Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

https://doc.rero.ch/record/9768/files/Renard_Ph.-_Calculating_equivalent_permeability_20080903.pdf

Calculating equivalent permeability: a review

In this paper we investigate the probability of failure of a cohesive slope using both simple and more advanced probabilistic analysis tools. The influence of local averaging on the probability of failure of a test problem is thoroughly investigated. In the simple approach, classical slope stability analysis techniques are used, and the shear strength is treated as a single random variable. The advanced method, called the random finite-element method (RFEM), uses elastoplasticity combined with random field theory. The RFEM method is shown to offer many advantages over traditional probabilistic slope stability techniques, because it enables slope failure to develop naturally by “seeking out” the most critical mechanism. Of particular importance in this work is the conclusion that simplified probabilistic analysis, in which spatial variability is ignored by assuming perfect correlation, can lead to unconservative estimates of the probability of failure. This contradicts the findings of other investigators w...

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Probabilistic slope stability analysis by finite elements

The concept of the suction stress characteristic curve (SSCC) for unsaturated soil is presented. Particle-scale equilibrium analyses are employed to distinguish three types of interparticle forces: (1) active forces transmitted through the soil grains; (2) active forces at or near interparticle contacts; and (3) passive, or counterbalancing, forces at or near interparticle contacts. It is proposed that the second type of force, which includes physicochemical forces, cementation forces, surface tension forces, and the force arising from negative pore-water pressure, may be conceptually combined into a macroscopic stress called suction stress. Suction stress characteristically depends on degree of saturation, water content, or matric suction through the SSCC, thus paralleling well-established concepts of the soil–water characteristic curve and hydraulic conductivity function for unsaturated soils. The existence and behavior of the SSCC are experimentally validated by considering unsaturated shear strength data for a variety of soil types in the literature. Its characteristic nature and a methodology for its determination are demonstrated. The experimental evidence shows that both Mohr–Coulomb failure and critical state failure can be well represented by the SSCC concept. The SSCC provides a potentially simple and practical way to describe the state of stress in unsaturated soil.

