Singular integrals in boundary element methods

CHAPTER 1 INTRODUCTORY NOTES ON SINGULAR INTEGRALS Introduction: Singular integrals Definitions Singular integrals in BEM formulations Regularization Direct limit approach with analytical integration of potentially singular integrals Regularization by infinitesimal deformation of the boundary Analytical regularization Boundary element approximations Collocation approach Galerkin approach Conclusions. CHAPTER 2 EVALUATION OF SINGULAR AND HYPERSINGULAR GALERKIN INTEGRALS: DIRECT LIMITS AND SYMBOLIC COMPUTATION Introduction: Symmetric Galerkin Condition Singular integrals: linear element Coincident integration G First derivative of G Second derivative of G Adjacent integration G and its first derivative Second derivative of G Accuracy Orthotropic elasticity Orthotropic boundary integral equations Singular integrals Fracture calculations Singular integrals: Curved elements Surface derivatives Example calculations Application: electromigration Conclusions. CHAPTER 3 FORMULATION AND NUMERICAL TREATMENT OF BOUNDARY INTEGRAL EQUATIONS WITH HYPERSINGULAR KERNELS Introduction:Some classical theorems General form of boundary integral identities Boundary integral equations Standard boundary integral equations (SBIE) Hypersingular boundary integral equations (HBIE) Evaluation of the free-term coefficients Direct evaluation of singular integrals Three-dimensional problems Limiting process and discretization of the geometry Semi-analytical treatment Final formula Less singular integrals Two-dimensional problems Limiting process and discretization of the geometry Semi-analytical treatment Final formula Numerical examples Strongly singular integrals (CPV) Hypersingular integrals Related works. CHAPTER 4 REGULARIZATION OF BOUNDARY ELEMENT FORMULATIONS BY THE DERIVATIVE TRANSFER METHOD Introduction: Notation Static elasticity Displacement equation 2D problems 3D problems Traction equation Collocation approach Variational approach Elastodynamics Displacement equation Laplace domain Time domain Traction equation Laplace domain Time domain Time domain: variational approach Kirchhoff plates Displacement and gradient equations Moment-shear equations Collocation approach Variation approach Numerical implementation and examples Brazilian test Retangular plate with a slanted crack Square plate with two opposite sides simply-supported and the other ones free Concluding remarks. CHAPTER 5 SINGULAR INTEGRALS AND THEIR TREATMENT IN CRACK PROBLEMS Introduction: The fundamental solution Continuity and discontinuity of the potentials Traction BIE's for fracture modeling Behavior of free term integrals on open surfaces BEM implementation Conclusion. CHAPTER 6 ACCURATE HYPERSINGULAR INTEGRAL COMPUTATIONS IN THE DEVELOPMENT OF NUMERICAL GREEN'S FUNCTIONS FOR FRACTURE MECHANICS Introduction: The boundary element method review Integral equations for displacements and tractions Boundary integral equations Boundary integral equations at crack surfaces Numerical Green's function Complementary solution Fundamental crack opening displacements Final numerical Green's function Stresses at internal points Implementation of NGF Geometric shape function Numerical treatment of the fundamental crack opening integral equation Interpolation to crack opening and its derivatives Examples Conclusions. CHAPTER 7 REGULARIZATION AND EVALUATION OF SINGULAR DOMAIN INTEGRALS IN BOUNDARY ELEMENT METHODS Introduction: 2D/3D - FBEM for plasticity at small strains Governing equations Field boundary integral equations for displacements Field boundary integral equations for displacement gradients Regularization for interior source points Regularization for source points on the boundary Discretization and numerical solution Numerical treatment of domain integrals Regular and nearly singular integrals Weakly singular integrals Example Extension to axisymmetric problems Field boundary integral equations for displacements Field boundary integral equations for displacement gradients Numerical treatment of axisymmetric problems FBEM for finite deformation problems Governing equations Field boundary integral equations for displacements Two- and three-dimensional problems Axisymmetric problems Field boundary integral equation for displacement gradients Two- and three-dimensional problems Axisymmetric problems Discretization and numerical solution Example FBEM for damage mechanics FBEM for non-linear fracture mechanics Dual field boundary element method (Dual - FBEM) Example Conclusions. CHAPTER 8 REGULARIZED BOUNDARY INTEGRAL FORMULATION FOR THIN ELASTIC PLATE BENDING ANALYSIS Introduction: Direct boundary integral formulation for thin elastic plate bending problem Governing equation Direct boundary integral formulation Singular boundary integral representations Regularized boundary integral representations Numerical treatment Discretization Applicability of C0 interpolation scheme for the deflection in the regularized boundary integral equation C1-continuous interpolation for the deflection Numerical examples Concluding remarks. CHAPTER 9 COMPLEX HYPERSINGULAR BEM IN PLANE ELASTICITY PROBLEMS Introduction: Advantages of functions of a complex variable Advantages of hypersingular integrals Combined advantages of complex variables and hypersingular equations Brief historical review Complex integral equations Real hypersingular integral equations Complex hypersingular integral equations (CHSIE) Scope of the paper Prerequisities Singular solutions Singular solutions in real variables Singular solutions in complex variables Particular case of Kelvin solution Employing K.M. functions to obtain singular solutions Complex potentials and their properties Complex potentials Particular case of Kelvin solution Limit values of complex potentials Physical meaning of densities Equations of the indirect approach General case Equations for Kelvin's solution Comment Equations of the direct approach General equations Equations for Kelvin's solution Equations for blocky systems and cracks Employing of K.M. functions Comment Complex hypersingular integrals Definitions of hypersingular integrals Connection of direct values of hypersingular integrals with limit values Method of solution: CVH-BEM approach BEM discretization and approximation Choice of approximating functions importance of conjugated polynomials and tip elements Evaluation of crucial integrals Tip elements Integrals from conjugated functions Evaluation of remaining (proper) integrals Straight element Circular arc element Comment Formulae for control Numerical examples Examples regarding approximations Importance of conjugated polynomials Importance of tip elements Influence of element sizes Approximation of boundaries Examples illustrating the range of applications Problems for cracks. CHAPTER 10 SOME COMPUTATIONAL ASPECTS ASSOCIATED WITH SINGULAR KERNELS Introduction: Approximation of boundary densities and geometry in regularized formulations Regularized boundary integral equations Ordinary boundary integral equations - OBIE Derivative boundary integral equations - DBIE Approximations by using standard elements Standard Lagrange-type elements (Slag) Standard Overhauser elements (Sov) Modified Overhauser elements Smooth contour at z Corner at z Numerical examples Conclusions Optimal transformations of the integration variable in numerical computation of nearly-singular integrals Polynomial transformations Optimal transformations Numerical experiments Conclusions Numerical integration of logarithmic and nearly-logarithmic singularity Numerical integrations (wt) - approach (pt) - approach (Tt) - approach (ln) - approach (an) - approach Numerical experiments Conclusions Weak - singularity in 3-d BEM formulations Weakly singular integral Nearly weakly singular integral Numerical experiments Conclusions.