In this article, we discuss how to perform a Monte Carlo Simulation (MCS) on a PMUT ultrasonic sensor in OnScale to obtain a full picture of the design space.

### What is Monte Carlo Simulation?

Monte Carlo Simulation is a statistical method which uses simulation to model the probability of outcomes of a complex model whose behavior cannot be easily determined due to a vast number of variables. This method is useful for multiple industries such as Design, Manufacturing, Finance and Hospitality. The three main purposes for using this method are:

- · Understand risk and uncertainty
- · Determine probability of outcomes
- · Optimize outcomes

This method is typically used in engineering for design optimization. MCS can provide engineers with a full picture of a design space as the simulation returns:

- · Probability of outcomes (how likely it is that an outcome will occur)
- · Correlation between inputs (the relationship between inputs)
- · Correlation between outcomes (the relationship between outcomes )
- · Sensitivity (which input is the most impactful on the outcomes)

All these metrics can be represented graphically so engineers can easily extract information.

### How does Monte Carlo Simulation work?

MCS is a bit like throwing darts on a dartboard. If you are not a very good player and just start throwing darts with the only aim of getting it on the board, after a certain amount darts thrown you will start to see where the darts are accumulating and get a picture of the distribution of outcomes.

[1] https://www.investopedia.com/articles/investing/093015/create-monte-carlo-simulation-using-excel.asp

So, like a bad dart player, MCS calculates the results (dart score) from multiple simulations (dart throws) and can do this for multiple inputs (e.g. player distance and level of intoxication). To obtain an idea of the distribution of outcomes, inputs are randomly generated from a probability distribution. The most commonly used probability distribution is Normal. This means that an input has a mean (expected) value and a standard deviation which is the amount that value deviates from the mean (in positive and negative direction).

Depending on the number of inputs and the input constraints, an MCS simulation usually involves thousands of calculations.

A typical MCS involves the following steps:

- Create a parametric model
- Define inputs and their constraints
- Generate N number of random inputs
- Run model with N random inputs
- Collect output data
- Analyse output data

### What is a PMUT?

Piezoelectric Micro-machined Ultrasonic Transducers (PMUTs) are MEMs-based piezoelectric transducers made up of a thin piezoelectric diaphragm typically formed on silicon substrates. Unlike bulk piezoelectric transducers which are typically thickness mode resonators, PMUTs are based on the flexural motion of a thin film membrane formed on silicon substrates. These devices are used in a wide range of broadband sensing applications.

PMUTs are a good alternative for conventional bulk piezoelectric transducers. This is due to their high operational frequencies, small element size, low power consumption, ease of fabrication of large array for imaging and communication applications due to their micro electromechanical system (MEMS) silicon-based fabrication.

For more information on PMUTs check out our whitepaper on accelerating ultrasonic fingerprint sensor R&D with cloud simulation here! (https://onscale.com/whitepapers/accelerating-ultrasonic-fingerprint-sensor-rd-with-cloud-simulation/)

### Why use MCS for PMUT Design?

Many industries including the MEMs industry use Design of Experiments (DoEs) to determine the influence of variation in fabrication parameters. However, due to the expense of fabrication, especially with complex MEMs devices, usually only a very limited amount of experiments can be performed from which a maximum amount of information must be gained. Therefore, Monte Carlo simulation is becoming a necessary step in the design process of PMUTs. Monte Carlo method allows engineers to get a full picture of the design space, reducing fabrication costs and time to market. PMUT dimensions are in the micron range so the fabrication of PMUTs is tricky. Monte Carlo Simulation can be used to simulate the device performance for machining inaccuracies, providing manufacturers with a clear picture of product yield.

### Monte Carlo Simulation in OnScale vs Legacy CAE

Monte Carlo Simulation requires as many simulations as possible to achieve accurate and enhanced predictions. This is a very computationally demanding task (Large CPU/RAM required) which would not be possible with legacy CAE packages, especially for large 3D models. There is also the complex aspect of automating the simulation execution. However, OnScale’s HPC Cloud Platform allows for 1000s of simulations to be run in parallel, accelerating the Monte Carlo simulation process.

### How to do Monte Carlo Simulation on PMUT Cell in OnScale

PMUTs can be easily modelled and simulated in OnScale in full 3D. For more information on how to build models in OnScale, check out our PMUT Array 3D Example (https://support.onscale.com/hc/en-us/articles/360006843711-PMUT-Array-3D). This example uses a pre-built 3D simple PMUT Unit Cell.

For this MCS we will analyse how the top silicon layer thickness (thk_elas) and piezoelectric layer thickness (thk_pzt) effects three key performance indicators (KPIs) of the PMUT:

- · Centre Frequency (Fc)
- · Maximum Displacement of the PMUT membrane
- · Frequency Bandwidth (FBW)

To keep the problem size down it is important to constrain the input parameters otherwise you have an infinite problem space. In this example, silicon layer thickness is constrained to 2.5um +/- 2.5% and the piezo layer thickness is constrained to 1.0um +/- 5%. This MCS uses 2000 random inputs from a normal distribution and calculates the Fc, membrane displacement and FBW for every input combination.

Step 1 – Generate Inputs

OnScale can run multiple simulations in parallel on the cloud, sweeping multiple variables at a time. To do this easily, batch simulations can be driven using a csv file. This file must contain the names of the variables at the top of the columns followed by the values underneath. The variables must also by defined using ‘symbx’ in the input file. Generating a csv file with randomly distributed numbers can be easily done in many software packages such as Octave, Python, MATLAB etc.

Figure 1: Use symbx for Monte Carlo input variables |
Figure 2: Csv file containing 2000 randomly distributed values for thk_elas and thk_pzt |

Step 2 – Run Simulations & Download Data

Jobs are ran in OnScale through the Cloud Scheduler. A parametric sweep can be driven with a csv file using the User Defined Variable File option. The Cloud Scheduler sets up a simulation for every row of variables in the csv file.

Loading in the PMUT model and selecting ‘Estimate’ then ‘Run’ uploads the 2000 simulations to the cloud to process in parallel.

When the 2000 simulations are complete, you must download the results for processing. For this PMUT MCS, the resulting files have the following information for each simulation:

- Voltage and Charge on top electrode
- Y Displacement of membrane

These results can be post-processed in Review or MATLAB.

Step 3 – Calculate Output Data

Simulation results can be easily batch processed in Review to obtain relevant KPIs. Similarly, this can be done in MATLAB. The steps are simple, read in the history files, perform the same KPI calculations on every dataset and output to a csv file to plot. For this study the KPIs are Fc, displacement and FBW. To calculate centre frequency, take the FFT of charge curve and select frequency at the maximum value. To get the maximum Y displacement take the maximum from the y displacement curve. Finally, the bandwidth can be calculated by halving the maximum amplitude of the FFT of charge and differencing the two frequencies at this point.

Step 4 – Analyse Results

To make the MCS results easier to understand and analyse, it is useful to plot them using software such as MATLAB.

*Figure 3: Inputs vs Inputs*

Figure 5: Inputs vs Outputs |
Figure 4: Outputs vs Outputs |

It is clear from this Monte Carlo Simulation that the piezo thickness is inversely correlated to the maximum displacement in the membrane, as expected. From the results we can also see that the silicon thickness is correlated to centre frequency and the bandwidth.

These graphs give a full picture of the design space and can be used for example, by manufacturers of PMUTs to assess product yield when a pass/fail criterion is applied. Especially for PMUT design, these studies are invaluable tools for analyzing how the structure effects devices performance and is key for optimizing designs.