JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

Index

Adjoint Map and Left and Right Translations, 839–842, 845, 850–854, 857–858, 126 861–862, 880, 885, 906, 913–914, Admissible Virtual Space, 91, 125, 524, 925, 934, 950–953 560–561, 564, 722, 788, 790, 793 Scaling, 37, 137, 161, 164, 166, 249 affine combinations, 166 Translation, 33–34, 42–43, 75, 77, 92, 127, Affine Space, 33–34, 41, 43, 136–137, 139, 136–137, 155, 164, 166–167, 280, 155, 248, 280, 352–353, 355–356 334, 355–356, 373, 397, 421, 533, Affine Transformation 565, 793 Identity, 21–22, 39, 47, 60–61, 63–64, Algorithm For Newton Method of Iteration, 67–68, 70, 81, 90, 98–100, 104–106, 441 116, 125–126, 137, 255, 259, 268, Algorithm: Numerical Iteration, 445 331, 347, 362–363, 367, 372, 375, Allowable Parametric Representation, 37, 377, 380, 398, 538, 550, 559, 562, 140, 249 564–565, 568, 575–576, 581, 599, An Important Lemma, 89, 605, 839 609, 616, 622, 628, 648, 655, 663, Angle between Vectors, 259 668, 670, 730, 733, 788, 790, Angular Momentum, 523–524, 537–538, 543, 792–793, 797–798, 805–806, 547–549, 553–554, 556, 558, 575, 585, 811, 825, 834, 852, 862, 871–872, 591–592, 594, 608, 622, 640, 647–648, 916 722, 748, 750, 755–756, 759–760, 762, Rotation, 1, 6, 9–15, 17, 22, 27, 33, 36, 39, 769–770, 776, 778, 780–783, 787, 805, 44, 64, 74, 90–94, 97–101, 104–108, 815, 822, 828, 842 110, 112, 114–116, 118–121, Angular Velocity Tensor, 125, 127, 526, 542, 123–127, 131, 137, 165–166, 325, 550, 565, 571, 723, 794, 800 334, 336–337, 353, 356, 359–360, Arc Length Constraint, 368, 438–439, 488, 490, 496, 498–499, 501, 504, 441–445, 643, 707, 714, 879 508, 525–527, 531, 533, 541–542, Arc Length Parameterization, 140, 270, 272, 555–556, 558–566,COPYRIGHTED 572, 574, 534, MATERIAL 735, 773 580–582, 584–585, 590, 596–597, augmented interpolation transformation 599, 601–602, 605–611, 613, 616, matrix, 679, 691, 906, 925 618, 623, 626–628, 643, 647, 679, Axiom #1: Conservation of Mass, 338 684–685, 688, 693–694, 698, 701, Axiom #2: Balance of Momentum, 339 707, 709–711, 715, 722–724, 745, Axiom #3: Balance of Moment of 750, 767, 782, 784, 786–788, Momentum, 342 790–794, 800, 803–804, 810–812, Axiom #4: Conservation of Energy 815, 819, 830, 832, 834, 836–837, Energy Conjugated Stress-strain, 348

Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells, First Edition. Debabrata Ray. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

967 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

968 Index

B-Rep, 8 linear precision, 150–151, 198, 285, 367, B-spline, 8, 14–16, 135–136, 150, 154, 163, 406, 411–412, 459, 514, 517 184–185, 190–191, 193–194, 198, Non-negativity and Unique Maximum, 218–219, 222, 224, 228, 230–231, 153, 309 234–235, 237, 239, 242–243, 245, Partition of Unity, 151, 153, 169, 190, 307, 247–248, 280, 296, 299–301, 304, 306, 309 367, 412, 414, 429–432, 434, 460, Power Basis Connection, 153 474–476, 478, 487–488, 492, 509, Product Rule, 153, 562, 592, 790, 822, 848 726 Recursive Property, 153 B-spline Curve Construction, 163 Subdivision Property, 153, 201 Balance Equations, 2D Symmetry Property, 10, 12, 154, 170, 181, For Linear Beam, 595 605, 839, 921 For Nonlinear Beam, 595 Bernstein polynomials, 8, 14, 148–151, Balance Principles, 74, 325–326, 337–338, 154–157, 164, 169–170, 174, 280, 286, 343, 351, 399, 523–524, 722 292–293, 307, 314–315, 406, 411, 413, Banach Space, 25 422–423, 428, 432, 434, 436, 486, 490, Barycentric Combination, 33, 35, 137, 164, 498, 513–514, 517, 520–522 166–167, 169, 190, 201, 206–207, 213, Bernstein-Bezier curves, 136, 148, 154, 157, 306, 313 198, 228, 406, 411, 459, 474 Basis Functions, 2–6, 14, 23, 135, 148–149, Bernstein-Bezier Generalized Displacement 151–155, 157, 163, 167, 170, 174, Element, 934 186–187, 189–190, 193–194, 201, 206, Bernstein-Bezier Geometric Element, 932 247, 280, 286, 293, 307, 309, 321, Bezier control vector, 8, 367, 413, 472, 489 367–368, 374, 376, 380, 403–404, Bezier form of B-spline curves, 154 406–407, 422–423, 433–435, 511, 513, Bezier form of B-spline surfaces, 280 517, 521, 687–688, 691, 715, 930, 932, Bezier-Bernstein Curves 934, 943 Affine Invariance, 164–165, 285 Basis Functions:, 433 Affine Reparameterization Invariance, Beam 1D Direct Engineering Representation, 164–166 532 global parameter, 165–167, 182, Beam 3D Continuum Representation 430–431 Beam Body: Interior and Boundary local parameter, 165–167, 182 Surfaces, 533 Cubic Bezier Curves, 182, 218, 475 Beam Angular Momentum Balance Bezier-Bernstein-de Casteljau, 135–136, 154, Equations, 553 247–248 Beam Balance Equations of Motion: bound vectors, 10 Dynamic, 554 bound vectors, 34, 51, 255 Beam Geometry: Definition and Assumptions Boundary or End Conditions, 230 Beam: Undeformed Geometry, 531 boundary representation, 8, 148, 176, 367, 407 Bernstein basis function, 149, 153 Bernstein Basis Functions: Properties C++ Code Snippets for Geometric Properties, Degree Elevation, 154, 164, 176, 178–179, 940 198, 292, 367, 406, 411–412, 423, C++ Code Snippets for Material and 498, 514, 517–518 Geometric Stiffness Matrices, 948 End Point Interpolation, 151, 156, 164, C++ Code: B-spline to Bezier, 243 169, 410 C++ Code: Generating Missing Control First Derivative, 5, 28–29, 154, 172, 174, Points, 304 202, 286, 691 c-type Curved Beam Element, 493–494, 496 Integral Property, 154 c-type Deep Beam: Plane Stress Element, 498 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

