By HAILONG CHEN

1

MIXED FORMULATION USING IMPLICIT BOUNDARY FINITE ELEMENT METHOD

By

HAILONG CHEN

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2012

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© 2012 Hailong Chen

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To my parents and wife

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ACKNOWLEDGMENTS

I would like to express my gratitude to my advisor and chairman of my supervisory

committee, Prof. Ashok V. Kumar, for his guidance, enthusiasm and constant support

throughout my master’s research. I would like to thank him for the numerous insights he

provided during every stage of the research. Without his assistance it would not have

been possible to complete this thesis.

I would like to thank the members of my advisory committee, Prof. Loc Vu-Quoc

and Prof. Bhavani V. Sankar. I’m grateful for their willingness to serve on my committee,

for providing help whenever required, for reviewing this thesis and valuable suggestions

provided.

I would like to thank my undergraduate mentor, Na Li, for her numerous

encouragement and support during my undergraduate study in China and graduate

study in University of Florida, US.

I would like to thank my wife and parents, for their constant love and support

without which this would not have been possible.

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TABLE OF CONTENTS page

ACKNOWLEDGMENTS .................................................................................................. 4

LIST OF TABLES ............................................................................................................ 8

LIST OF FIGURES ........................................................................................................ 10

LIST OF ABBREVIATIONS ........................................................................................... 14

ABSTRACT ................................................................................................................... 15

CHAPTER

1 INTRODUCTION .................................................................................................... 17

1.1 Overview ........................................................................................................... 17

1.2 Goals and Objectives ........................................................................................ 19 1.3 Outlines ............................................................................................................. 19

2 MESHLESS AND MESH INDEPENDENT METHOD ............................................. 21

2.1 Traditional FEM ................................................................................................ 21

2.2 Meshless and Mesh Independent Method ........................................................ 22

2.3 Implicit Boundary Finite Element Method .......................................................... 23

3 MIXED FORMULATION FOR NEARLY INCOMPRESSIBLE MEDIA .................... 26

3.1 Overview ........................................................................................................... 26 3.2 Mixed Formulation ............................................................................................ 27

3.2.1 Matrix Decomposition .............................................................................. 27

3.2.2 Weak Form .............................................................................................. 29 3.3 2D Plane Strain ................................................................................................. 33

3.4 3D Stress .......................................................................................................... 35 3.5 Numerical Examples and Results ..................................................................... 38

3.5.1 Bracket (Plane strain) .............................................................................. 38 3.5.2 Beam (3D stress)..................................................................................... 39

3.6 Concluding Remarks ......................................................................................... 39

4 CLASSICAL PLATE THEORIES ............................................................................ 43

4.1 Overview ........................................................................................................... 43

4.2 Classical (Kirchhoff) Plate Theory (CPT) .......................................................... 44 4.2.1 Assumptions ............................................................................................ 44 4.2.2 Strain-displacement Relationship ............................................................ 45 4.2.3 Governing Equations ............................................................................... 46

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4.3 Mindlin Plate Theory (First-order Shear Deformation Theory) (FSDT) .............. 46 4.3.1 Assumptions ............................................................................................ 46 4.3.2 Strain-displacement Relationship ............................................................ 47

4.3.3 Governing Equations ............................................................................... 48 4.3.4 Constitutive Relationship ......................................................................... 51

4.4 Analytical and Exact Solution ............................................................................ 52 4.4.1 Cantilever Plate ....................................................................................... 53 4.4.2 Square Plate ............................................................................................ 54

4.4.3 Circular Plate ........................................................................................... 58 4.4.4 30-degree Skew Plate ............................................................................. 60 4.4.5 60-degree Skew Plate ............................................................................. 61

5 MIXED FORMULATION FOR MINDLIN PLATE ..................................................... 66

5.1 Mixed Form ....................................................................................................... 66 5.2 Discrete Collocation Constraints Method .......................................................... 69

5.3 Applying EBC Using Implicit Boundary Method ................................................ 73 5.4 Numerical Results ............................................................................................. 78

5.4.1 Cantilever Plate ....................................................................................... 78 5.4.2 Square Plate ............................................................................................ 79 5.4.3 Circular Plate ........................................................................................... 79

5.4.4 30-degree Skew Plate ............................................................................. 80 5.4.5 60-degree Skew Plate ............................................................................. 80

5.4.6 Flange Plate ............................................................................................ 81

5.5 Concluding Remarks ......................................................................................... 81

6 MIXED FORMULATION FOR 2D MINDLIN SHELL ............................................. 101

6.1 Governing Equations ...................................................................................... 101 6.2 Mixed Formulation .......................................................................................... 103

6.3 Numerical Examples and Results ................................................................... 106 6.3.1 60-degree Skew Plate ........................................................................... 106

6.3.2 Square Plate .......................................................................................... 106 6.4 Concluding Remarks ....................................................................................... 107

7 CONCLUSION ...................................................................................................... 111

7.1 Summary ........................................................................................................ 111 7.2 Scope of Future Work ..................................................................................... 112

APPENDIX

A VOLUMETRIC LOCKING AND SHEAR LOCKING .............................................. 114

A.1 Volumetric locking .......................................................................................... 114 A.2 Shear locking .................................................................................................. 114

B EQUILIBRIUM EQUATIONS OF 3D ELASTOSTATIC CASE .............................. 116

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C DERIVATION OF SHEAR CORRECTION FACTOR ............................................ 118

D DERIVATION OF THE JACOBIAN MATRIX IN IBFEM ........................................ 121

E FORMULATION OF MINDLIN PLATE ELEMENTS ............................................. 123

E.1 Element Q4D4 ................................................................................................ 123 E.2 Element Q5D6 ................................................................................................ 126 E.3 Element Q8D8 ................................................................................................ 128 E.4 Element Q9D12 .............................................................................................. 130 E.5 Element Q16D24 ............................................................................................ 132

LIST OF REFERENCES ............................................................................................. 136

BIOGRAPHICAL SKETCH .......................................................................................... 138

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LIST OF TABLES

Table page 5-1 Location of three interpolation variables and the associated count conditions

for patch test ....................................................................................................... 82

5-2 Cantilever plate (Shear force applied at free end) .............................................. 83

5-3 Cantilever plate (Bending moment applied at free end) ...................................... 83

5-4 Uniformly loaded, clamped square plate [a/t = 10] ............................................. 83

5-5 Uniformly loaded, clamped square plate [a/t = 100] ........................................... 83

5-6 Uniformly loaded, simply-supported square plate [a/t = 10] ................................ 84

5-7 Uniformly loaded, simply-supported square plate [a/t = 100] .............................. 84

5-8 Uniformly loaded, clamped circular plate [a/t = 10] ............................................. 84

5-9 Uniformly loaded, clamped circular plate [a/t = 100] ........................................... 84

5-10 Uniformly loaded, simply-supported circular plate [a/t = 10] ............................... 85

5-11 Uniformly loaded, simply-supported circular plate [a/t = 100] ............................. 85

5-12 Uniformly loaded, clamped 30-degree skew plate [a/t = 10] ............................... 85

5-13 Uniformly loaded, clamped 30-degree skew plate [a/t = 100] ............................. 85

5-14 Uniformly loaded, simply-supported 30-degree skew plate [a/t = 10] ................. 86

5-15 Uniformly loaded, simply-supported 30-degree skew plate [a/t = 100] ............... 86

5-16 Uniformly loaded, clamped 60-degree skew plate [a/t = 10] ............................... 86

5-17 Uniformly loaded, clamped 60-degree skew plate [a/t = 100] ............................. 86

5-18 Uniformly loaded, simply-supported 60-degree skew plate [a/t = 10] ................. 87

5-19 Uniformly loaded, simply-supported 60-degree skew plate [a/t = 100] ............... 87

5-20 Uniformly loaded, arbitrary shape plate .............................................................. 87

6-1 Transverse deflection of 60-degree skew plate with one edge clamped .......... 107

6-2 In plane displacement of 60-degree skew plate with one edge clamped .......... 107

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6-3 Transverse deflection of square plate with one edge clamped ......................... 107

6-4 In plane displacement of square plate with one edge clamped ........................ 108

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LIST OF FIGURES

Figure page 2-1 Conforming mesh in traditional FEM .................................................................. 24

2-2 Scattered nodes in meshless methods ............................................................... 24

2-3 Nonconforming structured mesh in IBFEM ......................................................... 24

2-4 Step function configuration in IBFEM ................................................................. 25

3-1 Geometry of the 2D bracket................................................................................ 40

3-2 Transverse displacement distribution after deformation (Q9M 130x80 mesh density) ............................................................................................................... 40

3-3 Maximum transverse displacement w.r.t Poisson’s ratio (Q4M) ......................... 40

3-4 Maximum transverse displacement w.r.t Poisson’s ratio (Q9M) ......................... 41

3-5 Geometry of 3D beam ........................................................................................ 41

3-6 Transverse displacement distribution after deformation (Hexa8M 65x10x10 mesh density) ..................................................................................................... 41

3-7 Maximum transverse displacement w.r.t Poisson’s ratio (Hexa8M) .................... 42

3-8 Maximum transverse displacement w.r.t Poisson’s ratio (Hexa27M) .................. 42

4-1 Configuration for CPT ......................................................................................... 62

4-2 Configuration for FSDT ....................................................................................... 63

4-3 Definitions of variables for plate approximations ................................................ 63

4-4 Geometry of cantilever plate ............................................................................... 64

4-5 Geometry of square plate ................................................................................... 64

4-6 Geometry of circular plate .................................................................................. 64

4-7 Geometry of 30-degree skew plate ..................................................................... 65

4-8 Geometry of 60-degree skew plate ..................................................................... 65

5-1 A typical background mesh using 20x2 Q4 element ........................................... 87

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5-2 Distribution of transverse displacement after deformation for 100x10 Q4 element (L1/t = 100) ........................................................................................... 88

5-3 Convergence of total strain energy for cantilever when shear applied (L1/t = 10) ...................................................................................................................... 88

5-4 Convergence of total strain energy for cantilever when shear applied (L1/t = 100) .................................................................................................................... 88

5-5 Distribution of transverse displacement after deformation for 100x10 Q4 element (L1/t = 100) ........................................................................................... 89

5-6 Convergence of total strain energy for cantilever when bending moment applied (L1/t = 10) .............................................................................................. 89

5-7 Convergence of total strain energy for cantilever when bending moment applied (L1/t = 100) ............................................................................................ 89

5-8 A typical background mesh using 10x10 Q9 element ......................................... 90


5-10 Distribution of transverse displacement after deformation for 150x150 Q9 element (t = 0.1) ................................................................................................. 90


5-12 Convergence of total strain energy for clamped square plate (a/t = 10) ............. 91

5-13 Convergence of total strain energy for clamped square plate (a/t = 100) ........... 91

5-14 Convergence of total strain energy for simply-supported square plate (a/t = 10) ...................................................................................................................... 92

5-15 Convergence of total strain energy for simply-supported square plate (a/t = 100) .................................................................................................................... 92


5-17 Distribution of transverse displacement after deformation for 150x150 Q4 element (t = 1) .................................................................................................... 93

5-18 Convergence of total strain energy for clamped circular plate (a/t = 10) ............ 93

5-19 Convergence of total strain energy for clamped circular plate (a/t = 100) .......... 93

5-20 Convergence of total strain energy for simply-supported circular plate (a/t = 10) ...................................................................................................................... 94

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5-21 Convergence of total strain energy for simply-supported circular plate (a/t = 100) .................................................................................................................... 94



5-24 Distribution of bending moment using 200x200 Q4 element (t = 0.1) ................. 95

5-25 Convergence of total strain energy for clamped 30-degree skew plate (a/t = 10) ...................................................................................................................... 95

5-26 Convergence of total strain energy for clamped 30-degree skew plate (a/t = 100) .................................................................................................................... 96

5-27 Convergence of total strain energy for simply-supported 30-degree skew plate (a/t = 10) .................................................................................................... 96

5-28 Convergence of total strain energy for simply-supported 30-degree skew plate (a/t = 100) .................................................................................................. 96

5-29 A typical background mesh using 10x10 Q16 element ....................................... 97

5-30 Distribution of transverse displacement after deformation for 200x200 Q9 element (t = 1) .................................................................................................... 97

5-31 Distribution of bending moment using 200x200 Q4 element (t = 0.1) ................. 97

5-32 Convergence of total strain energy for clamped 60-degree skew plate (a/t = 10) ...................................................................................................................... 98

5-33 Convergence of total strain energy for clamped 60-degree skew plate (a/t = 100) .................................................................................................................... 98

5-34 Convergence of total strain energy for simply-supported 60-degree skew plate (a/t = 10) .................................................................................................... 98

5-35 Convergence of total strain energy for simply-supported 60-degree skew plate (a/t = 100) .................................................................................................. 99

5-36 Geometry of flange plate .................................................................................... 99


5-38 Distribution of transverse displacement after deformation for 150x150 Q9 element (t = 0.1) ............................................................................................... 100

5-39 Convergence of total strain energy for arbitrary shape plate (a/t = 10) ............. 100

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5-40 Convergence of total strain energy for arbitrary shape plate (a/t = 100) ........... 100

6-1 Convergence of total strain energy for 60-degree skew plate (a/t = 10) ........... 108

6-2 Convergence of total strain energy for 60-degree skew plate (a/t = 100) ......... 108

6-3 Distribution of transverse deflection after deformation for 60-degree skew plate using 150x150 Q4 element (t = 1)............................................................ 109

6-4 Geometry of the 60-degree skew plate ............................................................. 109

6-5 Convergence of total strain energy for square plate (a/t = 10) .......................... 109

6-6 Convergence of total strain energy for square plate (a/t = 100) ........................ 110

6-7 Distribution of transverse deflection after deformation for square plate using 100x100 Q9 element (t = 0.1) ........................................................................... 110

6-8 The geometry of the square plate ..................................................................... 110

B-1 Stresses notations and directions ..................................................................... 116

C-1 Distribution of transverse shear stress through the thickness........................... 119

D-1 Global coordinate and Local coordinate in IBFEM ............................................ 122

E-1 Collocation constraints on a 4-node Lagrange element .................................... 123

E-2 Interpolation nodes for Q4D4 element .............................................................. 124

E-3 Collocation constraints on a 5-node Serendipity element ................................. 126


E-5 Collocation constraints on an 8-node Serendipity element ............................... 128


E-7 Collocation constraints on a 9-node Lagrange element .................................... 130

E-8 Interpolation nodes for Q9D12 element ............................................................ 131

E-9 Collocation constraints on a 16-node Lagrange element .................................. 132

E-10 Interpolation nodes for Q16D24 element .......................................................... 135

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LIST OF ABBREVIATIONS

C3D8H 8-node hybrid hexahedral element, constant pressure

C3D20H 20-node hybrid hexahedral element, linear pressure

CPE4H 4-node hybrid plane strain quadrilateral element, constant pressure

CPE8H 8-node hybrid plane strain quadrilateral element, linear pressure

CPT Classical Plate Theory

EBC Essential Boundary Condition

FSDT First-Order Shear Deformable Theory

H27 27-node hexahedral element

H27M 27-node mixed hexahedral element

H8 8-node hexahedral element

H8M 8-node mixed hexahedral element

IBFEM Implicit Boundary Finite Element Method

LHS Left Hand Side

ODE Ordinary Differential Equation

Q4 4-node quadrilateral element

Q4M 4-node mixed quadrilateral element



Q9M 9-node mixed quadrilateral element


RHS Right Hand Side

S4R 4-node doubly curved thin or thick shell element with reduced integration, hourglass control and finite membrane strains

S8R5 8-node doubly curved thin shell element with reduced integration using 5 degrees of freedom per node

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science

MIXED FORMULATION USING IMPLICIT BOUNDARY FINITE ELEMENT METHOD

By

Hailong Chen

May 2012

Chair: Ashok V. Kumar Major: Mechanical Engineering

Mixed formulation analysis was proposed for the purpose of avoiding locking

phenomena that occurs in displacement-based finite element analysis. In displacement

based analysis, volumetric locking will inevitably happen when the material is almost

incompressible and the Poisson’s ratio is near 0.5, which results in an infinite bulk

modulus, Shear locking occurs in Mindlin plate formulation when the plate is very thin

but the shear strains in plate do not go to zero due to the limitation of the interpolation

functions. Aside from mixed formulation, several other techniques have also been

proposed in last three decades, such as reduced integration or selective reduced

integration method, assumed natural strain method.

Implicit Boundary Finite Element Method (IBFEM) is a mesh independent finite

element method, which is motivated by the desire to avoid mesh generation difficulties

in the traditional finite element method (FEM). Instead of generating a conforming mesh,

a background mesh that does not represent the geometry is constructed for

interpolating or approximating the trail and test functions. The geometry of the model is

exactly represented using equations obtained from CAD software. The essential

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boundary condition is imposed by using implicit boundary method, which uses

equations of the boundary and does not need to have nodes on the boundary.

IBFEM has been demonstrated for 2D and 3D displacement-based structural

analysis. In this thesis, the main goal is to extend this approach to structural analysis

using mixed formulation, to eliminate volumetric locking and shear locking. A three-field

mixed formulation for incompressible media analysis and a two-field mixed formulation

for Mindlin plates are used in this thesis. A Mindlin 2D shell, is also discussed can

model in-plane strains as well as bending and shear. Several benchmark problems are

utilized to evaluate the validity of this approach.

