The finite element method [51] was selected for this problem because it easily handles complexities in geometry, including changes in geometry with time and large variations in material properties over space. In this radial model, the sphere is discretized into N − 1 spherical shell elements, each of which may change its radial position and volume considerably from timestep to timestep, depending on the radial distribution of bubbles throughout the sphere and on losses from the original polymeric material. Temperatures are determined at every nodal position, and the temperature profile within each element is either linear or quadratic depending on the choice of basis function. Some details of the finite element method for a one-dimensional radial problem are given in Appendix B.
Since the elements are initially equal in volume and the outer elements shrink in size first, the radius of the innermost element is relatively large. Quadratic basis functions were thus used to counter difficulties in meeting the boundary condition at the center of the sphere. Additional nodes at the radial midpoint of each element were added to set up this problem.
Given the polymer volume fraction and bulk values of density, heat capacity, and thermal conductivity for each element, equation (7) is used to determine the temperature at each radial node i at the next time step:
|
| (63) |
The initial condition for temperature is uniformly at ambient temperature:
 | (64) |
The heat flux boundary condition at the surface of the sphere is
 | (65) |
The requirement of a smooth and finite temperature field at the sample center is imposed by
 | (66) |
The finite element formulation results in an easily solved tridiagonal system for linear basis functions and in a nearly block diagonal sparse system for quadratic basis functions. Solution of the quadratic problem is obtained using module DGEABD from the LAPACK scientific computing package.