A unified approach to the finite element method and error analysis procedures

Front Cover......Page 1
The Finite Element Method and Error Analysis Procedures......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 8
General Introduction......Page 12
Part l: Problem Definition and Development......Page 26
Introduction......Page 28
Lesson 1 Principle of Minimum Potential Energy......Page 34
Lesson 2 Elements of the Calculus of Variations......Page 46
Lesson 3 Derivation of the Plane Stress Problem......Page 66
Lesson 4 Rayleigh-Ritz Variational Solution Technique......Page 88
Part II: Physically Interpretable Notation......Page 104
Introduction......Page 106
Lesson 5 Strain Gradient Notation......Page 112
Lesson 6 Strain Gradient Representation of Discrete Structures......Page 132
Lesson 7 Strain Transformations......Page 158
Part III: The Strain Gradient Reformulation of the Finite Element Method......Page 180
Introduction......Page 182
Lesson 8 The Development of Strain-Gradient-Based Finite Elements......Page 194
Lesson 9 Four-Node Quadrilateral Element......Page 236
Lesson 10 Six-Node Linear Strain Element......Page 258
Lesson 11 Eight- and Nine-Node Elements......Page 274
Lesson 12 Shear Locking, Aspect Ratio Stiffening, and Qualitative Errors......Page 296
Part IV :The Strain Gradient Reformulation of the Finite Difference Method......Page 332
Introduction......Page 334
Lesson 13 Elements of the Finite Difference Method......Page 342
Lesson 14 Finite Difference Boundary Condition Models......Page 376
Lesson 15 Nonstandard Finite Difference Models......Page 396
Part V: A Posteriori Error Analysis Procedures: Pointwise Error Measures and a New Approach for Strain Extraction......Page 412
Introduction......Page 414
Lesson 16 The Zienkiewicz/Zhu Error Estimation Procedure......Page 430
Lesson 17 Error Estimation Based on Finite Difference Smoothing......Page 452
Lesson 18 Nonintegrated Pointwise Refinement Guides......Page 474
Lesson 19 A Theoretical Foundation for Pointwise Evaluation Measures......Page 492
Lesson 20 Application of a Universal Evaluation Post-Processor......Page 540
Index......Page 554