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CHAPTER 1 INTRODUCTION

1.1 Overview

The Finite Element Method (FEM) is a widely used numerical method solving

problems arising in the engineering analysis. Mesh generation is the first necessary

step in traditional FEM and mesh generation algorithms have been developed that work

acceptably for 2D problems but are still unreliable for complicated 3D geometries, often

leads to poor or distorted elements in some regions. Mesh generation is therefore often

the most challenging process in the analysis. In simulation of failure processes, due to

the propagation of cracks with arbitrary and complex paths, mesh regeneration is

needed in each step in traditional FEM. It becomes even more challenge because of the

discontinuity and complicated growing path of the cracks. Due to aforementioned

disadvantages in traditional FEM, there are challenges in its application in other fields

as well, such as manufacturing processes and fluid mechanics.

In order to better overcome these disadvantages of traditional FEM, a number of

meshless or mesh free analysis techniques have been proposed in last three decades.

Meshless methods use a scattered set of nodes for the analysis but the nodes are not

connected to form elements (Figure 2-2). Based on the method used to construct a

meshless approximation for the trial and test functions various meshless methods exist.

One of the popular meshless approximation schemes is based on moving least squares

method. Some other approaches are also used, such as kernels method and partition of

unity method, etc. [1]. Most methods used to represent trial functions for the meshless

approach do not have Kronecker delta properties, which results in difficulty to apply

boundary conditions precisely along the boundary.

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An alternative approach to reduce mesh generation difficulties is to use

nonconforming mesh, often a structured background mesh, to interpolate or

approximate functions in the analysis domain. This approach was first proposed by

Kantorovich and Krylov [10]. A typical solution structure for applying essential boundary

conditions is 0( , ) ( , ) ( , )u x y f x y U x y u , where ( , ) 0f x y is the implicit equation of the

boundary and 0u is the prescribed essential boundary condition. ,U x y is the unknown

function that is interpolated piecewise over a mesh. Several approaches were used to

construct the implicit equation. Rvachev and Shieko [15] have developed an R- function

to construct a single implicit equation ( , )f x y . All boundary conditions including

essential, natural, and convection boundary conditions are guaranteed in the solution

structures.

Belytschko et al. [4] has proposed extended finite element method (X-FEM) based

on a structured mesh and implicit boundary representation to remove mesh generation

process. In X-FEM, approximate implicit function of the model was constructed by fitting

a set of sample points on the boundary. Radial basis function was used for the implicit

equation construction. Clark and Anderson [5] have used the penalty method to satisfy

the prescribed EBC.

Another mesh independent method, the Implicit Boundary Finite Element Method

(IBFEM) [11]-[13], also utilizes a structured background mesh to interpolate or

approximate the trial and test function. The geometry of the model is exactly

represented by the equations as exported from CAD system. A solution structure,

similar to the one developed by Kantorovich and Krylov [10], is constructed to guarantee

the EBC. This approach has been tested to be valid for 2D and 3D structural

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displacement-based analysis. In this thesis, we extend this approach to three-field

mixed formulation for nearly incompressible media and two-field mixed formulation for

Mindlin plate theory, both pure bending and combination cases.

1.2 Goals and Objectives

The goal of this thesis is to implement mixed formulation using Implicit Boundary

Finite Element Method, so as to avoid volumetric locking for nearly incompressible

media and shear locking in thin Mindlin plates.

Volumetric locking and shear locking are the most common numerical phenomena

that occur in the displacement-based finite element analysis. For last three decades,

various finite element techniques have been proposed to take care of these problems,

such as reduced integration or selective reduced integration method, mixed/hybrid

method, assumed natural strain method, enhanced assumed strain method, etc. [8],

[20]. In this thesis, we will employ the mixed formulation to remove these locking

phenomena using Implicit Boundary Finite Element Method.

The main objectives of this thesis are

1. Extension of IBFEM to three-field mixed formulation for near incompressible media;

2. Extension of IBFEM to mixed formulation for Mindlin plates, pure bending;

3. Extension of IBFEM to 2D Mindlin shells, including both bending and in-plane stretching.

1.3 Outlines

The rest of this thesis is organized as follows:

In Chapter 1, a brief overview about FEM is presented, and the goals and

objectives of this thesis are clearly stated. In Chapter 2, we give some details and

reference to the meshless and mesh independent finite element methods. Two

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challenges while using Implicit Boundary Finite Element Method are described and

scheme to solve these issues is presented. In Chapter 3, a three-field mixed formulation

for plane strain and 3D stress using IBFEM so as to remove the volumetric locking for

near incompressible media is derived in details. Two examples are used to test the

performance of this method. In Chapter 4, before proceeding to mixed formulation for

Mindlin plates, we review the two classical plate theories, Kirchhoff-Love plate theory

and Mindlin plate theory. The analytical or exact solution of several benchmark

problems, which used later to evaluate the performance of IBFEM, are given in Chapter

4 too. The two-field mixed formulation using Mindlin plate theory is detailed in Chapter

5. The focus is on the derivation of mixed form and imposing EBC using IBFEM. An

extension of Mindlin plate including both in plane stretching and bending is presented in

Chapter 6. Two examples have been used in Chapter 5, the same geometry but

different boundary conditions, are employed two test the performance of IBFEM for 2D

Mindlin shell case. The conclusion and future work is in Chapter 7. An appendix is also

included in order to give more details about some specific topics and also aimed to

make this thesis more self-contained.

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CHAPTER 2 MESHLESS AND MESH INDEPENDENT METHOD

2.1 Traditional FEM

Finite Element Method ([1], [9], [21]) is a well established numerical technique and

is widely used in solving engineering problems such as stress analysis, heat transfer,

fluid flow and electromagnetics in academia as well as in industry.

According to Fish and Belytschko [6], the traditional Finite Element Method

consists of five procedures:

1. Preprocessing: subdividing the problem domain into finite elements and approximating the domain by these finite elements - mesh generation;

2. Element formulation: derivation of equations in the element level - discretization;

3. Element combination: obtaining the equation system for the approximated model from the equations of individual elements - assembly;

4. Solving the equations: using Gauss elimination, Cholesky decomposition or iterative schemes like Gauss-Siedel to solve the equation system;

5. Postprocessing: determining quantities of interest, such as displacement and force resultant, and visualizing the results for future evaluation - analysis.

Among above five procedures, mesh generation is the most challenge one, and

still most endeavors is spend on devising an effective automatic mesh generator for 3D

complex geometry, for which the generated mesh is unreliable nowadays. For finite

element analysis, the domain of interest is subdivided into small elements by mesh

generation techniques and the resulting element mesh approximates the geometry. A

typical mesh is shown in Figure 2-1. The mesh is also used to approximate the solution

by piece-wise interpolation within each element. In procedure 2, Galerkin’s approach is

employed to convert the strong from into weak form, namely, alleviate the interpolation

shape function degree requirement.

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In order to avoid the problems associated with mesh generation, several

approaches have been proposed which falls into two categories:

1. Meshless methods;

2. Mesh independent methods.

Implicit Boundary Finite Element (IBFEM), [11]-[13], falls into the mesh

independent method category, which utilizes a structured background mesh only for the

purpose of interpolation. We will present some details about meshless and mesh

independent method in following two sections.

2.2 Meshless and Mesh Independent Method

The objective of meshless methods is to eliminate at least part of mesh generation

by constructing the approximation entirely in terms of nodes. A set of scattered nodes is

used to construct the trial and test function, see Figure 2-2. Several schemes are

developed to approximate these functions, such as moving least square method, kernel

method and partition of unity method, etc. [3].

Mesh independent analysis is motivated by the desire to utilize accurate geometric

models presented by equations rather approximated by mesh while using a background

mesh solely for the purpose of piecewise approximation or interpolation of the trial and

test function. A solution structure is proposed by Kantorovich and Krylov [10],

0( , ) ( , ) ( , )u x y f x y U x y u , where ( , ) 0f x y is the implicit equation of the boundary and is

the essential boundary condition. ,U x y is the unknown function that interpolated

piecewise over a mesh. Rvachev and Shieko [15] have developed an R- function to

construct a single implicit equation to represent the entire boundary of a solid.

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2.3 Implicit Boundary Finite Element Method

Comparing to traditional finite element method, Implicit Boundary Finite Element

Method (IBFEM) is a mesh independent finite element method, in which the geometry of

the model is exactly presented by the equations as exported from CAD system.

Although there is no geometry approximation in IBFEM, a background mesh (Figure 2-

3) is still employed but solely for the purpose of approximation or interpolation of the trial

and test function.

Contrary to the R-function technique used in Rvachev and Shieko [15], an

approximate step function was used in IBFEM to construct the implicit equation of the

interested domain. The solution structure used in IBFEM is

0( ) ( ) ( )u H U u x x x (2-1)

where ( , )H x y is the step function. A typical approximate step function being used

in IBFEM is

0 0

(2 ), 0

1

iH

(2-2)

where i can be 1,2,3 , is the distance between point to the boundary lines in the

normal direction, and is the transition width, see Figure 2-4.

24

Figure 2-1. Conforming mesh in traditional FEM

Figure 2-2. Scattered nodes in meshless methods

Figure 2-3. Nonconforming structured mesh in IBFEM

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Figure 2-4. Step function configuration in IBFEM

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CHAPTER 3 MIXED FORMULATION FOR NEARLY INCOMPRESSIBLE MEDIA

3.1 Overview

It has been frequently noted that in certain constitutive laws, such as those of

viscoelasticity and associative plasticity, the material behaves in a nearly

incompressible manner. The incompressibility of these media, at certain critical stage,

e.g., metals at the range of plastic deformations (yielding), results in a locking

phenomenon, named volumetric locking or Poisson locking. This phenomenon results

from the fact that when the material is near incompressible status, the Poisson’s ratio, v ,

tends to 0.5, which makes the bulk modulus / (3 6 )E v tend towards infinity and

hence result in a ill-conditioned stiffness matrix in the finite element model.

Several approaches have been proposed in last three decades to reduce or

alleviate volumetric locking occurrence. Reduced or selective reduced integration

technique was the first successful irreducible form of solutions for volumetric locking

problems, although in the beginning not directed specially towards volumetric locking.

Later, other formulations succeeded in using augmented functional, when compared to

that obtained from displacement-based approaches, incorporating additional fields into

the formulation and leading to the onset of general mixed methods. In the 1990s, the

enhanced strain method was also applied to alleviate this locking phenomenon.

Generally, there are two choices of mixed formulation on additional fields,

displacement u and mean stress p , which named two-field formulation, and

displacement u , mean stress p and volume changev , termed three-field formulation

([1], [9], [21]). Which of these should be employed may depend on the form of the

constitutive equation used. For situations where changes in volume affect only the

27

pressure the two-field form can be easily used. However, for problems in which the

response may become coupled between the deviatoric and mean components of stress

and strain the three-field formulation leads to much simpler forms from which to develop

a finite element model. In this thesis, we will only focus on the later formulation, three-

field mixed formulation.

The layout of Chapter 3 is as follows:

In section 3.2, we derive the general mixed formulation, and the resultant

equations using three-field formulation;

In section 3.3 and 3.4, we specify our discussion to 2D plan strain and 3D stress

respectively. A further modification of the mixed forms for 4-node and 9-node Lagrange

elements in 2D case and 8-node and 27-node hexahedron elements for 3D case;

In section 3.5, we employ some benchmark problems to test the performance of

elements developed in IBFEM. The conclusion of Chapter 3 is made in section 3.6.

3.2 Mixed Formulation

As mentioned in the introduction of Chapter 3, a three-field mixed formulation is

adopted in this section to develop some valuable elements both for 2D plane strain and

3D stress using IBFEM. In this section, we will develop the general mixed form for both

cases.

3.2.1 Matrix Decomposition

Generally, in most cases, the strain and stress matrices can be split into the

deviator (isochoric) and mean parts. Since the separation slightly differs for 2D and 3D.

Here, we utilize 3D case for illustration. That for 2D will be presented in section 3.3.

Accordingly, the mean stress, pressure part, can be expressed as

28

11 22 33( )1 1

3 3 iip (3-1)

We use sum notation here. And 1,2,3i .

And the deviatoric part can be defined as

( )ij ij ijdp (3-2)

where ij is the Kronecker delta function, which when i j is one and else zero for

all , 1,2,3i j .

Similarly, the mean strain, volume change, can be defined as

11 22 33)(v ii (3-3)

and the deviatoric strain as

1( )

3 vij ij ijd (3-4)

Have above definitions for mean part and deviatoric part of stress and strain, the

strain and stress may now be expressed in a mixed form as

Strain 1

( )3 vd

ε I u m (3-5)

Stress dp σ I σ m (3-6)

where the mean matrix operator

(1 1 1 0 0 0)Tm (3-7)

and deviatoric matrix operator

29

2 / 3 1/ 3 1/ 3 0 0 0

1/ 3 2 / 3 1/ 3 0 0 0

1/ 3 1/ 3 2 / 3 0 0 01

0 0 0 1 0 03

0 0 0 0 1 0

0 0 0 0 0 1

T

d

I I mm (3-8)

the operator

0 0

0 0

0 0

0

0

0

x

y

z

z y

z x

y x

(3-9)

I is the identity matrix. σ is the set of stress obtained directly from the strain

rates, depending on the particular constitutive model form.

3.2.2 Weak Form

The governing equations of this topic are the same as those equilibrium equations

of static elastic problems, which only differs at which equation we should employ for 2D

plane strain and 3D stress scenarios. For detailed derivation of these governing

equations, one can refer to Appendix B.

We start our derivation of weak form using weighted Galerkin’s method. One can

use principle of minimization of potential energy to achieve the same purpose.

The strong form is

σ b 0 (3-10)

1

3

Tp m σ (3-11)

30

( )T

v m u (3-12)

go with natural and essential boundary conditions that

ij j in t on b (3-13)

ˆi iu u on u (3-14)

Using weighted Galerkin’s method and simplifying the results, we can have the

weak form as

( )t

T T Td d d

σ u b u tu (3-15)

1[ ] 03

T

v p d

m σ (3-16)

[ ( ) ] 0T

vp d

m u (3-17)

Introducing the finite element approximations of variables as

ˆˆ ˆ, ,u p v v vp u N u N p N ε (3-18)

And similar approximations to virtual quantities as

ˆˆ ˆ, ,u p v v vp u N u N p N ε (3-19)

Hence, the strain in an element becomes

1ˆ ˆ

3d v v ε I Bu mN ε (3-20)

In which B is the standard strain-displacement matrix. Similarly, the stresses in

each element may be computed by using

ˆd p σ I σ mN p (3-21)

31

Since strain εand stressσ are the element variables, we will keep it as constant

while constructing the weak form system.

Substitute Equations 3-18 and 3-19 into the weak form Equations 3-15 to 3-17, we

have

For Equation 3-15:

(

ˆ ˆ( ( ) ( )

)

ˆ )

t

t

t

T T T

T T T

u u u

T T T

u u

d d d

d d d

d d d

σ u b u t

N σ N u b N u t

B σ N b N t

u

u (3-22)

For Equation 3-16:

1ˆˆ [ ] 0

3

1ˆ( ) 0

3

T T

v v p

T T T

v v p

d

d d

N ε m σ N p

N m σ N N p

(3-23)

For Equation 3-17:

ˆ ˆ[ ( ) ] 0

ˆ( ) ( ) 0

ˆ

ˆ

T T

p u v v

T T T

p p v v

d

d d

N p m N N ε

N m B N N ε

u

u (3-24)

Writing Equations 3-22 to 3-24 in matrix form, the finite element equation system

becomes

ˆ ˆ

ˆ

T

vp v pu

p vp

P f

K ε K u 0

P K p 0 (3-25)

Where

T d

P B σ , 1

3

T T

p v d

P N m σ , T

vp v pd

K N N , T T

pu p d

K N m B

t

T T

u ud d

f N b N t

32

Above equation system cannot be solve in a global sense, since the stressσ in P is

not a global variable and not directly approximated. If the pressure and volumetric strain

approximations are taken locally in each element, it is possible to solve the above

second and third equation in each element individually. If we further make that v pN N in

each element, the array vpK is now symmetric positive definite. We can use the second

equation in 3-25 to solve for p and ˆvε in each element as

1ˆvp p

p K P (3-26)

1ˆ ˆ ˆv vp pu

ε K K u Wu (3-27)

The mixed strain in each element may now be computed as

1 1ˆ ˆ[ ]

3 3d v d

v

Bε BI mB u I m u

B (3-28)

Where v vB N W defines a mixed form of the volumetric strain-displacement

equation.

After solving the first two Equations of 3-25, the first equation can be write in an

alternative form as

1[ ]

3

1ˆ1

33

T

T T T

d v

dT T

v dTv

d

d

d

B σ f

B I B m Dε f

IB

B B D I m u fBm

(3-29)

where D is the general stress-strain matrix before using mixed formulation.

Equation 3-29 can be formatted in the same way as general displacement-based

formulation. For implementation purpose, we rewrite Equation 3-29 as

33

ˆT

M M M d

B D B u f

where

M

v

BB

B,

11

33

d

M dT

I

D D I mm

.

f is the same as defined in Equation 3-25.

