STOCHASTIC FINITE ELEMENTS:
A Spectral Approach

SYNOPSIS

In view of some copyright issues, only selected sections of this on-line edition have read permissions.

The first section of each chapter is readable
Chapters 1, 5, 6, and the references are readable
The references are readable
Some other random sections are readable

ISBN: 0-387-97456-3 Springer-Verlag New York Berlin Heidelberg
ISBN: 0-540-97456-3 Springer-Verlag Berlin Heidelberg New York

Roger G. Ghanem

Pol D. Spanos

To my Mother and my Father.

To my parents Demetri and Aicaterine, my first mentors in quantitative thinking; to my wife Olympia, my permanent catalyst in substantive living; and to my children Demetri and Evie, a perpetual source of delightful randomness.

`` ... The principal means for ascertaining truth - induction and analogy - are based on probabilities; so that the entire system of human knowledge is connected with the theory (of probability) ... ``
Pierre Simon de Laplace,
A Philosophical Essay on Probability, 1816.

`` ... Nature permits us to calculate only probabilities, yet science has not collapsed.''
Richard P. Feynman,
QED: The Strange Theory of Light and Matter, 1985.

• Preface
• Contents
• INTRODUCTION
  • Motivation
  • Review of Available Techniques.
  • The Mathematical Model
  • Outline
• REPRESENTATION OF STOCHASTIC PROCESSES
  • Preliminary Remarks
  • Review of the Theory
  • Karhunen-Loeve Expansion
    • Derivation
    • Properties
      • Error Minimizing Property
      • Uniqueness of the Expansion
      • Expansion of Gaussian Processes
      • Other Properties
    • Solution of the Integral Equation
      • Preliminary Remarks
      • Rational Spectra
      • Nonrational Spectra
      • Numerical Solution
  • Homogeneous Chaos
    • Preliminary Remarks
    • Definitions and Properties
    • Construction of the Polynomial Chaos
• STOCHASTIC FINITE ELEMENT METHOD: Response Representation
  • Preliminary Remarks
  • Deterministic Finite Elements
    • Problem Definition
    • Variational Approach
    • Galerkin Approach
    • p-Adaptive Methods, Spectral Methods and Hierarchical Finite Element Bases
  • Stochastic Finite Elements
    • Preliminary Remarks
    • Monte Carlo Simulation (MCS)
    • Perturbation Method
    • Neumann Expansion Method
    • Improved Neumann Expansion
    • Projection on the Homogeneous Chaos
    • Geometrical and Variational Extensions
      • Geometry of the Solution Process
      • Variational Extensions
• STOCHASTIC FINITE ELEMENT METHOD: Response Statistics
  • Reliability Theory Background
  • Statistical Moments
    • Moments and Cummulants Equations
    • Second Order Statistics
  • Approximation to the Probability Distribution
  • Reliability Index and Response Surface Simulation
• NUMERICAL EXAMPLES
  • Preliminary Remarks
  • One Dimensional Static Problem
    • Formulation
    • Results
      • Improved Neumann Expansion
      • Projection on the Homogeneous Chaos
      • Monte Carlo Simulation
  • Two Dimensional Static Problem
    • Formulation
      • Square Plate - Analytical Solution
      • Curved Plate - Numerical Solution
    • Results
      • Analytical Solution
      • Monte Carlo Simulation
  • One Dimensional Dynamic Problem
    • Description of the Problem
    • Implementation
    • Results
• SUMMARY AND CONCLUDING REMARKS
• References
• Index
• About this document ...