HEAD | PREVIOUS |

Next Steps

$\mathit{\nabla}.(D\mathit{\nabla}\mathit{\psi})+s(\mathit{x})=0.$ | $(13.1)$ |

$w\mathit{\nabla}.(D\mathit{\nabla}\mathit{\psi})=\mathit{\nabla}.(\mathit{wD}\mathit{\nabla}\mathit{\psi})-D(\mathit{\nabla}w).(\mathit{\nabla}\mathit{\psi}).$ | $(13.2)$ |

$\begin{array}{cc}\hfill 0=& \mathrm{\hspace{0.5em}\hspace{0.5em}}{\int}_{V}[w\mathit{\nabla}.(D\mathit{\nabla}\mathit{\psi})+ws]{d}^{3}x\hfill \\ \hfill =& \mathrm{\hspace{0.5em}\hspace{0.5em}}{\int}_{V}[-D(\mathit{\nabla}w).(\mathit{\nabla}\mathit{\psi})+ws]{d}^{3}x+{\int}_{\partial V}\mathit{wD}\mathit{\nabla}\mathit{\psi}.\mathit{dS}.\hfill \end{array}$ | $(13.3)$ |

$\mathit{\psi}(\mathit{x})={\mathit{\phi}}_{b}+\sum _{k=1}^{N}{a}_{k}{\mathit{\psi}}_{k}(\mathit{x}),$ | $(13.4)$ |

$\mathbf{K}\mathbf{a}=\mathbf{f}$ | $(13.5)$ |

${K}_{\mathit{jk}}={\int}_{V}\mathit{\nabla}{\mathit{\psi}}_{j}.\mathit{\nabla}{\mathit{\psi}}_{k}D\mathrm{\hspace{0.5em}\hspace{0.5em}}{d}^{3}x.$ | $(13.6)$ |

${f}_{j}={\int}_{V}[{\mathit{\psi}}_{j}s-\mathit{\nabla}{\mathit{\psi}}_{j}.\mathit{\nabla}{\mathit{\phi}}_{b}D]\mathrm{\hspace{0.5em}\hspace{0.5em}}{d}^{3}x.$ | $(13.7)$ |

Figure 13.1: Localized triangle functions in one dimension, multiplied by
coefficients, sum to a
piecewise linear total function. Their derivatives are box functions
with positive and negative parts. They overlap only with adjacent functions.

Such a set of functions will give rise to a matrix $\mathbf{K}$ that is
tridiagonal, just as was the case for finite
differences in one dimension. Using smoother, higher order, functions
representing the elements is sometimes advantageous. Cubic Hermite
functions, and cubic B-splines are two examples. They lead to a matrix
that is not quite as sparse, possessing typically seven non-zero
diagonals (in 1-dimension). Their ability to treat higher-order
differential equations efficiently compensates for the extra
computational complexity and cost.
In multiple dimensions, the geometry becomes somewhat more
complicated, but, for example, there is a straightforward extension of
the piecewise linear treatment to an unstructured mesh of
tetrahedra
in three dimensions.
Figure 13.2: In linear interpolation within a tetrahedron, the lines from
a point $p$ to the corner nodes of the tetrahedron, divide it into
four smaller tetrahedra, whose volumes sum to the total volume. The
interpolation weight of node $k$ is proportional to the
corresponding volume, ${V}_{k}$.

