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أخوانى فى الله أحضرت لكم كتاب Practical Aspects of Finite Element Modelling of Polymer Processing Vahid Nassehi Chemical Engineering Dept., Loughborough University
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Contents Preface xiii 1 THE BASIC EQUATIONS OF NON-NEWTONIAN FLUID MECHANICS 1 1.1 Governing Equations of Non-Newtonian Fluid Mechanics 2 1.1.l Continuity 2 1.1.2 Equation of motion 2 1.1.3 Thermal energy equation 3 1.2 Classification of Inelastic Time-Independent Fluids 1.2.1 Newtonian fluids 1.2.2 Generalized Newtonian fluids 1.3 Inelastic Time-Dependent Fluids 1.4 Viscoelastic Fluids 1.4.1 Model (material) parameters used in viscoelastic constitutive 1.4.2 Differential constitutive equations for viscoelastic fluids 1.4.3 Single-integral constitutive equations for viscoelastic fluids 1.4.4 Viscometric approach - the (CEF) model equations References 2 WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE 2.1 Finite Element 19 2.1.1 Interpolation models 20 2.1.2 Shape functions of commonly used finite elements 23 2.1.3 Non-standard elements 27 2.1.4 Local 29 2.1.S Order of continuity of finite elements 32 2.1.6 Convergence 33 2.1.7 Irregular and curved elements - isoparametric mapping 34 2.1.8 Numerical integration 38 2.1.9 Mesh refinement - h- and p-versions of the finite element method 40viii CONTENTS 2.2 Numerical Solution of Differential Equations by the Weighted Residual Method 2.2.1 Weighted residual statements in the context of finite element 2.2.2 The standard Galerkin method 2.2.2* Galerkin finite element procedure - a worked example 2.2.3 Streamline upwind Petrov-Galerkin method 2.2.3* Application of upwinding - a worked example 2.2.4 Least-squares finite element method 2.2.5 Solution of time-dependent problems discretizations References 3 FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES 3.1 Solution of the Equations of Continuity and Motion 3.1.1 The U-V-P scheme 3.1.2 The U-V-P scheme based on the slightly compressible continuity equation 3.1.3 Penalty schemes 3.1.4 Calculation of pressure in the penalty schemes - variational recovery method 3.1.5 Application of Green’s theorem - weak formulations 3.1.6 Least-squares scheme 3.2 Modelling of Viscoelastic Flow 3.2.1 Outline of a decoupled scheme for the differential constitutive models Derivation of the working equations models 3.2.2 Finite element schemes for the integral constitutive 3.2.3 Non-isothermal viscoelastic flow 3.3 Solution of the Energy Equation 3.4 Imposition of Boundary Conditions in Polymer Processing Models 3.4.1 Velocity and surface force (stress) components Inlet conditions Line of symmetry Solid walls Exit conditions 3.4.2 Slip-wall boundary conditions 3.4.3 Temperature and thermal stresses (temperature gradients) 3.5 Free Surface and Moving Boundary Problems 3.5.1 VOF method in ‘Eulerian’ frameworks 3.5.2 VOF method in ‘Arbitrary Lagrangian-Eulerian’ 3.5.3 VOF method in ‘Lagrangian’ frameworks frameworks References 108CONTENTS ix 4 WORKING EQUATIONS OF THE FINITE ELEMENT SCHEMES 4.1 Modelling of Steady State Stokles Flow of a Generalized Newtonian Fluid 4.1.1 Governing equations in two-dimensional Cartesian coordinate 4.1.2 Governing equations in two-dimensional polar coordinate 4.1.3 Governing equations in axisymmetric coordinate systems 4.1.4 Working equations of the L[-V-P scheme in Cartesian 4.1.5 Working equations of the W V - P scheme in polar coordinate 4.1.6 Working equations of the WV-P scheme in axisymmetric 4.1.7 Working equations of the continuous penalty scheme in 4.1.8 Working equations of the continuous penalty scheme in polar 4.1.9 Working equations