Three-Dimensional
High-Order Spectral Finite for Unstructured Grids
Yen Liu (To receive correspondence) NASA Ames Research Center,
Volume
Method
and Marcel Vinokur Mail Stop T27B-1
Moffett Field, CA 94035
[email protected]._ov Tel: 650-604-6667
Z.J. Wang Department 2555
of Mechanical Engineering
Engineering,
Building,
Michigan
East Lansing,
State
University
MI 48824
[email protected]
An extended
abstract
submitted
Orlando,
to 16th AIAA
Florida,
June
23-26,
CFD
Conference
2003
1. INTRODUCTION Many areas require a very high-order for complex shapes. This paper deals with Finite Volume (SV) method for unstructured
accurate numerical solution of conservation laws the extension to three dimensions of the Spectral grids, which was developed to solve such problems
[1-2]. We first summarize the limitations of traditional finite-volume for both structured and unstructured grids. of the spectral finite order finite-volume reconstruction. reconstruction, (SV),
Instead of using a large stencil of neighboring cells to perform a high-order the stencil is constructed by partitioning each grid cell, called a spectral volume
cells are partitioned the reconstructions unknowns can
be
sub-cells,
involving
just
pre-determined flexible.
In this paper
called
control
volumes
into polygonal or polyhedral for all the SVs become
or locations.
geometrically CVs.
such as finite-difference, and describe the basic formulation
volume method. What distinguishes the SV method from conventional highmethods [3-5] for unstructured triangular or tetrahedral grids is the data
into "structured"
orientations,
methods We then
It follows
that
a few simple once
The
for
most
we present
all. critical
(CVs).
One
adds
and multiplies,
The
method
is reduced and those
is thus
very
part of the SV method of a tetrahedral
code
to solve
Important
aspects
three-dimensional of the
data
by minimizing the optimized partitions. problems
structure
are
for
any
discussed.
to a weighted weights
accurate,
is the partitioning SV into polyhedral
of
accuracy
Comparisons
of and
and
yet
of the SV into CVs with
one
five. (Note that the order degree of precision.) The
Lebesgue constant of the The details of an efficient, order
sum
are universal
efficient,
free parameter for polynomial reconstructions up to degree of precision of accuracy of the method is one order higher than the reconstruction free parameter will be determined matrix or similar criteria to obtain
that if all the SV
CV sub-cells in a geometrically similar manner, universal, irrespective of their shapes, sizes,
the reconstruction
the partitioning
can show
are with
the
reconstruction parallelizable then
presented.
Discontinuous
Galerkin(DG) method[7-10]
are made.
Numerical
examples
for wave
propagation
problems
are
presented.
2. LIMITATIONS 2.1 Finite-difference The curvilinear
OF TRADITIONAL
METHODS
methods
most
widely
coordinate
used
method
system.
The
is the
finite-difference
limitations
for very
high
method order
applied
to a body-fitted
of accuracy
implementation
are:
a.
The spatial differencing directions. Thus a large
is essentially one-dimensional, carried out along coordinate number of data points near the unknown to be updated are
ignored. The are necessary.
has to be modified near boundaries, where one-sided methods, in order to maintain a necessary bandwidth,
of accuracy b.
The
large stencil For implicit must
metric
be reduced
terms
Since numerical accuracy of the
The unknowns
true
in highly
are values
stretched
A single,
structured
grid
methods
Finite-volume
b°
may be modified
The reconstruction The
surface
point
coordinates.
are often
by appropriate
area
is still done vectors
can
or boundaries
the differencing integral
for very
can
conservation
complex
shapes.
with very
be performed laws
can
Calculations
in a
only
be
must be between
grids employed
limiters
flux, using (approximate) Riemann solvers. limitations as the finite-difference method. a.
grid
or overlapping grids. At interface boundaries the high accuracy is generally degraded.
for structured
methods
true
unknowns are cell averages over quadrilaterals reconstruction in terms of neighboring unknowns which
the
or near corners
While
the
is not feasible
carried out over a set of patched patches, or in the overlap regions, 2.2 Finite-volume
differencing
areas,
at grid points.
numerically "conservative" manner, satisfied to second-order accuracy. d.
near the boundary.
by numerically
grid generators are mostly only second-order accurate, the overall solution can be severely degi'aded if the grid is not sufficiently smooth.
