# Perspectives on Incompressibility

10 Jan 2017

###### Notation

We primarily denote derivatives by some variable primarily through subscripts

but because temporal derivatives are so common we will also denote temporal derivatives by dot notation

$q$ will be used to represent Euclidean positions and $x$ will repsresnt general coordinates. $q_t$ and $u$ will both be the velocity field of particles $q$.

### Incompressibility

For a diffeomorphism of a domain $\Omega$ by some function $\Phi(q;t) = \Phi^t: \Omega: \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n$ we define incompressibility as

for volume defined in the usual way by

The volume of $S$ after being advected is given by

Obviously if incompressibility is maintained

The differential form of this statement is that

$\nabla \cdot \Phi^t(q;t) = 0\forall q \in \Omega$.

Usually one starts with some PDE

and incompressibility is added by the gradient of some scalar field $p$ that guarantees $\nabla q_t = 0$:

There are a few perspectives for what pressure $p$ is that are valuable to understand:

#### $L^2$ Projection

This is perhaps the most fundamental geometric perspective for incompressibility. Ignoring the fancy notation, all this really depends on is the fact that differentiation is a linear map and that there’s a nice orthogonal decomposition of arbitrary (but sufficiently smooth) vector fields using different differential operators. What this boils down to is that making a vector field incompressible is the removal of the part of it that has divergence, which is a projection.

Recall the Hodge-deRham decomposition of a $1$-form into three orthogonal components

$\omega = d\alpha + \delta \beta + \gamma$ for $\alpha$ a 0-form, $\beta$ a 2-form, and $\gamma$ harmonic.

Now note that $\delta \delta = 0$, $\delta \gamma = 0$, and $\left(\delta u^{\flat}\right)^{\sharp} = \nabla \cdot u$.

Therefore, so long as for the decomposition

the gradient will be

so long as $\delta d\alpha = 0$ the vector field is divergence free. By the definition of harmonic vector fields we see that $\alpha$ is orthogonal to the kernel of $\delta d$ $\alpha$ and so being incompressible is simply setting $\alpha = 0$. The required pressure is therefore just $-(d\alpha)^\sharp$ and pressure projection is

where $\alpha$ can be computed by

#### Variational energy minimization

The variational perspective for pressure is that its the vector field that its the arugment for the minimal kinetic energy necessary to be divergence-free:

This idea comes from orthogonoality in the sense that it’s finding a $p$ that removes as much $\nabla \cdot$ from $u$ as possible. The key advantage of this formulation is that it does not depend on splitting as projection does. That is, the previous formulation is more amenable for solving for $u$ and $p$ simultaneously (as well as other things if there is a need).

By doing some arithmetic we see

#### Augmented Lagrangian

Consider the Lagrangian mechanics perspective of dynamics where the general positions and general velocities $x,\dot x$ are defined by minimizing some Lagrangian $\mathcal{L}(x,\dot x)$ and the Euler-Lagrange equations

Here we assume that $x$ represents positions and densities in a piece of geometry, which implies that $\dot x$ represents a diffeomorphism of the geometry itself.

Incompressibility adding the constraint $\mathbf{D}\dot x = 0$ for some linaer operator $\mathbf{D}$ that satisfies

where $\Phi_x$ is the diffeomorphism induced by the general coordinates $x$.

###### Adjoint of $\mathbf{D}$

Note that the adjoint operator, given the condition that $f=0$ or $\frac{\partial \Phi_x}{\partial t}\cdot N = 0$ for any point along the boundary, can be defined by

##### Augmentation

Augmenting this Lagrangian for incompressibility is therefore done by

Necessary conditions for optimality dictate that

where the final condition is simply a restating of the incompresibility condition.

On hte other hand we can the adjoint operator from before

This $\lambda$ parameter is what is commonly called pressure.

###### An example with viscous fluids

Recall that incompressible Navier-Stokes can be written as

where $p$ is such that

If we ignore the term $u \cdot \nabla u$ (which can be dealt with by an ODE integrator) and solve for the future $u$ implicitly we obtain

Where the first and second rows represent $\frac{\partial \bar{\mathcal{L}}}{\partial u}$ and $\frac{\partial \bar{\mathcal{L}}}{\partial p}$ respectively for

which, is the augmented lagrangian for the energy

By applying Shur’s complement method to solve for $p$ one obtains from the above matrix

which simplifies to

In the inviscid case, or when we split viscosity and lump it into $g$, this naturally turns into

our standard Poisson problem.

TODO