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Microstructured Materials: Inverse Problems / by Jaan Janno, Jüri Engelbrecht
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Monographs in Mathematics | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2011Description: Online-Ressource (IX, 160p. 30 illus., 3 illus. in color, digital)ISBN:- 9783642215841
- 128347624X
- 9781283476249
- 620.11299
- 515.353
- QA370-380
- QA371
Contents:
Summary: Introduction -- 1 Inverse problems and non-destructive evaluation -- 2 Mathematical models of microstructured solids -- 3 Linear waves -- 4 Inverse problems for linear waves -- 5 Solitary waves in nonlinear models -- 6 Inverse problems for solitary waves -- 7 Summary -- References -- Index.Summary: Complex, microstructured materials are widely used in industry and technology and include alloys, ceramics and composites. Focusing on non-destructive evaluation (NDE), this book explores in detail the mathematical modeling and inverse problems encountered when using ultrasound to investigate heterogeneous microstructured materials. The outstanding features of the text are firstly, a clear description of both linear and nonlinear mathematical models derived for modelling the propagation of ultrasonic deformation waves, and secondly, the provision of solutions to the corresponding inverse problems that determine the physical parameters of the models. The data are related to nonlinearities at both a macro- and micro- level, as well as to dispersion. The authors’ goal has been to construct algorithms that allow us to determine the parameters within which we are required to characterize microstructure. To achieve this, the authors not only use conventional harmonic waves, but also propose a novel methodology based on using solitary waves in NDE. The book analyzes the uniqueness and stability of the solutions, in addition to providing numerical examples.PPN: PPN: 1651025460Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Microstructured Materials: Inverse Problems; Preface; Contents; Chapter 1: Introduction; Chapter 2: Inverse Problems and Non-destructive Evaluation; 2.1 Inverse Problems from a Mathematical Viewpoint; 2.2 Inverse Problems and Non-destructive Evaluation from a Practical Viewpoint; 2.2.1 General Remarks; 2.2.2 Practical Realisation; Chapter 3: Mathematical Models of Microstructured Solids; 3.1 Basic Principles; 3.2 Microstructured Solids; 3.3 General Formulation of Inverse Problems; Chapter 4: Linear Waves; 4.1 Dispersion Relations. Harmonic Waves; 4.1.1 Hierarchical Equation
4.1.2 Coupled SystemConvention; 4.1.3 Comparison of Models; 4.2 Other Linear Waves; 4.2.1 General Solution Formula; 4.2.2 Right-Propagating Waves; 4.2.3 Gaussian Wave Packets; 4.3 Proofs of Mathematical Statements; Chapter 5: Inverse Problems for Linear Waves; 5.1 Inverse Problems for Harmonic Waves; 5.1.1 Hierarchical Equation; 5.1.2 Coupled System; 5.1.3 General Consequences; 5.2 Inverse Problems for Gaussian Wave Packets; 5.3 Reconstruction of Parameters from Spectra of Waves; 5.3.1 The Case of Deformation Boundary Condition; 5.3.2 The Case of Displacement Boundary Condition
5.4 Stability and Examples5.4.1 Stability of Solutions; 5.4.2 Numerical Examples; 5.5 Proofs of Mathematical Statements; 5.5.1 Proof of Theorem 5.2; 5.5.2 Proofs of Sect. 5.2; Chapter 6: Solitary Waves in Nonlinear Models; 6.1 Solitary Waves; 6.2 Solitary Wave Solutions of Hierarchical Equation; 6.2.1 Reduction to Equation of First Kind. Canonical Description; 6.2.2 Existence and Basic Properties of Canonical Waves; 6.2.3 Physical and Geometrical Properties of Solitary Waves in General Form; 6.2.4 Series Expansion of Solitary Wave; 6.3 Solitary Wave Solutions of Coupled System
6.3.1 Separation of Unknowns. Reduction of System6.3.2 Existence and Basic Properties of Canonical Waves; 6.3.3 Properties of General Solitary Waves; 6.3.4 The Case nu=0; 6.3.5 Comparison with Hierarchical Equation; 6.4 Proofs of Mathematical Statements; 6.4.1 Proofs of Sect. 6.2; 6.4.2 Proofs of Sect. 6.3; Chapter 7: Inverse Problems for Solitary Waves; 7.1 Inverse Problems for Hierarchical Equation; 7.1.1 Formulation of Inverse Problems; 7.1.2 Uniqueness Issues; 7.1.3 Stability Estimates; 7.2 Inverse Problems for Coupled System; 7.2.1 Formulation of Inverse Problems; 7.2.2 Uniqueness Issues
7.3 Methods of Solution of Inverse Problems7.3.1 Minimisation of Cost Functional; 7.3.2 Application of Series Expansion. Linearisation; 7.3.3 Numerical Examples; 7.4 Proofs of Mathematical Statements; 7.4.1 Proofs of Sect. 7.1.2; 7.4.2 Proof of Theorem 7.5; 7.4.3 Proofs of Sect. 7.2.2; Chapter 8: Summary; 8.1 General Glance at Mathematical Methods; 8.2 From Mathematics to Physics; 8.3 Epilogue; References; Index;
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