Carlos Alberto Brebbia

Boundary elements : an introductory course

Boundary concepts are introduced and immediately applied in simple - but useful - computer codes. These codes facilitate the comprehension of boundary elements. This text also discusses basic concepts, potential problems, elastostatics and other topics of interest to engineers.

Boundary Element Techniques. Theory and Applications in Engineering

1 Approximate Methods.- 1.1. Introduction.- 1.2. Basic Definitions.- 1.3. Approximate Solutions.- 1.4. Method of Weighted Residuals.- 1.4.1. The Collocation Method.- 1.4.2. Method of Collocation by Subregions.- 1.5. Method of Galerkin.- 1.6. Weak Formulations.- 1.7. Inverse Problem and Boundary Solutions.- 1.8. Classification of Approximate Methods.- References.- 2 Potential Problems.- 2.1. Introduction.- 2.2. Elements of Potential Theory.- 2.3. Indirect Formulation.- 2.4. Direct Formulation.- 2.5. Boundary Element Method.- 2.6. Two-Dimensional Problems.- 2.6.1. Source Formulation.- 2.7. Poisson Equation.- 2.8. Subregions.- 2.9. Orthotropy and Anisotropy.- 2.10. Infinite Regions.- 2.11. Special Fundamental Solutions.- 2.12. Three-Dimensional Problems.- 2.13. Axisymmetric Problems.- 2.14. Axisymmetric Problems with Arbitrary Boundary Conditions.- 2.15. Nonlinear Materials and Boundary Conditions.- 2.15.1. Nonlinear Boundary Conditions.- References.- 3 Interpolation Functions.- 3.1. Introduction.- 3.2. Linear Elements for Two-Dimensional Problems.- 3.3. Quadratic and Higher-Order Elements.- 3.4. Boundary Elements for Three-Dimensional Problems.- 3.4.1. Quadrilateral Elements.- 3.4.2. Higher-Order Quadrilateral Elements.- 3.4.3. Lagrangian Quadrilateral Elements.- 3.4.4. Triangular Elements.- 3.4.5. Higher-Order Triangular Elements.- 3.5. Three-Dimensional Cell Elements.- 3.5.1. Tetrahedron.- 3.5.2. Cube.- 3.6. Discontinuous Boundary Elements.- 3.7. Order of Interpolation Functions.- References.- 4 Diffusion Problems.- 4.1. Introduction.- 4.2. Laplace Transforms.- 4.3. Coupled Boundary Element - Finite Difference Methods.- 4.4. Time-Dependent Fundamental Solutions.- 4.5. Two-Dimensional Problems.- 4.5.1. Constant Time Interpolation.- 4.5.2. Linear Time Interpolation.- 4.5.3. Quadratic Time Interpolation.- 4.5.4. Space Integration.- 4.6. Time-Marching Schemes.- 4.7. Three-Dimensional Problems.- 4.8. Axisymmetric Problems.- 4.9. Nonlinear Diffusion.- References.- 5 Elastostatics.- 5.1. Introduction to the Theory of Elasticity.- 5.1.1. Initial Stresses or Initial Strains.- 5.2. Fundamental Integral Statement.- 5.2.1. Somigliana Identity.- 5.3. Fundamental Solutions.- 5.4. Stresses at Internal Points.- 5.5. Boundary Integral Equation.- 5.6. Infinite and Semi-Infinite Regions.- 5.7. Numerical Implementation.- 5.8. Boundary Elements.- 5.9. System of Equations.- 5.10. Stresses and Displacements Inside the Body.- 5.11. Stresses on the Boundary.- 5.12. Surface Traction Discontinuities.- 5.13. Two-Dimensional Elasticity.- 5.14. Body Forces.- 5.14.1. Gravitational Loads.- 5.14.2. Centrifugal Load.- 5.14.3. Thermal Loading.- 5.15. Axisymmetric Problems.- 5.15.1. Extension to Nonaxisymmetric Boundary Values.- 5.16. Anisotropy.- References.- 6 Boundary Integral Formulation for Inelastic Problems.- 6.1. Introduction.- 6.2. Inelastic Behavior of Materials.- 6.3. Governing Equations.- 6.4. Boundary Integral Formulation.- 6.5. Internal Stresses.