Natural Element Method for the Simulation of Structures and Processes

, by Chinesta, Francisco; Cescotto, Serge; Cueto, Elias; Lorong, Philippe

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ISBN: 9781848212206 | 1848212208
Cover: Hardcover
Copyright: 2/21/2011

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This book presents a recent state of the art on the foundations and applications of the meshless natural element method in computational mechanics, including structural mechanics and material forming processes involving solids and Newtonian and non-Newtonian fluids.

Foreword	p. ix
Acknowledgements	p. xi
Introduction	p. 1
SPH method	p. 3
RKPM method	p. 5
Conditions of reproduction	p. 6
Correction of the kernel	p. 7
Discrete form of the approximation	p. 9
MLS based approximations	p. 10
Final note	p. 11
Basics of the Natural Element Method	p. 13
Introduction	p. 13
Natural neighbor Galerkin methods	p. 14
Interpolation of natural neighbors	p. 14
Discretization	p. 18
Properties of the interpolant based on natural neighbors	p. 19
Exact imposition of the essential boundary conditions	p. 22
Introduction to alpha shapes	p. 23
CNEM approaches	p. 25
Mixed approximations of natural neighbor type	p. 27
Considering the restriction of incompressibility	p. 28
Mixed approximations in the Galerkin method	p. 32
Natural neighbor partition of unity	p. 33
Partition of unity method	p. 33
Enrichment of the natural neighbor interpolants	p. 35
High order natural neighbor interpolants	p. 40
Hiyoshi-Sugihara interpolant	p. 40
The De Boor algorithm for B-splines	p. 43
B-spline surfaces and natural neighboring	p. 44
Some definitions	p. 44
Surface properties	p. 47
The case of repeated nodes	p. 47
Numerical Aspects	p. 49
Searching for natural neighbors	p. 49
Calculation of NEM shape functions of the Sibson type	p. 52
Stage-1: insertion of point x in the existing constrained Voronoi diagram (CVD)	p. 55
Look for a tetrahedron which contains point x	p. 55
Note concerning the problem of flat tetrahedrons	p. 55
Stage-2: calculation of the volume measurement common to c_x and c_v	p. 57
By the recursive Lasserre algorithm	p. 57
By means of a complementary volume	p. 60
By topological approach based on the CVD	p. 62
By topological approach based on the Constrained Delaunay tetrahedization (CDT)	p. 65
Using the Watson algorithm	p. 66
Comparative test of the various algorithms	p. 70
Numerical integration	p. 71
Decomposition of shape function supports	p. 71
Stabilized nodal integration	p. 73
Discussion in connection with various quadratures	p. 74
2D patch test with a technique of decomposition of shape function supports	p. 74
2D patch test with stabilized nodal integration	p. 75
3D patch tests	p. 77
NEM on an octree structure	p. 80
Structure of the data	p. 82
Description of the geometry	p. 82
Interpolation on a quadtree	p. 85
Numerical integration	p. 85
Application of the boundary conditions - interface conditions	p. 87
Dirichlet-type boundary conditions: use of R-functions	p. 87
Neumann-type boundary conditions	p. 90
Partition of unity method	p. 90
Applications in the Mechanics of Structures and Processes	p. 93
Two-and three-dimensional elasticity	p. 93
Indicators and estimators of error: adaptivity	p. 96
Meshless methods and adaptation	p. 96
Methodology of adaptive refinement for linear elasticity in statics	p. 98
Formulation in static linear elasticity	p. 99
The first indicator based on the rebuilding of discontinuous stress fields	p. 102
The second indicator based on the rebuilding of continuous fields of strain	p. 103
Refinement strategy based on Voronoi cells	p. 105
Metal extrusion	p. 107
Viscoplastic model for the extrusion of aluminum	p. 108
3D simulation of the extrusion of a cross-shaped profile	p. 111
Friction stir welding	p. 113
Constitutive equation	p. 117
Mechanical model	p. 118
Numerical results	p. 120
Models and numerical treatment of the phase transition: foundry and treatment of surfaces	p. 123
Introducing motion discontinuity	p. 123
Formulation of the thermal problem with phase transition	p. 124
CNEM discretization	p. 126
Adiabatic shearing, cutting, and high speed blanking	p. 136
General context of the implementation of the large transformations	p. 140
Updated Lagrangian formulation	p. 141
Processing the additional points	p. 141
Time integration scheme	p. 142
Processing of the contact	p. 142
Integration of the constitutive equation	p. 142
Applications	p. 145
Taylor's bar	p. 145
Adiabatic shearing	p. 148
Conclusions on the numerical simulation of shearing	p. 158
A Mixed Approach to the Natural Elements	p. 159
Introduction	p. 159
The Fraeijs de Veubeke variational principle for linear elastic problems	p. 161
Field decomposition	p. 164
Discretization	p. 166
Discretized equations	p. 170
Matrix solution for linear elastic problems	p. 172
Numerical integration	p. 176
Linear elastic patch tests	p. 178
Application 1: pure bending of a linear elastic beam	p. 182
Application 2: square domain with circular hole	p. 185
Mixed approach to nonlinear problems	p. 187
Step-by-step solution of the discretized nonlinear equations	p. 192
Example of an elastoplastic material	p. 195
Application: pure bending of an elastoplastic beam	p. 196
Conclusion	p. 199
Flow Models	p. 201
Natural element method in fluid mechanics: updated Lagrangian approach	p. 201
Mechanical model of a Newtonian fluid flow	p. 201
Free and moving surfaces	p. 202
Use of the characteristics method	p. 204
Short-fiber suspensions flow	p. 206
Flow kinematics	p. 207
Coupling of a particle method with ¿-NEM	p. 208
Breaking dam problem	p. 208
Multi-scale approaches	p. 209
Mechanical model	p. 214
FENE model	p. 216
Doi-Edwards model	p. 217
Integration of the model	p. 218
Functional approximation	p. 218
Discretization of the model	p. 218
Resolution algorithm	p. 219
Some results	p. 220
Startup of a simple shear flow of an FENE fluid model	p. 220
Flow from an extrusion die and FENE-fluid	p. 220
Startup of a simple shear flow of a Doi-Edwards fluid model	p. 223
Conclusion	p. 225
Bibliography	p. 227
Index	p. 239
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Natural Element Method for the Simulation of Structures and Processes

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