This is particularly high curvature. Co
for points
are evaluated
formulas the order
to overcome
where
necessary.
In practice,
one-dimensionally be exactly
limitations
b and c above.
calculated
the
along
These method
are used to calculate is subject
coordinate
in terms
of the
to the
the same
directions. cell vertices.
flux integral is approximated by a one-point quadrature at the computational which is equal to the face centroid only to second order. For a non-planar face centroid does not even exist.
2
The
(2D) or hexahedra (3D). A high order is used to obtain values at cell boundaries,
But the
face center, face in 3D, a
While
C.
the cell volume
to a computational d.
If the reduced
can be precisely
cell center,
which
grid is very unsmooth, to first order.
2.3 Finite-volume
methods
is equal
and
for unstructured
calculated,
to the cell centroid
highly
curved,
cells in all directions.
even
are implicitly
assigned
only to second
the
second
order.
order
accuracy
is
grids
The unknowns are cell averages over triangles of any desired order of accuracy is obtained in terms neighboring
the unknowns
The flux integral
(2D) or tetrahedra (3D). A reconstruction of unknowns at an appropriate number of
for each face is evaluated
using
a quadrature
approximation of the same order of accuracy as the reconstruction. point is obtained using the reconstructed solution for the two cells
The flux at each quadrature sharing that face. In principle,
one can in this manner
obtain
order
this method
computational
has severe
It is difficult overdetermined
a.
accuracy,
Each
b.
Due
of any desired
of accuracy.
requires
and
thus the
size
a different
reconstruction become
of the matrix
stencil.
prohibitive
cell at each time step would
to the unstructured
one is faced 'For very high to be inverted,
nature
If the
inversion
3.1 Basic
main
a single
motivation
non-singular
start with a relatively called spectral volumes sub-cells,
called
of precision. degree polygons
becomes
involve
of the data in physical
impractically
space,
SPECTRAL
FINITE
VOLUME
hand,
are
repeating
large
CPU times.
the data
from
neighboring
This
would
hamper
the
METHOD
formulation
The obtain
an of
coefficients
for 3D. On the other
cells required for the computation can be far apart in memory. efficiency of the code due to data gathering and scattering.
3. THE
with order
dimensions.
requirements
for every
In practice,
limitations.
of cells,
in three
the memory
the inversion C°
number large
cell
stored,
solution
to obtain a non-singular stencil. In general, problem which requires a least square inversion.
the
prohibitively
a numerical
control
The
of structure,
into triangular a geometrically
the spectral
finite
that can be applied
volume
method
to all the cells
coarse unstructured grid of cells, triangles (SVs). Each SV is then further subdivided volumes
unknowns making
or polyhedra.
behind stencil
(CVs),
that support
are now the cell averages use of all the symmetries
For 3D,
they
can have
a polynomial over
way to grid.
We
in 2D and tetrahedra in 3D, into a number of "structured" expansion
the CVs.
of the simplex
non-triangular
is to find a simple in an unstructured
faces,
The
of a desired
subdivision
geometry. which
The
must
degree
has a high CVs can be
be subdivided
facets in order to perform the required integrations. All the SVs are partitioned in similar manner. We thus obtain a single, universal reconstruction for all SVs.
3
Due to the symmetry of the subdivision, in terms of the CV unknowns. A CV face Riemann
solver
that
only a few distinct
lies on an SV boundary
is then necessary
to compute
coefficients
will have
appear
a discontinuity
in the expansion
on its two
the flux on that face. If the flux is a linear
sides.
A
function
of the unknowns, the integration can be performed analytically without invoking quadratures [5], and the result expressed as a weighted sum of all the CV unknowns in the two SVs. The weights for each type of CV face to the program.
are universal
numbers
If the flux is a non-linear
of the appropriate
degree
of precision
that are pre-calculated
function
of the unknowns,
is required
once
and read
a quadrature
[6]. The conservative
variable
in as input
approximation on one side of a
quadrature point can again be expressed as a weighted sum of the CV unknowns in the SV on that side. Since the quadrature points belong to just a few symmetry groups, the total number of distinct
weights
oscillations, or filters.
that need
to be stored
it may be necessary
is relatively
to modify
small.
