L.C. Wrobel

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.

Design and construction of a LiBr–water absorption machine

Abstract The objective of this paper is to present a method to evaluate the characteristics and performance of a single stage lithium bromide (LiBr)–water absorption machine. The necessary heat and mass transfer equations and appropriate equations describing the properties of the working fluids are specified. These equations are employed in a computer program, and a sensitivity analysis is performed. The difference between the absorber LiBr inlet and outlet percentage ratio, the coefficient of performance of the unit in relation to the generator temperature, the efficiency of the unit in relation to the solution heat exchanger area and the solution strength effectiveness in relation to the absorber solution outlet temperature are examined. Information on designing the heat exchangers of the LiBr–water absorption unit are also presented. Single pass, vertical tube heat exchangers have been used for the absorber and for the evaporator. The solution heat exchanger was designed as a single pass annular heat exchanger. The condenser and the generator were designed using horizontal tube heat exchangers. The calculated theoretical values are compared to experimental results derived for a small unit with a nominal capacity of 1 kW. Finally, a cost analysis for a domestic size absorber cooler is presented.

Measures used to lower building energy consumption and their cost effectiveness

This study uses the TRNSYS computer program, for the modelling and simulation of the energy flows of modern houses, to examine measures to reduce the thermal load. For the calculations, a typical meteorological year (TMY) and a typical model house are used. The measures examined are natural and controlled ventilation, solar shading, various types of glazing, orientation, shape of buildings, and thermal mass. In summer, ventilation leads to a maximum reduction of annual cooling load of 7.7% for maintaining the house at 25 °C. The effect depends on the construction type, with the better-insulated house saving a higher percentage. Window gains are an important factor and significant savings can result when extra measures are taken. The saving in annual cooling load, for a well-insulated house, may be as much as 24% when low-emissivity double glazing windows are used, which are recommended since the payback period is short (3.8 years). Overhangs may have a length over windows of 1.5 m. In this way, about 7% of the annual cooling load can be saved for a house constructed from single walls with no roof insulation. These savings are about 19% for a house constructed from walls and roof with 50 mm insulation. The shape of the building affects the thermal load. The results show that the elongated shape shows an increase in the annual heating load, which is between 8.2 and 26.7% depending on the construction type, compared with a square-shaped house. Referring to orientation, the best position for a symmetrical house is to face the four cardinal points and for an elongated house to have its long side facing south. In respect to thermal mass, the analysis shows that increasing the wall and roof masses and utilizing night ventilation is not enough to lower the house temperature to acceptable limits during summer. Also, the analysis shows that the roof is the most important structural element of the buildings in a hot environment. The roof must offer a discharge time of 6 h or more and have a thermal conductivity of less than 0.48 W/mK. The life-cycle cost analysis has shown that measures that increase the roof insulation, pay back in a short period of time, between 3.5 and 5 years. However, measures taken to increase wall insulation pay back in a long period of time, of about 10 years.

Modelling and simulation of an absorption solar cooling system for Cyprus

Abstract In this paper a modelling and simulation of an absorption solar cooling system is presented. The system is modelled with the TRNSYS simulation program and the typical meteorological year file containing the weather parameters of Nicosia, Cyprus. Initially a system optimisation is carried out in order to select the appropriate type of collector, the optimum size of storage tank, the optimum collector slope and area, and the optimum thermostat setting of the auxiliary boiler. The final optimised system consists of a 15-m2 compound parabolic collector tilted 30° from the horizontal and a 600-l hot water storage tank. The collector area is determined by performing the life cycle analysis of the system. The optimum solar system selected gives life cycle savings of C£1376 when a nonsubsidized fuel cost is considered. The system operates with maximum performance when the auxiliary boiler thermostat is set at 87°C. The system long-term integrated performance shows that 84,240 MJ required for cooling and 41,263 MJ for hot water production are supplied with solar energy.