We have detailed how to get the mixed weak form to get rid of volumetric locking

for both 2D plane strain and 3D stress. More attention should be focused on the solution

of the mixed equation system. Since an element variable, element stress σ , is involved

in the first equation of 3-25, it is impossible to globally solve this equation system. An

efficient approach is adopted in order to achieve solvability of the equation system, that

we solving the second and third equation of 3-25 in each element individually. The last

part of section 3.2.2 is focused on how to use this approach to solve these two

equations, and finally solve the whole system. Note should be made here that

alternatives is available to solve above mixed equation system, 3-25. Presented is just

one possible approach.

3.3 2D Plane Strain

In this section, we will provide details on the element variables needed for 2D

plane strain case.

For 2D Plain Strain, the B matrix can be expressed using the shape functions of

displacements as

34

1

1 1

1

11

0 00

0 00 0 0

0 0

x nx

x nx y ny

y ny

y nyx nx

N N

x xxN N N N

N Ny y y

N NN N

y x y x y x

B

The shape functions for displacement is the traditional Lagrange shape functions.

While, according to [1], [9] and [21], for 4-node Lagrange element the shape function

needed for volumetric strain and pressure are

1v p N N

And that for 9-node Lagrange element are

(1 )v p x y N N

The vpK matrix can be immediately obtained using above shape functions for mean

strain and stress.

(1 )T

vp v pd x y d

K N N

And the puK matrix can be simplified as

1

1

1 4

1 4

11

0

0 0[1 1 0] 0

0 0

0 0

0 0

x nxT T T

pu p p

y ny

x xT

p

y y

y nyT x nxp

xN N

d dN Ny

y x

N Nd

N Nx y

N NN Nd

x y x y

K N m B N

N

N

35

The m matrix only differs from that provided in Equation 3-7 in dimension.

(1 1 0)Tm

2 / 3 / 3 1/ 3 / 3 01

1/ 3 / 3 2 / 3 / 3 03

0 0 1

T

dp

v vv

v v

I I mm

where v is the Poisson’s ratio.

1 / 2 1 / 2 01

1 / 2 1 / 2 03

0 0 1

T

d

I I mm .

1

3

T T

p v

vd

P N m σ

The elastic stress-strain matrix is

1 01

(1 )1 0

(1 )(1 2 ) 1

1 20 0

2(1 )

v

v

E v v

v v v

v

v

D

The mixed stress-strain matrix is

1 1 10

2 2 6

1 1 110

1 2 2 621

11 12 10 0 03

23 6

10 0 0

3(1 2 )

dp dp d dp

M dTT T

d

Ev

v v v

v

v

I I DI I Dm

D D I mm

m DI m Dm

3.4 3D Stress

In this section, we continue our elaboration for 3D stress case.

36

For 3D stress, the B matrix can be expended using displacement shape functions

as

1

1

1

0 0

0 0

0 0 0 00 0

0 0 0 0

0 0 0 0 0

0

0

x nx

y ny

z nz

x

y

N Nz

N N

N Nz y

z x

y x

B

1

1

1

1 1

1 1

11

0 0 0 0

0 0 0 0

0 0 0 0

0 0

0 0

0 0

x nx

y ny

nzz

y ny nzz

x nx nzz

y nyx nx

N N

x x

N N

y y

NN

z z

N N NN

z y z y

N N NN

z x z x

N NN N

y x y x

The shape functions for displacement is the traditional Lagrange shape functions.

While, according to [1], [9] and [21], the shape functions for 8-node hexahedron element

of the mean strain and stress are

1v p N N

And for 27-node hexahedron element, the shape functions are

(1 )v p x y z xy yz xz N N

The vpK matrix can be obtained as

(1 )T

vp v pd x y z xy yz xz d

K N N

And the puK matrix is

37

1

1

1

1 1

1 1

11

0 0 0 0

0 0 0 0

0 0 0 0

[1 1 1 0 0 0]

0 0

0 0

0 0

x nx

y ny

nzz

T T T

pu p py ny nzz

x nx nzz

y nyx nx

N N

x x

N N

y y

NN

z zd d

N N NN

z y z y

N N NN

z x z x

N NN N

y x y x

K N m B N

N11 1y nyT x nx nzz

p

N NN N NNd

x y z x y z

The m matrix and dI matrix are already given in Equations 3-7 and 3-8 as

(1 1 1 0 0 0)Tm

2 / 3 1/ 3 1/ 3 0 0 0

1/ 3 2 / 3 1/ 3 0 0 0

1/ 3 1/ 3 2 / 3 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

d

I

The elastic stress-strain matrix for 3D stress is

1 0 0 01 1

1 0 0 01 1

1 0 0 01 1(1 )

1 2(1 )(1 2 ) 0 0 0 0 02(1 )

1 20 0 0 0 0

2(1 )

1 20 0 0 0 0

2(1 )

v v

v v

v v

v v

v v

v vE vvv vv

v

v

v

v

D

38

Hence, the mixed stress-strain matrix is

1

1 31

1 133

3 9

2 1 10 0 0 0

3 3 3

1 2 10 0 0 0

3 3 3

1 1 20 0 0 0

3 3 3

10 0 0 0 0 0

2(1 )

10 0 0 0 0 0

2

10 0 0 0 0 0

2

10 0 0 0 0 0

3(1 2 )

d d d d

M dTT T

d

E

v

v

v

I I DI I Dm

D D I mm

m DI m Dm

3.5 Numerical Examples and Results

In this section, two examples are used to test the performance of these elements

using IBFEM. The first example is a fixed bracket, which also has been used in [1]. 3D

clamped beam is utilized for 3D stress analysis. Also, we compare our results with

element CPE4H and CPE8H for plane strain, C3D8H and C3D20H for 3D stress, which

are all from ABAQUS package. Instead of tabulating those data, we give related plots to

help visualizing the volumetric locking and the difference between different elements.

3.5.1 Bracket (Plane strain)

The bracket is fixed at the two vertices, and a uniformly distributed load of value

6000 is applied at the top. The Young’s modulus is 5.5e10. The geometry of this bracket

is given in Figure 3-1. The distribution of transverse displacement after deformation

analyzed by Q9M is in Figure 3-2. The comparison between Q4 element and Q4M

39

element, Q9 element and Q9M element is plotted in Figure 3-3 and Figure 3-4,

respectively. From Figures 3-3 and 3-4, volumetric locking is very severe in Q4 element,

while with the increase of degree of shape function, using Q9 element, this

phenomenon can be better improved, but still cannot be avoided. See Figure 3-4.

3.5.2 Beam (3D stress)

A beam clamped at the left end is loaded with a distributed load of 6000 at the top

surface. The Young’s modulus is 5.5e10. The geometry of this beam is shown in Figure

3-5. The deformed shape of the beam with transverse displacement distribution is given

in Figure 3-6. The comparison between different elements also given in this example,

Figure 3-7 and Figure 3-8. From Figures 3-7 and 3-8, volumetric locking is also drastic

in H8 element, while with the increase of degree of shape function, using H27 element,

this phenomenon is better improved, but still cannot avoid. See Figure 3-8.

3.6 Concluding Remarks

A three-field mixed formulation scheme has been adopted in Chapter 3 to remove

or alleviate the volumetric locking in 2D plane strain and 3D stress. The detailed

derivation of such formulation is presented in section 3.1, 3.2 and 3.3. Two examples

are employed to test the validity of the developed elements using IBFEM. The

comparison between displacement based formulation and the three-field mixed

formulation is also presented in these two examples. Elements from Abaqus package

are also used in the comparison. From the results and plots given, it’s obviously that

these IBFEM elements can better remove or alleviate volumetric locking phenomenon in

both 2D plane strain and 3D stress, and performs the same, if not better than, as the

Abaqus elements.

40

Figure 3-1. Geometry of the 2D bracket

Figure 3-2. Transverse displacement distribution after deformation (Q9M 130x80 mesh

density)

Figure 3-3. Maximum transverse displacement w.r.t Poisson’s ratio (Q4M)

41

Figure 3-4. Maximum transverse displacement w.r.t Poisson’s ratio (Q9M)

Figure 3-5. Geometry of 3D beam

Figure 3-6. Transverse displacement distribution after deformation (Hexa8M 65x10x10

mesh density)

42

Figure 3-7. Maximum transverse displacement w.r.t Poisson’s ratio (Hexa8M)

Figure 3-8. Maximum transverse displacement w.r.t Poisson’s ratio (Hexa27M)

43

CHAPTER 4 CLASSICAL PLATE THEORIES

4.1 Overview

The plate element has attracted considerable amount of attentions due to its great

application in engineering field. Among numerous plate theories that have been

developed since the late nineteenth century, the first widely accepted and used in

engineering was proposed by Kirchhoff and developed by Love, named Kirchhoff-Love

(KL) plate theory, which is an extension of Euler-Bernoulli beam theory. In KL plate

theory, it’s assumed that the transverse normals remain straight and perpendicular to

the mid-plane of the plate before and after deformation, and the strain in the thickness

direction is negligible. In other word, the strains , ,xz yz zz are negligibly zero in KL plate

theory. These assumptions render KL plate theory only applicable for thin structures

and C1 continuity is required for shape functions. Mindlin and Reissner have relaxed

Kirchhoff assumption to include the shear flexibility in the plate theory, which called

Mindlin-Reissner (MR) plate theory or First-order shear deformation theory. In MR plate

theory, the normal not necessary perpendicular to the mid-plane and the transverse

shear strains should be taken into consideration. Inclusion of the shear flexibility in the

MR plate theory makes it comparatively suitable for moderately thick plates and only C0

continuity is required. However, MR plate elements exhibit a phenomenon termed shear

locking when the thickness of the plate tends to zero.

Over last three decades, extensive efforts have been spent to device an effective

finite element scheme to overcome shear locking phenomenon for MR plate element. A

lot of finite element techniques have been developed, including but not limited to:

Reduced Integration or Selective Reduced Integration method, Mixed/Hybrid method,

44

Assumed Natural Strain method, Enhanced Assumed Strain method, Discrete Shear

Gap method ([8], [20]). Recently, a large number of endeavors were focused on

developing new plate/shell elements using various new finite element techniques, such

as iso-geometric method, smoothed finite element method, discontinuous Galerkin finite

element method, Mesh free finite element method, implicit boundary finite element

method, using twist-Kirchhoff theory in finite element method, etc.

Implicit Boundary Finite Element Method (IBFEM) is a mesh independent Finite

Element Method. In IBFEM, the Dirichlet boundary conditions are imposed using implicit

boundary method where approximate step functions are used as weighting functions to

construct solution structures that enforce the boundary conditions. This method can

impose boundary conditions even when the boundary is not guaranteed to have nodes

on them or when the shape functions used for the analysis do not satisfy Kronecker-

delta property. Previous work by Kumar et al. [11]-[13] has illustrated its validity and

potential value by employing conventional Lagrange elements and B-Spline elements in

their analysis. In Chapter 5 we use the discrete shear collocation method and IBFEM to

develop a family of Mindlin plate elements, which can avoid shear locking in thin plates.

4.2 Classical (Kirchhoff) Plate Theory (CPT)

4.2.1 Assumptions

There are two assumptions for CPT:

1. Transverse normals remain straight and perpendicular to the mid-plane before and after deformation;

2. Strain in the transverse normal direction is negligible.

45

4.2.2 Strain-displacement Relationship

As depicted in Figure 4-1, taking the1 2x x plane of the Cartesian coordinate

system to coincide with the mid-plane of the plate, we can have the plate displacement

components represented as:

( , )( , , )

w x yu x y z z

x

(4-1)

( , )( , , )

w x yv x y z z

y

(4-2)

( , ) ( , )w x y w x y (4-3)

where ( , )w x y are regarded as the weighted average for the deflection, z is the

coordinate in the thickness direction.

Using above assumed displacements, the in-plane strains can be deduced by the

elasticity definition of strain components as following:

2

2

2

2

2

2

x

y

xy

wu

xx

v wz z

y y

u v w

x y x y

ε κ (4-4)

Usually, the matrixκ is termed as bending curvature matrix, which actually a

pseudo-curvature matrix.

Transverse shear strains, which actually are the appropriate averages, are

expressed in terms of deflection was

46

zx

yz

w

x

w

y

γ (4-5)

The zz is zero according to the assumption for Kirchhoff plate.

4.2.3 Governing Equations

Governing equations for CPT can be derived from equilibrium equations for static

elastic system. Since these equations are the same for both Kirchhoff plate theory and

Mindlin plate theory, we only provide the result here. Details on notation and derivation

will be elaborated in Mindlin plate section.

The governing equations of CPT are:

0xyx

xz

MMS

x y

(4-6)

0xy y

yz

M MS

x y

(4-7)

0yzxz

z

SSq

x y

(4-8)

So far, we have finished our brief discussion about Kirchhoff plate theory. From

now on, we will focus on the Mindlin plate theory and the development of valuable

Mindlin plate element using Implicit Boundary Method. Some content provided without

any derivation will be detailed in next section.

4.3 Mindlin Plate Theory (First-order Shear Deformation Theory) (FSDT)

4.3.1 Assumptions

All assumption made in Kirchhoff plate theory is applicable in Mindlin plate theory,

except one, that all plane sections normal to mid-plane remain plane, but no necessary

47

normal to the mid-plane after deformation. The configuration of Mindlin plate is shown in

Figure 4-2.

4.3.2 Strain-displacement Relationship

Assuming there is no in plane forces acting on the plate, the local displacement in

the directions of the x , y and z axes are presented as

( , , ) ( , )x xu x y z z x y (4-9)

( , , ) ( , )y yu x y z z x y (4-10)

( , ) ( , )zu x y w x y (4-11)

Where ( , )x x y , ( , )y x y are the rotation of the normals in the x and y directions

respectively. And this is the very distinction between CPT and FSDT.

Immediately the strains in the x , y and z axes are available as

In-plane strains:

0

0

)(

x

y

x x

xy y

y

xy

y yx x

z z

uz

x x xu

zy y y

uuz

y xy x y x

(4-12)

Transverse strains:

x zx

xz x

yyz yzy

w

u u w w

z x x xu w wu

y yz y

(4-13)

0z

w

z

(4-14)

48

This condition will be generally valid when we apply transverse load distributed

over a large area such that the normal stressz is negligible. However, in situations

where a very small area compared to the plate thickness is applied a transverse load,

surface indentations may be created and the assumption for transverse displacement

will not be valid. Low velocity impact due to external objects may incur such

phenomenon.

In above expression for strains, we have adopted a notation that

0

0

x

y

y x

4.3.3 Governing Equations

We will detail on the derivation of governing equations for both Kirchhoff plate and

Mindlin plate in this section.

The differential equilibrium equations of a static small deformation solid with body

force are:

0xyx xz

xbx y z

(4-15)

0xy y yz

ybx y z

(4-16)

0yzxz

zbx y

(4-17)

In above expansion, we omit the z term, since we have assumed at the very

beginning that this stress in negligible small. These three equations are not satisfied at

every point of z , instead we make them satisfied in an average sense through the

49

thickness of the plate. This is accomplished by requiring the equations and their

moments be satisfied on both side of the equation on an average sense. The notation

used is shown in Figure 4-3.

Integrating the Equation 4-15 in the thickness h direction, we have

2 2 2 2

2 2

2 2 2 2

( ) 0

h h h h

x xzx x xy xz h xz h x

h h h h

xyb dz dz dz b dz

x z x yy

0xyx

x

PPq

x y

(4-18)

Where2

2 2

2

h

x xz h xz h x

h

q b dz

,2

2

h

x x

h

P dz

, 2

2

h

xy xy

h

P dz

.

Multiply Equation 4-15 with z and integrate in the thickness h direction, we get

2 2 2 2 2

2 2

2 2 2 2 2

( ) 0

h h h h h

x xzx x xy xz xz h xz h x

h h h h h

xyz b dz z dz z dz dz z z zb dz

x z x yy

0xyx

xz x

MMS m

x y

(4-19)

Where2

2

h

x x

h

M z dz

, 2

2

h

xy xy

h

M z dz

and2

2 2

2

h

x xz h xz h x

h

m z z zb dz

.

Integrating the Equation 4-16 in the thickness h direction, we obtain

2 2 2 2

2 2

2 2 2 2

( ) 0

h h h h

xy y yz h yz h y

h h h h

xy y yz

y dz dz dz b dzx y

bx y z

0xy y

y

P Pq

x y

(4-20)

50

Where2

2 2

2

h

y yz h yz h y

h

q b dz

, 2

2

h

xy xy

h

P dz

,2

2

h

y y

h

P dz


2 2 2 2 2

2 2

2 2 2 2 2

( ) 0

h h h h h

xy y yz yz h yz h y

h h h h h

xy y yz

yz dz z dz z dz dz z z zb dzx y

bx y z

0xy y

yz y

M MS m

x y

(4-21)

Where 2

2

h

y y

h

M z dz

2

2

h

xy xy

h

M z dz

and 2

2 2

2

h

y yz h yz h y

h

m z z zb dz

Integrating the Equation 4-17 in the thickness h direction, we obtain

2 2 2 2

2 2 2 2

( ) 0

h h h h

xz yz z

h h h h

yzxzz dz dz dz b dz

x yb

x y

0yzxz

z

SSq

x y

(4-22)

Where2

2

h

z z

h

q b dz

, 2

2

h

xz xz

h

S dz

,2

2

h

yz yz

h

S dz


2 2 2 2

2 2 2 2

( ) 0

h h h h

xz yz z

h h h h

yzxzzz dz z dz z dz zb dz

x yb

x y

0yzxz

z

MMm

x y

(4-23)

51

Where2

2

h

xz xz

h

M z dz

, 2

2

h

yz yz

h

M z dz

and 2

2

h

z z

h

m zb dz

Omitting the in plane terms in Equations 4-18 to 4-23, we finally have the

governing equations as

0xyx

xz x

MMS m

x y

(4-24)

0xy yyz y

M MS m

x y

(4-25)

0yzxz

z

SSq

x y

(4-26)

4.3.4 Constitutive Relationship

In this section, we will derive the relationship between the plate resultants and the

displacement and rotations.