- 6.6. Alternative Boundary Element Formulations.- 6.6.1. Initial Strain.- 6.6.2. Initial Stress.- 6.6.3. Fictitious Tractions and Body Forces.- 6.7. Half-Plane Formulations.- 6.8. Spatial Discretization.- 6.9. Internal Cells.- 6.10. Axisymmetric Case.- References.- 7 Elastoplasticity.- 7.1. Introduction.- 7.2. Some Simple Elastoplastic Relations.- 7.3. Initial Strain: Numerical Solution Technique.- 7.3.1. Examples - Initial Strain Formulation.- 7.4. General Elastoplastic Stress-Strain Relations.- 7.5. Initial Stress: Outline of Solution Techniques.- 7.5.1. Examples: Kelvin Implementation.- 7.5.2. Examples: Half-Plane Implementation.- 7.6. Comparison with Finite Elements.- References.- 8 Other Nonlinear Material Problems.- 8.1. Introduction.- 8.2. Rate-Dependent Constitutive Equations.- 8.3. Solution Technique: Viscoplasticity.- 8.4. Examples: Time-Dependent Problems.- 8.5. No-Tension Materials.- References.- 9 Plate Bending.- 9.1. Introduction.- 9.2. Governing Equations.- 9.3. Integral Equations.- 9.3.1. Other Fundamental Solutions.- 9.4. Applications.- References.- 10 Wave Propagation Problems.- 10.1. Introduction.- 10.2. Three-Dimensional Water Wave Propagation Problems.- 10.3. Vertical Axisymmetric Bodies.- 10.4. Horizontal Cylinders of Arbitrary Section.- 10.5. Vertical Cylinders of Arbitrary Section.- 10.6. Transient Scalar Wave Equation.- 10.7. Three-Dimensional Problems: The Retarded Potential.- 10.8. Two-Dimensional Problems.- References.- 11 Vibrations.- 11.1. Introduction.- 11.2. Governing Equations.- 11.3. Time-Dependent Integral Formulation.- 11.4. Laplace Transform Formulation.- 11.5. Steady-State Elastodynamics.- 11.6. Free Vibrations.- References.- 12 Further Applications in Fluid Mechanics.- 12.1. Introduction.- 12.2. Transient Groundwater Flow.- 12.3. Moving Interface Problems.- 12.4. Axisymmetric Bodies in Cross Flow.- 12.5. Slow Viscous Flow (Stokes Flow).- 12.6. General Viscous Flow.- 12.6.1. Steady Problems.- 12.6.2. Transient Problems.- References.- 13 Coupling of Boundary Elements with Other Methods.- 13.1. Introduction.- 13.2. Coupling of Finite Element and Boundary Element Solutions.- 13.2.1. The Energy Approach.- 13.3. Alternative Approach.- 13.4. Internal Fluid Problems.- 13.4.1. Free-Surface Boundary Condition.- 13.4.2. Extension to Compressible Fluid.- 13.5. Approximate Boundary Elements.- 13.6. Approximate Finite Elements.- References.- 14 Computer Program for Two-Dimensional Elastostatics.- 14.1. Introduction.- 14.2. Main Program and Data Structure.- 14.3. Subroutine INPUT.- 14.4. Subroutine MATRX.- 14.5. Subroutine FUNC.- 14.6. Subroutine SLNPD.- 14.7. Subroutine OUTPT.- 14.8. Subroutine FENC.- 14.9. Examples.- 14.9.1. Square Plate.- 14.9.2. Cylindrical Cavity Problem.- References.- Appendix A Numerical Integration Formulas.- A.1. Introduction.- A.2. Standard Gaussian Quadrature.- A.2.1. One-Dimensional Quadrature.- A.2.2. Two- and Three-Dimensional Quadrature for Rectangles and Rectangular Hexahedra.- A.2.3. Triangular Domain.- A.3. Computation of Singular Integrals.- A.3.1. One-Dimensional Logarithmic Gaussian Quadrature Formulas.- A.3.3. Numerical Evaluation of Cauchy Principal Values.- References.- Appendix B Semi-Infinite Fundamental Solutions.- B.1. Half-Space.- B.2. Half-Plane.- References.- Appendix C Some Particular Expressions for Two-Dimensional Inelastic Problems.

