김경종
26 KiB
ZAVARISE, G., WRIGGERS, P., and SCHREFLER, B. A.
[A] "A Method for Solving Contact Problems," International Journal for Numerical Methods in Engineering, Vol. 42, pp. 473–498, 1998.
ZHAO, Q. C., CHEN, P., PENG, W. B., GONG, Y. C., and YUAN, M. W.
[A] “Accelerated Subspace Iteration with Aggressive Shift,” Computers & Structures, Vol. 85, pp. 1562–1578, 2007.
ZHONG, W., and QIU, C.
[A] "Analysis of Symmetric or Partially Symmetric Structures," Computer Methods in Applied Mechanics and Engineering, Vol. 38, pp. 1-18, 1983.
ZIENKIEWICZ, O. C., and CHEUNG, Y. K.
[A] The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, 1967; 4 ^{th} ed. by O. C. Zienkiewicz and R. L. Taylor, Vols. 1 and 2, 1989/1990.
ZIENKIEWICZ, O. C., and ZHU, J. Z.
[A] “The Superconvergent Patch Recovery and a Posteriori Error Estimates. Part 1: The Recovery Technique,” International Journal for Numerical Methods in Engineering, Vol. 33, pp. 1331–1364, 1992.
ŻYCZKOWSKI, M.
[A] Combined Loadings in the Theory of Plasticity, Polish Scientific, 1981.
Index
A
Accuracy of calculations (See also Errors in solution), 227
Acoustic fluid, 666
Almansi strain tensor, 585
Alpha ( \alpha ) integration method: in heat transfer analysis, 830 in inelastic analysis, 606
Amplitude decay, 811
Analogies, 82, 662
Angular distortion, 381
Approximation of geometry, 208, 342
Approximation operator, 803
Arbitrary Lagrangian-Eulerian formulation, 672
Aspect-ratio distortion, 381
Assemblage of element matrices, 78, 149, 165, 185, 983
Associated constraint problems, 728, 846
Augmented Lagrangian method, 146, 147
Axial stress member (See Bar; Truss element)
Axisymmetric element, 199, 202, 209, 356, 552
Axisymmetric shell element, 419, 568, 574
B
Bandwidth of matrix, 20, 714, 985
Bar, 108, 120, 124, 126
Base vectors: contravariant, 46 covariant, 46
Bathe method, 779, 791, 805, 809, 811
Beam element, 150, 199, 200, 397, 568
BFGS method, 759
Bilinear form, 228
Bisection method, 943
Body force loading, 155, 164, 165, 204, 213
Boundary conditions:
in analysis:
acoustic, 667
displacement and stress, 154
heat transfer, 643, 676
incompressible inviscid flow, 663
seepage, 662
viscous fluid flow, 675
convection, 644
cyclic, 192
displacement, 154, 187
essential, 110
force, 111
geometric, 110
natural, 110
phase change, 656
radiation, 644, 658
skew, 189
Boundary layer:
in fluid mechanics, 684
in Reissner/Mindlin plates, 434, 449
Boundary value problems, 110
Bubble function, 373, 432, 690
Buckling analysis, 90, 114, 630
Bulk modulus, 277, 297
C
\mathbf{C}^{m - 1} problem, 235
Cable element, 543
Cauchy stress tensor, 499
Cauchy-Green deformation tensor (left and right tensors), 506
Cauchy's formula, 516
Caughey series, 799
Cea's lemma, 242
Central difference method, 770, 815, 824
CFL number, 816
Change of basis, 43, 49, 60, 189, 786
Characteristic polynomial, 52, 888, 938
Characteristic roots (See Eigenvalues)
Cholesky factorization, 717
Collapse analysis, 630
Column heights, 708, 986
Column space of a matrix, 37
Compacted column storage, 21, 708, 985
Compatibility:
of elements/meshes, 161, 229, 377
of norms, 70
Compatible norm, 70
Complete polynomial, 244
Completeness condition:
of element, 229
of element assemblage, 263
Component mode synthesis, 875
Computer programs for: finite element analysis, 988
Gauss elimination equation solution, 708
isoparametric element, 480
Jacobi generalized eigensolution, 924
subspace iteration eigensolution, 964
Computer-aided engineering, 11
Concentrated loads, modeling of, 10, 228, 239
Condensation (See Static condensation)
