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| موضوع: كتاب Finite Element Methods for Engineers الأحد 05 سبتمبر 2021, 8:56 am | |
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أخواني في الله أحضرت لكم كتاب Finite Element Methods for Engineers Roger T. Fenner Department of Mechanical Engineering Imperial College of Science, Technology and Medicine London
و المحتوى كما يلي :
Contents Preface Noration Some Program Variable Names 1 Introduction and Structural Analysis 1 . 1 Computer Programming 1.2 Structural Analysis 1.3 Case Study: Bending of a Tapered Beam 2 Continuum Mechanics Problems 2.1 Continuum Mechanics Equations 2.2 Some Physical Problems Classification of Partial Differential Equations Methods for Solving Harmonic and Biharmonic Equations 3 Finite Element Analysis of Harmonic Problems Derivation of the Element Stiffness Matrix Assembly of the Overall Stiffness Matrix Comparison with the Finite Difference Approach 3.4 Variational Formulation 3.5 Boundary Conditions Solution of the Linear Equations Convergence of Finite Element Methods A Computer Program for Harmonic Problems 4 Finite Element Meshes 4.1 Choice of Mesh 4.2 Mesh Data in Numerical Form 72vi Contents 4.3 Generation of Mesh Data 4.4 Mesh Modification 5 Some Harmonic Problems Case Study: Downstream Viscous Flow in a Rectangular Case Study: Torsion of Prismatic Bars Channel 6 Finite Element Analysis of Biharmonic Problems 6.3 Variational Formulation 6.5 Boundary Conditions Derivation of the Element Stiffness Matrix Assembly of the Overall Stiffness Matrix Solution of the Linear Equations A Computer Program for Problems of the Biharmonic Plane Strain or Plane Stress Type 7 Some Biharmonic Problems Case Study: Plane Strain Compression Case Study: Stresses in Concentric Cylinders Case Study: Stress Concentration near a Hole in a Flat Plate 8 Further Applications Axi-symmetric Problems Higher-order Elements Three-dimensional Problems Biharmonic Problems Involving Incompressible Plate and Shell Problems Isoparametric Elements Nonlinear Problems A Summary of the Finite Element Approach Concluding Remarks Materials Appendix A Gaussian Elimination Appendix B The Gauss-Seidel Method Bibliogaphy x accuracy Airy’s stress function 27, 28 algebraic equations see simultaneous alphanumeric information 3, 66, analytical solution 1, 2, 8-9, 36, 13, 48, 71, 90, 97-8, 102-3, 137, 146, 153, 154 algebraic equations 122 89-90, 96, 99, 101, 135, 136, 137, 138-9, 146 argument of subprogram array, dimensioning of and order of aspect ratio 144-6 assembly of overall stiffness 6-7, axi-symmetric problems 135-8, 12, 69, 163 storage 11 12,43-5, 66-8, 110-1 1, 122 148-51, 154 back substitution 60, 159, I63 backing stores 60, 154 banded matrix bandwidth 56, 59, 72, 154 beam 4, 6, 8- 14 bending 4-14, 30, 32, 155-6 biharmonic equations 32, 33-6, 155 biharmonic operator 28 biharmonic problems 71, 72, 76 binary arithmetic 3 bits 3 body forces 13, 55-6, 60, 154 104-47, 148, 149-51, 154-5 17, 20, 22, 27, 28, 29, 31,104, 108, 110,111, 112, 118, 121, 122, 124, 125, 127, 137 113 bound for a solution 63, 97, 103, boundary conditions 9, 2 1, 22-3, 25, 26, 28, 29, 30-1, 33, 34-5,36, 48, 51, 53-5, 61, 89, 91, 93-5,96, 98, 101, 115-18, 122, 123,125-7, 129, 133, 136, 139, 144, 148, 155, 158 derivative 25, 26, 29, 30-1, 34-5, 51, 54-5,.93-5, 96, 158 boundary shape boundary-value problems 31-2, 33 36, 37, 71-2, 74, 86, 103, 139, 140, 152, 156, 158 card 2 Cartesian co-ordinates 15, 36, 38, centrifugal forces 