Singular Integrals in Boundary Element Methods [Kõva köide]

Preface xvii
Chapter 1 Introductory notes on singular integrals 1(32) V. Sladek J. Sladek 1 Introduction 1(2) 2 Singular integrals 3(4) 2.1 Definitions 3(1) 2.2 Singular integrals in BEM formulations 4(3) 3 Regularization 7(17) 3.1 Direct limit approach with analytical integration of potentially singular integrals 9(6) 3.2 Regularization by infinitesimal deformation of the boundary 15(5) 3.3 Analytical regularization 20(4) 4 Boundary element approximations 24(4) 4.1 Collocation approach 25(2) 4.2 Galerkin approach 27(1) 5 Conclusions 28(1) References 28(1) Appendix 29(4)
Chapter 2 Evaluation of singular and hypersingular Galerkin integrals: direct limits and symbolic computation 33(52) L.J. Gray 1 Introduction 33(4) 2 Symmetric Galerkin 37(2) 3 C1 Condition 39(2) 4 Singular integrals: Linear element 41(10) 4.1 Coincident integration 42(1) 4.1.1 G 43(1) 4.1.2 First derivative of G 43(2) 4.1.3 Second derivative of G 45(1) 4.2 Adjacent integration 46(2) 4.2.1 G and its first derivative 48(1) 4.2.2 Second derivative of G 48(1) 4.3 Accuracy 49(2) 5 Orthotropic elasticity 51(9) 5.1 Orthotropic boundary integral equations 51(1) 5.2 Singular integrals 52(3) 5.3 Fracture calculations 55(5) 6 Singular integrals: Curved elements 60(4) 7 Surface derivatives 64(3) 7.1 Example calculations 65(2) 8 Application: electromigration 67(3) 9 Conclusions 70(3) References 73(9) Appendix 82(3)
Chapter 3 Formulation and numerical treatment of boundary integral equations with hypersingular kernels 85(40) M. Guiggiani 1 Introduction 85(1) 2 Some classical theorems 86(1) 3 General form of boundary integral identities 87(3) 4 Boundary integral equations 90(5) 4.1 Standard boundary integral equations (SBIE) 91(2) 4.2 Hypersingular boundary integral equations (HBIE) 93(2) 5 Evaluation of the free-term coefficients 95(2) 6 Direct evaluation of singular integrals 97(13) 6.1 Three-dimensional problems 98(1) 6.1.1 Limiting process and discretization of the geometry 98(3) 6.1.2 Semi-analytical treatment 101(2) 6.1.3 Final formula 103(1) 6.1.4 Less singular integrals 104(1) 6.2 Two-dimensional problems 104(1) 6.2.1 Limiting process and discretization of the geometry 105(2) 6.2.2 Semi-analytical treatment 107(1) 6.2.3 Final formula 108(2) 7 Numerical examples 110(4) 7.1 Strongly singular integrals (CPV) 111(1) 7.2 Hypersingular integrals 112(2) 8 Related works 114(1) References 115(2) Appendices 117(8)
Chapter 4 Regularization of boundary element formulations by the derivative transfer method 125(40) A. Frangi 1 Introduction 125(2) 1.1 Notation 126(1) 2 Static elasticity 127(8) 2.1 Displacement equation 127(3) 2.1.1 2D problems 130(1) 2.1.2 3D problems 130(1) 2.2 Traction equation 131(2) 2.2.1 Collocation approach 133(1) 2.2.2 Variational approach 134(1) 3 Elastodynamics 135(7) 3.1 Displacement equation 135(2) 3.1.1 Laplace domain 137(1) 3.1.2 Time domain 138(1) 3.2 Traction equation 139(2) 3.2.1 Laplace domain 141(1) 3.2.2 Time domain 141(1) 3.2.3 Time domain: Variational approach 142(1) 4 Kirchhoff plates 142(6) 4.1 Displacement and gradient equations 143(2) 4.2 Moment-shear equations 145(2) 4.2.1 Collocation approach 147(1) 4.2.2 Variation approach 148(1) 5 Numerical implementation and examples 148(4) 5.1 Brazilian test 149(1) 5.2 Retangular plate with a slanted crack 150(2) 5.3 Square plate with two opposite sides simply-supported and the other ones free 152(1) 6 Concluding remarks 152(2) References 154(3) Appendices 157(8)
Chapter 5 Singular integrals and their treatment in crack problems 165(32) J.D. Richardson T.A. Cruse 1 Introduction 166(1) 2 The fundamental solution 167(2) 3 Continuity and discontinuity of the potentials 169(3) 4 Traction BIEs for fracture modeling 172(6) 5 Behavior of free term integrals on open surfaces 178(2) 6 BEM implementation 180(4) 7 Conclusion 184(1) References 184(2) Appendices 186(11)