Index 969

c-type FE Formulation: Dynamic Loading, computational element centrifugal stiffness 673, 891 matrix, 911 c-type FE Implementation and Examples: computational element symmetric mass Quasi-static Loading matrix, 681 c-type Beam Element Library, 687 Computational Equations of Motion, 590 Code Fragment for E Matrix, 693 Computational Real Strains Code Fragment for G Matrix, 697 Real Translational and Bending Strains, Code Fragment for Tangent Stiffness 864 Matrix, 711 computational tangent stiffness matrix, 701 Non-rational Elements, 688 Computational Virtual Work Equations Rational Elements, 688 External virtual work, 397, 402, 419, 557, c-type Finite Element Discretization, 687 613, 622–623, 845, 857, 886 c-type Finite Element Formulation, 460, 529, Inertial virtual work, 402, 529, 557, 613, 574, 677, 684, 725, 803, 901, 913, 921 622, 643, 725, 845, 880 c-type Finite Element Library, 435 Internal virtual work, 397, 402, 419, 500, c-type Finite Element Space, 434 525, 527, 529, 557–558, 560, 613, c-type Shell Element Library 618, 620, 622, 627, 631–632, Non-rational Tensor-product Quadrilateral 723–725, 788, 845, 847, 861, Elements, 930 864–866, 882 c-type Shell Elements and State Control computer-aided geometric design, 8, 15, 21, Vectors, 904, 923 137 c-type Solutions: Locking Problems Configuration Space, 9, 91–92, 125, 331, Remedy for shear locking, 511 524–526, 528, 539, 542–543, 555, Remedy for transverse shear locking, 520 558–561, 564, 570, 607, 626, 722–724, c-type Truss and Bar Element, 460 749, 782–783, 787–788, 793, 799, 806, CAGD, 8, 15, 137, 148, 152, 154, 195, 205, 841, 861, 898, 901, 918, 921 406 conforming, 6, 8–9, 434, 511, 516 Cartesian Product Bernstein-Bezier surfaces, conics, 135–136, 154, 198, 205–207, 248, 280 209–210, 247, 280, 687 Cartesian product patches, 280, 292 Conics as Implicit Functions, 205 Cartesian Product Surfaces Conics as Quadratic Rational Bezier, 207 Control Net Generation, 296 conics of revolution, 247 Cauchy Reciprocal Theorem, 340, 345 conservative system, 10, 13, 470, 606, 610, Cayley-Hamilton theorem, 66, 357 618, 840, 898, 901, 921 Centripetal Parameterization, 230 constitutive or material tensor, 363 characteristic polynomial, 66, 68, 70, 91, 106, Constitutive Theory: Hyperelastic 118, 357, 662 Stress-Strain, 351 Classes of Functions, 23 Construction of S-form from W-form, 381 Classical Rayleigh-Ritz-Galerkin Method, 403 Construction of W-form from S-form, 380 Classification of Conics, 210 constructive solid modeling, 367, 407 Component Force and Moment Vectors, Continuum Approach – 3D to 1D, 539 583–584, 774, 815, 819 Contravariant Base Vectors, 38–39, 46–47, Component or Operational Vector Form, 555, 53–54, 58, 60, 62, 75, 91, 94, 251–253, 580, 782, 810 257, 328, 343, 580, 592, 732–733, 805, composite Bezier curve, 181–182, 299 810–811, 822 Composite Bezier Form contravariant components, 47, 53, 55, 57–58, Quadratic and Cubic B-splines, 248, 300 60–61, 63, 67, 77, 85, 87, 252–254, 257, Computational Derivatives and Variations, 336, 584, 757, 815 596, 830 convected, 327, 329, 558, 575, 786, 804 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