Suction Stress Characteristic Curve for Unsaturated Soil

Preface. 1 Introduction. 1.1 Importance of Heat Transfer. 1.2 Heat Transfer Modes. 1.3 The Laws of Heat Transfer. 1.4 Formulation of Heat Transfer Problems. 1.4.1 Heat transfer from a plate exposed to solar heat flux. 1.4.2 Incandescent lamp. 1.4.3 Systems with a relative motion and internal heat generation. 1.5 Heat Conduction Equation. 1.6 Boundary and Initial Conditions. 1.7 Solution Methodology. 1.8 Summary. 1.9 Exercise. Bibliography. 2 Some Basic Discrete Systems. 2.1 Introduction. 2.2 Steady State Problems. 2.2.1 Heat flow in a composite slab. 2.2.2 Fluid flow network. 2.2.3 Heat transfer in heat sinks (combined conduction-convection). 2.2.4 Analysis of a heat exchanger. 2.3 Transient Heat Transfer Problem (Propagation Problem). 2.4 Summary. 2.5 Exercise. Bibliography. 3 The Finite Elemen t Method. 3.1 Introduction. 3.2 Elements and Shape Functions. 3.2.1 One-dimensional linear element. 3.2.2 One-dimensional quadratic element. 3.2.3 Two-dimensional linear triangular elements. 3.2.4 Area coordinates. 3.2.5 Quadratic triangular elements. 3.2.6 Two-dimensional quadrilateral elements. 3.2.7 Isoparametric elements. 3.2.8 Three-dimensional elements. 3.3 Formulation (Element Characteristics). 3.3.1 Ritz method (Heat balance integral method-Goodman's method). 3.3.2 Rayleigh-Ritz method (Variational method). 3.3.3 The method of weighted residuals. 3.3.4 Galerkin finite element method. 3.4 Formulation for the Heat Conduction Equation. 3.4.1 Variational approach. 3.4.2 The Galerkin method. 3.5 Requirements for Interpolation Functions. 3.6 Summary. 3.7 Exercise. Bibliography. 4 Steady State Heat Conduction in One Dimension. 4.1 Introduction. 4.2 Plane Walls. 4.2.1 Homogeneous wall. 4.2.2 Composite wall. 4.2.3 Finite element discretization. 4.2.4 Wall with varying cross-sectional area. 4.2.5 Plane wall with a heat source: solution by linear elements. 4.2.6 Plane wall with a heat source: solution by quadratic elements. 4.2.7 Plane wall with a heat source: solution by modified quadratic equations (static condensation). 4.3 Radial Heat Flow in a Cylinder. 4.3.1 Cylinder with heat source. 4.4 Conduction-Convection Systems. 4.5 Summary. 4.6 Exercise. Bibliography. 5 Steady State Heat Conduction in Multi-dimensions. 5.1 Introduction. 5.2 Two-dimensional Plane Problems. 5.2.1 Triangular elements. 5.3 Rectangular Elements. 5.4 Plate with Variable Thickness. 5.5 Three-dimensional Problems. 5.6 Axisymmetric Problems. 5.6.1 Galerkin's method for linear triangular axisymmetric elements. 5.7 Summary. 5.8 Exercise. Bibliography. 6 Transient Heat Conduction Analysis. 6.1 Introduction. 6.2 Lumped Heat Capacity System. 6.3 Numerical Solution. 6.3.1 Transient governing equations and boundary and initial conditions. 6.3.2 The Galerkin method. 6.4 One-dimensional Transient State Problem. 6.4.1 Time discretization using the Finite Difference Method (FDM). 6.4.2 Time discretization using the Finite Element Method (FEM). 6.5 Stability. 6.6 Multi-dimensional Transient Heat Conduction. 6.7 Phase Change Problems-Solidification and Melting. 6.7.1 The governing equations. 6.7.2 Enthalpy formulation. 6.8 Inverse Heat Conduction Problems. 6.8.1 One-dimensional heat conduction. 6.9 Summary. 6.10 Exercise. Bibliography. 7 Convection Heat Transfer 173 7.1 Introduction. 7.1.1 Types of fluid-motion-assisted heat transport. 7.2 Navier-Stokes Equations. 7.2.1 Conservation of mass or continuity equation. 7.2.2 Conservation of momentum. 7.2.3 Energy equation. 7.3 Non-dimensional Form of the Governing Equations. 7.3.1 Forced convection. 7.3.2 Natural convection (Buoyancy-driven convection). 7.3.3 Mixed convection. 7.4 The Transient Convection-diffusion Problem. 7.4.1 Finite element solution to convection-diffusion equation. 7.4.2 Extension to multi-dimensions. 7.5 Stability Conditions. 7.6 Characteristic-based Split (CBS) Scheme. 7.6.1 Spatial discretization. 7.6.2 Time-step calculation. 7.6.3 Boundary and initial conditions. 7.6.4 Steady and transient solution methods. 7.7 Artificial Compressibility Scheme. 7.8 Nusselt Number, Drag and Stream Function. 7.8.1 Nusselt number. 7.8.2 Drag calculation. 7.8.3 Stream function. 7.9 Mesh Convergence. 7.10 Laminar Isothermal Flow. 7.10.1 Geometry, boundary and initial conditions. 7.10.2 Solution. 7.11 Laminar Non-isothermal Flow. 7.11.1 Forced convection heat transfer. 7.11.2 Buoyancy-driven convection heat transfer. 7.11.3 Mixed convection heat transfer. 7.12 Introduction to Turbulent Flow. 7.12.1 Solution procedure and result. 7.13 Extension to Axisymmetric Problems. 7.14 Summary. 7.15 Exercise. Bibliography. 8 Convection in Porous Media. 8.1 Introduction. 8.2 Generalized Porous Medium Flow Approach. 8.2.1 Non-dimensional scales. 8.2.2 Limiting cases. 8.3 Discretization Procedure. 8.3.1 Temporal discretization. 8.3.2 Spatial discretization. 8.3.3 Semi- and quasi-implicit forms. 8.4 Non-isothermal Flows. 8.5 Forced Convection. 8.6 Natural Convection. 8.6.1 Constant porosity medium. 8.7 Summary. 8.8 Exercise. Bibliography. 9 Some Examples of Fluid Flow and Heat Transfer Problems. 9.1 Introduction. 9.2 Isothermal Flow Problems. 9.2.1 Steady state problems. 9.2.2 Transient flow. 9.3 Non-isothermal Benchmark Flow Problem. 9.3.1 Backward-facing step. 9.4 Thermal Conduction in an Electronic Package. 9.5 Forced Convection Heat Transfer From Heat Sources. 9.6 Summary. 9.7 Exercise. Bibliography. 10 Implementation of Computer Code. 10.1 Introduction. 10.2 Preprocessing. 10.2.1 Mesh generation. 10.2.2 Linear triangular element data. 10.2.3 Element size calculation. 10.2.4 Shape functions and their derivatives. 10.2.5 Boundary normal calculation. 10.2.6 Mass matrix and mass lumping. 10.2.7 Implicit pressure or heat conduction matrix. 10.3 Main Unit. 10.3.1 Time-step calculation. 10.3.2 Element loop and assembly. 10.3.3 Updating solution. 10.3.4 Boundary conditions. 10.3.5 Monitoring steady state. 10.4 Postprocessing. 10.4.1 Interpolation of data. 10.5 Summary. Bibliography. A Green's Lemma. B Integration Formulae. B.1 Linear Triangles. B.2 Linear Tetrahedron. C Finite Element Assembly Procedure. D Simplified Form of the Navier-Stokes Equations. Index.

