This project is dedicated to the implementation of a parallel Harmonic Balance Method within the Sparselizard framework for the analysis of a nonlinear mechanical system. The main objective is the accurate computation and tracing of the system’s nonlinear frequency responses (NLFRs) and nonlinear normal modes (NNMs). This is achieved through the use of a continuation strategy relying on a robust predictor–corrector scheme, ensuring numerical stability and efficiency when following solution branches.
The Sparselizard framework is used as a development and prototyping environment, enabling rapid implementation and validation of the proposed numerical formulations. Once validated, all developed algorithms have been integrated into Quanscient Allsolve (https://quanscient.com/), in order to benefit from advanced high-performance computing capabilities, such as domain decomposition methods and large-scale parallelisation. For clarity and reproducibility, the parts of the code transcribed from the Quanscient implementation are explicitly provided in the present project under the transcription_quanscient directory.
In addition, all numerical results presented and analysed in this work are fully reproducible using the test cases included in the testcase directory. These test cases cover a representative set of benchmark configurations, including a clamped–clamped beam, a cantilever beam, a fan blade, and a cantilever beam in rotation, thereby ensuring transparency and consistency between the numerical implementations and the reported results.
Finally, the governing formulation is expressed in a rotating reference frame, allowing centrifugal effects to be consistently taken into account in the system dynamics, which is essential for applications involving rotating mechanical structures.
This repository relies on Sparselizard. Please refer to the official documentation at this link for proper installation instructions. Additionally, this project requires specific Python packages, which are listed in the requirements.txt file. To install these dependencies, ensure that both Python and pip are installed on your system, then execute the following command:
pip install -r requirements.txtThe physical regions of the system must be clearly defined, including the full volume, clamped surfaces, excitation and measurement regions, together with the relevant material properties. For rotational effects, the distance to the axis of rotation must also be specified.
To compute the nonlinear frequency response of the system, execute the following command:
python3 src/main_NLFR.pyTo compute the Nonlinear Normal Mode (NNM) run:
python3 src/main_NNM.pyThe codebase is predominantly organized around a class-based architecture, distinguishing between the NLFR and NNM functionalities. Each of these main functionalities is encapsulated within a class that incorporates abstract methods. These abstract methods serve as guidelines for implementing custom predictor and corrector schemes, allowing for flexible extension of the framework.
The core of the code follows the continuation loop, which is represented by the continuation_loop_NNM and continuation_loop_NLFR functions. The adaptive adjustment of the predictor step size is managed by the StepSizeRule function, enabling efficient convergence. Furthermore, a dedicated function is implemented to store previous solutions, typically maintaining a history equivalent to the predictor's order plus one, which is crucial for the continuation process.
All numerical results obtained with this implementation are fully reported and analysed in:
Renkin, V. (2026). Parallel HarmonicBalance Method for the Analysis of
Nonlinear Mechanical Systems. Faculty of Applied Sciences, Université de Liège.
http://hdl.handle.net/2268.2/25188
@mastersthesis{renkin2026,
author = {Renkin, Victor},
title = {Parallel Harmonic Balance Method for the Analysis of Nonlinear Mechanical Systems},
school = {Université de Liège},
year = {2026},
month = jan,
url = {http://hdl.handle.net/2268.2/25188}
}