An adaptive model order reduction method for boundary element-based multi-frequency modeling of acoustic wave problems


Dynamical systems are common in many industrial sectors. The acoustic performance of these systems is generally a vital design criterion and has formed the basis of research in the past decades. To date, several methods for modeling the acoustic fields of dynamical systems have been developed. Among them, the boundary element method (BEM) is widely used because it provides detailed information regarding the acoustic qualities and can handle three-dimensional (3D) problems attributed to its semi-analytical characteristics. However, the practical applications of BEM are hindered by numerous limitations. The presence of non-unique solutions for exterior acoustic wave problems results in inaccurate results. In addition, classical BEM results in a dense and asymmetrical system matrix making acoustic modeling time-consuming, memory demanding and expensive. Therefore, developing efficient and low-cost computational approaches is necessary to overcome these drawbacks.

Frequency sweep analysis is a vital aspect of designing high-quality products and assessing noise emission levels. Nonetheless, it requires repeated assembly and solution of system equations, which increases the overall computational complexity. Previous research findings pointed out the elimination of frequency dependency as one promising way of overcoming the challenges. This can be achieved through different means, including empirical interpolation and series expansion methods. However, the required memory becomes more intensive. Model order reduction (MOR) can be another way to reduce the computational burden. Despite the extensive studies on the MOR methods, corresponding research on the dimension reduction of BEM is scarce, attributed to the difficulty in reducing the frequency-dependent boundary element models. Generally, model reduction processes aim to reduce the model size and to find an approximated solution. Therefore, it is extremely important to accurately obtain the order of the reduced model adaptively for practical applications.

Confronted by these problems, Dr. Xiang Xie and Professor Yijun Liu from Southern University of Science and Technology proposed an adaptive order reduction method for solving different problems associated with boundary element-based multi-frequency modeling of acoustic waves based on an offline-online solution framework. In their approach, the adaptive structure-preserving model was constructed by combining the ideas of series expansion such as Taylor’s theorem and MOR methods. The computational complexity and memory problem induced by the frequency-related decomposition were solved by constructing a global frequency independent approximation matrix based on the second-order Arnoldi (SOAR) algorithm. The feasibility of the presented approach was validated by investigating the acoustic features of a realistic problem and two academic benchmarks with different boundary conditions. The work is currently published in the journal, Computer Methods in Applied Mechanics and Engineering.

The results demonstrated that by incorporating the upper Hessenberg matrix into the SOAR process, the authors could automatically determine the desired low order. The memory problem induced by the frequency decomposition was successfully solved during the offline stage by column-by-column projection. This was attributed to factoring the frequency terms as scalar followed by setting the matrices through integration. In the online phase, the product of the summation of the offline matrices and frequency-dependent coefficients proves useful in recovering the reduced-order model. The algebraic manipulations of ROM with smaller systems significantly reduced the computational costs. Furthermore, it was worth noting that the automatic determination of the number of iterations required for converges was possible using the condition number of the Hessenberg matrix.

In summary, the Xie-Liu study presented a robust and adaptive offline-online Taylor-based SOAR computational method for overcoming the problems associated with frequency sweep analysis of the exterior acoustic problems. The model exhibited significantly improved performance by avoiding unnecessary iterations and lowering the memory allocation and computational costs. The high efficiency of the proposed method was validated in terms of runtime and decreased DOFs. It could perform multi-frequency analysis of complex systems with reduced simulation demands. In a statement to Advances in Engineering, the authors stated that the study widen the opportunities for improving the acoustic performance of dynamical systems.

About the author

Xiang XIE is currently a postdoctoral researcher in the Department of Mechanics and Aerospace Engineering at Southern University of Science and Technology (SUSTech), China. Prior to joining SUSTech in 2020, he received his double Ph.D. degrees in 2019 from KU Leuven (Belgium) and Shanghai Jiao Tong University (China). He is the author of over 20 refereed journal publications on model order reduction, topology optimization and vibro-acoustics. His research program is currently funded by National Natural Science Foundation of China and China Postdoctoral Science Foundation.

About the author

Yijun LIU has been a Chair Professor in the Department of Mechanics and Aerospace Engineering at SUSTech since 2018. Earlier, Dr. Liu was a professor at the University of Cincinnati of the USA for more than 20 years. Dr. Liu’s major research interests are in computational mechanics, especially the boundary element method and its fast solution methods, with applications in modeling composite materials, fracture, structural dynamics, and acoustics.

Dr. Liu has authored three books and published more than 100 research papers in technical journals and conference proceedings. His book Fast Multipole Boundary Element Method was published by Cambridge University Press in 2009 and has been well received in the research community. Dr. Liu is a fellow of ASME.


Xie, X., & Liu, Y. (2021). An adaptive model order reduction method for boundary element-based multi-frequency acoustic wave problemsComputer Methods in Applied Mechanics and Engineering, 373, 113532.

Go To Computer Methods in Applied Mechanics and Engineering

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