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Shaw F.S. Relaxation methods
Shaw F.S.  Relaxation methods









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: Relaxation methods

: Shaw F.S.

:

The main objective of this book is to provide an answer to the questions: Given a set of linear algebraic simultaneous equations, or a differential equation, how can a solution be obtained using relaxation methods? What are the computational steps involved?


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ed2k: ed2k stats

: 1953

: 403

: 12.07.2013

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Accuracy, difference corrections      313 314
Accuracy, differencing as guide to      67
Accuracy, general      311
Accuracy, improving by use of finer mesh      185
Accuracy, improving finite difference representation      312
Accuracy, increasing      14
Accuracy, integral equations      306
Accuracy, ordinary differential equations      63
Algebraic equations, linear simultaneous      7
Axisymmetric problems, infinite regions      150
Axisymmetric problems, second order partial differential equations, approximation to relaxation operator      154
Axisymmetric problems, second order partial differential equations, boundary conditions      152
Axisymmetric problems, second order partial differential equations, discrepancies due to coarse mesh      160
Axisymmetric problems, second order partial differential equations, incorrect relaxation operator      156
Axisymmetric problems, second order partial differential equations, irregular stars      166
Axisymmetric problems, second order partial differential equations, residual and relaxation operators      153
Bickley, W.G.      42
Biharmonic equation      196 223
Biharmonic equation, boundary conditions      199 230
Biharmonic equation, fictitious points      199
Biharmonic equation, group relaxation      219
Biharmonic equation, line and block relaxation operators      208
Biharmonic equation, residual and relaxation operators      197
Biharmonic operator, boundary conditions      230
Biharmonic operator, curved boundaries and irregular stars      223
Biharmonic operator, finite difference approximations      38 39
Biharmonic operator, irregular star second derivatives      231
Biharmonic operator, iteration      225 226 227
Biharmonic operator, multiply- connected regions      223
Biharmonic operator, selvedge points      224 227
Block      105
Block operators      5
Boundary conditions, axisymmetric second order partial differential equation      152
Boundary conditions, biharmonic equation      199 230
Boundary conditions, displacement, curved boundaries, Poissons equation      113
Boundary conditions, displacement, multiply-connected region, Laplace's equation      139
Boundary conditions, iteration, Poissons equation      117
Boundary conditions, line integral, Poissons equation      141 143
Boundary conditions, line integral: Laplaces equation      129
Boundary conditions, normal derivative, Laplaces equation      128 129 138
Boundary conditions, plane stress problems      192
Boundary conditions, two simultaneous second order partial differential equations      168 177
Build-up process      47
Coarse mesh      74
Colin, E.C.      143 144
Constraint      2 3
Constraint, forces on      3
Convergence      21
Convergence, slowing down      14
Convergence, speeding up      10
Crout, P.D.      24
Curved boundaries, biharmonic operator      223
Curved boundaries, irregular star operators      113
Curved boundaries, Laplaces equation      138
Curved boundaries, partial differential equation eigenvalue problems      272
Curved boundaries, Poissons equation, fictitious points      126
Curved boundaries, Poissons equation, graphical treatment      116
Curved boundaries, Poissons equation, irregular star operators      113 114 115 116 126
Curved boundaries, Poissons equation, linear relaxation operators      115
Curved boundaries, Poissons equation, second degree polynomial operators      117 119
Curved boundaries, two simultaneous second order partial differential equations      183
Difference corrections, accuracy      313 314
Differencing as guide to accuracy      67
Differential equations, linear ordinary      46
dimensions      320
Displacement      2 11
Displacement increment      3 8
Displacement unit      4
Eigenfunctions, orthogonality properties      244 250
eigenvalue problems      233
Eigenvalue problems, Algebraic equations      233
Eigenvalue problems, Algebraic equations, modal elimination      243
Eigenvalue problems, Algebraic equations, relaxation operators      235
Eigenvalue problems, optimal synthesis      277
Eigenvalue problems, ordinary differential equations      249
Eigenvalue problems, ordinary differential equations, method of solution      251
Eigenvalue problems, ordinary differential equations, Rayleigh Quotient      250
Eigenvalue problems, partial differential equations      266
Eigenvalue problems, partial differential equations, curved boundaries      272
Eigenvalue problems, partial differential equations, Rayleigh Quotient      267
Electrical networks      25
Elliptic equations      71 72
Emmons, H.W.      116
Equation, biharmonic      196 223
Equation, Laplaces      149
Equation, Poissons      150
Equations, differential, approximate finite difference solution      26
Equations, eigenvalue      233
Equations, eigenvalue, algebraic      233
Equations, eigenvalue, ordinary differential      249
