Parallel harmonic balance method for analysis of nonlinear mechanical systems

Abstract

Mechanical vibration analysis and modelling are essential tools used in the design of various mechanical components and structures. In the case of turbine engine design specifically, the ability to accurately predict vibration of various parts is crucial to ensure their safe operation while maintaining efficiency. As the designs become increasingly complex and margins for errors get smaller, high fidelity numerical vibration models are necessary for their analysis. Research of parallel algorithms has progressed significantly in the last decades, thanks to the exponential growth of the world's available computational resources. This work explores the possibilities for parallel implementations for solving large scale nonlinear vibration problems. A C++ code using MPI was developed to validate these implementations in practice. The harmonic balance method is used in combination with finite elements discretisation and applied to an elastic body with the Green-Lagrange nonlinear model for large deformations. A parameter continuation scheme using a predictor-corrector approach is included to compute frequency response functions. A Newton-Raphson solver is used to solve the bordered nonlinear system of equations in the frequency domain. Three different parallel algorithms for solving the linearised problem in each Newton iteration are analysed - a sparse direct solver (using MUMPS library), GMRES (using PETSc library) and an inhouse implementation of FETI. The performance of the solvers is analysed using beam testcases and a fan blade geometry. Scalability of MUMPS and the FETI solver is assessed. Full nonlinear frequency response functions with turning points are also computed. Use of artificial coarse space and preconditioning in FETI is discussed as it greatly impacts convergence properties of the solver. The presented parallel linear solvers show promising scalability results and an ability to solve nonlinear systems of several million degrees of freedom.