The Finite Element Method for Engineers

by Huebner, Kenneth H.; Dewhirst, Donald L.; Smith, Douglas E.; Byrom, Ted G.

Edition: 4th

ISBN13: 9780471370789

ISBN10: 0471370789

Format: Hardcover

Pub. Date: 2001-09-07

Publisher(s): Wiley-Interscience

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Summary

A useful balance of theory, applications, and real-world examples The Finite Element Method for Engineers, Fourth Edition presents a clear, easy-to-understand explanation of finite element fundamentals and enables readers to use the method in research and in solving practical, real-life problems. It develops the basic finite element method mathematical formulation, beginning with physical considerations, proceeding to the well-established variation approach, and placing a strong emphasis on the versatile method of weighted residuals, which has shown itself to be important in nonstructural applications. The authors demonstrate the tremendous power of the finite element method to solve problems that classical methods cannot handle, including elasticity problems, general field problems, heat transfer problems, and fluid mechanics problems. They supply practical information on boundary conditions and mesh generation, and they offer a fresh perspective on finite element analysis with an overview of the current state of finite element optimal design. Supplemented with numerous real-world problems and examples taken directly from the authors' experience in industry and research, The Finite Element Method for Engineers, Fourth Edition gives readers the real insight needed to apply the method to challenging problems and to reason out solutions that cannot be found in any textbook.

Author Biography

KENNETH H. HUEBNER, PhD, is retired from Ford Motor Company, where he was manager of the Computer-Aided Engineering Research Staff. He received his PhD from Purdue University in 1969.<BR>

Preface

xvii

PART I

Meet the Finite Element Method

(14)

What Is the Finite Element Method?

(2)

How the Finite Element Method Works

(3)

A Brief History of the Method

(3)

Range of Applications

(2)

Commercial Finite Element Software

(1)

The Future of the Finite Element Method

(3)

References

(2)

The Direct Approach: A Physical Interpretation

(57)

Introduction

(1)

Defining Elements and Their Properties

(22)

Linear Spring Systems

(3)

Flow Systems

(4)

Simple Elements from Structural Mechanics

(12)

Coordinate Transformations

(3)

Assembling the Parts

(16)

Assembly Rules Derived from an Example

(6)

General Assembly Procedure

(1)

Features of the Assembled Matrix

(1)

Introducing Boundary Conditions

(8)

Solver Technology

(7)

Linear Direct Solvers

(1)

Iterative Solvers

(2)

Eigensolvers

(1)

Nonlinear Equation Solvers

(3)

Closure

(11)

References

(1)

Problems

(10)

The Mathematical Approach: A Variational Interpretation

(39)

Introduction

(1)

Continuum Problems

(4)

Introduction

(1)

Problem Statement

(2)

Classification of Differential Equations

(1)

Some Methods for Solving Continuum Problems

(6)

An Overview

(1)

The Variational Approach

(1)

The Ritz Method

(4)

The Finite Element Method

(23)

Relation to the Ritz Method

(1)

Generalizing the Definition of an Element

(1)

Example of a Piecewise Approximation

(4)

Element Equations from a Variational Principle

(2)

Requirements for Interpolation Functions

(6)

Domain Discretization

(3)

Example of a Complete Finite Element Solution

102

(6)

Closure

108

(5)

References

108

(1)

Problems

109

(4)

The Mathematical Approach: A Generalized Interpretation

113

(24)

Introduction

113

(1)

Deriving Finite Element Equations from the Method of Weighted Residuals

114

(17)

Example: One-Dimensional Poisson Equation

119

(6)

Example: Two-Dimensional Heat Conduction

125

(4)

Example: Time-Dependent Heat Conduction

129

(2)

Closure

131

(6)

References

131

(1)

Problems

132

(5)

Elements and Interpolation Functions

137

(86)

Introduction

138

(1)

Basic Element Shapes

139

(5)

Terminology and Preliminary Considerations

144

(2)

Types of Notes

144

(1)

Degrees of Freedom

144

(1)

Interpolation Functions-Polynomials

144

(2)

Generalized Coordinates and the Order of the Polynomial

146

(5)

Generalized Coordinates

146

(1)

