Finite and boundary element methods in engineering / O.P. Gupta.

Format:

Book

Author/Creator:

Gupta, O. P.

Language:

English

Subjects (All):

Finite element method.

Boundary element methods.

Physical Description:

xxviii, 466 pages : illustrations ; 24 cm

Place of Publication:

Rotterdam, Netherlands ; Brookfield, Vt. : A.A. Balkema, [1999]

Summary:

Gupta (mechanical engineering, Indian Institute of Technology) provides detailed explanation of the basics of the two modeling techniques and their applications areas, and prepares practicing engineers to tackle advanced applications of FEM and BEM analysis in their fields of interest using the software packages available on the market. Topics include elastic stress analysis, steady state and transient thermal conductivity, fluid flow, and plasticity. Annotation copyrighted by Book News, Inc., Portland, OR

Contents:

1.2 Finite Element Method 3

1.3 Boundary Element Method 5

1.4 Finite Element Implementation 6

1.4.1 Force equilibrium approach 6

1.4.2 Assembly procedure 12

1.4.3 Formulation using potential energy minimization 13

1.4.4 Other approaches 20

2. Elastic Stress Analysis Using Linear Elements 22

2.1 Nature of Loading 22

2.1.1 Concentrated or distributed loads 23

2.1.2 Body force (gravity etc.) 24

2.1.3 Loading due to thermal strains etc. 25

2.1.4 Residual stresses 25

2.2 Two-Dimensional Analysis 25

2.2.1 Strain displacement relation 30

2.2.2 Stress-strain relation 33

2.2.3 Potential energy 36

2.3 Three-Dimensional Analysis 44

2.3.1 Numbering sequence for nodes of elements 47

2.3.2 Shape function 48

2.3.3 Strain displacement relation 49

2.3.4 Stress-strain relation 51

2.3.5 Solution 52

2.4 Axi-symmetric Analysis 55

2.4.1 Shape function 57

2.4.2 Strain displacement relation 58

2.4.3 Stress-strain relation 61

2.4.4 Solution 62

2.4.5 Nature of expressions 63

2.4.6 Numerical integration 64

2.5 Illustrative Examples 68

2.5.1 Specifying loads and restraints 69

2.5.2 Stress analysis in crane hook 72

3. Some Mathematical Fundamentals and Computer Algorithms 80

3.2 Scalar, Vector and Tensor 80

3.2.1 Products of vectors 82

3.2.2 Summation convention and Kronecker delta 85

3.2.3 Gradient (or operator [Delta]) 86

3.2.4 Tensors 87

3.3 Gauss' and Green's Theorems 88

3.4 Matrices 92

3.4.1 Transpose of a matrix, square matrix 93

3.4.2 Matrix multiplication 94

3.4.3 Inverse of a matrix, solution of simultaneous equations 95

3.5 Solution of Matrix Equation 97

3.5.1 Gauss' elimination method 97

3.5.2 Boundary restraints 99

3.6 Banded Matrix Solver 101

3.6.1 Principle of banded solver 103

3.7 Computer Implementation 104

4. Variational Approach and Heat-Flow Analysis (Potential Problem) 113

4.1 Introduction and Application Examples 113

4.1.1 General procedure 117

4.2 Fundamentals of Variational Calculus 118

4.2.1 Minimization of functional 119

4.2.2 Euler-Lagrange equation 122

4.3 Steady-State Analysis 126

4.3.1 Element characteristics 126

4.3.2 Solution 130

4.3.3 Two-dimensional analysis 131

4.3.4 Axi-symmetric case 131

4.4 Illustrative Examples 133

4.4.1 Cutting tool 133

4.4.2 Continuously cast steel billet 133

4.4.3 Auto-engine analysis and design 137

4.5 Heat-transfer coefficient 138

5. Weighted Residue Technique and Unsteady-State Heat-Flow Analysis 139

5.2 Weighted Residue Technique 141

5.2.1 Form of weighting function 142

5.3 Application to Steady-State Heat Flow 143

5.4 Unsteady-State Heat Flow 146

5.4.1 Shape function in time domain 149

5.4.2 Matrix relation 149

5.5 Illustrative Examples 150

5.5.1 Resistance spot welding 150

5.5.2 Heat transfer in piston-cylinder assembly 151

6. Beams, Plates and Shells 155

6.2 Bending of Beams 155

6.2.1 Analysis of beam element 158

6.2.2 Interelement continuity of displacement and slope

C[subscript 1] continuity 159

6.2.3 Displacement function 161

6.2.4 Strain energy of deformation 163

6.2.5 Potential energy due to external loads 164

6.2.6 Stiffness relation 165

6.2.7 Beam element with general orientation in 3D space 170

6.3 Bending of Plates 171

6.3.1 Theory of plate bending 171

6.4 Finite Element Implementation 179

6.4.1 External work done 182

6.5 Other Types of Elements 183

6.6 Application Example 184

7. Non-linear, Curved, Isoparametric Elements and Advanced Plate, Shell Elements 186

7.2 Basic Requirement of Displacement Function 186

7.3 Natural Coordinate System 190

7.3.1 Higher order element shape functions 192

7.4 Area Coordinates 194

7.4.1 Higher order elements 195

7.4.2 Completeness requirement 196

7.4.3 Continuity requirement 196

7.5 Curved Elements 197

7.5.1 An alternative relation 197

7.5.2 Generalization of alternative relation to curved elements 198

7.6 Isoparametric Elements 200

7.6.1 Area coordinates 201

7.7 Stiffness Matrix 202

7.8 Numerical Integration 203

7.8.1 Gauss-Legendre quadrature 206

7.8.2 Extension to two or three dimensions 207

7.8.3 Area coordinates 208

7.8.4 Stiffness matrix in area coordinates 212

