Hybrid and Incompatible Finite Element Methods

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.

Table of Contents:
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.