For any point within a particular tetrahedron,
the interpolated value of the function is taken to be equal to a
weighted sum of the values at the four corner nodes. The weight of
each corner is equal to the volume of the smaller tetrahedron obtained
by replacing the corner with the point of interest, divided by the
total volume of the original tetrahedron, which Fig. 13.2 illustrates${}^{81}$. The ${\mathit{\psi}}_{k}$ function associated with a node $k$ is unity at
that node and decreases linearly to zero along every connection leg
to an adjacent node. The
interpolation formula within a single
tetrahedral element is then
$\mathit{\psi}(p)=\sum _{1}^{4}{V}_{k}{a}_{k}/V.$ | $(13.8)$ |

$f(t)=\sum _{n=-\mathit{\infty}}^{\mathit{\infty}}{e}^{i{\mathit{\omega}}_{n}t}{F}_{n},$ | $(13.9)$ |

${F}_{n}=\frac{1}{T}{\int}_{0}^{T}{e}^{-i{\mathit{\omega}}_{n}t}f(t)\mathit{dt}.$ | $(13.10)$ |

${F}_{n}=\frac{1}{T}{\int}_{0-\mathit{\epsilon}}^{T-\mathit{\epsilon}}{e}^{-i{\mathit{\omega}}_{n}t}f(t)\mathit{dt}\approx \frac{1}{T}\sum _{j=0}^{N-1}{e}^{-i{\mathit{\omega}}_{n}{t}_{j}}{f}_{j}\mathit{\Delta}t=\frac{1}{N}\sum _{j=0}^{N-1}{e}^{-i2\mathit{\pi}nj/N}{f}_{j}.$ | $(13.11)$ |

${f}_{j}=\sum _{n=0}^{N-1}{e}^{i2\mathit{\pi}nj/N}{F}_{n}\hspace{0.5em};\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{F}_{n}=\frac{1}{N}\sum _{j=0}^{N-1}{e}^{-i2\mathit{\pi}nj/N}{f}_{j}.$ | $(13.12)$ |

${\int}_{0}^{T}{e}^{-{\mathit{\omega}}_{n}t}{\int}_{0}^{T}f(t\text{\'})g(t-t\text{\'})\mathit{dt}\text{\'}\mathit{dt}={\int}_{0}^{T}{e}^{-{\mathit{\omega}}_{n}t\text{\'}}f(t\text{\'})\mathit{dt}\text{\'}{\int}_{0}^{T}{e}^{-{\mathit{\omega}}_{n}t"}g(t")\mathit{dt}".$ | $(13.13)$ |

${K}_{n}={\int}_{-\mathit{\Delta}t}^{\mathit{\Delta}t}{e}^{-i{\mathit{\omega}}_{n}t}(1-|t|/\mathit{\Delta}t)\mathit{dt}/\mathit{\Delta}t=\frac{{\mathrm{sin}}^{2}({\mathit{\omega}}_{n}\mathit{\Delta}t/2)}{({\mathit{\omega}}_{n}\mathit{\Delta}t/2{)}^{2}}.$ | $(13.14)$ |

$L(u)+M(u)N(u)=s(x).$ | $(13.15)$ |

$\frac{1}{X}{\int}_{0}^{X}[L(u)+M(u)N(u)]{e}^{-{\mathit{ik}}_{m}x}\mathit{dx}={S}_{m}.$ | $(13.16)$ |

$L}_{m}{U}_{m}{e}^{{\mathit{ik}}_{m}x}+\sum _{l+n=m}{M}_{l}{U}_{l}{e}^{{\mathit{ik}}_{l}x}{N}_{n}{U}_{n}{e}^{{\mathit{ik}}_{n}x}={S}_{m$ | $(13.17)$ |

$\mathbf{A}\mathbf{x}=\mathbf{b}$ | $(13.18)$ |

${\mathbf{r}}_{k+1}={\mathbf{r}}_{k}-{\mathit{\alpha}}_{k}\mathbf{A}{\mathbf{p}}_{k}.$ | $(13.19)$ |

$\mathbf{p}}_{k}=\mathbf{P}{\mathbf{r}}_{k}+\sum _{j=0}^{k-1}{\mathit{\beta}}_{\mathit{kj}}{\mathbf{p}}_{j$ | $(13.20)$ |