Modelling, simulation and warming impact assessment of a domestic-size absorption solar cooling system

In this paper the modelling, simulation and total equivalent warming impact (TEWI) of a domestic-size absorption solar cooling system is presented. The system consists of a solar collector, storage tank, a boiler and a LiBr–water absorption refrigerator. Experimentally determined heat and mass transfer coefficients were employed in the design and costing of an 11 kW cooling capacity solar driven absorption cooling machine which, from simulations, was found to have sufficient capacity to satisfy the cooling needs of a well insulated domestic dwelling. The system is modelled with the TRNSYS simulation program using appropriate equations predicting the performance of the unit. The final optimum system consists of 15 m2 compound parabolic collector tilted at 30° from horizontal and 600 l hot water storage tank. The total life cycle cost of a complete system, comprising the collector and the absorption unit, for a lifetime of 20 years will be of the order of C£ 13,380. The cost of the absorption system alone was determined to be C£ 4800. Economic analysis has shown that for such a system to be economically competitive compared to conventional cooling systems its capital cost should be below C£ 2000. The system however has a lower TEWI being 1.2 times smaller compared to conventional cooling systems.

Review of solar and low energy cooling technologies for buildings

The objective of this paper is to examine solar cooling and low energy cooling technologies. A brief review of various cooling systems is presented, including solar sorption cooling, solar-mechanical systems, solar related air conditioning, and other low energy cooling technologies. The relative efficiencies and applications of the various technologies are presented. These technologies can be utilized to reduce both the energy consumption and environmental impact of mechanical cooling systems.

Modeling of the modern houses of Cyprus and energy consumption analysis

This study uses the TRNSYS computer program for the modeling and simulation of the energy flows of the modern houses of Cyprus followed by an energy consumption analysis. For the calculations, a Typical Meteorological Year for the Nicosia area and a typical model house are used. Initially, the Cyprus energy scene and an analysis of the number of houses employing heating and cooling equipment is presented from which it is observed that the number of systems installed has increased tremendously during the last decade. The results of the simulation show that the inside house temperature, when no air-conditioning is used, varies between 10–20°C for winter and between 30–50°C for summer. The effect on the temperature and the heating and cooling loads that various wall and roof constructions present is determined. This investigation indicates the importance of the roof insulation, which results in a reduction up to 45.5% of the cooling load and up to 75% of the heating load. The effect of mechanical ventilation, window shading, as well as that of the inclined concrete roof used for aesthetic reasons, is also examined. The life cycle analysis is used for the economic analysis of the various constructions. The results indicate that the wall insulation pays back in a twenty year period with marginal savings, whereas the roof insulation has considerable economic benefit, with life cycle savings up to EUR 22374 depending on the type of construction.

On the convergence of the dual reciprocity boundary element method

Abstract This paper presents a study of the convergence properties of the dual reciprocity method (DRM). DRM is one of the most popular techniques used to transform volume integrals that arise, for example, from the inhomogeneous term of Poissons equation, into equivalent boundary integrals in the boundary element method (BEM). The transformation is carried out by expanding the inhomogeneous term into approximating functions whose particular solutions can be easily obtained. In the present paper, interpolation functions are derived from the approximating functions, and their properties are studied theoretically and numerically. The results obtained confirm that the interpolation converges to the real function.

COMPUTATIONAL MODELLING OF FREE AND MOVING BOUNDARY PROBLEMS

Part 1 Basic theory: phase interface phenomena phase distribution and separation instabilities solid-liquid phase change stefan problems. Part 2 Applications: combustion chemically reacting flow surface and internal wave propagation in solids and liquids. Part 3 Numerical methods: finite elements boundary elements spectral methods.

A BEM formulation using B-splines: I-uniform blending functions

Abstract This work develops a new element formulation for boundary element analysis using uniform cubic B-splines. These functions can be employed to provide higher degrees of continuity along the geometric boundary of the region, and also as interpolation functions for the problem variables. Applications are presented related to potential problems governed by Laplaces equation but there are no restrictions in the formulation regarding its extension to other physical problems.