Numerical analysis of a fluid droplet subject to acoustic waves
Creators: name: Meng, Xuan
Publisher: name: Zenodo
Year: 2018
Subjects: acoustics
microfluidics
finite elements

description: Efficient and rigorous acoustic solvers that enable high frequency sweep application over a wide range of frequencies are of great interest due to their practical importance in many engineering, physical problems or life science research that involve acoustic radiation, such as engine noise analysis, acoustic simulation in micro-fluidics and the design of lab device, etc. There is room for reduction of cost on experimental systems that can be investigated and optimised through numerical modelling of physical processes on the micro-scale level. The major difficulty that arises is the inconsistency of materials, time scales and fast oscillation nature of the solution that leads to unstable results for conventional numerical methods. However, analytical solutions are infeasible for large problems with complex geometries and sophisticated boundary conditions. Hence, the vital need for efficient solvers.

In this research the development of computational methods for acoustic application is presented. The proposed method is applied to the study of propagating waves in particular to simulate acoustic phenomena in micro-droplet actuated by leaky Surface Acoustic Waves on a lithium niobate (LiNbO3) substrate. Explicitly, we introduce a new computational method for the analysis of fluids subjected to high frequency mechanical forcing. Here we solve the Helmholtz equation in the frequency domain, applying higher order Lobatto hierarchical finite element approximation in H1 space [10], where both pressure field and geometry are independently approximated with arbitrary and heterogeneous polynomial order. Meanwhile, a time dependent acoustic solver with arbitrary input signals is also proposed and implemented. The development of extended computational methods for the solution of the Helmholtz equation with polychromatic waves is presented, where Fourier transformation is applied to switch the incident wave and solution space from the frequency domain to the temporal domain. Consequently, the implementation and convergence rate of the numerical methods are demonstrated with benchmark problems. The numerical method is an extension of the conventional higher order finite element method and as such it relies on the definition of basis functions. In this

work we implement a set of basis functions using integrated Legendre polynomials (Lobatto polynomial). Two type of basis functions are presented and compared. Therefore, the significant improvements in efficiency is demonstrated using a Lobatto hierarchical basis compared with a Legendre type basis. Moreover, a novel error estimation and automatic adaptivity scheme is outlined based on an existing a priori error estimator [256]. The accuracy and efficiency of the proposed object oriented (predefined error level) a priori error estimator is validated through numerical assessments on a three-dimensional spherical problem and compared with uniformly h and p adaptivities. The simple and generic features of the proposed scheme allow fast frequency sweeps with low computational cost for multiple frequencies acoustic application. The current finite element approach is executed in parallel with pre-partitioned domain, which guarantees the optimal computational speed with minimal computational effort for large problems. Overall, the benefits of using the proposed acoustic solver is explained in detail.

Finally, we illustrate the model’s performance using an example of a micro- droplet actuated by a surface acoustic wave (SAW), which has vast applications in micro-fluidics and micro-rheology at high frequency. Conclusions are drawn, and future directions are pointed out.

The proposed finite element technology is implemented in the University of Glasgow in-house open-source finite element parallel computational code, MoFEM (Mesh Oriented Finite Element Method). All algorithms and examples are publicly available for download and testing.