124 coding 3, 97, 130 coefficient of thermal expansion 19, 123, 133, 135, 144 comment statement 3 COMMON block 118, 121, 124, 127 compatibility 103, 138 65, 66, 68, 76, 18, 22, 23, 27, 29, 30, 36,40, 52,61-2, 106, 151 inter-element 40, 52, 61-2, 106, 113, 152-3, 154-5,156 complementary stresses 16, 23 computer conduction 25, 32, 124 conforming (compatible) elements constant strain triangles (CST) 37, 1, 2, 36, 55, 56, 59, 60, 71, 73, 158 62,63, 106, 112, 150, 152-3, 155 40, 48, 52, 92, 104, 113, 116, 134, 151, 153Index constitutive equations 19-20, 22, 28, 29, 42,45, 61, 108-9, 149-50 continuity continuum mechanics 2, 14, 15,-36, convergence, finite element methods Gauss-Seidel method 57-9, 19, 24, 26, 29, 30, 154 49,61 48, 52,61-2, 113 61, 69, 70, 73, 93, 95-6, 102, 114, 131, 144, 164-5 co-ordinate systems 15, 36, 103, core store 3, 60 136, 138, 158 amount of 3, 12, 55,56, 57, 59-60, 97, 98, 113, 147, 154, 158 cost of computing couple 23, 31, 101 2,97, 154, 158 data 125 deformation 155, 156 density dependent variable 30-2, 36, 39, 2, 12, 66,68, 71-88, 90, 121, 15, 21, 106, 131, 135, 17, 19, 133, 137, 142 40, 103 choice of 30, 35, 36, 103, 104-5, 154-5 determinant 114, 160 diagonal dominance 57-9,60.114,164 diagonal of a matrix differential equations 17, 20, 23, 12, 54, 55, 57, 163 26, 30-6, 37, 49, 51, 53, 63, 89, 91, 101, 104, 112, 148 diffusion problems 25-6, 32 dimension matrix 67, 106, 108, 122, 151 DIMENSION statement 1 1 direct equilibrium formulation 37, 48, 53, 89, 104, 112, 150, 153, 157 39, 40, 42, 53,66, discretisation 33 displacement 1, 4, 5, 6, 7, 8, 9, 11, 12, 15, 16, 33, 36, 103, 104, 105, 106, 109, 112, 113, 116, 150, 151, 153, 154, 157 displacement vector 6, 7, 106, 110, 111, 114 distributed external force 8, 112, 116, 125-6, 133, 144 domain of a solution 3 1, 33, 35, 36, 37, 38, 42,49, 50, 51, 61.65, 71, 72, 86, 90, 93, 97, 98, 99, 102, 104, 105, 112, 123, 124, 125, 132, 133, 134, 135, 137, 138, 139, 140, 144, 148, 156, 158 downstream viscous flow 20-1, 32, 37, 39, 46, 52-3, 54, 61, 63, 89- 100 drag flow 90, 98- 100 eigenvalue problems 148 elastic property matrix elastic solid 149 109, 122, 150 19, 49, 104, 108, 148, linear 19, 148 nonlinear 156 element, area of 73, 104, 153 39, 65, 66, 68, 72, aspect ratio of 144-6 centroid of 42, 108, 124, 133, 134, 137, 144, 146, 150, 152 distribution of 71-2, 81, 86, 97, 99, 131, 139-47, 157, 158 numbering of shape of 4, 13, 38, 72, 75, 77-8, 81, 84, 104, 142 58-9, 72, 76, 87, 100, 102, 114, 131, 140-7, 157, 158 size of 71-2, 81, 87, 97, 99, 102, 131, 140-7, 157, 158 element as subregion 104, 149, 157 element force vector element stiffness matrix 1, 3, 36, 37, 6, 7, 42, 109 6, 7, 11, 12, 37-43,47, 58-9,66,67, 104-10, 111, 114, 121, 122, 155 elimination see gaussian elimination elliptic differential equation 31-2 energy equation equilibrium equations 17- 18, 25, 6 1 6, 17, 22, 23, 24, 27, 28, 29, 30, 36, 37, 43, 45, 47-8, 53,61, 110-11, 112 equilibrium problems 2, 3 1, 148 EQUIVALENCE statement 11 error, convergence 96, 144, 164 data 73, 74 relative 164 ~ roundoff 3, 160 truncation 34, 35, 48, 62, 96, 97,105 .