Finite Elements

A. J. Baker is Professor Emeritus, Engineering Science and Computational Engineering, The University of Tennessee, USA. He is an elected Fellow of the International Association for Computational Mechanics (IACM) and the US Association for Computational Mechanics (USACM) and an Associate Fellow of the American Institute of Aeronautics and Astronautics (AIAA).

• Preface viii Notation xi1 COMPUTATIONAL ENGINEERING SCIENCE 11.1 Engineering simulation 11.2 A problem solving environment 21.3 Problem statements in engineering 41.4 Decisions on forming WSN 61.5 Discrete approximate WSh implementation 81.6 Chapter summary 91.7 Chapter references 102 PROBLEM STATEMENTS 112.1 Engineering simulation 112.2 Continuum mechanics viewpoint 122.3 Continuum conservation law forms 122.4 Constitutive closure for conservation law PDEs 142.5 Engineering science continuum mechanics 182.6 Chapter references 203 SOME INTRODUCTORY MATERIAL 213.1 Introduction 213.2 Multi-dimensional PDEs, separation of variables 223.3 Theoretical foundations, GWSh 273.4 A legacy FD construction 283.5 An FD approximate solution 303.6 Lagrange interpolation polynomials 313.7 Chapter summary 323.8 Exercises 343.9 Chapter references 344 HEAT CONDUCTION354.1 A steady heat conduction example 354.2 Weak form approximation, error minimization 374.3 GWSN discrete implementation, FE basis384.4 Finite element matrix statement 414.5 Assembly of {WS}e to form algebraic GWSh 434.6 Solution accuracy, error distribution 454.7 Convergence, boundary heat flux 474.8 Chapter summary 474.9 Exercises 484.10 Chapter reference 485 STEADY HEAT TRANSFER, n =1495.1 Introduction 495.2 Steady heat transfer, n = 1 505.3 FE k = 1 trial space basis matrix library 525.4 Object-oriented GWSh programming 555.5 Higher completeness degree trial space bases585.6 Global theory, asymptotic error estimate 625.7 Non-smooth data, theory generalization 665.8 Temperature dependent conductivity, non-linearity 695.9 Static condensation, p-elements 725.10 Chapter summary 755.11 Exercises 765.12 Computer labs 775.13 Chapter references 786 ENGINEERING SCIENCES, n =1 796.1 Introduction 796.2 The Euler-Bernoulli beam equation 806.3 Euler-Bernoulli beam theory GWSh reformulation 856.4 The Timoshenko beam theory 926.5 Mechanical vibrations of a beam 996.6 Fluid mechanics, potential flow 1066.7 Electromagnetic plane wave propagation1106.8 Convective-radiative finned cylinder heat transfer 1126.9 Chapter summary 1206.10 Exercises1226.10 Computer labs 1236.11 Chapter references 1247 STEADY HEAT TRANSFER, n > 1 1257.1 Introduction 1257.2 Multi-dimensional FE bases and DOF 1267.3 Multi-dimensional FE operations 1297.4 The NC k = 1,2 basis FE matrix library 1327.5 NC basis {WS}e template, accuracy, convergence 1367.6 The tensor product basis element family 1397.7 Gauss numerical quadrature, k = 1 TP basis library 1417.8 Convection-radiation BC GWSh implementation 1467.9 Linear basis GWSh template unification 1507.10 Accuracy, convergence revisited 1527.11 Chapter summary 1537.12 Exercises1557.13 Computer labs 1557.14 Chapter references 1568 FINITE DIFFERENCES OF OPINION 1598.1 The FD-FE correlation1598.2 The FV-FE correlation1628.3 Chapter summary 1678.4 Exercises1689 CONVECTION-DIFFUSION, n = 1 1699.1 Introduction1699.2 The Galerkin weak statement 1709.3 GWSh completion for time dependence1729.4 GWSh + qTS algorithm templates 1739.5 GWSh + qTS algorithm asymptotic error estimates 1759.6 Performance verification test cases 1779.7 Dispersive error characterization 1809.8 A modified Galerkin weak statement 1849.9 Verification problem statements revisited 1879.10 Unsteady heat conduction 1909.11 Chapter summary 1939.12 Exercises 1939.13 Computer labs 1949.14 Chapter references 19510 CONVECTION-DIFFUSION, n > 1 19710.1 The problem statement 19710.2 GWSh + qTS formulation reprise 19810.3 Matrix library additions, templates 20010.4 mPDE Galerkin weak forms, theoretical analyses 20210.5 Verification, benchmarking and validation 20710.6 Mass transport, the rotating cone verification 20810.7 The gaussian plume benchmark 21110.8 The steady n-D Peclet problem verification 21310.9 Mass transport, a validated n = 3 experiment 21510.10 Numerical linear algebra, matrix iteration 22210.11 Newton and AF TP jacobian templates 22710.12 Chapter summary 22910.13 Exercises23110.14 Computer labs 23110.15 Chapter references23211 ENGINEERING SCIENCES, n > 1 23511.1 Introduction 23511.2 Structural mechanics23611.3 Structural mechanics, virtual work FE form 24011.4 Plane stress/strain, GWSh implementation 24211.5 Elasticity computer lab 24611.6 Fluid mechanics, incompressible-thermal flow 25111.7 Vorticity-streamfunction GWSh + qTS algorithm 25411.8 An isothermal INS validation experiment 25811.9 Multi-mode convection heat transfer26211.10 Mechanical vibrations, normal mode GWSh 26711.11 Normal modes of a vibrating membrane27011.12 Multi-physics solid-fluid mass transport 27611.13 Chapter summary 28011.14 Exercises 28211.15 Computer labs28311.14 Chapter references 28412 CONCLUSION 287Index 289