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Luiz C. Wrobel

Bio: Luiz C. Wrobel is an academic researcher from Brunel University London. The author has contributed to research in topics: Boundary element method & Boundary value problem. The author has an hindex of 36, co-authored 270 publications receiving 10779 citations. Previous affiliations of Luiz C. Wrobel include Federal University of Rio de Janeiro & University of Brasília.


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Book
28 Feb 1984
TL;DR: In this article, the authors propose a method of approximate boundary refinement based on the theory of elasticity, and apply it to two-dimensional problems with different types of boundary conditions.
Abstract: 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.

1,424 citations

Book
01 Jan 1991
TL;DR: In this article, the authors present the boundary element method for Laplace's Equation, which is used to solve the Torsion Problem with different approximations of functions.
Abstract: 1 Introduction.- 2 The Boundary Element Method for Equations ?2u = 0 and ?2u = b.- 2.1 Introduction.- 2.2 The Case of the Laplace Equation.- 2.2.1 Fundamental Relationships.- 2.2.2 Boundary Integral Equations.- 2.2.3 The Boundary Element Method for Laplace's Equation.- 2.2.4 Evaluation of Integrals.- 2.2.5 Linear Elements.- 2.2.6 Treatment of Corners.- 2.2.7 Quadratic and Higher-Order Elements.- 2.3 Formulation for the Poisson Equation.- 2.3.1 Basic Relationships.- 2.3.2 Cell Integration Approach.- 2.3.3 The Monte Carlo Method.- 2.3.4 The Use of Particular Solutions.- 2.3.5 The Galerkin Vector Approach.- 2.3.6 The Multiple Reciprocity Method.- 2.4 Computer Program 1.- 2.4.1 MAINP1.- 2.4.2 Subroutine INPUT1.- 2.4.3 Subroutine ASSEM2.- 2.4.4 Subroutine NECMOD.- 2.4.5 Subroutine SOLVER.- 2.4.6 Subroutine INTERM.- 2.4.7 Subroutine OUTPUT.- 2.4.8 Results of a Test Problem.- 2.5 References.- 3 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y).- 3.1 Equation Development.- 3.1.1 Preliminary Considerations.- 3.1.2 Mathematical Development of the DRM for the Poisson Equation.- 3.2 Different f Expansions.- 3.2.1 Case f = r.- 3.2.2 Case f = 1+ r.- 3.2.3 Case f = 1 at One Node and f = r at Remaining Nodes.- 3.3 Computer Implementation.- 3.3.1 Schematized Matrix Equations.- 3.3.2 Sign of the Components of r and its Derivatives.- 3.4 Computer Program 2.- 3.4.1 MAINP2.- 3.4.2 Subroutine INPUT2.- 3.4.3 Subroutine ALFAF2.- 3.4.4 Subroutine RHSVEC.- 3.4.5 Comparison of Results for a Torsion Problem using Different Approximating Functions.- 3.4.6 Data and Output for Program 2.- 3.5 Results for Different Functions b = b(x,y).- 3.5.1 The Case ?2u = ?x.- 3.5.2 The Case ?2u = ?x2.- 3.5.3 The Case ?2u = a2 ? x2.- 3.5.4 Results using Quadratic Elements.- 3.6 Problems with Different Domain Integrals on Different Regions.- 3.6.1 The Subregion Technique.- 3.6.2 Integration over Internal Region.- 3.7 References.- 4 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y, u).- 4.1 Introduction.- 4.2 The Convective Case.- 4.2.1 Results for the Case ?2u = ??u/?x.- 4.2.2 Results for the Case ?2u = ?(?u/?x+ ?u/?y).- 4.2.3 Internal Derivatives of the Problem Variables.- 4.3 The Helmholtz Equation.- 4.3.1 DRM Formulations.- 4.3.2 DRM Results for Vibrating Beam.- 4.3.3 Results for Non-Inversion DRM.- 4.4 Non-Linear Cases.- 4.4.1 Burger's Equation.- 4.4.2 Spontaneous Ignition: The Steady-State Case.- 4.4.3 Non-Linear Material Problems.- 4.5 Computer Program 3.- 4.5.1 MAINP3.- 4.5.2 Subroutine ALFAF3.- 4.5.3 Subroutine RHSMAT.- 4.5.4 Subroutine DERIVXY.- 4.5.5 Results of Test Problems.- 4.6 Three-Dimensional Analysis.- 4.6.1 Equations of the Type ?2u = b(x, y, z).- 4.6.2 Equations of the Type ?2u = b(x, y, z, u).- 4.7 References.- 5 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y, u, t).- 5.1 Introduction.- 5.2 The Diffusion Equation.- 5.3 Computer Program 4.- 5.3.1 MAINP4.- 5.3.2 Subroutine ASSEMB.- 5.3.3 Subroutine VECTIN.- 5.3.4 Subroutine BOUNDC.- 5.3.5 Results of a Test Problem.- 5.3.6 Data Input.- 5.3.7 Computer Output.- 5.3.8 Further Applications.- 5.3.9 Other Time-Stepping Schemes.- 5.4 Special f Expansions.- 5.4.1 Axisymmetric Diffusion.- 5.4.2 Infinite Regions.- 5.5 The Wave Equation.- 5.5.1 Infinite and Semi-Infinite Regions.- 5.6 The Transient Convection-Diffusion Equation.- 5.7 Non-Linear Problems.- 5.7.1 Non-Linear Materials.- 5.7.2 Non-Linear Boundary Conditions.- 5.7.3 Spontaneous Ignition: Transient Case.- 5.8 References.- 6 Other Fundamental Solutions.- 6.1 Introduction.- 6.2 Two-Dimensional Elasticity.- 6.2.1 Static Analysis.- 6.2.2 Treatment of Body Forces.- 6.2.3 Dynamic Analysis.- 6.3 Plate Bending.- 6.4 Three-Dimensional Elasticity.- 6.4.1 Computational Formulation.- 6.4.2 Gravitational Load.- 6.4.3 Centrifugal Load.- 6.4.4 Thermal Load.- 6.5 Transient Convection-Diffusion.- 6.6 References.- 7 Conclusions.- Appendix 1.- Appendix 2.- The Authors.

1,010 citations

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Cited by
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01 May 1993
TL;DR: Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems.
Abstract: Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

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