$(\mathbf{A}{\mathbf{p}}_{k}{)}^{T}\mathbf{R}({\mathbf{r}}_{k}-{\mathit{\alpha}}_{k}\mathbf{A}{\mathbf{p}}_{k})={\mathbf{p}}_{k}^{T}{\mathbf{A}}^{T}\mathbf{R}{\mathbf{r}}_{k+1}=0,$ | $(13.21)$ |

${\mathit{\alpha}}_{k}={\mathbf{p}}_{k}^{T}{\mathbf{A}}^{T}\mathbf{R}{\mathbf{r}}_{k}/{\mathbf{p}}_{k}^{T}{\mathbf{A}}^{T}\mathbf{R}\mathbf{A}{\mathbf{p}}_{k},$ | $(13.22)$ |

${\mathbf{p}}_{j}^{T}{\mathbf{A}}^{T}\mathbf{R}\mathbf{A}{\mathbf{p}}_{k}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{for}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}j\ne k$ | $(13.23)$ |

$\mathit{\beta}}_{\mathit{kj}}=-{\mathbf{p}}_{j}^{T}{\mathbf{A}}^{T}\mathbf{R}\mathbf{A}\mathbf{P}{\mathbf{r}}_{k}/{\mathbf{p}}_{j}^{T}{\mathbf{A}}^{T}\mathbf{R}\mathbf{A}{\mathbf{p}}_{j$ | $(13.24)$ |

${\mathbf{p}}_{j}^{T}{\mathbf{A}}^{T}\mathbf{R}{\mathbf{r}}_{k}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{forall}\hspace{0.5em}k\hspace{0.5em}\text{andall}\hspace{0.5em}jk.$ | $(13.25)$ |

${\mathbf{p}}_{j}^{T}{\mathbf{A}}^{T}\mathbf{R}{\mathbf{r}}_{k+1}={\mathbf{p}}_{j}{\mathbf{A}}^{T}\mathbf{R}{\mathbf{r}}_{k}-{\mathit{\alpha}}_{k}{\mathbf{p}}_{j}{\mathbf{A}}^{T}\mathbf{R}\mathbf{A}{\mathbf{p}}_{k}.$ | $(13.26)$ |

$(\mathbf{P}{\mathbf{r}}_{j}{)}^{T}{\mathbf{A}}^{T}\mathbf{R}{\mathbf{r}}_{k}={\mathbf{r}}_{k}^{T}{\mathbf{R}}^{T}\mathbf{A}\mathbf{P}{\mathbf{r}}_{j}=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{forall}\hspace{0.5em}k\hspace{0.5em}\text{andall}\hspace{0.5em}jk.$ | $(13.27)$ |

${\mathbf{r}}_{k}^{T}{\mathbf{R}}^{T}\mathbf{A}{\mathbf{p}}_{k}={\mathbf{r}}_{k}^{T}{\mathbf{R}}^{T}\mathbf{A}\mathbf{P}{\mathbf{r}}_{k}.$ | $(13.28)$ |

${\mathbf{r}}_{k}^{T}{\mathbf{R}}^{T}\mathbf{A}\mathbf{P}{\mathbf{r}}_{j+1}={\mathbf{r}}_{k}^{T}{\mathbf{R}}^{T}\mathbf{A}\mathbf{P}{\mathbf{r}}_{j}-{\mathit{\alpha}}_{j}{\mathbf{r}}_{k}^{T}{\mathbf{R}}^{T}\mathbf{A}\mathbf{P}\mathbf{A}{\mathbf{p}}_{j}.$ | $(13.29)$ |

(1) | $\mathbf{P}$ and $\mathbf{A}$ are symmetric, |

(2) | $(\mathbf{A}\mathbf{P})$ and $(\mathbf{A}\mathbf{R})$ commute. |

${\mathit{\beta}}_{k,k-1}={\mathbf{r}}_{k}^{T}{\mathbf{R}}^{T}\mathbf{A}\mathbf{P}{\mathbf{r}}_{k}/{\mathbf{r}}_{k-1}^{T}{\mathbf{R}}^{T}\mathbf{A}\mathbf{P}{\mathbf{r}}_{k-1},$ | $(13.30)$ |

