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Copyrighted Material 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,
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