Condition number, 738
Conditional stability, 773
Conductivity matrix, 651
Conforming (See Compatibility)
Conjugate gradient method, 749
Connectivity array, 185, 984
Consistent load vector, 164, 213, 814
Consistent mass matrix, 165, 213
Consistent tangent stiffness matrix, 758
Consistent tangent stress-strain matrix, 583, 602, 758
Constant increment of external work criterion, 763
Constant strain:
one-dimensional (truss) element, 150, 166, 339
two-dimensional 3-node element, 205, 364, 373
three-dimensional 4-node element, 366, 373
Constant-average-acceleration method, 780
Constitutive equations (See Stress-strain relations)
Constraint equations, 190
Constraint function method, 626
Contact analysis, 622
Contactor, 623
Continuity of a bilinear form, 237
Contravariant:
base vectors, 46
basis, 46
Convection boundary conditions, 644
Convergence criteria in:
conjugate gradient method, 750
eigensolutions, 892, 914, 920, 949, 963
finite element discretization, 254
for iterative processes using norms, 67
Gauss-Seidel iteration, 747
mode superposition solution, 795
nonlinear analysis, 764
Convergence of:
conjugate gradient iteration, 750
finite element discretization, 225, 376
Gauss-Seidel iteration, 747
Jacobi iteration, 914, 920
Lanczos method, 949, 953
mode superposition solution, 795, 814
Newton-Raphson iteration, 756
QR iteration, 935
Rayleigh quotient iteration, 904
subspace iteration, 959, 963
vector forward iteration, 897
vector inverse iteration, 892
Coordinate interpolation, 342
Coordinate systems:
area, 371
Cartesian, 40
global, 154
local, 154, 161
natural, 339, 342, 372
skew, 189
volume, 373
Coulomb's law of friction, 624
Coupling of different integration operators, 782
Covariant:
base vectors, 46
basis, 46
Creep, 606
Critical time step for use of:
\alpha integration method, 831
central difference method, 772, 808, 817
Cyclic symmetry, 192
D
d'Alembert's principle, 134, 165, 402
Damping, 165, 796
Damping ratio, 796, 802
Dc network, 83
Deflation of:
matrix, 906
polynomial, 942, 945
vectors, 907
Deformation dependent loading, 527
Deformation gradient, 502
Degree of freedom, 161, 172, 273, 286, 329, 345, 413, 981
Determinant:
of associated constraint problems, 729, 850
calculation of, 31
of deformation gradient, 503
of Jacobian operator, 347, 389
Determinant search algorithm, 938
Digital computer arithmetic, 734
Dimension of:
space, 36
subspace, 37
Direct integration in:
dynamic stress analysis, 769, 824
fluid flow analysis, 680, 835
heat transfer analysis, 830
Direct stiffness method, 80, 151, 165
Director vectors, 409, 438, 570, 576
Displacement interpolation, 161, 195
Displacement method of analysis, 149
Displacement/pressure formulations:
basic considerations, 276
elements, 292, 329
u / p formulation, 287
u / p - c formulation, 287
Distortion of elements (effect on convergence), 382, 469
Divergence of iterations, 758, 761, 764
Divergence theorem, 158
Double precision arithmetic, 739
Drucker-Prager yield condition, 604
Duhamel integral, 789, 796
Dyad, 44
Dyadic, 44
Dynamic buckling, 636
Dynamic load factor, 793
Dynamic response calculations by: mode superposition solution, 785
step-by-step integration, 769
E
Effective
creep strain, 607
plastic strain, 599
stress, 599
Effective stiffness matrix, 775, 778, 781
Effective-stress-function algorithm, 600, 609, 611, 616
Eigenpair, 52
Eigenproblem in:
buckling analysis, 92, 632, 839
heat transfer analysis, 105, 836, 840
vibration mode superposition analysis, 786, 839
Eigenspace, 56
Eigensystem, 52
Eigenvalue problem, 51
