Normal view

# The Finite Element Method : Its Basis and Fundamentals.

Publisher: Jordan Hill : Elsevier Science & Technology, 2005Copyright date: ©2005Edition: 6th edDescription: 1 online resource (753 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9780080472775Genre/Form: Electronic books. Additional physical formats: Print version:: The Finite Element Method : Its Basis and FundamentalsDDC classification: 620/.00151825 LOC classification: TA640.2.Z52 2005Online resources: Click to View
Contents:
Front Cover -- The Finite Element Method: Its Basis and Fundamentals -- Copyright Page -- Contents -- Preface -- Chapter 1. The standard discrete system and origins of the finite element method -- 1.1 Introduction -- 1.2 The structural element and the structural system -- 1.3 Assembly and analysis of a structure -- 1.4 The boundary conditions -- 1.5 Electrical and fluid networks -- 1.6 The general pattern -- 1.7 The standard discrete system -- 1.8 Transformation of coordinates -- 1.9 Problems -- Chapter 2. A direct physical approach to problems in elasticity: plane stress -- 2.1 Introduction -- 2.2 Direct formulation of finite element characteristics -- 2.3 Generalization to the whole region- internal nodal force concept abandoned -- 2.4 Displacement approach as a minimization of total potential energy -- 2.5 Convergence criteria -- 2.6 Discretization error and convergence rate -- 2.7 Displacement functions with discontinuity between elements - non-conforming elements and the patch test -- 2.8 Finite element solution process -- 2.9 Numerical examples -- 2.10 Concluding remarks -- 2.11 Problems -- Chapter 3. Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches -- 3.1 Introduction -- 3.2 Integral or 'weak' statements equivalent to the differential equations -- 3.3 Approximation to integral formulations: the weighted residual-Galerkin method -- 3.4 Virtual work as the 'weak form' of equilibrium equations for analysis of solids or fluids -- 3.5 Partial discretization -- 3.6 Convergence -- 3.7 What are 'variational principles'? -- 3.8 'Natural' variational principles and their relation to governing differential equations -- 3.9 Establishment of natural variational principles for linear, self-adjoint, differential equations -- 3.10 Maximum, minimum, or a saddle point?.
3.11 Constrained variational principles. Lagrange multipliers -- 3.12 Constrained variational principles. Penalty function and perturbed lagrangian methods -- 3.13 Least squares approximations -- 3.14 Concluding remarks - finite difference and boundary methods -- 3.15 Problems -- Chapter 4. 'Standard' and 'hierarchical' element shape functions: some general families of C0 continuity -- 4.1 Introduction -- 4.2 Standard and hierarchical concepts -- 4.3 Rectangular elements- some preliminary considerations -- 4.4 Completeness of polynomials -- 4.5 Rectangular elements- Lagrange family -- 4.6 Rectangular elements- 'serendipity' family -- 4.7 Triangular element family -- 4.8 Line elements -- 4.9 Rectangular prisms - Lagrange family -- 4.10 Rectangular prisms - 'serendipity' family -- 4.11 Tetrahedral elements -- 4.12 Other simple three-dimensional elements -- 4.13 Hierarchic polynomials in one dimension -- 4.14 Two- and three-dimensional, hierarchical elements of the 'rectangle' or 'brick' type -- 4.15 Triangle and tetrahedron family -- 4.16 Improvement of conditioning with hierarchical forms -- 4.17 Global and local finite element approximation -- 4.18 Elimination of internal parameters before assembly - substructures -- 4.19 Concluding remarks -- 4.20 Problems -- Chapter 5. Mapped elements and numerical integration- 'infinite' and 'singularity elements' -- 5.1 Introduction -- 5.2 Use of 'shape functions' in the establishment of coordinate transformations -- 5.3 Geometrical conformity of elements -- 5.4 Variation of the unknown function within distorted, curvilinear elements. Continuity requirements -- 5.5 Evaluation of element matrices. Transformation in ξ,η,ζ coordinates -- 5.6 Evaluation of element matrices. Transformation in area and volume coordinates -- 5.7 Order of convergence for mapped elements -- 5.8 Shape functions by degeneration.
5.9 Numerical integration- one dimensional -- 5.10 Numerical integration- rectangular (2D) or brick regions (3D) -- 5.11 Numerical integration - triangular or tetrahedral regions -- 5.12 Required order of numerical integration -- 5.13 Generation of finite element meshes by mapping. Blending functions -- 5.14 Infinite domains and infinite elements -- 5.15 Singular elements by mapping - use in fracture mechanics, etc. -- 5.16 Computational advantage of numerically integrated finite elements -- 5.17 Problems -- Chapter 6. Problems in linear elasticity -- 6.1 Introduction -- 6.2 Governing equations -- 6.3 Finite element approximation -- 6.4 Reporting of results: displacements, strains and stresses -- 6.5 Numerical examples -- 6.6 Problems -- Chapter 7. Field problems - heat conduction, electric and magnetic potential and fluid flow -- 7.1 Introduction -- 7.2 General quasi-harmonic equation -- 7.3 Finite element solution process -- 7.4 Partial discretization- transient problems -- 7.5 Numerical examples- an assessment of accuracy -- 7.6 Concluding remarks -- 7.7 Problems -- Chapter 8. Automatic mesh generation -- 8.1 Introduction -- 8.2 Two-dimensional mesh generation- advancing front method -- 8.3 Surface mesh generation -- 8.4 Three-dimensional mesh generation- Delaunay triangulation -- 8.5 Concluding remarks -- 8.6 Problems -- Chapter 9. The patch test, reduced integration, and non-conforming elements -- 9.1 Introduction -- 9.2 Convergence requirements -- 9.3 The simple patch test (tests A and B) - a necessary condition for convergence -- 9.4 Generalized patch test (test C) and the single-element test -- 9.5 The generality of a numerical patch test -- 9.6 Higher order patch tests -- 9.7 Application of the patch test to plane elasticity elements with 'standard' and 'reduced' quadrature -- 9.8 Application of the patch test to an incompatible element.