Geometric Isotropy

147

(2)

Deriving Interpolation Functions

149

(2)

Natural Coordinates

151

(10)

Natural Coordinates in One Dimension

152

(1)

Natural Coordinates in Two Dimensions

153

(4)

Natural Coordinates in Three Dimensions

157

(4)

Interpolation Concepts in One Dimension

161

(5)

Lagrange Polynomials

161

(2)

Hermite Polynomials

163

(3)

Internal Nodes--Condensation/Substructuring

166

(4)

Two-Dimensional Elements

170

(14)

Elements for C0 Problems

170

(8)

Elements for C1 Problems

178

(6)

Three-Dimensional Elements

184

(5)

Elements for C0 Problems

184

(5)

Elements for C1 Problems

189

(1)

Isoparametric Elements for C0 Problems

189

(8)

Coordinate Transformation

190

(3)

Evaluation of Element Matrices

193

(3)

Example of Isoparametric Element Matrix Evaluation

196

(1)

Numerical Integration

197

(13)

Newton-Cotes

199

(3)

Gauss-Legendre

202

(3)

Numerical Example for Element Matrix Evaluation

205

(5)

Closure

210

(13)

References

211

(2)

Problems

213

(10)

PART II

Elasticity Problems

223

(65)

Introduction

224

(1)

General Formulation for Three-Dimensional Problems

224

(14)

Problem Statement

224

(2)

The Variational Method

226

(6)

The Galerkin Method

232

(4)

The System Equations

236

(2)

Application to Plane Stress and Plane Strain

238

(8)

Displacement Model for a Triangular Element

238

(2)

Element Stiffness Matrix for a Triangle

240

(3)

Element Force Vectors for a Triangle

243

(3)

Application to Axisymmetric Stress Analysis

246

(8)

Displacement Model for Triangular Toroid

247

(1)

Element Stiffness Matrix for Triangular Toroid

247

(3)

Element Force Vectors for Triangular Toroid

250

(4)

Application to Plate-Bending Problems

254

(8)

Requirements for the Displacement Interpolation Functions

257

(1)

Rectangular Plate-Bending Elements

258

(4)

Three-Dimensional Problems

262

(2)

Introduction

262

(1)

Formulation for the Linear Tetrahedral Element

262

(1)

Higher-Order Elements

263

(1)

Introduction to Structural Dynamics

264

(14)

Formulation of Equations

264

(3)

Free Undamped Vibrations

267

(2)

Finding Transient Motion via Mode Superposition

269

(3)

Finding Transient Motion via Recurrence Relations

272

(6)

Closure

278

(10)

References

279

(2)

Problems

281

(7)

General Field Problems

288

(60)

Introduction

289

(1)

Equilibrium Problems

289

(14)

Quasi-Harmonic Equations

289

(1)

Boundary Conditions

290

(3)

Variational Principle

293

(1)

Element Equations

294

(3)

Element Equations in Two Dimensions

297

(6)

Eigenvalue Problems

303

(8)

Helmholtz Equations

304

(1)

Variational Principle

305

(1)

Element Equations

306

(1)

Examples

307

(2)

Sample Problem

309

(2)

Propagation Problems

311

(9)

General Time-Dependent Field Problems

311

(5)

Finite Element Equations

316

(2)

Element Equations in One Space Dimension

318

(2)

Solving the Discretized Time-Dependent Equations

320

(15)

Solution Methods for First-Order Equations

321

(1)

Finding Transient Response via Mode Superposition

321

(3)

Finding Transient Response via Recurrence Relations

324

(3)

Oscillation and Stability of Transient Response

327

(3)

Algorithm Order

330

(2)

Sample Problem

332

(3)

Closure

335

(13)

References

335

(2)

Problems

337

(11)

Heat Transfer Problems

348

(74)

Introduction

349

(1)

Conduction

349

(30)

Problem Statement

349

(2)

Finite Element Formulation

351

(5)

Element Equations

356

(9)

Linear Steady-State and Transient Solutions

365

(6)

Nonlinear Steady-State Solutions

371

(4)

Nonlinear Transient Solutions

375

(4)

Conduction with Surface Radiation

379

(11)

