Topology optimization of MEMS resonators with target eigenfrequencies and modes

Highlights

• An efficient topology optimization approach for MEMS resonators is proposed.

• Focus is given to industrially relevant layouts with suspended proof masses.

• The first structural eigenmodes are controlled considering different target eigenfrequencies.

• Numerical efficiency is obtained through the use of reduced order models.

• The method is demonstrated for single mass and tuning fork MEMS gyroscopes.

Abstract

In this paper we present a density based topology optimization approach to the synthesis of industrially relevant MEMS resonators. The methodology addresses general resonators employing suspended proof masses or plates, where the first structural vibration modes are typically of interest and have to match specific target eigenfrequencies. As a significant practical example we consider MEMS gyroscope applications, where target drive and sense eigenfrequencies are prescribed, as well as an adequate distance of spurious modes from the operational frequency range. The 3D dynamics of the structure are analysed through Mindlin shell finite elements and a numerically efficient design procedure is obtained through the use of model order reduction techniques based on the combination of multi-point constraints, static approximations and static reduction. Manufacturability of the optimized designs is ensured by imposing a minimum length scale to the geometric features defining the layout. Using deterministic, gradient-based mathematical programming, the method is applied to the design of both single mass and tuning fork MEMS resonators. It is demonstrated that the proposed methodology is capable of meeting the target frequencies and corresponding modes fulfilling common industrial requirements.

Introduction

Micro Electro Mechanical Systems (MEMS) are one of the most disruptive technologies of the 21st century. To a large extent, they form the basis for the forthcoming Internet-of-Things (IoT) revolution, that will increase the interaction between electronic devices and between devices and real world, facilitating smart living for humans and machine-to-machine communication (M2M). In this framework, MEMS are used as ultra-compact and high performance sensors and actuators in different domains, including consumer electronics, automotive safety, autonomous driving, avionics, smart transportation systems, energy grids, and healthcare facilities.

MEMS resonators are small electromechanical structures that vibrate at high frequencies. They are key components that are extensively employed in many of the aforementioned applications, such as frequency selection (e.g. MEMS radio-frequency (RF) and intermediate-frequency (IF) filters, Nguyen et al., 1998, Colombo et al., 2018, Kochhar et al., 2020), timing (i.e. MEMS oscillators, Partridge et al., 2013, Mussi et al., 2018, Mussi et al., 2019), inertial detection (i.e. MEMS accelerometers and gyroscopes, Acar and Shkel, 2009, Kempe, 2011, Larkin et al., 2021), optical signals manipulation (e.g. scanning micromirrors, Frangi et al., 2017, Brunner et al., 2021), energy harvesting (Challa et al., 2008, Iannacci et al., 2015), and mass/chemical sensing purposes (Lu et al., 2013, Schwartz et al., 2020, Fort et al., 2021). MEMS resonators are manufactured as extruded planar geometries with a given thickness, usually relying on silicon etching processes (Laerme et al., 1999): the structure has therefore an essentially 2D layout, but in general it exhibits relevant 3D dynamics. In typical applications, the micromechanical structure is forced into vibrations by converting an input electrical signal into a force which is used to excite the device. Vibrations of the structure are then picked up and often converted back into an electrical signal through various capacitive, piezoelectric, thermal or piezoresistive transduction techniques.

The mechanical characteristics of MEMS resonators, and MEMS devices in general, are typically designed merging predefined simple elements, usually masses and flexural or torsional springs made by simple or folded beams. The traditional design procedure is trial and error, and the design engineer has to iterate the manual tuning of the position and dimensions of the mechanical elements in order to satisfy the device requirements given by the application. This results in a much time-consuming and expensive design process, that also depends on the design engineer experience (Benkhelifa et al., 2010, Farnsworth et al., 2017). As recent market developments push towards continuous miniaturization of MEMS devices and reduction of power consumption, the design procedure of the devices becomes more and more crucial, and the development of automatic design tools based on structural optimization becomes fundamental to ensure industrial progress (Benkhelifa et al., 2010). Structural optimization methods can, crudely, be put into two different categories, depending on the level of design freedom they provide.

