Computing diffraction integrals with the numerical method of steepest descent

نویسندگان

  • U. Peter Svensson
  • Andreas Asheim
چکیده

A common type of integral to solve numerically in computational room acoustics and other applications is the diffraction integral. Various formulations are encountered but they are usually of the Fourier-type, which means an oscillating integrand which becomes increasingly expensive to compute for increasing frequencies. Classical asympotic solution methods, such as the stationary-phase method, might have limited accuracy across the relevant frequency range. The method of steepest descent is known to offer efficient evaluation of such integrals but for most diffraction integrals, the optimum deformed integration path might be impossible to find analytically. A recent numerical version of the method of steepest descent finds an approximate path numerically and this paper will show the application of this method to one specific edge diffraction integral which is valid for finite and infinite edges. The required integration path sections are found numerically via applying a Taylor expansion of the integrand oscillator function, involving up to the fourth-order derivative for this example, and a subsequent series inversion. Once the path is avaliable, two efficient quadrature methods are used for the exponentially decaying integrands, Gauss-Laguerre and Gauss-Hermite. The method is compared with brute-force numerical integration using Gauss-Kronrod quadrature in the Matlab implementation. Numercial examples demonstrate that the new method has a computation time which is independent of frequency and of edge length, whereas that of the brute-force method depends heavily on frequency as well as edge length. It is shown that the accuracy of the new method decreases for low frequencies and for geometrical cases where the receiver point is near a zone boundary. Methods to tackle these limitations are outlined.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

A Free Line Search Steepest Descent Method for Solving Unconstrained Optimization Problems

In this paper, we solve unconstrained optimization problem using a free line search steepest descent method. First, we propose a double parameter scaled quasi Newton formula for calculating an approximation of the Hessian matrix. The approximation obtained from this formula is a positive definite matrix that is satisfied in the standard secant relation. We also show that the largest eigen value...

متن کامل

A new Levenberg-Marquardt approach based on Conjugate gradient structure for solving absolute value equations

In this paper, we present a new approach for solving absolute value equation (AVE) whichuse Levenberg-Marquardt method with conjugate subgradient structure. In conjugate subgradientmethods the new direction obtain by combining steepest descent direction and the previous di-rection which may not lead to good numerical results. Therefore, we replace the steepest descentdir...

متن کامل

Asymptotic Analysis of Numerical Steepest Descent with Path Approximations

We propose a variant of the numerical method of steepest descent for oscillatory integrals by using a low-cost explicit polynomial approximation of the paths of steepest descent. A loss of asymptotic order is observed, but in the most relevant cases the overall asymptotic order remains higher than a truncated asymptotic expansion at similar computational effort. Theoretical results based on num...

متن کامل

Hybrid steepest-descent method with sequential and functional errors in Banach space

Let $X$ be a reflexive Banach space, $T:Xto X$ be a nonexpansive mapping with $C=Fix(T)neqemptyset$ and $F:Xto X$ be $delta$-strongly accretive and $lambda$- strictly pseudocotractive with $delta+lambda>1$. In this paper, we present modified hybrid steepest-descent methods, involving sequential errors and functional errors with functions admitting a center, which generate convergent sequences ...

متن کامل

The Uniform geometrical Theory of Diffraction for elastodynamics: Plane wave scattering from a half-plane.

Diffraction phenomena studied in electromagnetism, acoustics, and elastodynamics are often modeled using integrals, such as the well-known Sommerfeld integral. The far field asymptotic evaluation of such integrals obtained using the method of steepest descent leads to the classical Geometrical Theory of Diffraction (GTD). It is well known that the method of steepest descent is inapplicable when...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2010