Eigenvalue separation property (See also Sturm sequence property), 63, 728, 846
Eigenvalues and eigenvectors:
of associated constraint problems, 64
basic definitions, 52
calculation of, 52, 887
Electric conduction analysis, 83, 662
Electro-static field analysis, 662
Element matrices, definitions in:
displacement-based formulations, 164, 347, 540
displacement/pressure formulations, 286, 388, 561
field problems, 661
general mixed formulations, 272
heat transfer analysis, 651
incompressible fluid flow, 677
Elliptic equation, 106
Ellipticity condition, 304
Ellipticity of a bilinear form, 237
Energy norm, 237
Energy-conjugate (work-conjugate) stresses and strains, 515
Engineering strain, 155, 486
Equations of finite elements:
assemblage, 185, 983
in heat transfer analysis, 651
in incompressible fluid flow analysis, 678
in linear dynamic analysis, 165
in linear static analysis, 164
in nonlinear dynamic analysis, 540
in nonlinear static analysis, 491, 540
Equilibrium:
on differential level, 160, 175
on element level, 177
Equilibrium iteration, 493, 526, 754
Equivalency of norms, 67, 238
Error bounds in eigenvalue solution, 880, 884, 949
Error estimates for:
MITC plate elements, 432
displacement-based elements, 246, 380, 469
displacement/pressure elements, 312
Error measures in:
eigenvalue solution, 884, 892
finite element analysis, 254
mode superposition solution, 795
Errors in solution, 227
Euclidean vector norm, 67
Euler backward method, 602, 831, 834
Euler forward method, 831, 834
Euler integration method, use in: creep, plasticity, 607 heat transfer, 831
Eulerian formulation, 498, 672
Existence of inverse matrix, 27
Explicit integration, 770
Explicit-implicit integration, 783
Exponential scheme of upwinding, 686
F
Field problems, 661
Finite difference method: approximations, 132 differential formulation, 129 in dynamic response calculation, 769 energy formulation, 135
Finite elements: elementary examples, 79, 124, 149, 166 history, 1 an overview of use, 2
Finite strip method, 209
First Piola-Kirchhoff stress tensor, 515
Fluid flow analysis: incompressible viscous flow, 671 irrotational (potential) flow, 663
Fluid-structure interactions, 668, 672, 690
Folded plate structure, 208
Forced vibration analysis (See Dynamic response calculations by)
Forms: bilinear, 228 linear, 228
Fracture mechanics elements, 369
Free vibration conditions, 95, 786
Friction (See Coulomb's law of friction)
Frontal solution method, 725
Full Newton-Raphson iteration, 756, 834
Full numerical integration, 469
Functionals (See also Variational indicators), 110, 115
G
Galerkin least squares method, 688
Galerkin method (See also Principle of virtual displacements; Principle of virtual temperatures; and Principle of virtual velocities), 118, 126
Gauss elimination: computational errors in, 734 a computer program, 715 introduction to, 697 number of operations, 714 physical interpretation, 699
Gauss quadrature, 461
Gauss-Seidel iteration, 747
Generalized coordinates, 171, 195
Generalized displacements, (See Generalized coordinates)
Generalized eigenproblems (See also Eigenproblem in): definition, 53 various problems, 839
Generalized formulation, 125
Generalized Jacobi method, 919
Ghost frequencies (See Phantom frequencies)
GMRes (Generalized minimal residual) method, 752
Gram-Schmidt orthogonalization, 907, 952, 956
Green-Lagrange strain tensor, 50, 512
Guyan reduction, 875, 960
H
Half-bandwidth (See Bandwidth of matrix)
Hardening in:
creep, 607
plasticity, 599
viscoplasticity, 610
Hat function, 131, 692
Heat capacity matrix, 89, 655
Heat conduction equation, 106, 107, 643
