# Sample Project: FEM for 1-D Transmission and Crack Problem.

The students will study how to numerically solve the following class of linear problems:

(a(x) u'(x))'= f(x), \qquad 0< x < L, \qquad u(0) = u(L) = 0, (1)
where f and a are prescribed functions, with a(x) a positive, piecewise constant function, and f sufficiently smooth. The case where a has a jump at a given point x_0 in (0,L) and f is a simple polynomial will be the main focus of the project. This problem can be seen as a 1D analog of a strongly elliptic problem, e.g. for the system of elasticity in a composite material, on a domain \Omega with a single interface \Gamma. To be a weak solution, any locally regular solution of (1) must also satisfy transmission conditions at x_0 of the form, where the subscripts \pm refer to one-sided limits:
u(x_0^+)=u(x_0^-), \qquad a(x_0^+) u'(x_0^+) = a(x_0^-) u'(x_0^-). (2)

Above u mimics the displacement, while a(x) u'(x) mimics the traction in elasticity. The students will solve (1) using the FEM and piece-wise linear functions. They will investigate different choices of nodal points with respect to the position of the jump point x_0. They will compute the stiffness matrix for the problem and solve the resulting linear system using MATLAB.

To model a 1D analog of an elastic crack, the two sides" of the crack (i.e., x_0^+ and x_0^-) are set traction free, while u itself is allowed to jump at x_0. The Generalized Finite Element method (GFEM), which uses local basis functions combined with a partition of unity, is well suited for this type of problems, as inherently non-conforming. A possible project is an implementation of the GFEM in this 1D setting, again using piece-wise linear functions. The students will compute the stiffness matrix and find its condition number (the matrix may be nearly singular, depending on node arrangement).