Structural Topology Optimization for Frequency Response Problems Using Adaptive Second-Order Arnoldi Method

For topology optimization problems under harmonic excitation in a frequency band, a large number of displacement and adjoint displacement vectors for different frequencies need to be computed. This leads to an unbearable computational cost, especially for large-scale problems. An effective approach, the Second-Order Arnoldi (SOAR) method, effectively solves the response and adjoint equations by projecting the original model to a reduced order model. The SOAR method generalizes the well-known Krylov subspace in a specified frequency point and can give accurate solutions for the frequencies near the specified point by using only a few basis vectors. However, for a wide frequency band, more expansion points are needed to obtain the required accuracy. This brings up the question of how many points are needed for an arbitrary frequency band. The traditional reduced order method improves the accuracy by uniformly increasing the expansion points. However, this leads to the redundancy of expansion points, as some frequency bands require more expansion points while others only need a few. In this paper, a bisection-based adaptive SOAR method (ASOAR), in which the points are added adaptively based on a local error estimation function, is developed to solve this problem. In this way, the optimal number and position of expansion points are adaptively determined, which avoids the insufficient efficiency or accuracy caused by too many or too few points in the traditional strategy where the expansion points are uniformly distributed. Compared to the SOAR, the ASOAR can deal with wide low/mid-frequency bands both for response and adjoint equations with high precision and efficiency. Numerical examples show the validation and effectiveness of the proposed method.