Parallel Harmonic Balance Method for Analysis of Nonlinear Dynamical Systems

Designers of modern aircraft engine face two opposite challenges: increasing specific performance  while assuring a higher level of safety compared to the previous generation of engine. High levels of vibration in the different components of the engine may lead to high cycle fatigue.

For this reason the study of vibration in gas turbines is of extremely high importance. Computer simulations are one of the essential tools in these studies. Contrary to an experiment, a simulation provides a cheap and fast way to assess the quality of a design. Ideally, a high-fidelity model of a whole engine should be used. Historically, this has not been possible due to the high complexity of a turbine with all its components. Only individual parts have been modelled separately and techniques like reduced order modelling (ROM), cyclic symmetry and others have often been used to further reduce the problem complexity. However, due  to the massive increase in computational power of high-performance computers in the last decades, running a simulation of a whole turbine structure at once on a fine scale is within reach. 

This paper presents a parallel implementation of Harmonic Balance Method (HBM) for 3d non-linear dynamics problems. Then, preliminary results and validations are shown, followed by a discussion about parallel scalability of proposed  algorithms.

For spatial discretisation, a standard 3d finite element technique is applied. HBM is used on top of the standard equations of motions to express steady state periodic motion. HBM is a common tool for vibration analysis. It avoids the need for time stepping schemes which massively reduces computational cost and may introduce numerical damping. An accurate description of the non-linear behaviour is achieved using higher harmonics and the alternating frequency time (AFT) procedure.

The HBM formulation leads to  a nonlinear algebraic system where the unknowns are the Fourier coefficients of the harmonic motion. It  can be solved using a Newton-Raphson algorithm. The linearised system at each Newton-Raphson iteration produces a non-symmetric and indefinite matrix, which makes the implementation of efficient linear iterative solvers more difficult. For instance, Conjugate Gradient algorithm can not be used to solve the algebraic system. Furthermore, the non-linear term creates coupling between harmonics leading to increased number of non-zero entries in the sparse matrix. This requires the whole system to be solved as one instead of solving N separate systems, one for each harmonic. However, an alternative approach that uses an approximation matrix which neglects the interharmonic coupling could be used in the Newton-Raphson iteration or at least as a preconditioner for the iterative linear solver.

Most parts of the proposed  code are parallelised and their implementation will be introduced in the paper. The input/output part remains sequential for now. The code is designed to take  advantage of distributed memory architecture. A standard spatial partition of the input mesh into parts of equal size is performed. Each partition is handled by one MPI process attached to one computational core. This allows for efficient parallelisation of the problem assembly even for large problems. A direct decomposition is used to solve the linear system.  Efficiency of the proposed implementation will be discussed, in particular the memory usage.

Numerical examples will illustrate the proposed parallel Harmonic Balance Method. Two numerical examples will be used - a simple 3D finite element clamped-clamped beam and a fan blade.