Page 6 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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CONTENTS
9.3
9.3.1
Backward-facing step . ... .. .. .. .. .. .. ... .. .. .. 281
9.4 Non-isothermal Benchmark Flow Problem . . . . . . . . . . . . . . . . . . 280
Thermal Conduction in an Electronic Package . . . . . . . . . . . . . . . . 283
9.5 Forced Convection Heat Transfer From Heat Sources . . . . . . . . . . . . 286
9.6 Summary .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 294
9.7 Exercise. .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 294
Bibliography . .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 296
10 Implementation of Computer Code 299
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
10.2 Preprocessing .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 300
10.2.1 Mesh generation .. .. ... .. .. .. .. .. .. ... .. .. .. 300
10.2.2 Linear triangular element data . . . . . . . . . . . . . . . . . . . . . 302
10.2.3 Element size calculation ... .. .. .. .. .. .. ... .. .. .. 303
10.2.4 Shape functions and their derivatives . . . . . . . . . . . . . . . . . 304
10.2.5 Boundary normal calculation . . . . . . . . . . . . . . . . . . . . . 305
10.2.6 Mass matrix and mass lumping . .. .. .. .. .. ... .. .. .. 306
10.2.7 Implicit pressure or heat conduction matrix . . . . . . . . . . . . . . 307
10.3 Main Unit .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 309
10.3.1 Time-step calculation .. ... .. .. .. .. .. .. ... .. .. .. 310
10.3.2 Element loop and assembly .. .. .. .. .. .. .. . .. .. .. . 313
10.3.3 Updating solution . .. ... .. .. .. .. .. .. ... .. .. .. 314
10.3.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 315
10.3.5 Monitoring steady state ... .. .. .. .. .. .. ... .. .. .. 316
10.4 Postprocessing . .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 317
10.4.1 Interpolation of data .. ... .. .. .. .. .. .. ... .. .. .. 317
10.5 Summary .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 317
Bibliography . .. .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 317

A Green’s Lemma 319
B Integration Formulae 321
B.1 Linear Triangles . .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. 321
B.2 Linear Tetrahedron . .. .. .. ... .. .. .. .. .. .. ... .. .. .. 321
C Finite Element Assembly Procedure 323

D Simplified Form of the Navier–Stokes Equations 326

Index 329