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  Finite Elements and Differential Equations

Finite Elements:

Triangulating a Region:

A drawback of finite difference methods is that they require a very regular grid as well as thus a very regular region either rectangular or a regular part of a rectangle. Finite elements is a manner that works for any shape region because it isn’t built on a grid but on triangulation of the region that is cutting the region up into triangles as we did in a previous lecture.

The following figure depicts a triangularization of a region.

2419_triangulization of region.jpg

An annular region with a triangulation note down that the nodes and triangles are very evenly spaced.

This figure was formed by the script program mywasher.m. Notice that the nodes are consistently distributed. This is better for the finite element process where we will use it.

Open the program mywasher.m. This program describes a triangulation by defining the vertices in a matrix V in which every row contains the x and y coordinates of a vertex. Notice that we list the interior nodes first afterwards the boundary nodes.

Triangles are described in the matrix T. Every row of T has three integer numbers indicating the indices of the nodes that form a triangle. For example the first row is 43 42 25, so T1 is the triangle.

1028_finite element of triangulization.jpg

The finite elements for the triangulation of a one dimensional object.

with vertices v43, v42 and v25. The matrix T in this case was produced by the Mat lab commands Delaunay. The command produced more triangles than want and the unwanted ones were deleted.

We a three dimensional plot of the region as well as triangles is produced by having the last line be trimesh (T,x,y,z).

What is a finite element?

The restricted element method is a mathematically complicated process. Nevertheless a finite element is actually a very simple object. To every node vj we associate a function Φj that has the properties Φj(vj) = 1 and Φj(vi) = 0 for i /= j. Between nodes, Φj is a linear function. This function is the finite element. There are fancier kinds of finite elements that we will not discuss.

If we consider one dimension subsequently a triangulation is just a subdivision into subintervals. We illustrate an uneven subdivision of an interval and the finite elements corresponding to every node in the above figure.

In two dimensions Φj is calm of triangular piece of planes. Therefore Φj is a function whose graph is a pyramid with its peak over node vj.

What is a finite element solution?

A finite element (approximate) explanation is a linear combination of the elements:

1754_finite element solution.jpg

Therefore finding a finite element solution amounts to finding the best values for the constants {Cj}nj=1. In the program mywasher.m the vector z contains the node values Cj. These values give the height at every node in the graph. For illustration if we set all equal to 0 except one equal to 1then the function is a finite element. Do this for one boundary node afterwards for one interior node.

Notice that a total of linear functions is a linear function. Therefore the solution using linear elements is a piecewise linear function.

As well notice that if we denote the j-th vertex by vj, then

U(vj) = Cj

Therefore we see that the constants Cj are just the values at the nodes.

Take the one-dimensional case since we know that the solution is linear on every subinterval knowing the values at the endpoints of the subintervals that are the nodes gives us complete knowledge of the solution. Could be a restricted element solution since it is piecewise linear.

1337_piecewise linear.jpg

A probable finite element solution for a one dimensional object. Values are assigned at every node and a linear interpolant is used in between.

In the two-dimensional case the solution is linear on every triangle and so again if we know the values {Cj}nj=1 at the nodes afterwards we know everything.

Experiment with finite elements:

By changing the values in c in the program we are able to produce different three dimensional shapes based on the triangles. The point subsequently of a finite element solution is to find the values at the nodes that best approximate the true solution. This task is able to be subdivided into two parts- (1) assigning the values at the boundary nodes and (2) assigning the values at the interior nodes.

Values at boundary nodes:

Once a triangulation as well as a set of finite elements is set up the next step is to incorporate the boundary conditions of the problem. Presume that we have fixed boundary conditions that is of the form

u(x) = g(x), for x ∈ ∂D,

Where D is the object (domain) as well as ∂D is its boundary. After that the boundary condition directly determines the values on the boundary nodes.

In particular presume that vlis a boundary node. Since Φl(vl) = 1 and all the other elements are zero at node vl, then to create

2068_boundary nodes.jpg

We should choose:

Cl= g(vl).

Therefore the constants Cjfor the boundary nodes are set at exactly the value of the boundary condition at the nodes.

Therefore if there are m interior nodes then,

Cj= g(vj), for all m + 1 ≤ j ≤ n.

In the program mywasher the initial 32 vertices correspond to interior nodes and the last 32 correspond to boundary nodes. By setting the last 32 values of z we get the boundary conditions. We could do this by adding up the following commands to the program

z(33:64) = .5;

Or more elaborately we might utilize functions:

z(33:48) = x(33:48).^2 - .5*y(33:48).^2
z(49:64) = .2*cos(y(49:64))

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