3. How can hulls reduce the degrees of freedom of Finite
Elements?
DOF = 18
DOF = 7
DOF = 3
DOF = 36
DOF = 19
DOF = 6
DOF = 60
DOF = 37
DOF = 10
Figure 1: The DOF requirement of three methods; (Top-row)
Discontinuous FEM. (Middle-row) Continuous FEM and (Bottom-row)
Spectral Hull in P space (see Eq.(30) for definition of P space)
4. How can hulls reduce the degrees of freedom of Finite
Elements?
0 5 10 15 20
0
200
400
600
800
1000
1200
1400
P (Polynomial Degree)
DegreeofFreedom
Standard DG
Standard CG
SHull in Q space
SHull in P space
Figure 2: The DOF requirement of three methods discussed in Fig. 1.
5. How can hulls reduce the degrees of freedom of Finite
Elements?
The truncation errors:
α4 × h × O(h3
) ≤ 1/6α1O(h3
)
α4 × h × O(h3
) ≤ 1/6α2O(h3
), (1)
or equivalently,
T.E.SHull
DOF=10
p=3
≤ T.E.DG
DOF=36
p=2
,
T.E.SHull
DOF=10
p=3
≤ T.E.CG
DOF=19
p=2
,
Figure 3: hp DOFs comparison
7. Spectral Hulls in Time Only
We first consider agglomerating the time steps (elements) only.
The space is discretized by a given method and kept unchanged.
8. General Volterra System of Integral Equations for PDEs
The general system of nonlinear time-dependent PDEs in the
residual form
v
j=0
σv−j
∂ju
∂tj
= R (u, t) (2)
Take v times integral of both sides and use:
t t
. . .
t
n times
A(ξ)dξ =
1
(n − 1)! t
(t − ξ)n−1
A(ξ)dξ, (3)
to write
u = u0+
v−1
j=0
γjtj
+
t
v
j=1
(t − ξ)(j−1)
(j − 1)!
σju −
(t − ξ)(v−1)
(v − 1)!
R (u, t)
dξ,
(4)
is a system of nonlinear Volterra equations of the second kind with
a nonlinear non-separable kernel K (u, t, σj).
9. General Volterra System of Integral Equations for PDEs
For
u = u0+
v−1
j=0
γjtj
+
t
v
j=1
(t − ξ)(j−1)
(j − 1)!
σju −
(t − ξ)(v−1)
(v − 1)!
R (u, t)
dξ,
(5)
With the following definitions
K (u, t, σj) =
v
j=1
(t − ξ)(j−1)
(j − 1)!
σju −
(t − ξ)(v−1)
(v − 1)!
R (u, t) , (6)
and
˜u0 = u0 +
v−1
j=0
γjtj
, (7)
Eq.(5) can be written as
u = ˜u0 +
t
K (u, t, σj) dξ. (8)
10. Implicit Discrete Picard Iteration
Thus
u = ˜u0 +
t
K (u, t, σj) dξ. (9)
can be discretized using the integration operator S:
I−
a∆t
a + b
(S ⊗
∂Kn
∂u
) un+1
= ˜u0 + ∆t ¯f1 ⊗ K(u0, t0, σj)
+ ∆tS ⊗ Kn
+
a
a + b
∂Kn
∂t
[δt] −
∂Kn
∂u
un
.
Note:
No need to discretize ∂/∂t: Example BDF2
No need to discretize ∂2/∂t2: Example Newmark
Also covers ∂3/∂t3, ∂4/∂t4 ...
The Volterra form constructs a unifying framework of arbitrary
order accuracy in time
11. The concept of space-time amplification factor
Assume the residual vector R(t, u) is not time-dependent and is
linear, i.e., R = Cu. Then, the general form reduces to:
(I − ∆tS ⊗ C) un+1
= u0. (10)
or
u1
u2
...
uM
= (I − ∆tS ⊗ C)−1
u0
u0
...
u0
. (11)
Introduce I1 and I2 sub-section operators:
uM
uM
...
uM
=
I
I
...