9.9 Higher order patch test- assessment of robustness -- 9.10 Concluding remarks -- 9.11 Problems -- Chapter 10. Mixed formulation and constraints- complete field methods -- 10.1 Introduction -- 10.2 Discretization of mixed forms - some general remarks -- 10.3 Stability of mixed approximation. The patch test -- 10.4 Two-field mixed formulation in elasticity -- 10.5 Three-field mixed formulations in elasticity -- 10.6 Complementary forms with direct constraint -- 10.7 Concluding remarks - mixed formulation or a test of element 'robustness' -- 10.8 Problems -- Chapter 11. Incompressible problems, mixed methods and other procedures of solution -- 11.1 Introduction -- 11.2 Deviatoric stress and strain, pressure and volume change -- 11.3 Two-field incompressible elasticity (u-p form) -- 11.4 Three-field nearly incompressible elasticity (u-p-εv form) -- 11.5 Reduced and selective integration and its equivalence to penalized mixed problems -- 11.6 A simple iterative solution process for mixed problems: Uzawa method -- 11.7 Stabilized methods for some mixed elements failing the incompressibility patch test -- 11.8 Concluding remarks -- 11.9 Problems -- Chapter 12. Multidomain mixed approximations- domain decomposition and 'frame' methods -- 12.1 Introduction -- 12.2 Linking of two or more subdomains by Lagrange multipliers -- 12.3 Linking of two or more subdomains by perturbed lagrangian and penalty methods -- 12.4 Interface displacement 'frame' -- 12.5 Linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements -- 12.6 Subdomains with 'standard' elements and global functions -- 12.7 Concluding remarks -- 12.8 Problems -- Chapter 13. Errors, recovery processes and error estimates -- 13.1 Definition of errors -- 13.2 Superconvergence and optimal sampling points -- 13.3 Recovery of gradients and stresses.
13.4 Superconvergent patch recovery - SPR -- 13.5 Recovery by equilibration of patches - REP -- 13.6 Error estimates by recovery -- 13.7 Residual-based methods -- 13.8 Asymptotic behaviour and robustness of error estimators - the Babuška patch test -- 13.9 Bounds on quantities of interest -- 13.10 Which errors should concern us? -- 13.11 Problems -- Chapter 14. Adaptive finite element refinement -- 14.1 Introduction -- 14.2 Adaptive h-refinement -- 14.3 p-refinement and hp-refinement -- 14.4 Concluding remarks -- 14.5 Problems -- Chapter 15. Point-based and partition of unity approximations. Extended finite element methods -- 15.1 Introduction -- 15.2 Function approximation -- 15.3 Moving least squares approximations - restoration of continuity of approximation -- 15.4 Hierarchical enhancement of moving least squares expansions -- 15.5 Point collocation- finite point methods -- 15.6 Galerkin weighting and finite volume methods -- 15.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement -- 15.8 Concluding remarks -- 15.9 Problems -- Chapter 16. The time dimension- semi-discretization of field and dynamic problems and analytical solution procedures -- 16.1 Introduction -- 16.2 Direct formulation of time-dependent problems with spatial finite element subdivision -- 16.3 General classification -- 16.4 Free response - eigenvalues for second-order problems and dynamic vibration -- 16.5 Free response - eigenvalues for first-order problems and heat conduction, etc. -- 16.6 Free response- damped dynamic eigenvalues -- 16.7 Forced periodic response -- 16.8 Transient response by analytical procedures -- 16.9 Symmetry and repeatability -- 16.10 Problems -- Chapter 17. The time dimension- discrete approximation in time -- 17.1 Introduction.
17.2 Simple time-step algorithms for the first-order equation.
Summary: The Sixth Edition of this influential best-selling book delivers the most up-to-date and comprehensive text and reference yet on the basis of the finite element method (FEM) for all engineers and mathematicians. Since the appearance of the first edition 38 years ago, The Finite Element Method provides arguably the most authoritative introductory text to the method, covering the latest developments and approaches in this dynamic subject, and is amply supplemented by exercises, worked solutions and computer algorithms. The classic FEM text, written by the subject's leading authors Enhancements include more worked examples and exercises, plus a companion website with a solutions manual and downloadable algorithms With a new chapter on automatic mesh generation and added materials on shape function development and the use of higher order elements in solving elasticity and field problems Active research has shaped The Finite Element Method into the pre-eminent tool for the modelling of physical systems. It maintains the comprehensive style of earlier editions, while presenting the systematic development for the solution of problems modelled by linear differential equations. Together with the second and third self-contained volumes (0750663219 and 0750663227), The Finite Element Method Set (0750664312) provides a formidable resource covering the theory and the application of FEM, including the basis of the method, its application to advanced solid and structural mechanics and to computational fluid dynamics. * The classic introduction to the finite element method, by two of the subject's leading authors * Any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in this key text * Enhancements include more worked examples, exercises, plus a companionSummary: website with a worked solutions manual for tutors and downloadable algorithms.
Holdings
Item type Current library Call number Status Date due Barcode Item holds Ebrary Afghanistan
Available EBKAF00015869 Ebrary Algeria
Available Ebrary Cyprus
Available Ebrary Egypt
Available Ebrary Libya
Available Ebrary Morocco
Available Ebrary Nepal
Available EBKNP00015869 Ebrary Sudan

