Operators

All discrete operators are implemented as matrix-free kernels using KernelAbstractions.jl and Cartesian indices, similar to WaterLily.jl. This allows for dimension- and backend-agnostic code; the same kernels run on CPU and GPU. See this blog post for how such kernels are written.

Each operator comes in two variants:

• A fast mutating variant with an exclamation mark (e.g. divergence!), which writes its result into a preallocated output array.

• A non-mutating variant (e.g. divergence), which allocates its output and is differentiable with reverse-mode automatic differentiation (see Differentiating code).

The operators act on fields that include ghost volumes, so boundary conditions must be applied first with apply_bc_u and friends (see Problem setup). Sparse matrix versions of the linear operators are also available, see Sparse matrices.

API

IncompressibleNavierStokes.Offset Type

struct Offset{D}

Cartesian index unit vector in D = 2 or D = 3 dimensions. Calling Offset(D)(i) returns a Cartesian index with 1 in the dimension i and zeros elsewhere.

See https://b-fg.github.io/research/2023-07-05-waterlily-on-gpu.html for writing kernel loops using Cartesian indices.

Fields

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IncompressibleNavierStokes.apply! Method

apply!(kernel, setup, args...; ndrange, offset)

Apply kernel to args with offset offset. By default, it is applied everywhere except for at the outermost boundary.

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IncompressibleNavierStokes.applygravity! Method

applygravity!(f, temp, setup, gdir, gravity) -> Any

Compute gravity term (in-place version). add the result to F.

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IncompressibleNavierStokes.applygravity Method

applygravity(temp, setup, gdir, gravity) -> Any

Compute gravity term (differentiable version).

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IncompressibleNavierStokes.applypressure! Method

applypressure!(u, p, setup) -> Any

Subtract pressure gradient (in-place version).

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IncompressibleNavierStokes.avg Method

avg(ϕ, Δ, I, i) -> Any

Average scalar field ϕ in the i-direction.

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IncompressibleNavierStokes.convection! Method

convection!(f, u, setup) -> Any

Compute convective term (in-place version). Add the result to F.

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IncompressibleNavierStokes.convection Method

convection(u, setup) -> Any

Compute convective term (differentiable version).

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IncompressibleNavierStokes.convection_diffusion_temp! Method

convection_diffusion_temp!(
    c,
    u,
    temp,
    setup,
    conductivity
) -> Any

Compute convection-diffusion term for the temperature equation. (in-place version). Add result to c.

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IncompressibleNavierStokes.convection_diffusion_temp Method

convection_diffusion_temp(
    u,
    temp,
    setup,
    conductivity
) -> Any

Compute convection-diffusion term for the temperature equation. (differentiable version).

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IncompressibleNavierStokes.convection_diffusion_temp_adjoint_temp! Method

convection_diffusion_temp_adjoint_temp!(
    tempbar,
    φ,
    u,
    setup,
    conductivity
) -> Any

Adjoint of convection_diffusion_temp! with respect to temp (add result to tempbar).

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IncompressibleNavierStokes.convection_diffusion_temp_adjoint_u! Method

convection_diffusion_temp_adjoint_u!(
    ubar,
    φ,
    temp,
    setup
) -> Any

Adjoint of convection_diffusion_temp! with respect to u (add result to ubar).

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IncompressibleNavierStokes.convectiondiffusion! Method

convectiondiffusion!(f, u, setup, viscosity) -> Any

Compute convective and diffusive terms (in-place version). Add the result to f.

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IncompressibleNavierStokes.diffusion! Method

diffusion!(f, u, setup, viscosity) -> Any

Compute diffusive term (in-place version). Add the result to F.

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IncompressibleNavierStokes.diffusion Method

diffusion(u, setup, viscosity) -> Any

Compute diffusive term (differentiable version).

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IncompressibleNavierStokes.dissipation! Method

dissipation!(diss, u, setup, coeff) -> Any

Compute dissipation term for the temperature equation (in-place version). Add result to diss.

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IncompressibleNavierStokes.dissipation Method

dissipation(u, setup, coeff) -> Any

Compute dissipation term for the temperature equation (differentiable version).

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IncompressibleNavierStokes.divergence! Method

divergence!(div, u, setup) -> Any

Compute divergence of velocity field (in-place version).

