A Differential Quadrature Hierarchical Finite Element Method

✍ Scribed by Bo Liu, Cuiyun Liu, Yang Wu, Yufeng Xing

• Publisher: World Scientific Publishing Company
• Year: 2021
• Tongue: English
• Leaves: 651
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents
Preface
About the Authors
Chapter 1. An Overview of High-Order Methods for Structural Mechanics
1.1 The Differential Quadrature Method
1.1.1 The quadrature rules
1.1.2 Weighting coefficients and sampling points
1.1.3 The method of imposing initial/boundary conditions
1.1.4 Strong formulation finite element method
1.1.5 A few comments on the DQM
1.2 Weak Form Quadrature Element Method
1.2.1 Weak form quadrature rules
1.2.2 A few comments on the weak form QEM
1.3 The Hierarchical Finite Element Method
1.3.1 The HFEM on 1D and its tensor product domains
1.3.2 The HQEM on quadrilateral and hexahedral domains
1.3.3 The HFEM on simplex domains
1.3.4 The DQM and HQEM on simplex domains
1.3.5 The HFEM on pyramid domains
1.3.6 The HQEM on pyramid domains
1.4 The p-version Finite Element Method for Nonlinear Problems
1.5 The p-version Finite Element Method for Plates and Shells
1.6 A Review of High-Order Mesh Generation
1.7 Outlook
Chapter 2. A Differential Quadrature Finite Element Method
2.1 The Reformulated Differential Quadrature Rule
2.2 Gauss–Lobatto Quadrature Rule
2.3 The Differential Quadrature Finite Element Method for Kirchhoff Thin Plates
2.3.1 The quadrature element method (QEM)
2.3.2 The differential quadrature finite element method for Kirchhoff thin plates
2.4 The Differential Quadrature Finite Element Method for Elasticity
2.4.1 Rod element
2.4.2 Euler beam element
2.4.3 Plane element
2.4.4 Mindlin plate element
2.4.5 3D element
2.4.6 Thickness-shear vibration analysis of rectangular quartz plates
2.5 Numerical Comparisons
2.5.1 Vibration and bending of plates and 3D solids
2.5.2 Thickness-shear vibration analysis of rectangular quartz plates
2.6 Conclusions
Chapter 3. The Differential Quadrature Hierarchical Finite Element Method for Mindlin Plates
3.1 The Reformulated Differential Quadrature Rule
3.2 The DQHFEM on Quadrilateral Domain
3.2.1 The method of geometric mapping
3.2.2 The construction of shape functions
3.2.3 Gauss–Lobatto quadrature rule
3.3 The DQHFEM on Triangular Domain
3.3.1 The construction of DQHFEM basis on triangles
3.3.2 Geometric mapping of triangular elements
3.4 The DQHFEM for Euler–Bernoulli Beam
3.5 The DQHFEM for Mindlin Plate
3.6 The DQHFEM for Quartz Crystal Plate
3.7 Results and Discussion
3.7.1 The DQHFEM for isotropic Mindlin plates
3.7.2 The DQHFEM for crystal plates
3.8 Conclusions
Chapter 4. The Differential Quadrature Hierarchical Finite Element Method for Plane Problems
4.1 The Differential Quadrature Hierarchical Method
4.1.1 Rectangular elements
4.1.2 Triangular elements
4.1.3 The Fekete points and the DQ rules on triangles
4.1.4 The differential quadrature hierarchical rules
4.2 Applications of the DQHFEM to Plane Problems
4.3 Results for Free Vibrations
4.3.1 Rectangular elements
4.3.2 Triangular elements
4.4 Results for Static Analysis
4.4.1 Thick-walled cylinder under uniform boundary pressure
4.4.2 Stress concentration problem of circular hole in plate
4.4.3 The interaction of grains and grain boundaries of metals
4.4.4 The interface between nanoparticles and the matrix of nanoparticle composites
4.5 Conclusions
Chapter 5. The Hierarchical Quadrature Element Method for 3D Solids
5.1 Element Energy Functions
5.2 Shape Function Construction
5.2.1 Tetrahedral elements
5.2.2 Wedge elements
5.2.3 Hexahedral elements
5.2.4 Pyramid elements
5.3 Strain Matrix
5.4 Element Matrices
5.4.1 Isotropic solids
5.4.2 Thermo-mechanical cross-ply laminates
5.5 Results and Discussions
5.5.1 3D vibration analyses
5.5.2 Applications of the pyramid elements
5.5.3 The interaction of grains and grain boundaries of metals
5.5.4 Three-dimensional analysis of nanoparticulate polymer nanocomposites
5.5.5 Thermo-mechanical analysis of cross-ply laminated plates
5.6 Comments on Gauss Integration
5.7 Conclusions
Chapter 6. The Hierarchical Quadrature Element Method for Kirchhoff Plates
6.1 Hermite Blending Function Interpolation
6.1.1 Blending function interpolation on a unit square domain
6.1.2 Blending function interpolation on a unit triangular domain
6.2 Hierarchical Bases for Quadrilateral Elements
6.2.1 Vertex functions
6.2.2 Edge functions
6.2.3 Face functions
6.3 Hierarchical Bases for Triangular Elements
6.3.1 Vertex functions
6.3.2 Edge functions
6.3.3 Face functions
6.4 Numerical Implementation
6.4.1 Basis transformation
6.4.2 Node collocation
6.4.3 FEM discretization
6.5 Results and Discussion
6.5.1 Complete element order and computational efficiency
