Finite Element Method
Finite Element Method
Consider an ODE of the form:
[pic]
with the following Boundary Conditions:
[pic] and [pic]
1.) Substitute for U an unspecified trial function into governing equation, i.e.
[pic]
[pic]
where
n = Number of nodes in system (or, locally, in element).
Uj = Unknown coefficients (to be determined), which are not a function of space, in general.
Nj = Known (user-specified) Basis function, which are a function of space, but not a function of time, in general.
2.) Force the Residual to zero in the global sense, i.e. satisfy the governing equation in the "weak form".
a.) Multiply by a Weighting function:
[pic]
[pic]
where Wi = Known (user-specified) Weight function.
b.) Integrate (take inner product) over global domain and force to zero.
Definition of an Inner Product: [pic]
[pic] Implies that W is orthogonal to R
< ( ) > = [pic]
(dimensionally dependent)
[pic]
If you can solve this equation then you have solved the governing equations, at least in the weak form.
3.) Discretize global domain into subdomains (Elements) such that Σ (Elements) = contiguous, non-overlapping representation of the global domain.
[pic]
4.) Select a basis function for [pic] - a few helpful hints in this selection are
a.) select an orthogonal series basis function
b.) choose a computationally efficient series
Ex. Lagrange Polynomials applied locally on each element
[pic]
[pic]
Assume Linear Element (N = 2)
[pic]
Assume Quadratic Element (N = 3)
[pic]
[pic]
5.) Integration by parts:
Required for [pic] order PDEs using linear basis functions; Provides a direct means to apply Boundary Conditions
[pic]
[pic]
more generally,
[pic]
Adding in the other components of the ODE yields:
[pic]
6.) Select Weighting functions
Ex: Choose "Galerkin" [pic]
(However, for clarity retain the W symbol. It will help when assembling matrices later.)
Note that for each weighting function, Wi, one must assemble all contributions from each basis function Nj.
7.) Assemble Matrices – consider only 2 elements, and assemble all components associated with Row I (for Wi):
[pic]
One row of equations for each Wi
Note: the local definitions of Trial and Weighting functions
[pic] for all [pic]
= 0 for all [pic]
[pic] for row i only contributions from [pic]
Left Elem Right Elem
[pic] = [pic] + 0 = [pic]
[pic] = [pic] + [pic] = [pic]
[pic] = 0 + [pic] = [pic]
[pic] = [pic] + 0 = [pic]
[pic] = [pic] + [pic] = [pic]
[pic] = 0 + [pic] = [pic]
[pic] = [pic] + [pic] = [pic]
Assembly of row I
[pic]
If uniform spacing then h1 = h2 = h
[pic]
[pic] [pic] + [pic] Simpson’s Rule = g
Note the similarities to F.D.
In general, we are forming an expression
[pic]
[pic]
In a FE code: Loop over each element and assemble the local (elemental) contributions for the Lhs matrix [A] and the Rhs vector.
After this loop is completed, then (and only then) apply the boundary conditions to the set of equations.
FE codes contain a lot of bookkeeping. One common mapping array is the element connectivity (or Incidence) list. It has the form:
IN(K,L) where
L = Element Number, L = 1, NE ( and NE = Number of Elements)
K = Local Node Number, K = 1, 2, … to the number of nodes in an element.
IN(K,L) = global node number ( maps global node number to the local node number within element L)
Consider the following 1-D example:
The Element number can have significance if using a frontal matrix solver.
The Node numbering can have significance if using a banded matrix solver.
Node and Element numberings have less significance if using a sparse, iterative matrix solver.
The mapping for this example is:
IN(1,1) = 3 IN(2,1) = 1
IN(1,2) = 4 IN(2,2) = 6
IN(1,3) = 1 IN(2,3) = 4
IN(1,4) = 7 IN(2,4) = 2
IN(1,5) = 5 IN(2,5) = 7
IN(1,6) = 6 IN(2,6) = 5
The FE approximation to the governing equation is accumulated (summed) in 3 nested loops (L, I, and J)
Do L = 1, NE (Loop over all elements)
h = x(in(2,L)) – x(in(1,L)) (Calculate element length)
Do I = 1, 2 (Loop for all Weighting Functions)
dWdx = (1/h)
if (I = 1) dWdx = - dWdx
Do J = 1, 2 (Loop for all Trial Functions)
dNdx = (1/h)
if (J = 1) dNdx = - dNdx
term1 = - dNdx * dWdx * h
term2 = f * h/6
if (I = J) term2 = 2 * term2
AL(J, I) = term1 + term2 (Local 2x2 construction)
Irow = IN(I,L) (Global Mapping)
Jcol = IN(J,L)
AG(Jcol, Irow) += AL(J, I) (Global Matrix Construction)
End Do (End of J loop)
Rhs(Irow) += g * h/2 (Rhs Construction)
End Do (End of I loop)
End Do (Finished with all Elements)
Everything done at the element level!
Easy to automate
a) One Element (problem-specific)
b) Assembly (problem-independent)
8) Add in the Boundary Conditions
a.) Type 1 BC : Satisfy Exactly U(0) = 1
(Note this is at global node number 3 for our example.)
One less unknown in algebraic system.
Remove row_3 and then move column_3 of matrix to Rhs since it is known (not usually done, but possible)
[pic]
This process is somewhat cumbersome. It can change the bandwidth or structure of your system. It can cause other solution-solving anomalies if one is not careful.
Cleaner process:
Remove row 3 completely, i.e. place zeros in all entries.
Then place a 1 on diagonal of [A]
Put the solution (i.e. U(0)=1) into the Rhs
[pic]
b.) Type II B.C. Satisfy approximately in {BCs} vector.
Recall: [pic] and for node 2 (in our example) the flux at the boundary is 5. Also note: [pic]
At the boundary node Wi = 1 for I = 2 (global) and Wi = 0 for all other nodes.
Therefore to apply Type II Boundary Conditions in our example:
Rhs(2) = Rhs(2) – 5
The entire formulation is complete. Call a matrix solver and obtain the solutions for Uj.
-----------------------
[pic]
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- finite volume method cfd
- solidworks finite element analysis tutorial
- finite element analysis basics
- finite element method book pdf
- finite element analysis book pdf
- finite element analysis textbook pdf
- finite element structural analysis pdf
- finite element analysis
- finite element analysis tutorial pdf
- finite element analysis training
- finite element analysis services
- what is finite element analysis