As in last section, the shell resultants have been defined as

x x

y xh

xy xy

M

M z dz

M

, xz xz

hyz yz

Sdz

S

Using the stress-strain relationship and the strain-displacement relation, we can

write the resultants in term of displacement and rotations as

2 2

2 2

h hx x x

y y y

h h

xy xy xy

M

M z dz z dz

M

M C D (4-27)

Where 2

1 0

1 0(1 )

0 0 (1 ) / 2

vE

vv

v

C , 3

12

hD C .

52

2 2 2

2 2 2

h h h

xz xz

yz yzh h h

xz

yz

dz dz kG w dz wS

S

S (4-28)

where kGh . Constant k is added here to account for the fact that shear stresses

aren’t constant across the section. A value of 5 / 6k is the exact for rectangular,

homogeneous section and corresponds to a parabolic shear stress distribution. Details

about how to derive this constant can be found in Appendix C.

In next section, the analytical solution using above governing equation for various

Kirchhoff plate and Mindlin plate is presented. The purpose of this work is to compare

the analytical result with numerical result using developed Mindlin plate elements.

Hence, evaluate the performance of those elements.

4.4 Analytical and Exact Solution

Before proceeding to mixed formulation of Mindlin plate problem, the analytical or

exact solutions for those benchmark problems are derived for the purpose of comparing

results and hence testing the validity of our numerical solution. Since this thesis is

mainly focus on the finite element analysis of various kinds of Mindlin plates, we provide

the analytical or exact solution without any further derivation. For more details, one can

refer to standard textbooks on plates and shells, such as Timoshenko et al. [18] and

Reddy et al. [19].

The benchmark problems will be studied here are: cantilever plate, clamped and

simply-supported square plate, clamped and simply-supported circular plate and

clamped and simply-supported 30 degree and 60 degree skew plates.

For simplicity, we utilize following non-dimensional constants for all testing

examples.

53

Young’s modulus E = 1.12e10; Poisson’s ratio v = 0.3;

Uniform load p = 100 or bending moment m = 100;

Notations for geometries are as following:

Lx: Length of edge x; a: Edge length; r: radius; t: thickness.

The thickness varies among 0.1 and 1, so as to test the validation of the element

for different length/thickness ratio.

4.4.1 Cantilever Plate

Cantilever plate is studied here for the purpose of testing the performance of new

elements when only shear force or bending moment is applied. The cantilever in Figure

4-4 is clamped at the left edge and constant shear force or bending moment, both with

value of 100, is applied separately at the free end.

Due to the loading condition and the length to width ratio, Cantilever plate can be

viewed as 1-D plate undertaking cylindrical bending. In these cases, when shear force

or bending moment applied separately, the transversal deflection is derived here using

1-D assumption. And the results from using Timoshenko beam theory are also provided

here without derivation. For more details, one can refer to Timoshenko et al. [18].

Shear force applied

Using plate theory:

The deflection of the cantilever is

32 1

1

11

ˆ ˆ( ) ( )

2 3

x xS S Lxw x L x

D (4-29)

where 11D is the component of stiffness matrix D in row one and column one.

Using beam theory:

The deflection of the beam using beam theory is

54

32

1

ˆ( ) ( )

2 3

xS xw x L x

EI (4-30)

where E is the Young’s modulus, and I is the moment of inertia, 3

2

12

L hI .

Bending moment applied

A positive bending moment of 100 is applied at the free end.

Using plate theory:

The deflection of the cantilever is

2

11

ˆ( )

2

xMw x x

D

(4-31)

Using beam theory:

From Timoshenko beam theory, we can have the deflection of the beam is

2ˆ

( )2

xMw x x

EI (4-32)

4.4.2 Square Plate

Square plate is conventionally used to assess the element performance. Both

camped and simply-supported cases are studied here. The plate is subjected to a

uniformly distributed load with value of 100. The dimension is shown in Figure 4-5.

Clamped

The general analytical solution for problem of this case is not easy to be obtained

using any series solution. But the exact solution of deflection and bending moments at

the center can be found in numerous in literatures. Below, same valuable results are

given both for thin and thick case. For details, one can refer to the references.

The transverse displacement and bending moment solution:

Thin case:

55

Here, we present several solutions, analytical and exact, so as to fully study the

performance of elements using IBFEM.

Analytical solution by Timoshenko et al. [18]:

The central deflection is

40.00126c

r

qaw

D (4-33)

The central bending moments are

20.0231x yM M qa (4-34)

The exact solution by ZienKiewicz et al. [22]:


40.00126748c

r

qaw

D (4-35)


20.02290469x yM M qa (4-36)

Exact solution by Taylor et al. [17]:


40.001265319087c

r

qaw

D (4-37)


20.02290508352x yM M qa (4-38)

Thick case:

Exact solution by ZienKiewicz et al. [22]:


56

40.00150442c

r

qaw

D (4-39)


20.02319536x yM M qa (4-40)

Simply-supported


Thin case:

From Timoshenko et al. [18], the deflection of Kirchhoff plate of this case is

2 24 21 1

2

( , ) sin sin

( )

ijK

i j

r

q i x j yw x y

i j a aD

a

(4-41)

where

2

0 0

4( , )sin sin

a a

ij

i x j yq q x y dxdy

a a a

, and flexural rigidity

3

212(1 )r

EhD

v

The bending moments are

2 2

222 2

1 1

2 2

222 2

1 1

222 2 2

1 1

1sin sin ;

1sin sin ;

1cos cos

K

x ij

i j

K

y ij

i j

K

xy ij

i j

i vj i x j yM q

a ai j

vi j i x j yM q

a ai j

ij i x j yM q

a aa i j

(4-42)

An approximate result was also given by Timoshenko et al. [18] as


40.00406237c

r

qaw

D (4-43)

57


20.0479x yM M qa (4-44)

The exact solution by ZienKiewicz et al. [22]:


40.00410658c

r

qaw

D (4-45)


20.04825772x yM M qa (4-46)

Thick case:

According to Reddy et al. [19], the deflection of Mindlin plate of this case is the

sum of the deflection of Kirchhoff plate and the Marcus moments. That’s

( , ) ( , )K

M Kw x y w x y

(4-47)

where

K is the Marcus moment of Kirchhoff plate.

Of this case, the Marcus moment is given by

2

2 221 1

2

( , ) , sin sin

( )

ijK K

r

i j

q j x i yx y D w x y

i j a a

a

(4-48)

Using above deflection relationship between Kirchhoff plate and Mindlin plate of

this case, we have the deflection for Mindlin plate as

2 24 21 1

2

( , ) sin sin

( )

ijM

i j

Q j x i yw x y

i j a a

a

(4-49)

58

where

2 22

2( )

16

ij ij

i jh

aQ q

The bending moments for thick case are

;

;

.

M K

x x

M K

y y

M K

xy xy

M M

M M

M M

(4-50)

The solution also given by ZienKiewicz et al. [22] as


40.00461856c

r

qaw

D (4-51)


20.05096269x yM M qa (4-52)

4.4.3 Circular Plate

A circular plate under uniformly distributed load with clamped and simply

supported boundary conditions is considered here. It’s a good problem to test the

validity of imposing essential boundary condition using Implicit Boundary Method. The

geometry of this circular plate is shown in Figure 4-6.

Clamped


Thin case:

In Timoshenko et al. [18], the deflection in the form as

59

2 2 2

0( ) ( )64

K

r

qw r r r

D (4-53)

And the bending moment

2 2

0

2 2

0

(1 ) (3 ) ;16

(1 ) (1 3 )16

K

r

K

qM v r v r

qM v r v r

(4-54)

Thick case:

For simply-supported circular plate, according to Reddy et al. [19], the relationship

of deflection and bending moments between Mindlin plate and Kirchhoff plate is

( ) ( ) ;

( ) ( );

( ) ( )

KM K

r

M K

r r

M K

Cw r w r

D

M r M r

M r M r

(4-55)

where

0 0

;1

( ) ( )

(1 )

K KK r

K K

r

M M

v

M r M rC

v

Simply-supported


Thin case:

From Timoshenko et al. [18], the deflection of a simply-supported plate is of the

form as

2 22 20

0

( ) 5( )

64 1

K q r r vw r r r

D v

(4-56)

The bending moments are

60

2 2

0

2 2

0

( ) (3 )( );16

( ) (3 ) (1 3 )16

K

r

K

qM r v r r

qM r v r v r

(4-57)

Thick case:

The relation between Mindlin plate and Kirchhoff plate is the same as that in the

clamped case:

( ) ( ) ;

( ) ( );

( ) ( )

KM K

r

M K

r r

M K

Cw r w r

D

M r M r

M r M r

(4-58)

where

0

;1

( )

(1 )

K KK r

K

M M

v

M rC

v

4.4.4 30-degree Skew Plate

Skew plate usually being utilized in testing the performance of distorted element.

In order to test the validity of the Implicit Boundary Method for imposing essential

boundary condition, skew plate is a valuable problem for this purpose. We directly

provide some results for skew plate according to Liew [14] and Sengupta [16]. For more

details, one can refer to Liew [14] and Sengupta [16]. The geometry of this plate is

shown in Figure 4-7.

Clamped

The transverse displacement solution:

Thin case:

According to Sengupta [16], the deflection at the center is

61

40.000109125c

r

w qaD

(4-59)

For thick case, there is no analytical solution or exact solution available in

literature.

Simply-supported


Thin case:

According to Liew [14] et al., the deflection at the center is

40.0004174375c

r

w qaD

(4-60)

Thick case:

According to Liew et al. [14], the deflection at the center is

40.0005176875c

r

w qaD

(4-61)


The geometry of this plate is shown in Figure 4-8.

Clamped

Transverse displacement solution:

Thin case:

According to Sengupta [16], the deflection at the center is

40.0007709375c

r

w qaD

(4-62)

For thick case, there is no analytical solution or exact solution available in

literature.

62

Simply-supported


Thin case:


40.00261925c

r

w qaD

(4-63)

Thick case:


40.0029833125c

r

w qaD

(4-64)

Figure 4-1. Configuration for CPT

63

Figure 4-2. Configuration for FSDT

Figure 4-3. Definitions of variables for plate approximations

64

Figure 4-4. Geometry of cantilever plate

Figure 4-5. Geometry of square plate

Figure 4-6. Geometry of circular plate

65

Figure 4-7. Geometry of 30-degree skew plate

Figure 4-8. Geometry of 60-degree skew plate

66

CHAPTER 5 MIXED FORMULATION FOR MINDLIN PLATE

We will present the derivation of mixed form for Mindlin plate and validate our

IBFEM elements in Chapter 5. Details will focus on deriving the mixed weak form and

imposing essential boundary condition (EBC) using Implicit Boundary Method. The

governing equations for Mindlin plate are already derived in Chapter 4. So we begin the

finite formulation with the matrix form in Chapter 5.

5.1 Mixed Form

Writing in matrix form, the governing equation system is

0

0

Mx S mx y xz xM y S myz yMx y xy

0 (5-1)

T M S m 0 (5-2)

0

Sxz

qzSx y

yz

(5-3)

0T q S (5-4)

Generally, m is not included in the plate theory.

Equations 5-1 to 5-4 constitute the governing equations in the plate theory.

Eliminating the bending moment M , we have the three governing equations as:

0T q S (5-5)

T D S 0 (5-6)

67

1w

S 0 (5-7)

Using the weighted residual form and Green’s formula, we can derive the weak

form.

Green’s formula:

( )T T Tw d w d w d

q q n q (5-8)

For Equation 5-5:

[ ] 0

0

( ) 0

ˆ ( ) 0s

T

T

T T

T T

n

w q d

w d wqd

w d w d wqd

w d w d wqd

S

S

S n S

S S

(5-9)

ˆnS is the prescribed shear on the boundary s .

For Equation 5-6:

( ) 0

( ) 0

( ) ( ) ( ) 0

ˆ ( ) ( ) 0

M

T T

T T T

T T T

T T T

d

d d

d d d

d d d

D S

D S

D n D S

M D S

(5-10)

M is the prescribed moment on the boundary M .

For Equation 5-7:

1( ) 0

10

T

T T T

w d

wd d d

S S

S S S S

(5-11)

We use following approximations for different variables:

68

ˆ ˆˆ , ,

ˆ ˆˆ , ,

w s

w s

w

w

N w N S N S

N w N S N S

(5-12)

Substitute above approximation 5-12, into Equations 5-9 to 5-11, the weak form

system can be represented as

ˆ

ˆ

ˆ

T

wT

b

w0 0 E f

0 K C f

E C H 0S

(5-13)

Where

( ) ( )

1

ˆ

ˆs

M

Tb

Ts w

Ts

Ts s

T Tw w w n

T

d

d

d

d

qd S d

d

K N D N

E N N

C N N

H N N

f N N

f N M

For fairly thin plate, and1T

s sd

H N N 0 , where sN is the shape

function for shearS . As the thin plate limit is approached, there are certain necessary

criteria should be satisfied in order to achieve the solution stability.

According to Zienkiewicz et al. [21], certain necessary but not sufficient count

conditions must be satisfied in order to develop useful and robust elements. The count

conditions are:

w sn n n (5-14)

s wn n (5-15)

69

Where n , sn and

wn are the numbers of , S and w parameters. When this count

conditions are not satisfied then the equation system will be either locked or singular.

Several elements satisfy these criterions are presented in Table 5-1.

In Table 5-1, we can see that the 4-node element ([2]) fails the count condition for

four-element patch test and the 9-node element ([7]) fails the count condition for single

element patch test, but the margin is small. If we increase the element number for the

patch test, this count condition will be satisfied. The 5-node element, depicted in above

table, passes the count condition for both single element and four element patch test,

but it results in singular stiffness matrix, which illustrate the fact that the count condition

is just a necessary, but not sufficient condition to ensure the system solvable and

stable.

In the next section, we will employ discrete collocation method to impose the

above count condition and finally arrive at the mixed weak form.

5.2 Discrete Collocation Constraints Method

To some degree, using three-field mixed formulation to achieve satisfactory

performance is limited. But a different approach uses collocation constraints for the

shear approximation on the element boundaries, thus limiting the number of S

parameters and making the count condition more easily satisfied. We will focus on this

method to derive a useful family of Mindlin elements using IBFEM from now on.

The shear strain xz x

w

x

is uniquely determined given and w at all the

nodes. And hence xz xzS is also uniquely determined there. Using this relationship

and imposing the count conditions, the prescribed values of shear at certain nodes are

70

ˆ ˆˆ[ ]w S Q w Q , where wQ and Q are two constant matrices for each interpolation type.

For simplicity, we directly give this expression here. A detailed derivation is presented in

Appendix E.

We will re-derive the two-field mixed weak form by above discrete shear field

collocation method in this section. Instead of using Galerkin’s weighted residual form,

we adopt the principle of minimum potential energy method to derive the weak form for

the static system.

The potential energy of the system is

st fU V (5-16)

Where

2

1 1

2 2

1

2

1 1

2 2

1 1( ) ( )

2 2

1 1[ ( ) ( )]

2 2

1 1 1( ) ( )

2 2

T T

stV h

kT

k Sh

S

T T

k k S Sh h

T T

S Sh

T T

S Sh

T T

U dV dzd

dzdkG

dzd kG dzd

z z dzd d

z dzd d

d

ε Qε ε Qε

εC 0ε γ

γ0

ε Cε γ γ

C γ γ

C γ γ

D S Sd

(5-17)

ˆ ˆs

M

T T

f nV d wqd w d d

u q S M (5-18)

In order to make the potential energy minimum and hence a stable status of the

system, we let

71

1 ˆ ˆ( ) ( ) 0s

M

st f

T T T T T

n

U V

d d w qd w d d

D S S S M (5-19)

Plugging the relation for the shear term, ˆ ˆˆ[ ( )]s w S N Q w Q , into equation 5-19

and interpolate , w .using their shape function, 5-12, respectively, the variation of

potential energy becomes

ˆ ˆ ˆ ˆˆ ˆ( ) ( ) ( ( )) ( )T T T

st s w s wU d d

N D N N Q w Q N Q w Q (5-20)

ˆ ˆ ˆˆ ˆ( ) ( ) ( )

ˆ ˆˆ( )

T T T T T

st w s s w

T T T

s s w

U d d

d

N D N w Q N N Q w Q

Q N N Q w Q

(5-21)

ˆ ˆ ˆ ˆˆ[ ( ) ( ) ( ) ]

ˆ ˆˆ ˆ[ ( ) ] 0

M

s

T T T T T

s s w

T T T T T

w s s w w w n

d d d

d qd d

N D N Q N N Q w Q N M

w Q N N Q w Q N N S

(5-22)

In the above deduction, we only introduced the variation of total strain energy

terms in the first two expressions, 5-20 and 5-21, since for the external force term, there

are no shear collocation terms, and naturally, it follows a traditional routine.