7.9 ARea and Volume Integral Using Numerical Integration 214

7.9.1 Surface integral 217

7.10 Advanced Plate Elements 218

7.11 Quadrilateral Plate Bending Element 218

7.11.1 Continuity and completeness requirements 219

7.11.2 Elemental stiffness matrix 220

7.11.3 In-plane loading and shell element 226

7.11.4 Global stiffness matrix 229

7.12 9DOF Triangular Plate Bending Element 233

7.12.1 Displacement formulation 234

7.12.2 Slope formulation 237

7.13 Other Shell Elements 241

7.14 Application Examples 242

8. Fluid Flow 247

8.2 Governing Equations in Fluid Mechanics 248

8.2.1 Continuity condition 248

8.2.2 Momentum conservation or force equilibrium 249

8.2.3 Energy equation 251

8.2.4 Irrotationally condition 255

8.2.5 Constitutive equations 258

8.3 Special Forms of Governing Equation 263

8.3.1 Viscous flow: Navier-Stokes equation 263

8.3.2 Creeping viscous flow: Stokes flow 264

8.4 General Approach to Solution 265

8.5 Inviscid, Incompressible, Steady Flow 267

8.6 Inviscid, Incompressible, Irrotational Steady Flow 268

8.6.1 Two-dimensional flow: stream function 269

8.6.2 Finite element implementation 270

8.6.3 General remarks 276

8.7 Compressible Flow 276

8.8 Viscous Flow 276

8.8.1 Stokes flow and penalty function 277

8.8.2 Finite element formulation 278

8.9 Illustrative Examples 281

8.9.1 Cooling water flow in engine cylinder 283

8.9.2 Molten metal flow in tundish during steel melting 283

9. Material Non-linearity Including Plasticity 285

9.2 Reversible Non-linearity 285

9.2.1 Direct iteration 286

9.2.2 Improving convergence through use of relaxation factor 289

9.2.3 Newton-Raphson method 291

9.2.4 Newton-Raphson method for multivariable case 292

9.2.5 Modified Newton-Raphson method 295

9.2.6 Tangent matrix for heat conduction problem 295

9.3 Irreversible Non-Linearity (Plasticity) 298

9.3.1 General elastoplastic behaviour 298

9.3.2 Three-dimensional plasticity 300

9.3.3 Post-yield behaviour 303

9.3.4 Approach to finite element analysis 307

9.3.5 Another method of presenting experimental stress-plastic strain relation 311

9.3.6 Incremental elastoplastic stress-strain analysis 313

9.3.7 Iterative elastoplastic analysis and initial stress method 315

9.3.8 Radial return method 322

9.3.9 Mixed incremental and iterative approach 325

9.3.10 Conclusion: Elastoplastic analysis 325

9.4 Illustrative Examples 326

10. Creeping Viscous Flow and Metal Forming 329

10.2 Boundary Conditions 330

10.2.1 Forging 332

10.3 Constitutive Equations 334

10.3.1 Stress-strain rate relationship 335

10.4 Finite Element Formulation 336

10.4.1 Alternative formulation 338

10.4.2 Special boundary conditions 342

10.4.3 Global to local transformation 342

10.5 Iterative Solution and Special Procedures 345

10.5.1 Rigid regions 346

10.6 Illustrative Examples 346

11. Boundary Element Method: Potential Problems 350

11.2 Boundary Element Approach 350

11.2.1 Fundamental solution 353

11.2.2 Another form of boundary integral equation 355

11.2.3 Volume integral of [Delta][superscript 2]w at source point 355

11.3 Numerical Implementation 358

11.3.1 Determination of C[subscript i] 362

11.3.2 Final Relation 363

11.3.3 Consideration of internal heat generation (body force term) 366

11.3.4 Three-dimensional analysis 367

11.3.5 Tackling kernel singularity 370

11.3.6 Axi-symmetric kernel 371

11.3.7 Mixed boundary

condition 373

11.4 Analysing Time Domain (Transient Case) 374

11.4.1 Three-dimensional formulation 378

11.4.2 Numerical implementation 379

11.5 Illustrative Examples 385

11.5.1 Temperature distribution in cutting tool 385

11.5.2 Thermal design of blast furnace bottom 387

11.5.3 Laser heating and hardening 388

12. Boundary Element Formulation for Elastostatic Problems 391

12.2 Basic Relation 391

12.2.1 Boundary condition 393

12.2.2 Other relations 395

12.3 Boundary Integral Relation 397

12.4 Fundamental Solution 400

12.5 Discretization and Matrix Formulation 402

12.5.1 Determination of term C (P)[subscript m] 406

12.6 Determination of Stresses 407

12.7 Other Cases 411

12.8 Illustrative Examples 412

12.8.1 Loose-fit, loaded pin in hole 412

12.8.2 Cam-tappet contact problem 412

13. Adaptive Mesh Refinement and Large Problem Solvers 416

13.2 Automatic Mesh Generation 417

13.2.1 Isoparametric coordinate mapping 417

13.2.2 Automatic triangulation 422

13.2.3 Octree-based approach 423

13.2.4 Element type conversion 426

13.3 Adaptive Mesh Refinement 426

13.3.1 Error norm 427

13.3.2 Estimating error norm in FE analysis 429

13.3.3 Energy error norm used for adaptive mesh refinement 432

13.4 Frontal Solver 434

1. Area of Triangle and Volume of Tetrahedron 439

2. Augmented Matrix and Its Use 442

3. Vector Representation for Bending Moment and Rotation 445

4. Penalty Function and Its Application 450

5. Stokes Flow: Elemental Stiffness Matrix 456

6. Higher Order Shape Functions 462.

Notes:

Includes bibliographical references and index.

ISBN:

9054107650

OCLC:

42774966