Velocity extension

abstract type AbstractVelocityExtension end

An abstract type for velocity extension structures. Structs that inherit from this type must implement the project! method.

struct IdentityVelocityExtension <: AbstractVelocityExtension

A velocity-extension method that does nothing.

struct VelocityExtension{A,B} <: AbstractVelocityExtension

Wrapper to hold a stiffness matrix and a cache for the Hilbertian extension-regularisation. See Allaire et al. 2022 (link).

The Hilbertian extension-regularisation method involves solving an identification problem over a Hilbert space $H$ on $D$ with inner product $\langle\cdot,\cdot\rangle_H$: Find $g_\Omega\in H$ such that $\langle g_\Omega,w\rangle_H =-J^{\prime}(\Omega)(w\boldsymbol{n})~ \forall w\in H.$

This provides two benefits:

1. It naturally extends the shape sensitivity from $\partial\Omega$ onto the bounding domain $D$; and
2. ensures a descent direction for $J(\Omega)$ with additional regularity (i.e., $H$ as opposed to $L^2(\partial\Omega)$)

Properties

• K::A: The discretised inner product over $H$.
• U_reg::B: The trial space used for the Hilbertian extension-regularisation.
• cache::C: Cached objects used for project!

VelocityExtension(biform,U_reg,V_reg;assem,ls)

Create an instance of VelocityExtension given a bilinear form biform, trial space U_reg, and test space V_reg.

Optional

• assem: A matrix assembler
• ls::LinearSolver: A linear solver

project!(vel_ext::VelocityExtension,dF::AbstractVector,V_φ,uhd) -> dF_reg::Vector

Project dFh onto a function space described by the vel_ext.

Note:

• We expect that dF is a vector resulting from assembly on V_φ.
• uhd should be an FEFunction on V_φ.