For any arbitrary ˆand ˆw , above Equation 5-22, must satisfied. Hence we obtain

the weak form as

ˆ ˆˆ( )s

T T T T

w s s w w w nd qd d

Q N N Q w Q N N S (5-23)

ˆ ˆ ˆˆ( ) ( ) ( )

M

T T T T

s s wd d d

N D N Q N N Q w Q N M (5-24)

Rearranging terms in Equations 5-23 and 5-24 and being sorted in matrix form, we

have the weak form for two-field mixed formulation as

T d

X fB DB (5-25)

72

Where

, , ,ˆ

ˆs w s

w

D 0B D X f

0

0 N

N N

w f

fQ Q.

So far, we have finished the derivation of two-field mixed formulation using

discrete shear field collocation method. Some post-processing issues are also

discussed in following content.

Strains and Stresses

After solving the equation system for wand at each node, the associated strains

and stresses are of our interest sometimes. Hence, it’s important to give a hint on how

to obtain these quantities for the implementation.

ˆ[ ]

ˆ ( )

x

y

xy

s w s s

xz

yz

w0 NB X

N Q N Q N w

(5-26)

We can calculate the in plane strains at any layer on the condition that Equation 5-

26 is calculated ahead. The in plane strains on the top surface of the plate are

2

x x

t y y

xy xy

h

ε (5-27)

For bottom in plane strains, due to the linearity with respect to thickness h , it has

the same magnitude as those of the top surface of the plate, but in the opposite

direction.

After computing the in plane strains, the corresponding stresses on the top surface

of the plate also can be calculated by constitutive relationship as

73

x

t y t

xy

σ Cε (5-28)

where C is the isotropic elasticity matrix.

The average transverse shear stresses can be computed as

xz xz

yz yz

(5-29)

Equation 5-29 for the transverse shear stresses is just in an average sense. Most

of times, it is of little use to predict the stress loading condition. In order to get a more

accurate transverse stresses distribution along the thickness, the equilibrium equations

of the static system is used for this purpose, just as what we did to derive the shear

correction factor in Appendix C. For more details, one can refer to Appendix C.

Those bending moments and shear forces also can be calculated by

ˆ

[ ][ ]ˆ ( )

x

y

xy

s w s sx

y

M

M

M

S

S

w0 N DD 0D B X

N Q N Q N w0

(5-30)

Again, the shear forces are also in an average sense. One can calculate the exact

shear force distribution once given the transverse stresses distribution.

5.3 Applying EBC Using Implicit Boundary Method

Two important issues in developing IBFEM for mixed formulations is how to

constructing the step function and solution structure and imposing the essential

boundary condition. Various ways of constructing these functions are already presented

in the Chapter 1. One can refer to those references for details. In this section, we will

focus on the Implicit Boundary Method utilized by Kumar et al. [11]-[13].

74

Solution structure and step function

The solution structure to ensure the imposed essential boundary condition for the

transverse displacement and rotations are

ˆ ˆ ˆ( ) ( ) , ( ) ( )

ˆ ˆ ˆ( ) ( ) , ( ) ( )

w g a w g

g a g

w H w w w H w

x x x x

x H x x H x (5-31)

where H is the step functions to be constructed, ˆˆ ,a aw are the prescribed essential

boundary condition, and ˆˆ ,g gw are the nodal values.

H can be constructed as

0 0

(2 ), 0

1

iH

(5-32)

where is the distance between points to the boundary lines in the bi-normal

direction, and is the transition width in order to avoid computation problem of sharp

change from 0 to1. For transverse displacement, 1i , for rotations, 1,2i .

Total strain energy

In order to impose EBC, the two-field mixed form needs to be modified. We use

the principle of potential energy method to re-derive the weak form using IBFEM.

The potential energy of the system is

1 1 1 ˆ ˆ( ) ( )2 2 s

M

T T T T T

nd d w qd w d d

D S S S M (5-33)

In order to make the potential energy minimum hence make the system stable, we

let the variation of the potential energy to be zero. That is

75

1 ˆ ˆ( ) ( )

ˆ ˆ( ) ( ) 0

s

M

s

M

T T T T T

n

T T T T T

s s n

d d w qd w d d

d d w qd w d d

D S S S M

D γ γ S M

(5-34)

Rewriting above equation in a matrix form, the equation becomes

0 ˆ ˆ( ) 0

0 s

M

T T T T T

s n

s

d w qd w d d

Dγ S M

γ

(5-35)

For convenience of elaboration, we define

s

εγ

(5-36)

0

0

DD (5-37)

Hence the total strain energy term of Equation 5-35 can be rewritten as

0

( )0

T T

st s

s

Td dU

Dγ

γε Dε

(5-38)

Knowing that s w γ , and plugging in the trial solution and test function,

Equation 5-31, the integrant of 5-38 can be expressed as

) ( ))

( )( )

T

g w g g

g aT Tg g a

a aw g g

H wwH w

H HH

ε Dε D ε D ε εH

(5-39)

Where

),

( )

g ag a

a aw g g wH w

Hε ε

H

(5-40)

76

aε results from the prescribed essential boundary conditions. For implementation

purpose, we can move this term to the RHS, considered to be a force term together with

the original external force terms. This fictitious force term is

a

Tg ad

F ε Dε (5-41)

aε can be rewrite as

ˆ

ˆ a

aaa a

a a w a

w

w

0 Nε B X

N N

(5-42)

The LHS can be further simplified as follows:

Separating gε into two parts as,

gg

gg g w gw H w

Hε

(5-43)

Also, we define here

1

g

gg gw

ε

(5-44)

2

g

gw gH w

Hε

(5-45)

Above two terms also can be expressed in terms of nodal displacement as:

1 1

ˆ

ˆ a

gg

g gg gg g w g

w

w

0 Nε B X B X

N N

(5-46)

2 2

ˆ

ˆ

gg

gg gw g w w g

w

H w H

H 0 H Nε B X

N 0

(5-47)

Expanding g g

Tε Dε , we can have

77

1 2 1 2 1 1 1 2 2 1 2 2( ) ( )g g

T T T T T Tg g g g g g g g g g g g ε Dε ε ε D ε ε ε Dε ε Dε ε Dε ε Dε (5-48)

Apparently, 1 1Tg gε Dε is what we have used in the collocation method for inner

elements. For boundary elements, there are two more additional terms, that 2gε and

aε ,

due to the essential boundary condition.

Integration issue

For terms not including the step functionwH and H , we use volume integration.

That

1

1 1 1

12

e T

g g g

A

hd dA

K B DB (5-49)

1

12

e T

q w

A

hq d dA

F N (5-50)

1

1

ˆ2

e T

s w

A

hd dA

F N S (5-51)

1

1

ˆ2

e T

m

A

hd dA

F N M (5-52)

For terms contain these two step functions, instead of using volume integration,

we can use boundary integration, since the step functions are define as constant except

the boundary transit region.

1

0

1

1 02

e

e

e T T

g g g g g

hd d d d

K B DB B DB (5-53)

1

0

1

1 02

e

e

T

a g a

hd d d

F B DB (5-54)

78

So far, we have successfully imposed the EBC using Implicit Boundary Method.

And details about formulation and implementation are given. In next section, we will

compare our numerical solution with analytical or exact solution provided in Chapter 4.

5.4 Numerical Results

Numerical results for the benchmark problems in section 4.3 will be studied in this

section. S4R and S8R5 elements in commercial package ABAQUS were also used in

this analysis to test the performance of the developed elements. Comparison among

numerical results, exact solutions and analytical solutions are presented. Within each

example, the convergence of total strain energy is also studied.

5.4.1 Cantilever Plate

A cantilever plate with background mesh is shown in Figure 5-1.

Shear applied

Shear is applied at the free end of the cantilever plate. The relevant result of this

case is provided in Table 5-2. And the distribution of the transverse displacement after

deformation of thin case is presented in Figure 5-2. The convergence plots of the total

strain energy for both thin and thick cases are provided in Figure 5-3 and Figure 5-4

respectively.

Bending moment applied

Only bending is applied at the free end in this case. The relevant result of this case

is presented in Table 5-3, both thin and thick cases. The distribution of the transverse

displacement after deformation for thin case is presented in Figure 5-5. And the total

strain energy plots are in Figure 5-6 and Figure 5-7.

79

5.4.2 Square Plate

We utilize two kinds of background mesh for the interpolation for this very

conventional benchmark problem, as shown in Figure 5-8 and Figure 5-9. The results

obtained from both these two approximations using IBFEM converges to the exact

solution from Zienkiewicz et al. [22]. Data tabulated in Table 5-4 to Table 5-7 are

collected using background pattern one (Figure 5-8).

Fully clamped

The relevant data of this case is given in Table 5-4 and Table 5-5. The deformed

transverse displacements for both two mesh patterns are provided in Figure 5-10 and

Figure 5-11 respectively. And the convergence plots for the total strain energy are in

Figure 5-12 and Figure 5-13.

Simply-supported

The relevant data of this case is given in Table 5-6 and Table 5-7. The

convergence plots for the total strain energy is in Figure 5-14 and Figure 5-15.

5.4.3 Circular Plate

A typical background mesh for this circular plate is depicted in Figure 5-16. Table

5-8 to Table 5-11 give the relevant testing results.

Fully clamped

The relevant data of this case is given in Table 5-8 and Table 5-9. And the

convergence plot for the total strain energy is in Figure 5-18 and Figure 5-19. A graph

for the transverse displacement of the thick case is given in Figure 5-17.

Periphery is simply-supported

The relevant data of this case is given in Table 5-10 and Table 5-11. And the

convergence plot for the total strain energy is in Figure 5-20 and Figure 5-21.

80


A typical background mesh for this 30-degree skew plate is provided in Figure 5-

22. Table 5-12 to Table 5-15 give the relevant testing results.

Fully clamped

We found that the analytical or exact solution for a comparatively thick plate, such

as length/thickness ratio is 0.1, is very rare, even impossible to refer to in literature. Due

to this limitation, we will not provide any analytical solution or exact solution for this case

for comparison. We only compare our results with S4R and S8R5 elements. The

relevant data are given in Table 5-12 and Table 5-13. The deformed transverse

displacement distribution is presented in Figure 5-23, and the convergence plot for the

total strain energy is in Figure 5-25 and Figure 5-26.

Fully simply-supported

Since there is bending moment singularity at the obtuse corners for simply

supported thin plate, we choose not to provide the bending moment for this case here,

only with one figure, Figure 5-24, to demonstrate the singularity of bending moment at

the obtuse corners. The relevant data of this case are given in Table 5-14 and Table 5-

15. And the convergence plot for the total strain energy is in Figure 5-27 and Figure 5-

28.


A typical background mesh using 10X10 Q16 element is shown in Figure 5-29.

Fully clamped

The analytical or exact solution for a comparatively thick skew plate is also very

rare, even impossible to find in literature. For this case, we only compare the result with

S4R and S8R5 elements. The relevant data of this case are formatted in Table 5-16 and

81

Table 5-17. And the convergence plot for the total strain energy is in Figure 5-32 and

Figure 5-33. The deformed transverse displacement distribution is in Figure 5-30.

Fully simply-supported

The bending moment singularity also occurs in this case at small thickness. This

singularity is demonstrated in Figure 5-31. The relevant data of this case are given in

Table 5-18 and Table 5-19. And the convergence plot for the total strain energy is in

Figure 5-34 and Figure 5-35.

5.4.6 Flange Plate

In addition to the above conventional problems, we also present the analysis result

using IBFEM for an arbitrary shaped plate. It also aims to test the performance of these

elements for arbitrary shapes. The plate is clamped at all four corner holes. The

geometry of this plate is provided in Figure 5-36, and a typical background mesh is

presented in Figure 5-37. The result data tabulated in Table 5-20 is for the four inner

holes clamped scenario. The convergence plot for the total strain energy and transverse

displacement plot are in Figure 5-39, Figure 5-40 and Figure 5-38 respectively.


Implicit Boundary Finite Element method was employed in Chapter 5 to develop a

family of Mindlin plate elements. Discrete shear field collocation method is used to make

the count condition satisfied. Implicit Boundary Method is adopted to impose the

essential boundary condition. Several benchmark problems and one arbitrary shape

problem are tested in order to fully assess the performance of these elements. From the

obtained results and comparison with the exact solution, analytical solution and that

from S4R and S8R5 of commercial package, these developed elements are valid and

robust. In conclusion, employing the Implicit Boundary Finite Element Method can solve

82

plate problem for both thin and thick cases using a background mesh and equations of

the boundary to avoid generating a conforming mesh.

Table 5-1. Location of three interpolation variables and the associated count conditions

for patch test

Element

Clamped edges Relaxed edges

One - element patch Four - element patch One - element patch Four - element patch

s

w

n

n

w

s

n n

n

s

w

n

n

w

s

n n

n

s

w

n

n

w

s

n n

n

s

w

n

n

w

s

n n

n

0

0

0

0

4

1

3

4 (F)

4

3

9

4

12

8

24

12

2

1

3

2

12

5

15

12

6

4

12

6

20

12

36

20

0

0

0

0

8

5

15

8

8

7

21

8

24

20

60

24

4

1

3

4 (F)

24

9

27

24

12

8

24

12

40

24

72

40

12

4

12

12

60

25

75

60

24

15

45

24

84

48

144

84

: deflection w interpolation node; : rotations ,x y interpolation node; : Shear xS

collocation point; : Shear yS collocation point.

83

Table 5-2. Cantilever plate (Shear force applied at free end)

Mesh density

a/t = 10 tip displacement w

( 0510 )


( 0210 )

Q4 Q8 Q9 Q16 S4R S8R5 Q4 Q8 Q9 Q16 S4R S8R5

10X1 3.558 3.604 3.583 3.583 3.581 3.583 3.530 3.575 3.536 3.539 3.554 3.538 20X2 3.575 3.588 3.584 3.584 3.585 3.586 3.530 3.552 3.539 3.539 3.542 3.537 50X5 3.582 3.585 3.585 3.585 3.584 3.587 3.536 3.542 3.539 3.539 3.539 3.538

100X10 3.584 3.585 3.586 3.585 3.585 3.588 3.538 3.540 3.539 3.539 3.539 3.539

Analytical 3.571 3.571

Table 5-3. Cantilever plate (Bending moment applied at free end)

Mesh density


( 0610 )


( 0310 )


10X1 5.324 5.342 5.339 5.342 5.348 5.342 5.324 5.342 5.321 5.326 5.348 5.325 20X2 5.337 5.342 5.342 5.343 5.347 5.345 5.320 5.334 5.325 5.325 5.332 5.323 50X5 5.341 5.343 5.343 5.344 5.344 5.344 5.323 5.327 5.326 5.326 5.326 5.325

100X10 5.343 5.344 5.344 5.344 5.344 5.344 5.325 5.326 5.326 5.326 5.326 5.326


Table 5-4. Uniformly loaded, clamped square plate [a/t = 10]

Mesh density

Central displacement w

( 0610 )

Central bending moment x yM M

( 0210 )


10x10 1.457 1.642 1.482 1.467 1.470 1.466 2.335 2.539 2.368 2.298 2.278 2.375 20x20 1.464 1.515 1.470 1.467 1.468 1.467 2.324 2.375 2.332 2.315 2.310 2.333 50x50 1.467 1.475 1.468 1.467 1.467 1.467 2.321 2.329 2.322 2.319 2.318 2.322

100x100 1.467 1.469 1.467 1.467 1.467 1.467 2.320 2.322 2.320 2.320 2.320 2.321 150x150 1.467 1.468 1.467 1.467 1.467 1.467 2.320 2.321 2.320 2.320 2.320 2.320

Exact 1.467 2.320

Table 5-5. Uniformly loaded, clamped square plate [a/t = 100]

Mesh density


( 0310 )


( 0210 )


10x10 1.227 1.417 1.251 1.236 1.239 1.236 2.316 2.516 2.338 2.269 2.253 2.346 20x20 1.234 1.286 1.240 1.236 1.237 1.236 2.297 2.349 2.302 2.286 2.281 2.304 50x50 1.236 1.245 1.237 1.236 1.236 1.236 2.292 2.300 2.293 2.290 2.289 2.293

100x100 1.236 1.238 1.236 1.236 1.236 1.236 2.291 2.293 2.291 2.291 2.291 2.291 150x150 1.236 1.237 1.236 1.236 1.236 1.236 2.291 2.292 2.291 2.291 2.291 2.291

Exact 1.236 2.290

84

Table 5-6. Uniformly loaded, simply-supported square plate [a/t = 10]

Mesh density


( 0610 )


( 0210 )