Equations, eigenvalue, partial differential      272
Equations, elliptic      71 72
Equations, fourth order partial differential      196
Equations, hyperbolic      71 72
Equations, ill-conditioned      22
Equations, integral      299
Equations, linear algebraic simultaneous      7
Equations, linear ordinary differential      47
Equations, non-linear, free surface problems      283 284
Equations, parabolic      71 72
Equations, second order linear partial differential      149 167
Errors, finite difference approximations to ordinary derivatives      29
Errors, sources of, ordinary differential equations      66
Fictitious points, biharmonic equation      219
Fictitious points, curved boundaries and irregular stars      113 114 115 116
Fictitious points, two simultaneous second order partial differential equations      178 183
Finite-difference approximations      26
Finite-difference approximations, biharmonic operator      38 39
Finite-difference approximations, error involved      29
Finite-difference approximations, improvements      312
Finite-difference approximations, integrals      41 42 43
Finite-difference approximations, Laplace operator, Cartesian coordinates      38 40 41 107
Finite-difference approximations, Laplace operator, rotational symmetry      38 40
Finite-difference approximations, linear ordinary differential expressions      33
Finite-difference approximations, linear partial differential expressions      38
Finite-difference approximations, ordinary derivatives, polynomial representation      29 31
Finite-difference approximations, ordinary derivatives, simple derivation      27
Finite-difference approximations, ordinary derivatives, Taylor series      29
Finite-difference approximations, partial derivatives, hexagonal and triangular meshes      40
Finite-difference approximations, partial derivatives, more correct expressions      41
Finite-difference approximations, partial derivatives, shift operator      37
Finite-difference approximations, partial derivatives, square mesh      33 34
Finite-difference approximations, Poissons equation      74
Forces, on constraints      3
Fourth order linear partial differential equations      196
Fox, L.      21 23 126 137 138 224 226 227 312 313 320
frameworks      1 25
Free surface problems      281
Free surface problems, Elasto-plastic torsion problem      281
Free surface problems, non-linear differential equation      283 284
Free surface problems, relaxation-iteration procedure      285 299
Girders      24
Graphical treatment, curved boundaries, Poissons equation      116
Group relaxation operators      19 21
Group relaxation, biharmonic equation      219
Hexagonal mesh      40
Howard, W.      178
Hyperbolic equations      71 72
Ill-conditioned equations      22
Increments of displacement      3 8
Infinite regions, axisymmetric problems      150
Infinity, position of      152
Instability, strut, non-dimensional treatment      323
Integral equations      299
Integral equations, accuracy      306
Integral equations, residual and relaxation operators      301
Integrals, finite difference approximations      41 42 43
Irregular stars, axisymmetric problems      166
Irregular stars, biharmonic operator      223 231
Irregular stars, curved boundaries      113 114 115 116
Irregular stars, first derivative expressions      325
Irregular stars, tables, first derivative coefficients      328
Irregular stars, tables, fourth derivative coefficients      372
Irregular stars, tables, second derivative coefficients      350
Iteration, biharmonic equation      225 226 227
Iteration, free surface problems      285 299
Iteration, Poissons equation      117 142
Laplace operator, finite-difference approximation      38 40 41
Laplaces equation      106 149
Laplaces equation, block relaxation      109
Laplaces equation, finite-difference approximation      107
Laplaces equation, line integral boundary condition      129
Laplaces equation, lines of symmetry      107
Laplaces equation, multiply-connected region, displacement boundary condition      139
Laplaces equation, normal derivative boundary condition      128 129
Laplaces equation, normal derivative boundary condition, strings      138
Laplaces equation, residual and relaxation operators      107
Laplaces equation, special symmetry operators      108
Line relaxation operators      56
Liquidation of residuals      3 10 55
Loading term, Poissons equation      186
Localized fine mesh      186
Magnitude of unit block      105
Maxwell reciprocal relations      21
Mesh, coarse      74
Mesh, coarse, axisymmetric problem fine localized      186
Mesh, hexagonal      40
Mesh, mixed, special operators, Poissons equation      187 190
Mesh, refining of      66 185
Mesh, square      34
Mesh, triangular      40
Modal elimination, eigenvalue problems      243
Morris, J.      24
Multiply-connected region, biharmonic operator      223
Multiply-connected region, Laplaces equation      139
Multiply-connected region, Poissons equation      139
Networks, electrical      25
Neville, E.H.      24
Newmark, N.M.      143 144
Non-dimensional treatment      320 321 323
Operators, block      5
Operators, displacement      5
Operators, irregular star      113 115 117 119 126 325 328 350 372
Operators, relaxation      5 7 11 17
Operators, relaxation, algebraic eigenvalue problems      235
Operators, relaxation, approximation to      154
Operators, relaxation, biharmonic equation      197
Operators, relaxation, block, biharmonic equation      208
