Finite Elements : Computational Engineering Sciences

Approaches computational engineering sciences from the perspective of engineering applications Uniting theory with hands-on computer practice, this book gives readers a firm appreciation of the error mechanisms and control that underlie discrete approximation implementations in the engineering sciences. Key features: Illustrative examples include heat conduction, structural mechanics, mechanical vibrations, heat transfer with convection and radiation, fluid mechanics and heat and mass transport Takes a cross-discipline continuum mechanics viewpoint Includes Matlab toolbox and .m data files on a companion website, immediately enabling hands-on computing in all covered disciplines Website also features eight topical lectures from the author's own academic courses It provides a holistic view of the topic from covering the different engineering problems that can be solved using finite element to how each particular method can be implemented on a computer. Computational aspects of the method are provided on a companion website facilitating engineering implementation in an easy way. Finite Elements: Computational Engineering Sciences......Page 1 Contents......Page 9 Preface......Page 13 About the Author......Page 17 Notations......Page 19 1.1 Engineering Simulation......Page 23 1.2 A Problem-Solving Environment......Page 25 1.3 Weak Formulation Essence......Page 26 1.4 Decisions on Forming WSN......Page 28 1.5 Discrete WSh Implementations......Page 30 1.6 Chapter Summary......Page 32 References......Page 33 2.1 Engineering Simulation......Page 35 2.3 Continuum Conservation Principle Forms......Page 36 2.4 Constitutive Closure for Conservation Principle PDEs......Page 38 2.5 Engineering Science Continuum Mechanics......Page 42 References......Page 43 3.1 Example Linear Heat Conduction Solutions......Page 45 3.2 Multidimensional PDEs, Separation of Variables......Page 46 3.3 Mathematical Foundation Essence for GWSN......Page 50 3.4 A Legacy FD Construction......Page 51 3.5 An FD Approximate Solution......Page 53 3.6 Lagrange Interpolation Polynomials......Page 54 3.7 Chapter Summary......Page 55 References......Page 56 4.1 A Steady Heat Conduction Example......Page 57 4.2 Weak Form Approximation, Error Extremization......Page 58 4.3 GWSN Discrete Implementation, FE Trial Space Basis......Page 60 4.4 Finite Element Matrix Statements......Page 62 4.5 Assembly of {WS} to form Algebraic GWSh......Page 64 4.6 Solution Accuracy, Error Distribution......Page 66 4.7 Convergence, Boundary Heat Flux......Page 67 Reference......Page 68 5.1 Introduction......Page 69 5.2 Steady Heat Transfer, n =1......Page 70 5.3 FE k = 1 Trial Space Basis Matrix Library......Page 72 5.4 Object-Oriented GWSh Programming......Page 75 5.5 Higher Completeness Degree Trial Space Bases......Page 78 5.6 Global Theory, Asymptotic Error Estimate......Page 81 5.7 Nonsmooth Data, Theory Generalization......Page 85 5.8 Temperature-Dependent Conductivity, Nonlinearity......Page 87 5.9 Static Condensation, p-Elements......Page 90 5.10 Chapter Summary......Page 92 Computer Labs......Page 94 References......Page 95 6.1 Introduction......Page 97 6.2 The Euler–Bernoulli Beam Equation......Page 98 6.3 Euler–Bernoulli Beam Theory GWSh Reformulation......Page 103 6.4 Timoshenko Beam Theory......Page 109 6.5 Mechanical Vibrations of a Beam......Page 114 6.6 Fluid Mechanics, Potential Flow......Page 122 6.7 Electromagnetic Plane Wave Propagation......Page 126 6.8 Convection–Radiation Finned Cylinder Heat Transfer......Page 127 6.9 Chapter Summary......Page 135 Computer Labs......Page 137 References......Page 138 7.1 Introduction......Page 139 7.2 Multidimensional FE Bases and DOF......Page 140 7.3 Multidimensional FE Operations for {Nk(ζ α)}......Page 143 7.4 The NCk = 1,2 Basis