10x10 4.395 4.602 4.499 4.501 4.472 4.476 5.012 5.204 5.117 5.075 4.968 5.127 20x20 4.465 4.518 4.503 4.501 4.493 4.498 5.066 5.115 5.102 5.091 5.063 5.106 50x50 4.495 4.504 4.502 4.501 4.500 4.501 5.090 5.098 5.097 5.095 5.090 5.098

100x100 4.500 4.502 4.502 4.501 4.501 4.502 5.094 5.096 5.096 5.096 5.094 5.096 150x150 4.501 4.502 4.502 4.501 4.501 4.502 5.095 5.096 5.096 5.096 5.095 5.096

Exact 4.503 5.096

Table 5-7. Uniformly loaded, simply-supported square plate [a/t = 100]

Mesh density


( 0310 )


( 0210 )


10x10 3.954 4.146 3.982 3.978 3.977 3.969 4.793 4.981 4.832 4.781 4.706 4.849 20x20 3.967 4.016 3.982 3.988 3.978 3.975 4.795 4.843 4.813 4.806 4.779 4.813 50x50 3.979 3.987 3.992 3.996 3.988 3.987 4.803 4.811 4.816 4.818 4.808 4.813

100x100 3.988 3.990 3.996 3.997 3.994 3.995 4.811 4.813 4.819 4.819 4.816 4.818 150x150 3.992 3.993 3.997 3.997 3.995 3.996 4.815 4.816 4.819 4.819 4.818 4.819

Exact 4.004 4.826

Table 5-8. Uniformly loaded, clamped circular plate [a/t = 10]

Mesh density


( 0610 )

Central bending moment rM M

( 0210 )


10x10 0.737 1.194 1.091 1.109 1.110 1.128 1.573 2.199 2.048 1.992 1.970 2.058 20x20 0.970 1.143 1.111 1.113 1.119 1.126 1.860 2.065 2.029 2.012 2.000 2.029 50x50 1.099 1.119 1.112 1.113 1.126 1.129 2.003 2.025 2.020 2.017 2.029 2.035

100x100 1.108 1.115 1.113 1.113 1.126 1.125 2.013 2.020 2.019 2.018 2.030 2.031 150x150 1.111 1.114 1.113 1.113 1.126 1.124 2.016 2.019 2.018 2.018 2.031 2.031


Table 5-9. Uniformly loaded, clamped circular plate [a/t = 100]

Mesh density


(0410 )

Central bending moment rM M


10x10 4.745 9.100 9.017 9.313 9.374 9.502 1.348 2.075 2.033 1.985 1.970 2.043 20x20 7.027 9.535 9.371 9.399 9.471 9.504 1.731 2.049 2.026 2.011 2.002 2.028 50x50 8.895 9.459 9.409 9.410 9.535 9.537 1.959 2.025 2.019 2.017 2.029 2.032

100x100 9.253 9.428 9.413 9.413 9.538 9.539 2.000 2.020 2.018 2.018 2.030 2.032 150x150 9.360 9.419 9.413 9.413 9.538 9.539 2.012 2.019 2.018 2.018 2.031 2.031


85

Table 5-10. Uniformly loaded, simply-supported circular plate [a/t = 10]

Mesh density


( 0610 ) Central bending moment

rM M


10x10 1.076 3.127 3.046 3.742 3.978 4.053 2.008 4.257 4.180 4.825 5.049 5.182 20x20 1.788 3.535 3.848 3.950 4.029 4.050 2.788 4.536 4.972 5.047 5.114 5.153 50x50 3.497 3.922 3.959 3.982 4.053 4.059 4.591 5.019 5.075 5.096 5.152 5.159

100x100 3.805 3.986 3.989 3.991 4.055 4.055 4.914 5.098 5.106 5.107 5.155 5.157 150x150 3.920 3.986 3.989 4.028 4.056 4.054 5.034 5.102 5.106 5.107 5.156 5.156


Table 5-11. Uniformly loaded, simply-supported circular plate [a/t = 100]

Mesh density


( 0310 ) Central bending moment

rM M


10x10 0.737 1.557 1.114 1.532 3.805 3.875 2.002 4.202 4.143 4.795 5.049 5.168 20x20 0.966 1.683 1.706 2.393 3.857 3.875 2.767 4.506 4.962 5.034 5.115 5.153 50x50 1.637 2.215 2.517 3.073 3.881 3.883 4.481 5.013 5.073 5.092 5.152 5.157

100x100 3.120 3.211 3.417 3.743 3.883 3.884 4.912 5.098 5.102 5.107 5.155 5.157 150x150 3.406 3.690 3.708 3.743 3.883 3.884 5.033 5.100 5.106 5.107 5.156 5.156


Table 5-12. Uniformly loaded, clamped 30-degree skew plate [a/t = 10]

Mesh density


( 0710 )

Central bending moment xM

( 0110 )


10x10 0.806 2.576 1.717 1.726 1.642 1.767 2.545 7.479 5.183 4.635 4.684 5.087 20x20 1.320 2.106 1.729 1.729 1.695 1.735 3.711 5.767 4.999 4.802 4.626 4.928 50x50 1.662 1.812 1.728 1.728 1.724 1.730 4.624 5.003 4.894 4.859 4.861 4.868

100x100 1.723 1.755 1.728 1.728 1.727 1.728 4.847 4.895 4.877 4.868 4.861 4.880 150x150 1.726 1.742 1.728 1.728 1.728 1.726 4.859 4.878 4.873 4.869 4.868 4.885


Mesh density


( 0410 )


( 0110 )


10x10 0.143 1.480 0.971 1.055 0.957 1.062 1.107 6.430 4.685 4.578 4.801 4.880 20x20 0.664 1.362 1.060 1.062 1.027 1.059 3.628 5.731 4.903 4.708 4.554 4.782 50x50 0.967 1.126 1.063 1.062 1.058 1.062 4.490 4.945 4.802 4.764 4.755 4.780

100x100 1.047 1.079 1.062 1.062 1.062 1.062 4.743 4.818 4.782 4.772 4.763 4.776 150x150 1.057 1.070 1.062 1.062 1.062 1.062 4.760 4.794 4.778 4.773 4.772 4.776

Exact 1.064 -

86

Table 5-14. Uniformly loaded, simply-supported 30-degree skew plate [a/t = 10]

Mesh density


( 0710 )


( 0210 )


10x10 1.217 6.258 4.759 5.004 4.796 5.081 0.366 1.554 1.225 1.217 1.148 1.271 20x20 2.556 5.456 5.001 5.043 4.945 5.050 0.651 1.337 1.267 1.235 1.225 1.269 50x50 4.429 5.140 5.043 5.047 5.031 5.053 1.075 1.277 1.265 1.258 1.259 1.263

100x100 5.001 5.080 5.046 5.047 5.044 5.053 1.247 1.267 1.263 1.262 1.261 1.265 150x150 5.024 5.066 5.047 5.047 5.046 5.049 1.255 1.265 1.263 1.262 1.262 1.264

Exact 5.047 -


Mesh density


( 0410 )

Q4 Q8 Q9 Q16 S4R S8R5

10x10 0.255 4.584 2.707 3.451 3.897 4.069 20x20 0.867 4.036 3.334 3.802 4.013 4.083 50x50 1.671 4.032 3.909 4.081 4.105 4.118

100x100 3.111 3.909 4.080 4.132 4.130 4.135 150x150 3.410 4.122 4.122 4.138 4.135 4.138

Exact 4.070


Mesh density


( 0710 )


( 0210 )


10x10 5.358 11.31 9.446 9.355 9.315 1.033 1.257 2.031 1.751 1.581 1.635 1.764 20x20 8.793 9.916 9.339 9.320 9.302 1.034 1.624 1.766 1.692 1.650 1.665 1.727 50x50 9.237 9.413 9.305 9.304 9.299 1.034 1.669 1.688 1.677 1.670 1.673 1.717

100x100 9.288 9.331 9.300 9.300 9.298 1.034 1.674 1.677 1.675 1.674 1.674 1.715 150x150 9.294 9.315 9.299 9.299 9.298 1.034 1.674 1.675 1.674 1.674 1.674 1.720


Mesh density


( 0410 )


( 0210 )


10x10 2.764 8.688 7.539 7.525 7.539 7.527 1.283 1.897 1.728 1.989 1.623 1.704 20x20 5.643 8.072 7.546 7.519 7.521 7.528 1.526 1.750 1.672 1.644 1.645 1.666 50x50 6.872 7.622 7.523 7.517 7.518 7.529 1.622 1.670 1.656 1.652 1.652 1.656

100x100 7.357 7.545 7.519 7.517 7.517 7.529 1.647 1.657 1.654 1.653 1.653 1.654 150x150 7.456 7.530 7.518 7.517 7.517 7.529 1.651 1.655 1.653 1.653 1.653 1.654

Analytical 7.517 -

87


Mesh density


( 0610 )


( 0210 )


10x10 0.809 3.103 2.898 2.923 2.881 2.999 1.721 4.082 3.895 3.798 3.743 3.937 20x20 2.349 2.975 2.912 2.914 2.900 3.009 3.201 3.932 3.875 3.845 3.829 3.907 50x50 2.837 2.925 2.910 2.910 2.907 3.010 3.768 3.875 3.863 3.858 3.855 3.896

100x100 2.904 2.916 2.909 2.909 2.908 3.010 3.854 3.867 3.861 3.860 3.859 3.895 150x150 2.907 2.914 2.909 2.909 2.909 3.010 3.858 3.865 3.860 3.860 3.860 3.894

Exact 2.909 -


Mesh density


( 0310 )

Q4 Q8 Q9 Q16 S4R S8R5

10x10 0.352 2.722 2.399 2.489 2.507 2.510 20x20 0.868 2.596 2.487 2.520 2.511 2.517 50x50 1.951 2.551 2.527 2.535 2.524 2.530

100x100 2.430 2.537 2.535 2.537 2.532 2.536 150x150 2.483 2.539 2.536 2.537 2.534 2.537

Exact 2.554

Table 5-20. Uniformly loaded, arbitrary shape plate

Mesh density

a/t = 10 Central displacement w

( 0710 )

a/t = 100 Central displacement w

( 0410 )


20x20 5.606 7.350 6.927 7.310 7.540 7.379 3.235 4.539 4.234 4.551 4.770 4.698 50x50 5.754 7.440 7.275 7.458 7.423 7.356 3.635 4.668 4.610 4.604 4.739 4.698

100x100 7.260 7.502 7.423 7.475 7.368 7.356 4.542 4.749 4.671 4.701 4.707 4.697 150x150 7.070 7.467 7.430 7.475 7.356 7.346 4.505 4.754 4.738 4.704 4.699 4.697

Figure 5-1. A typical background mesh using 20x2 Q4 element

88

Figure 5-2. Distribution of transverse displacement after deformation for 100x10 Q4

element (L1/t = 100)

Figure 5-3. Convergence of total strain energy for cantilever when shear applied (L1/t =

10)

Figure 5-4. Convergence of total strain energy for cantilever when shear applied (L1/t =

100)

89


element (L1/t = 100)

Figure 5-6. Convergence of total strain energy for cantilever when bending moment

applied (L1/t = 10)

Figure 5-7. Convergence of total strain energy for cantilever when bending moment

applied (L1/t = 100)

90




element (t = 0.1)

91


element (t = 0.1)

Figure 5-12. Convergence of total strain energy for clamped square plate (a/t = 10)

Figure 5-13. Convergence of total strain energy for clamped square plate (a/t = 100)

92

Figure 5-14. Convergence of total strain energy for simply-supported square plate (a/t =

10)

Figure 5-15. Convergence of total strain energy for simply-supported square plate (a/t =

100)


93


element (t = 1)

Figure 5-18. Convergence of total strain energy for clamped circular plate (a/t = 10)

Figure 5-19. Convergence of total strain energy for clamped circular plate (a/t = 100)

94

Figure 5-20. Convergence of total strain energy for simply-supported circular plate (a/t =

10)

Figure 5-21. Convergence of total strain energy for simply-supported circular plate (a/t =

100)


95


element (t = 0.1)

Figure 5-24. Distribution of bending moment using 200x200 Q4 element (t = 0.1)

Figure 5-25. Convergence of total strain energy for clamped 30-degree skew plate (a/t =

10)

96


100)

Figure 5-27. Convergence of total strain energy for simply-supported 30-degree skew

plate (a/t = 10)


plate (a/t = 100)

97



element (t = 1)

Figure 5-31. Distribution of bending moment using 200x200 Q4 element (t = 0.1)

98


10)


100)


plate (a/t = 10)

99


plate (a/t = 100)

Figure 5-36. Geometry of flange plate


100


element (t = 0.1)

Figure 5-39. Convergence of total strain energy for arbitrary shape plate (a/t = 10)

Figure 5-40. Convergence of total strain energy for arbitrary shape plate (a/t = 100)

101

CHPATER 6 MIXED FORMULATION FOR 2D MINDLIN SHELL

In plate elements, we are mainly concerned about bending and transverse loading.

Plates subjected to both transverse loading and in-plane stretching is a special case of

shells. Chapter 6 is intended to model this special case, and hence prepare for future

work on curved 3D shell-like structures.

6.1 Governing Equations

The derivation of the governing equations for Mindlin plate has already been done

in Chapter 4. In Chapter 4, we assume that there is no in-plane stretching, therefore,

terms associated with stretching are omitted. In Chapter 6, we will consider in-plane

stretching also. The derivation is the same as in Chapter 4, except that two more

equations are added into the governing equations.

The governing equations are

0xyx

x

PPq

x y

(6-1)

0xy y

y

P Pq

x y

(6-2)

0yzxz

z

SSq

x y

(6-3)

0xyx

xz

MMS

x y

(6-4)

0xy yyz

M MS

x y

(6-5)

Writing in matrix form, the governing equation system can be expressed as

102

0

0

xx

yy

xy

qP

qP

Px y

y x

0 (6-6)

T P q 0 (6-7)

0

0

xxz

yyz

xy

MSx y

MS

My x

0 (6-8)

T M S 0 (6-9)

0xz

zyz

Sq

Sx y

(6-10)

0T q S (6-11)

For clarity, notations used in above equations are rewritten here.

2 2

2 2

h hx x x

y y y

h h

xy xy xy

P

P dz dz

P

P C A u (6-12)

2 2

2 2

h hx x x

y y y

h h

xy xy xy

M

M z dz z dz

M

M C D (6-13)

2 2 2

2 2 2

h h h

xz xz

yz yzh h h

xz

yz

dz dz kG w dz wS

S

S (6-14)

2

1 0

1 0(1 )

0 0 (1 ) / 2

vE

vv

v

C

hA C

103

3

12

hD C

kGh

Equations 6-6 to 6-11 constitute the governing equation system for Chapter 6.

Substituting notation 6-12 to 6-14 into Equations 6-6 and 6-11, respectively, we

can eliminate the in plane forces P and moment M and finally obtain the four governing

equations as:

0

10

T

T

T

q

A u q 0

S

D S 0

S

(6-15)

So far, we have obtained the governing equation for plates including in-plane

stretching. In next section, we will formulate the mixed form for this plate.

6.2 Mixed Formulation

We have already derived the governing equations for Mindlin plate including in-

plane stretching. Next, we will focus on the derivation of the mixed form.

As we have mentioned in the last section, except the first governing equation, the

remaining three are the same as those in Chapter 5. And the weak mixed form for these

three equations are still the same since there is no coupling between in-plane stretching

and transverse bending in our formulation. For simplicity, we will only derive the weak

form for the first equation. For the other three, one can refer to Chapter 5 for details.

Using integration by part and Green’s formula, the weak form of 6-6 to 6-11 can be

deduced as

104

[ ( ) ]

[ ( )]

( ) ( ) ( )

ˆ ( ) ( )P

T T

T T T

T T T

T T T

d

d d

d d d

d d d

u A u q 0

u A u u q 0

u A u n u A u u q 0

u P u A u u q 0

(6-16)

P is the prescribed traction on the boundary P .

The final weak system of the combined Mindlin plate is

ˆ( ) ( )

ˆ ˆˆ( ) ( )

ˆ ˆˆ( ) [ ( ) ( ) ( ) ]

P

s

M

T T T

u u u u

T T T T

s w s w s w s w w n

T T T T

s s w s s

d d d

d d qd d

d d d d

N A N N q N P

N Q N Q w N Q N Q N N S

N Q N Q w N D N N Q N Q N M

(6-17)

Writing in matrix form, the mixed weak form can be expressed as

T d

X fB DB (6-18)

Where

, , ,

ˆ

ˆ

ˆ

u

s w s

u

w

A 0 0

B D 0 D 0 X f

0 0

N 0 0

0 0 N

0 N N

u f

w f

Q Q f

(6-19)

Strains and Stresses

After solving the equations for u , wand at each node, we need to calculate the

associated strains and stresses.

0

0

0

[ ]( )

ˆˆˆ

x

y

xy

x

ys

xy

xz

yz

u

s w s

uB X

N w

N 0 00 0 N

0 N N

uw

Q Q (6-20)

105

Once the membrane strain and curvature are obtained, the in plane strains on any

layer can be immediately calculated. At the height equal to 2

h, the membrane strains on

the top surface of the plate is

0

0

02

x x x

t y y y

xy xy xy

h

ε (6-21)

Using constitutive relationship, we can obtain the corresponding stresses on the

top surface of the plate as

x

t y t

xy

σ Cε (6-22)

where C is the stiffness matrix.