Heat transfer analysis, 80, 89, 642
Hellinger-Reissner functional 274, 285, 297, 477
Hencky (logarithmic) strain tensor, 512, 614
Hierarchical functions, 252, 260, 692
Hierarchy of mathematical models, 4
History of finite elements, 1
h-method of finite element refinement, 251
Houbolt method, 774, 804, 809, 811
Householder reduction to tridiagonal form, 927
h/p method of finite element refinement, 253
Hu-Washizu functional, 270, 297
Hydraulic network, 82
Hyperbolic equation, 106
Hyperelastic, 582, 592
Hypoelastic, 582
I
Identity matrix, 19
Identity vector, 19
Imperfections (on structural model), 634
Implicit-explicit integration, 783
Incompatible modes, 262
Incompressibility, 276
Incremental potential, 561
Indefinite matrix, 60, 731, 939, 944
Indicial notation:
definition, 41
use, 41, 499
Infinite eigenvalue, 853
Inf-sup condition for incompressible analysis: derivation, 304, 312 general remarks, 291
Inf-sup condition for structural/beam elements, 330
Inf-sup test, 322
Initial calculations in: central difference method, 771 Houbolt method, 775 Newmark method, 778 Bathe method, 780
Initial stress load vector, 164
Initial stress method, 758
Initial value problems, 110
Instability analysis of: integration methods, 806 structural systems, 630
Integration of: dynamic equilibrium equations (See Direct integration in)
finite element matrices (See Numerical integration of finite element matrices) stresses (See Stress integration)
Interelement continuity conditions (See Compatibility of elements/meshes)
Interpolant of solution, 246
Interpolation functions, 338, 343, 344, 374
Inverse of matrix, 27
Inviscid (acoustic) fluid, 666
Inviscid flow, 663
Isobands of stresses, 255
Isoparametric formulations: computer program implementation, 480 definition, 345 interpolations (See Interpolation functions) introduction, 338
Iteration (See Gauss-Seidel iteration; Conjugate gradient method; Quasi-Newton methods; Eigenvalues and eigenvectors)
J
Jacobi eigensolution method, 912
Jacobian operator, 346
Jaumann stress rate tensor, 591, 617
Joining unlike elements, 377
Jordan canonical form, 808
K
Kernel:
definition, 39
use in analysis of stability, 318
Kinematic assumptions, 399, 420, 437
Kirchhoff hypothesis, 420
Kirchhoff stress tensor, 515
Kronecker delta, 45, 46
L
L^2 space, 236
Lagrange multipliers, 144, 270, 286, 626, 744
Lagrangian formulations: linearization, 523, 538
total Lagrangian (TL), 523, 561, 586
updated Lagrangian (UL), 523, 565, 586
updated Lagrangian Hencky (ULH), 614
Lagrangian interpolation, 456
Lamé constants, 298, 584
Lanczos method, 945
Laplace equation, 106, 107
Large displacement/strain analysis, 487, 498
Latent heat, 656
LDL^{T} factorization (See also Gauss elimination), 705
Least squares averaging (smoothing), 256
Legendre polynomials, 252
Length of vector (See Euclidean vector norm)
Linear acceleration method, 777, 780
Linear dependency, 36
Linear equations (See Equations)
Linear form, 228
Lipschitz continuity, 757
Load-displacement-constraint methods, 761
Load operator, 803
Loads in analysis of:
fluid flows, 678
heat transfer, 652
structures, 164
Locking:
in (almost) incompressible analysis, 283, 303, 308, 317
in structural analysis, 275, 332, 404, 408, 424, 444
Logarithmic strain tensor, 512, 614
Loss of orthogonality, 952
Lumped force vectors, 213
Lumped mass matrix, 213
M
Mass matrix, 165
Mass proportional damping, 798
Master-slave solution, 740
Materially-nonlinear-only analysis, 487, 540
Mathematical model:
accuracy, 7
effectiveness, 4, 6
Mathematical model (cont.)