970 Index

Conventional Finite Element Methods 261, 328, 343, 543, 580, 732, 737, 748, Bubble Mode, 7, 367, 509 810–811 Drilling Freedom, 7, 509 covariant components, 53–54, 56–58, 60, 63, h-type, 2–4, 6–8, 406, 433, 438, 509, 511, 66–67, 77, 83, 88–89, 252–253, 257, 522 345, 582, 812, 869 Incomplete Strain States, 5 covariant derivative, 11, 13, 32, 76–77, 83, 85, Induced Anisotropy, 5 87, 91, 125–126, 331, 346–347, 545, interpolation, 2–3, 6–9, 15, 23, 123–124, 560–562, 588, 590–591, 644, 762, 772, 135, 151, 155–156, 164, 166–167, 778, 791, 818–819, 822, 882 169, 182, 185, 187, 189, 193, 198, Covariant Linearization of Virtual Work 218, 228, 230–233, 235, 237, 245, Geometric stiffness functional, 652, 660, 247, 280, 282, 285, 307, 313–314, 672–674, 685–686, 888, 892, 898, 410–411, 429, 486, 493, 509, 511, 901–902, 914–915, 918, 921 514, 517, 522, 679, 685, 687–688, linearized dynamic virtual work equations, 691, 904, 906, 914, 923, 925, 930 652, 673, 892 Locking Problems, 8, 14, 460, 494, 509, linearized quasi-static virtual work 511 equations, 685, 914 Shear and Membrane Locking, 7, 9 Material stiffness functional, 652, 656, non-conforming, 6, 8, 511, 516 672–674, 685–686, 888, 892, p-type, 3–5, 406, 433, 438, 509, 511, 522 897–898, 902, 914–915, 917–918 Patch Test, 6–7 residual force functional, 644, 652, reduced or selective integration, 7, 407, 459 673–674, 676, 685–686, 888, 892, selective under-integration, 8, 509 914–915, 917 Convex Hull, 23, 164, 169–170, 198, 201, unbalanced force functional, 652, 673–674, 206, 285 685–686, 888, 892, 914–917 Convex Set, 23 covariant metric tensor, 39 Coons’ Patch, 297–299, 303–304 covariant Riemann curvature tensor, 279 Coordinate Frame Rotation, 566, 794 Curvature, 3, 7, 9–10, 13, 16, 90–91, coordinate systems 125–127, 136, 145–147, 173, 180, 187, 4D homogeneous coordinate system, 139 189, 202, 248, 261, 263, 265–270, 272, Cartesian, 5, 10, 13, 23, 30, 36–37, 39, 42, 277, 279, 337, 433, 470, 474, 488, 496, 55, 71, 74–76, 136–137, 142–144, 524–526, 560–562, 565, 568, 571–572, 248–250, 280, 285–286, 290, 574, 588, 591, 597–599, 601, 605, 610, 292–293, 296, 298–299, 304, 306, 620, 628, 693–694, 697–698, 703–704, 319, 321, 327–329, 336, 367, 408, 706, 713, 722–723, 728, 730, 733, 735, 435, 498, 508, 774, 815, 942, 946 738–739, 746, 763, 772–773, 778, Base or Coordinate Vectors for 787–788, 790, 794, 797, 800, 803, 810, Cartesian Coordinate Systems, 36 818, 825, 832, 834, 836, 839, 847, curvilinear, 10, 13, 16, 22–23, 36, 42, 852–853, 855, 862, 867, 869, 897, 901, 75–76, 327–329, 331, 336–338, 352, 904, 907, 917, 921, 923, 925–926, 929, 488–489, 524–526, 532, 537, 539, 938, 941, 947–948, 950–954 542, 580, 582, 584, 722–723, curved beam, 9–10, 12, 15–17, 327, 438, 460, 727–728, 733, 774, 778, 810, 812, 486–487, 492–494, 496, 498, 500, 522, 815, 930 531, 687, 715–718 spinning coordinate system, 12, 327, 568, Curvilinear (Convected) Coordinate System, 571, 580, 620, 786, 796, 800, 810, 327, 329 847 Covariant Base Vectors, 38–39, 45, 47, 53, 58, Darboux’s vector, 146, 588 60–61, 74, 77, 91, 94, 251–253, 257, de Boor-Cox, 8, 135–136, 195, 247–248 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

