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E-raamat: Differential Quadrature Hierarchical Finite Element Method, A

(Beihang Univ, China), (Inst Of Applied Physics And Computational Mathematics, China), (Beihang Univ, China), (Beihang Univ, China)
  • Formaat: 652 pages
  • Ilmumisaeg: 03-Aug-2021
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811236778
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Differential Quadrature Hierarchical Finite Element Method, ASuurem pilt
  • Formaat: 652 pages
  • Ilmumisaeg: 03-Aug-2021
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811236778
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"The differential quadrature hierarchical finite element method (DQHFEM) was proposed by Bo Liu. This method incorporated the advantages and the latest research achievements of the hierarchical finite element method (HFEM), the differential quadrature method (DQM) and the isogeometric analysis (IGA). The DQHFEM also overcame many limitations or difficulties of the three methods. This unique compendium systemically introduces the construction of various DQHFEM elements of commonly used geometric shapes like triangle, tetrahedrons, pyramids, etc. Abundant examples are also included such as statics and dynamics, isotropic materials and composites, linear and nonlinear problems, plates as well as shells and solid structures. This useful reference text focuses largely on numerical algorithms, but also introduces some latest advances on high order mesh generation, which often has been regarded as the major bottle neck for the wide application of high order FEM"--
Preface v
About the Authors xi
Chapter 1 An Overview of High-Order Methods for Structural Mechanics 1(84)
1.1 The Differential Quadrature Method
3(10)
1.1.1 The quadrature rules
3(6)
1.1.2 Weighting coefficients and sampling points
9(2)
1.1.3 The method of imposing initial/boundary conditions
11(1)
1.1.4 Strong formulation finite element method
12(1)
1.1.5 A few comments on the DQM
13(1)
1.2 Weak Form Quadrature Element Method
13(4)
1.2.1 Weak form quadrature rules
13(4)
1.2.2 A few comments on the weak form QEM
17(1)
1.3 The Hierarchical Finite Element Method
17(45)
1.3.1 The HFEM on 1D and its tensor product domains
17(9)
1.3.2 The HQEM on quadrilateral and hexahedral domains
26(2)
1.3.3 The HFEM on simplex domains
28(14)
1.3.4 The DQM and HQEM on simplex domains
42(14)
1.3.5 The HFEM on pyramid domains
56(5)
1.3.6 The HQEM on pyramid domains
61(1)
1.4 The p-version Finite Element Method for Nonlinear Problems
62(6)
1.5 The p-version Finite Element Method for Plates and Shells
68(6)
1.6 A Review of High-Order Mesh Generation
74(9)
1.7 Outlook
83(2)
Chapter 2 A Differential Quadrature Finite Element Method 85(30)
2.1 The Reformulated Differential Quadrature Rule
85(5)
2.2 Gauss-Lobatto Quadrature Rule
90(1)
2.3 The Differential Quadrature Finite Element Method for Kirchhoff Thin Plates
91(4)
2.3.1 The quadrature element method (QEM)
91(2)
2.3.2 The differential quadrature finite element method for Kirchhoff thin plates
93(2)
2.4 The Differential Quadrature Finite Element Method for Elasticity
95(10)
2.4.1 Rod element
95(2)
2.4.2 Euler beam element
97(2)
2.4.3 Plane element
99(1)
2.4.4 Mindlin plate element
100(1)
2.4.5 3D element
101(1)
2.4.6 Thickness-shear vibration analysis of rectangular quartz plates
102(3)
2.5 Numerical Comparisons
105(8)
2.5.1 Vibration and bending of plates and 3D solids
105(6)
2.5.2 Thickness-shear vibration analysis of rectangular quartz plates
111(2)
2.6 Conclusions
113(2)
Chapter 3 The Differential Quadrature Hierarchical Finite Element Method for Mindlin Plates 115(54)
3.1 The Reformulated Differential Quadrature Rule
116(1)
3.2 The DQHFEM on Quadrilateral Domain
117(6)
3.2.1 The method of geometric mapping
117(2)
3.2.2 The construction of shape functions
119(2)
3.2.3 Gauss-Lobatto quadrature rule
121(2)
3.3 The DQHFEM on Triangular Domain
123(10)
3.3.1 The construction of DQHFEM basis on triangles
123(7)
3.3.2 Geometric mapping of triangular elements
130(3)
3.4 The DQHFEM for Euler-Bernoulli Beam