-Index 169 error in analytical solution 139 error in axi-symmetric analysis 150 execution, control of execution time externally applied forces 68, 70, 123 termination of 12, 67, 122 3, 162, see also time, computing 7, 11, 12, 42,43, 66,67-8, 104, 110-1 1, 112, 115, 116, 117, 118, 125, 129, 148, 156 factor of safety 1 finite difference methods 15, 33-6, 45-8, 53-4, 59, 62, 89, 90, 97, 103, 139, 158 flexural rigidity 30 flow rate fluid flow 148 fluid mechanics forces on an element FORMAT statements 3, 68 functional 49-53, 61, 63, 112 21, 25, 30, 31, 89, 90, 92, 95,97,98-9 1, 15, 17, 18, 30, 36, 37, 1, 2, 17, 18, 37 4, 6, 7, 40-2, 43, 107-8, 110 FORTRAN 2-3,9,11 Gauss-Seidel method 55, 56-61, 63, 67, 68-70, 72, 73, 98, 102, 113-14, 116, 118, 122, 131, 144, 146, 147, 154, 156, 164-5 subprograms for 63, 68-70, gaussian elimination method 9, 12, 118, 127-8 55-6, 59-61, 72,98, 113, 118, 147, 154, 156, 159-63 subprogram for 9, 12, 161-3 generation of meshes global co-ordinates 7 1, 74-86, 6, 38, 65, 68, 73, 74, 77, 81, 83, 87, 115, 131, 138, 141 90, 139, 140-2 graph plotting 73, 130 Green’s theorem 24, 50 grid 33, 36 hZ extrapolation 97-8, 102 harmonic equations 32-6, 46, 49 harmonic operator 20 harmonic problems 148, 149, 154 37-70, 7 1, 72, 89-103, 104, 111, 114, 116, 121, heat transfer heat transfer coefficient 25 higher-order elements 151-3, 156, homogeneous materials 19, 24, 123, hydrodynamic lubrication 32 hyperbolic differential equation 3 1 ideal fluid 24-5, 26, 32 ill-conditioned equations 160, 162 incompressibility 19, 24, 30, 104, independent variables 30-2 inertia forces 17, 24 infinite series 33, 90 integration 1, 2, 17-18, 25, 148, 157 157 138, 158 109, 150, 154-5 9, 21, 24, 31, 49, 50, 52, 61, 92, 98, 102, 112, 113, 150, 153, 158 irrotational flow 25 isoparametric elements 156 isotropic materials 19, 24 Lam6 equations 136 laminarflow 20 Laplace’s equation 32 line printer 3, 68 linear algebraic equations see simultaneous algebraic equations linearity of algebraic equations 156, 159 linearity of differential equations linearity of material properties 19, local co-ordinates local variable 69, 127 magnetic tape 2, 60 main program matrix notation matrix transposition mesh 31,32 20,45, 156-7 4, 38, 45, 62, 72, 75, 78, 81, 84, 105, 155 9, 11, 12, 63-8, 86, 6, 7, 53, 159, 160 42, 108, 113 92, 118-23 37, 43, 44, 46, 47, 55, 56, 59, 63, 66, 67, 68, 71-88, 90, 97, 99-100, 102, 103, 104, 140-2, 156, 158 Circular 81-6, 102, 135, 137 irregular 47 square 74-80, 90, 99- 100, 133, 140 triangular 80-1, 103, 137Index mesh data mesh plotting 73, 130 modification of meshes 63, 66, 68, 7 1-88, 90, 118, 122, 125, 133, 139, 140-2 7 1, 74, 76, 86-8, 90, 99-100, 102, 103, 133, 139, 140-2 molecule, computing 3 5 moment 4, 5, 7, 40, 107 Navier-Stokes equations 17 newtonian fluid node (nodal point) 19-20, 25, 42, 148 4, 5, 7, 11, 36, 37, 38, 54, 55, 59, 71, 72, 104, 152-3 numbering of 4, 13, 38, 71, 72, 74-5, 76-8, 80-1, 82-5, 104, 141-2 nonlinear problems 60, 148, 156-7, non-newtonian fluid 32, 156 numerical methods 158 1, 33, 139, 158 one-dimensional problems 2, 4, 37, overall stiffness matrix 155, 157 7, 11, 12, 43-5, 54, 55, 56, 57, 58, 59, 60, 66, 67, 72, 110-11, 114, 116, 118, 122, 154, 155, 156 144, 165 over-relaxation 69, 93, 95-6, 133, paper tape 2 parabolic differential equation 3 1 permeability 26 photoelasticity 1 pin-jointed structures 3 pivotal coefficient 162 plane strain plane stress Poisson’s equation 32 Poisson’sratio 19, 123, 133, 135, 142, 154 polynomial 5, 51, 151, 153, 155, 157 porous medium 25 potential energy precision of a stored number pressure pressure flow 90- 100 26-8, 29, 30, 32, 26, 28-9, 32, 104-30 104-30, 131-8, 149, 150, 154 131, 138-47, 149 49, 112, 113, 150 3 1, 17, 20, 24, 25, 26, 32, 42,43 pressure forces pressure gradient program 17, 24, 42, 43, 45, 20, 25, 29, 42, 4 7 , 4 8 45, 63, 67, 90, 98 71, 73, 89, 90, 118, 158 2, 36, 48, 53, 55, 56, 63, biharmonic problems 71, 86, harmonic problems 59, 63-70, rigid-jointed structures 9- 13 programming 2-3, 14, 36, 61, 63, propagation problems 31, 32, 148 quasi-harmonic equations 32 READ statement 2 recirculating viscous flow 29-30, 32, 35, 104, 154, 156 rectangularised matrix 56, 57, 59, 60, 66, 67, 98, 114, 118, 122 restraint conditions 7, 11, 12, 13, 54, 116, 117, 127, 144, 158 rigid-jointed structures 155 roundoff error 3. 