Index 971

de Casteljau Algorithm Effective External Applied Moment, 553–554 linear interpolations, 161, 166, 282, 307, Einstein summation convention, 23, 39, 47, 314 51, 63, 137, 345, 347, 551 Decomposition of Tensors Elastic Materials Polar Decomposition, 72, 91, 334, 359–360 Homogeneous Elastic Material, 359 Spherical and Deviatoric Decomposition, Isotropic, Homogeneous, 358, 360, 362, 72 882 Deformation Geometry, 326 Principle of Determinism, 326, 358 deformation gradient, 91, 325, 327, 329, Principle of Local Action, 326, 351, 358, 331–334, 337, 339, 347, 349, 351, 356, 632, 866 358–362, 575, 632–633, 805, 866–867, End Point First Derivatives (Tangents), 288 869, 872 energy conjugate, 326, 338, 349, 359, 399, deformation gradient tensor 557–558, 560, 632, 785–786, 788, Tangent Map, 331–332 866 deformation map, 327–329, 331–332, 336, Energy inner product, 27–28, 376–378, 338, 524, 541, 545, 547, 574–575, 722, 382–385, 392, 421 749–751, 755, 766–767, 804–805 energy space Deformation Power, 524, 574, 576, 722, 785, Essential Boundary Conditions: Dirichlet 803, 806 Type, 379 Deformed Geometry, 325, 523, 534, 722, Natural Boundary Conditions: Neumann 772–773 Type, 379 Derivative of a Curve, 140 Solution Set, 372–373, 378, 381, 407, 421 Derivatives of Bezier Curves Equivalence of S-form and W-form, 380 End Point First Derivatives, 172, 182, 288 Equivalence of S-form, W-form and V-form, End Point Higher Derivatives, 173 382 First Derivative, 5, 28–29, 154, 172, 174, Euclid’s geometry 202, 286, 691 fifth or the parallel postulate, 42 General Derivatives, 172, 174 Euclidean spaces, 22, 28, 39, 42, 51 Derivatives, Angular Velocity and Variations, Euclidean to Riemannian Space, 36 125 Euler Theorem, 100 determinant, 37, 39, 51, 58, 66, 68, 70–71, Eulerian, 325, 327, 336, 338, 345, 367, 399 93–94, 98, 148, 205, 249, 254–255, 265, External applied power, 557, 785 279, 343, 417, 445, 485, 488, 507, 562, Extremum Values of Curvature 677, 687, 691, 703–704, 735, 776, 790, Gaussian Curvature, 269–270, 279 872, 926, 938 Mean Curvature, 269–270 Differential Area Element, 332 direct stiffness, 11, 406, 478, 529, 677, 687, Finite Difference Methods, 371 726 finite element method Divergence of Vector Functions, 84–85 c-type finite element method, 1, 7,9, Divergence Operator, 84–85, 87, 755 14–15, 91, 148, 325–326, 337, Dual Base vectors, 38 367–368, 371, 373, 375, 377, 379, Dynamic Balance Equations, 537, 543, 590, 381, 383, 385, 391, 393, 395, 397, 592, 607, 748, 750, 763, 770, 778, 781, 399, 403, 405–407, 411, 413, 417, 825, 841 419, 421, 423, 427, 429, 431, 433, Dynamic Momentum Balance Equation, 547 435, 438–439, 441, 443, 445, 450–451, 453, 455–456, 459, 509, Effective angular momentum, 548–549, 554, 522, 524, 526, 528–529, 559, 561, 608, 640 626, 717, 722–724, 726, 773, 788, Effective External Applied Force, 547 860 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

972 Index

First Fundamental form Groups Riemannian Metric, 39, 252–253 Abelian, 22 flat-spaces, 10 Lie, 2, 10, 12, 22, 91–92, 98–100, 104–105, For Linear Circular Beam, 595 115, 118, 120, 125–126, 265–266, Forward Difference Operator, 32, 172, 174, 561–562, 564–566, 568, 571, 598, 178, 202 601–602, 605–606, 609, 616, 648, Frame Indifference, 326, 351–352, 359–360, 709, 790, 792–793, 796–798, 800, 632, 866 825, 832, 836, 839–840, 852 free vectors, 10 Frenet Frame Heat Conduction and Potential Flow bi-normal vector, 143–145 Problems, 428 osculating plane, 144, 147–148, 263, Hellinger-Reissner Variational Theorem, 391 485 Hessian Symmetry Lemma, 83 Principal Normal Vector, 143–144 Higher-order Tensor Component, 55 Rectifying Plane, 144–145 Hilbert Space, 25, 33, 71 Unit Tangent Vector, 142, 144, 147 Horner’s Algorithm, 157 Frenet-Serret Derivatives, 136, 145 Hyperelastic Material Property Principle of Equivalence, 632, 866 Gateaux or Directional Derivative, 28–29 Thin, Linear, Homogeneous, Isotropic, Gauss-Legendre Quadrature Formulas 635, 869 Composite Formulas, 448 Hyperelastic, Isotropic, Homogeneous Higher Dimensional Gauss-Legendre, Material, 362 452 Interpolatory Quadratures, 446 Important Computational Identities, 127 Linear Transformation of Interval, 448, incremental generalized displacement, 450 528–529, 656, 679, 687, 701, 707, Newton-Cotes Formulas, 446, 448 725–726, 879, 897, 904, 916–917, 929 Tabular Examples of Gauss-Legendre incremental strain-displacement, 656, 898, Nodes and Weights, 453 918 Gauss-Weingarten Formulas Inertial power, 785 Christoffel symbol, 77, 272, 745 Inner Product Space, 25, 27–28, 33, 405, 421 first kind, 272–273 Integral Transforms: Green-Gauss Theorems, second kind, 33, 74, 77, 272, 277 87 Gauss Formula, 272 Integration by Parts, 28, 87, 367, 372, Normal Coordinate System, 272, 580, 584, 375–377, 380–381, 383–385, 557 728, 810–811, 815, 871 Inter-element Continuity Weingarten Formula, 248, 272, 275, 733 inner control points, 184 Geometric stiffness matrix, 12–13, 528, 655, junction control point, 184 667, 672, 681, 707, 725, 897, 901, 917, Inter-element Continuity (IEC): Bezier to 921, 948 B-spline, 429 Geometric Stiffness Matrix with Distributed Internal deformation power, 785 Moment Loading, 672 internal energy, 338, 348–349, 631, 866 global basis function, 368 internal energy density, 348 Gradient of Tensor Functions, 84 Iso-curves, 250–251, 266, 306, 343, 728, 730, Gradient of Vector Functions, 83 773 Gram-Schmidt Orthogonalization, 14, Isotropic Tensor Functions, 326, 356–357 143–144 Green’s identity or the divergence theorem, Jacobian determinant, 39, 677, 687 372, 375, 380 Jacobian matrix, 37, 51, 249, 255, 417, 691 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