The average transverse shear stresses can be computed as

xz xz

yz yz

(6-23)

We also can calculate the in-plane forces, bending moments and shear forces as

[ ][ ]( )

ˆˆˆ

x

y

xy

x

ys

xy

x

y

u

s w s

PP

P

MM

M

SS

A 0 0 A uD B X 0 D 0 D

N w0 0

N 0 00 0 N

0 N N

uw

Q Q (6-24)

106

6.3 Numerical Examples and Results

The way to impose the EBC is almost the same as what have been done in

Chapter 5. We will not redo the procedures for combined Mindlin plates. One can refer

to Chapter 5 for details.

Examples employed in Chapter 6 are the 60 degree skew plate and square plate

that have been used in Chapter 5. The geometry and elastic constants are exactly the

same as in Chapter 5.


The plate is clamped only at the left edge. A transverse pressure and in plane

stretching force in the x-direction, both with value of 100, are applied to this plate. See

Figure 6-4. Since there is no analytical or exact solution for this loaded plate, we only

compare our results with S4R and S8R5 element from Abaqus. Both thin case and thick

case are studied in this section. The testing date for transverse deflection is in Table 6-1

and Table 6-2. The convergence plot of total strain energy is in Figure 6-1 and Figure 6-

2. A plot of the transverse deflection for thin skew plate is also given as Figure 6-3.

6.3.2 Square Plate

The square plate is clamped at the left edge. See Figure 6-8. A transverse

pressure and in plane stretching force in the x-direction, both with value of 100, are

applied on this plate. Test data for displacement are tabulated in Table 6-3 and Table 6-

4. The convergence plot of total strain energy is in Figure 6-5 and Figure 6-6. The

deformed transverse deflection is in Figure 6-7.

From above testing data and convergence plots of two examples, we can see that

the new elements performance the same as S4R element, but better than S8R5

element.

107


We have successfully extended Mindlin plate theory to include in plane stretching

in Chapter 6. Since we study the elastic static problem, there is no coupling between in

plane stretching and bending in our formulation. The mixed weak form is a linear

combination of plane stress and Mindlin plate theory. Examples validate the

performance of these IBFEM elements that we have developed. The content of Chapter

6 can be viewed as a special case of shells problem. Future work is to extend this into a

general shell analysis.

Table 6-1. Transverse deflection of 60-degree skew plate with one edge clamped

Mesh density

a/t = 10 Maximum deflection w

( 0410 )


( 0110 )


10x10 0.970 1.233 1.218 1.219 1.218 1.230 0.585 1.198 1.184 1.184 1.185 1.182 20x20 1.176 1.222 1.218 1.219 1.218 1.232 0.934 1.188 1.184 1.185 1.183 1.183 50x50 1.209 1.219 71.218 1.219 1.218 1.232 1.068 1.185 1.185 1.185 1.184 1.184

100x100 1.216 1.219 1.219 1.219 1.219 1.232 1.160 1.185 1.185 1.185 1.184 1.185 150x150 1.217 1.219 1.219 1.219 1.219 1.232 1.174 1.185 1.185 1.185 1.184 1.185

Table 6-2. In plane displacement of 60-degree skew plate with one edge clamped

Mesh density

a/t = 10

Maximum displacement xu

( 0710 )

a/t = 100


( 0610 )


10x10 1.034 1.034 1.057 1.033 1.033 1.033 1.034 1.034 1.057 1.033 1.034 1.033 20x20 1.035 1.033 1.045 1.033 1.033 1.033 1.035 1.033 1.045 1.033 1.033 1.033 50x50 1.034 1.033 1.038 1.033 1.033 1.033 1.034 1.034 1.038 1.033 1.033 1.033

100x100 1.034 1.033 1.036 1.033 1.033 1.033 1.034 1.034 1.036 1.033 1.033 1.033 150x150 1.034 1.033 1.033 1.033 1.033 1.033 1.034 1.034 1.033 1.033 1.033 1.033

Table 6-3. Transverse deflection of square plate with one edge clamped

Mesh density


( 0410 )


( 0110 )


10x10 1.279 1.295 1.281 1.281 1.281 1.281 1.258 1.276 1.260 1.259 1.259 1.259 20x20 1.280 1.285 1.281 1.281 1.281 1.281 1.258 1.264 1.259 1.259 1.259 1.259 50x50 1.281 1.282 1.281 1.281 1.281 1.281 1.259 1.260 1.259 1.259 1.259 1.259

100x100 1.281 1.281 1.281 1.281 1.281 1.281 1.259 1.259 1.259 1.259 1.259 1.259

108

Table 6-4. In plane displacement of square plate with one edge clamped

Mesh density

a/t = 10


( 0810 )

a/t = 100


( 0710 )


10x10 8.855 8.861 8.862 8.863 8.854 8.863 8.855 8.861 8.862 8.863 8.893 8.863 20x20 8.861 8.863 8.863 8.864 8.862 8.864 8.861 8.863 8.863 8.864 8.862 8.864 50x50 8.863 8.864 8.864 8.864 8.863 8.864 8.863 8.864 8.864 8.864 8.863 8.864

100x100 8.864 8.864 8.864 8.864 8.864 8.864 8.864 8.864 8.864 8.864 8.864 8.864

Figure 6-1. Convergence of total strain energy for 60-degree skew plate (a/t = 10)

Figure 6-2. Convergence of total strain energy for 60-degree skew plate (a/t = 100)

109

Figure 6-3. Distribution of transverse deflection after deformation for 60-degree skew

plate using 150x150 Q4 element (t = 1)

Figure 6-4. Geometry of the 60-degree skew plate

Figure 6-5. Convergence of total strain energy for square plate (a/t = 10)

110

Figure 6-6. Convergence of total strain energy for square plate (a/t = 100)

Figure 6-7. Distribution of transverse deflection after deformation for square plate using

100x100 Q9 element (t = 0.1)

Figure 6-8. The geometry of the square plate

111

CHAPTER 7 CONCLUSION

7.1 Summary

In summary, we have formulated and evaluated elements that use mixed

formulation to avoid locking and the implicit boundary method to apply boundary

conditions. These elements avoid volumetric locking and shear locking phenomena in

near incompressible media and thin Mindlin plate respectively. These elements enables

mesh independent analysis where the geometry is represented as equations and a

background mesh is used to construct the trial solution. The background mesh can

consist of uniform elements and need not conform to the shape of the analysis domain

because the elements developed here use implicit boundary method to apply boundary

conditions which does not need nodes on the boundary to apply boundary conditions.

For volumetric locking, since this phenomenon is totally due to the fact that bulk

modulus tends to infinity when the Poisson’s ratio tends to 0.5, three-field mixed

formulation is adopted for the plane strain and 3D stress in Chapter 3. From the

convergence plot and transverse displacement distribution plot of 3D stress, we can

conclude that these elements perform very well and the volumetric locking is removed.

Before proceeding to the formulation of Mindlin plate theory, two classical plate

theories were reviewed in Chapter 4. The difference between these two theories is

clearly stated in Chapter 4. For future evaluation purpose, the analytical or exact

solutions of several benchmark problems are provided in Chapter 4 also.

Mixed formulation of Mindlin plates using discrete shear field collocation method

and the imposition of the essential boundary condition using implicit boundary method

are detailed in Chapter 5. From the comparison of the results from analytical solution,

112

exact solution, IBFEM element solution and Abaqus element analysis, the performance

of the IBFEM elements can be easily assessed. It is clear that these elements are

useful and robust, and at least as good as, if not better than, the elements from the

commercial package ABAQUS.

2D Mindlin shell, that includes in-plane stretching and transverse pressure, is also

studied in this thesis. From the results, these elements also perform well. The objective

of this extension of Mindlin plate is to prepare the formulation of general curved shell

elements using IBFEM.

In conclusion, we have successfully implemented the mixed formulation using

IBFEM so as to avoid volumetric locking and shear locking phenomena.

7.2 Scope of Future Work

The three-field mixed formulation we have employed in this thesis is only valid for

nearly incompressible media, since there is still bulk modulus term in the modified

stiffness matrix. Hence, in order to remove volumetric locking for completely

incompressible media, new finite element techniques should be used for this purpose.

Two-field ( u p ) mixed formulation is a possible way according to Hughes [9] and Bathe

[1].

The mixed formulation using discrete shear field collocation method and IBFEM is

developed in this thesis only for elastic static Mindlin plates. The validity of this method

is already evaluated in Chapter 5. Future work can extend this method to nonlinear and

dynamic analysis of Mindlin plates.

The mixed formulation of 2D Mindlin shell, including both in-plane stretching and

bending, is derived in Chapter 6. This can be viewed a special 2D case of shell

113

formulation. Hence, generalizing this method for 3D shell analysis can be a prospective

future direction.

114

APPDENDIX A VOLUMETRIC LOCKING AND SHEAR LOCKING

Volumetric locking and shear locking are the most common numerical phenomena

in the displacement-base finite element formulation. We will give some details in order

to better understand these two frequent phenomena in FEM.

A.1 Volumetric locking

Volumetric locking is very obvious as we already presented at the beginning of

Chapter 3. This phenomenon results from the fact that when the material is near

incompressible status, the Poisson’s ratio, v , tends to 0.5, which makes the bulk

modulus / (3 6 )E v tend to infinity and hence a ill-conditioned stiffness matrix in the

finite element model.

A.2 Shear locking

Shear locking is an error that occurs in finite element analysis. It can be explained

as following.

Let us consider a beam with thickness h under combined shear and bending

deformations. As we have done in Chapter 5, the total strain energy can be expressed

as

2

1 1

2 2

1

2

1 1

2 2

1 1[ ( ) ( )]

2 2

1 1( ) ( )

2 2

T T

stV h

kT

k Sh

S

T T

k k S Sh h

T T

S Sh

T T

S S

U dV dzd

dzdkG

dzd kG dzd

z dzd d

d d

ε Qε ε Qε

εC 0ε γ

γ0

ε Cε γ γ

C γ γ

D γ γ

(A-1)

Where

115

3

2

1 0

1 012 (1 )

0 0 (1 ) / 2

vh E

vv

v

D (A-2)

h G (A-3)

From (A-1), the portion of the strain energy associated with the bending

deformation is proportional to 3h , and the portion related to shear deformations is

proportional to h . If we reduce the thickness h , the value of 3h approaches to zero much

faster than h and as a result, the total strain energy of the beam will largely come from

shear deformation. This is not correct because for a thin beam it is always the bending

deformation which provides most part of the strain energy. So any shear strain resulting

from numerical errors and due to the use of low order shape functions can lead to

wrong results for very thin beam, small displacement.

116

APPENDIX B EQUILIBRIUM EQUATIONS OF 3D ELASTOSTATIC CASE

Figure B-1. Stresses notations and directions

Comparing to these stresses, the body force is of high order, so we neglect it in

above figure B-1. But in the process of deriving the equilibrium equation, we should take

the body force into consideration, since it’s the same order as the variation terms of

stresses.

The equation of equilibrium of force in the x-direction is

' ' ' 0zxyxx x yx zx xdxdydzdydz dydz dxdz dxdz dxdy dxdy b (B-1)

Dividing by dxdydz ,

'' '

0yxx zxyxx zx

xdx dy dz

b

(B-2)

If the block is become smaller and smaller, i.e.,

0, 0, 0dx dy dz (B-3)

By definition of derivative, we have

117

'

0lim x

dx

x x

dx x

(B-4)

'

0limdy

yx yx yx

dy y

(B-5)

'

0limdz

zx zx zx

dz z

(B-6)

Plugging B-4 to B-6 into B-2, we have the first differential equilibrium equation as

0yxx zx

xbx y z

(B-7)

Using the same manner, we can derive the left two equations as

0xy y zy

ybx y z

(B-8)

0yzxz z

zbx y z

(B-9)

118

APPENDIX C DERIVATION OF SHEAR CORRECTION FACTOR

The transverse shear correction coefficient can be derived using equivalence of

shear strain energy before and after using this factor.

Equivalence of Shear strain energy

The first two equilibrium equations of a static case without body force are

0xyx xz

x y z

(C-1)

0xy y yz

x y z

(C-2)

Integrating C-1 and C-2, we can obtain the transverse stress at any point through

the thickness as

/2/2

( )

z

xz

h

xyxxz h

dzx y

(C-3)

/2/2

( )

z

yz

h

xy y

yzh

dzx y

(C-4)

/2xz h

and

/2yz

h

are the shear stresses at the bottom of the plate and both are

equal to zero.

Given the in-plane strains, the in-plane stresses can be obtained using the stress-

strain relationship as

11 12 11 12

12 22 12 22

33 33

0 0

0 0

0 0 0 0

)(

x

x xy

y y

xy xy

yx

C C C C

C C C C

C C

zx

zy

zy x

(C-5)

119

Substituting Equation C-5 into Equations C-3 and C-4, and performing the

integration, we obtain the following results

2 22 22 2

11 12 332 2 2

4[ ( )](1 )

8

y yx xxz

h zC C C

x x y y x y h

(C-6)

2 22 22 2

33 22 122 2 2

4[ ( ) ](1 )

8

y yx xyz

h zC C C

x y x y x y h

(C-7)

For rectangular plate, the transverse stresses parabolically vary through the

thickness h .

Figure C-1. Distribution of transverse shear stress through the thickness

In Equations C-6 and C-7, we have following relations

22

11 12 11 122 3

22

33 332 3

2 2

22 12 22 122 3

22

33 332

12( )

12( ) ( ( ))

12( )

( ) ( (

y yx x x

y y xyx x

y y yx x

yx x

MC C C C

x x y x x y h x

MC C

y x y y y x h y

MC C C C

y x y y y x h y

C Cx y x x y

3

12))

y xyM

x h x

(C-8)

Thus, xz and yz can be further expressed as

2

2

3 4[ ](1 )

2

xyxxz

MM z

h x y h

(C-9)

120

2

2

3 4[ ](1 )

2

y xy

yz

M M z

h y x h

(C-10)

From Equation C-8, C-9 and C-10, we finally have

2

2

3 4(1 )

2xz x

zS

h h

(C-11)

2

2

3 4(1 )

2yz y

zS

h h (C-12)

Generally, the total strain energy per unit area due to transverse shears is

2/2 /2 2

/2 /2

1 1( ) ( )

2 2

h hyzxz

xz xz yz yzSAh h

U dz dzG G

(C-13)

Plugging Equations C-11 and C-12 into Equation C-13, we have

2 21 6( )

2 5x ySA S S

G hU (C-14)

In shear deformable plate theory, we assumed at the beginning that

xz xz

yzyz

SGhS

(C-15)

Hence, we also have the total strain energy per unit area due to transverse shears

as

2 21 1( ) ( )

2 2SA xz xz yz yz xz yzS S S S

GhU

(C-16)

Comparing Equation C-14 and Equation C-16 for total strain energy due to the

same transverse shears, we have

5

6 (C-17)

121

APPENDIX D DERIVATION OF THE JACOBIAN MATRIX IN IBFEM

The Jacobian matrix for 2D is in the format as following:

x y

x y

J (D-1)

We can modified above expression for J of 9-node element as

9 9

1 19 9

1 1

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

I I

w wI I

eI I

w wI I

w w w w w w w w w

w w w w w w w w w

N Nx yx y

x y N Nx y

N N N N N N N N N

N N N N N N N N N

J

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

x yx yx yx yx yx yx yx yx y

(D-2)

In IBFEM, we have following relations

1 8 4 1 5 9 7 2 2 6 3 3

1 5 2 1 8 9 6 2 4 7 3 3

2 1 3

2 1 3

, ,

, ,

2

2

x x x X x x x X x x x X

y y y Y y y y Y y y y Y

Y Y Y

X X X

(D-3)

Using these relations in above expression for J , we can get

1 3

1 3

02

02

X X

Y Y

J (D-4)

122

Figure D-1. Global coordinate and Local coordinate in IBFEM

123

APPENDIX E FORMULATION OF MINDLIN PLATE ELEMENTS

E.1 Element Q4D4

Figure E-1. Collocation constraints on a 4-node Lagrange element

Shape function for displacement and rotations

The shape functions for rotation are the same as those for the displacement with

corresponding node.

These shape functions for displacement are

1 2

3 4

1 1(1 )(1 ), (1 )(1 ),

4 4

1 1(1 )(1 ), (1 )(1 ),

4 4

w w

w w

N N

N N

(E-1)

The shape functions for rotations are

1 1 1 2 1 2

3 1 3 4 1 4

, ,

, ,

x y w x y w

x y w x y w

N N N N N N

N N N N N N

(E-2)

Shape function for shear fields

124

The shape functions for shear fields can be derived using the same structure of

Lagrange polynomials. And these shape functions are

1 1 2 2

3 3 4 4

1 1, 0, 0, ,

2 2

1 1, 0, 0, ,

2 2

s s s s

s s s s

N N N N

N N N N

(E-3)

Derivation of the prescribed shear force value

Figure E-2. Interpolation nodes for Q4D4 element

For above Q4D4 element, the displacement and the rotation can be approximated

as ˆwwN w and ˆ

N

Using

ˆ ˆˆ ˆ( , )

ˆ ˆˆ ˆ( , )

w wx

w wy

Sx x

Sy y

N Nw N w N

N Nw N w N

, (E-4)

we can derive the prescribed values for shear force at certain nodes using prescribed w

and as

125

1 4

3 4

1 22 1

01

1 4

10

3

01

2

ˆ ˆ

2

ˆ ˆ

2

ˆ ˆˆ ˆˆ ˆˆ [( ) ]2

ˆ ˆˆ ˆˆ [( ) ]

2

ˆˆ ˆˆ [( )

I w x xx

y yII wy

III w xx

w w

w w

w wS

x x

Sy y

Sx x

Nw N

Nw N

Nw N

2 3

4

2 3

10

ˆ ˆ

2

ˆ]

2

ˆ ˆˆ ˆˆ [( ) ]

2

x

y yIV wy

w wS

y y

Nw N

(E-5)

Where N , N , ˆ and ˆ

are the and components of N and .