A new approach to free vibration analysis using boundary elements

Abstract A boundary element method for the analysis of free vibrations in solid mechanics is developed using a non-standard boundary integral approach. It is shown that, utilizing the statical fundamental solution and employing a special class of coordinate functions, the algebraic eigenvalue problem results. The method has been realized numerically for the two-dimensional elastodynamics, and a number of examples demonstrating its accuracy have been included.

Computational methods in water resources X

1. Groundwater and Flow in Porous Media. 2. Subsurface Transport. 3. Scaling and Heterogeneity. 4. Geostatistics. 5. Reactive Flow. 6. Fractured Porous Media. 7. Parameter Estimation. 8. Remediation and Optimization. 9. Subsurface Multiphase Flow. 10. Saltwater Intrusion. 11. Shallow Water Equations. 12. Flow and Transport in Rivers. 13. Navier--Stokes Equations. 14. Coastal Flow. 15. Sediment Transport. 16. Algebraic Methods. 17. Software Development. 18. Parallel Methods.

The dual reciprocity boundary element formulation for diffusion problems

Abstract This paper presents an extension of the dual reciprocity boundary element method (DRBEM) to deal with nonlinear diffusion problems in which thermal conductivity, specific heat, and density coefficients are all functions of temperature. The DRBEM, recently applied to the solution of problems governed by parabolic and hyperbolic equations, consists in the transformation of the differential equation into an integral equation involving boundary integrals only, the solution of which is achieved by employing a standard boundary element discretization coupled with a two-level finite difference time integration scheme. Contrary to previous formulations for the diffusion equation, the dual reciprocity BEM utilizes the well-known fundamental solution to Laplaces equation, which is space-dependent only. This avoids complex time integrations that normally appear in formulations employing time-dependent fundamental solutions, and permits accurate numerical solutions to be obtained in an efficient way. For nonlinear problems, the integral of conductivity is introduced as a new variable to obtain a linear diffusion equation in the Kirchhoff transform space. This equation involves a modified time variable which is itself a function of position. The problem is solved in an iterative way by using an efficient Newton-Raphson technique which is shown to be rapidly convergent.

Dual reciprocity method using compactly supported radial basis functions

Compactly supported radial basis functions have been used to interpolate the forcing term in the DRM. The resulting DRM matrix is sparse and our approach is very accurate and efficient. Our proposed method is especially attractive for large scale problems. Copyright

Boundary element solution for half-plane problems

Abstract The complete fundamental solution due to unit point loads within the half-plane is presented together with its application to the boundary element method. This solution procedure has proved to be highly accurate and computationally more efficient than the analogue implementation of Kelvin fundamental solution for problems concerning the half-plane. Expressions for stresses at internal points are also presented and the application of the formulation to some classical problems is included.

Boundary element methods for potential problems

Abstract The boundary element formulation of potential problems is presented using weighted residual techniques. The paper emphasizes the simplicity of the boundary methods and the way in which they can be applied in engineering. The advantage of using this method in preference to finite elements is discussed in the applications.