The simplest, and most restrictive structural optimization approaches, are size and/or shape optimization: in this case, the dimensions, position and shapes of the mechanical elements constituting the MEMS device are parametrized by a set of design variables. Numerical optimization is then employed to obtain the desired performance, usually relying on FEM simulations. The application of size/shape optimization to the design of MEMS resonators covers a wide extent of purposes, such as extending the operational frequency range of piezoelectric MEMS energy harvesters (Hoffmann et al., 2016), tailoring mechanical nonlinearities through non-uniform beam profiles (Li et al., 2017), or improving temperature stability of tuning fork resonators through slots in the resonator beams (Zega et al., 2018). Another interesting application is the design of MEMS gyroscopes, that are microsensors able to measure the external angular rate, and are based on the interaction of different natural modes due to Coriolis effects. Early works are focused on the tuning of the structural natural frequencies related to specific mode shapes (Xia et al., 2015a, Xia et al., 2015b), while an automatic design environment for MEMS resonators is presented in Giannini et al. (2020a): it allows the parametric generation of the structure geometry, the simulation of its behaviour via 3D FEM, and the layout optimization by means of gradient based techniques, focusing both on target natural frequencies and mode shapes, and on the maximization of the sensor response to the external angular rate. However, both size and shape optimization does not allow for new features, such as additional springs or new holes, to appear, i.e. the initial conceptual configuration remains unchanged.

Another approach to structural optimization is topology optimization: this method does not require any initial parameterization of the layout by the design engineer, but directly looks for the best way to distribute the material within the available design space. This makes the design procedure much more free to explore different structural shapes and able to go beyond usual design concepts, fostering innovation. Since its introduction for stiffness maximization subject to a constraint on the available material (Bendsøe and Sigmund, 1999), the method has evolved to be able to treat problems related to dynamics and vibrations, such as frequency optimization (Olhoff, 1989, Díaaz and Kikuchi, 1992, Tenek and Hagiwara, 1993, Ma et al., 1995), also in cellular structures (Wang et al., 2018, Zhang et al., 2020), dynamic compliance optimization (Jog, 2002, Tcherniak, 2002) and frequency constraints (Ma et al., 1994). The application of topology optimization to MEMS resonators ranges from the control of the first eigenfrequencies (He et al., 2012), to thermo-elastic damping reduction (Gerrard et al., 2017), to electric power maximization in energy harvesters (Wein et al., 2013). Also, in Bruggi et al. (2016) optimal auxetic structures are designed to properly propagate the motion in the resonator from actuated to non actuated masses, while in Philippine et al. (2013) the reliability of capacitive RF MEMS switches is improved by reducing the impact of intrinsic biaxial stresses and stress gradients on the switch’s membrane.

In a recent publication, the authors presented a topology optimization methodology for MEMS resonators, tailored for the design of MEMS gyroscopes (Giannini et al., 2020b). The focus was on method development and the design task was confined to the suspending structure of single mass resonators limited to in-plane 2D dynamics. The paper proposed three different optimization formulations with varying numerical complexity, which were evaluated on an academic benchmark example and then translated to a more realistic design case.

The focus of the present paper is to extend the topology optimization approach developed in Giannini et al. (2020b) to the design of industrially relevant MEMS resonators. More general planar structures employing suspended proof masses or plates are therefore here addressed, and their complete 3D dynamics are considered. The method is applied to control the first eigenfrequencies of both single mass and double mass tuning fork MEMS resonators for gyroscope applications, when imposing different target eigenfrequencies in an industrially acceptable range. In particular, we employ a discretization by Mindlin shell finite elements in the analysis, and we propose the use of reduced order models in order to achieve a numerically efficient optimization procedure. Also, we ensure manufacturability through minimum length-scale constraints.

The manuscript is organized as follows. Section 2 summarizes the working principle of MEMS gyroscopes and their design requirements. Also, it introduces the considered design problems for the single mass and the tuning resonators, followed by the employed physical models and model reduction techniques. The proposed topology optimization approach is presented in Section 3, discussing the employed regularization strategies, material properties interpolation and optimization problem formulations, as well as the choice of the algorithm parameters. Section 4 presents and discusses the optimization results for the single mass and tuning fork MEMS resonator layouts, as well as their validation with full model simulations. Finally, Section 5 concludes the paper with a summary of the obtained findings.