Chapter 6 Accurate hypersingular integral computations in the development of numerical Greens functions for fracture mechanics 197(26) N.P.P. Silveira S. Guimaraes J.C.F. Telles 1 Introduction 197(1) 2 The boundary element method review 198(2) 2.1 Integral equations for displacements and tractions 198(1) 2.2 Boundary integral equations 199(1) 2.3 Boundary integral equations at crack surfaces 199(1) 3 Numerical Greens function 200(7) 3.1 Complementary solution 200(2) 3.2 Fundamental crack opening displacements 202(3) 3.3 Final numerical Greens function 205(2) 4 Stresses at internal points 207(1) 5 Implementation of NGF 207(9) 5.1 Geometric shape function 208(1) 5.2 Numerical treatment of the fundamental crack opening integral equation 209(7) 5.3 Interpolation to crack opening and its derivatives 216(1) 6 Examples 216(5) 7 Conclusions 221(1) 8 References 221(2)
Chapter 7 Regularization and evaluation of singular domain integrals in boundary element methods 223(40) G. Kuhn P. Partheymuller O. Kohler 1 Introduction 223(1) 2 2D/3D - FBEM for plasticity at small strains 224(13) 2.1 Governing equations 224(2) 2.2 Field boundary integral equations for displacements 226(1) 2.3 Field boundary integral equations for displacement gradients 227(1) 2.3.1 Regularization for interior source points 227(1) 2.3.2 Regularization for source points on the boundary 228(3) 2.4 Discretization and numerical solution 231(2) 2.5 Numerical treatment of domain integrals 233(1) 2.5.1 Regular and nearly singular integrals 233(1) 2.5.2 Weakly singular integrals 234(1) 2.6 Example 235(2) 3 Extension to axisymmetric problems 237(4) 3.1 Field boundary integral equations for displacements 238(1) 3.2 Field boundary integral equations for displacement gradients 239(2) 3.3 Numerical treatment of axisymmetric problems 241(1) 4 FBEM for finite deformation problems 241(10) 4.1 Governing equations 242(2) 4.2 Field boundary integral equations for displacements 244(1) 4.2.1 Two- and three-dimensional problems 244(1) 4.2.2 Axisymmetric problems 245(1) 4.3 Field boundary integral equation for displacement gradients 246(1) 4.3.1 Two-and three-dimensional problems 246(1) 4.3.2 Axisymmetric problems 247(1) 4.4 Discretization and numerical solution 247(2) 4.5 Example 249(2) 5 FBEM for damage mechanics 251(1) 6 FBEM for non-linear fracture mechanics 251(7) 6.1 Dual field boundary element method (Dual-FBEM) 252(3) 6.1.1 Example 255(3) 7 Conclusions 258(1) References 258(5)
Chapter 8 Regularized boundary integral formulation for thin elastic plate bending analysis 263(36) T. Matsumoto M. Tanaka 1 Introduction 263(2) 2 Direct boundary integral formulation for thin elastic plate bending problem 265(11) 2.1 Governing equation 265(1) 2.2 Direct boundary integral formulation 266(1) 2.2.1 Singular boundary integral representations 266(5) 2.2.2 Regularized boundary integral representations 271(5) 3 Numerical treatment 276(9) 3.1 Discretization 276(6) 3.2 Applicability of CO interpolation scheme for the deflection in the regularized boundary integral equation 282(1) 3.3 C1-continuous interpolation for the deflection 283(2) 4 Numerical examples 285(4) 5 Concluding remarks 289(5) References 294(5)