I
I2
× [0 0 . . . 0 I]
I1
×
u1
u2
...
uM
, (12)
12. The concept of space-time amplification factor
The first time span, i.e., T:
u1
u2
...
uM
= (I − ∆tS ⊗ C)
−1
I2 × I1
A
×
u0
u0
...
u0
, (13)
The second time span, i.e., 2T:
u1
u2
...
uM
= (I − ∆tS ⊗ C)
−1
I2 × I1
A
× (I − ∆tS ⊗ C)
−1
I2 × I1
A
×
u0
u0
...
u0
,
(14)
Continuing to the s time span yields the amplification matrix (factor):
u1
u2
...
uM
= A s
u0
u0
...
u0
, (15)
13. The concept of space-time amplification factor
Thus
u1
u2
...
uM
= A s
u0
u0
...
u0
, (16)
can be rewritten using eigenvalue decomposition:
u1
u2
...
uM
= RA Λs
A R−1
A
u0
u0
...
u0
, (17)
15. One-dimensional periodic linear convection
∂u
∂t
+
∂f
∂x
= 0, t = [0, T]
u(x, t = 0) = u0 = cos(x), x = [0, 2π], (18)
Combination of T = 2π
10 , 2π, 4π, 16π, 32π, 64π for
M = [8, 19, 50, 200]
Linear flux is used f = cu
FFT in space with N = 20 points This yields the following
semi-discrete form:
∂u
∂t
+
c
∆x
Du = 0,
u(x, t = 0) = u0. (19)
For comparison exact-in-time solution:
u = exp
−c t
∆x
D u0.
26. 2D Wave Propagation in Complex Geometries
x
y
-4 -2 0 2 4 6
-4
-2
0
2
4
xy
-4 -2 0 2 4
-4
-2
0
2
4
Figure 15: (Left) Fixed spatial mesh exact Fekete points. (Right) The
adapted FEM-P1 solution having the same interpolation error.
27. 2D Wave Propagation in Complex Geometries
x
y -4 -2 0 2 4
-4
-2
0
2
4
u1
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
-0.05
-0.1
-0.15
-0.2
Figure 16: The space-time solution of time-accurate wave propagation
about the cylinder at t = 1. (Left) exact Fourier-Bessel solution. (Right)
numerical solution.
28. 2D Wave Propagation in Complex Geometries
Figure 17: (Left) exact solution and (Right) space-time DPI at t = 10.
29. 2D Wave Propagation in Complex Geometries
Figure 18: The comparison between space-time DPI (dots) and exact
solution (solid line) at t = 10.
30. 2D Wave Propagation in Complex Geometries
0 100 200 300 400 500 600
10
−20
10
−15
10
−10
10
−5
10
0
Arnoldi Iterations
GmresResidual
Comparision of unpreconditioned and band LU preconditioned Space−Time
Unpreconditioned Matrix Free
band LU preconditioned Matrix Free
Figure 19: Unpreconditioned versus band-LU preconditioned GMRES
solution of space-time DPI of the benchmark problem.
31. 2D Wave Propagation in Complex Geometries
Figure 20: Only spatial reordering was performed. (Left) Original (Right)
Reordered
32. 2D Wave Propagation in Complex Geometries
0 100 200 300 400 500 600
10
−20
10
−15
10
−10
10
−5
10
0
Arnoldi iterations
GmresResidual
The effect of Spatial Reordering Using Cuthill−Mckee algorithm
Natural Ordering − 400 off diagonal band LU preconditioner
Reordered − 400 off diagonal band LU preconditioner
Figure 21: 400 off-diagonals of the exact space-time Jacobian matrix are
stored in the LAPACK band storage format and exact LU is performed
and used as the preconditioner.
34. 2D Wave Propagation in Complex Geometries
Figure 23: The increase in convergence by increasing time points in
space-time formulation.
35. 2D Wave Propagation in Complex Geometries
The method is general; replace cylinder of the previous example
with the following:
36. 2D Wave Propagation in Complex Geometries
Figure 25: The scattered wave from the 30P30N geometry obtained using
high-order exact Fekete elements. The adapted P1 elements are used for
smooth visualization.
37. 2D Wave Propagation in Complex Geometries
Figure 26: The scattered wave from the 30P30N geometry obtained using
high-order exact Fekete elements. The adapted P1 elements are used for
smooth visualization.
