CONTENTS -- PREFACE -- Chapter 1 THE BOUNDARY ELEMENT METHOD FOR GEOMETRICALLY NON-LINEAR ANALYSES OF PLATES AND SHELLS -- 1.1. Introduction -- 1.2. Basic Definitions of Shear Deformable Plates and Shallow Shells -- 1.2.1. Forces and moments for stress resultants and stress couples -- 1.2.2. Linear kinematic equations -- 1.2.3. Linear constitutive equations -- 1.2.4. Linear equilibrium equations -- 1.3. Large Deflection Theory -- 1.4. Integral Representations -- 1.4.1. Rotations and out-of-plane integral representations -- 1.4.2. In-plane integral representations -- 1.5. Boundary Integral Equations -- 1.6. Stresses in Shear Deformable Plates/Shallow Shells -- 1.6.1. Internal stress resultants -- 1.6.2. Boundary stress resultants: indirect approach -- 1.6.3. Boundary stress resultants: direct approach -- 1.7. Non-linear Terms -- 1.7.1. Boundary non-linear terms -- 1.7.2. Domain non-linear terms -- 1.7.3. Derivatives of non-linear terms -- 1.8. Transformation of Domain Integrals (The Dual Reciprocity Method) -- 1.9. Numerical Implementation -- 1.9.1. Discretization -- 1.9.2. Treatment of the integrals -- 1.10. Solution of the Non-linear Integral Equations for Shallow Shells and Plates -- 1.10.1. System of equations -- 1.10.2. Load increment technique -- 1.11. Numerical Examples -- 1.11.1. Clamped circular plate subjected to uniform load q0 -- 1.11.2. Simply supported square plate subjected to uniform load q0 -- 1.11.3. Clamped shallow cylindrical shells: uniformly loaded -- 1.11.4. Simply supported shallow spherical shells: uniformly loaded -- 1.12. Conclusions -- References -- Chapter 2 TIME-DOMAIN BEM TECHNIQUES -- 2.1. Introduction -- 2.2. Standard TD-BEM Integral Equations -- 2.2.1. Potential integral equation -- 2.2.2. Numerical solution -- 2.2.3. Velocity integral equation -- 2.3. Step-by-Step Procedure: SSTD-BEM Formulation.

2.3.1. Examples -- 2.3.1.1. One-dimensional rod under a Heaviside-type forcing function -- 2.3.1.2. One-dimensional rod under sinusoidal initial conditions -- 2.4. D-BEM Formulation -- 2.4.1. Numerical procedure -- 2.4.1.1. Time marching with the Houbolt method -- 2.4.1.2. Time marching with the Newmark method -- 2.4.2. Examples -- 2.4.2.1. Square Membrane under prescribed initial displacement over the entire domain -- 2.4.2.2. Square membrane under prescribed initial velocity over part of the domain -- 2.5. Conclusions and Outlook -- References -- Chapter 3 THE BOUNDARY ELEMENT METHOD FOR THE FRACTURE ANALYSIS OF THE GENERAL PIEZOELECTRIC SOLIDS -- 3.1. Introduction -- 3.2. Basic Equations in Piezoelectricity -- 3.2.1. Piezoelectric equations in 3-D -- 3.2.2. Piezoelectric equations in 2-D -- 3.2.3. Lekhnitskii formalism for 2-D piezoelectricity -- 3.3. Physical Interpretation of Extended Somigliana's Identity in 2-D Piezoelectricity -- 3.4. Direct Formulation of the BEM in 2-D -- 3.4.1. Fundamental solutions for 2-D piezoelectricity -- 3.4.2. Implementation -- 3.5. Numerical Green's Functions for Crack Modeling -- 3.6. Crack Surface Electric BCs and Solution Algorithms -- 3.6.1. Impermeable and permeable cracks -- 3.6.2. Semipermeable cracks -- 3.7. Material Constants -- 3.8. Numerical Results -- 3.8.1. Upper and lower bound analysis -- 3.8.2. Semipermeable cracks -- 3.8.2.1. Single crack in an infinite body -- 3.8.2.2. Single crack in a finite body -- 3.8.2.3. Three parallel cracks -- 3.9. Concluding Remarks -- References -- Chapter 4 BOUNDARY INTEGRAL ANALYSIS FOR THREE-DIMENSIONAL EXPONENTIALLY GRADED ELASTICITY -- 4.1. Introduction -- 4.2. Graded Green's Function U -- 4.3. Traction Fundamental Solution T -- 4.4. Boundary Element Method -- 4.5. Numerical Examples -- 4.5.1. Pressurized thick-walled tube.

