New Trends in Thin Structures: Formulation, Optimization and Coupled Problems (eBook)

Title Page 
3 
Copyright Page 
4 
PREFACE 
5 
Table of Contents 
7 
A plate theory as a mean to compute precise 3D solution includung edge effects and related issues 
8 
1 Introduction 
8 
2 The classical theory of Laminated Plated or CLT 
10 
2.1 Description of the plate and of the constitutive relation 
10 
2.2 Basic assumptions of the CLT 
12 
2.3 The equations of the CLT in general forces 
13 
2.4 Formulation of the plate problem 
15 
3 Reconstruction of an approximation of the inner 3D solution from the plate solution 
16 
3.1 Reconstruction of a quasi-admissible stress field 
16 
3.2 Reconstruction of an associated approximation ofthe displacement field 
17 
3.3 Example of an isotropic plate loaed only on its lateral surfaces 
18 
3.4 Expression and properties of the plated for an isotropic plate loaed only on its lateral surfaces 
19 
3.5 Iterative reconstruction of the 3D expression of the displacement and stresses: principle adn notations 
19 
4 Edge effects analysis 
23 
4.1 Analysis of boundary conditions 
24 
4.2 Edge effects computation 
25 
4.3 Example with isotropic plate 
29 
5 Conclusions 
32 
Bibliography 
33 
A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type 
36 
1 Introduction 
36 
2 Nonlinear Kirchhoff shell theory 
37 
2.1 Kinematics 
37 
2.2 Strains 
39 
2.3 Stresses 
44 
2.4 Power 
45 
2.5 Weak form of the local equilibrium equation 
47 
2.6 Statics 
49 
2.7 Boundary conditions 
50 
2.7.1 Free and unloaded edges 
54 
2.7.2 Fixed but free to rotate edges 
54 
2.7.2 Clamped edges 
54 
2.8 Complete weak form of the equilibrium equations 
55 
2.9 Tangent weak form of the equilibrium equations 
55 
3 Elastic constitutive equations 
57 
3.1 Plane stress condition 
57 
3.2 General elastic isotropic material 
58 
3.3 Neo-Hookean material 
59 
3.4 Incompressible isotropic elastic material 
60 
4 Conclusions 
61 
Bibliography 
61 
A Bean Finite Element for Nonlinear Analysis of Shape Memory Alloy Devices 
66 
1 Introduction 
66 
2 Geometrically exact beam theory 
69 
2.1 Preliminaries: the rotation manifold 
69 
2.2 Reissner's beam model: kinematics 
72 
2.3 Reissner's beam model equilibrium 
74 
3 Total lagrangian finite element scheme 
79 
3.1 Consistent linearization 
81 
3.2 Remarks on the chosen finite element model 
82 
4 Shape Menory Alloy Constitutive Model 
82 
4.1 Time-discrete frame and solutions algorithm 
87 
5 Cross section integration algorithm 
89 
6 Numerical tests 
91 
6.1 Calibration of the proposed scheme 
91 
6.2 Cantilever beam 
93 
6.3 Elbow beam 
94 
6.4 Spring actuator 
94 
7 Consluding remarks 
98 
Bibliography 
99 
A Unifield Approach for the Nonlinear Dynamics of Rods and Shells Using an Exact Conserving Integration Algorithm 
105 
1 Introduction 
105 
2 Parameterization of the rotation field 
106 
3 Rod and Shell Dynamics 
108 
3.1 Kinematics 
108 
3.2 Strains 
110 
3.3 Stresses 
111 
3.4 External forces 
112 
3.5 Linear and angular momentum 
113 
3.6 Equations of motion 
113 
3.7 Weak form of the equations of motion 
115 
3.8 Internal and kinetic energy 
116 
4 Time increment 
121 
4.1 Incremental displacements and rotations 
122 
4.2 Incremental strains and strain rates 
123 
4.3 Increments of momentum, kinetic and strain energy 
124 
5 Time integration Algorithm 
125 
5.1 Time collocation of the equations of motion 
125 
5.2 Time approximations and algorithmic weak form 
126 
5.3 Conservation of linear and angular momentum 
127 
5.4 Conservation of energy 
127 
5.5 Tangent of the weak form 
129 
6 Finite element implementation and numerical examples 
129 
6.1 Large overall motion of an inclined beam 
132 
6.2 Free vibration of a beam in 3-D space 132
6.3 Dynamics of a satellite-like structure 
133 
6.4 Free vibration of a hemispherical shell 
133 
7 Conclusions 
136 
Bibliography 
137 
Advanced Numerical Methods for the Form Finding and Patterning of Membrane Structures 
139 
1 Introduction 
139 
2 "Correct" Continnum Mechanical Description of Membrane Structures 
141 
3 "Classical" Formfinding 
142 
3.1 Updated Reference Strategy 
144 
3.2 Prestress as shaping parameter 
146 
4 Optimezed cutting pattern generation 
148 
4.1 Method I: Least-squares optimization 
149 
4.2 Method II: Minimization of "stress difference" energy 
150 
4.3 Comparison of both methods 
152 
5 Intruducing cutting patterns in form finding and structural analysis 
152 
5.1 Form finding through cutting patterns 
153 
5.2 Structural analysis in consideration of cutting patterns 
154 
5.3 Conclusions 
157 
Bibliography 
158 
Contact between Beams and Shells 
161 
1 Introduction 
161 
2 Kinematics 
162 
2.1 Nomarl contact 
162 
2.2 Tangential contact 
166 
3 Variation of the Gap in Normal and Tangential Directions 
168 
4 Contact Contribution to Weak Form 
168 
4.1 Weak form 
168 
4.2 Constitutive relations 
169 
4.3 Algorithm for friction 
170 
5 Finite Element Formulation 
172 
6 Contact Search 
176 
7 Concluding Remarks 
178 
Bibliography 
178 
Advances in Computatonal Fluid-Thin-Walled-Structure Interaction Formulations and Solvers 
181 
1 Introduction 
181 
2 Problem definition 
182 
3 Artificial added mass effect 
185 
4 Efficient solver for FSI-block preconditionaled Newton-Krylov schemes 
190 
4.1 Block Gauss-Seidel process 
191 
4.2 Preconditioning based on Gauss-Seidel 
192 
4.3 AMG with block Gauss-Seidel smoothing 
192 
5 Examples of ALE-based FSI simulations 
194 
5.1 Elexible flag behind rigid obstacle 
194 
5.2 Abdominal Arotic Aneurysm 
197 
6 Robust formulation for large deformation FSI - an XFEM based fixed -gird approach 
197 
6.1 Introduction 
197 
6.2 Velocity and stress discontinuity across the fluid-structure interface 
200 
6.3 Fluid-interface coupling and interface discretization 
203 
6.4 Interface-structure coupling and FSI formulation 
205 
7 Conclusion 
207 
Bibliography 
207 
Advances in Subdivision Finite Elements for Thin Shells 
210 
1 Introduction 
210 
2 Subdivision Schemes for Smooth Interpolation 
211 
2.1 Univatiate B-Splines and Subdivision 
212 
2.2 Subdivision Surfaces 
215 
3 Thin-Shell Equations 
221 
3.1 Kinematics 
221 
3.2 Thin-shell Energy Functional and its Discretization 
222 
4 Examples 
224 
4.1 Pinched Cylinder 
224 
4.2 Pinched Square Tube 
225 
4.3 Spherical Panel with Stiffeners 
225 
4.4 Composite Plate 
227 
5 Conclusions 
229 
Bibliography 
230