38. 2D Wave Propagation in Complex Geometries
Figure 27: The scattered wave from the 30P30N geometry obtained using
high-order exact Fekete elements. The adapted P1 elements are used for
smooth visualization.
40. Master Element to Physical Space Transformation
Example 1:
Figure 28: A curved prism on the
CAD boundary
Reconstruction of the top face of
the prism
xT =
6
j=4
xjψj (r, s) (24)
xL =
3
j=1
xjψj (r, s) (25)
is the reconstructed lower face
which is then projected on the
CAD model to yield the final map
x = t xT + (1 − t) S¯k u , v
(26)
where (u , v , ¯k) = Φ (xL) for xL
given in Eq. (25).
41. Master Element to Physical Space Transformation
Table 1: The closed-form analytical map x = f (xj, r, s, t) for different
types of curved elements
Element Type Closed-Form Analytical Map
Triangle 1−r−s
1−r
S¯k [Φ (rx2 + (1 − r) x1)]
+ rs
1−r
x2 + s x3
Quadrilateral (1 − s) S¯k [Φ (rx2 + (1 − r) x1)]
+rs x3 + s (1 − r) x4
Curved-face Tetrahedron t x4 + (1 − t)
S¯k Φ 3
j=1 xj ψj (Ω(r, t), Ω(s, t))
Curved-edge Tetrahedron t x4 + (1 − t) Ω(s, t)x3 + (1 − Ω(s, t))
S¯k [Φ (x1 + Ω (Ω(r, t), Ω(s, t)) (x2 − x1))]
Curved-face Hexahedron t 8
j=5 xj φj (r, s) + (1 − t)
S¯k Φ 4
j=1 xj φj (r, s)
Curved-face Prism t 6
j=4 xj ψj (r, s) + (1 − t)
S¯k Φ 3
j=1 xj ψj (r, s)
Curved-face Pyramid t x5 + (1 − t) S¯k Φ 4
j=1 xj φj (r, s)
and Ω defined below
Ω(α, β) =
α
1−β
0 ≤ β < 1
α β = 1
(27)
removes the singularity of the tetrahedron element.
42. Physical Space To Parametric Space Map Φ(x)
(a) Step 1 : Surface Triangulation (b) Step 2 : Find Bounding Boxes
Figure 29: Steps required before using the Physical Space To Parametric
Space Map Φ(x)
43. Physical Space To Parametric Space Map Φ(x)
Input: x: Physical point coordinates; Sk=1...kmax : CAD entities; Box:
Bounding box data structure; : CAD tolerance; (u0, v0) Some initial
value for the parametric space; n: Newton’s iteration tolerance
Output: (u , v ): The parametric coordinates of the projected point on the ¯kth
CAD entity; ¯k: The tag of the CAD entity where the projected point
has minimum distance
begin
dmin ← 10 ;
for k ← 1 to kmax do
if x ≥ Box(k).xmin and x ≤ Box(k).xmax then
else
cycle;
end
find the nearest (us
, vs
) by minimizing the distance
d ← Sk (us
, vs
) − x ;
if d ≤ dmin then
dmin ← d;
¯k ← k;
(u , v ) ← (us
, vs
);
end
end
end
44. Results
Figure 30: A fifth-order curved finite element grid; near-boundary
elements are mapped first
49. Improvements in load balancing
0 20 40 60 80 100 120 140 160 180 200
0
50
100
150
200
Number of MPI Processes
SpeedUp
Ideal
METIS
Current Method
4 8 1216 32 44 64 84 96 128 180 200
0
2
4
6
x 10
4
Number of MPI Processes
TotalGridGenerationTime(s)
METIS
Current Method
180 200
(a)
(b)
Figure 35: (a) The speedup of generating a 13th
-order accurate mesh for
civil aircraft geometry. (b) The total computation time (in fraction of the
net bar).
50. Improvements in load balancing
0 100 200 300 400 500
0
500
1000
Process Rank
ProcessTime(s)
METIS
0 100 200 300 400 500
0
500
1000
New Load Balance
Figure 36: Comparison of wall time of 500 processes using two domain
decomposition algorithms measured for 13th
-order accurate mesh for civil
aircraft geometry.