Problem Statement

379

(2)

Element Equations with Radiation

381

(3)

Steady-State Solutions

384

(2)

Transient Solutions

386

(4)

Convective-Diffusion Equation

390

(8)

Problem Statement

391

(1)

Finite Element Formulation

391

(2)

One-Dimensional Problem

393

(4)

Two-Dimensional Solutions

397

(1)

Free and Forced Convection

398

(5)

Problem Statement

398

(1)

Finite Element Formulation

399

(2)

Solution Techniques

401

(1)

Free-Convection Example

402

(1)

Forced Convection

403

(1)

Closure

403

(19)

References

406

(4)

Problems

410

(12)

Fluid Mechanics Problems

422

(73)

Introduction

422

(1)

Inviscid Incompressible Flow

423

(11)

Problem Statement

424

(1)

Finite Element Formulation

425

(3)

Velocity Component Smoothing

428

(1)

Example with Unstructured Mesh

429

(4)

The Kutta Condition

433

(1)

Viscous Incompressible Flow without Inertia

434

(7)

Problem Statement

434

(2)

Stream Function Formulation

436

(1)

Velocity and Pressure Formulation

437

(4)

Viscous Incompressible Flow with Inertia

441

(18)

Mixed Velocity and Pressure Formulation

442

(6)

Penalty Function Formulation

448

(3)

Equal-Order Velocity and Pressure Formulation

451

(8)

Compressible Flow

459

(20)

Problem Statement

460

(1)

Low-Speed Flow with Variable Density

461

(6)

High-Speed Flow

467

(12)

Closure

479

(16)

References

480

(6)

Problems

486

(9)

Boundary Conditions, Mesh Generation, and Other Practical Considerations

495

(75)

Introduction

496

(1)

Physical Singularities

496

(2)

Benchmark Problems

498

(4)

Symmetry

502

(15)

Definition of Types of Symmetry

503

(7)

Antisymmetry

510

(2)

A Complex Loading Example: Two Axes of Symmetry

512

(1)

Axisymmetry and Rotational Symmetry

512

(1)

The Torsion Problem

513

(3)

Heat Transfer

516

(1)

Dimensional Analysis

517

(2)

Mesh Generation

519

(22)

Mapped Meshing

520

(1)

Free Meshing

521

(5)

Mesh Topology Cleanup and Mesh Smoothing

526

(3)

Mesh Refinement Methods

529

(2)

Error Indicators

531

(4)

Adaptive Remeshing

535

(2)

p and h/p Methods

537

(4)

Lumped Mass versus Consistent Mass

541

(2)

Modeling Fasteners

543

(3)

Connecting Rod Analysis

546

(3)

Crankshaft and Flywheel Analysis

549

(2)

Disc Brake Analysis

551

(7)

An Automotive Brake Primer

552

(1)

Rotor Analysis Using Rotational Symmetry

553

(3)

Rotor Coning Analysis Using Axisymmetry

556

(1)

Simulating Friction

557

(1)

Hot Spotting

558

(1)

Closure

558

(12)

References

559

(6)

Problems

565

(5)

Finite Elements in Design

570

(66)

Introduction

571

(1)

Design Optimization

571

(11)

The Optimization Problem

572

(2)

Practical Aspects of Numerical Optimization

574

(3)

Optimization Algorithms

577

(4)

Software Packages for Optimal Design

581

(1)

Finite Element-Based Optimal Design

582

(12)

Design Parameterization

583

(3)

Structural Optimization

586

(3)

Topology Optimization

589

(3)

Approximation Techniques

592

(1)

Multidisciplinary Design Optimization

593

(1)

Design Sensitivity Analysis

594

(12)

Finite Difference Approximations

594

(1)

Analytical Methods for Design Sensitivity Analysis

595

(5)

Design Sensitivities for Eigenproblems

600

(5)

Semianalytical Approach

605

(1)

Other Advancements in Design Sensitivity Analysis

606

(1)

Examples of Design Sensitivity Analysis

606

(20)

Numerical Example: Steady-State Equilibrium Analysis

606

(5)

Numerical Example: Eigenvalue and Eigenvector Analysis

611

(4)