Computational methods and experimental measurements

ContentsSection 1: Computational and experimental methodsTransmitted field in the lossy ground from ground penetrating radar (GPR) dipole antenna The Matrix Pencil method applied to smart monitoring and radar Redesign of ZIGZAG Chair by fiber reinforced plastics: fusing product design and engineering Development of a new segment to improve the dispersion of a nanofiller by extensional flow in a co-rotating twin-screw extruder Stochastic collocation analysis of the transient current induced along the wire buried in a lossy medium Service life assessment of exterior lime-pozzolan renders containing ceramic powderSection 2: Advances in computational methodsStrong form collocation: input for optimization New two-step iteration process for solving the semiconductor plasma generation problem with arbitrary BC in 2D case Determination of the positive weather year for application in hygrothermal simulations A higher-order finite volume method with collocated grid arrangement for incompressible flowsSection 3: Computer modellingNumerical simulation of coupled problems Modeling inside building electromagnetic wave propagation using a finite difference time domain method Control of the objects with a single output and with two or more input channels of influence Validation of the direct numerical simulation code based on the criteria of force coefficient convergence Simulation of nonorganic scintillation detector response for the problems of active interrogation by tagged neutron technology Modeling the economic growth of Arctic regions in RussiaSection 4: Simulation and forecastingThe building blocks of risk theory and methodology Integrated material process simulation for lightweight metal products Prediction of the outlet temperature of an experimental heat exchanger using artificial neural networksSection 5: Fluid flowComplementary studies on supersonic nozzle flow: Heterodyne interferometry, high-speed video shadowgraphy, and numerical simulation Investigation of cross flow over a circular cylinder at low Re using the Immersed Boundary Method (IBM) Control of flow around a circular cylinder using a patterned surface Effect of structural forms on the performance of a jet pump for a deep well jet pumpSection 6: Materials characterisationComputational modelling of elastomeric materials to fit experimental data Fatigue-induced grain growth and formation of slip bands in Cu processed by equal channel angular pressing Characterization of the effect of brick-powder application in lime-based plasters Impact of coarse aggregate type and angularity on permanent deformation of asphalt concrete Impact of metakaolin additive on residual mechanical properties of cement based composites loaded by high temperature Scaling resistance of special high performance composites with burnt clay additive Damage of porous stones by salt crystallization Eutectic Na-Tl and Pb-Mg alloys as liquid-metal coolants for fast nuclear reactors Section 7: Structural analysisDynamic characterization of the Basilica of S. Maria di Collemaggio after the earthquake of 2009 by means of operational modal analysis The tri-harmonic plate bending equationSection 8: Heat transferFinite element modelling of the resistive heating of disposable molecular diagnostics devices Approximate analytical method for thermal analysis of an annular fin having a step profile Numerical model analysis and experimental verification of a thermal regulation strategy for thermal vacuum test On-orbit temperature field calculation and analysis for large net-shape deployable antennas The design of a small-scale multi-stage desalination plant driven by the exhaust of diesel generators Numerical simulations of heat and mass transfer in the MOVPE process for obtaining high-quality nitride-based semiconductors Section 9: Environmental modelling and applicationsMathematical models supporting the monitoring of Civitavecchia harbour (Rome) Application of IRF-Kriging to the mapping of environmental noise generated by quarrying plantsSection 10: Air pollution modellingExperimental research and numerical modelling of the movement of explosive and deflagrable clouds Atmospheric dispersion modeling to simulate rocket exhaust cloudsSection 11: Composite materials and structuresPerformance of fibre-reinforced high-strength concrete with steel sandwich composite panels under static loading The post-buckling response of stiffened CFRP panels using a single-stringer compression specimen (SSCS): a numerical investigationSection 12: Applications in industryModel of a system for the capacitive measurement of the height of liquids in metallic tanks Interoperability test methodology for a train control system using interface channels

Formulation of the boundary element method for transient problems governed by the scalar wave equation

An integral formulation for solving two- and three-dimensional problems governed by the scalar wave equation is presented. The three-dimensional case which has already been the subject of many analytical and numerical works is briefly discussed. The paper concentrates on finding an integral expression for the two-dimensional case, involving source density, initial and boundary conditions, which can be used for time-stepping numerical analysis.