prez

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

Weakly electric fish are able to detect and localize prey based on microvolt-level perturbations in the fish’s self-generated electric field [fn:1]

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

Why do it?

• Electrosensory robots [fn:2]
• understand & test tech
• understand fish behaviour
• Test analytical approx for sphere

[fn:1] Chen 2005

[fn:2] Lebastard et al., 2016; Solberg et al., 2008

How it works

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

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

Formulation

\begin{equation*} ε _e ∇ ²\underbrace{ψ _e}_{\text{Empty field}} =\underbrace{∑ q_i δ (x_i )}_{\text{point sources}} \end{equation*}

• sum of \(δ(x_i)\)s approximates generated field [fn:chens]
• Field perturbed by dielectric object $S$ ($ψ _o$): $$∇ \cdotp (ε ∇ ψ _o )=∑ q_i δ (x_i ), \qquad ε =\begin{cases} ε _i & \text{internal to } S
  ε _e & \text{external to } S \end{cases}$$

$$\underbrace{δ ψ }_{\text{Perturbation}} =ψ _o -ψ _e$$ [fn:chens] Chen 2005

Methods

FEniCS

FEniCS

• 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)

FEM

Variational Formulation [fn:fen]

• Multiply the PDE by a function $v$
• Integrate the resulting equation over the domain $Ω$

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

• Integrate by parts

\begin{equation*} ∫ _Ω ε ∇ ψ _o \cdotp ∇ v\ \mathrm{d} x=∑ q_i v(x_i ) \end{equation*}

• Weaker continuity requirement on $ε\psi _o$

[fn:fen] The FEniCS Tutorial

FE Basis

• finite element function $u$ is expressed as a linear combination of basis functions $φ _j$

\begin{equation*} u≈ ∑ U_j φ _j \end{equation*}

• 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

• 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

• 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?