reliability, 4
very-comprehensive, 6
Matrix:
addition and subtraction, 21
bandwidth, 19, 985
definition, 18
determinant, 31
identity matrix, 19
inverse, 27
multiplication by scalar, 22
norms, 68
partitioning, 28
products, 22
storage, 20
symmetry, 19
trace, 30
Matrix deflation, 906
Matrix shifting, 851, 899, 943, 964
Membrane locking, 408, 444
Mesh (from a sequence of meshes): compatible, 377
regular, 380
quasi-uniform, 382, 434
uniform, 243, 245
Metric tensor, 47
Mindlin (Reissner/Mindlin) plate theory, 420
Minimax characterization of eigenvalues, 63
Minimization of bandwidth, 714, 986
MITC elements:
error estimates, 432
plate elements, 425
shell elements, 445, 577
Mixed finite element formulations, 268
Mixed interpolations:
for continuum elements (See Displacement/pressure formulations)
for structural elements
beam, 274, 330, 406
plate, 424
shell, 444
Mode shape, 786
Mode superposition:
with damping included, 796
with damping neglected, 789
Modeling:
constitutive relations, 582
linear/nonlinear conditions, 487
type of problems, 196
Modeling in dynamic analysis, 813
Modified Newton-Raphson iteration, 759
Monotonic convergence, 225, 376
Mooney-Rivlin material model, 592
Multiple eigenvalues:
convergence to, 895, 944, 951, 960
orthogonality of eigenvectors, 55
N
Nanson's formula, 516
Natural coordinate system, 339, 342
Navier-Stokes equations, 676, 680
Newmark method, 777, 805, 809, 811
Newton identities, 939
Newton-Raphson iterations, 493, 755
Nodal point:
information, 981
numbering, 986
Nonaxisymmetric loading, 209
Nonconforming elements (See Compatibility of elements/meshes; Incompatible modes)
Nondimensionalized variables, 676
Nonlinear analysis:
classification, 487
introduction to, 485
simple examples, 488
Nonproportional damping, 799
Nonsymmetric coefficient matrix, 528, 628, 678, 696, 744, 752
Norms of:
matrices, 68
vectors, 67
Numerical integration of finite element matrices:
composite formulas, 459
effect on order of convergence, 469
Gauss quadrature, 461
in multiple dimensions, 464
Newton-Cotes formulas, 457
for quadrilateral elements, 466
recommended (full) order, 469
Simpson rule, 457
trapezoidal rule, 457
for triangular elements, 467
0
Ogden material model, 592, 594
Order of convergence of:
finite element discretizations, 247, 312, 432, 469
Newton-Raphson iteration, 757
polynomial iteration, 941
Rayleigh quotient iteration, 904
subspace iteration, 959
vector iteration, 895, 898
Orthogonal matrices:
definition, 43
use, 44, 73, 189, 913, 927, 931
Orthogonal similarity transformation, 53
Orthogonality of:
eigenspaces, 55
eigenvectors, 54
Orthogonalization:
by Gram-Schmidt, 907, 952, 956
in Lanczos method, 946
in subspace iteration, 958
Orthonormality, 55
Ovalization, 413
Overrelaxation, 747
P
Parabolic equation, 106
Partitioning (See Matrix, partitioning)
Pascal triangle, 246, 380
Patch test, 263
Peclet number, 677
Penalty method:
connection to Timoshenko beam theory (and Reissner/Mindlin plate theory), 404
elementary concepts, 144
to impose boundary conditions, 190
relation to Lagrange multiplier method, 147
Period elongation, 811
Petrov-Galerkin method, 687
Phantom frequencies/modes, 472
Phase change, 656
Pipe elements, 413
Plane reflection matrix, 73
Plane rotation matrix, 43, 913, 932
Plane strain element, 199, 351
Plane stress element, 170, 199, 351
Plasticity, 597
Plate bending, 200, 205, 420
Plate/shell boundary conditions, 448
p -method of finite element refinement, 251
Poincaré-Friedrichs inequality, 237
Polar decomposition, 508
Polynomial displacement fields, 195, 246, 385
Polynomial iteration:
explicit iteration, 938
implicit iteration, 939
Positive definiteness, 60, 726
