERAL CHIRAL OBJECTS

In this paper, transient electromagnetic scattering by general Chiral objects is investigated using time-domain integral equations with the Poggio, Miller, Chang, Harrington, Wu, and Tsai (PMCHWT) formulations. By introducing a pair of equivalent electric and magnetic currents, electromagnetic flelds inside a homogeneous Chiral region can be represented by these sources over its boundary. The uncoupled equations are solved numerically by the Galerkin's method that involves separate spatial and temporal testing procedures. The scaled Laguerre functions are used as the temporal basis and testing functions. The use of the Laguerre functions completely removes the time variable from computation, and the results are stable even at late times. Numerical results are presented and compared with analytical results, and good agreements are observed.

Size reduction technique for the marching-on-in-order time-domain integral equation method in analysis of transient electromagnetic scattering

The aggregative basis functions (ABFs) are introduced to construct a size-reduced system for the marching-on-in-order (MOO) time-domain integral equation (TDIE) method to analyse transient electromagnetic scattering from conducting objects. Based on the previously developed characteristic basis function method (CBFM), a set of orthogonal vectors that expand the original unknown current coefficients are obtained via the singular value decomposition (SVD). The ABF method can be considered as an application and counterpart of CBFM in TDIE with some differences. The ABFs are aggregations of the weighted Laguerre polynomials and RWG basis functions, which are the elemental temporal and spatial basis functions, respectively. The ABFs are defined over the entire geometry and effective in each order of the MOO scheme. The proposed method gives significant reduction to matrix size and also the storage by several orders of magnitude. This is achieved because of the much less number of ABFs or the orthogonal vectors than the inner edges of the geometry. Several numerical results are presented to illustrate the validity of the proposed method.

An Improved Marching-on-in-Degree Method Using a New Temporal Basis

The marching-on-in-degree (MOD) method has been presented earlier for solving time domain electric field integral equations in a stable fashion. This is accomplished by expanding the transient responses by a complete set of orthogonal entire domain associated Laguerre functions, which helps one to analytically integrate out the time variable from the final computations in a Galerkin methodology. So, the final computations are carried out using only the spatial variables. However, the existing MOD method suffers from low computational efficiency over a marching-on-in-time (MOT) method. The two main causes of the computational inefficiency in the previous MOD method are now addressed using a new form of the temporal basis functions and through a different computational arrangement for the Green's function. In this paper, it is shown that incorporating these two new concepts can speed up the computational process and make it comparable to a MOT algorithm. Sample numerical results are presented to illustrate the validity of these claims in solution of large problems using the MOD method.

Marching-on-in-order time-domain volume-surface integral equations in analysis of transient electromagnetic scattering from composite conducting-dielectric objects

In this letter, the matching-on-in-order time-domain volume-surface integral equation method is used to analyze transient electromagnetic scattering from objects comprising both conductor and dielectric material. SWG and RWG basis functions are used as the spatial basis functions in dielectric region and on the conducting surface, respectively. Different orders of weighted Laguerre polynomials are used as temporal basis functions, and the relative expansion coefficients are solved recursively order by order. The spatial and temporal testing procedures are separate, and both of them are carried on with the Galerkin's method. Stable solutions are ensured, and several numerical results are given to verify the proposed method. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1315–1319, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.25987

Accuracy-efficiency tradeoff of temporal basis functions for time-marching solvers

An investigation of the impact of the temporal discretization on the marching-on-in-time solution of integral equations is presented. Numerical results that quantify the efficiency–accuracy tradeoff for causal piecewise polynomial and band-limited interpolatory functions are presented. It is observed that the former is more efficient for low to moderate accuracy levels, and the latter achieves higher, but extrapolation-limited, accuracy levels. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1343–1348, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.25960

Time-domain augmented EFIE and its marching-on-in-degree solution

A time-domain augmented electric field integral equation (TDAEFIE) and its marching-on-in-degree (MOD) solution are presented for analysis of transient electromagentic responses from three-dimensional closed conducting bodies of arbitrary shape.By enforcing a condition on the normal component of the electric flux density, the TDAEFIE eliminates the potential internal resonance problem of the time-domain electric field integral equation (TDEFIE) algorithm. With the use of weighted Laguerre polynomials as entire-domain temporal basis functions, the MOD solution overcomes the late-time instability problem that often occurs in the marching-on-in-time (MOT) approach. Compared with the MOD solution of the time-domain combined field integral equation (TDCFIE), the MOD solution of the TDAEFIE is more efficient because it takes less computational time for calculating the matrix elements and the matrix-vector multiplications related to the excitation at the right-hand side of the matrix equation. Numerical results are presented to illustrate the good performance of the TDAEFIE algorithm. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1439–1444, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26015

Efficient TDFEM schemes with second-order PML based on different temporal basis functions

The time domain finite element method (TDFEM) formulations have been presented using two kinds of temporal basis functions, B-spline and Lagrange interpolation polynomial, and the computational domain is truncated with second-order perfectly matched layer (PML). This is motivated by utilising good interpolation features of these temporal basis functions and better absorption performances of second-order PMLs. Numerical cases demonstrate that the proposed TDFEM schemes achieve more accurate results with about 5% to 7% increase of run time and 1.5% increase of memory usage compared with the TDFEM scheme with Sacks PML. However, the presented TDFEM schemes merely require 1/6th of the total run time for frequency domain FEM.

