Practical Aspects of Finite Element Modelling of Polymer Processing

Contents......Page 8
Preface......Page 14
1 THE BASIC EQUATIONS OF NON-NEWTONIAN FLUID MECHANICS......Page 16
1.1.2 Equation of motion......Page 17
1.1.4 Constitutive equations......Page 18
1.2.1 Newtonian fluids......Page 19
1.2.2 Generalized Newtonian fluids......Page 20
1.3 Inelastic Time-Dependent Fluids......Page 23
1.4.1 Model (material) parameters used in viscoelastic constitutive equations......Page 24
1.4.2 Differential constitutive equations for viscoelastic fluids......Page 26
1.4.3 Single-integral constitutive equations for viscoelastic fluids......Page 28
1.4.4 Viscometric approach – the (CEF) model......Page 29
References......Page 30
2 WEIGHTED RESIDUAL FINITE ELEMENT METHODS – AN OUTLINE......Page 32
2.1 Finite Element Approximation......Page 34
2.1.1 Interpolation models......Page 35
2.1.2 Shape functions of commonly used finite elements......Page 38
2.1.3 Non-standard elements......Page 42
2.1.4 Local coordinate systems......Page 44
2.1.5 Order of continuity of finite elements......Page 47
2.1.6 Convergence......Page 48
2.1.7 Irregular and curved elements – isoparametric mapping......Page 49
2.1.8 Numerical integration......Page 53
2.1.9 Mesh refinement – h- and p-versions of the finite element method......Page 55
2.2 Numerical Solution of Differential Equations by the Weighted Residual Method......Page 56
2.2.1 Weighted residual statements in the context of finite element discretizations......Page 57
2.2.2 The standard Galerkin method......Page 58
2.2.2* Galerkin finite element procedure – a worked example......Page 59
2.2.3 Streamline upwind Petrov–Galerkin method......Page 68
2.2.3* Application of upwinding – a worked example......Page 70
2.2.5 Solution of time-dependent problems......Page 79
References......Page 83
3.1 Solution of the Equations of Continuity and Motion......Page 86
3.1.1 The U–V–P scheme......Page 87
3.1.2 The U–V–P scheme based on the slightly compressible continuity equation......Page 89
3.1.3 Penalty schemes......Page 90
3.1.5 Application of Green's theorem – weak formulations......Page 92
3.2 Modelling of Viscoelastic Flow......Page 94
3.2.1 Outline of a decoupled scheme for the differential constitutive models......Page 96
Derivation of the working equations......Page 98
3.2.2 Finite element schemes for the integral constitutive models......Page 101
3.2.3 Non-isothermal viscoelastic flow......Page 104
3.3 Solution of the Energy Equation......Page 105
3.4.1 Velocity and surface force (stress) components......Page 108
Inlet conditions......Page 110
Solid walls......Page 111
Exit conditions......Page 112
3.4.2 Slip-wall boundary conditions......Page 113
3.4.3 Temperature and thermal stresses (temperature gradients)......Page 114
3.5.1 VOF method in 'Eulerian' frameworks......Page 116
3.5.2 VOF method in 'Arbitrary Lagrangian–Eulerian' frameworks......Page 117
3.5.3 VOF method in 'Lagrangian' frameworks......Page 119
References......Page 123
4.1.1 Governing equations in two-dimensional Cartesian coordinate systems......Page 126
4.1.2 Governing equations in two-dimensional polar coordinate systems......Page 127
4.1.3 Governing equations in axisymmetric coordinate systems......Page 128
4.1.4 Working equations of the U–V–P scheme in Cartesian coordinate systems......Page 129
4.1.5 Working equations of the U–V–P scheme in polar coordinate systems......Page 131
4.1.6 Working equations of the U–V–P scheme in axisymmetric coordinate systems......Page 132
4.1.7 Working equations of the continuous penalty scheme in Cartesian coordinate systems......Page 133
4.1.8 Working equations of the continuous penalty scheme in polar coordiante systems......Page 135
4.1.9 Working equations of the continuous penalty scheme in axisymmetric coordinate systems......Page 136