160 104, 118-30, 131, 133, 151 71, 86, 89, 110 110, 141, 165 3, 4- 14, 54, St Venant’s theory of torsion scale factor 141-7 second moment of area 12, 13 self-flexibility 122, 127 self-stiffness 127 shape factor, channel flow shape function shear 5, 16, 17, 40, 54, 149 shear modulus 19, 103 simultaneous algebraic equations 2, 21 4, 8, 9, 11, 1 14, 1 16, 1 17, 1 18, 55, 66, 67, 114, 122, 90- 100 39, 42, 62, 63, 92, 105, 150, 151-5, 156, 157 9, 13, 33, 36, 53, 55-61, 63, 66, 68, 72, 90, 97, 110, 113-14, 118, 146-7, 156, 158, 159-65 162 singular set of equations slip 21, 29, 132, 134-5 slow flows 17 solid medium sparse matrix 1 18, 160, 1, 15, 17, 19, 30, 36, 7, 12, 44, 55, 56, 59, 37, 49, 104, 108, 148, 149 63Index 171 specific heat 18 statement numbering 2-3 storage requirement strain 12, 55, 56, 57, 59-60, 97, 98, 113, 147, 154, 158 4, 15-17, 18, 19, 20, 23, 105, 107, 129, 149 element 129 volumetric 19 strain rate 156 stream function 154, 157 stress 149 15- 17, 18, 19, 20, 42, 24-5, 26, 29, 30, 1, 15-16, 19, 33, 36, 54, 107, element 129, 134, 146, 158 nodal point 130, 146, 158 stress analysis 1 , 17, 18, 148 stress concentration 131, 138-47 stress function 23, 24, 27, 30, 49, structural analysis 1, 2, 3-14, 37, submatrix 110, 111, 114, 162 subprogram 101, 102, 103, 112, 157 49, 54, 55, 60, 72, 104 3, 9, 12, 63, 7 1, 76 boundary conditions 9 1, 123, Gauss -Seidel method 63, gaussian elimination 9, 12, mesh data 72-3, 76-7, mesh data output 68, 125 mesh modification 9 1, output of results 92-3, 125-7 68-70, 123, 127-8 161-3 78-80, 81-2, 85-6 142-4 129-30 subregions of continua subscripts 11, 15-16 symmetry of a matrix 54, 56, 110, 111 symmetry of problems 1, 36, 37, 51, 104, 157 6, 43, 45, 90, 93, 98, 116, 131, 136, 137, 139, 148-51 Taylor series 34, 62 temperature 18, 19, 25, 27, 28, 61, 104, 108-10, 118, 121, 122, 124, 125, 131, 133, 135, 157 testing of programs thermal conductivity 18 thermal convection thermalforces 110, 111, 118, 121, 122, 127 thermal strain 121, 122, 135, 149 three-dimensional problems 2, 15, time, computing 146, 154, 162 time-dependent problems 17, 148 tolerance, convergence 69, 93, torsion triangular elements 3, 89, 131, 134, 158 1, 17- 18, 25 19, 31, 104, 109, 30, 37,49, 148, 150, 153-4, 157 3, 57, 59-60, 97, 95-6, 133, 144, 164, 165 3, 4, 21-4, 31, 32, 100-3, 37, 47, 71, 74, 148, 151 76, 80, 90, 99, 104, 132, 140, 149, 152, 153, 154, 155, 156 tridiagonal matrix 55-6 truncation error two-dimensional problems 2, 15, 34, 35, 48, 62, 96, 97, 105 24, 26, 29, 33, 37,49, 63, 71, 104, 108, 112, 118, 148, 151, 154, 155, 157 variable names 3, 11, 65-6, 69-70, 76, 78, 85, 118-22, 123, 124, 125-6, 127, 129-30, 142 variational formulation 37, 40 48-53, 54,61-2,63, 112-13, 150-1, 153, 157 velocity 1 , 15, 16, 18, 20, 21, 25, 33, 39, 40, 52, 89, 103, 151, 154-5, 157 viscoelastic materials 19 viscosity viscous forces vorticity 36 20, 32, 45, 63, 156, 157 17, 24, 25, 40, 43, 46-7 word 3 WRITE statement 2 Young’s modulus
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