133(3)
3.5 The DQHFEM for Mindlin Plate
136(6)
3.6 The DQHFEM for Quartz Crystal Plate
142(4)
3.7 Results and Discussion
146(20)
3.7.1 The DQHFEM for isotropic Mindlin plates
146(14)
3.7.2 The DQHFEM for crystal plates
160(6)
3.8 Conclusions
166(3)
Chapter 4 The Differential Quadrature Hierarchical Finite Element Method for Plane Problems 169(58)
4.1 The Differential Quadrature Hierarchical Method
170(16)
4.1.1 Rectangular elements
170(4)
4.1.2 Triangular elements
174(5)
4.1.3 The Fekete points and the DQ rules on triangles
179(5)
4.1.4 The differential quadrature hierarchical rules
184(2)
4.2 Applications of the DQHFEM to Plane Problems
186(5)
4.3 Results for Free Vibrations
191(23)
4.3.1 Rectangular elements
191(13)
4.3.2 Triangular elements
204(10)
4.4 Results for Static Analysis
214(11)
4.4.1 Thick-walled cylinder under uniform boundary pressure
214(2)
4.4.2 Stress concentration problem of circular hole in plate
216(1)
4.4.3 The interaction of grains and grain boundaries of metals
217(4)
4.4.4 The interface between nanoparticles and the matrix of nanoparticle composites
221(4)
4.5 Conclusions
225(2)
Chapter 5 The Hierarchical Quadrature Element Method for 3D Solids 227(74)
5.1 Element Energy Functions
227(3)
5.2 Shape Function Construction
230(24)
5.2.1 Tetrahedral elements
231(6)
5.2.2 Wedge elements
237(1)
5.2.3 Hexahedral elements
238(2)
5.2.4 Pyramid elements
240(14)
5.3 Strain Matrix
254(2)
5.4 Element Matrices
256(6)
5.4.1 Isotropic solids
256(1)
5.4.2 Thermo-mechanical cross-ply laminates
257(5)
5.5 Results and Discussions
262(37)
5.5.1 3D vibration analyses
262(7)
5.5.2 Applications of the pyramid elements
269(7)
5.5.3 The interaction of grains and grain boundaries of metals
276(3)
5.5.4 Three-dimensional analysis of nanoparticulate polymer nanocomposites
279(3)
5.5.5 Thermo-mechanical analysis of cross-ply laminated plates
282(17)
5.6 Comments on Gauss Integration
299(1)
5.7 Conclusions
300(1)
Chapter 6 The Hierarchical Quadrature Element Method for Kirchhoff Plates 301(48)
6.1 Hermite Blending Function Interpolation
302(6)
6.1.1 Blending function interpolation on a unit square domain
302(3)
6.1.2 Blending function interpolation on a unit triangular domain
305(3)
6.2 Hierarchical Bases for Quadrilateral Elements
308(5)
6.2.1 Vertex functions
308(2)
6.2.2 Edge functions
310(2)
6.2.3 Face functions
312(1)
6.3 Hierarchical Bases for Triangular Elements
313(5)
6.3.1 Vertex functions
313(2)
6.3.2 Edge functions
315(1)
6.3.3 Face functions
316(2)
6.4 Numerical Implementation
318(10)
6.4.1 Basis transformation
318(2)
6.4.2 Node collocation
320(6)
6.4.3 FEM discretization
326(2)
6.5 Results and Discussion
328(16)
6.5.1 Complete element order and computational efficiency
328(2)
6.5.2 Analysis using conforming elements
330(3)
6.5.3 Plate with a singularity
333(3)
6.5.4 Free vibration
336(3)
6.5.5 Analysis using quasi-conforming elements
339(1)
6.5.6 Bending of a square plate
340(1)
6.5.7 Bending of a circular plate
341(2)
6.5.8 Plate with an irregular cutout
343(1)
6.6 Conclusions
344(5)
Chapter 7 The Hierarchical Quadrature Element Method for Shells in Orthogonal Curvilinear Coordinate System 349(44)
7.1 Element Energy Functions for Deep Shell Element
350(6)
7.2 The Configuration of Double-Curved Sandwich Shell
356(2)
7.3 Estimation of Material Properties
358(3)
7.3.1 ROM model
359(1)
7.3.2 Mori-Tanaka model
360(1)
7.4 Solution of Temperature Field
361(4)
7.5 Layerwise Theory of Functionally Graded Shells
365(6)
7.5.1 Linear strain energy
366(1)
7.5.2 Nonlinear strain energy
367(3)
7.5.3 The kinetic energy
370(1)
7.5.4 The governing equation
371(1)
7.6 The Differential Quadrature Hierarchical Finite Element Method
371(9)
7.6.1 Approximation of the displacement field
371(4)
7.6.2 Integral scheme
375(1)
7.6.3 Linear stiffness matrix
376(1)
7.6.4 Geometric stiffness matrix
377(1)
7.6.5 Mass matrix
378(1)
7.6.6 Element assembly
378(2)
7.6.7 The dynamic equation
380(1)
7.7 Results and Discussion
380(11)
7.7.1 Free vibration of functionally graded single-layer shell in non-thermal environment
381(3)
