ERCOFTAC Bulletin - Centre Acoustique

marks indicate, that the scheme does not directly lead
to a decision between ‘self’ and ‘neighbor’ at this point.
Since the modal basis is chosen orthogonal, the integral
Q Φl Φk dV vanishes for l = k. This allows the direct
calculation of ûl without the inversion of a mass matrix.
If every element is mapped to a reference triangle
or tetrahedron. which is a linear transformation for
non-curved elements, every term gets the mapping determinant
as additional factor. Then the space integrals
can be precomputed analytically or by quadrature and
stored. The resulting reduction of the computational
effort and its speed-up is a great advantage of the scheme.
2.2 Variable Jacobi matrices
For space dependent, but cell constant mean flows, the
values of the Jacobi matrices A and B are ambiguous
at the cell interface. For stronger spatial changes of the
matrices this can lead to serious problems, as reported
in [9]. To avoid this problem, the scheme is extended
with space dependent Jacobi matrices within each cell.
Therefore the modal representation A = nDegF rJ
i=1 Â Φi
i
is introduced with a polynomial basis of the same type as
used for the state variables. The maximum polynomial
degree may be lower depending on the demand and rates
of change of the problem under consideration.
The resulting scheme then reads as
dûl
dt
= Â i ûl
+ ˆ
B ûl
i
Φl Φk dV
Q
Φl
Q
Φl
Q
+
− Â
i ? + ˆ B +
i ?
−
− Â
i ? + ˆ B −
i ?
∂
∂x (Φi Φk) dV
∂
∂y (Φi Φk) dV
û self
l
∂Q
û neighbor
l
Φ ? i Φ self
l
∂Q
Φk dS
Φ ? i Φ neighbor
l
Φk dS (2)
After transferring the equation to a reference element the
volume integrals can be precomputed. However, this is
not possible for the surface integrals, because the mean
flow velocity might change between inbound and outbound
along the surface and consequently the steadiness
of the matrices A + (ξ) and A −
? (ξ) is not guaranteed any
more. A quadrature formula is used instead.
2.3 Nodal integration
The non-constant Jacobi matrices are fundamental to
overcome the problems, that can result from a jump of
the Jacobi matrices at the cell interface. But in 2D and
3D it is very cumbersome to enforce unambiguous values
at the common surface. A further step ahead is an integration
scheme, that combines the values within the cell,
represented by the modal basis of the cell, with surface
nodes, that can be forced to have identical values for both
the adjacent cells. On the other hand, the modal scheme
with its hierarchical basis functions allows an easy application
of order based filters. Gassner et. al. [2] present
a nodal-modal DG scheme, that combines both sets of
basis functions.
We adopt this approach here and create additionally to
the existing modal representation a nodal one for the
state u as well as the Jacobi matrices A and B:
u =
nDegF r
i=1
ûi Φi =
nNodes
i=1
ũi ψi
Hence, the surface flux integral in (Eq. (2)) can be written
as
=
∂Qj
F h Φk dS
nBndNds
k=1
· ψ self
(ξNdj,k l
+
nBndNds
k=1
+
Ã
m + ˜ B +
ψm(ξNdj,k ) ũself
m
l
) Φself
k (ξNdj,k
à −
m + ˜ B −
m
) wk
ψm(ξNdj,k ) ũneighbor
l
· ψ neighbor
(ξNdj,k l
) Φself(ξNdj,k
k ) wk
where A, B, ûl, Φl and ωk are the Jacobi matrices,
degrees of freedom of the state variables, basis functions
and integration weights in the modal scheme, respecitvely,
and à m , ˜ B m , ũl, ψl and wk their equivalents
in the nodal scheme. The latter is constructed in the
way, that its degrees of freedom are equal to the state at
the location of the correspondent node. The distribution
of the nodes is based on Gauss-Lobatto points, which, in
contrast to the Gauss quadrature points, also include
nodes on the interval boundaries and consequently on
the element surfaces in 2D or 3D.
The transfer between the modal degrees of freedom and
their nodal counterparts can be done by a matrix multiplication
ũ = V û and û = V −1 ũ, respectively, where
Vij = Φj(ξNdi) is the Vandermonde matrix and V −1 its
pseudo-inverse.
To achieve the full order of the scheme, the integration
of A u, each with a modal representation of the degree
nP oly, has to be of the order 2 nP oly with the corresponding
number of Gauss points or the product has to
be projected on a nP oly-basis which corresponds to an
“order truncation”. Both of these approaches result in a
performance drawback. However, in the nodal scheme,
the multiplication is done at each of the nodal points
and consequently the product is automatically a projection
on a nP oly-basis and needs only the original number
of integration points for a full order integration.
2.4 Superparametric elements
For complex shaped domains the spatial discretization
with straight edged elements leads to a high number of
elements to capture the geometry and to guarantee a
good approximation. This may strongly reduce in addition
the time step due to their small size. To avoid
this, we use curved elements at the curved wall boundaries,
which should in principle be based on a mapping
ansatz of the same spatial order as the underlying numerical
scheme. For practical simulations such a highly
accurate representation of the boundary is usually not
needed and the acoustic simulation rarely exceeds 4th
order of accuracy, an ansatz with p ≤ 3 is used for the
mapping:
x(ξ, η) =
3 3−i
i=0 j=0
γij ξ i η j (1 − ξ − η) 3−i−j
which can be rewritten as x(ξ, η) = L(ξ, η) E with an
space dependent part L(ξ, η) and an element dependent,
ERCOFTAC Bulletin 90 29