Fast MR parameter mapping using k‐t principal component analysis

Quantification of magnetic resonance parameters plays an increasingly important role in clinical applications, such as the detection and classification of neurodegenerative diseases. The major obstacle that remains for its widespread use in clinical routine is the long scanning times. Therefore, strategies that allow for significant decreases in scan time are highly desired. Recently, the k-t principal component analysis method was introduced for dynamic cardiac imaging to accelerate data acquisition. This is done by undersampling k-t space and constraining the reconstruction of the aliased data based on the k-t Broad-use Linear Acquisition Speed-up Technique (BLAST) concept and predetermined temporal basis functions. The objective of this study was to investigate whether the k-t principal component analysis concept can be adapted to parameter quantification, specifically allowing for significant acceleration of an inversion recovery fast imaging with steady state precession (TrueFISP) acquisition. We found that three basis functions and a single training data line in central k-space were sufficient to achieve up to an 8-fold acceleration of the quantification measurement. This allows for an estimation of relaxation times T(1) and T(2) and spin density in one slice with sub-millimeter in-plane resolution, in only 6 s. Our findings demonstrate that the k-t principal component analysis method is a potential candidate to bring the acquisition time for magnetic resonance parameter mapping to a clinically acceptable level.

A Time-Domain Volume Integral Equation and Its Marching-On-in-Degree Solution for Analysis of Dispersive Dielectric Objects

A marching-on-in-degree (MOD)-based scheme for analyzing transient electromagnetic scattering from three-dimensional dispersive dielectric objects is proposed. A time-domain volume integral equation (TDVIE) for the electric flux density throughout the object is first formulated and then solved using the MOD scheme. With the use of weighted Laguerre polynomials as entire-domain temporal basis functions, the convolution of the electric flux density and the medium susceptibility and its derivatives can be handled analytically. By employing the Galerkin temporal testing procedure, the time variable is eliminated in the resultant recursive matrix equation so that the proposed algorithm overcomes the late-time instability problem that may occur in the conventional marching-on-in-time (MOT) approach. Some complex dispersive media, such as the Debye, Lorentz, and Drude media, are simulated to illustrate the validity of the TDVIE-MOD algorithm.

Accelerated Dynamic MRI Exploiting Sparsity and Low-Rank Structure: k-t SLR

We introduce a novel algorithm to reconstruct dynamic magnetic resonance imaging (MRI) data from under-sampled k-t space data. In contrast to classical model based cine MRI schemes that rely on the sparsity or banded structure in Fourier space, we use the compact representation of the data in the Karhunen Louve transform (KLT) domain to exploit the correlations in the dataset. The use of the data-dependent KL transform makes our approach ideally suited to a range of dynamic imaging problems, even when the motion is not periodic. In comparison to current KLT-based methods that rely on a two-step approach to first estimate the basis functions and then use it for reconstruction, we pose the problem as a spectrally regularized matrix recovery problem. By simultaneously determining the temporal basis functions and its spatial weights from the entire measured data, the proposed scheme is capable of providing high quality reconstructions at a range of accelerations. In addition to using the compact representation in the KLT domain, we also exploit the sparsity of the data to further improve the recovery rate. Validations using numerical phantoms and in vivo cardiac perfusion MRI data demonstrate the significant improvement in performance offered by the proposed scheme over existing methods.

TIME-DOMAIN INTEGRAL EQUATION SOLVER FOR RADIATION FROM DIPOLE ANTENNA LOADED WITH GENERAL BI-ISOTROPIC OBJECTS

Electromagnetic radiation by dipole antenna loaded with general bi-isotropic objects is investigated using time-domain integral equations. By introducing pairs of equivalent electric and magnetic sources, electromagnetic flelds inside a homogeneous bi-isotropic region can be represented by these sources over its boundary. A series of coupled surface integral equations are obtained after imposing boundary conditions. These equations are solved numerically by the Galerkin's method that involves separate spatial and temporal testing procedures. The scaled Laguerre functions are used as the temporal basis and testing functions. The use of the Laguerre functions completely removes the time variable from computation, and the results are stable even at late times. Numerical results are presented and compared with analytical results, and similarities and difierences are observed.

Analysis of transient electromagnetic scattering using UV method enhanced time‐domain integral equations with Laguerre polynomials

AbstractIn this article, the multilevel UV algorithm is utilized to reduce both the memory requirement and CPU time in the time‐domain combined field integral equation (TD‐CFIE) method to analyze transient electromagnetic scattering from conducting structures.The UV method is kernel‐independent, and is based on the rank‐deficiency feature of the impedance submatrix which represents far interaction. To obtain an unconditionally stable solution, finite orders of weighted Laguerre polynomials are used as temporal basis functions and the unknown expansion coefficients of the time variable are solved recursively order by order, as the so‐called matching‐on‐in‐order (MOO) procedure. RWG basis function is used as the spatial basis function for its flexibility in modeling arbitrary geometries. The spatial and temporal testing procedures are separate, and both of them are carried on with the Galerkin's method. Numerical results are given to verify the proposed method. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:158–163, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.25646