MultiPhysicsVault/.raw/FiniteElementProcedures/FiniteElementProcedures_106.md

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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.

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# 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

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# 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

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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

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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

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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

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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

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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

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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

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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