【MEMS】🧠 02. Visualizing Piezoelectric Hysteresis and Butterfly Displacement with mems-ana

topics: [“MEMS”, “piezoelectric”, “visualization”, “Python”, “simulation”]

📌 Introduction

In piezoelectric MEMS devices,
it is critically important to develop an intuitive understanding of
“what actually moves when a voltage is applied.”

Even when tracking equations and coefficients (such as $d_{33}$),
it is common to lose a clear mental picture of the resulting behavior.

In this article, we use the lightweight analysis tool mems-ana to:

• Apply piezoelectric hysteresis inputs
• Observe butterfly-type displacement–voltage (V–u) characteristics
• Visualize the time evolution of spatial displacement distributions

The goal is not numerical accuracy, but to visually observe and internalize the behavior.

This article is the second installment in the mems-ana series.

🧪 Demo Overview

In this demonstration, we perform the following:

• Input
  • A P–$E_z$ hysteresis loop driven by applied voltage $V$
• Model
  • A $d_{33}$-dominant piezoelectric actuation model
  • Thin-film rectangular diaphragm represented by a ROM
• Output
  • Central displacement $u_z$
  • Spatial displacement distribution $u_z(x,y)$
  • Time evolution along the voltage cycle

The emphasis is on behavioral understanding, not precision fitting.

🎞 Demo Animation (Spatial Displacement)

At the link below, the spatial displacement
$u_z(x,y)$ is displayed as a real-time animation along the voltage cycle.

Interactive Demo (GitHub Pages)
https://samizo-aitl.github.io/mems-ana/mems-ana_demo/

• Displayed quantity: signed displacement distribution $u_z(x,y)$
• Color scale: fixed across all frames
• Voltage sweep: follows the hysteresis input

Because Zenn only supports static images,
please refer to the link above for the actual dynamic behavior.

🔁 P–$E_z$ Hysteresis Input

In piezoelectric materials, polarization $P$ exhibits hysteresis
with respect to the electric field $E_z$.

In this demo, we use:

• A hysteresis loop modeled after measured behavior
• A $P(E_z)$ input converted from voltage drive

The key points are:

• The input is explicitly nonlinear
• The ROM-based structural model still responds smoothly

This separation of nonlinear input and interpretable structural response is intentional.

🦋 Butterfly-Type Displacement–Voltage Characteristics

When a hysteresis input is applied,
the central displacement $u_z$ exhibits a characteristic
butterfly-shaped curve.

• Asymmetry between positive and negative voltage regions
• History-dependent displacement response

This behavior is extremely common in:

• Piezoelectric devices
• MEMS actuators

Before understanding it mathematically,
it is valuable to recognize the shape visually.

🗺 Why Spatial Distributions Matter

By looking not only at scalar quantities (such as center displacement)
but also at the spatial distribution $u_z(x,y)$, one can immediately see:

• Where deformation is concentrated
• How mode shapes invert or evolve
• Whether symmetry is preserved

This is also excellent training for interpreting FEM results later.

⚙️ How to Run the Demo (Brief)

The demo can be executed using the code published in the following repository:

https://github.com/Samizo-AITL/mems-ana

The basic workflow is:

1. Clone the repository
2. Prepare a Python environment
3. Run the scripts under mems-ana_demo

The setup is intentionally kept simple.

🚫 Scope and Limitations

This demo intentionally makes several simplifications:

• No strict reproduction of material nonlinearities
• No higher-order vibration modes
• No dynamic resonance analysis

It is designed purely to answer the question:

“When piezoelectric hysteresis is applied,
what kind of behavioral face does a MEMS structure show?”

🧭 Position in the Series

• 01: Design philosophy and overall architecture
• 02: Visualization demo (this article)
• 03: ROM mathematical structure and design policy

The next article will dig deeper into
the equations and ROM structure that generate this behavior.

📝 Closing Remarks

Viewing analysis results not as static numbers, but as
moving, evolving shapes, dramatically changes:

• Speed of understanding
• Design intuition

The mems-ana_demo is designed as an entry point for visualization.

First, observe.
Then, design.

That is where MEMS design truly begins.