Index 973

Kinematic and Configuration Space, 555, 559, 746, 763, 772–773, 778, 787, 803, 810, 783 836, 852, 867, 869, 897, 901, 904, 907, Kinematic Parameters 917, 921, 923, 925–926, 929, 938, 941, displacement vector field, 91, 125, 948 524–526, 559, 561, 564, 580, 687, local basis functions, 3, 368, 404, 687 722–723, 787, 790, 793, 804, 810 Local Gauss Frame, 261 rotational tensor field, 91, 125, 524, 526, Love-Kirchhoff, 11, 542, 767 559, 574, 722, 724, 787–788, 804 Love-Kirchhoff-type, 11, 542, 767 Kinetic Energy Balance, 349, 351 knot sequence, 182, 186, 190, 192–193, Mainardi-Codazzi conditions, 787 234–235, 245 material coordinate system, 327–328, 335, knot vector, 182, 192, 194 338–339, 558, 568, 574, 786, 796, 803 Knots of Multiplicity, 187, 190, 235 Material Curvature Tensor, 127 Kronecker delta tensor, 38, 252 Material stiffness matrix, 527–528, 655, 674, 686, 693, 707, 725, 918, 926 Lagrange equations of motion, 346 material symmetry, 351, 360, 632, 866 Lagrangian, 2–3, 7, 16, 135, 148, 247, Material Tensor, 361, 363 325–329, 331, 333–343, 345–347, 349, Matrix Form of a Bezier Curve, 157, 164 351, 353, 355–357, 359, 361, 363, 367, Matrix Representation, 46–47, 49, 529, 556, 399, 543, 550–551, 738, 746, 748 585, 592, 597, 725, 733, 784, 822, 830, Lagrangian inertia tensor, 551 832 Lame constants, 362–363 mega-element, 531 Least Square Form (L–form), 385 mesh generation, 1, 8–10, 13, 133, 135, 145, Length of a Curve, 259 148, 155–156, 163–164, 170, 173, Length of a Vector on a Surface, 257 180–182, 189, 247, 280–282, 286, 290, Linear Balance Equations, 3D, 595 292, 312, 407–408, 417, 423, 434, 460, Linear Elasto-dynamic Problems, 427 486, 498, 507, 717, 719, 923, 932 Linear Interpolation, 123–124, 166–167, 198, metric tensor, 39, 44, 56, 58, 60, 85, 87, 91, 237, 282, 307, 313–314, 685, 914 253–254, 259, 267–268, 273, 485, 730, Linear Interpolation or Approximation on 735, 778, 940, 946 3-Sphere, 123, 914 Meusnier’s Theorem Linear Momentum, 537, 545, 547, 549, 554, Normal Section of a Surface, 263 556, 585, 591, 594, 608, 640, 751, Minimal Support Property, 193 753–755, 759, 767, 776, 780, 783, 815, Minimum Potential Energy Principle, 189, 828 368, 395, 399, 419, 470 Linear Transformations Minimum Potential Energy Theorem, 391 Linear Functionals, 14, 26–27, 42, 44–45, 248 Nanson’s Formula, 332 Linear Operators, 26–27, 42, 44, 174, 248 New Finite Element Method: c-type Orthogonal transformation, 27, 33, 64, Coerced Isoparametry, 406, 411–412, 423 91–93, 97, 100, 118, 125, 136, 353, Element (Local) Level Formulation, 414 358 Finite Element Basis Functions, 422 Linear Vector Space, 22, 26–27, 41, 44, 46, Finite Element Mesh, 407, 422, 429 62, 75, 126, 135–136, 155, 247–248, Finite Element Trial and Virtual Subspace, 280, 560, 563, 628, 792, 862 422 Linear, Hyperelastic, Isotropic, Homogeneous Root Element, 4, 6, 282, 285, 408, 410, Material, 362 417, 422, 433, 435, 488, 500, Lines of Curvature, 91, 125, 266–270, 272, 507–508, 513, 517, 520, 677, 337, 723, 728, 730, 733, 735, 738–739, 687–688, 904, 923, 930 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