In order
the reconstructed
to suppress
solution
using
The reconstruction within each spectral volume is continuous. over a CV face that lies in the interior of a SV can be evaluated directly, type of face
can be stored.
quadrature regions.
point.
3.2 Details
of the spectral
We present a vector
notation
Again
For a non-linear a modification
finite
further
volume
details
for brevity.
flux,
a similar
procedure
involving
limiters
or filters
spurious
numerical
appropriate
limiters
Therefore a linear flux and the weights for each can be carried
out for each
may 'be required
in certain
method
of the formulation
A conservation
for a general
law is written
conservation
law. We employ
as
_U
--+V.F
= O,
Ot
where
the conservative
tensor.
Integrating
variable
where
Vj._ is the
bounding averages
Vj.,.
u can be a scalar
or a vector,
and the flux
F can be a vector
or a
(1) over each CV, we obtain
udV+ dt
(1)
vj.,
=
volume
(In 2D,
of u, defined
dS.F _._.,
=0,
(2)
of the jth CV in the i th SV, each
facet
is actually
a line
and
S,.j._ is the
segment.)
The
area
unknowns
of planar are the
facet
k
volume
as
Ujd
uaV. Vj, i
).i
(3)
The partitioning reconstruction. n requires
of each SV into CVs depends on the choice of For a complete polynomial basis, a reconstruction
a subdivision
N=
In the
present
work
into (at least)
N CVs, where
(n+1)(n+2)/2 (n + 1) (n+l)(n+2)(n+3)/6
2D. 1D 3D
we
partition
(4)
SV into N CVs, ourselves
reconstruction
experiments
five for both 2D and 3D have
basis is substituted in the form
value. been
Partitions obtained.
into (3), and the resulting
involving
valid
one
for reconstruction
If the expansion
matrix
only
the
numerical
an optimum
to partitions
involves
The choice of parameter for each degree of precision is determined by minimizing the constant of the reconstruction matrix or similar convergence criteria. We will conduct to determine
restrict
the
of a square
of precision
also
so that
parameter. Lebesgue degree
We
for the precision
inversion
the polynomial can be written
matrix.
the
basis functions of degree of
equation
free
up to
of u in terms
is solved,
of
the result
u;(r)= _ Lj,,(r) _j,:,
(5)
J
where
Lj, i (r) are known
shape functions
function
The
surface
or cardinal
expression
integrations
For non-linear
approximation
functions.
in (2) for a given
(5), and the result
of the shape
functions
be calculated and stored in advance. Flux integrals Riemann solver, and the expansion is now a weighted that facet.
basis
Details
for obtaining
the
in the final paper.
of u, the flux integral
by substituting
unknowns.
functions
L/. i (r) will be given
If F is a linear be evaluated
as shape
flux functions,
facet in the interior
written
as a weighted
per unit area
of an SV can sum of the CV
are universal,
which
can
for facets on the SV boundaries require a sum of CV unknowns in both SVs sharing
the flux integral
is evaluated
by a consistent
quadrature
of the form
fs,,., aS. F = _
wqn. F(u i (rq))S_,j. i ,
(6)
q
where
the
Wq are known
quadrature
u, (rq) = _
weights.
Using
(5), we can evaluate
u, (rq) as
Lj, i (rq) u_,_.
(7)
J
The
above
weighted functions advance.
equation sum
indicates
of the
CV
that the value unknowns.
of u at a quadrature
These
weights
are
at the quadrature point, which are also universal, For facets on the SV boundaries, the flux is replaced
point
the
can also be evaluated
functional
values
of the
as a shape
can be calculated and stored by a Riemann flux of the form
in
n.
3.3 Data
F(u
(rq))
=
FRiem (U L (rq),
u R (rq),
(8)
n).
Structure There
are
parallelizable
several
code.