7.7.2 Vibration of functionally graded sandwich shells in non-thermal environment
384(1)
7.7.3 Free vibration of functionally graded single-layer shell in thermal environment
384(2)
7.7.4 Free vibration of functionally graded sandwich shells in thermal environment
386(5)
7.8 Conclusions
391(2)
Chapter 8 The Hierarchical Quadrature Element Method for Isotropic and Composite Laminated General Shells 393(32)
8.1 Geometry Representation
394(2)
8.2 The Layerwise Shell Model
396(4)
8.3 The Hierarchical Quadrature Elements
400(8)
8.3.1 Modified high-order bases
400(3)
8.3.2 FEM discretization
403(5)
8.4 Numerical Examples
408(16)
8.4.1 Analyses of plates
409(7)
8.4.2 Analyses of shells
416(8)
8.5 Conclusions
424(1)
Chapter 9 Hierarchical Quadrature Element Method for Geometrically Nonlinear Problems 425(32)
9.1 The Hierarchical Quadrature Element Method
426(5)
9.1.1 Shape functions for quadrilateral elements
426(2)
9.1.2 Shape functions for hexahedral elements
428(3)
9.2 Measures of Stress and Strain
431(1)
9.3 Geometrically Nonlinear Formulation of Hierarchical Quadrature Elements
432(14)
9.3.1 Formulation for two- and three-dimensional elements
432(6)
9.3.2 Formulation for shallow shell elements
438(6)
9.3.3 Solution of the system of nonlinear equations
444(2)
9.4 Numerical Tests
446(10)
9.4.1 A cantilever beam in planar configuration
446(2)
9.4.2 A cantilever beam in three-dimensional configuration
448(4)
9.4.3 An extension spring in three-dimensional configuration
452(2)
9.4.4 A cylindrical shallow shell in Mindlin formulation
454(2)
9.5 Conclusions
456(1)
Chapter 10 The Hierarchical Quadrature Element Method for Incremental Elasto-Plastic Analysis 457(30)
10.1 Classical J2 Flow Theory with Nonlinear Isotropic Hardening
457(9)
10.1.1 Classical three-dimensional elasto-plastic theory
457(3)
10.1.2 Numerical algorithm for three dimensional elasto-plastic problems
460(5)
10.1.3 Return-mapping algorithm for plane stress elasto-plastic problems
465(1)
10.1.4 Numerical calculation process of elasto-plastic problems
466(1)
10.2 The Hierarchical Quadrature Element Method
466(6)
10.3 Numerical Examples and Discussions
472(12)
10.3.1 A thick-walled tube under uniform internal pressure
472(5)
10.3.2 Perforated square plate under plane stress condition
477(5)
10.3.3 Thick perforated square plate
482(2)
10.4 Conclusions
484(3)
Chapter 11 Curved p-version Cl- Finite Elements for the Finite Deformation Analysis of Isotropic and Composite Laminated Thin Shells 487(40)
11.1 Thin Shell Model
488(7)
11.1.1 Kinematics
488(3)
11.1.2 Weak form
491(1)
11.1.3 Constitutive equation
492(3)
11.2 Mesh Generation
495(4)
11.3 Finite Element Implementation
499(9)
11.3.1 Hierarchical bases
499(2)
11.3.2 Nodal variable collocation
501(5)
11.3.3 Discretization and linearization
506(1)
11.3.4 Boundary condition imposition
506(2)
11.4 Numerical Examples
508(13)
11.4.1 Cantilever beam
509(1)
11.4.2 Slit annular plate
510(3)
11.4.3 Pinched hemispherical shell
513(2)
11.4.4 Post buckling of shallow cylindrical shell
515(3)
11.4.5 Shell with irregular shape and material discontinuity
518(3)
11.5 Conclusions
521(2)
Appendix A.1. Hierarchical Bases of Quadrilateral Elements
523(2)
Appendix A.2. The Interpolation Points
525(2)
Chapter 12 Incorporation of the Hierarchical Quadrature Element Method with Isogeometric Analysis 527(54)
12.1 B-Splines and NURBS
528(6)
12.2 Non-uniform Rational Lagrange Functions
534(10)
12.3 Isogeometry Analysis of Rods
544(6)
12.4 Isogeometry Analysis of In-Plane Vibrations and Static Deformation by NURL
550(11)
12.4.1 Differential and integration rules
550(2)
12.4.2 In-plane vibration and static deformation of plates by NURL
552(7)
12.4.3 Vibration of membranes by the NURL
559(2)
12.5 Surface Intersection Algorithms
561(9)
12.5.1 Nonlinear polynomial solvers
562(3)
12.5.2 Surface/surface intersections
565(5)
12.6 Mesh Generation and Optimization
570(4)
12.7 Geometric Mapping of Triangular Patch
574(2)
12.8 High-Order Mesh Generation through Gmsh and Open CASCADE
576(3)
12.9 Conclusions
579(2)
References 581(34)
Index 615
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