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Finite Element Theory: Some Essential Analysis Implications

The Isoparametric Element formulation An example derivation for a simple "bar" element to demonstrate the part played by shape functions, Jacobians and strain-displacement matrices in generating a finite element stiffness matrix
The Local Coordinate System An explanation of the local coordinate system used by  element shape functions
Shape Functions A description of shape functions and their effect on the evaluation of element displacements, the choice of linear versus quadratic elements, nodal temperature loading with temperature dependent materials and element thickness interpolation
Equivalent Force Distribution A description of how the applied loads are actually applied to finite elements, including the effects on multiple element loading, loading visualisation arrows in reverse directions, obtaining planar displacement fields, non-constant reactions along a supported boundary, equivalent nodal loading for beam/shell/plate elements and joining lower and higher order elements
Numerical Integration Describes the reasons for, and methods available to implement numerical integration. Also discusses explicit integration for beam elements
Element Mechanisms Explains what an element mechanism is and what to do if such behaviour occurs in an analysis
Evaluating The Mass Matrix Describes how to create a mass matrix for an isoparametric finite element. Discusses the difference between lumped and consistent mass formulations
Finite Element Equilibrium How is equilibrium achieved and when are results correct? This section answers such questions and specifically deals with smoothed and unsmoothed stress contours, limitations of the averaging scheme and why the removal of elements affects the stress contours
Element Quality Describes the effect that poor element quality has on the analysis results, including aspect ratios, warping, distortion (translation and angular) and issues with the Jacobian matrix
The Jacobian Matrix Explains what it is, why is it important and how to eliminate problems arising from it
Element Aspect Ratio Considerations Describes the element aspect ratio test and considerations should this test be violated
Ill-Conditioned Matrices Explains ill-conditioning in finite elements and the typical causes
Links to Useful Reference Material Links to the NAFEMS website as well as recommended reference books


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