Q4 element Practical considerations for applying the

FEM

Formulation

Displacements and shape functions Strains and gradient matrix Element stiffness matrix

Can we factor out B matrix? Can we factor out D matrix?

Comparison with CST

Compared to Q4 element, CST: 1. Provides less accuracy 2. Produces stiffer FE model 3. Converges more slowly

General considerations for applying the FEM

General considerations Modeling

› Symmetry › Selection of finite element type › Mesh refinement › Mixing different element types

Convergence and the patch test

1. Check input 2. Check output 3. Check with your colleagues

Nodes Elements Boundary conditions Applied loads

Always try to develop an expectation of the correct answer whenever possible › Self-weight vs vertical reactions › Crude models › Experimental data

Modeling in the FEM • Symmetry • Selection of element type • When to utilize more elements • Mesh refinement • Mixing different element types

1. In general, a good idea 2. Exceptions include problems involving:

1. Buckling 2. Vibration modes

1. Minimize aspect ratio 2. Aim for compact and regular shapes

Generally, use more elements near abrupt changes in: 1. Loads 2. Material properties 3. Geometry

Methods of refining a mesh 1. H method 2. P method 3. R method

1. Compatibility between adjacent elements

2. Expected physical behavior

Accuracy of FEM solution • Equilibrium • Compatibility

In a FE-based solution, equilibrium always satisfied: › At the nodes › Globally

FEM based solution may still be acceptable even though equilibrium is not satisfied: › Within element › Between two adjacent elements

In a FE-based solution, compatibility should generally be satisfied: › At the nodes › Within elements › Between adjacent elements

However, there are cases for which incompatibility may be desirable

Convergence of FEM solution; the patch test

Assumed trial function should be: 1. Simple 2. Continuous within element 3. Lead to compatibility between adjacent

elements 4. Complete All terms with order below highest must be

included Equivalently, trial function should be capable

of predicting rigid body mode and constant strain behaviors

How to check if any given element type will lead to convergence?

Apply practical test on this single element type, known as the patch test

In essence, we check for ability of small ‘patch’ of elements of the same type to produce rigid body mode and constant strain behaviors

1. Create patch of elements of same type 2. Assume rigid body motion at

boundaries; verify rigid body motion internally

3. Assume constant strain at boundaries; verify assumption internally

4. Assume constant stress; verify assumption internally