Numerical Methods in Sensitivity Analysis and Shape Optimization

by Laporte, Emmanuel; Le Tallec, Patrick

• ISBN13: 9780817643225
• ISBN10: 0817643222
• eBook ISBN(s): 9781461200697
• Edition: CD
• Format: Hardcover
• Copyright: 2002-12-01
• Publisher: Birkhauser
• More Book Details
• Purchase Benefits

Summary

Sensitivity analysis and optimal shape design are key issues in engineering that have been affected by advances in numerical tools currently available. This book/interactive CD-ROM presents basic optimization techniques that can be used to compute the sensitivity of a given design to local change or to improve its performance by local optimization of these data. The relevance and scope of these techniques have improved dramatically in recent years because of progress in discretization strategies, optimization algorithms, automatic differentiation, software availability, and the power of personal computers.Key features of this original, progressive, and comprehensive approach:* description of mathematical background and underlying tools* up-to-date review of grid construction and control, optimization algorithms, software differentiation and gradient calculations* practical solutions for implementation in many real-life problems* solution of illustrative examples and exercises* basic mathematical programming techniques used to solve constrained minimization problems are presented; these fairly self-contained chapters can serve as an introduction to the numerical solution of generic constrained optimization problems* companion CD-ROM contains source files and data; readers can test different solution strategies to determine their relevance and efficiency* CD-ROM also offers software building, updating computational grids, performing automatic code differentiation, and computing basic aeroelastic solutions.Numerical Methods in Sensitivity Analysis and Shape Optimization will be of interest to graduate students involved in mathematical modeling and simulation, as well as engineers and researchers in applied mathematics looking for an up-to-date introduction to optimization techniques, sensitivity analysis, and optimal design. The work is suitable as a textbook for graduate courses in any of the topics mentioned above, and as a reference text.

Table of Contents

Outline and Notation	ix
Acknowledgements	xv
Basic Formulations	1	(24)
A generic example	1	(3)
Abstract formulation of a shape optimization problem	4	(3)
General framework and notation	4	(1)
Discretization	5	(2)
Sensitivity analysis	7	(1)
Shape parametrization	8	(4)
Cubic spline interpolation	8	(2)
Wing profile parametrization by cubic splines	10	(1)
Parametrization of a shape deformation	11	(1)
Mesh construction and deformation	12	(7)
Mesh construction	13	(2)
A simple mesh deformation algorithm	15	(1)
A sophisticated mesh deformation algorithm	16	(3)
Exercises	19	(6)
Mesh generation around a wing profile	19	(2)
A software package for mesh deformation	21	(1)
Cubic spline shape parametrization	21	(4)
Finite Dimensional Optimization	25	(10)
Basic problem and notation	25	(1)
Necessary conditions of optimality	26	(2)
Optimality conditions of Euler--Lagrange	28	(6)
Equality constraints	28	(2)
Inequality constraints	30	(4)
Exercises	34	(1)
Newton's Algorithms	35	(18)
The problem to solve	35	(1)
Newton's algorithm	36	(2)
Unconstrained optimization	38	(3)
Basic algorithms	38	(1)
Quasi-Newton variant	38	(1)
GMRES solution	39	(2)
Line search	41	(1)
Constrained optimization	41	(4)
Basic algorithm	41	(1)
Han's penalized cost function	42	(1)
Line search	43	(2)
Complement: gradient based algorithms	45	(2)
Complement: detailed description of the GMRES algorithm	47	(4)
Description	47	(1)
The main steps	47	(2)
Final algorithm	49	(1)
Convergence analysis	50	(1)
Exercises	51	(2)
Constrained Optimization	53	(18)
Optimality conditions	53	(1)
Interior point algorithms	54	(2)
Introduction	54	(1)
Basic philosophy	54	(1)
A simplified interior point algorithm	55	(1)
Interior point algorithms with deflection	56	(8)
Introduction	56	(3)
Central trajectories	59	(1)
Centered trajectory algorithms	60	(1)
Herskovits' algorithm without equality constraints	61	(2)
Introduction of equality constraints	63	(1)
Using an interior point algorithm	64	(7)
Code's description	64	(3)
Example	67	(1)
Application to optimum design problems	67	(1)
Exercises	68	(3)
Automatic Differentiation	71	(16)
Introduction	71	(1)
Computing gradients by finite differences	71	(1)
Schematic structure of a computer code	72	(2)
Automatic differentiation	74	(8)
Examples of automatic differentiation by Odyssee	82	(2)
Exercises	84	(3)
Computing Gradients by Adjoint States	87	(10)
Structure of the cost function	87	(1)
A Lagrangian approach	88	(2)
Application to automatic differentiation in adjoint mode	90	(1)
Numerical calculation of the gradient of the cost function	91	(1)
Calculation of the Hessian	92	(5)
Applications	97	(40)
Introduction	97	(1)
Inverse problem in diffusion	97	(10)
Shape optimization in aerodynamics	107	(17)
Navier--Stokes equations	107	(1)
Euler's equations	108	(1)
NSC2KE solver	109	(3)
Input-Output data files	112	(12)
Computing the gradient in aeroelasticity	124	(13)
One Shot Methods	137	(24)
Introduction	137	(2)
Global algorithm	139	(1)
Reduction of the linear systems	140	(2)
Line search	142	(2)
Detailed algorithm	144	(5)
Numerical examples	149	(3)
An example without inequality constraints	150	(1)
Constrained torsion	151	(1)
Conclusion	152	(9)
Conclusions	161	(2)
A Subroutine cubspl	163	(2)
B Prototype Programmes for the Optimization Code	165	(8)
Main programme	165	(5)
Cost function and constraints	170	(1)
Gradients	171	(1)
Typical values of control parameters	172	(1)
C Odyssee User's Manual (short version)	173	(2)
D A Subroutine computing the Gradient with respect to the Grid for the Steady Aerodynamic Example	175	(14)
A typical aerodynamic cost function	175	(7)
A full subroutine for computing the gradient with respect to the grid	182	(7)
Bibliography	189	(4)
Index	193