Uncertain Input Data Problems and the Worst Scenario Method
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Table of contents
- Uncertain Input Data Problems and the Worst Scenario Methodiii
- Copyrightiv
- Prefacev
- Contentsvii
- List of Figuresxiii
- List of Tablesxv
- Introductionxvii
- Acknowledgmentsxxv
- I Reality, Mathematics, and Computation1
- 1 Modeling, Uncertainty, Verification, and Validation1
- 2 Various Approaches to Uncertainty16
- II General Abstract Scheme and the Analysis of the Worst Scenario Method51
- 3 Formulation, Solvability, Approximation, Convergence51
- III Quasilinear Elliptic Boundary Value Problems61
- 4 Uncertain Thermal Conductivity Problem61
- 5 Uncertain Nonlinear Newton Boundary Condition89
- IV Parabolic Problems103
- 6 Linear Parabolic Problems103
- 7 Parabolic Problems With a Unilateral Obstacle120
- V Elastic and Thermoelastic Beams129
- 8 Transverse Vibration of Timoshenko Beams with an Uncertain Shear Correction Factor129
- 9 Buckling of a Timoshenko Beam on an Elastic Foundation144
- 10 Bending of a Thermoelastic Beam with an Uncertain Coupling Coefficient153
- VI Elastic Plates and Pseudoplates163
- 11 Pseudoplates163
- 12 Buckling of Elastic Plates188
- VII Contact Problems in Elasticity and Thermoelasticity207
- 13 Signorini Contact Problem with Friction207
- 14 Unilateral Frictional Contact of Several Bodies in Quasi-Coupled Thermoelasticity229
- VIII Hencky’s and Deformation Theories of Plasticity241
- 15 Timoshenko Beam in Hencky’s Model with Uncertain Yield Function241
- 16 Torsion in Hencky’s Model with Uncertain Stress-Strain Law and Uncertain Yield Function253
- 17 Deformation Theory of Plasticity265
- IX Flow Theories of Plasticity281
- 18 Perfect Plasticity281
- 19 Flow Theory with Isotropic Hardening302
- 20 Flow Theory with Isotropic Hardening in Strain Space322
- 21 Combined Linear Kinematic and Isotropic Hardening345
- 22 Validation of an Elasto-Plastic Plane Stress Model350
- X Domains With Uncertain Boundary357
- 23 Neumann Boundary Value Problem358
- 24 Dirichlet Boundary Value Problem379
- XI Essentials of Sensitivity and Functional Analysis391
- 25 Essentials of Sensitivity Analysis391
- 26 Essentials of Functional and Convex Analysis399
- Appendix413
- V&V in Computational Engineering413
- Bibliography427
- Subject Index449
- List of Symbols455
This book deals with the impact of uncertainty in input data on the outputs of mathematical models. Uncertain inputs as scalars, tensors, functions, or domain boundaries are considered. In practical terms, material parameters or constitutive laws, for instance, are uncertain, and quantities as local temperature, local mechanical stress, or local displacement are monitored. The goal of the worst scenario method is to extremize the quantity over the set of uncertain input data.
A general mathematical scheme of the worst scenario method, including approximation by finite element methods, is presented, and then applied to various state problems modeled by differential equations or variational inequalities: nonlinear heat flow, Timoshenko beam vibration and buckling, plate buckling, contact problems in elasticity and thermoelasticity with and without friction, and various models of plastic deformation, to list some of the topics. Dozens of examples, figures, and tables are included.
Although the book concentrates on the mathematical aspects of the subject, a substantial part is written in an accessible style and is devoted to various facets of uncertainty in modeling and to the state of the art techniques proposed to deal with uncertain input data.
A chapter on sensitivity analysis and on functional and convex analysis is included for the reader's convenience.
· Rigorous theory is established for the treatment of uncertainty in modeling
· Uncertainty is considered in complex models based on partial differential equations or variational inequalities
· Applications to nonlinear and linear problems with uncertain data are presented in detail: quasilinear steady heat flow, buckling of beams and plates, vibration of beams, frictional contact of bodies, several models of plastic deformation, and more
· Although emphasis is put on theoretical analysis and approximation techniques, numerical examples are also present
· Main ideas and approaches used today to handle uncertainties in modeling are described in an accessible form
· Fairly self-contained book
A general mathematical scheme of the worst scenario method, including approximation by finite element methods, is presented, and then applied to various state problems modeled by differential equations or variational inequalities: nonlinear heat flow, Timoshenko beam vibration and buckling, plate buckling, contact problems in elasticity and thermoelasticity with and without friction, and various models of plastic deformation, to list some of the topics. Dozens of examples, figures, and tables are included.
Although the book concentrates on the mathematical aspects of the subject, a substantial part is written in an accessible style and is devoted to various facets of uncertainty in modeling and to the state of the art techniques proposed to deal with uncertain input data.
A chapter on sensitivity analysis and on functional and convex analysis is included for the reader's convenience.
· Rigorous theory is established for the treatment of uncertainty in modeling
· Uncertainty is considered in complex models based on partial differential equations or variational inequalities
· Applications to nonlinear and linear problems with uncertain data are presented in detail: quasilinear steady heat flow, buckling of beams and plates, vibration of beams, frictional contact of bodies, several models of plastic deformation, and more
· Although emphasis is put on theoretical analysis and approximation techniques, numerical examples are also present
· Main ideas and approaches used today to handle uncertainties in modeling are described in an accessible form
· Fairly self-contained book