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This book provides an in-depth background to better understanding of finite element results and techniques for improving accuracy of finite element methods. Thus, the reader is able to identify and eliminate errors contained in finite element models. Three different error analysis techniques are systematically developed from a common theoretical foundation: 1) modeling erros in individual elements; 2) discretization errors in the overall model; 3) point-wise errors in the final stress or strain results. Thoroughly class tested with undergraduate and graduate students. A Unified Approach to the Finite Element Method and Error Analysis Procedures is sure to become an essential resource for students as well as practicing engineers and researchers. * New, simpler element formulation techniques, model-independent results, and error measures * New polynomial-based methods for identifying critical points * New procedures for evaluating sheer/strain accuracy * Accessible to undergraduates, insightful to researchers, and useful to practitioners * Taylor series (polynomial) based * Intuitive elemental and point-wise error measures * Essential background information provided in 12 appendices 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 Advances in the field of computational mechanics coupled with the widespread adoption of the finite element method and advanced computational software have revolutionized the field of structural engineering. Civil, mechanical, and aerospace engineers now rely on these new computational techniques and programs to design and test the stress, strain, and load capability of a wide variety of structural elements including construction elements, plates, and shells. Because each element or point is subject to a wide array of variables, the possibility for errors in input data can be high; errors can cause critical mistakes in the design of structures and components and can lead to dangerous failure.This book is the first comprehensive reference to teach the finite element method with an emphasis on a robust procedure for analyzing errors. The text combines elementary finite element and finite difference methods with advanced computational and error-analysis techniques previously unapproachable by advanced undergraduates, graduate students, professional engineers and researchers. The author is an award-winning engineering educator, as well as a respected researcher and industry consultant. His background in industry and education have provided valuable insight into this book that will sell as both an advanced textbook and professional tutorial/reference.

A Unified Approach to the Finite Element Method and Error Analysis Procedures provides an in-depth background to better understanding of finite element results and techniques for improving accuracy of finite element methods. Thus, the reader is able to identify and eliminate errors contained in finite element models. Three different error analysis techniques are systematically developed from a common theoretical foundation: 1) modeling erros in individual elements; 2) discretization errors in the overall model; 3) point-wise errors in the final stress or strain results. Thoroughly class tested with undergraduate and graduate students.

A Unified Approach to the Finite Element Method and Error Analysis Procedures is sure to become an essential resource for students as well as practicing engineers and researchers.

• New, simpler element formulation techniques, model-independent results, and error measures
• New polynomial-based methods for identifying critical points
• New procedures for evaluating sheer/strain accuracy
• Accessible to undergraduates, insightful to researchers, and useful to practitioners
• Taylor series (polynomial) based
• Intuitive elemental and point-wise error measures
• Essential background information provided in 12 appendices

A Unified Approach to the Finite Element Method and Error Analysis Procedures provides an in-depth background to better understanding of finite element results and techniques for improving accuracy of finite element methods. Thus, the reader is able to identify and eliminate errors contained in finite element models. Three different error analysis techniques are systematically developed from a common theoretical foundation: 1) modeling erros in individual elements; 2) discretization errors in the overall model; 3) point-wise errors in the final stress or strain results. Thoroughly class tested with undergraduate and graduate students. A Unified Approach to the Finite Element Method and Error Analysis Procedures is sure to become an essential resource for students as well as practicing engineers and researchers. New, simpler element formulation techniques, model-independent results, and error measures New polynomial-based methods for identifying critical points New procedures for evaluating sheer/strain accuracy Accessible to undergraduates, insightful to researchers, and useful to practitioners Taylor series (polynomial) based Intuitive elemental and point-wise error measures Essential background information provided in 12 appendices