4.5.2. Spherical cavity in a cube subjected to uniaxial tension -- 4.6. Analytic Expansion -- 4.7. Conclusions -- Acknowledgements -- References -- Chapter 5 FAST HIERARCHICAL BOUNDARY ELEMENT METHOD FOR LARGE-SCALE 3-D ELASTIC PROBLEMS -- 5.1. Introduction -- 5.2. The 3-D Dual Boundary Element Method -- 5.2.1. DBEM systems of equations -- 5.3. Hierarchical Dual Boundary Element Method -- 5.3.1. Boundary subdivision and cluster tree -- 5.3.2. Block tree and admissibility condition -- 5.3.3. Low rank blocks and ACA algorithm -- 5.3.4. Block recompression and tree coarsening -- 5.3.5. Hierarchical arithmetic -- 5.3.5.1. Addition -- 5.3.5.2. Multiplication -- 5.3.5.3. LU decomposition -- 5.3.5.4. Inversion -- 5.3.6. System solution -- 5.3.7. Some details about code implementation -- 5.4. Numerical Experiments -- 5.4.1. Uncracked isotropic elastic bracket -- 5.4.2. Embedded crack in isotropic bar -- 5.4.3. Embedded crack in anisotropic bar -- 5.5. Conclusions -- Appendix A. Fundamental Solutions -- Appendix B. Numerical Scheme -- References -- Chapter 6 MODELLING OF PLATES AND SHALLOW SHELLS BY MESHLESS LOCAL INTEGRAL EQUATION METHOD -- 6.1. Introduction -- 6.2. The MLS Approximation -- 6.3. Analyses of Orthotropic FGM Plates Under Mechanical and Thermal Load -- 6.3.1. Meshless local integral equations for a Reissner-Mindlin plate -- 6.3.2. Meshless local integral equations for heat conduction problems in plates -- 6.4. Analyses of Orthotropic FGM Shells Under Mechanical and Thermal Load -- 6.5. Numerical Examples -- 6.5.1. Plates -- 6.5.2. Shells -- 6.6. Conclusions -- References -- Chapter 7 BOUNDARY ELEMENT TECHNIQUE FOR SLOW VISCOUS FLOWS ABOUT PARTICLES -- 7.1. Introduction -- 7.2. Stokes Equations. First Properties and Simple Solutions -- 7.2.1. Assumptions. Navier-Stokes equations. Reynolds number and Stokes approximation.

7.2.2. First properties and noticeable solutions -- 7.3. General Solution. Singularities. Singularity Method and Illustrating Examples -- 7.3.1. General form of a steady Stokes flow -- 7.3.2. Examples of singularity solutions in an unbounded domain -- 7.3.2.1. Potential velocity fields -- 7.3.2.2. Rotational velocity fields -- 7.3.3. Singularity method and examples -- 7.3.4. General properties -- 7.3.4.1. Reciprocal identity -- 7.3.4.2. Energy dissipation and unicity of the solution -- 7.3.4.3. Basic application -- 7.4. Integral Representations for the Pressure and Velocity Fields. Resulting Boundary Integral Equations -- 7.4.1. Green solution -- 7.4.1.1. Free-space Green tensor -- 7.4.1.2. Case of a bounded fluid domain -- 7.4.2. General integral representation for the velocity field -- 7.4.3. Integral representation for the pressure field -- 7.4.4. Relevant boundary-integral equations -- 7.4.4.1. Basic steps and key preliminary results -- 7.4.4.2. Relevant general boundary integral equations -- 7.4.4.3. Case of a N−bubble cluster -- 7.4.4.4. Case of a N−particle cluster involving at least a solid particle -- 7.5. Numerical Implementation -- 7.5.1. Mesh and boundary-elements -- 7.5.2. Associated functions -- 7.5.3. Computation of regular and weakly singular boundary integrals: advocated scheme -- 7.5.4. Computation of regular and weakly singular boundary integrals: accuracy issues -- 7.5.5. Implementation for a N−particle cluster -- 7.6. Illustrating Comparisons and Results -- 7.6.1. Sedimentation of a solid spheroid near a plane wall -- 7.6.2. Solid particle in a spherical cavity -- 7.7. Concluding Remarks -- References -- Chapter 8 BIT FOR FREE SURFACE FLOWS -- 8.1. The Nature of Free Surface Flows -- 8.2. Mathematical Formulation -- 8.2.1. Three-dimensional BIT -- 8.2.2. Two-dimensional BIT -- 8.2.3. Open, periodic geometry.

8.2.4. Linear stability analysis -- 8.3. Numerical Approximation -- 8.3.1. Numerical stability -- 8.4. Applications with a Single Surface -- 8.4.1. Waves on deep water -- 8.4.2. Rising bubbles -- 8.5. Applications with Two Surfaces -- 8.5.1. Rayleigh-Taylor instability of a liquid layer of finite thickness -- 8.5.2. Water waves in finite depth -- 8.6. Some Challenges and Improvements -- Acknowledgments -- References -- Chapter 9 SIMULATION OF CAVITATING AND FREE SURFACE FLOWS USING BEM -- 9.1. Introduction -- 9.2. 2-D Hydrofoil -- 9.2.1. Formulation -- 9.2.2. The boundary element method -- 9.2.3. Cavity extent for given cavitation number -- 9.2.4. Effects of flow unsteadiness -- 9.2.5. Effects of viscosity -- 9.2.6. The cavity detachment point -- 9.3. 3-D Hydrofoil -- 9.3.1. The green's third identity -- 9.3.2. The dynamic boundary condition -- 9.3.3. The cavity thickness distribution -- 9.3.4. The cavity planform -- 9.3.5. Numerical aspects -- 9.3.6. The split panel technique -- 9.3.7. Multiplicity of solutions -- 9.4. Propellers -- 9.4.1. Super-cavitating propellers -- 9.5. Comparisons with Experiments -- 9.6. Surface-Piercing Flows -- 9.6.1. Surface-piercing propellers -- 9.6.2. Surface-piercing 2-D hydrofoils: entry stage -- 9.6.2.1. Formulation -- 9.6.2.2. Algorithm of marching free-surface in time -- 9.6.2.3. Results -- 9.7. Concluding Remarks -- 9.8. Acknowledgements -- References -- Chapter 10 CONDITION NUMBERS AND LOCAL ERRORS IN THE BOUNDARY ELEMENT METHOD -- 10.1. Introduction -- 10.2. BEM Formulation for Potential Problems -- 10.3. Local Errors in Potential Problems -- 10.3.1. Numerical examples -- 10.3.2. A detailed study of local errors -- 10.4. Large Condition Numbers -- 10.5. Condition Numbers in Potential Problems -- 10.5.1. Logarithmic capacity -- 10.5.2. Dirichlet problem -- 10.5.3. Mixed problem.

10.6. Condition Numbers in Flow Problems.