# Uncertain Input Data Problems and the Worst Scenario Method, Volume 46

## 1st Edition

**Series Volume Editors:**Jan Achenbach

**Authors:**Ivan Hlavacek Jan Chleboun Ivo Babuska

**eBook ISBN:**9780080543376

**Hardcover ISBN:**9780444514356

**Imprint:**Elsevier Science

**Published Date:**9th December 2004

**Page Count:**484

**View all volumes in this series:**North-Holland Series in Applied Mathematics and Mechanics

## Table of Contents

Preface List of Figures List of Tables Introduction Acknowledgments

I Reality, Mathematics, and Computation 1 Modeling, Uncertainty, Verification, and Validation 1.1 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 Method 3 Formulation, Solvability, Approximation, Convergence 3.1 Worst Scenario Problem 3.2 Approximate Worst Scenario Problem 3.3 Convergence Analysis

III Quasilinear Elliptic Boundary Value Problems 4 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 Problem 5.3 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 Problem 10 Bending of a Thermoelastic Beam with an Uncertain Coupling Coefficient 10.1 Approximations Bibliography and Comments on Chapter V

VI 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 Data
14.3 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 Plasticity 17.1 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 Problems 17.6 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 Analysis 25.1 Matrix-Based State Problems 25.2 Weakly Formulated Elliptic State Problems 25.3 General Theorem 26 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

## Description

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.

## Key Features

· 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

## Readership

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.

## Details

- No. of pages:
- 484

- Language:
- English

- Copyright:
- © Elsevier Science 2004

- Published:
- 9th December 2004

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780080543376

- Hardcover ISBN:
- 9780444514356