Space-time finite element path-following for periodic solutions in nonlinear vibration problems

An efficient path-following method is presented for computing periodic solutions of nonlinear dynamical systems with a large number of degrees of freedom. The approach is illustrated on structural dynamics problems, such as trusses and shells. The method employs a displacement-based Galerkin finite element discretization, continuous in both space and time. Displacement periodicity is enforced by identifying the first and last temporal nodes in the assembly, which naturally ensures velocity periodicity in weak form. The space-time equilibrium is solved iteratively together with an arc-length constraint, enabling stepwise tracing of periodic solution paths with respect to a continuation parameter, such as the excitation frequency, including multiple solutions and unstable branches. The weak form is formulated in a dimensionless time domain so that the temporal discretization does not depend on the loading frequency. Exact Jacobians, including derivatives with respect to the continuation parameter, are assembled very efficiently from the finite element contributions. By exploiting this feature, along with the sparsity of the space-time stiffness matrix and a modified Newton solver, the method remains computationally efficient for high-dimensional models. Finally, a reduced temporal quadrature is implemented as a stabilization technique for coarse time discretizations, enabling the robust and accurate reconstruction of solution curves using a limited number of quadratic or cubic time elements.