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Electrolocation with Active Electric Field

Intro

Electrolocation

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• Some fish can detect object in murky water
• How do they do it? ‘see’ change in electric field
• Pick up perturbation on their skin (receptor plane)
• Same ability, different methods

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Weakly electric fish are able to detect and localize prey based on microvolt-level perturbations in the fish’s self-generated electric field [fn:: Chen 2005]

• We can simulate that
• with finite elements method (FEM) and FEniCS

Why do it?

• Electrosensory robots [fn:: Lebastard et al., 2016; Solberg et al., 2008]
• Develop tool for researchers
• Test analytical approx for sphere

How it works

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• Conductors/dielectrics perturbs its generated field
• Electrical properties of different objects: diff perturbations

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Model

Idea

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• Compute perturbation in field (diff in electrostatic potential)
• Variable coefficient Poisson equation in 3D
• Infinite domain, approximate on finite mesh
• Natural (Neumann) BC: (\( ∇ ψ \cdotp \vec n = 0 \)) on boundary

Model field as

• Self-generated field: plane of point charges
• Flat fish in 3D

Assumptions

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• Linear, homogeneous, isotropic sphere and cube
• No fish body, just receptor and charge plane
• If uniform body, change magnitude only not shape

Formulation

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\[ ε _e ∇ ²\underbrace{ψ _e}_{\text{Empty field}} = \underbrace{∑ q_i δ (x_i )}_{\text{point sources}} \]

• sum of \( δ(x_i) \)s approximates generated field [fn:: Chen 2005]
• Field perturbed by dielectric object \( S \) (\( ψ _o \)):

\[ ∇ \cdotp (ε ∇ ψ _o )=∑ q_i δ (x_i ), \qquad ε =\begin{cases} ε _i & \text{internal to } S
ε _e & \text{external to } S \end{cases} \]

\[ \underbrace{δ ψ }_{\text{Perturbation}} =ψ _o -ψ _e \]

Methods

FEniCS

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• Open-source FEM solver for PDEs
• Components for:
  • C++ backend, Python interface (DOLFIN)
  • Compiler for math expressions (FFC)
  • Generating FE basis functions (FIAT)
  • Language for variational problems (UFL)
  • Mesh generation (mshr)

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FEM

Variational Formulation [fn:: The FEniCS Tutorial]

• Multiply the PDE by a function \( v \)
• Integrate the resulting equation over the domain \( Ω \)

\[ ∫ _Ω [ ∇ \cdotp (ε ∇ ψ _o )] v\ \mathrm{d} x = ∫ _Ω fv\ \mathrm{d} x = ∑ q_i∫ _Ω δ (x_i )v\ \mathrm{d} x \]

• Integrate by parts

\[ ∫ _Ω ε ∇ ψ_o \cdotp ∇ v\ \mathrm{d} x = ∑ q_i v(x_i ) \]

• Weaker continuity requirement on \( ε\psi_o \)

FE Basis

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• finite element function \( u \) is expressed as a linear combination of basis functions \( φ_j \)

\[ u≈ ∑ U_j φ _j \]

• sub \( u \) into variational problem and solve for \( U_j \)

Example basis \( φ_j \)

Example basis \( φ_j \) approximates discrete delta: exact accuracy on grid functions

Verification

Refinement

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• error near singularity and object
• mark then refine

Convergence

• No analytical solution on finite domain
• Test problem: single point charge

\begin{equation*} ∇ ² ψ =qδ ( 0) \end{equation*}

• Analytical solution on infinite domain:

\begin{equation*} ψ =\frac{q}{ε | \mathbf{r}| } \end{equation*}

• Choose R to approximate infinite domain
• Point-wise error where perturbation occurs
• 3 parameters: R, initial cells, refinements

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• Dir BC: 2 refinements
• 64: more points on boundary
• interpolation error

				initial cells
/			<
			16	32	64
		50	3.141E-02	-1.336E-04	-8.438E-04
	R	75	3.371E-02	-8.959E-04	-2.021E-03
		100	1.043E-02	-9.038E-03	-2.091E-03

Conclusion

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• Works on test problems
• Perturbations happen

Future Work

• Viable for test sensitivity to geometry
• Use HPC
• Modelling package
• Time dependent simulations
• Size & geometry of object in non uniform field?