Higher-order Pade Approximations Research Articles

An effective method to suppress numerical dispersion in 2D acoustic and elastic modelling using a high-order Padé approximation

In this paper, we propose a method that can effectively reduce the numerical dispersion for solving the acoustic and elastic-wave equations. The method is a fourth-order Pade approximation scheme, in which the time difference operator is a rational function and a block tridiagonal system needs to be solved at each step. On the one hand, to efficiently solve a large linear system of equations we propose an explicit method for this implicit algrithom. On the other hand, to approximate the high-order spatial derivatives we use an eighth-order stereo-modelling method using wavefield displacements and their gradients simultaneously. For this new method, we investigate some mathematical properties including the stability, errors and the numerical dispersion relationship for 1D and 2D cases. We also present numerical results computed by the Pade approximation and compare them with the eighth-order Lax–Wendroff correction method and the eighth-order staggered-grid method. Numerical results show that the high-order Pade approximation scheme can effectively suppress the numerical dispersion caused by discretizing the wave equations when coarse spatial grids are used or models have strong velocity contrasts between adjacent grids. In contrast to other high-order finite-difference methods, the new method takes substantially less computational time and requires less memory because large spatial and time increments can be used. Thus the high-order Pade approximation method can potentially be used to solve large-scale wave propagation problems and seismic tomography based on the wave equations.

Continued Fraction Expansion을 이용한 Dead Time 근사의 새로운 접근

Dead times appear often in describing process dynamics and raise some difficulties in simulating process dynamics or analyzing process control systems. To relieve these difficulties, it is needed to approximate the infinite dimensional dead time by the finite dimensional transfer function and, for this, the Pade approximation method is often used. For the accurate approximation of the dead time, high order Pade approximation is needed and the high order Pade approximation is not easy to memorize and is not stable numerically. We propose a method based on the continued fraction expansion that provides the same transfer functions. The method is excellent numerically as well as systematic to be memorized easily. It can be used conveniently for the process control lecture and computations.

Analysis of on-chip distributed interconnects based on Pade expansion

In this paper, on-chip interconnects are modeled as distributed parameter RLCG transmission lines, based on which the matrix ABCD of interconnects is deduced. With help of the ABCD matrix, a voltage transfer function of an interconnect system, consisting of a driver, interconnect line and load, is obtained analytically in the form of a transcendental function, and it is reduced to a finite order system based on high order Pade approximation. With the reduced-order transfer function, response waveforms with step input can be obtained, and signal delay can be calculated consequently. Two numerical experiments are conducted to demonstrate its efficiency.

Observer-Based Smith Prediction Scheme for Unstable Plus Time Delay Processes

This work considers the stabilization problem for unstable linear input-delay systems. The main idea of the paper is to use a finite-dimensional approximation for the delay operator, which is based on non-overlapping partitions of the time delay. Subsequently, each individual delay is approximated by means of a classical Pade approximation, where the overall approximation results in a high-order Pade approximation that converges to the original delay operator. By departing from a state-space realization of the approximate process, a linear observer is used to estimate the delay-free output, which is used within a compensation scheme to stabilize the process output. The resulting control strategy has the structure of an observer-based Smith prediction scheme. Numerical results on three examples show that (i) the finer the time delay partition, the better the control performance and (ii) high-order compensators can be required to stabilize certain unstable processes.

Large strain elastic-plastic theory and nonlinear finite element analysis based on metric transformation tensors

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate-independent finite strain analysis of solids undergoing large elastic-plastic deformations. The formulation relies on the introduction of a mixed-variant metric transformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure whose rate is shown to be additively decomposed into elastic and plastic strain rate tensors. The mixed-variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response in the elastic-plastic solid. Additionally, the plastic material behavior is assumed to be governed by a generalized J2 yield criterion and rate-independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on 1st and higher order Pade approximations. Estimates of the stress and strain histories are obtained via a highly stable and accurate explicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step. The development of a consistent elastic-plastic tangent operator as well as its implementation into a nonlinear finite element program will also be discussed. Finally, the numerical solution of finite strain elastic-plastic problems is presented to demonstrate the efficiency of the algorithm.

Formulation and numerical treatment of incompressibility constraints in large strain elastic-plastic analysis

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate-independent finite strain analysis of solids undergoing large elastic-isochoric plastic deformations. The formulation relies on the introduction of a mixed-variant metric deformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure which can be additively decomposed into the exact isochoric (deviatoric) and volumetric (spheric) strain measures. This fact may be seen as the basic idea in the formulation of appropriate mixed finite elements which guarantee the accurate computation of isochoric strains. The mixed-variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response whereas the plastic material behavior is assumed to be governed by a generalized J2 yield criterion and rate-independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on higher-order Pade approximations. To be able to take into account the plastic incompressibility constraint a modified mixed variational principle is considered which leads to a quasi-displacement finite element procedure. Finally, the numerical solution of finite strain elastic-plastic problems is presented to demonstrate the efficiency and the accuracy of the algorithm. Copyright © 1999 John Wiley & Sons, Ltd.

Wide-Angle Parabolic Equation Solution of Ocean Acoustic Propagation with Lossy Penetrable Bottom

Accurate and efficient analysis methods of underwater sound propagation are essential for the development of ocean acoustic tomography. In this paper, the outlines of the derivation of two recently developed finite-difference parabolic equation methods based on the higher-order Pade approximations are given to predict the sound propagation in ocean acoustics. In order to show the validity and usefulness of these approaches, numerical examples are presented for the shallow water models considering the attenuation of sound field in ocean-bottom sediment.

Continuous-time identification of SISO systems using Laguerre functions

This paper looks at the problem of estimating the coefficients of a continuous-time transfer function given samples of its input and output data. We first prove that any nth-order continuous-time transfer function can be written as a fraction of the form /spl Sigma//sub k=0//sup n/b~/sub k/L/sub k/(s)//spl Sigma//sub k=0//sup n/a~/sub k/L/sub k/(s), where L/sub k/(s) denotes the continuous-time Laguerre basis functions. Based on this model, we derive an asymptotically consistent parameter estimation scheme that consists of the following two steps: (1) filter both the input and output data by L/sub k/(s), and (2) estimate {a~/sub k/, b~/sub k/} and relate them to the coefficients of the transfer function. For practical implementation, we require the discrete-time approximation of L/sub k/(s) since only sampled data is available. We propose a scheme that is based on higher order Pade approximations, and we prove that this scheme produces discrete-time filters that are approximately orthogonal and, consequently, a well-conditioned numerical problem. Some other features of this new algorithm include the possibility to implement it as either an off-line or a quasi-on-line algorithm and the incorporation of constraints on the transfer function coefficients. A simple example is given to illustrate the properties of the proposed algorithm.

Wide-angle finite-element beam propagation methodusing Padé approximation

A wide-angle beam propagation method using a finite-element scheme is presented. The method is based on Pade approximations expressed with polynomial matrix functions. Tilted-waveguide propagation characteristics are calculated, and an analysis for a steeper waveguide is shown to be allowed when proposed higher-order Pade approximations are used.

Higher-order parallel methods for a model of percutaneous drug absorption

A number of parallel algorithms based on higher-order Pade approximations are developed for the numerical solution of a mathematical model of percutaneous drug absorption. These methods are L o-stable and require only the application of tridiagonal solvers with complex arithmetic.