51. Addition of Hulls to the Standard Grid
Once the standard grid is generated, hulls can be added
52. Addition of Hulls to the Standard Grid
Once the standard grid is generated, hulls can be added
Case Tris Hulls Reduction
Cylinder 274 165 %40
Triangle 2758 1442 %48
NACA0012 1820 958 %47
Table 2: The number of triangles in
the original mesh versus the number
of generated dual hulls
Edges
3
4
5
6
7
8
9
10
Figure 38: The dual mesh of a
NACA0012 triangulation.
53. Near-Optimal Convex Tessellation
Objective : We need to tessellate a complex domain with
minimum number of elements.
A p-refinement on such tessellation yields more efficiency
compared to a fine grid FEM. (in the smooth regions)
The minimum number of convex partitions of a domain is an
optimization problem.
However, Hertel-Mehlhorn procedure yields a quick estimation
of the near-optimal number of convex partitions of an
arbitrary domain.
Number of convex hulls = 1+2 + 2 + . . . + 2
r reflex vertices
= 1+2r (28)
54. Near-Optimal Convex Tessellation - Example
(a) Triangulation, 24 tris (b) Quadrilateralization, 7 quads
(c) Hertel-Mehlhorn, 4 hulls (d) Optimum, 2 hulls
Figure 39: The number of convex partitions
56. Near-Optimal Convex Tessellation - Complex Geometry
−8 −6 −4 −2 0 2 4 6
−2
0
2
4
6
8
10
h−refinement of convex hulls for Lake Superior geometry
Figure 41: Lake Superior with further h-refinements
58. Basic Terminology
Definition of Q Space:
fQ =
d
i=1
xi
di
, di = 0, . . . , (ni − 1), (29)
Example : fQ = {1, r, s, rs} for d = 1
Definition of P Space:
fP =
d
i=1
xi
di
,
d
k=1
dk ≤ (n − 1), (30)
Example : fP = {1, r, s} for d = 1; Generates Pascal Triangle for
higher P.
59. Basic Terminology
Candidate Points:
X = {ˆxi} ⊂ Ω
1 ≤ i ≤ M
M N (31)
Figure 42: Example of candidate
points inside a concave hull Ω
Vandermonde Matrix:
VT
= [[f(ˆx1)], [f(ˆx2)], . . . , [f(ˆxM )]] ∈ RN×M
. (32)
60. Basic Terminology
The Moment of Monomials:
mj =
Ω
fj(x)dµ, (33)
Approximate Fekete Points:
Quadrature points are known to yield small Lebesgue constant
Therefore instead of finding Exact Fekete Points we find
Gauss-Legendre quadrature points
Mathematically speaking:
j
wjfi (ˆxj) = VT
w = mi, 1 ≤ i ≤ N, 1 ≤ j ≤ M.
(34)
or simply
VT
w = m. (35)
61. Computation of Moments
The RHS of
j
wjfi (ˆxj) = VT
w = mi =
Ω
fi(x)dµ, 1 ≤ i ≤ N, 1 ≤ j ≤ M.
(36)
can be computed on arbitrary convex/concave domain by
introducing
Fk =
1
d
d
i=1
xi
(di+δik)
di + δik
, (37)
satisfies .F = ∂Fk/∂xk = f
reduces the computational complexity of moment calculations
to one dimension smaller, i.e.
m =
Ω
fdΩ =
∂Ω
Fk ˆnkd∂Ω ≈
∆Ωl
Jl
m k
Fk (˜xlm) ˆnklWlm,
(38)
62. Conditioning the Vandermonde Matrix using a SVD
algorithm
Once the RHS of the following is computed
VT
w = m (39)
We need to condition V before solving the above
Data: Vk=0 is the Vandermode matrix defined in Eq.(32)
Result: Matrix Ps and well-conditioned Vandermonde matrix Vs+1
for k = 0 . . . s do
Vk = UkSkV T
k , Note : Economy size SVD ;
Pk = VkS−1
k ;
Vk+1 = VkPk Note : Vk+1 =
Uk is well-conditioned since Uk is unitary ;
end
Algorithm 2: Reducing the condition number of the Vandermonde
matrix using only one iteration (s = 0)
63. Advantages of the SVD algorithm
QR-based algorithm of Sommariva and Vianello (Sommariva
and Vianello, 2009) needs at least 2 iterations (the rule of
“twice is enough”, see Ref. (Giraud et al., 2005))
Algorithm 2 needs only one evaluation!
