Standard Finite Element (FE) models, especially those that incorporate multiple physical domains, consist of detailed representations of a device that include a large number of Degrees of Freedom (DoF). The Degrees of Freedom in a design are the number of independent variables or parameters needed to describe the motion or state of the device. Generally, the larger the number of Degrees of Freedom, the greater the requirements for analysis both in terms of computational resources and simulation time. Finite Element models can achieve a high level of confidence in the predictive accuracy of a design, due to the ability to systematically increase the number of Degrees of Freedom until a sufficiently dense representation is achieved for all physical domains under investigation. However, this strategy can come at the cost of long simulation times.

Accordingly, a common practice in the MEMS industry is to rely on Reduced Order Models (ROMs, alternatively described as Compact Models), to decrease the number of Degrees of Freedom and subsequently reduce simulation times. [1] Compact modeling is a common technique employed by circuit engineers to generate models that are sufficiently simple to be used during simulation, but also accurate enough to allow engineers to trust the results. [2] Although several Reduced Order Model approaches include mechanical nonlinearities [3], these ROMs often neglect nonlinear characteristics critical for MEMS inertial sensor design, such as electrostatic spring softening effects associated with electromechanical coupling or contact mechanics.

Recently, Coventor worked with our colleagues at NXP Semiconductors to help them verify the design of an automotive High-G MEMS capacitive accelerometer. The design was comprised of a lateral accelerometer surrounded by a flexible stop frame. NXP designers used a nonlinear, compact model developed in *MEMS+* to model this design. *MEMS+* models display the accuracy found in a detailed, standard Finite Element model but with a significant reduction in the number of Degrees of Freedom normally required during Finite Element modeling. [4,5]

To improve their design, the NXP design team developed a coupled, multi-physics compact model that included relevant nonlinear physics using Coventor*MP*^{®} (*MEMS+*). These compact models incorporated all coupled multi-physical nonlinearities essential to the design of their accelerometer and the surrounding frame, including electrostatic softening and contact. The models replicated the complete behavior of the accelerometer system, without the uncertainty associated with generating custom ROMs or the lengthy simulation times seen using a Finite Element approach.

The capacitive accelerometer developed by NXP is shown in Figure 1. The flexible model was built using nonlinear Timoshenko and Bernoulli beam components available in the *MEMS+* library. The model was able to capture all relevant mechanical nonlinearities in the springs and proof-mass. The gap-closing sense and self-test electrodes in the design (Figure 2) are capable of modeling sense capacitance changes, along with contact and squeezed-film damping effects. The frame surrounding the accelerometer acts as a flexible motion stop, and is included in the *MEMS+* design (Figure 2). The ability to model the interaction between the transducer and its enclosure is a critical engineering requirement for MEMS designers, and can be completed using *MEMS+*.

NXP performed parametric studies during their design process, varying the Critical Dimensions (CDs) of spring element widths and proof-mass thicknesses, to model what might happen if the device were over-etched or under-etched during production. The etch loss effects on device capacitance that were seen in the model correlated well with actual measured capacitance for static low and high bias conditions. In addition, the dynamic behavior of the device was accurately predicted by the compact modeling approach used in *MEMS+*. The model provided a complete transient simulation response for the accelerometer and the surrounding stop frame, with only a few minutes needed for each simulation. NXP was able to simulate expected dynamic impact forces on the flexible stops using a sinusoidal shock force with peak magnitudes of 25e^{3} g’s, 50e^{3} g’s and 75e^{3} g’s (Figure 3). The compact model was subsequently transferred to CoventorWare^{®}, a complementary standard FE solver within the Coventor*MP* suite. Using CoventorWare, a static stress analysis was performed that subjected the surrounding frame to the maximum contact force extracted during the transient shock simulations completed in *MEMS+* (Figure 4). This static analysis indicated that the displacement of the frame would not create a risk of device failure under the maximum contact force expected.

NXP’s models could accurately predict the transient, dynamic response of their design to shock stimuli. Their compact model could also produce modeling results of transient, nonlinear behavior within practical simulation times, unlike Finite Element approaches. By employing compact models to represent the transducer and flexible lateral overload stoppers in the accelerometer, NXP was able to perform comprehensive design exploration that would have been essentially impossible using standard Finite Element (FE) modeling. In addition, NXP was able to easily transfer their compact model to Coventor’s standard finite element modeling tool (CoventorWare), to verify that the design was not subject to shock-based failures. Using Coventor*MP*, the team at NXP was able to perform a large number of design iterations in a reasonable timeframe. This allowed them to reduce dynamic contact forces in their final accelerometer design, and to ultimately improve the reliability of the final product.

**Acknowledgment: This article is based upon a joint publication authored by Tousif Shaikh and Aaron Geisberger of NXP Semiconductors, Inc., and Brian van Dyk and Arnaud Parent of Coventor, entitled “AUTOMOTIVE MEMS ACCELEROMETER DESIGN VERIFICATION USING NONLINEAR COMPACT MODELING”,** **Hilton Head 2022, Solid-State Sensors, Actuators and Microsystems Workshop, June 5-9, 2022 **

- L. Feng, P. Benner, J. Korvik, “System-Level Modelling of MEMS by Means of Model Order Reduction (Mathematical Approximation) – Mathematical Background”, in System-level Modeling of MEMS, WILEY-VCH Verlag GmbH & Co. KGaA, 2013, pp. 53-93.
- Gennady Gildenblat.
*Compact Modeling: Principles, Techniques and Applications*https://link.springer.com/book/10.1007%2F978-90-481-8614-3 - T. Mähne, K. Kehr, A. Franke, J. Hauer and B. Schmidt, “Creating Virtual Prototypes of Complex Micro-Electro-Mechanical Transducers Using Reduced-Order Modelling and VHDL-AMS,” in Proc. of the 8th International Forum on Specification and Design Languages, 2005.
- A. Parent, A. Krust, G. Lorenz, I. Favorskiy and T. Piirainen, “Efficient nonlinear Simulink models of MEMS gyroscopes generated with a novel model order reduction method”, IEEE Proceedings, June 2015.
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