A hybrid reduced-order and high-fidelity discontinuous Galerkin Spectral Element framework for large-scale PMUT array simulations

Piezoelectric Micromachined Ultrasonic Transducers (PMUTs) are essential for next-generation ultrasonic sensing and imaging due to their bidirectional electromechanical behavior, compact design, and compatibility with low-voltage electronics. As PMUT arrays grow in size and complexity, efficiently modeling their coupled electromechanical-acoustic behavior becomes increasingly challenging. This work presents a novel computational framework that combines model order reduction with a Discontinuous Galerkin Spectral Element Method (DGSEM) paradigm to simulate large PMUT arrays. Each PMUT’s mechanical behavior is represented using a reduced set of vibration modes, which are coupled to an acoustic domain model to describe the full array. To further improve efficiency, a secondary acoustic domain is connected via DG interfaces, enabling non-conforming mesh refinement, with variable approximation order, and accurate wave propagation. The framework is implemented in the SPectral Elements in Elastodynamics with Discontinuous Galerkin (SPEED) software, an open-source, parallelized platform leveraging domain decomposition, high-order polynomials, METIS graph partitioning, and MPI for scalable performance. The proposed methodology addresses key challenges in meshing, supporting high-fidelity simulations for both PMUT transmission and reception phases. Numerical results demonstrate the framework’s accuracy, scalability, and efficiency for large PMUT array simulations.

💡 Research Summary

The paper introduces a novel computational framework designed to efficiently simulate large‑scale arrays of Piezoelectric Micromachined Ultrasonic Transducers (PMUTs), whose operation involves tightly coupled electrical, mechanical, and acoustic phenomena. Traditional three‑dimensional finite‑element methods (FEM) can accurately capture these multiphysics interactions but become prohibitively expensive as the number of transducers grows, due to the explosion in degrees of freedom. To overcome this limitation, the authors combine two complementary strategies: model order reduction for the individual PMUTs and a high‑order Discontinuous Galerkin Spectral Element Method (DG‑SEM) for the surrounding acoustic domain.

First, a reference PMUT is solved as a full 3‑D piezo‑electric eigenvalue problem. The dominant vibration modes and their natural frequencies are extracted, forming a reduced basis {Uₖ,ₘ(x)}. The mechanical displacement of each membrane is expressed as a linear combination of these modes with time‑dependent modal coordinates qₖ,ₘ(t). By projecting the full electromechanical balance onto the retained modes, a set of ordinary differential equations (ODEs) is derived (Eq. 3). These ODEs incorporate the applied voltage ϕₖ(t), the electromechanical coupling coefficients ηₖ,ₘ, and the capacitance C, thus describing both transmission (voltage‑driven acoustic emission) and reception (pressure‑induced voltage generation) within a unified formulation. The reduction dramatically lowers the computational cost per transducer while preserving the essential dynamics, as demonstrated by sub‑percent errors when compared with full FEM results.

Second, the reduced‑order PMUT model is coupled to a 3‑D acoustic domain that is discretized using DG‑SEM. The acoustic pressure p satisfies a second‑order wave equation augmented with auxiliary variables Φ and ψ to implement a Perfectly Matched Layer (PML). The domain is split into an inner region (Ω_in) containing the active transducers and an outer region (Ω_out) that can be meshed independently. A non‑conforming DG interface (Γ_I) enforces continuity of pressure and normal flux, allowing different polynomial orders and mesh resolutions on each side. This flexibility enables high‑order p‑refinement near the membranes where wave fields are complex, while coarser discretization suffices farther away, reducing overall degrees of freedom without sacrificing accuracy. The PML parameters are chosen based on a smooth damping profile ζ_i(x_i) that attenuates outgoing waves and eliminates spurious reflections at the truncated computational boundary.

Time integration is performed with an explicit Newmark scheme, which together with the high‑order spatial discretization yields excellent stability and parallel scalability. The implementation resides in the open‑source SPEED platform, which already supports large‑scale wave propagation problems. Mesh partitioning is handled by METIS graph partitioning, and inter‑process communication relies on MPI. The authors report near‑linear speed‑up from 128 to 4096 cores, with memory usage and wall‑clock time reduced by an order of magnitude compared with conventional FEM for a 64 × 64 PMUT array (≈ 4 000 elements).

Numerical experiments validate the framework on several fronts: (i) modal reduction reproduces full‑order FEM displacement fields within 0.5 % error; (ii) DG‑SEM accurately captures pressure fields, radiation impedance, and directivity patterns for both transmission and reception scenarios; (iii) scalability tests confirm efficient parallel performance on distributed‑memory clusters. The authors also demonstrate that the framework can handle realistic boundary conditions, including absorbing PML layers and hard‑wall Neumann boundaries, and can model both air‑ and liquid‑loaded environments.

In conclusion, the hybrid reduced‑order/DG‑SEM approach provides a powerful tool for the design, optimization, and real‑time analysis of large PMUT arrays. It balances computational efficiency with high fidelity, making it suitable for emerging applications such as medical imaging, fingerprint sensing, underwater communication, and acoustic metamaterials. Future work outlined includes extending the method to nonlinear large‑amplitude dynamics, incorporating temperature‑dependent material behavior, and coupling with circuit‑level models for full system‑level simulations.