Asymptotic expansions for efficient and accurate numerical evaluation of integral transforms

✍ Scribed by Fry, G. T. ;Robinson, A. R.

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 239 KB
• Volume: 15
• Category: Article
• ISSN: 1069-8299
• DOI: 10.1002/(sici)1099-0887(199912)15:12<901::aid-cnm304>3.0.co;2-5

✦ Synopsis

Analyses of practical engineering problems often require the repeated evaluation of semi-in®nite integral transforms whereby a single variable is changed incrementally through a large range of values. Common examples include time±history analyses of dynamical systems, and fatigue analyses of solid bodies. In such situations, any savings in the time required to evaluate the integral once could amount to substantial savings in the time required to perform the overall analysis. Usually these integrals are evaluated numerically from zero to some number that is deemed suciently large to capture most of the value of the integral: the larger the value, the more accurate the integration. An alternative approach is presented herein which enhances both the eciency and accuracy of evaluating such integrals by approximating the tail of the integrand by its ®rst-order asymptotic expansion. The method is presented in the context of determining the static stresses within the two-layered elastic half-plane subjected to normal and tangential Hertzian contact tractions. With the aid of Fourier transforms, analytic expressions are derived for the displacement, strain and stress functions in the frequency domain. Detailed stress distributions beneath the contact patch are computed using the proposed method.