Chapter 9 Complex hypersingular BEM in plane elasticity problems 299(66) A. Linkov 1 Introduction 299(6) 1.1 Advantages of functions of a complex variable 300(1) 1.2 Advantages of hypersingular integrals 301(1) 1.3 Combined advantages of complex variables and hypersingular equations 302(1) 1.4 Brief historical review 302(1) 1.4.1 Complex integral equations 302(1) 1.4.2 Real hypersingular integral equations 303(1) 1.4.3 Complex hypersingular integral equations (CHSIE) 304(1) 1.5 Scope of the paper 304(1) 2 Prerequisities 305(4) 3 Singular solutions 309(7) 3.1 Singular solutions in real variables 309(1) 3.2 Singular solutions in complex variables 310(2) 3.3 Particular case of Kelvin solution 312(1) 3.4 Employing K.-M. functions to obtain singular solutions 313(3) 4 Complex potentials and their properties 316(3) 4.1 Complex potentials 316(1) 4.2 Particular case of Kelvin solution 317(1) 4.3 Limit values of complex potentials 318(1) 4.4 Physical meaning of densities 319(1) 5 Equations of the indirect approach 319(3) 5.1 General case 319(2) 5.2 Equations for Kelvins solution 321(1) 5.2.1 Comment 321(1) 6 Equations of the direct approach 322(10) 6.1 General equations 322(4) 6.2 Equations for Kelvins solution 326(2) 6.3 Equations for blocky systems and cracks 328(2) 6.4 Employing of K.-M. functions 330(1) 6.4.1 Comment 331(1) 7 Complex hypersingular integrals 332(7) 7.1 Definitions of hypersingular integrals 332(5) 7.2 Connection of direct values of hypersingular integrals with limit values 337(2) 8 Method of solution: CVH-BEM approach 339(11) 8.1 BEM discretization and approximation 340(2) 8.2 Choice of approximating functions; importance of conjugated polynomials and tip elements 342(1) 8.2.1 Ordinary (not tip) elements (Fig. 5a) 342(1) 8.2.2 Tip elements (Fig. 5b) 343(1) 8.3 Evaluation of crucial integrals 344(1) 8.3.1 Tip elements 345(1) 8.3.2 Integrals from conjugated functions 346(1) 8.4 Evaluation of remaining (proper) integrals 346(1) 8.4.1 Straight element 347(1) 8.4.2 Circular arc element 348(1) 8.4.3 Comment 349(1) 8.5 Formulae for control 349(1) 9 Numerical examples 350(19) 9.1 Examples regarding approximations 350(1) 9.1.1 Importance of conjugated polynomials 350(1) 9.1.2 Importance of tip elements 351(1) 9.1.3 Influence of element sizes 352(1) 9.1.4 Approximation of boundaries 353(1) 9.2 Examples illustrating the range of applications 354(1) 9.2.1 Problems for cracks 354(1) 9.2.2 Two straight edge cracks in a circular disk (Fig. 11) 354(1) 9.2.3 A straight crack within a circular inclusion (Fig. 13) 355(1) 9.2.4 A straight crack between two holes (Fig. 14) 355(1) 9.2.5 Circular arc cracks along the contour of a circle (Fig. 16a and b) 355(3) 9.2.6 A branched crack in an infinite plane (Fig. 16c) 358(1) 9.2.7 Blocky systems (Fig. 17) 359(1) References 359(6)
Chapter 10 Some computational aspects associated with singular kernels 365 V. Sladek J. Sladek 1 Introduction 365(3) 2 Approximation of boundary densities and geometry in regularized formulations 368(16) 2.1 Regularized boundary integral equations 368(1) 2.1.1 Ordinary boundary integral equations - OBIE 368(1) 2.1.2 Derivative boundary integral equations - DBIE 369(3) 2.2 Approximations by using standard elements 372(1) 2.2.1 Standard Lagrange-type elements (SLag) 372(1) 2.2.2 Standard Overhauser elements (SOv) 373(2) 2.3 Modified Overhauser elements 375(1) 2.3.1 Smooth contour at XXX 376(2) 2.3.2 Corner at XXX 378(1) 2.4 Numerical examples 379(4) 2.5 Conclusions 383(1) 3 Optimal transformations of the integration variable in numerical computation of nearly-singular integrals 384(12) 3.1 Polynomial transformations 385(2) 3.2 Optimal transformations 387(2) 3.3 Numerical experiments 389(6) 3.4 Conclusions 395(1) 4 Numerical integration of logarithmic and nearly-logarithmic singularity 396(13) 4.1 Numerical integrations 397(1) 4.1.1 (wt) - approach 397(1) 4.1.2 (pt) - approach 397(1) 4.1.3 (Tt) - approach 398(1) 4.1.4 (ln) - approach 398(1) 4.1.5 (an) - approach 399(8) 4.2 Numerical experiments 407(1) 4.3 Conclusions 408(1) 5 Weak - singularity in 3-d BEM formulations 409(15) 5.1 Weakly singular integral 410(5) 5.2 Nearly weakly singular integral 415(2) 5.3 Numerical experiments 417(6) 5.4 Conclusions 423(1) References 424