Inverse Problems in Engineering Mechanics IV (eBook)

Proceedings of the International Symposium on

Mana Tanaka (Herausgeber)

eBook Download: PDF

2003 | 1. Auflage
544 Seiten
Elsevier Science (Verlag)
978-0-08-053517-3 (ISBN)

Infos

"This latest collection of proceedings provides a state of the art review of research on inverse problems in engineering mechanics.

Inverse problems can be found in many areas of engineering mechanics, and have many successful applications. They are concerned with estimating the unknown input and/or the characteristics of a system given certain aspects of its output. The mathematical challenges of such problems have to be overcome through the development of new computational schemes, regularization techniques, objective functionals, and experimental procedures. The papers within this represent an excellent reference for all in the field.

* Providing a state of the art review of research on inverse problems in engineering mechanics
* Contains the latest research ideas and related techniques
* A recognized standard reference in the field of inverse problems?
* Papers from Asia, Europe and America are all well represented"

Inhalt

Front Cover 1
Inverse Problems in Engineering Mechanics IV 4
Copyright Page 5
Contents 10
Preface 6
Symposium Chair 8
International Organizing Committee 8
International Scientific Committee 8
Local Organizing Committee 9
Part 1: Inverse Thermal Problems 14
Chapter 1. Application of the proper orthogonal decomposition in steady state inverse problems 16
Chapter 2. Estimation of thermophysical properties of a drying body at high mass transfer Biot number 26
Chapter 3. Boundary and geometry inverse thermal problems in continuous casting 34
Chapter 4. Boundary value identification analysis in the unsteady heat conduction problem 46
Chapter 5. A hyper speed boundary element-based inverse convolution scheme for solution of IHCP 56
Chapter 6. Numerical method for backward heat conduction problems using an arbitrary-order finite difference method 66
Part 2: Parameter Identification and Design 76
Chapter 7. Analysis of inverse transient thermoelasticity problems by filtration methods 78
Chapter 8. Estimation of unknown boundary values from inner displacement and strain measurements and regularization using rank reduction method 88
Chapter 9. Modal measurements using strain sensors and application to impact force identification 98
Chapter 10. Material coefficients identification of bone tissues using evolutionary algorithms 108
Chapter 11. Estimation of material properties on tactile warmth of wood by human's hand 116
Chapter 12. Inverse analysis method for identification of local elastic properties by using displacement data 124
Chapter 13. Determination of the mass density of the layer deposited on the surface of the resonator in QCM (Quartz Crystal Microbalance) 134
Chapter 14. Identifications of source distributions using BEM with dual reciprocity method 140
Chapter 15. Numerical source identification for Poisson equation 150
Chapter 16. A human-like optimization method for constrained parametric design 160
Part 3: Damage or Defect Detection 170
Chapter 17. Cross-sectional imaging of three-dimensional flaw from waveforms in a restricted measurement surface 172
Chapter 18. Computational inverse techniques for crack detection using dynamic responses 180
Chapter 19. Inverse scattering analysis for cavities in anisotropic solid 188
Chapter 20. Characterization of multiple cracks from eddy current testing signals by a template matching method and inverse analysis 198
Chapter 21. Identification of delamination defect in laminated composites by passive electric potential CT method 208
Chapter 22. Damage identification analysis on large scaled floating offshore structure model by parametric projection filter 218
Chapter 23. Corrosion pattern detection by multi-step genetic algorithm 226
Chapter 24. Crack identification in a Timoshenko beam from frequency change using genetic algorithm 234
Part 4: Shape or Object Detection 242
Chapter 25. Inverse determination of smelter wall erosion shapes using a Fourier series method 244
Chapter 26. An efficient singular superposition technique for cavity detection and shape optimization 254
Chapter 27. Detection of subsurface cavities and inclusions using genetic algorithms 264
Chapter 28. A remote sensing method of ground mines using time difference IR images 274
Part 5: Applications in Solid Mechanics 284
Chapter 29. A CRE model updating method in structural dynamics with uncertain measurements 286
Chapter 30. Conjugate Gradient and L-curve like methods for large inverse problem 296
Chapter 31. Modeling and identification in a dynamic viscoelastic contact problem with normal damped response and friction 308
Chapter 32. Solving equationless problems in elasticity using only boundary data 318
Chapter 33. Incremental approach for inverse analysis of 3D photoelasticity 328
Chapter 34. Inverse stress analysis of pinned connections using strain gages and airy stress function 336
Chapter 35. Considerations of multidimensional consolidation inverse analysis 346
Chapter 36. Optimum grain size based on a grain pullout model of polycrystalline alumina 356
Chapter 37. Adjoint method for the problem of coefficient identification in linear elastic wave equation 366
Part 6: Applications in Acoustics and Electromagnetics 374
Chapter 38. Finding optimal shapes of the sound-insulating wall by means of BEM and cellular automata 376
Chapter 39. Identification of speech source wave by inverse-filtering of vocal tract transfer characteristics 388
Chapter 40. A magnetostatic reconstruction of permeability distribution in material 396
Chapter 41. A use of BEM and NN system to estimate density distribution of plasma 402
Part 7: Applications in Fluid Mechanics and Aeronautics 412
Chapter 42. Solution to shape optimization problem of viscous flow fields considering convection term 414
Chapter 43. Application of inverse design method to estimation of wind tunnel models 422
Chapter 44. Implementation of the boundary conditions for cascade airfoil shape design using the discretized Navier-Stokes equations 432
Chapter 45. A study to estimate control surface deflections from flight data 440
Part 8: Mathematical and Numerical Aspects 448
Chapter 46. Linear ill-posed problems on sets of functions convex along all lines parallel to coordinate axes 450
Chapter 47. Optimization method for solving ill-posed boundary value problems for elliptic and hyperbolic equations 460
Chapter 48. Regularization parameters choosing for discrete ill-posed problems 470
Chapter 49. Error estimation of the reconstruction of symmetry velocity profiles using Abel type integral equation 478
Part 9: Computational Algorithms 488
Chapter 50. A priori information in image reconstruction 490
Chapter 51. Boundary element based solution for the Cauchy problem associated with the Helmholtz equation by the Tikhonov regularisation method 498
Chapter 52. Large molecular systems: computational modeling of geometry, force field parameters and intermolecular potential on a base of stable numerical methods 508
Chapter 53. Comparative analysis of boundary control and Gel'fand-Levitan methods of solving inverse acoustic problem 516
Chapter 54. 3D vector tomography reconstruction by series expansion method 526
Chapter 55. Tomographic reconstruction of the vector fields by Doppler spectroscopy measurements 534
Author Index 544

