Time-dependent and Vibration Problems

This series has been developed in response to the interest shown in boundary ele­ ments by scientists and engineers. Whilst Volume 1 was dedicated to basic principles and applications, this book is concerned with the state of the art in the solution of time-dependent problems. Since papers have recently been published on this im­ portant topic it is time to produce a work of a more permanent nature. The volume begins with a chapter on the Fundamentals of Boundary Integral Equation Methods in Elastodynamics. After reviewing the basic equations of elasto­ dynamics, the wave equation and dynamic reciprocal theorems are stated and the direct and indirect boundary element formulations are presented. Eigenvalue problems are discussed together with the case of the Fourier transformations. Several applications illustrate the effectiveness of the technique for engineering. Chapter 2 examines some of the various boundary integral equation formulations available for elastodynamic problems. In particular the displacement-traction for­ mulation is compared with the displacement-potential case. The special character­ istics of the elastodynamics fundamental solutions are discussed in detail and a criti­ cal comparison with the elastostatics case is presented. While the chapter is not meant to be a complete review of the work in the field, the original presentation of the problem and the suggestions for further work make an important contribu­ tion to the development of the method.

1 Fundamentals of Boundary Integral Equation Methods in Elastodynamics.- 1.1 Introduction.- 1.2 Elastodynamic Problems.- 1.2.1 Basic Equations.- 1.2.2 Elastodynamic Problems.- 1.2.3 Reciprocal Theorem.- 1.3 BIE Formulations in Time-Space Domain.- 1.3.1 Integral Representation and Fundamental Solutions.- 1.3.2 Two-Dimensional Representation.- 1.3.3 Boundary Integral Equations.- 1.4 BIE Formulations in the Transformed Domain of Integral Transforms.- 1.4.1 Fourier Transformed Problems of Elastodynamics.- 1.4.2 BIE Formulations of Transformed Problems.- 1.4.3 Laplace Transformed Domain BIE.- 1.5 Integral Equation Formulations for Inhomogeneous Domain.- 1.5.1 Basic Equations.- 1.5.2 Integral Equation Formulations.- 1.6 Eigenfrequency Problems.- 1.7 Some Remarks on Inherent Problems of BIEM in Elastodynamics.- 1.7.1 Fictitious Eigenfrequencies in the Time-Harmonic Elastodynamics.- 1.7.2 Half-Plane Problems.- 1.7.3 Treatment of Singularities.- 1.7.4 BIE-FE Hybrid Method.- 1.7.5 Miscellaneous in Numerical Treatment.- 1.8 Application Examples.- 1.8.1 Transient Response Analysis by the Time-Space Domain BIEM.- 1.8.2 Applications of Integral Transformed Domain BIEM.- 1.9 Concluding Remarks.- References.- 2 Elastic Potentials in BIE Formulations.- 2.1 Introduction.- 2.2 Elastodynamic Formulations.- 2.3 Elastostatic Formulations.- 2.4 Solution Methods.- 2.5 Comments and Suggestions.- References.- 3 Time Dependent Non-Linear Potential Problems.- 3.1 Introduction.- 3.2 Governing Equations.- 3.3 Homogeneous Parabolic Equation.- 3.4 Constant and Linear Time Interpolation.- 3.5 Non-Linear Boundary Conditions for the Case of Constant Conductivity.- 3.6 Non-Linear Boundary Conditions for the Case of Temperature Dependent Conductivity.- 3.7 Applications.- 3.8 Conclusions.- References.- 4 Further Developments on the Solution of the Transient Scalar Wave Equation.- 4.1 Introduction.- 4.2 The Boundary Initial Value Problem.- 4.3 Dirac Delta and Heaviside Functions.- 4.4 Fundamental Solution in Three Dimensions.- 4.5 Kirchhoff Integral Representation.- 4.6 Two-Dimensional Boundary Integral Equation.- 4.7 Additional Transformations to Volterra’s Integral Representation.- 4.8 Numerical Implementation.- 4.8.1 Boundary Integrals.- 4.8.2 Domain Integrals.- 4.8.3 Double Nodes.- 4.9 Examples.- 4.9.1 One-Dimensional Rod under a Heaviside Type Forcing Function.- 4.9.2 One-Dimensional Rod under Prescribed Initial Velocity and Displacement.- 4.9.3 Square Membrane under Prescribed Initial Velocity.- References.- 5 Transient Elastodynamics.- 5.1 Introduction.- 5.2 Basic Theory.- 5.3 The Initial Value Problem of Elastodynamics.- 5.4 One-Dimensional Motions.- 5.5 Plane Motions.- 5.6 Fundamental Solutions for Transient Elastodynamics.- 5.7 Time Domain Elastodynamic Boundary Integral Representation.- 5.8 Additional Transformations to the Two-Dimensional Boundary Integral Equation of Elastodynamics.- 5.9 Numerical Implementation for Two Dimensions.- 5.10 Examples―Two-Dimensional Elastodynamics.- 5.11 Conclusions.- References.- 6 Propagation of Surface Waves.- 6.1 Introduction.- 6.2 Three-Dimensional Formulation.- 6.2.1 Particular Fundamental Solutions.- 6.2.2 Numerical and Computational Aspects.- 6.3 Floating Bodies.- 6.4 Vertical Axisymmetric Bodies.- 6.5 Vertical Cylinders of Arbitrary Section.- 6.6 Horizontal Cyclinders of Arbitrary Section.- 6.6.1 Obliquely Incident Waves.- 6.7 Transient Problems.- 6.8 Nonlinear Problems.- References.- 7 Boundary Integral Formulation of Mass Matrices for Dynamic Analysis.- 7.1 Introduction.- 7.2 Formulation of the Dynamical Problem.- 7.3 Various Boundary Integral Formulations.- 7.4 Boundary Integral Formulation Using the Statical Fundamental Solution.- 7.5 The Numerical Solution Procedure.- 7.6 Derivation of Different Types of Dynamical Problems.- 7.7 Two-Dimensional Formulation.- 7.8 Computer Implementation.- 7.9 Applications.- Conclusions.- References.- 8 Boundary Element Method for Laminar Viscous Flow and Convective Diffusion Problems.- 8.1 Introduction.- 8.2 Governing Equations.- 8.2.1 Field Equations.- 8.2.2 Boundary Conditions.- 8.3 Boundary Integral Equations.- 8.3.1 BIE of the Stream Function.- 8.3.2 BIE of the Vorticity.- 8.3.3 BIE of the Temperature.- 8.4 Boundary Element Approximation.- 8.4.1 Discretization of the Stream Function.- 8.4.2 Discretization on Vorticity and Temperature.- 8.5 Computational Scheme.- 8.6 Numerical Examples.- 8.6.1 Isothermal Channel Flow.- 8.6.2 Isothermal Flow Past a Cylinder.- 8.6.3 Isothermal Driven-Cavity Flow.- 8.6.4 Convection-Diffusion in Wind-Driven Flow.- 8.6.5 Natural Convection in a Compartment.- 8.6.6 Natural Convection around a Heated Cylinder.- 8.7 Conclusion.- References.- 9 Asymptotic Accuracy and Convergence for Point Collocation Methods.- 9.1 Introduction.- 9.2 Examples of Boundary Integral Equations.- 9.3 Standard Collocation for Two-Dimensional Problems.- 9.4 Standard Collocation for Three-Dimensional Problems with Fredholm Boundary Integral Equations of the Second Kind.- References.