Positive semidefiniteness, 60, 726
Postbuckling response (postcollapse), 630, 762
Potential:
incremental, 561
total, 86, 160, 268
Prandtl number, 677
Preconditioning, 749
Pressure modes:
checkerboarding, 319
physical, 315
spurious, 316, 318, 325
Principle of virtual displacements (or principle of virtual work):
basic statement, 125, 156, 499
derivation, 126, 157
linearization of continuum mechanics equations, 523
linearization with respect to finite element variables, 538
relation to stationarity of total potential, 160
Principle of virtual temperatures:
basic statement, 644, 678
derivation, 645
Principle of virtual velocities, 677
Products of matrices (See Matrix, products)
Projection operator, 308
Proportional damping, 796
Q
QR iteration, 931
Quadrature (See Numerical integration of finite element matrices)
Quarter-point elements, 370
Quasi-Newton methods, 759
Quasi-uniform sequence of meshes, 382, 434
R
Radial return method, 598
Radiation boundary conditions, 644, 658
Rank, 39
Rate of convergence of finite element discretizations, 244, 247, 312, 432, 469
Rate-of-deformation tensor, 511
Rayleigh damping, 797
Rayleigh quotient, 60, 868, 904
Rayleigh quotient iteration, 904
Rayleigh-Ritz analysis, 868, 960
Rayleigh's minimum principle, 63
Reaction calculations, 188
Reduced order numerical integration, 476
Reduction of matrix to:
diagonal form (See also Eigenvalues and eigenvectors), 57
upper triangular form, 699, 705
Reflection matrix (See Plane reflection matrix)
Regular mesh, 380
Reissner plate theory, 420
Relative degrees of freedom, 739, 741
Reliability:
of finite element methods, 12, 296, 303, 469
of mathematical model, 4
Residual vector:
in eigensolution, 880
in solution of equations, 736
Resonance, 793
Response history (See Dynamic response calculations by)
Reynolds number, 677
Rigid body modes, 230, 232, 704, 726
Ritz analysis, 119, 234
Robustness of finite element methods, 12, 296
Rotation matrix (See Plane rotation matrix)
Rotation of axes, 189
Rotation of director vectors, consistent/full linearization:
beams, 570, 579
shells, 577, 580
Round-off error, 735
Row space of a matrix, 39
Row-echelon form, 38
Rubber elasticity, 561, 592
S
Schwarz inequality, 238
Secant iteration, 941
Second Piola-Kirchhoff stress tensor, 515
Seepage, 662
Selective integration, 476
Shape functions (See Interpolation functions)
Shear correction factor, 399
Shear locking, 275, 332, 404, 408, 424, 444
Shell elements, 200, 207, 437, 575
Shifting of matrix (See Matrix shifting)
Similarity transformation, 53
Simpson's rule, 457
Single precision arithmetic, 734
Singular matrix, 27
Skew boundary displacement conditions, 189
Skyline of matrix, 708, 986
Snap-through response, 631
Sobolev norms, 237
Solution of equations in:
dynamic analysis, 768
static analysis, 695
Solvability of equations, 313
Spaces:
L^2 ,236
V. 236
V_{h} , 239
Span of vectors, 36
Sparse solver, 714
Spatial isotropy, 368
Spectral decomposition, 57
Spectral norm, 68
Spectral radius, 58, 70, 808
Spherical constant arc length criterion, 763
Spin tensor, 511
Spurious modes, 472
Stability constant of matrix, 71
Stability of formulation, 72, 308, 313
Stability of step-by-step:
displacement and stress analysis, 806
fluid flow analysis, 835
heat transfer analysis, 831
Standard eigenproblems:
definition, 51
use, 58, 231, 726
STAP (STructural Analysis Program), 988
Starting iteration vectors, 890, 909, 952, 960
Static condensation, 717
Static correction, 795
Step-by-step integration methods (See Direct integration in)
Stiffness matrix:
assemblage, 79, 149, 165, 185, 983
definition, 164
elementary example, 166
Stiffness proportional damping, 798