Access a wide range of magazines and books using Pressreader and Ebook central.

Available Ebrary Tunisia
Available
Total holds: 0

Front Cover -- The Finite Element Method: Its Basis and Fundamentals -- Copyright Page -- Contents -- Preface -- Chapter 1. The standard discrete system and origins of the finite element method -- 1.1 Introduction -- 1.2 The structural element and the structural system -- 1.3 Assembly and analysis of a structure -- 1.4 The boundary conditions -- 1.5 Electrical and fluid networks -- 1.6 The general pattern -- 1.7 The standard discrete system -- 1.8 Transformation of coordinates -- 1.9 Problems -- Chapter 2. A direct physical approach to problems in elasticity: plane stress -- 2.1 Introduction -- 2.2 Direct formulation of finite element characteristics -- 2.3 Generalization to the whole region- internal nodal force concept abandoned -- 2.4 Displacement approach as a minimization of total potential energy -- 2.5 Convergence criteria -- 2.6 Discretization error and convergence rate -- 2.7 Displacement functions with discontinuity between elements - non-conforming elements and the patch test -- 2.8 Finite element solution process -- 2.9 Numerical examples -- 2.10 Concluding remarks -- 2.11 Problems -- Chapter 3. Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches -- 3.1 Introduction -- 3.2 Integral or 'weak' statements equivalent to the differential equations -- 3.3 Approximation to integral formulations: the weighted residual-Galerkin method -- 3.4 Virtual work as the 'weak form' of equilibrium equations for analysis of solids or fluids -- 3.5 Partial discretization -- 3.6 Convergence -- 3.7 What are 'variational principles'? -- 3.8 'Natural' variational principles and their relation to governing differential equations -- 3.9 Establishment of natural variational principles for linear, self-adjoint, differential equations -- 3.10 Maximum, minimum, or a saddle point?.