Therefore, the new algorithm yields a closed-form relation for
approximate Fekete points, i.e., solve for indices of the
following:
UT
0 w = µ, if w = 0, (40)
where µ = PT m and P = P0 = V0S−1
0 .
This underdetermined sensitivity problem can be solved using:
Orthogonal Matching Pursuit (OMP) (Mallat and Zhang,
1993; Bruckstein et al., 2008).
QR-based sorting of strong weights and selecting the first N
most influential weights
64. Method I : Fill Pattern Method
Figure 43: The generation of the candidate points using the fill pattern
method.
69. Example 1 : Approximation of Fekete points
Figure 48: Step 1 : fill candidate points
70. Example 1 : Approximation of Fekete points
Figure 49: Step 2: Solve for UT
0 w = µ, if w = 0
71. Example 2 : Comparison with QR-based Reference Method
10
0
10
1
10
2
10
−8
10
−6
10
−4
10
−2
10
0
10
2
Multivariate Polynomial Degree
L2
(error)
SVD method (current)
QR method (reference)
(a)
0 5 10 15 20 25 30
10
−2
10
0
10
2
10
4
10
6
10
8
10
10
10
12
Multivariate Polynomial Degree
Σ
i
w
i
, QR
Σi
wi
, SVD
|w|2
, QR
|w|2
, SVD
ΛT
, QR
ΛT
, SVD
Σi
wi
= 4
(b)
Figure 50: The comparison between SVD and QR approaches to find
approximate Fekete points using only one iteration. a) The interpolation
error versus polynomial degree. b) Various measures.
72. Nodal Spectral Hull Basis Functions
A polynomial of degree at most N in d-dimensional space can be
represented by
ψj (x1, x2, . . . , xd) = 1, x1, x2
1, . . . , x2, x2
2, . . . , xd, x2
d, . . . (1,N)
either fP orfQ
[a]j(N,1)
= faj. (41)
Evaluating at XF = ˆxi, i = 1 . . . N yields the nodal basis functions
Ψ = [[ψ1], [ψ2], . . . , [ψN ]](N,N) =
[f(ˆx1)](1,N)
[f(ˆx2)](1,N)
...
[f(ˆxN )](1,N)
(N,N)
[a](N,N).
(42)
or in compact form
Ψ = Va, (43)
73. Nodal Spectral Hull Basis Functions
At Approximate Fekete points, we have:
Ψ = Va = I, (44)
Replace Vandermonde with its full SVD, i,e.,
USV T
a = I, → a = V S−1
UT
. (45)
Therefore for x ∈ Ω, nodal basis are
ψj = [ψ1, ψ2, . . . , ψN ](1,N) = [f(x)](1,N)V S−1
UT
(46)
74. Modal Spectral Hull Basis Functions
The modal basis are
¯ψj = ψjU = [f(x)](1,N)V S−1
, (47)
For a function u(x) for x ∈ Ω ⊂ Rd
u =
km
k=1
¯ψkwk, 1 ≤ (km = kmax) ≤ N, (48)
where wk are the generalized Fourier coefficients (GFCs).
Theorem
¯ψj forms an orthogonal set of basis functions for i = 1 . . . N.
Theorem
GFCs wk decay.
75. Modal Spectral Hull Basis Functions
Theorem
The GFCs of orthogonal hull expansion
u = km
k=1
¯ψkwk, 1 ≤ (km = kmax) ≤ N, are given by
w = UT
u (49)
This constitutes Generalized Discrete Transform similar to
Discrete Fourier Transform (DFT) by multiplying the given
function u with the unitary matrix U.