6.5.2 Analysis using conforming elements
6.5.2.1 Bending of a square plate
6.5.3 Plate with a singularity
6.5.4 Free vibration
6.5.5 Analysis using quasi-conforming elements
6.5.6 Bending of a square plate
6.5.7 Bending of a circular plate
6.5.8 Plate with an irregular cutout
6.6 Conclusions
Chapter 7. The Hierarchical Quadrature Element Method for Shells in Orthogonal Curvilinear Coordinate System
7.1 Element Energy Functions for Deep Shell Element
7.2 The Configuration of Double-Curved Sandwich Shell
7.3 Estimation of Material Properties
7.3.1 ROM model
7.3.2 Mori–Tanaka model
7.4 Solution of Temperature Field
7.5 Layerwise Theory of Functionally Graded Shells
7.5.1 Linear strain energy
7.5.2 Nonlinear strain energy
7.5.3 The kinetic energy
7.5.4 The governing equation
7.6 The Differential Quadrature Hierarchical Finite Element Method
7.6.1 Approximation of the displacement field
7.6.1.1 The boundary shape functions
7.6.1.2 The hierarchical shape functions
7.6.2 Integral scheme
7.6.3 Linear stiffness matrix
7.6.4 Geometric stiffness matrix
7.6.5 Mass matrix
7.6.6 Element assembly
7.6.7 The dynamic equation
7.7 Results and Discussion
7.7.1 Free vibration of functionally graded single-layer shell in non-thermal environment
7.7.2 Vibration of functionally graded sandwich shells in non-thermal environment
7.7.3 Free vibration of functionally graded single-layer shell in thermal environment
7.7.4 Free vibration of functionally graded sandwich shells in thermal environment
7.8 Conclusions
Chapter 8. The Hierarchical Quadrature Element Method for Isotropic and Composite Laminated General Shells
8.1 Geometry Representation
8.2 The Layerwise Shell Model
8.3 The Hierarchical Quadrature Elements
8.3.1 Modified high-order bases
8.3.2 FEM discretization
8.4 Numerical Examples
8.4.1 Analyses of plates
8.4.2 Analyses of shells
8.5 Conclusions
Chapter 9. Hierarchical Quadrature Element Method for Geometrically Nonlinear Problems
9.1 The Hierarchical Quadrature Element Method
9.1.1 Shape functions for quadrilateral elements
9.1.2 Shape functions for hexahedral elements
9.2 Measures of Stress and Strain
9.3 Geometrically Nonlinear Formulation of Hierarchical Quadrature Elements
9.3.1 Formulation for two- and three-dimensional elements
9.3.2 Formulation for shallow shell elements
9.3.3 Solution of the system of nonlinear equations
9.4 Numerical Tests
9.4.1 A cantilever beam in planar configuration
9.4.2 A cantilever beam in three-dimensional configuration
9.4.3 An extension spring in three-dimensional configuration
9.4.4 A cylindrical shallow shell in Mindlin formulation
9.5 Conclusions
Chapter 10. The Hierarchical Quadrature Element Method for Incremental Elasto-Plastic Analysis
10.1 Classical J2 Flow Theory with Nonlinear Isotropic Hardening
10.1.1 Classical three-dimensional elasto-plastic theory
10.1.2 Numerical algorithm for three dimensional elasto-plastic problems
10.1.3 Return-mapping algorithm for plane stress elasto-plastic problems
10.1.4 Numerical calculation process of elasto-plastic problems
10.2 The Hierarchical Quadrature Element Method
10.3 Numerical Examples and Discussions
10.3.1 A thick-walled tube under uniform internal pressure
10.3.2 Perforated square plate under plane stress condition
10.3.3 Thick perforated square plate
10.4 Conclusions
Chapter 11. Curved p-version C1 Finite Elements for the Finite Deformation Analysis of Isotropic and Composite Laminated Thin Shells
11.1 Thin Shell Model
11.1.1 Kinematics
11.1.2 Weak form
11.1.3 Constitutive equation
11.2 Mesh Generation
11.3 Finite Element Implementation
11.3.1 Hierarchical bases
11.3.2 Nodal variable collocation
11.3.3 Discretization and linearization
11.3.4 Boundary condition imposition
11.4 Numerical Examples
11.4.1 Cantilever beam
11.4.2 Slit annular plate
11.4.3 Pinched hemispherical shell
11.4.4 Post buckling of shallow cylindrical shell
11.4.5 Shell with irregular shape and material discontinuity
11.5 Conclusions
Appendix A.1 Hierarchical Bases of Quadrilateral Elements
Appendix A.2 The Interpolation Points
Chapter 12. Incorporation of the Hierarchical Quadrature Element Method with Isogeometric Analysis
12.1 B-Splines and NURBS
12.2 Non-uniform Rational Lagrange Functions
12.3 Isogeometry Analysis of Rods
12.4 Isogeometry Analysis of In-Plane Vibrations and Static Deformation by NURL
12.4.1 Differential and integration rules
12.4.2 In-plane vibration and static deformation of plates by NURL
12.4.3 Vibration of membranes by the NURL
12.5 Surface Intersection Algorithms
12.5.1 Nonlinear polynomial solvers
12.5.2 Surface/surface intersections
12.6 Mesh Generation and Optimization
12.7 Geometric Mapping of Triangular Patch
12.8 High-Order Mesh Generation through Gmsh and Open CASCADE
12.9 Conclusions
References
Index