Textauszug

Application of the Proper Orthogonal Decomposition in Steady State Inverse Problems

Ryszard A. Białecki1 [e-mail]; Alain J. Kassab2 [e-mail]; Ziemowit Ostrowski1 [e-mail]    1 Institute of Thermal Technology, Silesian University of Technology, Gliwice, Poland
2 Department of Mechanical Material and Aerospace Engineering University of Central Florida, Orlando, USA

ABSTRACT

A novel inverse analysis technique for retrieving unknown boundary conditions has been developed. The first step of the approach is to solve a sequence of forward problems made unique by defining the missing boundary condition as a function of some unknown parameters. Taking several combinations of values of these parameters produces a sequence of solutions (snapshots) which are then sampled at a predefined set of points. Proper Orthogonal Decomposition (POD) is used to produce a truncated sequence of orthogonal basis functions, being appropriately chosen linear combinations of the snapshots. The solution of the forward problem is then written as a linear combination of the basis vectors. The unknown coefficients of this combination are evaluated by minimizing the discrepancy between the measurements and the POD approximation of the field. Two numerical examples show the robustness and numerical stability of the proposed scheme.

KEYWORDS

inverse problems

Proper Orthogonal Decomposition

steady state conduction

INTRODUCTION

Reduction of the number of the degrees of freedom in an inverse problem is a well known technique of filtering out the higher frequency error. The Proper Orthogonal Decomposition (POD) offers an elegant way to cut down the number of unknowns without loosing the accuracy. The method has been developed about 100 years ago as a tool of processing statistical data [1–4]. It has been also used in signal processing, pattern recognition, control theory, fluid flow and dynamics [5–7]. Another important area of application is in the turbulence where the technique, know also as Karhunen-Loeve method, has been used to detect the spatial large scale organized motions [8]. Some more recent description of POD theory can be found in [9,10].

The present paper presents a technique of applying POD in inverse analysis and, to the best of the knowledge of the authors, it is the first attempt to use POD in this context. The proposed technique uses the Proper Orthogonal Decomposition (POD) as a regularization technique. The idea is to solve a sequence of forward problems within the body under consideration. The solution of each problem is sampled at a predefined set of points. Each sampled field corresponds to a certain set of assumed values of the parameters defining the distribution of the function to be retrieved. POD detects the correlation between the discretized fields leading to a significant reduction of the degrees of freedom necessary to describe the field with a high accuracy.

