Finite Element Method - Contents _toc This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.
Contents Preface to Volume 3 Xlll Introduction and the equations of fluid dynamics 1.1 General remarks and classification of fluid mechanics problems discussed in this book 1.2 The governing equations of fluid dynamics 1.3 Incompressible (or nearly incompressible) flows 1.4 Concluding remarks References 10 12 12 Convection dominated problems - finite element approximations to the convection-diffusion equation 2.1 Introduction 2.2 The steady-state problem in one dimension 2.3 The steady-state problem in two (or three) dimensions 2.4 Steady state - concluding remarks 2.5 Transients - introductory remarks 2.6 Characteristic-based methods 2.7 Taylor-Galerkin procedures for scalar variables 2.8 Steady-state condition 2.9 Non-linear waves and shocks 2.10 Vector-valued variables 2.11 Summary and concluding remarks References 13 13 15 26 30 32 35 47 48 48 52 59 59 A general algorithm for compressible and incompressible flows - the characteristic-based split (CBS) algorithm 3.1 Introduction 3.2 Characteristic-based split (CBS) algorithm 3.3 Explicit, semi-implicit and nearly implicit forms 3.4 ‘Circumventing’ the BabuSka-Brezzi (BB) restrictions 3.5 A single-step version 3.6 Boundary conditions 64 64 67 76 78 80 81 viii Contents 3.7 3.8 The performance of two- and single-step algorithms on an inviscid problem Concluding remarks References 85 87 87 Incompressible laminar flow - newtonian and non-newtonian fluids 91 4.1 Introduction and the basic equations 91 4.2 Inviscid, incompressible flow (potential flow) 93 4.3 Use of the CBS algorithm for incompressible or nearly incompressible flows 97 4.4 Boundary-exit conditions i00 4.5 Adaptive mesh refinement 102 4.6 Adaptive mesh generation for transient problems 113 4.7 Importance of stabilizing convective terms 113 4.8 Slow flows - mixed and penalty formulations 113 4.9 Non-newtonian flows - metal and polymer forming 118 4.10 Direct displacement approach to transient metal forming 132 4.1 Concluding remarks 133 References 134 Free 5.1 5.2 5.3 5.4 surfaces, buoyancy and turbulent incompressible flows Introduction Free surface flows Buoyancy driven flows Turbulent flows References 143 143 144 153 161 165 Compressible high-speed gas flow 6.1 Introduction 6.2 The governing equations Boundary conditions - subsonic and supersonic flow 6.3 6.4 Numerical approximations and the CBS algorithm Shock capture 6.5 6.6 Some preliminary examples for the Euler equation 6.7 Adaptive refinement and shock capture in Euler problems 6.8 Three-dimensional inviscid examples in steady state 6.9 Transient two and three-dimensional problems 6.10 Viscous problems in two dimensions 6.11 Three-dimensional viscous problems 6.12 Boundary layer-inviscid Euler solution coupling 6.13 Concluding remarks References 169 169 170 171 173 174 176 180 188 195 197 207 209 212 212 Shallow-water problems 7.1 Introduction The basis of the shallow-water equations 7.2 7.3 Numerical approximation 218 218 219 223 Contents ix 7.4 7.5 7.6 Examples of application Drying areas Shallow-water transport References 224 236 237 239 Waves Introduction and equations 8.1 8.2 Waves in closed domains - finite element models 8.3 Difficulties in modelling surface waves 8.4 Bed friction and other effects 8.5 The short-wave problem 8.6 Waves in unbounded domains (exterior surface wave problems) Unbounded problems 8.7 8.8 Boundary dampers 8.9 Linking to exterior solutions 8.10 Infinite elements 8.11 Mapped periodic infinite elements 8.12 Ellipsoidal type infinite elements of Burnett and Holford 8.13 Wave envelope infinite elements 8.14 Accuracy of infinite elements 8.15 Transient problems 8.16 Three-dimensional effects in surface waves References 242 242 243 245 245 245 250 253 253 255 259 260 26 262 264 265 266 270 Computer implementation of the CBS algorithm 9.1 Introduction 9.2 The data input module 9.3 Solution module 9.4 Output module 9.5 Possible extensions to CBSflow References 274 274 275 278 289 289 289 Appendix A: Non-conservative form of Navier-Stokes equations 29 Appendix B: Discontinuous Galerkin methods in the solution of the convection-diffusion equation 293 Appendix C: Edge-based finite element formulation 298 Appendix D: Multigrid methods 300 Appendix E: Boundary layer-inviscid flow coupling 302 Author index 307 Subject index 315 Volume 1: The basis Some preliminaries: the standard discrete system A direct approach to problems in elasticity Generalization of the finite element concepts Galerkin-weighted residual and variational approaches Plane stress and plane strain Axisymmetric stress analysis Three-dimensional stress analysis Steady-state field problems - heat conduction, electric and magnetic potential, fluid flow, etc ‘Standard’ and ‘hierarchical’ element shape functions: some general families of Co continuity Mapped elements and numerical integration - ‘infinite’ and ‘singularity’ elements 10 The patch test, reduced integration, and non-conforming elements 1 Mixed formulation and constraints - complete field methods 12 Incompressible problems, mixed methods and other procedures of solution 13 Mixed formulation and constraints - incomplete (hybrid) field methods, boundary/Trefftz methods 14 Errors, recovery processes and error estimates 15 Adaptive finite element refinement 16 Point-based approximations; element-free Galerkin - and other meshless methods 17 The time dimension - semi-discretization of field and dynamic problems and analytical solution procedures 18 The time dimension - discrete approximation in time 19 Coupled systems 20 Computer procedures for finite element analysis Appendix A Matrix algebra Appendix B Tensor-indicia1 notation in the approximation of elasticity problems Appendix C Basic equations of displacement analysis Appendix D Some integration formulae for a triangle Appendix E Some integration formulae for a tetrahedron Appendix F Some vector algebra Appendix G Integration by parts Appendix H Solutions exact at nodes Appendix I Matrix diagonalization or lumping Volume 2: Solid and structural mechanics 10 1 12 13 General problems in solid mechanics and non-linearity Solution of non-linear algebraic equations Inelastic materials Plate bending approximation: thin (Kirchhoff) plates and C , continuity requirements ‘Thick’ Reissner-Mindlin plates - irreducible and mixed formulations Shells as an assembly of flat elements Axisymmetric shells Shells as a special case of three-dimensional analysis - Reissner-Mindlin assumptions Semi-analytical finite element processes - use of orthogonal functions and ‘finite strip’ methods Geometrically non-linear problems - finite deformation Non-linear structural problems - large displacement and instability Pseudo-rigid and rigid-flexible bodies Computer procedures for finite element analysis Appendix A: Invariants of second-order tensors ... 8.10 Infinite elements 8.11 Mapped periodic infinite elements 8.12 Ellipsoidal type infinite elements of Burnett and Holford 8.13 Wave envelope infinite elements 8.14 Accuracy of infinite elements... Semi-analytical finite element processes - use of orthogonal functions and finite strip’ methods Geometrically non-linear problems - finite deformation Non-linear structural problems - large displacement... error estimates 15 Adaptive finite element refinement 16 Point-based approximations; element- free Galerkin - and other meshless methods 17 The time dimension - semi-discretization of field and