974 Index

New Moving or Spinning Coordinate System, Patch by 786 Extrusion of One Boundary Curve, 296 Newton Iteration, 438 Lofting of Two Boundary Curves, 296 Newton’s Method, 438–439, 442 Revolution of a Curve, 299 Newton-type iteration, 11, 368, 439, 643, 714, Permutation symbol, 33, 71 880, 901, 921 Piola Transformation, 332, 346, 359 Non-conservative Systems, 12 Points on a Surface non-rational, 136, 139, 154, 163, 198–199, Elliptical Point, 265 201, 205–207, 222, 224, 228, 230, 232, Hyperbolic Point, 266 243, 248, 280–281, 283, 286, 296, Parabolic Point, 266, 270 299–301, 306, 322, 508–509, 687–688, Position or radius vector, 137 930, 932 Potential Energy Functional, 368, 392, non-rational Bernstein-Bezier surfaces, 394–395, 419, 470, 488–489 280 Projective Map and Cross Ratio Invariance, Non-rational Bezier Surfaces 198 hyperbolic paraboloid surface, 282–283 Properties of Metric or Fundamental Tensor, Normal Vector for a Degenerate Patch, 253 290 Pseudo-vector Representation of Tensors, 50 Surface Normal Vectors, 290 Non-rational Curves, 154, 198 Quadratic Rational Bezier Curves and Conics, Non-symmetric Arc Length Constraint 206 Method, 441 quadrature rules, 414, 438 Norm, 5, 25–26, 33, 70–71, 81, 98, 115–116, Quasi-static Proportional Loading, 439, 529, 195, 376, 385, 399, 404–405, 421, 442, 622, 639, 643, 725, 877, 879–880 486, 496, 711, 872 Quotient Rule and Invariance, 57 normal curvature, 266–267 Normal Plane, 142–143 Radius of Curvature, 146–147, 261, 267, 869 NURB, 235 Ratio Invariance, 198 Rational B-spline Curves, 135, 230 Objective Functions, 355 rational B-spline curves, 135, 230 Objective Tensors, 351, 353 Rational Bernstein-Bezier curves, 154 objective vector, 355 rational Bernstein-Bezier surfaces, 280 One-dimensional Strain Energy, 525 rational Bezier, 135–136, 163, 198–199, Open Curve Interpolation, 218 201–210, 213, 232–234, 248, 280–281, Operators, 17, 26–29, 42, 44, 83, 104–105, 292–293, 295–296, 321–322, 486, 488 135, 151, 170, 172, 174, 247–248, 286, Rational Bezier Curves, 163, 198–199, 201, 371, 385 205–206, 210, 280, 292, 321 Orthogonal Transformation and the Rotation Rational Bezier Surfaces, 280, 292, 295 Tensor Real Coordinate Space, 22–23, 33, 41, 43–44, determinant of an orthogonal 136, 248 transformation, 93 Real Parametric Curves, 139 Real Parametric Surface, 248–249 Parametric Mapping Real Strain and Strain Rates isoparametry, 6, 406, 411–413, 417, 423, Real Strain Field, 527, 572, 724, 800 688 Recursion Relation, 190, 315 subparametric, 6, 368, 407, 411, 435–436, Recursive Algorithm: de Boor-Cox spline, 195 904, 923 Reduced Internal Energy, 631, 866 superparametric, 6 Reference Configuration, 325, 353, 356, 360, Pascal triangle, 5 362, 574, 803, 805 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

Index 975

Regular Curve, 140, 250 rotating coordinate system, 327, 574, 803 regular points, 250 Rotation Matrix: Components of a Rotation Regular Reparameterization, 140 Tensor, 94 Reissner-Mindlin-type, 542, 574, 750, 767 Rotation Tensor, 11, 14, 39, 74, 90–94, 97, represent a curve 99–101, 105, 107, 115, 119, 121, 123, implicit form, 135, 205, 247 125–127, 131, 325, 334, 337, 353, 356, parametric form, 135, 150, 247, 406, 459 359–360, 525–526, 533, 541–542, 556, Representation of the 3D Rotation Matrix 558–564, 566, 574, 580–581, 584–585, Cayley’s Representation, 100 596–597, 601, 605, 608, 626, 643, 684, Cayley’s Theorem, 100–101, 105–107, 707, 723–724, 750, 784, 786–788, 564, 596, 609, 616, 792, 830, 851 790–791, 793–794, 803, 810–811, 815, Instantaneous Rotation, 560, 596, 608, 830, 832, 836, 839, 841, 851, 861, 880 788, 830, 851 Rotational Variation, 126–127, 565, 793, 850 Euler–Rodrigues Parameters, 112 4D Euler-Rodrigues Vector, 112 Schoenberg, 8, 17, 135, 148, 185, 188, 193, Hamilton’s Quaternions, 115 247 Four-dimensional Quaternions, 115 Schoenberg-deBoor B-spline forms, 247 Quaternion Properties, 116 Schoenberg-Whitney Theorem, 188 inverse quaternion, 115–116 screw motion, 92 multiplication rule, 116, 118 Second Fundamental form pure or vector quaternion, 116 Surface Curvatures, 261, 947 unit quaternion, 115–116, 118 Second-Order Tensor, 44, 46–47, 50, 54–58, Hamilton-Rodrigues Quaternion, 115, 60–63, 71–72, 80–81, 83–84, 87–89, 94, 118–121, 685 253, 334, 345, 570 4D Hamilton-Rodrigues Quaternion, Second-order Tensor Component 119 Contravariant Components of a Rodrigues Parameters, 107, 112, 114, 560, Second-order Tensor, 55 597, 609, 616, 709, 788, 852 Covariant Components of a Second-order, 3D Rotation Vector, 108 54 Geometry of the Rotation Matrix and Mixed Components of a Second-order the Rodrigues Vector, 110 Tensor, 54–55 Rodrigues Vector, 92, 100, 107–108, Shear, 7–12, 16–17, 137, 185, 326–327, 337, 110, 112, 114, 119–120, 127, 560, 363, 395, 438, 460, 475, 478, 485–488, 564, 580, 596–597, 599, 608–609, 493–497, 499, 502, 509, 511–512, 516, 616, 788, 792, 810, 830, 851–852 520, 522, 529, 531–532, 539, 541–542, Rotation Matrix, 91–92, 94, 97–101, 555, 572, 575, 583, 595, 628, 633, 656, 104–108, 110, 112, 114, 116, 715–716, 726, 746, 750, 767, 774, 782, 118–121, 125, 127, 137, 560, 804, 814, 828, 850, 862, 864, 867 562–564, 596–597, 607–609, 616, Shell analysis 647, 709–711, 788, 790–792, 810, derived approach, 11 830, 841, 850–851 direct approach, 11, 524, 722 Residual Error Form: (R-Form), 371 Shell Geometry: Definition and Assumptions Riemann curvature tensor 3D Shell-like Body, 722, 727, 750, 754, mixed Riemann curvature tensor, 279 766, 782, 803–804, 842, 866 Riemannian geometry, 39, 253 Optimal Normal Coordinate System, 728, Riemannian spaces, 36, 39, 44 871 Rigid Body Dynamics, 549 Reference Surface Oriented Differential Rigid Body Rotation, 6, 91–92, 94, 97, 100, Area, 738 334, 336, 533 Shell Boundary, 735–738, 751, 756–758 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