of the continuous penalty scheme in 4.1.10 Working equations of the discrete penalty scheme in Cartesian 4.1.11 Working equations of the least-squares scheme in Cartesian systems systems coordinate systems systems coordinate systems Cartesian coordinate systems coordinate systems axisymmetric coordinate systems coordinate systems coordinate systems 4.2 Variations of Viscosity 4.3 Modelling of Steady-State Viscometric Flow - Working Equations of the Continuous Penalty Scheme in Cartesian Coordinate Systems 4.4.1 Working equations of the streamline upwind (SU) scheme for 4.4 Modelling of Thermal Energy Balance the steady-state energy equation in Cartesian, polar and axisymmetric coordinate sys,tems schemes 4.4.2 Least-squares and streamline upwind Petrov-Galerkin (SUPG) 4.5 Modelling of Transient Stokes Flow of Generalized Newtonian References and Non-Newtonian Fluids 5 RATIONAL APPROXIMATIONS AND ILLUSTRATIVE EXAMPLES 5.1 Models Based on Simplified Domain Geometry 5.1.1 Modelling of the dispersion stage in partially filled batch internal mixers Flow simulation in a single blade partially jilled mixer Flow simulation in a partially jlled twin blade mixer 146x CONTENTS Models Based on Simplified Governing Equations 5.2.1 Simulation of the Couette flow of silicon rubber - generalized 5.2.2 Simulation of the Couette flow of silicon rubber - viscoelastic Newtonian model model Models Representing Selected Segments of a Large Domain 5.3.1 Prediction of stress overshoot in the contracting sections of a 5.3.2 Simulation of wall slip in a rubber mixer Models Based on Decoupled Flow quations - Simulation of the Flow Inside a Cone-and-Plate Rheometer 5.4.1 Governing equations 5.4.2 Finite element discretization of the governing equations Models Based on Thin Layer Approximation 5.5.1 Finite element modelling of flow distribution in an 5.5.2 Generalization of the Hele-Shaw approach to flow in thin symmetric flow domain extrusion die curved layers Asymptotic expansion scheme Stiffness Analysis of Solid Polymeric Materials 5.6.1 Stiffness analysis of polymer composites filled with spherical particles References 6 FINITE ELEMENT SOFTWARE - MAIN COMPONENTS General Considerations Related to Finite Element Mesh Generation 6.1.1 Mesh types Block-structured grids Overset grids Hybrid grids 6.1.2 Common methods of mesh generation Main Components of Finite Element Processor Programs Numerical Solution of the Global Systems of Algebraic Equations 6.3.1 Direct solution methods Pivoting Guussiun elimination with partial pivoting Number of operations in the Guussiun elimination method Solution algorithms based on the Gaussian elimination method 6.4.1 LU decomposition technique 6.4.2 Frontal solution technique 205CONTENTS xi 6.5 Computational Errors 6.5.1 Round-off error 6.5.2 Iterative improvement of the solution of systems of linear equations References 7 COMPUTER SIMULATIONS - FINITE ELEMENT PROGRAM 7.1 Program Structure and Algorithm 7.2 Program Specifications 7.3 Input Data File 7.4 Extension of PPVN.f to Axisyrnmetric Problems 7.5 Circulatory Flow in a Rectangular Domain 7.6 Source Code of PPVN.f References 8 APPENDIX - SUMMARY OF VECTOR AND TENSOR ANALYSIS 8.1 Vector Algebra 8.2 Some Vector Calculus Relations 8.2.1 Divergence (Gauss’) theorem 8.2.2 Stokes theorem 8.2.3 Reynolds transport theorem 8.2.4 Covariant and contravariant vectors 8.2.5 Second order tensors 8.3 Tensor Algebra 8.4 Some Tensor Calculus Relatiolns 8.3.1 Invariants of a second-order tensor ( T ) 8.4.1 Covariant, contravariant and mixed tensors 8.4.2 The length of a line and metric tensor Author Index Subject Index
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