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IncompressibleNavierStokes.divergence Method

divergence(u, setup) -> Any

Compute divergence of velocity field (differentiable version).

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IncompressibleNavierStokes.gridsize_vol Method

gridsize_vol(setup, I::CartesianIndex{2}) -> Any

Grid size based on the volume of the cell.

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IncompressibleNavierStokes.interpolate_u_p! Method

interpolate_u_p!(up, u, setup) -> Any

Interpolate velocity to pressure points (in-place version).

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IncompressibleNavierStokes.interpolate_u_p Method

interpolate_u_p(u, setup) -> Any

Interpolate velocity to pressure points (differentiable version).

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IncompressibleNavierStokes.interpolate_ω_p! Method

interpolate_ω_p!(ωp, ω, setup) -> Any

Interpolate vorticity to pressure points (in-place version).

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IncompressibleNavierStokes.interpolate_ω_p Method

interpolate_ω_p(ω, setup) -> Any

Interpolate vorticity to pressure points (differentiable version).

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IncompressibleNavierStokes.kinetic_energy! Method

kinetic_energy!(ke, u, setup; interpolate_first) -> Any

Compute kinetic energy field (in-place version).

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IncompressibleNavierStokes.kinetic_energy Method

kinetic_energy(u, setup; kwargs...) -> Any

Compute kinetic energy field $k$ (in-place version). If interpolate_first is true, it is given by

$$k_I=\frac{1}{8}\sum _{\alpha }(u_{I+h_{\alpha }}^{\alpha }+u_{I-h_{\alpha }}^{\alpha }{)}^2.$$

Otherwise, it is given by

$$k_I=\frac{1}{4}\sum _{\alpha }((u_{I+h_{\alpha }}^{\alpha }{)}^2+(u_{I-h_{\alpha }}^{\alpha }{)}^2),$$

as in [14].

Differentiable version.

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IncompressibleNavierStokes.laplacian! Method

laplacian!(L, p, setup) -> Any

Compute Laplacian of pressure field (in-place version).

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IncompressibleNavierStokes.laplacian Method

laplacian(p, setup) -> Any

Compute Laplacian of pressure field (differentiable version).

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IncompressibleNavierStokes.left Method

left(I::CartesianIndex{D}, i) -> Any
left(I::CartesianIndex{D}, i, n) -> Any

Left index n times away in direction i.

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IncompressibleNavierStokes.pressuregradient! Method

pressuregradient!(G, p, setup) -> Any

Compute pressure gradient (in-place version).

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IncompressibleNavierStokes.pressuregradient Method

pressuregradient(p, setup) -> Any

Compute pressure gradient (differentiable version).

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IncompressibleNavierStokes.qcrit Method

qcrit(u, setup) -> Any

Compute $Q$, the second invariant of the velocity gradient tensor.

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IncompressibleNavierStokes.right Method

right(I::CartesianIndex{D}, i) -> Any
right(I::CartesianIndex{D}, i, n) -> Any

Right index n times away in direction i.

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IncompressibleNavierStokes.scalewithvolume! Method

scalewithvolume!(p, setup) -> Any

Scale scalar field with volume sizes (in-place version).

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IncompressibleNavierStokes.scalewithvolume Method

scalewithvolume(p, setup) -> Any

Scale scalar field p with volume sizes (differentiable version).

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IncompressibleNavierStokes.total_kinetic_energy Method

total_kinetic_energy(u, setup; kwargs...) -> Any

Compute total kinetic energy. The velocity components are interpolated to the volume centers and squared.

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IncompressibleNavierStokes.unit_cartesian_indices Method

unit_cartesian_indices(D) -> Any

Get tuple of all unit vectors as Cartesian indices.

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IncompressibleNavierStokes.vorticity! Method

vorticity!(ω, u, setup) -> Any

Compute vorticity field (in-place version).

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IncompressibleNavierStokes.vorticity Method

vorticity(u, setup) -> Any

Compute vorticity field (differentiable version).

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IncompressibleNavierStokes.δ Method

δ(setup, p, i, I) -> Any

Differentiate scalar $p$ in direction $e_i$.

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IncompressibleNavierStokes.δ Method

δ(setup, u, i, j, I) -> Any

Differentiate vector $u_i$ in direction $e_j$. Make sure that i and j are known at compile-time to remove the if-statement.

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