1st Edition

Hybrid and Incompatible Finite Element Methods

By Theodore H.H. Pian, Chang-Chun Wu Copyright 2005
    394 Pages 167 B/W Illustrations
    by Chapman & Hall

    While the theory and application of finite elements methods can be extended to incompatible, hybrid, and mixed element methods, important issues, such as determining the reliability of the solution of incompatible multivariable elements, along with a common perception of impracticality, have hindered the widespread implementation of these methods. Today, however, recent advances--many directly attributable to these authors--have allowed the development of the stability theory and abstract mathematics to useful tools.

    Hybrid and Incompatible Finite Element Methods introduces these advances in the theory and applications of incompatible and multivariable finite element methods. After an overview of the variation formulation of finite element methods in solid mechanics, the authors discuss the fundamental theory and systematically demonstrate the theoretical foundations of incompatible elements and their application to different problems in the theory of elasticity. They also introduce new ideas in the development of hybrid finite elements, study the numerical stability of the hybrid and mixed element, and establish the theory of zero energy deformation modes. The final chapters, explore applications to fracture problems, present a bound analysis for fracture parameters, and demonstrate an implementation of a finite element analysis program.

    VARIATIONAL FORMULATION OF FINITE ELEMENT METHODS IN SOLID MECHANICS
    Introduction
    Equations for 3-D Elasticity
    Conventional Variational Principles in Solid Mechanics
    Modified Variational Principles for Relaxed Continuity or Equilibrium Conditions Along Interelement Boundaries
    Assumed-Displacement Finite Elements
    Assumed-Stress Hybrid-Finite Elements
    Hybrid-Strain Finite Elements
    Hybrid Finite Elements by the Hu–Washizu Principle
    Hybrid-Displacement Finite Elements
    FOUNDATION OF INCOMPATIBLE ANALYSIS
    Introduction
    Energy Inequality and Elliptic Conditions
    Weak Connection Condition of Incompatible Elements
    Numerical Stability of Incompatible Elements
    Consistency and Patch Test Condition (PTC)
    Generation of Incompatible Functions: General Formulation
    Relaxation of PTC by the Revise-Stiffness Approach
    The PTC in Curvilinear Coordinates
    Equivalent Nodal Load and Calculation of Stresses
    ELEMENTS FOR THE THEORY OF ELASTICITY
    Introduction
    Four-Node Plane-Incompatible Elements: NQ6
    P2-Linked Incompatible Methods with the Fewest Degrees of Freedom (DOF)
    Eight-Node 3-D Solid Incompatible Element
    Axisymmetric Incompatible Elements
    Hermite Type Incompatible Plate Elements
    Bending Model Under Reasonable w-• Constraint
    FOUNDATION IN MECHANICS OF HYBRID STRESS ELEMENTS
    Introduction
    Energy Consistency Analysis for Incompatible Hybrid Elements
    Patch Test and Element Optimization Condition (OPC)
    Optimization Method for Hybrid-Stress Finite Elements
    Matching Multivariable Parameters
    OPTIMIZATION OF HYBRID-STRESS FINITE ELEMENTS
    Four-Node Plane Hybrid Element
    Penalty Equilibrium Hybrid Element P-S(a)
    Three-Dimensional Body 18b-Optimization Hybrid Element
    Axisymmetric 8b-Optimization Hybrid Element
    Model Optimization of Hybrid-Stress General-Shell Element
    Appendix
    NUMERICAL STABILITY: ZERO ENERGY MODE ANALYSIS
    Introduction
    Definition of ZEM
    Rank Conditions for Two-Field Hybrid-Mixed Elements
    Determination of the Zero Energy Modes
    Control of the Zero-Energy Displacement Modes
    Control of the Zero Energy Stress Modes
    Patch Stability Test
    Examples
    PLASTIC ANALYSIS OF STRUCTURES
    Introduction
    Form of Incompressible Elements and Analysis of
    Plane-Stress Plastic Analysis
    Incompatible Elements in Plasticity Analysis
    Deviatoric Hybrid Model for the Incompressible Medium
    COMPUTATIONAL FRACTURE
    Introduction
    Dual Path-Independent Integral and Bound Theorem
    Numerical Strategy and Error Measure
    Numerical Tests of Crack Estimation
    Incompatible Numerical Simulation of an Axisymmetric Cracked Body
    Extension of J to Dynamic the Fracture of a Functional Graded Material
    Evaluation of Electro-Mechanical Crack Systems
    COMPUTATIONAL MATERIALS
    Hybrid Element Analysis of Composite Laminated Plates
    Bimaterial Interface Hybrid Element for Piezoelectric Laminated Analysis
    Numerical Solutions on Fractures of Piezoelectric Materials
    Homogenization-based Hybrid Element for Torsion of Composite Shafts
    A Study of 3-D Braided Piezoceramic Composites
    FINITE ELEMENT IMPLEMENTATION
    Overview
    Description of Variables and Subroutines
    Instructions for Input Data
    Examples
    Each chapter also contains a complete section of References.

    Biography

    Pian\, Theodore H.H.; Wu\, Chang-Chun

    “… is useful for graduate students in computational mechanics.”
    ­ Mathematical Reviews, Issue 2006m.