Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods

Abstract

11R9. Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods. - LG Stanley (Montana State Univ, Bozeman MT) and DL Stewart (Air Force Inst of Tech, Wright Patterson Air Force Base, WPAFB OH). SIAM, Philadelphia. 2002. 139 pp. ISBN 0-89871-524-5. $65.00.Reviewed by Yu-Tai Lee (David Taylor Model Basin, 9500 MacArthur Blvd, W Bethesda MD 20817).Design Sensitivity Analysis is a condensed book that introduces the mathematical aspects of the continuous sensitivity equation methods (CSEMs) for partial differential equations (PDEs). It may serve as a reference book for graduate students or scientists working in the areas of numerical analysis and computational mathematics. A familiarity with real or functional analysis would make reading easier for some chapters of the book. Sensitivity analysis is an important element in the context of systems optimization and optimal designs. It provides the necessary and critical assessment of the influence of the system/design parameters on the state of the systems or the design evaluations. The book covers only topics related to the construction and analysis of algorithms for computing sensitivities and is not application specific. The authors start with an overview of the early-developed algorithms including the discretize-then-differentiate approach, which approximates sensitivities by first employing some discretization scheme to approximate the solution to a PDE and then implicitly differentiating this result to obtain a sensitivity approximation scheme. By pointing out the shortcoming of these early techniques, they lead the readers into the concept of CSEM (ie, a differentiate-then-discretize scheme) with a mathematical interpretation of the sensitivity analysis. Examples of simplified linear and nonlinear one-dimensional problems are used to demonstrate the procedures and formulation of the CSEM. Coordinate transformation is discussed for the purpose of practical computational requirements. Both the hybrid SEM (H-SEM) and an abstract version of the semianalytic method (A-SAM) are introduced from the coordinate transformation. Discussion of both CSEMs is included and numerical results for the linear and nonlinear one-dimensional problems are obtained in illustrating the methodologies. The authors devote themselves in the last part of the book to the mathematical framework for the Navier-Stokes equations. They use a finite-element approach to demonstrate methodologies that continuously solve the sensitivity equation with a remeshing strategy and provide improved solutions to the Navier-Stokes equations. The book is presented in a well-thought-out order and the simplified examples used are properly selected to convey the concepts. Since the topic of the sensitivity analysis serves as the basis to many engineering shape optimization applications, a chapter or two demonstrating the connection of the two subjects seems to be a strategy to increase readership. The finite-element approach is used as the basic discretization scheme for all the examples used in the book. Finite-difference or finite-volume methods, however, are used more frequently in most of the CFD software. The development of the sensitivity analysis calculation for the later schemes is not demonstrated. Instead of using a nomenclature section for the mathematical symbols used, the authors have chosen to refer them to a reference, ie, Wloka 54, quoted in the bibliography section. This may increase readers’ difficulty in following the context. Design Sensitivity Analysis serves as a good reference book for students and researchers to understand the concept of the CSEMs. In order to adapt the methodology for other problems, it would require further detailed formulation. Continuous growth in adapting the CSEM in complex CFD software is, however, envisioned in the near future.