The
aspects global
locations, numbers,
and cell numberings. in an order indicating
make
of the universal
use
of the
data
grid data
structure
consists
The topology its orientation,
nature
of the
which
of face
can
lead
numberings,
to a very
vertex
efficient
numberings
and
is specified by listing for each face its vertex and the two adjacent cell numbers. In order to
partitioning,
all global
cells
are
mapped
into
a single
standard 2D, and
SV. Thus, each global face can have three possible orientations in the standard SV for twelve for 3D. All the information connecting the local CV face numbering for each
possible
orientation
shall
be given
desired
is pre-determined
in the final
order
of accuracy.
There
is an aspect
paper.
and read in as input We thus can write
inherent
in the spectral
to the program.
a single
finite
code
volume
given
cell, all the data required
from
the cell is found
valid
method
use of cache memory, resulting in great computational efficiency Since all unknowns in a single SV cell are packed together, when contiguous
Detail
of this mapping
for 2D or 3D, and any
that permits
on modem performing in memory.
supercomputers. calculations for a Since
most two SVs is involved in any single computation, data communication between memory is minimized. All the needed data can be located in cache memory, and into L1 cache.
This results
3.4 Comparison The
in great reductions
with the Discontinuous
spectral
finite
volume
Galerkin method [7-10]. We method solves the conservation The unknowns
in the DG
in memory
Galerkin method
are point
The
evaluated. surfaces.
DG
method
In the DG In contrast,
requires
bears
values,
an integration
from at
the CPU and may even fit
method certain
similarities
by parts,
method, flux calculations they are evaluated over
to
of the directly
and for time-dependent
together, thus requiring an expensive mass matrix inversion the weak formulation, N test functions are needed, resulting one.
data
time.
point out some of the advantages law in a weak form, rather than
method
an optimum
the
Discontinuous
SV method. The DG as in the SV method. problems
are coupled
to maintain high accuracy. Due in N coupled equations instead
which
results
in additional
terms
to of
to be
are carried out at quadrature points on the SV CV faces for the SV method, which include
additional faces in the interior of the SV. It appears that evaluate the flux integrals than the DG method. However,
the SV method has more faces to in a (n+l) th order formulation, the
former onl_ involves n th order surface integrations, while the latter requires 2n th order surface and (2n-l) u' order volume integrations. Finally, the order of accuracy of the DG method is based on the size of the SV. In contrast, the accuracy in the SV method is based on the size of the much smaller CV.
4. PARTITIONING 4.1 Details
OF THE SPECTRAL
of the partitioning
6
VOLUME
The most critical part of the SFV method is the partitioning of the SV into Partitions
for
partitions
for the 3D case.
parameter.
the 2D
If N,
analytically,
case
up
the number
using
to degree
3 have
In the present
work
been
of CVs, is not too large,
a symbolic
language
inversion is performed numerically. studies to determine optimum values
presented
we restrict
such
in [2].
ourselves
one can invert
as Mathematica.
Here
we
to partitions a matrix
For
CVs.
present
with
one
the free
with one parameter
higher
values
of N,
the
In the final paper we will present results of convergence of the free parameters for various degrees of precision.
For partitions with degrees of precision no greater than three, all the CVs have at least one face on the SV boundary. There are no CVs in the interior of the SV. All the CVs then consist of the vertices of the 2D CVs for each face of the SV connected to the SV centroid with straight edges. In Fig. la, we show the partition The four CVs are hexahedra, with all faces being
of an SV into CVs planar quadrilaterals.
for degree of precision 1. The partition for degree
of precision 2 is depicted in Fig. 2a. The CVs are here members of two symmetry groups. One consists of the four hexahedra at the comers of the SV. They are shown in Fig. 2b. Note that the four
quadrilateral
interior
faces
are no longer
planar.
When
required
for integration
purposes,
they are subdivided into two triangular facets by means of a straight line connecting comer to the SV centroid. The other group consists of the six mid-edge tetrahedra. shown
in Fig. 2c. Note
quadrilateral degrees
interior
of precision
4.2 Shape
CV,
determining
it shares
consists
of two exterior
with the comer
up to 5 will be presented
CVs.