In IBFEM, we have

2

2

x a

y b

, where a andb are the length of edges of an element

in x and y direction, respectively.

Writing in the matrix form, we have

ˆ ˆˆw

S Q w Q

where

ˆ ˆ ˆ ˆ ˆT

I II III IV

x y x yS S S SS (E-6)

0 0

0 0

0 0

0 0

1 1

1 1

1 1

1 1

w

a a

b b

a a

b b

Q (E-7)

126

1 0 1 0 0 0 0 0

0 1 0 0 0 0 0 11

0 0 0 0 1 0 1 02

0 0 0 1 0 1 0 0

Q (E-8)

The same procedure is used to get the collocated shear term and its shape

function, and we will not give more details in the following content about the derivation

for each element. And also, the shape functions for the displacement are the same as

traditional ones. The shape functions for rotations are the same as those of

displacement at the corresponding nodes.

E.2 Element Q5D6

Figure E-3. Collocation constraints on a 5-node Serendipity element


1 1 2 2

3 3 4 4

5 5 6 6

( 1) ( 1), 0, 0, ,

2 2

( 1) ( 1), 0, 0, ,

2 2

( 1)( 1), 0, 0, ( 1)( 1),

s s s s

s s s s

s s s s

N N N N

N N N N

N N N N

(E-9)

127


wQ and Q matrices

ˆ ˆ ˆ ˆ ˆ ˆ ˆT

I II III IV V VI

x y x y x yS S S S S SS (E-10)

0 0 0

0 0 0

0 0 0

0 0 0

0

0

1 1

1 1

1 1

1 1

1 1 1 1

2 2 2 2

1 1 1 1

2 2 2 2

w

a a

b b

a a

b b

a a a a

b b b b

Q (E-11)

1 0 1 0 0 0 0 0 0 0

0 0 0 1 0 1 0 0 0 0

0 0 0 0 1 0 1 0 0 01

0 1 0 0 0 0 0 1 0 02

0 0 0 0 0 0 0 0 2 0

0 0 0 0 0 0 0 0 0 2

Q (E-12)

128

E.3 Element Q8D8

Figure E-5. Collocation constraints on an 8-node Serendipity element

Shape functions for shear fields

1 1 2 2

3 3 4 4

5 5 6 6

7 7 8 8

1 1 1 1( )( 1), 0, ( )( 1), 0,

2 2 2 2

1 1 1 10, ( 1)( ), 0, ( 1)( ),

2 2 2 2

1 1 1 1( )( 1), 0, ( )( 1), 0,

2 2 2 2

1 1 10, ( 1)( ), 0,

2 2 2

s s s s

s s s s

s s s s

s s s s

N N N N

N N N N

N N N N

N N N N

1

( 1)( ),2

(E-13)


wQ and Q matrices

129

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆT

I II III IV V VI VII VIII

x x y y x x y yS S S S S S S SS (E-14)

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

1 1

1 10

1 1

1 10 0

21 1

1 1

1 10 0

1 1

w

a a

a a

b b

b b

a a

a a

b b

b b

Q (E-15)

3 0 1 0 0 0 0 0 6 0 0 0 0 0 0 0

1 0 3 0 0 0 0 0 6 0 0 0 0 0 0 0

0 0 0 3 0 1 0 0 0 0 0 6 0 0 0 0

0 0 0 1 0 3 0 0 0 0 0 6 0 0 0 01

0 0 0 0 3 0 1 0 0 0 0 0 6 0 0 08

0 0 0 0 1 0 3 0 0 0 0 0 6 0 0 0

0 1 0 0 0 0 0 3 0 0 0 0 0 0 0 6

0 3 0 0 0 0 0 1 0 0 0 0 0 0 0 6

Q (E-16)

130

E.4 Element Q9D12


Shape functions for shear fields

1 1 2 2

3 3 4 4

5 5 6 6

1 1 1 1( ) ( 1), 0, ( ) ( 1), 0,

2 2 2 2

1 1 10, ( 1)( ), 0, ( 1)( 1)( ),

2 2 2

1 1 10, ( 1)( ), ( )( 1)( 1), 0,

2 2 2

s s s s

s s s s

s s s s

N N N N

N N N N

N N N N

(E-17)

7 7 8 8

9 9 10 10

11 11 12 12

1 1 1( )( 1)( 1), 0, 0, ( 1)( ),

2 2 2

1 1 10, ( 1)( 1)( ), 0, ( 1)( ),

2 2 2

1 1 1 1( ) ( 1), 0, ( ) ( 1), 0,

2 2 2 2

s s s s

s s s s

s s s s

N N N N

N N N N

N N N N

(E-18)

wQ and Q matrices

131


ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆT

I II III IV V VI VII VIII IX X XI XII

x x y y y x x y y y x xS S S S S S S S S S S SS (E-19)

1

8

1 0 3 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0

3 0 1 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0

0 0 0 3 0 1 0 0 0 0 0 6 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 3 0 0 0 1 0 0 0 6

0 3 0 0 0 0 0 1 0 0 0 0 0 0 0 6 0 0

0 0 0 0 0 0 0 0 0 0 3 0 0 0 1 0 6 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 3 0 6 0

0 0 0 1 0 3 0 0 0 0 0 6 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 3 0 0 0 6

0 1 0 0 0 0 0 3 0 0 0 0 0 0 0 6 0 0

0 0 0 0 3

Q

0 1 0 0 0 0 0 6 0 0 0 0 0

0 0 0 0 1 0 3 0 0 0 0 0 6 0 0 0 0 0

(E-20)

132

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

1 1

1 1

1 1

1 1

1 1

1 1

21 1

1 1

1 1

1 1

1 1

1 1

w

a a

a a

b b

b b

b b

a a

a a

b b

b b

b b

a a

a a

Q

(E-21)

E.5 Element Q16D24



133

1 1

2 2

3 3

4 4

5 5

81 1 1 2( )( )( 1) ( ), 0,

128 3 3 3

81 1 1 2 2( )( )( 1)( )( ), 0,

64 3 3 3 3

81 1 1 2( )( )( 1) ( ), 0,

128 3 3 3

81 1 1 20, ( )( )( 1) ( ),

128 3 3 3

81 1 10, ( )(

64 3

s s

s s

s s

s s

s s

N N

N N

N N

N N

N N

6 6

2 2)( 1)( )( ),

3 3 3

81 1 1 20, ( )( )( 1) ( ),

128 3 3 3s sN N

(E-22)

7 7

8 8

9 9

10 10

11 11

81 1 1 2( )( )( 1) ( ), 0,

128 3 3 3

81 1 1 2 2( )( )( 1)( )( ), 0,

64 3 3 3 3

81 1 1 2( )( )( 1) ( ), 0,

128 3 3 3

81 1 1 20, ( )( )( 1) ( ),

128 3 3 3

81 10, (

128

s s

s s

s s

s s

s s

N N

N N

N N

N N

N N

12 12

1 2 2)( )( 1)( )( ),

3 3 3 3

81 1 1 20, ( )( )( 1) ( ),

128 3 3 3s sN N

(E-23)

13 13

14 14

15 15

16 16

17 17

243 1 20, ( )( 1)( 1) ( ),

128 3 3

243 1 20, ( )( 1)( 1) ( ),

128 3 3

243 1 2( )( 1)( 1) ( ), 0,

128 3 3

243 1 2( )( 1)( 1) ( ), 0,

128 3 3

2430, (

128

s s

s s

s s

s s

s s

N N

N N

N N

N N

N N

18 18

1 2)( 1)( 1) ( ),

3 3

243 1 20, ( )( 1)( 1) ( ),

128 3 3s sN N

(E-24)

134

19 19

20 20

21 21

22 22

23

243 1 2( )( 1)( 1) ( ), 0,

128 3 3

243 1 2( )( 1)( 1) ( ), 0,

128 3 3

243 1 2 2( )( 1)( 1)( )( ), 0,

64 3 3 3

243 1 2 20, ( )( 1)( 1)( )( ),

64 3 3 3

243(

64

s s

s s

s s

s s

s

N N

N N

N N

N N

N

23

24 24

1 2 2)( 1)( 1)( )( ), 0,

3 3 3

243 1 2 20, ( )( 1)( 1)( )( ),

64 3 3 3

s

s s

N

N N

(E-25)

wQ and Q matrices

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

TI II III IV V VI VII VIII IX X XI XII

x x x y y y x x x y y yXIII XIV XV XVI XVII XVIII XIX XX XXI XXII XXIII XXIV

y y x x y y x x x y x y

S S S S S S S S S S S S

S S S S S S S S S S S S

S (E-26)

23 / 1 / 0 0 21 / 3 / 0 0 0 0 0 0 0 0 0 0

1 / 1 / 0 0 27 / 27 / 0 0 0 0 0 0 0 0 0 0

1 / 23 / 0 0 3 / 21 / 0 0 0 0 0 0 0 0 0 0

0 23 / 1 / 0 0 0 21 / 0 0 0 3 / 0 0 0 0 0

0 1 / 1 / 0 0 0 27 / 0 0 0 27 / 0 0 0 0 0

0 1 / 23 / 0 0 0 3 / 0 0 0 21 / 0 0 0 0 0

0 0 23 / 1 / 0 0 0 0 0 0 0 0 0 0 21 / 3 /

1

8w

a a a a

a a a a

a a a a

b b b b

b b b b

b b b b

a a a

Q

0 0 1 / 1 / 0 0 0 0 0 0 0 0 0 0 27 / 27 /

0 0 1 / 23 / 0 0 0 0 0 0 0 0 0 0 3 / 21 /

1 / 0 0 23 / 0 0 0 0 0 3 / 0 0 0 21 / 0 0

1 / 0 0 1 / 0 0 0 0 0 27 / 0 0 0 27 / 0 0

23 / 0 0 1 / 0 0 0 0 0 21 / 0 0 0 3 / 0 0

0 0 0 0 23 / 0 0 0 21 / 0 0 0 3 / 0 0 1 /

0 0 0 0 0 23 / 0 21 / 0 0 0 3 / 0 0 1 / 0

0 0 0

a

a a a a

a a a a

b b b b

b b b b

b b b b

b b b b

b b b b

0 0 0 23 / 21 / 3 / 1 / 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 23 / 21 / 3 / 1 / 0 0

0 0 0 0 0 1 / 0 3 / 0 0 0 21 / 0 0 23 / 0

0 0 0 0 1 / 0 0 0 3 / 0 0 0 21 / 0 0 23 /

0 0 0 0 0 0 0 0 0 0 1 / 3 / 21 / 23 / 0 0

0 0 0 0 0 0 1 / 3 / 21 / 23 / 0 0 0 0 0 0

0 0 0 0 0 0 1 / 27 / 27 / 1 / 0 0 0 0 0 0

0 0 0 0 0 1

a a a a

a a a a

b b b b

b b b b

a a a a

a a a a

a a a a

/ 0 27 / 0 0 0 27 / 0 0 1 / 0

0 0 0 0 0 0 0 0 0 0 1 / 27 / 27 / 1 / 0 0

0 0 0 0 1 / 0 0 0 27 / 0 0 0 27 / 0 0 1 /

b b b b

a a a a

b b b b

(E-27)

135

5 0 1 0 0 0 0 0 15 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 1 0 0 0 0 0 9 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 5 0 0 0 0 0 5 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 5 0 1 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 00 0 0 1 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 00 0 0 1 0 5 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 15 0 0 0 0 0

1

16

Q

0 0 0 00 0 0 0 5 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 50 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 90 0 0 0 1 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 150 1 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 15 0 0 00 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 00 5 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 5 0

0 0

0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 5 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 5 0 0 0 15 0 0 0 0 0 0 0 5 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 5 0 15 0 5 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 15 0 5 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 5 0 0 0 0 0 0 0 15 0 0 0 0 0 5 00 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 15 0 0 0 0 0

000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 5 0 15 0 5 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 0 5 0 15 0 5 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 0 9 0 9 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 9 0 0 0 0 0 0 0 9 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 9 0 9 0 1 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 0 0 0

000000000100005000001

(E-28)


136

LIST OF REFERENCES

[1] Bathe, K.J. (1996). Finite Element Procedures, 2nd ed., Prentice-Hall, New Jersey.

[2] Bathe, K.J., Dvorkin, E.N. (1985). A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. International Journal of Numerical Methods in Engineering, 21(2), 367-383.

[3] Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P. (1996). Meshless Methods: An Overview and Recent Developments. Computer Methods in Applied Mechanics and Engineering, 139, 3-47.

[4] Belytschko, T., Parimi, C., Moes, N., Usui, S., Sukumar, N. (2003). Structured extended finite element methods for solids defined by implicit surfaces. International Journal for Numerical Methods in Engineering, 56(4), 609-635.

[5] Clark, B.W., Anderson, D.C. (2002). Finite Element Analysis in 3D Using Penalty Boundary Method. Proceedings of Design Engineering Technical Conferences, Montreal, Canada.

[6] Fish, J., Belytschko, T. (2007). A first course in finite elements, John Wiley and Sons.

[7] Hinton, E., Huang, H.C. (1986). A family of quadrilateral Mindlin plate elements with substitute shear strain fields. Computers and Structures, 23 (3), 409-431.

[8] Hrabok, M.M., Hrudey, T.M. (1984). A review and catalogue of plate bending finite elements. Computers and Structures, 19(3), 479-495.

[9] Hughes, T.J.R. (2000). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, 2nd edition. Dover Editions, New Jersey.

[10] Kantorovich, L.W., Krylov, W.I. (1956). Näherungsmethoden der Höheren Analysis. VEB Deutscher Verlag der Wissenschaften, Berlin.

[11] Kumar, A.V., Burla, R., Padmanabhan, S., Gu, L. (2008). Finite element analysis using nonconforming mesh. Journal of Computing and Information Science in Engineering, 8, 031005-1.

[12] Kumar, A.V., Lee, J. (2003). Step function representation of solid models and application to mesh free engineering analysis, Journal of Mechanical Design. Transactions of the ASME, 128(1), 46-56.

[13] Kumar, A.V., Padmanabhan, S., Burla, R. (2008). Implicit boundary method for finite element analysis using non-conforming mesh or mesh. International Journal for Numerical Methods in Engineering, 74 (9), 1421-1447.

137

[14] Liew, K.M., Han, J.-B. (1997). Bending analysis of simply supported shear deformable skew plates. Engineering Mechanics, 123(3), 214-221.

[15] Rvachev, V.L., Shieko, T.I. (1995). R-functions in Boundary Value Problems in Mechanics. Applied Mechanics Review, 48, 151-188.

[16] Sengupta, D. (1995). Performance study of a simple finite element in the analysis of skew rhombic plates. Computers and Structures, 54 (6), 1173-1182.

[17] Taylor, R.L., Govindjee, S. (2004. )Solution of clamped rectangular problems. Communications in Numerical Methods in Engineering. 20, 757-765.

[18] Timoshenko, S., Woinowsky-Kreiger, S. (1987). Theory of Plates and Shells, second edition. McGraw-Hill, New York.

[19] Wang, C.M., Reddy, J.N., Lee, K.H. (2000). Shear Deformable Beams and Plates: Relationships with Classical Solutions. Elsevier, Oxford.

[20] Yang, T.Y., Saigal, S., Masud, A., Kapania, R.K. (2000). A survey of recent shell finite elements. International Journal for Numerical Methods in Engineering, 47, 101-127.

[21] Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z. (2005). The Finite Element Method for Solid and Structural Mechanics, sixth edition. Butterworth-Heinemann.

[22] Zienkiewicz, O.C., Xu, Z., Zeng, L.F., Samuelson, Wilbert, N.E. (1993). Linked interpolation for Reissner – Mindlin plate elements: Part I - A simple quadrilateral. International Journal for numerical Methods in Engineering, 36, 3043 – 3056.

138

BIOGRAPHICAL SKETCH

Hailong Chen was born and brought up in Jiangxi, China. He graduated with a

Bachelor of Science in engineering degree in mechanical engineering from Shanghai

Normal University, Shanghai, China in June 2010 with honors. After graduation, he

directly enrolled in the master’s program in Mechanical and Aerospace Engineering at

the University of Florida in fall 2010 with Achievement Award for New Engineering

Graduate Students. In 2011, he and Xiao Zhou got married in China. His area of interest

includes application of FEM in plates and shells, numerical methods and uncertainty

quantification.

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