Storage of matrices, 20, 985
Strain hardening (tangent) modulus, 601
Strain measures:
Almansi, 585, 586
engineering, 155
Green-Lagrange, 512
Hencky, 512, 614
logarithmic, 512, 614
Strain-displacement matrix:
definition, 162
elementary example, 168
Strain singularity, 369
Stress calculation, 162, 170, 179, 254
Stress integration, 583, 596
Stress jumps, 254
Stress measures:
Cauchy, 499
first Piola-Kirchhoff, 515
Kirchhoff, 515
second Piola-Kirchhoff, 515
Stress-strain (constitutive) relations, 109, 161, 194, 297, 581
Stretch matrix:
left, 510
right, 508, 510
Strong form, 125
Structural dynamics, 813
Studying finite element methods, 14
Sturm sequence check, 953, 964
Sturm sequence property, 63, 728, 846
application in calculation of eigenvalues, 943, 964
application in solution of equations, 731
proof for generalized eigenvalue problem, 859
proof for standard eigenvalue problem, 64
Subdomain method, 119
Subparametric element, 363
Subspace, 37
Subspace iteration method, 954
Substructure analysis, 721
SuPG method, 691
Surface load vector, 164, 173, 205, 214, 355, 359
Symmetry of:
bilinear form, 228
matrix, 19
operator, 117
T
Tangent stiffness matrix, 494, 540, 755
Tangent stress-strain matrix, 524, 583, 602
Target, 623
Temperature gradient interpolation matrix, 651
Temperature interpolation matrix, 651
Tensors, 40
Thermal stress, 359
Thermoelastoplasticity and creep, 606
Torsional behavior, 664
Total Lagrangian formulation, 523, 538, 561, 587
Total potential (or total potential energy), 86, 160, 268
Trace of a matrix, 30
Transformations:
to different coordinate system, 189
of generalized eigenproblem to standard form, 854
in mode superposition, 789
Transient analysis (See Dynamic response calculations by)
Transition elements, 415
Transpose of a matrix, 19
Transverse shear strains, 424
Trapezoidal rule:
in displacement and stress dynamic step-by-step solution, 625, 780
in heat transfer transient step-by-step solution, 831
Newton-Cotes formula, 457
Triangle inequality, 67
Triangular decomposition, 705
Triangular factorization, 705
Truncation error, 735
Truss element, 150, 184, 199, 342, 543
Turbulence, 676, 682
Tying:
of in-layer strains, 408, 444
of transverse shear strains, 408, 430, 444
U
Unconditional stability, 774, 807
Uniqueness of linear elasticity solution, 239
Unit matrix (See Identity matrix)
Unit vector (See Identity vector)
Updated Lagrangian formulation, 523, 565, 587, 614
Upwinding, 685
V
Vandermonde matrix, 456
Variable-number-nodes elements, 343, 373
Variational formulations (introduction to), 110
Variational indicators:
Hellinger-Reissner, 274, 285, 297
Hu-Washizu, 270, 285, 297
incremental potential, 561
total potential energy, 86, 125, 160, 242, 268
Vector:
back-substitution, 707, 712
cross product, 41
definition, 18, 40
deflation (See Gram-Schmidt orthogonalization)
dot product, 41
forward iteration, 889, 897
inverse iteration, 889, 890
norm, 67
reduction, 707, 712
space, 36
subspace, 37
Velocity gradient, 511
Velocity strain tensor, 511
Vibration analysis (See Dynamic response calculations by)
Virtual work principle (See Principle of virtual displacements)
Viscoplasticity, 609
Von Mises yield condition, 599
Vorticity tensor, 511
W
Warping, 413
Wave equation, 106, 108, 114
Wave propagation, 772, 783, 814
Wavefront solution method (See Frontal solution method)
Weak form, 125
Weighted residuals:
collocation method, 119
Galerkin method (See also Principle of virtual displacements; Principle of virtual temperatures; and Principle of virtual velocities), 118, 126, 688
least squares method, 119, 257, 688
variational formulation, 116
z
Zero mass effects, 772, 852, 862