where s(j) is the predictor stepsize. The tangent vector p(j) = [p ∗ z,(j) pT,(j)] ∗ to the branch defined by (6) is solution of the system ∂H ∂z p0 (z p0,(j),T(j)) ∂H ∂T (z p0,(j),T(j)) ∂h ∂z p0 ∗ (z p0,(j)) 0 pz,(j) pT,(j) = 0 0 (8) with the condition p(j) = 1. The star denotes the transpose operator. This normalization can be taken into account by fixing one component of the tangent vector and solving the resulting overdetermined system using the Moore-Penrose matrix inverse; the tangent vector is then normalized to 1. Corrector step The prediction is corrected by a shooting procedure in order to solve (6) in which the variations of the initial conditions and the period are forced to be orthogonal to the predictor step. At iterationk, the corrections z (k+1) p0,(j+1) = z (k) p0,(j+1) +∆z (k) p0,(j+1) T(k+1) (j+1) = T (k) (j+1) +∆T (k) (j+1) (9) are computed by solving the overdetermined linear system using the Moore-Penrose matrix inverse ∂H ∂z p0 (z (k) p0,(j+1) ,T (k) (j+1) ) ∂H ∂T (z (k) p0,(j+1) ,T (k) (j+1) ) ∂h ∂z p0 ∗ (z (k) p0,(j+1) ) 0 p∗ z,(j) pT,(j) " ∆z (k) p0,(j+1) ∆T(k) (j+1) # = −H(z (k) p0,(j+1),T (k) (j+1)) −h(z (k) p0,(j+1)) 0 (10) where the prediction is used as initial guess, i.e, z (0) p0,(j+1) =˜zp0,(j+1) andT (0) (j+1) = ˜T(j+1). The last equation in (10) corresponds to the orthogonality condition for the corrector step. This iterative process is carried out until convergence is achieved. The convergence test is based on the relative error of the periodicity condition: kH(zp0,T)k kzp0k = kzp(T, zp0) −zp0k kzp0k <ǫ (11) whereǫ is the prescribed relative precision. 3.3 Sensitivity Analysis z (k) p0 ,T (k) = z (k) p (T (k), z (k) p0 ) −z (k) p0 . As evidenced by equation (10), the method also requires the evaluation of the2n×2nJacobian matrix ∂H ∂z0 (z0,T)= ∂z(t, z0) ∂z0 t=T −I (12) whereI is the2n×2nidentity matrix. The classical finite-difference approach requires to perturb successively each of the 2n initial conditions and integrate the nonlinear governing equations of motion. This approximate method therefore relies on extensive numerical simulations and may be computationally intensive for large-scale finite element models. 0 0 instead of a numerical finite-difference procedure. The sensitivity analysis consists in differentiating the equations of motion (2) with respect to the initial conditionsz0 which leads to d dt ∂z(t, z0) ∂z0 = ∂g(z) ∂z z(t,z0) ∂z(t, z0) ∂z0 (13) Each shooting iteration involves the time integration of the equations of motion to evaluate the current shooting residueH Targeting a reduction of the computational cost, a significant improvement is to use sensitivity analysis for determining ∂z(t, z )/∂z 226
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