The cost function remains unchanged, and is still defined by (2.10), as the idea is to find some cracks that minimize the difference between the two solutions and . We assume that the crack is equal to , where is the point of insertion of the crack, is the size of the inserted crack (assumed to be small), and is a reference crack, of unit normal vector . Then, we can rewrite the cost function defined by equation (2.10) as a function of . The asymptotic expansion is then the following:
(2.14) |
The topological gradient defined by equation (2.15) can be rewritten in the following way:
(2.18) |
From this equation, we can deduce that the minimum of is reached when is the eigenvector associated to the lowest eigenvalue of .