cupc

\titlepage

About Me

Computational scientist

• Numerical partial differential equations (PDEs)

Programming language enthusiast

• Dependent types
• Lisps/Schemes
• Functional programming/Theorem provers

Outline

\tableofcontents

Physical Problem

Electroreception

Electric Sensing

• Characterize environment from its electric field

Weakly electric fish

• generates field with EOD[fn:: Electric Organ Discharge]
• senses perturbation on skin
• locates object

Fish fig

Formulate Model

Translate to Physics

English → Physics

• EOD → Plane of point charges
• Object → Dielectric constant
• Perturbation = Object field - Empty field

Now what

\pause How do we solve this?

\pause FEniCSx can help!

Simulate/Solve

Simulate it!

FEM DUMP

FEniCSx can …

1. Make a mesh
   • where we save our solution
2. Object → Dielectric constant
   • mark location & value on mesh
3. EOD → Plane charge
   • define as source term
4. Solve → Object field
5. Same thing without object

Mesh Fig

./pix/un-refn-mesh.jpeg \pause

Perturbation

Object field - Empty field

Results and Discussion

Visualize

Cross section of 3D mesh \pause

./pix/pert-sphere.jpeg

Convergence

Empty field (point charges) has exact solution

• finer mesh \(\implies\) lower error, hopefully
• even finer mesh → high performance computing

\pause

./pix/refn-mesh.jpeg

Conclusion

Recap

\tableofcontents

\secname

2 key players

• EOD
• Dielectric object

\pause Now what?

\pause Your turn, I told you all the steps

\pause Also my thesis with Greg Lewis

\pause https://faculty.uoit.ca/lewis

https://github.com/MaxCan-Code/thesis

Questions?

Appendix

Future work

• Stuck: Apptainer on Graham
  • Read-only container → MPI breaks
  • MPI works outside

FEniCSx

• Easy way to solve PDEs with FEM
• DSL for variational form (UFL)
• Python interface
• FOSS

Learn more: "FEniCSx doc/tutorial"

HPC

• Apptainer
• Slurm
• Unix
  • shell
  • ssh

Pix

./pix/empty-field.jpeg

./pix/pert-box.jpeg

FEM

BVP

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

• Field perturbed by dielectric object \( S\ ( ψ _o ) \):

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

• Natural (Neumann) BC: (\( ∇ ψ \cdotp \vec{n} = 0 \)) on boundary

Variational Formulation

\[ ∇ \cdotp (ε ∇ ψ _o )=∑ q_i δ (x_i ) \]

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

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

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