4.1.10 Working equations of the discrete penalty scheme in Cartesian coordinate systems......Page 138
4.1.11 Working equations of the least-squares scheme in Cartesian coordinate systems......Page 140
4.2 Variations of Viscosity......Page 141
4.3 Modelling of Steady-State Viscometric Flow – Working Equations of the Continuous Penalty Scheme in Cartesian Coordinate Systems......Page 142
4.4 Modelling of Thermal Energy Balance......Page 143
4.4.1 Working equations of the streamline upwind (SU) scheme for the steady-state energy equation in Cartesian, polar and axisymmetric coordinate systems......Page 144
4.4.2 Least-squares and streamline upwind Petrov–Galerkin (SUPG) schemes......Page 146
4.5 Modelling of Transient Stokes Flow of Generalized Newtonian and Non-Newtonian Fluids......Page 147
References......Page 154
5.1 Models Based on Simplified Domain Geometry......Page 156
Flow simulation in a single blade partially filled mixer......Page 157
Flow simulation in a partially filled twin blade mixer......Page 161
5.2 Models Based on Simplified Governing Equations......Page 165
5.2.1 Simulation of the Couette flow of silicon rubber – generalized Newtonian model......Page 166
5.2.2 Simulation of the Couette flow of silicon rubber – viscoelastic model......Page 167
5.3.1 Prediction of stress overshoot in the contracting sections of a symmetric flow domain......Page 171
5.3.2 Simulation of wall slip in a rubber mixer......Page 173
5.4 Models Based on Decoupled Flow Equations – Simulation of the Flow Inside a Cone-and-Plate Rheometer......Page 175
5.4.1 Governing equations......Page 177
5.4.2 Finite element discretization of the governing equations......Page 181
5.5 Models Based on Thin Layer Approximation......Page 185
5.5.1 Finite element modelling of flow distribution in an extrusion die......Page 188
5.5.2 Generalization of the Hele-Shaw approach to flow in thin curved layers......Page 190
Asymptotic expansion scheme......Page 192
5.6 Stiffness Analysis of Solid Polymeric Materials......Page 198
5.6.1 Stiffness analysis of polymer composites filled with spherical particles......Page 199
References......Page 203
6.1 General Considerations Related to Finite Element Mesh Generation......Page 206
6.1.1 Mesh types......Page 207
Overset grids......Page 208
Hybrid grids......Page 209
6.1.2 Common methods of mesh generation......Page 210
6.2 Main Components of Finite Element Processor Programs......Page 211
6.3 Numerical Solution of the Global Systems of Algebraic Equations......Page 214
Pivoting......Page 215
Gaussian elimination with partial pivoting......Page 216
Number of operations in the Gaussian elimination method......Page 217
6.4.1 LU decomposition technique......Page 218
6.4.2 Frontal solution technique......Page 220
6.5.1 Round-off error......Page 221
References......Page 222
7.1 Program Structure and Algorithm......Page 224
7.2 Program Specifications......Page 225
7.3 Input Data File......Page 228
7.4 Extension of PPVN.f to Axisymmetric Problems......Page 230
7.5 Circulatory Flow in a Rectangular Domain......Page 232
7.6 Source Code of PPVN.f......Page 235
References......Page 265
8 APPENDIX – SUMMARY OF VECTOR AND TENSOR ANALYSIS......Page 266
8.1 Vector Algebra......Page 268
8.2 Some Vector Calculus Relations......Page 270
8.2.1 Divergence (Gauss) theorem......Page 271
8.2.3 Reynolds transport theorem......Page 272
8.2.5 Second order tensors......Page 273
8.3 Tensor Algebra......Page 274
8.3.1 Invariants of a second-order tensor (T)......Page 276
8.4.1 Covariant, contravariant and mixed tensors......Page 277
8.4.2 The length of a line and metric tensor......Page 278
K......Page 282
Z......Page 283
C......Page 284
J......Page 285
R......Page 286
V......Page 287
Z......Page 288