# The Finite Element Method: Its Basis and Fundamentals

## The Finite Element Method: Its Basis and Fundamentals

O. C. Zienkiewicz,

Preference :

The limitations of the human mind are such that it cannot grasp the behavior of its complex surroundings and creations in one operation. Thus the process of subdividing all systems into their individual components or “elements,” whose behavior is readily understood, and then rebuilding the original system from such components to study its behavior is a natural way in which the engineer, the scientist, or even the economist proceeds. In many situations, an adequate model is obtained using a finite number of well-defined components. We shall term such problems discrete. In others, the subdivision is continued indefinitely and the problem can only be defined using the mathematical fiction of an infinitesimal. This leads to differential equations or equivalent statements which imply an infinite number of elements. We shall term such systems continuous. With the advent of digital computers, discrete problems can generally be solved readily even if the number of elements is very large. As the capacity of all computers is finite, continuous problems can only be solved exactly by mathematical manipulation. The available mathematical techniques for exact solutions usually limit the possibilities of oversimplified situations.

• 1. The Standard Discrete System and Origins of the Finite Element Method
• 2. Problems in Linear Elasticity and Fields
• 3. Weak Forms and Finite Element Approximation
• 4. Variational Forms and Finite Element Approximation
• 5. Field Problems: A Multidimensional Finite Element Method
• 6. Shape Functions, Derivatives, and Integration
• 7. Elasticity: Two- and Three-Dimensional Finite Elements
• 8. The Patch Test, Reduced Integration, and Nonconforming Elements
• 9. Mixed Formulation and Constraints: Complete Field Methods
• 10. Incompressible Problems, Mixed Methods, and Other Procedures of Solution
• 11. Multidomain Mixed Approximations
• 12. The Time Dimension: Semi-Discretization of Field and Dynamic Problems
• 13. Plate Bending Approximation: Thin and Thick Plates
• 14. Shells as a Special Case of Three-Dimensional Analysis
• 15. Errors, Recovery Processes, and Error Estimates
• 16. Adaptive Finite Element Refinement
• 17. Automatic Mesh Generation
• 8. Computer Procedures for Finite Element Analysis