The truncated GFCs using w1...km = UT
(N,1...km)u yields a
generalized error estimator. The norm of the eliminated tail,
i.e. UT
(N,km+1...N)u
2
is an error estimator of the sum of the
eliminated energy according to Parseval theorem.
76. Calculating the Lebesgue constant
The interpolation operator
I(u) = LN u =
km
k=1
¯ψkwk. (50)
It can be shown that
E (I(u)) ≤ (1 + LN ) u − u∗
, (51)
where u∗ is the optimal interpolation and the Lebesgue constant is
ΛN (T) = LN . (52)
It is shown in the full paper
LN max =
Ω (fT f) dΩ
σmin
(53)
77. Lebesgue Constant Remains Small for Spectral Hull Basis
LN max =
Ω (fT f) dΩ
σmin
The transformation of the physical hull leads to Ω
fT
f dΩ ≤ 1.
x1
x2
a−a
a−a
Ω
(a) 2D (b) 3D
The SVD-based algorithm (2) forces the Vandermonde matrix to mimic
the unitary matrix U. Hence, σmin is around unity. As a combined
result, the Lebesgue constant is small.
78. The approximation theory of the spectral hull expansion
The normalized version of the orthogonal basis ¯ψk = f
V(k)
σk
is
˜ψk =
f
f 2
V(k), (54)
Theorem (Parseval theorem for orthogonal ¯ψj)
Ω
m
k=1
¯ψkwk
2
dΩ = f
2
2
m
k=1
wk
σk
2
. (55)
Theorem (Parseval theorem for the orthonormal ˜ψk)
Ω
m
k=1
˜ψk ˜wk
2
dΩ =
m
k=1
˜w2
k. (56)
79. The approximation theory of the spectral hull expansion
Theorem (Weierstrass Approximation Theorem for Ω arbitrary
subset of Rd)
Assume that u is a bounded real-valued function on Ω ⊂ Rd then
for every > 0, there exists a polynomial p such that for all x ∈ Ω,
u − p < . Particularly, we also give an explicit relation for the
polynomial p below.
p =
Ω
˜ψ˜kudΩ ˜ψ˜k,
˜k = perm sort↓
Ω
f1udΩ, . . . ,
Ω
fN udΩ
× V(1), V(2), . . . , V(N) (57)
80. The approximation theory of the spectral hull expansion
Proof Sketch.
Use the Parseval’s Theorem 5, multiplying u = ˜ψk ˜wk with ˜ψl
Ω
˜ψludΩ =
Ω
˜ψl
˜ψk ˜wkdΩ = δlk ˜wk = ˜wl (58)
Therefore
u = ˜ψk ˜wk = ˜ψk
Ω
˜ψkudΩ , (59)
which is the Gram-Schmidt process for the error vector u⊥ which is
normal to the span of all orthonormal hull basis functions, i.e.
u =< u, ˜ψ1 > ˜ψ1+ < u, ˜ψ2 > ˜ψ2+. . . + < u, ˜ψN > ˜ψN +u⊥ (60)
we need to show that the magnitude of the projections < u, ˜ψk >
can be made monotonically decreasing.
81. The approximation theory of the spectral hull expansion
Proof Sketch.
Write
Ω
˜ψkudΩ = [u(˚x1)µ1, u(˚x2)µ2, . . .]
˜ψk(˚x1)
˜ψk(˚x2)
.
.
.
= [u(˚x1)µ1, u(˚x2)µ2, . . .]
f1(˚x1)
f 2
f2(˚x1)
f 2
. . .
fN (˚x1)
f 2
f1(˚x2)
f 2
f2(˚x2)
f 2
. . .
fN (˚x2)
f 2
.
.
.
.
.
.
.
.
.
.
.
.
× V(1), V(2), . . . , V(N) . (61)
The first two terms of the RHS of Eq. (61) can be combined to
Ω
˜ψkudΩ =
1
f 2
∞
i=1
u(˚xi)f1(˚xi)µi,
∞
i=1
u(˚xi)f2(˚xi)µi, . . . V(1), V(2), . . . , V(N) , (62)
or
Ω
˜ψkudΩ =
1
f 2 Ω
f1udΩ,
Ω
f2udΩ, . . . ,
Ω
fN udΩ V(1), V(2), . . . , V(N) . (63)
82. The approximation theory of the spectral hull expansion
Proof Sketch.