FE Matrix Library......Page 146 7.5 NC Basis {WS}e Template, Accuracy, Convergence......Page 149 7.6 The Tensor Product Basis Element Family......Page 152 7.7 Gauss Numerical Quadrature, k = 1 TP Basis Library......Page 154 7.8 Convection–Radiation BC GWSh Implementation......Page 158 7.9 Linear Basis GWSh Template Unification......Page 162 7.10 Accuracy, Convergence Revisited......Page 163 7.11 Chapter Summary......Page 165 Exercises......Page 166 References......Page 167 8.1 The FD–FE Correlation......Page 169 8.2 The FV–FE Correlation......Page 172 8.3 Chapter Summary......Page 176 Exercises......Page 177 9.1 Introduction......Page 179 9.2 The GalerkinWeak Statement......Page 180 9.3 GWSh Completion for Time Dependence......Page 182 9.4 GWSh + θTS Algorithm Templates......Page 183 9.5 GWSh + θTS Algorithm Asymptotic Error Estimates......Page 185 9.6 Performance Verification Test Cases......Page 186 9.7 Dispersive Error Characterization......Page 189 9.8 A Modified Galerkin Weak Statement......Page 193 9.9 Verification Problem Statements Revisited......Page 196 9.10 Unsteady Heat Conduction......Page 199 9.11 Chapter Summary......Page 201 Computer Labs......Page 202 References......Page 203 10.1 The Problem Statement......Page 205 10.2 GWSh + θTS Formulation Reprise......Page 207 10.3 Matrix Library Additions, Templates......Page 208 10.4 mPDE Galerkin Weak Forms, Theoretical Analyses......Page 210 10.5 Verification, Benchmarking, and Validation......Page 214 10.6 Mass Transport, the Rotating Cone Verification......Page 215 10.7 The Gaussian Plume Benchmark......Page 218 10.8 Steady n-D Peclet Problem Verification......Page 219 10.9 Mass Transport, a Validated n = 3 Experiment......Page 222 10.10 Numerical Linear Algebra, Matrix Iteration......Page 225 10.11 Newton and AF TP Jacobian Templates......Page 232 10.12 Chapter Summary......Page 233 Computer Labs......Page 235 References......Page 236 11.1 Introduction......Page 237 11.2 Structural Mechanics......Page 238 11.3 Structural Mechanics, Virtual Work FE Implementation......Page 241 11.4 Plane Stress/Strain GWSh Implementation......Page 243 11.5 Plane Elasticity Computer Lab......Page 246 11.6 Fluid Mechanics, Incompressible-Thermal Flow......Page 248 11.7 Vorticity-Streamfunction GWSh + θTS Algorithm......Page 253 11.8 An Isothermal INS Validation Experiment......Page 256 11.9 Multimode Convection Heat Transfer......Page 260 11.10 Mechanical Vibrations, Normal Mode GWSh......Page 264 11.11 Normal Modes of a Vibrating Membrane......Page 267 11.12 Multiphysics Solid-Fluid Mass Transport......Page 270 11.13 Chapter Summary......Page 276 Computer Labs......Page 279 References......Page 280 12 Conclusion......Page 283 Index......Page 285

Approaches computational engineering sciences from the perspective of engineering applications

Uniting theory with hands-on computer practice, this book gives readers a firm appreciation of the error mechanisms and control that underlie discrete approximation implementations in the engineering sciences.

Key features:

• Illustrative examples include heat conduction, structural mechanics, mechanical vibrations, heat transfer with convection and radiation, fluid mechanics and heat and mass transport
• Takes a cross-discipline continuum mechanics viewpoint
• Includes Matlab toolbox and.m data files on a companion website, immediately enabling hands-on computing in all covered disciplines
• Website also features eight topical lectures from the author's own academic courses

It provides a holistic view of the topic from covering the different engineering problems that can be solved using finite element to how each particular method can be implemented on a computer. Computational aspects of the method are provided on a companion website facilitating engineering implementation in an easy way.