Flexible Body Node#

Module: mime.nodes.robot.flexible_body Stability: experimental Algorithm ID: MIME-NODE-006 Version: 1.0.0 Verification Mode: Mode 2 (Independent)

Summary#

Euler-Bernoulli beam dynamics for flexible flagellar microrobots. Models transverse bending waves along a filament under the balance of elastic restoring force and viscous drag.

Governing Equations#

\[ K\frac{\partial^4 y}{\partial x^4} = -\xi_{\perp} \frac{\partial y}{\partial t} + f_{\text{fluid}}(x,t) \]

where \(K = EI\) is the bending stiffness and \(\xi_{\perp}\) is the perpendicular RFT drag coefficient (analytical fallback; replaced by IB-LBM forces in Phase 2+).

4th-order FD stencil:

\[ \frac{\partial^4 y}{\partial x^4} \approx \frac{y_{n+2} - 4y_{n+1} + 6y_n - 4y_{n-1} + y_{n-2}}{\Delta x^4} \]

Discretization#

4th-order central finite differences for \(\partial^4 y/\partial x^4\). Implicit Euler for time integration (required due to stiffness of the 4th-order spatial operator).

Implementation Mapping#

Equation Term

Implementation

Notes

\(K \partial^4 y / \partial x^4\)

mime.nodes.robot.flexible_body.build_beam_stiffness_matrix

Stencil [1,-4,6,-4,1] / dx^4

Implicit Euler

mime.nodes.robot.flexible_body.FlexibleBodyNode.update

jnp.linalg.solve(A, rhs)

Clamped BC at x=0

mime.nodes.robot.flexible_body.FlexibleBodyNode.update

Row substitution in A matrix

Assumptions and Simplifications#

  1. Small-amplitude transverse deflections (linearised beam)

  2. Inertia negligible (overdamped, low Re)

  3. Uniform material properties along filament

  4. Clamped-free boundary conditions

Validated Physical Regimes#

Parameter

Verified Range

Notes

Sperm number \(S_p\)

0.5–3.0

Optimal propulsion near \(S_p \approx 2.1\)

Known Limitations and Failure Modes#

  1. Small-deformation only — fails for large curvatures (\(S_p \gg 2.1\))

  2. 1D transverse deflection — no torsion or 3D shape

  3. RFT drag coefficient is a scalar approximation

State Variables#

Field

Shape

Units

Description

deflection

(N,)

m

Transverse displacement y(x)

velocity

(N,)

m/s

Transverse velocity dy/dt

Parameters#

Parameter

Type

Default

Units

Description

n_nodes

int

20

-

Discretisation nodes

length_m

float

100e-6

m

Filament length

bending_stiffness_nm2

float

4e-21

N.m^2

EI

drag_coeff_perp

float

auto

Pa.s

RFT \(\xi_{\perp}\)

References#

  • [@Lighthill1976] Lighthill, J. (1976). Flagellar Hydrodynamics. — Slender body theory and RFT for flagellar filaments.

Verification Evidence#

  • Unit tests: tests/nodes/test_flexible_body.py (7 tests)

Changelog#

Version

Date

Change

1.0.0

2026-03-20

Initial implementation — small-deformation Euler-Bernoulli beam