976 Index

Shell Geometry: Definition and Assumptions Symmetric Tensors, 66–68, 72, 357, 565, (Continued) 794, 825 Shell Coordinates, 736 Eigen Properties of Symmetric Tensors, Shell Interior Differential Volume, 739 66 Shell Optimal Normal Coordinate System, Third Order Permutation Tensor, 61 728 Transposed Tensors, 63 Shell Reference Surface, 91, 125, 609, spheres as rational Bezier surfaces, 295 722–723, 728, 730, 732, 735–738, spherical coordinates, 250, 253 745–746, 748, 750–751, 756, 763, Spherical Displacement, 92, 97 773–774, 776, 778, 782, 784–787, Spline Function Representation, 188 798, 800, 803–804, 810, 815, 819, Spline Space, 186, 189–190, 193, 195, 434 828, 830, 836, 841, 844, 848, 852, splines 861, 866, 872, 923 B-splines (Basis Functions), 189 Shell Shifter Tensor, 732 Cardinal splines, 189 Shell Triads of Covariant Base Vectors, 737 Natural splines, 189, 195 Shell Undeformed Geometry, 727 Periodic splines, 189, 195 Shoulder Point and Parameterization, 209 Physical splines, 185–186, 189 simple knot, 190 Standard Bezier Curve, 166, 169 singular points, 250 standard form, 165, 167, 201, 207–213, 231, Singularity-free Update, 959 233, 235, 245 Sobolev Space, 14, 25–26, 367, 372–373, 376 Static and Dynamic Equations: Continuum spatial coordinate system, 327, 558, 795 Approach - 3D to 2D Special Orthogonal Group, SO(3) External applied moment, 553–554, 760, Adjoint Map, 104, 106, 126–127, 564–565, 762, 770 793 Internal resistive stress couple, 556, 757, Exponential Map, 104–106, 126, 562–564, 770, 783 609, 616, 790–793, 852 Internal resistive stress resultant, 545, 556, Lie algebra, 22, 99–100, 105, 125–126, 783 564–565, 609, 616, 792–793, 852 shell resultant stress tensor, 751 Matrix Lie Group, SO(3), 98 Shell total angular momentum, 759 Tangent Operators of SO (3), 104 Shell total linear momentum, 753 tangent space at identity, so (3), 100 Stress couple tensor, 756–758 Special Tensors Surface angular momentum, 756, 759, 762, Fourth-order Identity Tensor, 61, 81 769 Inverse Tensors, 63 Surface external applied force, 754 Metric or Fundamental Tensor, 58, 253 Surface linear momentum, 751, 753, 755, Orthogonal and Rotation Tensors, 64 767 Second-order Identity Tensor, 39, 60–61, Static and Dynamic Equations: Engineering 63 Approach, 534 Skew-symmetric Tensors, 66–68, 72, 565, Static Balance Equations, 524, 538, 594, 721, 794, 825 762–763, 781–782, 828 Axial Vectors, 67–68, 91, 100, 105, 107, Strain Tensors 568, 598, 605, 610, 628, 648, 772, Green-Lagrange Strain Tensor, 325, 797–798, 818, 825, 832, 839, 853, 334–335, 351, 356, 359, 361–363, 862 635, 869 Eigenproperties of Skew-symmetric Stress Couple, 527, 535, 548, 554, 556, 570, Tensors, 67 574, 583–584, 631, 633, 724, 756–758, Important Identities of Skew-symmetric 762, 770, 774, 783, 799, 803, 815, 865, Tensors, 68 867, 871 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