3.11 Constrained variational principles. Lagrange multipliers -- 3.12 Constrained variational principles. Penalty function and perturbed lagrangian methods -- 3.13 Least squares approximations -- 3.14 Concluding remarks - finite difference and boundary methods -- 3.15 Problems -- Chapter 4. 'Standard' and 'hierarchical' element shape functions: some general families of C0 continuity -- 4.1 Introduction -- 4.2 Standard and hierarchical concepts -- 4.3 Rectangular elements- some preliminary considerations -- 4.4 Completeness of polynomials -- 4.5 Rectangular elements- Lagrange family -- 4.6 Rectangular elements- 'serendipity' family -- 4.7 Triangular element family -- 4.8 Line elements -- 4.9 Rectangular prisms - Lagrange family -- 4.10 Rectangular prisms - 'serendipity' family -- 4.11 Tetrahedral elements -- 4.12 Other simple three-dimensional elements -- 4.13 Hierarchic polynomials in one dimension -- 4.14 Two- and three-dimensional, hierarchical elements of the 'rectangle' or 'brick' type -- 4.15 Triangle and tetrahedron family -- 4.16 Improvement of conditioning with hierarchical forms -- 4.17 Global and local finite element approximation -- 4.18 Elimination of internal parameters before assembly - substructures -- 4.19 Concluding remarks -- 4.20 Problems -- Chapter 5. Mapped elements and numerical integration- 'infinite' and 'singularity elements' -- 5.1 Introduction -- 5.2 Use of 'shape functions' in the establishment of coordinate transformations -- 5.3 Geometrical conformity of elements -- 5.4 Variation of the unknown function within distorted, curvilinear elements. Continuity requirements -- 5.5 Evaluation of element matrices. Transformation in ξ,η,ζ coordinates -- 5.6 Evaluation of element matrices. Transformation in area and volume coordinates -- 5.7 Order of convergence for mapped elements -- 5.8 Shape functions by degeneration.

5.9 Numerical integration- one dimensional -- 5.10 Numerical integration- rectangular (2D) or brick regions (3D) -- 5.11 Numerical integration - triangular or tetrahedral regions -- 5.12 Required order of numerical integration -- 5.13 Generation of finite element meshes by mapping. Blending functions -- 5.14 Infinite domains and infinite elements -- 5.15 Singular elements by mapping - use in fracture mechanics, etc. -- 5.16 Computational advantage of numerically integrated finite elements -- 5.17 Problems -- Chapter 6. Problems in linear elasticity -- 6.1 Introduction -- 6.2 Governing equations -- 6.3 Finite element approximation -- 6.4 Reporting of results: displacements, strains and stresses -- 6.5 Numerical examples -- 6.6 Problems -- Chapter 7. Field problems - heat conduction, electric and magnetic potential and fluid flow -- 7.1 Introduction -- 7.2 General quasi-harmonic equation -- 7.3 Finite element solution process -- 7.4 Partial discretization- transient problems -- 7.5 Numerical examples- an assessment of accuracy -- 7.6 Concluding remarks -- 7.7 Problems -- Chapter 8. Automatic mesh generation -- 8.1 Introduction -- 8.2 Two-dimensional mesh generation- advancing front method -- 8.3 Surface mesh generation -- 8.4 Three-dimensional mesh generation- Delaunay triangulation -- 8.5 Concluding remarks -- 8.6 Problems -- Chapter 9. The patch test, reduced integration, and non-conforming elements -- 9.1 Introduction -- 9.2 Convergence requirements -- 9.3 The simple patch test (tests A and B) - a necessary condition for convergence -- 9.4 Generalized patch test (test C) and the single-element test -- 9.5 The generality of a numerical patch test -- 9.6 Higher order patch tests -- 9.7 Application of the patch test to plane elasticity elements with 'standard' and 'reduced' quadrature -- 9.8 Application of the patch test to an incompatible element.