Operators, relaxation, block, Poissons equation      84 87
Operators, relaxation, group      19 21 219
Operators, relaxation, handle of      17
Operators, relaxation, incorrect, use of      156
Operators, relaxation, integral equations      301
Operators, relaxation, Laplaces equation      7
Operators, relaxation, line, biharmonic equation      208
Operators, relaxation, line, ordinary differential equations      56
Operators, relaxation, line, Poissons equation      84 86
Operators, relaxation, ordinary differential equations      49
Operators, relaxation, ordinary differential equations, eigenvalue problems      252257
Operators, relaxation, partial differential equations, eigenvalue problems      268 269
Operators, relaxation, partial differential equations, two simultaneous      170 182
Operators, relaxation, symmetry      95 108
Operators, relaxation, tuning up      17
Operators, relaxation, unit      11 17
Operators, relaxation, unit, axisymmetric problems      153 154
Operators, relaxation, unit, Poissons equation      75 77
Operators, relaxation, wedge      61
Operators, residual      7 12
Operators, residual, axisymmetric problems      153
Operators, residual, biharmonic equation      197
Operators, residual, integral equations      301
Operators, residual, Laplaces equation      107
Operators, residual, ordinary differential equations      49
Operators, residual, Poissons equation      75 77
Operators, residual, two simultaneous second order partial differential equations      170 182
Operators, special, mixed meshes, Poissons equation      187 190
Operators, special, symmetry, Laplaces equation      108
Optimal synthesis      277
Ordinary derivatives, finite-difference approximations      27 29 31
Ordinary differential equations, accuracy      63
Ordinary differential equations, reduction to finite-difference equations      47
Ordinary differential equations, relaxation operators      49 56
Ordinary differential equations, residual operators      49
Ordinary differential equations, sources of error      66
Ordinary differential equations, terminal conditions      69
Ordinary differential expressions, finite-difference approximations      33
Orthogonality of eigenfunctions      244 250
Over-relaxation      14 21 55 156
Parabolic equations      71 72
Partial derivative expressions, finite-difference approximations      38
Partial derivatives, finite-difference approximations      33
Plane stress problem, boundary conditions      192
Poissons equation      72 73 96 150
Poissons equation, block relaxation operators      84 87
Poissons equation, boundary conditions, iteration      117
Poissons equation, curved boundaries and irregular stars      113 114 115 116 117 119 126
Poissons equation, displacement boundary condition      113
Poissons equation, fictitious points      126
Poissons equation, finite-difference approximation      74
Poissons equation, line relaxation operators      84 86
Poissons equation, loading term      186
Poissons equation, multiply-connected region      139 141 142 143
Poissons equation, residual operator      75 77
Poissons equation, special operators for mixed meshes      187 190
Poissons equation, unit relaxation operator      75 77
Quadratic function, positive definite      21
Rayleigh quotient      250 267
Reciprocal relations, Maxwell      21
Refining meshes      66
Relaxation operators      5 7 11 17
Relaxation operators, algebraic eigenvalue problems      235
Relaxation operators, approximation to      154
Relaxation operators, biharmonic equation      197
Relaxation operators, block, biharmonic equation      208
Relaxation operators, block, Poissons equation      84 87
Relaxation operators, group      19 21 219
Relaxation operators, handle of      17
Relaxation operators, incorrect, use of      156
Relaxation operators, integral equations      301
Relaxation operators, Laplaces equation      107
Relaxation operators, line, biharmonic equation      208
Relaxation operators, line, ordinary differential equations      56
Relaxation operators, line, Poissons equation      84 86
Relaxation operators, ordinary differential equations      49
Relaxation operators, ordinary differential equations, eigenvalue problems      252257
Relaxation operators, partial differential equations, eigenvalue problems      268 269
Relaxation operators, symmetry      95 108
Relaxation operators, tuning up      17
Relaxation operators, unit      11 17
Relaxation operators, unit, axisymmetric problems      153 154
Relaxation operators, unit, Poissons equation      75 77
Relaxation operators, wedge      61
Relaxation, criterion for sufficiency of      14
Relaxation-iteration, free surface problems      285 299
Residual forces (residuals)      3 7
Residual operators      7 12
Residual operators, axisymmetric problems      153
Residual operators, biharmonic equation      197
Residual operators, integral equations      301
Residual operators, Laplaces equation      107
Residual operators, ordinary differential equations      49
Residual operators, Poissons equation      75 77
Residual operators, two simultaneous second order partial differential equations      170 182
Residuals, checking of      22
Residuals, liquidation of      3 10 55
Residuals, smearing of      84
Residuals, washback of      56 84 88 117 143
Rotational symmetry, finite-difference approximation      38 40
Second order partial differential equations, rotational symmetry      149
Selvedge points      224 277
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