AdvancedTopic 97 of 104Engineering Mathematics
Finite element basics
Splits a domain into small elements with simple shape functions and assembles a global system.
In depth
FEM divides the domain into elements, approximates the unknown within each with shape functions, and assembles element stiffness matrices into a global system KU=F. Boundary conditions are applied and the system solved. Mesh refinement increases accuracy.
Key points
- Nodes, elements and degrees of freedom
- Triangles, quads, tetrahedra, hexahedra
- Convergence checks are essential
ExampleStress analysis of a bracket with a hole using a refined mesh near the hole.
Finite elements handle complex shapes and loads, so they are the standard tool for structural, thermal and fluid analysis in industry.
What is Finite element basics?
Splits a domain into small elements with simple shape functions and assembles a global system.
Why is it important to understand finite element basics?
Finite elements handle complex shapes and loads, so they are the standard tool for structural, thermal and fluid analysis in industry.
What is a shape function?
A simple function defined on each element that interpolates the unknown field from the values at the element nodes.
How do I know if my finite element result is right?
Refine the mesh and see whether results converge, compare with a simple hand check, and confirm the boundary conditions match the real support.
How is Finite element basics used in practice?
Typical applications include checking the stress around a hole in a machine bracket, predicting the deflection of a car chassis under load and simulating heat flow through a heat sink with fins.
What should I learn before Finite element basics?
Study "Finite difference method" first (topic 96 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Finite element basics?
Nodes, elements and degrees of freedom. Triangles, quads, tetrahedra, hexahedra. Convergence checks are essential.
Can you give an example of Finite element basics?
Stress analysis of a bracket with a hole using a refined mesh near the hole.
What comes after Finite element basics?
The next topic is "Optimisation and Lagrange multipliers", which builds directly on this one.
How difficult is Finite element basics?
It is classed as Advanced in Engineering Mathematics (topic 97 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.