# Uncertain Input Data Problems and the Worst Scenario Method

**By**

- Ivan Hlavacek
- Jan Achenbach
- Jan Chleboun
- Ivo Babuska

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.

**Audience**

* Researchers and graduate students working in applied mathematics with emphasize on problems described by differential equations or variational inequalities.* Researchers and graduate students working in computational science related to engineering problems.* Researchers and graduate students working in the area of numerical methods.

Hardbound, 484 Pages

Published: December 2004

Imprint: Elsevier

ISBN: 978-0-444-51435-6

## Contents

- Preface List of FiguresList of Tables Introduction Acknowledgments I Reality, Mathematics, and Computation1 Modeling, Uncertainty, Verification, and Validation1.1 Modeling 1.2 1.2 Verification and Validation 1.3 Desirable Features of a Mathematical Model 2 Various Approaches to Uncertainty 2.1 Coupling the Worst Scenario Method with Fuzzy Sets, Evidence Theory, and Probability 2.1.1 Worst Scenario and Fuzzy Sets I 2.1.2 Worst Scenario and Evidence Theory 2.1.3 Worst Scenario and Probabilistic Methods 2.1.4 Worst Scenario and Fuzzy Sets II 2.2 Key Point: Admissible Set 2.3 How to Formulate Worst Scenario Problems 2.4 On the Origin of Data 2.5 Conclusions II General Abstract Scheme and the Analysis of the Worst Scenario Method3 Formulation, Solvability, Approximation, Convergence 3.1 Worst Scenario Problem 3.2 Approximate Worst Scenario Problem 3.3 Convergence Analysis III Quasilinear Elliptic Boundary Value Problems4 Uncertain Thermal Conductivity Problem 4.1 Setting of the Problem 4.2 Approximate Worst Scenario Problem 4.3 Convergence Analysis 4.4 Sensitivity Analysis 4.5 Numerical Examples 4.6 Heat Conduction: Special Case 5 Uncertain Nonlinear Newton Boundary Condition 5.1 Continuous Problem 5.2 Approximate Problem5.3 Convergence of Approximate Solutions IV Parabolic Problems 6 Linear Parabolic Problems 6.1 Stability of Solutions to Parabolic Problems 6.2 Worst Scenario Problem 6.3 Approximate Worst Scenario Problem 6.4 Convergence Analysis 7 Parabolic Problems With a Unilateral Obstacle 7.1 Worst Scenario for a General Variational Inequality 7.2 Applications to Fourier Obstacle Problems V Elastic and Thermoelastic Beams 8 Transverse Vibration of Timoshenko Beams with an Uncertain Shear Correction Factor 8.1 Eigenvalue Problems 8.2 Worst Scenario Problems, Sensitivity Analysis 9 Buckling of a Timoshenko Beam on an Elastic Foundation 9.1 Buckling of a Timoshenko Beam 9.2 Buckling of a Simply Supported Timoshenko Beam on an Elastic Foundation 9.3 Singular and Negative Values of the Shear Correction Factor 9.4 Summary of the Analysis 9.5 Worst Scenario Problem10 Bending of a Thermoelastic Beam with an Uncertain Coupling Coefficient10.1 Approximations Bibliography and Comments on Chapter VVI Elastic Plates and Pseudoplates 11 Pseudoplates 11.1 Formulation of a State Problem 11.2 Stability of the Solution for a Class of Variational Inequalities 11.3 Application to a Unilateral Pseudoplate Problem 11.4 Criterion-Functionals and Worst Scenario Problems 11.5 Approximate State Problem 11.6 Approximate Worst Scenario Problems 11.7 Convergence of Approximate Solutions 12 Buckling of Elastic Plates 12.1 Buckling of a Rectangular Plate 12.2 Worst Scenario Problem 12.3 Initial Imperfection Combined from One and Two Halfsinewaves Bibliography and Comments on Chapter VI VII Contact Problems in Elasticity and Thermoelasticity 13 Signorini Contact Problem with Friction 13.1 Setting of the Worst Scenario Problems 13.2 Existence of a Worst Scenario 13.3 Approximate Worst Scenario Problems 13.4 Convergence Analysis 14 Unilateral Frictional Contact of Several Bodies in Quasi-Coupled Thermoelasticity 14.1 Setting of Thermoelastic Contact Problems 14.2 Sets of Uncertain Input Data14.3 Worst Scenario Problems 14.4 Stability of Weak Solutions 14.5 Existence of a Solution 14.6 Comments on Unilateral Contact with Coulomb Friction Bibliography and Comments on Chapter VII VIII Hencky's and Deformation Theories of Plasticity 15 Timoshenko Beam in Hencky's Model with Uncertain Yield Function 15.1 Setting of the Problem in Terms of Bending Moment and Shear Forces 15.2 Worst Scenario Problems 15.3 Numerical Examples: von Mises Yield Function 16 Torsion in Hencky's Model with Uncertain Stress-Strain Law and Uncertain Yield Function 16.1 Problem Setting and Stability of the Solution 16.2 Worst Scenario Problems 16.3 Approximate Worst Scenario Problems 16.4 Convergence Analysis 17 Deformation Theory of Plasticity17.1 Setting of the State Boundary Value Problem 17.2 Admissible Material Functions and the Unique Solvability of the State Problem 17.3 Continuous Dependence of the Solution 17.4 Worst Scenario Problems 17.5 Approximate Worst Scenario Problems17.6 Convergence Analysis Bibliography and Comments on Chapter VIII IX Flow Theories of Plasticity 18 Perfect Plasticity 18.1 State Problem 18.2 Worst Scenario Problems 18.3 Approximate Problems 19 Flow Theory with Isotropic Hardening 19.1 Formulation of the State Problem 19.2 Uncertain Input Data 19.3 Approximate State Problem 19.4 Approximate Worst Scenario Problems 20 Flow Theory with Isotropic Hardening in Strain Space 20.1 Variational Formulation of the State Problem 20.2 Uncertain Input Data 20.3 Regularizations of Problem P by Kinematic Hardening 20.4 Stability of the Solution of the Regularized Problem 20.5 Stability of the Stress Tensor 20.6 Worst Scenario Problems 21 Combined Linear Kinematic and Isotropic Hardening 21.1 Variational Formulation of the State Problem 21.2 Uncertain Input Data 21.3 Stability of the State Solution 21.4 Worst Scenario Problems 22 Validation of an ElastoPlastic Plane Stress Model Bibliography and Comments on Chapter IX X Domains With Uncertain Boundary 23 Neumann Boundary Value Problem 23.1 Instability of Solutions 23.2 Reformulated Newton Boundary Value Problem 23.3 Convergence with Respect to Sequences of Domains 23.4 Difference Between Two Solutions 23.5 Closing Remarks 24 Dirichlet Boundary Value Problem 24.1 Stability of Solutions 24.2 Difference Between Two Solutions 24.3 Numerical Example XI Essentials of Sensitivity and Functional Analysis 25 Essentials of Sensitivity Analysis25.1 Matrix-Based State Problems 25.2 Weakly Formulated Elliptic State Problems 25.3 General Theorem26 Essentials of Functional and Convex Analysis 26.1 Functional Analysis 26.2 Function Spaces Appendix V&V in Computational Engineering Introduction Definitions A View of V&V Process and Rules for Model Selection Summary Bibliography Subject Index List of Symbols