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.
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.
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
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