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201. Zienkiewicz OC, Zhu JZ (1992) "The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique", Int. J. Num. Methods in Eng. Vol. 33: 1331-1364

202. Zienkiewicz OC, Zhu JZ (1992) "The superconvergent patch recovery and a posteriori error estimates, Part 2: Error estimates and adaptivity", Int. J. Num. Methods in Eng. Vol. 33: 1365-1382

Index

CO -elements, 252 C1-elements, 334 Gj,75 L2 scalar product, 411 Ph, 27 S,21 V,21 Vh, 6 Vh+, 139 00,68 oi,68 div, 409 div2 M, 336 tr, trace, 308 \7,409 C[], elasticity tensor, 307

a posteriori error indicators, 112 accuracy, 75 actio = reactio?, 132 adaptive mesh refinement, 296 adaptive methods, 112 adaptivity, dimension, 118 adaptivity, model, 118 approximations for stiffness matrices,

237 asymptotic error estimates, 453 Aubin-Nitsche trick, 436, 457

Babuska-Brezzi condition, 312 backward error analysis, 152 Bathe-Dvorkin element, 349 beam girders, 372 beam models, 364 beam-like elements, 270 Bessel's inequality, 10 bilinear elements, 254 bilinear form, 410

boolean matrix, 243 boundary layer effect, 346 Boussinesq, 198 brachistochrone, 129 bulk modulus, 306

capacity, 54 Castigliano's Theorem, 50 Cauchy-Schwarz inequality, 412 circular arches, 397 circular slabs, 369 columns, 353 composition, 22 concentrated forces, 43 condition number, 173 continuity, 164 corner force, 331 coupling degrees of freedom, 167 critical angles, plane problem, 250 CST element, 252 cubic elements, 252 cycloid, 129

d-adaptivity, 118 deformation space, 6 degenerate shell elements, 396 dimension argument, 44 dipole, 92 Dirac delta, 68 direct design method, 384 discontinuities, 302 displaced point supports, 268 displacement methods, 307 displacement model of slabs, 331 displacements, 40 distance, 36 DKT element, 350 drilling degrees of freedom, 258 drop panels and column capitals, 361

482 Index

DST element, 351 dual load, 67 dual quantities, 68 dual weighted residual error estimate,

428 duality, 40 duality techniques, 121, 122, 423

effective width, 376 efficiency of estimator, 460 eigenvalue, 173 eigenvector, 173 eigenwork, 17 element stresses, 271 elements, 163 elements, plane problems, 252 elements, slabs, 332 ellipticity, 21 energy method, 3 energy norm, 115 equilibrium, 81, 138, 330 equilibrium of resultant forces, 292 equilibrium point, 20 error estimators, 459 error in energy, 116 error indicator, 459 essential boundary conditions, 463 Euler equation, 24, 39 Euler-beam I, 144 explicit error estimators, 460

finite element, 149 first fundamental form, 392 flat elements, 399 flexibility matrix, 149 folded plates, 373 forces, 40 foundation slabs, 378 fundamental solution, 183

Galerkin method, 145 Galerkin orthogonality, 11 Gateaux derivative, 315 Gaussian curvature, 394 geometric matrix, 239 global equilibrium, 139 goal-oriented recovery, 122, 423 Green's first identity, 414 Green's function, 50, 55

Green's identities, 414 Green-Lagrangian strain tensor, 314

h-methods, 118 half-space model, 198, 379 hard support, 346 hat functions, 148 Hellinger-Reissner principle, 309 Hermite Polynomials, 152 higher-order elements, 258 homogeneous deflection curve, 227 hourglass modes, 173 hp-method, 118 HSM element, 335 hybrid elements, 335 hyperelastic material, 314

implicit error estimators, 461 incompressible material, 306 infinite energy, 158 influence functions, 41, 67 integral operators, 77 integration by parts, 414 interpolation, 148 isotropy, 164

kernel of a matrix, 174 kernel of an influence function, 55 kinematic constraints, 261 Kirchhoff plates, 326 Kirchhoff shear, 330 Kronecker delta, 151

Lagrange element, 151, 257 Lagrange multipliers, 169 Lagrange polynomials, 151 least action, 20 least squares, 33 linear elements, 252 linear strain triangle, 255 lintels, 270 load case 8, 17 load case p, 17 loads, 197 local equilibrium, 139 LST element, 255

m-adaptivity, 118 mapped polynomials, 154 master degree of freedom, 169

membrane locking, 397, 398 membrane stresses, 393 membranes, 404 Mindlin shell elements, 396 mixed methods, 307 Mohr's integral, 182

natural boundary conditions, 463 nodal stresses, 271 nonlinear problems, 314, 428 normal equations, 34

order of the singularity, 45 order of the strain energy, 45 output, 135

p-methods, 118 particular deflection curve, 227 partition of unity, 161 Pascal's triangle, 151 patch test, 258 peaks, 96 piecewise polynomials, 148 piled raft foundations, 200 plane elements, 252 plane strain, 166, 248 plane stress, 166, 248 point supports, 267, 386 Poisson's ratio, 326, 329 polynomials, 151 potential ef\ergy, 5 primary load case, 300 principal stresses, 250 principle of minimum potential energy,

6 principle of virtual displacements, 24 principle of virtual forces, 182 projection method, 2 Pythagorean theorem, 466

Q4,254 Q4 + 2, 255 Q8,257 quadratic elements, 252

r-adaptivity, 118 Rayleigh quotient, 146 recovery based error estimators, 460 reduced global stiffness matrix, 171

Index 483

reduced integration, 172 reduced stiffness matrix, 139, 167 Reissner-Mindlin plates, 341 remeshing, 118 residual, 179 residual forces, 15 restriction, 1 Riesz representation theorem, 423 rigid-body motions, 163 rope, 5 rotational invariance, 164, 252 rounding errors, 173

scalar product, 32, 39, 409 scalar product of matrices, 410 second fundamental form, 392 second Piola-Kirchhoff stress tensor,

314 self-equilibrated stresses, 300 serendipity elements, 151, 257 set theory, 16 settlement of supports, 141 settlements, 301 shape functions, 251 shear forces, 361 shear locking, 347 shear modulus, 250, 306 shearing strain, 342 shell equations, 391 shells, 391 shells of revolution, 394 shift, 76 Shifted Green's function theorem, 443 shifted Green's functions, 437 singular matrices, 170 singularities, 129 skew projection, 12, 21 slabs, Reissner-Mindlin, 341 slave degree of freedom, 169 smoothing process, 276 Sobolev norms, 12 Sobolev's Embedding Theorem, 46 soft support, 346 St. Venant's principle, 127 stability problems, 144 stiffness matrices, 164 stiffness matrices for 1-D elements, 231 stiffness matrices, slabs, 334 stiffness matrices, three properties, 166

484 Index

stiffness matrix, 2-D, 253 strain energy product, 31, 34, 165, 253,

308 stream model, 244 study, slabs, 386 substitute load case, 2 superconvergence, 157 support conditions, 137 support reactions, 94, 267 supports, 351 symmetries, 268

T beams, 169 tangential supports, 268 Taylor series, 151 temperature, 141 tension, 6 test range, 35 test space, 21 three-moment equation, 149

Tottenham's equation, 62 translations, 162 trial space, 6, 21 truss models, 277

values at a point, 60 variable thickness, 362 virtual displacements, 24 volume elements, 396

weak convergence, 39 weak equal sign, 4 weak solution, 39 weighted least squares, 34 weighted residual method, 145 wheel loads, 369 Wilson's Element, 255 Winkler model, 197, 378

zero-energy modes, 173