Finite Element method is a numerical method, rather than analytic method of solving PDEs.
Interpolation function
- Defines the type of polynomials used approximate the real solution
- Higher order polynomials have some flexibilities to approximate the actual solution more accurately
- However it requires more information and operations to compute a higher order element. For example, in 1D, a 2nd order function requires information from 3 nodes to determine the 3 constants:
- 2 nodes
- 3 nodes
- In 2D, a simple linear equation would have requried 3 nodes to inform the 3 parameters:
- 3 nodes
- 6 nodes
Example problem: Plane Stress/Strain and Isotropic Elasticity
We try to minimise the potential energy of the system:
where is the strain energy of the structure and is the work done by external force.
Strain Energy
To calculate the strain energy of the structure , we have .
- For 1D linear elasticity, we have .
- For 2D linear elasticity, we have .
- Inside this equation:
- For Plane stress:
- For Plane strain:
At each node:
- Remember the following:
- After differentiation,
- ie. for a 2-D triangular element:
For simplex elements, return constants.
Then it sums up to the element stiffness matrix :
where .
Work done by external force
For simplicity, we assume a nodal load:
Assembling the equations of the entire system:
To get:
is called the global stiffness matrix.
2D Simplex elements
Nodal information:
Shape function:
Nodal Strain:
- Boom!
- Assemble these equations to give :
Get :
where is the thickness, and is the element area.
Now we get , how do we construct $latex \mathbf{K} = \sum^{E}_{e=1} \mathbf{K_e}$?
We do a numerical integration using Gaussian quadrature:
which means that we numerically integrate function by summing up the function output at sample points multiplied by the corresponding weights . is the number of sample points.
The weight is calculated by with being the Legendre polynomial. Or (a look-up table):
No. of points | Points | Weights |
---|---|---|
1 | 0 | 2 |
2 | 1 | |
3 | 0 | |
4 | ||
5 | 0 | |
This sets up the global stiffness matrix. Together with boundary conditions, initial conditions and the finite element mesh, we can solve the linear/non-linear matrix to obtain nodal results. This current example is a linear model. We will soon explore a non-linear mechanical model.
Sources:
- Endre Suli’s Lecture notes on Finite Element Methods for Partial Differential Equations
- Endre Suli and David Mayers – An Introduction to Numerical Analysis
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