${\mathit{\alpha}}_{k}={\mathbf{r}}_{k}^{T}{\mathbf{R}}^{T}\mathbf{A}\mathbf{P}{\mathbf{r}}_{k}/{\mathbf{p}}_{k}^{T}{\mathbf{A}}^{T}\mathbf{R}\mathbf{A}{\mathbf{p}}_{k}.$ | $(13.31)$ |

$\mathbf{f}(\mathbf{v})=0.$ | $(13.32)$ |

$\mathbf{J}(\mathbf{v})=\frac{\partial \mathbf{f}}{\partial \mathbf{v}}.$ | $(13.33)$ |

$\mathbf{J}\mathit{\delta}\mathbf{v}=-\mathbf{f}(\mathbf{v})$ | $(13.34)$ |

$\mathbf{J}\mathbf{u}\approx [-\mathbf{f}(\mathbf{v}+\mathit{\epsilon}\mathbf{u})-\mathbf{f}(\mathbf{v})]/\mathit{\epsilon},$ | $(13.35)$ |

$\mathbf{C}}^{-1}\mathbf{A}\mathbf{x}={\mathbf{C}}^{-1}\mathbf{b$ | $(13.36)$ |

1. | Update the residual $\mathbf{r}(=\mathbf{b}-\mathbf{A}\mathbf{x})$, via ${\mathbf{r}}_{k}={\mathbf{r}}_{k-1}-{\mathit{\alpha}}_{k-1}\mathbf{A}{\mathbf{p}}_{k-1}$. |

2. | Solve the system $\mathbf{C}{\mathbf{z}}_{k}={\mathbf{r}}_{k}$ to find the preconditioned residual ${\mathbf{z}}_{k}$. |

3. | Update the search direction using ${\mathbf{z}}_{k}$, via ${\mathbf{p}}_{k}={\mathbf{z}}_{k}-{\mathit{\beta}}_{k,k-1}{\mathbf{p}}_{k-1}$. |

4. | Increment $k$ and repeat from 1. |

$\frac{\partial}{\partial t}(\mathit{\rho}\mathit{v})+\nabla p=\nabla .(\mathit{\rho}\mathit{v}\mathit{v})-\mathit{\mu}{\nabla}^{2}\mathit{v}+\mathit{F}=\mathit{G}.$ | $(13.37)$ |

$(\mathit{\rho}\mathit{v}{)}^{(n+1)}-(\mathit{\rho}\mathit{v}{)}^{(n)}=\mathit{\Delta}t(-\mathit{\nabla}{p}^{(n)}+{\mathit{G}}^{(n)}).$ | $(13.38)$ |

${\nabla}^{2}{p}^{(n)}=\mathit{\nabla}.{\mathit{G}}^{(n)}+\mathit{\nabla}.(\mathit{\rho}\mathit{v}{)}^{(n)}/\mathit{\Delta}t.$ | $(13.39)$ |

Figure 13.3: Schematic energy spectrum $E(k)$ of turbulence as a function
of wave number $k=2\mathit{\pi}/L$. There is an inertial range where theory
(and experiments) indicate that a cascade of energy towards smaller
scales gives rise to a power law $E(k)\propto {k}^{-5/3}$. Eventually
viscosity terminates the cascade. LES artificially cuts it off at
lower $k$.

In fact, the finite resolution of a discrete grid representation gives
rise to an effective $k$ cut-off in any case. If a
difference scheme is used that introduces sufficient dissipation (not
just aliasing and dispersion), it is sometimes presumed that no explicit
additional filtering is essential.
The Reynolds Averaged Navier Stokes (RANS) is an even more
approximate treatment, which averages over all relevant time scales
leaving only the steady part of the flow. Therefore essentially
HEAD | NEXT |