Finite element solution for the Heat equation[edit | edit source]
Let us now try to create a finite element approximation for the
variational initial boundary value problem for the heat equation . This
problem can be stated as
Let be a finite dimensional approximation to and let
be a finite dimensional approximation to . Let the
weighting functions be such that on .
Similarly, any other function also goes to zero on the
Assume that the trial solutions can be represented as
The additional quantity results in the satisfaction of the
boundary condition on .
If we substitute these quantities into the variational initial boundary value
problem, we get the Galerkin formulation. The steps in this process
are shown below.
The approximate form of the variational IBVP is
Replacing with we get
After expanding and rearranging terms, we get
In compact notation
Similarly, the initial condition can be written as
Substituting for and expanding out terms, we have
In compact form,
Therefore the approximate form of the variational BVP can be written
Note that the derivatives with respect to time are still continuous in
this approximate variational BVP.
Finite element approximation[edit | edit source]
We now discretize our domain into elements where
and is the total number of elements. Nodes may exist anywhere within
the element domains. But the most common locations are at the element
vertices and along the interelement boundaries.
Let the set of global node numbers be
Let the set nodes at which a temperature (essential BC) is prescribed be
Then the set of nodes at which a temperature is to be determined is
Let be the number of nodes in . Then the number of
equations to be solved is also . See Figure 1
to get an idea about what these sets of nodes refer to.
Figure 1. FEM discretization for the heat conduction problem.
As we have seen before, a typical weighting function
is assumed to have the form
Note that only if for every . On
the other hand for every node in .
Similarly, a typical trial solution can be written as
where is the unknown temperature at node at time .
We also often represent the approximate form of the essential BCs in a
similar way, i.e.,
Substitute equations (42), (43), and
(44) into the first of equations (40).
You will get
Assuming that does not change with time, we can take the
constants and time-dependent parameters outside the integrals to get
Using the usual argument about the arbitrary nature of , we get
In compact notation,
where , and .
Let us rename parts of the above equation as follows:
Then we get
In matrix form,
As before, we can break up the global matrices into sums over elements.
and is the number of nodes in an element.
We can do the same for the initial conditions. However, a more common
approach is to specify the values of directly at the nodes
instead of going through an assembly process.
Computing , , [edit | edit source]
The finite element system of equations at time :
Integrating Stiffness Matrix[edit | edit source]
Stiffness matrix terms:
Expanded out (in 2D) (assume ):
Returning to the integration of the stiffness matrix terms:
In parent element coordinates:
Gaussian Quadrature Example[edit | edit source]
Find the constants Co, C1, and X so that the quadrature formula
This has the highest possible degree of precision.
Since there are three unknowns, Co, C1 and X1, we will expect the formula to be exact for
Equation 2 and 3 will yield.
Thus the degree of the precision is 2
A similar approach can be used when w:Robin boundary conditions are needed to be applied. Let us assume that the Robin boundary conditions take the form
Recall from the previous section that the boundary term in the weak form of the heat equation is
Let us assume, for simplicity, that the material is isotropic. In that case becomes a scalar quantity and we can write
Now, we can write the Robin boundary condition as
Using this expression we can get of the flux term in to get
Then the boundary term takes the form
If we ignore the contributions to the trial function from the Dirichlet boundary conditions we can write
Plugging these expressions into the boundary term leads to
Now recall that spatially discretized equation for transient heat conduction can be expressed as (in simplified form; after getting rid of contributions due to Dirichlet boundary conditions and with a scalar )
Substituting the last term on the right hand side with the new expression for the boundary term, we have
Invoking the arbitrariness of and rearranging, we get
Then the finite element system of equations is
or, in matrix form
Note that the matrix and the vector are different from the case where there are no Robin boundary conditions. The boundary terms in the discretized system of equations can be computed in the usual manner and will add terms to the diagonal of the matrix.