diff --git a/scripts/Embedded/MWEs/zach_embedded.jl b/scripts/Embedded/MWEs/zach_embedded.jl index 9d5a049..429f473 100644 --- a/scripts/Embedded/MWEs/zach_embedded.jl +++ b/scripts/Embedded/MWEs/zach_embedded.jl @@ -164,4 +164,14 @@ writevtk( writevtk( Triangulation(cutgeo,PHYSICAL_OUT),"results/trian_out3d" ) -end \ No newline at end of file +end + +## Surface functionals +# We should have something like the following for J(φ)=∫_∂Ω(φ) f ds: +# dJ(φ)(w) = lim_{t->0} (J(φₜ)-J(φ))/t = -∫_∂Ω (n⋅∇f)w/|∂ₙφ| dS - ∑(...) +# The first term is easy, the second term not so much... + +# Need [[fm]]=f₁m₁ + f₂m₂, where mₖ is the co-normal for τ = 0 with +# mₖ = tₖˢ×n_{∂Ω(φ(0))∩S} +# where tₖˢ is the tangent vector along the edge ∂Ω(φ(0))∩S and fₖ is +# the limit of f on S defined by fₖ(x)=lim_{ϵ->0⁺} f(x-ϵmₖ) for x ∈ ∂Ω∩S. \ No newline at end of file