BASICS OF POD

The fundamental notion of POD is the snapshot being a collection of N sampled values of the field under consideration. The snapshot is stored in a vector Ui,i = 1,2,…, M. A collection of all snapshots is a rectangular N by M matrix U. The snapshots are generated by changing the values of some parameter(s) upon which the field depends on. Time, parameters entering the boundary conditions, internal heat generation or material properties are examples of such parameters. The snapshots may be obtained either from a mathematical model of the phenomenon or from experiments. The aim of POD is to construct a set of vectors (basis) Φj resembling the original matrix U. The basis is stored in another rectangular matrix Φ of the same dimensionality as U. The elements of the basis are defined as

=U⋅V

  (1)

where V is a modal matrix defined as a nontrivial solution of a problem

⋅V=ΛV

  (2)

In the above Λ is a diagonal matrix storing the eigenvalues λi of the positive definite covariance matrix C. The entries of the latter are defined as

ij=∑k=1NUkiUkj

  (3)

the eigenvalues are real, positive and distinct and should be sorted in an descending order. The basis vectors associated with the eigenvalues are orthogonal ie

T⋅Φ=Λ.

  (4)

It can be shown [10], that the jth eigenvalue is a measure of the kinetic energy transferred within jth basis mode Φj (strictly speaking this is only true, when the field under consideration is the velocity field). Typically, this energy decreases rapidly with the increased number of the mode, which permits discarding the majority of modes. This can be done by deciding which fraction of the energy may be neglected in further calculations. The resulting POD basis ¯ consists of K < N elements.

This basis captures, in an optimal way, the spatial variation of the field. To have a full picture of the field, dependence on additional parameters used in generating the snapshots needs to be built into the approximation formula. This is accomplished by expressing the field represented by an arbitrary snapshot as

≈∑j=1KαjΦ¯j

  (5)

with the unknown scalars αj depending on the parameters. The αj’s are found by an appropriate procedure, say the least square fit or weighted residuals.

INVERSE PROBLEM

If the points at which the snapshots were evaluated coincide with the location of the sensors, the values of the unknown parameters αj in eq. (5) can be found by least square curve fitting. Note that only few terms would typically be retained in eq. (5), thereby significantly reducing the number of unknowns, which is a very desirable feature in any inverse analysis algorithm.

To make the discussion more concrete, assume that the aim of inverse problem is to retrieve a distribution of a heat flux on a portion of the boundary. Steady state heat conduction with other boundary conditions known is considered. Both heat conductivity and distribution of the internal heat generation are known. The functional form of the unknown heat flux is postulated, say as a polynomial of a given degree. The coefficients of the polynomial (parameters of heat flux distribution) are unknown. Additional information is produced by temperature sensors located at some points of the domain.

To generate the POD basis, a sequence of direct problems is solved using any analytical or numerical technique. For each problem, other combination of values of the unknown parameters is taken. All solutions are sampled at the same set of points. When a numerical technique is used, a natural snapshot is the set of all nodal temperatures. Adding normal heat fluxes at nodes located on the surface where the heat flux distribution is to be retrieved, significantly improves the stability of the algorithm. In the numerical examples discussed hereafter, snapshots gathering both temperatures and heat fluxes have been used.

To solve an inverse problem, some measured values are needed. The simplest way to approach this problem, is to locate the sensors at a set of points being a subset of points used to create the snapshots. Otherwise, some additional interpolation would be necessary.

Solution of each direct problems, ie., each combination of the unknown parameters generates one snapshot. From the collection of all snapshots, the truncated POD bases in eq.(5) can be constructed, based on the user specified fraction of the energy to be neglected. The values of the unknown multipliers in expansion (5) are calculated by a least square fit of the measurements and the model. Once the values of αj are known, the values of temperatures at all snapshot points can be determined. As the snapshot definition encompasses also the sought for heat fluxes, the procedure also yields the values the retrieved heat flux. Two simple numerical example are used to demonstrate the effectiveness of this technique.

NUMERICAL EXAMPLE 1. ANALYTICAL METHOD USED TO SOLVE THE FORWARD PROBLEM

Steady state heat conduction in a unit square 0 < x, y < 1 with unit conductivity is considered. Edges x = 0 and x = 1 are insulated. At y = 0 temperature is zero. The unknown distribution of the normal heat flux q at y = 1 is to be retrieved. It assumed that this distribution can be described by a cubic polynomial

¯=a+bx+cx2+dx3

  (6)

where a, b, c, d are unknown parameters. The sketch of the problem has been shown in Fig 1. Every of these parameters has been given a value of -1., -1.5,...

Produkt-Details

• Erscheint lt. Verlag: 19.11.2003
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-053517-8 / 0080535178
• ISBN-13: 978-0-08-053517-3 / 9780080535173