Sensitivity Analysis for Steady-State Conduction in a Solid

615

(8)

Numerical Example for Steady-State Conduction in a Solid

623

(3)

Case Study: Finite Element-Based Design

626

(2)

Closure

628

(8)

References

628

(4)

Problems

632

(4)

Appendix A Matrices

636

(12)

A.1 Definitions

637

(1)

A.2 Special Types of Square Matrices

637

(1)

A.3 Matrix Operations

638

(2)

A.4 Special Matrix Products

640

(1)

A.4.1 Product of a Square Matrix and a Column Matrix

640

(1)

A.4.2 Product of a Row Matrix and a Square Matrix

640

(1)

A.4.3 Product of a Row Matrix and a Column Matrix

641

(1)

A.4.4 Product of the Identity Matrix and Any Other Matrix

641

(1)

A.5 Matrix Transpose

641

(1)

A.6 Quadratic Forms

641

(2)

A.7 Matrix Inverse

643

(1)

A.8 Matrix Partitioning

643

(2)

A.9 The Calculus of Matrices

645

(1)

A.9.1 Differentiation of a Matrix

645

(1)

A.9.2 Integration of a Matrix

645

(1)

A.9.3 Differentiation of a Quadratic Functional

645

(1)

A.10 Norms

646

(2)

Appendix B Variational Calculus

648

(9)

B.1 Introduction

648

(1)

B.2 Calculus--The Minima of a Function

648

(2)

B.2.1 Definitions

648

(1)

B.2.2 Functions of One Variable

649

(1)

B.2.3 Functions of Two or More Variables

650

(1)

B.3 Variational Calculus--The Minima of Functionals

650

(7)

B.3.1 Definitions

650

(1)

B.3.2 Functionals of One Variable

651

(4)

B.3.3 More General Functionals

655

(1)

References

656

(1)

Appendix C Basic Equations from Linear Elasticity Theory

657

(17)

C.1 Introduction

657

(1)

C.2 Stress Components

658

(1)

C.3 Strain Components

658

(1)

C.4 Generalized Hooke's Law (Constitutive Equations)

659

(3)

C.5 Static Equilibrium Equations

662

(1)

C.6 Compatibility Conditions

663

(1)

C.7 Differential Equations for Displacements

664

(1)

C.8 Minimum Potential Energy Principle

665

(2)

C.9 Plane Strain and Plane Stress

667

(3)

C.10 Thermal Effects

670

(1)

C.11 Thin-Plate Bending

671

(3)

References

673

(1)

Appendix D Basic Equations from Fluid Mechanics

674

(17)

D.1 Introduction

674

(1)

D.2 Definitions and Concepts

675

(2)

D.3 Laws of Motion

677

(6)

D.3.1 Differential Continuity Equation

678

(1)

D.3.2 Differential Momentum Equation (Navier-Stokes Equations)

678

(2)

D.3.3 Thermal Energy Equation

680

(1)

D.3.4 Conservative Form of Equations

680

(2)

D.3.5 Supplementary Equations

682

(1)

D.3.6 Problem Statement

682

(1)

D.4 Stream Functions and Vorticity

683

(2)

D.5 Potential Flow

685

(1)

D.6 Viscous Incompressible Flow

686

(3)

D.6.1 Primitive Variable Formulation

686

(1)

D.6.2 Vorticity and Stream Function Formulation

687

(2)

D.7 Boundary Layer Flow

689

(2)

References

690

(1)

Appendix E Basic Equations from Heat Transfer

691

(18)

E.1 Introduction

691

(1)

E.2 Conduction

692

(4)

E.2.1 Heat Conduction Equation

693

(1)

E.2.2 Boundary Conditions

694

(2)

E.2.3 Nondimensional Parameters

696

(1)

E.3 Convection

696

(5)

E.3.1 Convection Equations

697

(2)

E.3.2 Boundary Conditions

699

(1)

E.3.3 Nondimensional Parameters

700

(1)

E.4 Radiation

701

(6)

E.4.1 Surface Radiation

702

(2)

E.4.2 Radiation Exchange between Surfaces

704

(3)

E.5 Heat Transfer Units

707

(2)

References

708

(1)

Index

709

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