Fundamentals of the Finite Element Method for Heat and Fluid Flow

The majority of slope stability analyses performed in practice still use traditional limit equilibrium approaches involving methods of slices that have remained essentially unchanged for decades. This was not the outcome envisaged when Whitman & Bailey (1967) set criteria for the then emerging methods to become readily accessible to all engineers. The finite element method represents a powerful alternative approach for slope stability analysis which is accurate, versatile and requires fewer a priori assumptions, especially, regarding the failure mechanism. Slope failure in the finite element model occurs 'naturally' through the zones in which the shear strength of the soil is insufficient to resist the shear stresses. The paper describes several examples of finite element slope stability analysis with comparison against other solution methods, including the influence of a free surface on slope and dam stability. Graphical output is included to illustrate deformations and mechanisms of failure. It is argue...

Slope stability analysis by finite elements

From the Publisher:
In the third edition of this highly-successful book, readers are shown how to develop computer programs which use the finite element method to solve specific problems which they may encounter. This third edition is completely revised, and has all programs and subroutine libraries written in FORTRAN 90. The programs and most subroutines are listed in full in the text, and are all available on the Internet via anonymous ftp. A wide variety of new problem solving analyses are presented, including classical structural analysis, elasticity and plasticity, steady state and transient fluid flow, linear and non-linear solid dynamics and construction processes in geomechanics.

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Programming the finite element method

