The Stochastic Perturbation Method for Computational Mechanics

Marcin Kaminski is a professor within the Faculty of Civil Engineering, Architecture and Environmental Engineering at the Technical University of Lodz, Poland. Having obtained a PhD in the field of stochastic finite elements in 1997 he has continued his research work in the area, winning the John Argyris award in computational mechanics of solids and fluids in 2001 at ECCOMAS. He currently lectures in the stochastic perturbation method at Lodz. His monograph Computational Mechanics of Composite Materials: Sensitivity, Randomness and Multiscale Behaviour was published in 2002 by Springer, and he has authored over 150 research papers.

Introduction 3

1. Mathematical considerations 14

1.1. Stochastic perturbation technique basis 14

1.2. Least squares technique description 34

1.3. Time series analysis  47

2. The Stochastic Finite Element Method (SFEM) 73

2.1. Governing equations and variational formulation 73

2.1.1. Linear potential problems  73

2.1.2. Linear elastostatics 75

2.1.3. Nonlinear elasticity problems 78

2.1.4. Variational equations of elastodynamics 79

2.1.5. Transient analysis of the heat transfer 80

2.1.6. Thermo-piezoelectricity governing equations 82

2.1.7. Navier-Stokes equations 86

2.2. Stochastic Finite Element Method equations 89

2.2.1. Linear potential problems  89

2.2.2. Linear elastostatics 91

2.2.3. Nonlinear elasticity problems 94

2.2.4. SFEM in elastodynamics 98

2.2.5. Transient analysis of the heat transfer 101

2.2.6. Coupled thermo-piezoelectrostatics SFEM equations 105

2.2.7. Navier-Stokes perturbation-based equations 107

2.3. Computational illustrations  109

2.3.1. Linear potential problems  109

2.3.1.1. 1D fluid flow with random viscosity 109

2.3.1.2. 2D potential problem by the response function 114

2.3.2. Linear elasticity 118

2.3.2.1. Simple extended bar with random stiffness 118

2.3.2.2. Elastic stability analysis of the steel telecommunication tower 123

2.3.3. Nonlinear elasticity problems 129

2.3.4. Stochastic vibrations of the elastic structures 133

2.3.4.1. Forced vibrations with random parameters for a simple 2 d.o.f. system 133

2.3.4.2. Eigenvibrations of the steel telecommunication tower with random stiffness 138

2.3.5. Transient analysis of the heat transfer 140

2.3.5.1. Heat conduction in the statistically homogeneous rod 140

2.3.5.2. Transient heat transfer analysis by the RFM 145
3. The Stochastic Boundary Element Method (SBEM) 152

3.1. Deterministic formulation of the Boundary Element Method  151

3.2. Stochastic generalized perturbation approach to the BEM 156

3.3. The Response Function Method into the SBEM equations  158

3.4. Computational experiments  162

4. The Stochastic Finite Difference Method (SFDM) 186
4.1. Analysis of the unidirectional problems with Finite Differences 186

4.1.1. Elasticity problems  186

4.1.2. Determination of the critical moment for the thin-walled elastic structures  199

4.1.3. Introduction to the elastodynamics using difference calculus 204

4.1.4. Parabolic differential equations  210

4.2. Analysis of the boundary value problems on 2D grids  214

4.2.1. Poisson equation  214

4.2.2. Deflection of elastic plates in Cartesian coordinates 219

4.2.3. Vibration analysis of the elastic plates 227

5. Homogenization problem 230

5.1. Composite material model 232

5.2. Statement of the problem and basic equations 237

5.3. Computational implementation 244

5.4. Numerical experiments 246

6. Concluding remarks  284

7. References 289

8. Index 300

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