【MEMS】🧠 03. Mathematical Structure and Design Policy of mems-ana_core Built with ROM

topics: [“MEMS”, “ROM”, “modeling”, “mathematics”, “Python”]

📌 Introduction

In the previous article (MEMS:02),
we visually explored the behavior of a MEMS structure under piezoelectric hysteresis input
through a visualization demo.

In this article, we focus on what lies beneath that behavior:
the mathematical structure and design policy of mems-ana_core.

The purpose of this article is not:

• A line-by-line explanation of implementation code
• A strictly accurate reproduction of physical phenomena

Instead, the goal is to clarify—explicitly and from a designer’s perspective—

Why this mathematical structure was chosen
What is intentionally included in the model, and what is intentionally excluded

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

🧩 Positioning of mems-ana_core

mems-ana_core corresponds to:

• The computational core of mems-ana
• A concrete implementation of a Reduced Order Model (ROM)

Its defining characteristics are:

1. Dominant modes are explicitly assumed
2. A clear correspondence between physical quantities and equations
3. Calibration parameters with limited and explicit meaning

Rather than prioritizing raw speed,
the design prioritizes structural readability and interpretability.

🧱 Modeling Assumptions

The intended targets are typical:

• Thin-film MEMS structures
• Rectangular diaphragms
• Small-deformation, quasi-static responses

Accordingly, the following are outside the model scope:

• Geometric nonlinearity
• Large deformation
• Local plasticity

If these assumptions do not hold,
mems-ana_core should not be used.

📐 Basic Form of the Reduced Order Model (ROM)

The displacement field $u(x,y,t)$ is expressed as
a product of spatial modes $\phi_n(x,y)$
and time-dependent coefficients $q_n(t)$:

The key design choices here are:

• Intentionally keeping $N$ small
• Selecting only physically meaningful modes

In mems-ana_core, mode selection is guided by:

• Symmetry
• Dominant deformation behavior
• Boundary conditions

The result is a model with the minimum number of modes required to capture behavior.

⚙️ Structure of the Governing Equations

The general ROM-based equation of motion takes the form:

where:

• $\mathbf{q}$: vector of modal coordinates
• $M$: mass matrix
• $C$: damping matrix
• $K$: stiffness matrix

Because mems-ana_core primarily targets:

• Quasi-static behavior
• Low-frequency response

the inertial and damping terms
($\ddot{\mathbf{q}}$, $\dot{\mathbf{q}}$)
are often simplified or omitted.

⚡ Treatment of Piezoelectric Actuation

Piezoelectric actuation is introduced as
an external force term $\mathbf{f}(t)$.

The critical modeling decision is that:

• The piezoelectric constant $d_{33}$ is treated as an
  effective proportional coefficient
• The exact electric field distribution is not solved

This yields a simplified, design-oriented causal chain:

• Voltage → polarization → equivalent force

which preserves interpretability and controllability.

🔁 Positioning of Hysteresis Input

The P–$E_z$ hysteresis used in MEMS:02 is:

• Not a detailed internal material model

Instead, it represents:

• A representative nonlinear input waveform

Thus, mems-ana_core:

• Does not solve hysteresis as an internal state
• Treats it as a history-dependent external input

This choice:

• Prevents excessive mathematical complexity
• Maintains ROM transparency and clarity

🎯 Calibration Philosophy

Calibration in mems-ana_core can reference:

• FEM results
• Experimental measurements

but the philosophy remains consistent:

Fit numerical values without destroying parameter meaning

In practice, calibration is limited to parameters such as:

• Global stiffness scaling
• Effective piezoelectric coupling coefficient
• Damping coefficient

Arbitrary parameter fitting is explicitly avoided.

🔗 Relationship with FEM (Revisited)

mems-ana_core is not a replacement for FEM.

The intended division of roles is:

• FEM
  • Local stress analysis
  • Complex geometries
  • Higher-order modes
• mems-ana_core
  • Global behavior
  • Sensitivity and trends
  • Design reasoning

FEM results are used as
reference points to reason about “why” behavior occurs.

🗂 GitHub Repository

The implementation of mems-ana_core discussed here
is available in the following GitHub repository:

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

Please refer to the repository for
the correspondence between equations and code.

🧭 Series Summary

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

Together, these three articles present:

• Thought process
• Observed behavior
• Mathematical foundation

as a single, coherent narrative.

📝 Closing Remarks

mems-ana_core is designed not as:

A model that produces exact answers

but as:

A model that helps designers think

Rather than minimizing equations,
the priority is not to minimize meaning.

ROMs are approximations—
but good approximations accelerate design thinking.

If this article helps clarify
how and why models are constructed in MEMS design,
it has achieved its purpose.