Further
triangular details
faces
and the two
of the partitionings
for
in the final paper.
functions
For a given each
that each tetrahedron
faces
the SV They are
as defined
partition,
there
by Eq.
the surface
exists
(5). These
flux integrals.
only two sets of weights
needed
a unique functions
function
are necessary
For degree
to be stored,
shape
or cardinal to calculate
one reconstruction,
{-23/52,
basis
function
the weights
due to symmetry,
5/4, 5152, 5152 } and
{29/52,
for
used there
29/52,
in are
-3/52,
-3/52}, corresponding to the boundary CV faces and the interior CV faces, respectively. Since the shape functions are functions of three variables, they cannot be easily depicted. We choose to represent
them
by showing
contour
plots on the surface
(a) SV partition Fig. 1. SV partition
and shape
of the SV. These
(b) shape function
for polynomial
7
are shown
in Fig. lb for
function
reconstruction
of degree
one
a representative CV for the partition of degree1.The contourplots for the two typesof CVs in the partitions of degree2, correspondingto Figs. 2b and 2c, are plotted in Figs. 2d and2e, respectively.Contourplots for higherdegreesof precision,aswell asthe analyticexpressions for theshapefunctionsin termsof local coordinates,will begiven in thefinal paper.
(b) comerCVs
(a) SVpartition
(d) shapefunctionfor comerCV
(c) mid-edgeCVs
(e) shapefunctionfor mid-edgeCV
Fig. 2. SV partitionandshapefunctionsfor polynomialreconstructionof degreetwo
5. Numerical In order phase
to demonstrate
to choose
electromagnetic Those
problems wave
for two space
field components
the high for
which
equations. variables
triangles, shows
and contour
refinements. used. the
The
successive
plots
excellent
solutions
We first carry
refinement
involving
To
decided this
two and three
we solve space
since they involve
out a convergence
in this initial
end
the
variables.
electromagnetic
study by calculating
the
region at 45 degrees. A non-reflecting boundary of the square. The coarsest grid consists of 200 multiplies grid,
and
the
of 1, leading has propagated
boundaries
number
of triangles
Fig. 3b shows
the wave
open
it was
solutions.
of precision
are after
at the
solutions
a square boundaries
of degree
shown
exact
three-dimensional,
of E, for the coarsest
A partition solutions
of the method,
exist
We present
in all three directions.
each
accuracy there
are actually
propagation of a wave through condition is applied at the four
Results
demonstrating
to second through the
by four.
the solution order
after
accuracy,
two periods effectiveness
Fig.
3a
two grid has been
in time. Note of the
non-
reflecting boundarycondition. In the final paper we will verify the order of accuracyby calculatingthe errorsfor five successive grid refinements.Similar calculationswill bepresented for higherordersof accuracy. In Fig.4,we presenta three-dimensional solution for a planewavepropagatingthrougha rectangularparallelepiped,which hasbeendiscretiziedby a tetrahedralgrid. Only the second orderresultfor the coarsestgrid is shown.Resultsfor variousordersof accuracyas well asgrid refinementswill begiven in thefinal paper. We next presentresultsfor a plane wave incident on a perfectly conductingcircular cylinder.The grid usedis shownin Fig. 5.The averagedgrid sizeis about1/12of a wavelength. Figures.6 showcontoursof E. for a TM wave. The exact solution is presented in Fig.6a. The numerical order
(cubic
solution,
for second
reconstruction)
order
(linear
reconstruction)
in Fig. 6c. Fig. 7 shows
is shown
in Fig.
6b, and
results
for H:
for a TE wave.
analogous
the final paper we will present results up to sixth order, the exact solution. We will also show results for a plane
0.2_5
as well as quantitative comparisons wave incident on a sphere.
0
-0,5
X
(a) coarse
Fig. 4. Contour
0.5
1
x
grid (200 triangles)
plot of E: for a plane
0
(b) fine grid (3200
wave propagating
plot of E, for a plane wave
propagating
9
through
through
a square
triangles) region
a rectangular
(second
In
with
0.25
-0,5
Fig. 3. Contour
for fourth
order)
parallelepiped
Fig. 5. Grid for a regionexteriorto a circularcylinder
(a) exact Fig. 6. Contour
(a) exact Fig. 7. Contour
solution
(b) second order
plot of E z for a TM plane wave
solution
(b) second
plot of H_ for a TE plane
incident
order
wave incident
10
(c) fourth on a perfectly
conducting
(c) fourth on a perfectly
order cylinder
order
conducting
cylinder
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