Conventional Numerical Approaches Research Articles

The interaction between a high-frequency gust and a blade row

The ingestion of convected vorticity by a high-solidity rotating blade row is a potent noise source in modern aeroengines, due largely to the high level of mutual aerodynamic interactions between adjacent blades. In order to model this process we solve the problem of determining the unsteady lift on an infinite cascade of finite-chord flat plates due to an incident vorticity wave. The method of solution is the Wiener–Hopf technique, and we consider the case of the reduced frequency, Ω, being large, allowing application of asymptotic analysis in the formal limit Ω → ∞. This approach yields considerable simplification, both in allowing the truncation of an infinite reflection series to just two terms, and in allowing algebraic expressions for the Wiener–Hopf split functions to be found. The unsteady lift distribution is derived in closed form, and the accuracy of the asymptotic Wiener–Hopf factorization demonstrated for even modest values of Ω by comparison with exact (but less tractable) methods. Our formulae can easily be incorporated into existing noise prediction codes: the advantage of our scheme is that it handles a regime in which conventional numerical approaches become unwieldy, as well as providing significant physical insight into the underlying mechanisms.

Three-dimensional representation of articulatory kinematics from x-ray microbeam data

Records of speech-related movements of flesh points and anatomic landmarks obtained from the University of Wisconsin x-ray microbeam system are fundamentally two dimensional in the sense that they provide time histories of relative positions of points within a plane. However, the third spatial dimension associated with each point is implicit in its two-space representation and, under special conditions, can be reconstructed mathematically. For example, three points attached to a rigid body such as the mandible determine a system of three polynomial equations, which, when solved, provides an account of the location of each point with respect to sagittal and coronal plane axes. Both the feasibility and resolution limits of a conventional numerical approach to this problem have been established by analysis of calibrated movements of three-point arrays attached to a machine-driven phantom and tracked by the microbeam system. Moreover, the same analysis procedure has been applied to kinematic data for multipoint arrays attached to the head and mandible of nine adult subjects to determine whether and to what extent their respective movements, during speech and other selected oral motor behaviors, are restricted to the sagittal plane. The advantages of a three-dimensional representation of speech motor behavior for at least these structures, within the context of recent microbeam experiments, will be discussed. [Work supported by NIH Grant NS-16373.]

A Computational Model for Predicting the Life of Tubes Used in Petrochemical Heater Service

This paper describes a model developed for computer simulation of stresses in heater tubes used in petrochemical service. The model predicts the damage to the tube material and thus provides an assessment of both tube design life and residual-life of tubes that have been subjected to service exposure. The analysis procedure has been incorporated into a computer code (TUBE), which has been applied to a number of applications for heat-resistant alloys. The focus of these applications has been to address the effect of sustained and cyclic loadings on the predicted service lives of tubes used in elevated temperature service. The model was originally developed at Battelle-Columbus in the 1970s, as part of an industrial group research program on “Materials for Steam Reformer Furnaces.” The computer model uses conventional numerical approaches [1–4] to solve finite element models of two-dimensional creep problems. This paper addresses the practical difficulties of applying such models to real service conditions and real commercial alloys.

Summary Representation Applied to Seepage Problems

The method of summary representation, which is based on applying the classical finite-difference approach to certain operators, is presented and applied to the numerical solution of certain boundary value problems associated with steady-state seepage. An example problem is given to illustrate how this technique reduces the number of algebraic equations obtained from conventional numerical approaches and how it decreases the necessary assumptions related to such analyses. Accordingly, computer efficiency can be increased, because the operating matrices are reduced in size, and problems which are defined over an infinite domain can be handled without introducing artificial boundary conditions.

Direct method approximation to the state regulator control problem using a Ritz-Trefftz suboptimal control

The linear quadratic cost control problem \dot{x}(t) = A(t)x(t) + B(t)u(t) x(0) = x_{0} with a cost functional J[u] = \frac{1}{2} \int\min{0}\max{T} [\langlex, Q(t)x\rangle + \langleu, R(t)u\rangle] dt is considered, supposing S is a suitable space of piecewise cubic polynominals on a mesh of norm h on the interval [0, T] . Then a Ritz type algorithm is developed for minimizing J [\cdotp] over S . The authors have previously discussed [3] certain convergence properties of the algorithm. Here the algorithm is discussed in a form suitable for real-time implementation and additional convergence criteria are presented. In [3] it was shown that the Ritz-Treffiz suboptimal control \bar{u} converges to the optimal control u\ast with order 0(h^{3}) . If x_{\bar{u}} is the trajectory generated by \bar{u} , then it is shown that x_{\bar{u}} approximates the optimal trajectory x\ast to 0(h^{3}) . Finally, it is shown that J[\bar{u}] approximates J[u\ast] to order 0(h^{6}) . The numerical properties of the algorithm, including speed and accuracy comparisons with the conventional numerical approach, are presented in a forthcoming paper.