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[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