Index 977

stress power, 348–349, 361, 785 Partial Derivatives of Scalar Functions of Stress Resultant, 10, 13, 17, 526, 535, 545, Tensors, 80 547, 556, 570, 583, 633, 724, 726, 751, Partial Derivatives of Tensor Functions of 753, 755, 774, 783, 799, 865, 867 Tensors, 81 Stress Tensors Partial Derivatives of Tensors, 74 Piola-Kirchhoff Stress Tensors, 343 Tensor Operations first Piola-Kirchhoff stress tensor (PK Contractions, 56, 69 I), 345, 359 double contraction, 55–56, 60–61, 64, second Piola-Kirchhoff stress tensor 69–73, 81, 85, 87, 349, 848 (PK II), 345, 359 simple contraction, 55–57, 67, 69–71, Subdivision Algorithm, 179–180, 184, 198, 89 201–202, 286, 293, 498, 509, 717 Tensor Product Surface Area, 259, 327, 340–343, 543, 739, Tensor Product (Dyadic) Notation, 45 746, 753–755, 759–760, 762, 769–770, Tensorial Description of Rotational 784, 844, 848, 871 Transformation, 94 Surface curvature tensors, 277 Tensors as Linear Transformation, 44 Surface curvature vector, 277 Thin and Very Flexible Beam: Deformation surface Jacobian, 255, 732 Map, 541 surface traction vector, 338, 341–342 Thin and Very Flexible Shell: Specialization symmetrization of the geometric stiffness, 12 Effective surface angular momentum, 769 Symmetry of the Tangent Operator, 12–13, 16, Effective surface linear momentum, 767 701 Surface rotary inertia, 769 Symmetry Preserving Arc Length Constraint Time-dependent Dynamic Loading, 623, Iteration Method, 443 857 Symmetry: the Geometric Stiffness Matrix Timoshenko, 9, 11, 17, 496, 504–505 Rotational Part, 667, 897, 916 Torsion, 147–148, 202, 595, 693–694, Translational Part, 897, 916 697–698, 704–706, 716 Total angular momentum, 549, 759 Tangent Plane, 250–252, 301, 345, 433, 751, total current “curvature” vectors, 855 756–757, 784, 844, 848 Trace, 56, 60–61, 69–71, 73, 80, 91, 112, 206, Taylor series, 29–30, 143, 358, 643, 880 438, 623, 709–712, 714, 857, 948–949, Tensor by Component Transformation 951 Allowable Coordinate Transformation, Triangular Bezier-Bernstein Surfaces 41–42, 51, 53, 255, 257 Axis-Directional Tangents, 319 Transformation Law, 44, 53–55, 57, 257 Derivatives and Normal to Triangular Contravariant Components of a Vector, Surface, 315 53, 55, 77, 257, 336 Normals on a Bezier Triangle, 319 Covariant Base Vector, 47, 53, 257, Rational Bezier Triangle and Derivatives, 345 321 Covariant Components of a Vector, Triangle Number, 307, 309 53–54, 77, 257 Triangles and Barycentric Coordinates, Tensor Calculus 306 Covariant or Absolute Derivative, 75 Triangular Bernstein Polynomials, 307, Covariant or Absolute Derivatives of the 436 Components of Vectors, 77 triangular patches or surfaces, 280 Differential Operators, 371 Twist, 2, 10–11, 13, 136, 145, 147–148, 161, Partial Derivatives of Base Vectors, 74 248, 272, 286, 288, 292–293, 301, Partial Derivatives of Parametric Functions 303–304, 306, 326, 460, 496, 500, 595, of Tensors, 81 717–718 JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm

978 Index

Twist or Second Mixed Partial Derivative, Virtual Work Principle, 11, 368, 384, 391, 397, 288, 293 399, 402, 416, 419, 421, 428, 470, 486, Two- or Three-dimensional Scalar-valued 489, 527, 559, 561, 564, 574, 607, 613, Functions, 88 632, 724, 788, 793, 803, 841, 845, 866 Virtual Work Principle for Linear Systems, Vainberg Principle, 14, 32, 419, 626, 628, 860, 397 862, 864 Variational form (V-form), 381 Weak Form of the Vector Balance Equations variational formulations, 368, 371 External virtual work rate, 557 Variational Methods, 15–17, 371–372, Inertial virtual work rate, 557 386 Internal virtual work rate, 557 Vector product, 33, 663, 668, 670 Weak or Galerkin Form: (W- Form) vector space, 22, 26–27, 34, 41, 44–47, 62, 75, Admissible Boundary Conditions, 373, 105, 126, 135–136, 155, 199, 247–248, 379, 392, 396 280, 307, 355–356, 421, 528, 560–564, Collocation Method, 374 628, 647, 707, 791–792, 862 Galerkin Methods, 375 Vector triple product, 33, 91, 135, 144, 247, Green’s Function Method, 375 550, 556–557, 560, 571, 596, 611, Rayleigh-Ritz-Galerkin Method, 3, 5, 368, 613, 784–785, 788, 800, 830, 842, 376, 379, 403–404, 428 844 test space, 368, 372–377, 379–381, Vector-valued Functions: Divergence or Gauss 383–385 Theorem, 88 trial space, 368, 372–377, 379–381, Vectors in Affine Space, 34 383–385, 421, 434 Vectors on a Surface, 250, 252, 277, 772 Weierstrass Approximation Theorem, 149 Virtual (Displacement) Work Principle for Weight, 207–208, 210, 212–213, 233–234, Nonlinear Systems, 399 242, 295, 299–300, 304