13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- * 0 ( / (r)) i i i Pi t t d v , (1) where (r) Pi v is the wave velocity field in the specimen or structure. If the material is homogeneous, Eq. (1) can be simplified as: 0 * 0 0 0 i i i p p x x x t t t v v . (2) For each sensor i there will be residual ir between the detected onset time it and the calculated onset time * it : * i i i r t t . (3) If j t is the onset time at another sensor j x , the measured onset time difference between sensors i and j is used. Usually we have: 0 0 * , ( ) i j ij ij p x x r t i j v . (4) If there are more than four onset times available for one event, the problem is overdetermined. These residuals are minimized using the least square method, in which the total error for (n-1) equations is simply the sum of all squared time-residuals: 2 * 2 2 ( ) n i i r . (5) Residuals are reduced by applying corrections x and pv to the source parameters, which can be written as: A( , ) T T p A Ar x v . (6) Thereby, r is the data vector with the residuals for n observations of one event. A, which is a ( 1) 4 n -matrix, contains the partial derivatives of the calculated travel times with respect to the source coordinates, calculated at 0x : 2 2 0 * * * * 2 2 * * * * p n n n n p x r r r r x y z v A r r r r x y z v . (7) Due to the linearization of Eq. (6) the problem is solved iteratively until convergence, starting with an initial guess for the crack source location.
RkJQdWJsaXNoZXIy MjM0NDE=