Preface. Acknowledgements. PART 1: THEORY. Chapter 1: Review of Probability Theory. 1.1 Introduction. 1.2 Basic Set Theory. 1.3 Probability. 1.4 Conditional Probability. 1.5 Random Variables and Probability Distributions. 1.6 Measures of Central Tendency, Variability, and Association. 1.7 Linear Combinations of Random Variables. 1.8 Functions of Random Variables. 1.9 Common Discrete Probability Distributions. 1.10 Common Continuous Probability Distributions. 1.11 Extreme-Value Distributions. Chapter2: Discrete random Processes. 2.1 Introduction. 2.2 Discrete-Time, Discrete-State Markov Chains. 2.3 Continuous-Time Markov Chains. 2.4 Queueing Models. Chapter 3: Random Fields. 3.1 Introduction. 3.2 Covariance Function. 3.3 Spectral Density Function. 3.4 Variance Function. 3.5 Correlation Length. 3.6 Some Common Models. 3.7 Random Fields in Higher Dimensions. Chapter 4: Best Estimates, Excursions, and Averages. 4.1 Best Linear Unbiased Estimation. 4.2 Threshold Excursions in One Dimension. 4.3 Threshold Excursions in Two Dimensions. 4.4 Averages. Chapter 5: Estimation. 5.1 Introduction. 5.2 Choosing a Distribution. 5.3 Estimation in Presence of Correlation. 5.4 Advanced Estimation Techniques. Chapter 6: Simulation. 6.1 Introduction. 6.2 Random-Number Generators. 6.3 Generating Nonuniform Random Variables. 6.4 Generating Random Fields. 6.5 Conditional Simulation of Random Fields. 6.6 Monte carlo Simulation. Chapter 7: Reliability-Based Design. 7.1 Acceptable Risk. 7.2 Assessing Risk. 7.3 Background to Design Methodologies. 7.4 Load and Resistance Factor Design. 7.5 Going Beyond Calibration. 7.6 Risk-Based Decision making. PART 2: PRACTICE. Chapter 8: Groundwater Modeling. 8.1 Introduction. 8.2 Finite-Element Model. 8.3 One-Dimensional Flow. 8.4 Simple Two-Dimensional Flow. 8.5 Two-Dimensional Flow Beneath Water-Retaining Structures. 8.6 Three-Dimensional Flow. 8.7 Three Dimensional Exit Gradient Analysis. Chapter 9: Flow Through Earth Dams. 9.1 Statistics of Flow Through Earth Dams. 9.2 Extreme Hydraulic Gradient Statistics. Chapter 10: Settlement of Shallow Foundations. 10.1 Introduction. 10.2 Two-Dimensional Probabilistic Foundation Settlement. 10.3 Three-Dimensional Probabilistic Foundation Settlement. 10.4 Strip Footing Risk Assessment. 10.5 Resistance Factors for Shallow-Foundation Settlement Design. Chapter 11: Bearing Capacity. 11.1 Strip Footings on c-o Soils. 11.2 Load and Resistance Factor Design of Shallow Foundations. 11.3 Summary. Chapter 12: Deep Foundations. 12.1 Introduction. 12.2 Random Finite-Element Method. 12.3 Monte Carlo Estimation of Pile Capacity. 12.4 Summary. Chapter 13: Slope Stability. 13.1 Introduction. 13.2 Probabilistic Slope Stability Analysis. 13.3 Slope Stability Reliability Model. Chapter 14: Earth Pressure. 14.1 Introduction. 14.2 Passive Earth Pressures. 14.3 Active Earth Pressures: Retaining Wall Reliability. Chapter 15: Mine Pillar Capacity. 15.1 Introduction. 15.2 Literature. 15.3 Parametric Studies. 15.4 Probabilistic Interpretation. 15.5 Summary. Chapter 16: Liquefaction. 16.1 Introduction. 16.2 Model Size: Soil Liquefaction. 16.3 Monte Carlo Analysis and Results. 16.4 Summary PART 3: APPENDIXES. APPENDIX A: PROBABILITY TABLES. A.1 Normal Distribution. A.2 Inverse Student t -Distribution. A.3 Inverse Chi-Square Distribution APPENDIX B: NUMERICAL INTEGRATION. B.1 Gaussian Quadrature. APPENDIX C. COMPUTING VARIANCES AND CONVARIANCES OF LOCAL AVERAGES. C.1 One-Dimensional Case. C.2 Two-Dimensional Case C.3 Three-Dimensional Case. Index.

Risk Assessment in Geotechnical Engineering

Soils with spatially varying shear strengths are modeled using random field theory and elasto-plastic finite element analysis to evaluate the extent to which spatial variability and cross-correlati...

D. V. Griffiths

Papers

Slope stability analysis by finite elements

Programming the finite element method

Probabilistic slope stability analysis by finite elements

Risk Assessment in Geotechnical Engineering

Bearing-capacity prediction of spatially random c ϕ soils