9.9 Higher order patch test- assessment of robustness -- 9.10 Concluding remarks -- 9.11 Problems -- Chapter 10. Mixed formulation and constraints- complete field methods -- 10.1 Introduction -- 10.2 Discretization of mixed forms - some general remarks -- 10.3 Stability of mixed approximation. The patch test -- 10.4 Two-field mixed formulation in elasticity -- 10.5 Three-field mixed formulations in elasticity -- 10.6 Complementary forms with direct constraint -- 10.7 Concluding remarks - mixed formulation or a test of element 'robustness' -- 10.8 Problems -- Chapter 11. Incompressible problems, mixed methods and other procedures of solution -- 11.1 Introduction -- 11.2 Deviatoric stress and strain, pressure and volume change -- 11.3 Two-field incompressible elasticity (u-p form) -- 11.4 Three-field nearly incompressible elasticity (u-p-εv form) -- 11.5 Reduced and selective integration and its equivalence to penalized mixed problems -- 11.6 A simple iterative solution process for mixed problems: Uzawa method -- 11.7 Stabilized methods for some mixed elements failing the incompressibility patch test -- 11.8 Concluding remarks -- 11.9 Problems -- Chapter 12. Multidomain mixed approximations- domain decomposition and 'frame' methods -- 12.1 Introduction -- 12.2 Linking of two or more subdomains by Lagrange multipliers -- 12.3 Linking of two or more subdomains by perturbed lagrangian and penalty methods -- 12.4 Interface displacement 'frame' -- 12.5 Linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements -- 12.6 Subdomains with 'standard' elements and global functions -- 12.7 Concluding remarks -- 12.8 Problems -- Chapter 13. Errors, recovery processes and error estimates -- 13.1 Definition of errors -- 13.2 Superconvergence and optimal sampling points -- 13.3 Recovery of gradients and stresses.

13.4 Superconvergent patch recovery - SPR -- 13.5 Recovery by equilibration of patches - REP -- 13.6 Error estimates by recovery -- 13.7 Residual-based methods -- 13.8 Asymptotic behaviour and robustness of error estimators - the Babuška patch test -- 13.9 Bounds on quantities of interest -- 13.10 Which errors should concern us? -- 13.11 Problems -- Chapter 14. Adaptive finite element refinement -- 14.1 Introduction -- 14.2 Adaptive h-refinement -- 14.3 p-refinement and hp-refinement -- 14.4 Concluding remarks -- 14.5 Problems -- Chapter 15. Point-based and partition of unity approximations. Extended finite element methods -- 15.1 Introduction -- 15.2 Function approximation -- 15.3 Moving least squares approximations - restoration of continuity of approximation -- 15.4 Hierarchical enhancement of moving least squares expansions -- 15.5 Point collocation- finite point methods -- 15.6 Galerkin weighting and finite volume methods -- 15.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement -- 15.8 Concluding remarks -- 15.9 Problems -- Chapter 16. The time dimension- semi-discretization of field and dynamic problems and analytical solution procedures -- 16.1 Introduction -- 16.2 Direct formulation of time-dependent problems with spatial finite element subdivision -- 16.3 General classification -- 16.4 Free response - eigenvalues for second-order problems and dynamic vibration -- 16.5 Free response - eigenvalues for first-order problems and heat conduction, etc. -- 16.6 Free response- damped dynamic eigenvalues -- 16.7 Forced periodic response -- 16.8 Transient response by analytical procedures -- 16.9 Symmetry and repeatability -- 16.10 Problems -- Chapter 17. The time dimension- discrete approximation in time -- 17.1 Introduction.

17.2 Simple time-step algorithms for the first-order equation.

The Sixth Edition of this influential best-selling book delivers the most up-to-date and comprehensive text and reference yet on the basis of the finite element method (FEM) for all engineers and mathematicians. Since the appearance of the first edition 38 years ago, The Finite Element Method provides arguably the most authoritative introductory text to the method, covering the latest developments and approaches in this dynamic subject, and is amply supplemented by exercises, worked solutions and computer algorithms. The classic FEM text, written by the subject's leading authors Enhancements include more worked examples and exercises, plus a companion website with a solutions manual and downloadable algorithms With a new chapter on automatic mesh generation and added materials on shape function development and the use of higher order elements in solving elasticity and field problems Active research has shaped The Finite Element Method into the pre-eminent tool for the modelling of physical systems. It maintains the comprehensive style of earlier editions, while presenting the systematic development for the solution of problems modelled by linear differential equations. Together with the second and third self-contained volumes (0750663219 and 0750663227), The Finite Element Method Set (0750664312) provides a formidable resource covering the theory and the application of FEM, including the basis of the method, its application to advanced solid and structural mechanics and to computational fluid dynamics. * The classic introduction to the finite element method, by two of the subject's leading authors * Any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in this key text * Enhancements include more worked examples, exercises, plus a companion