Uncertain Input Data Problems and the Worst Scenario Method

Uncertain Input Data Problems and the Worst Scenario Method, 1st Edition

Uncertain Input Data Problems and the Worst Scenario Method, 1st Edition,Ivan Hlavacek,Jan Chleboun,Ivo Babuska,ISBN9780444514356
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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


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.


* 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.

Information about this author is currently not available.
Information about this author is currently not available.
Information about this author is currently not available.

Uncertain Input Data Problems and the Worst Scenario Method, 1st Edition

List of Figures
List of Tables

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

V&V in Computational Engineering
A View of V&V
Process and Rules for Model Selection
Subject Index
List of Symbols
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