The vector
W =
Ω
f1udΩ,
Ω
f2udΩ, . . . ,
Ω
fN udΩ , (64)
exists and is finite. Therefore
Ω
˜ψkudΩ =
1
f 2
[W1, W2, . . . , WN ] V(1), V(2), . . . , V(N) , (65)
can be made monotonically decreasing by a matching pursuit procedure. First define the vector product of the W
with the kth
column of the unitary matrix V as below
˜Wk = W.V(k), k = 1 . . . N. (66)
This is the projection of W into the unitary space (matrix) V . Then find the permutation of integer indices k by
sorting the result of vector product as follows
˜k = permutation sort↓( ˜W ) . (67)
Therefore ˜k is always monotonically decreasing.
83. The approximation theory of the spectral hull expansion
wm = σm max u
2
f
2
, ˘u
˜w˜m
N → ∞
Mode m
Amplitudewm
Figure 51: The decay of Generalized Fourier Coefficients (GFCs).
84. Numerical results of spectral hull basis functions
Figure 52: The orthogonal spectral hull basis Eq. (47) evaluated on
Chebyshev points. (a) ¯ψ1, (b) ¯ψ2, (c) ¯ψ60, (d) last mode, i.e., ¯ψ400.
85. Numerical results of spectral hull basis functions
Figure 53: The spectral filtering of u = cos 4π x2 + y2 in Q space by
eliminating higher frequency orthogonal hull basis Eq. (48). (a)
km = 400, i. e. full rank, (b) km = 200, i. e. half rank, (c) km = 360, i.
e. ninety percent rank, (d) Exact
86. Numerical results of spectral hull basis functions
(a) (b) (c) (d)
Figure 54: The first and the 200th shape functions of nodal (top row)
and modal (bottom row) Approx. Fekete basis. (a) ψ1, (b) ψ200. (c) ¯ψ1.
(d) ¯ψ200.
87. Numerical results of spectral hull basis functions
0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
x
0.6
0.4
0.2
0.0
0.2
0.4
0.6
1.338
0.869
0.400
0.069
0.538
1.007
1.476
(a) 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
x
0.6
0.4
0.2
0.0
0.2
0.4
0.6
y
(b)
Figure 55: Comparison of spectral hull basis functions and RBF (with
same DOF) in the reconstruction the solution. (a) RBF exponential
Kernel. (b) Spectral Hull Basis ¯Ψ.
88. Numerical results of spectral hull basis functions
1.5 1.0 0.5 0.0 0.5 1.0 1.5
x
1.5
1.0
0.5
0.0
0.5
1.0
1.5
0.856
0.571
0.285
0.000
0.285
0.571
0.856
(a)
1.5 1.0 0.5 0.0 0.5 1.0 1.5
x
1.5
1.0
0.5
0.0
0.5
1.0
1.5
y
(b)
Figure 56: Comparison of spectral hull basis functions and RBF (same L2
error) in the reconstruction the solution. (a) RBF exponential Kernel. (b)
Spectral Hull Basis ¯Ψ.
89. Numerical results of spectral hull basis functions
1.0
0.5
0.0
0.5
1.0
y
Figure 57: Resolution of six wavelengths of u = sin(2πx) sin(2πy)
90. Numerical results of spectral hull basis functions
(a) (b)
Figure 58: Superlinear convergence of spectral hull basis on the concave
T-hull. (a) Reconstructed function using 16th
-order spectral hull basis
constructed on approximate Fekete points. (b) L2 error compared to
analytical reconstruction.
91. Numerical results of spectral hull basis functions
0 10 20 30 40 50
0
0.5
1
1.5
2
2.5
3
3.5
4
Approx. Fekete Modes From 1 to 49
GeneralizedFourierAmplitude
(a)
0 20 40 60 80 100 120 140
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Approx. Fekete Modes from 1 to 121GeneralizedFourierAmplitude
(b)
0 50 100 150 200 250
0
0.5
1
1.5
2
2.5
3
3.5
4
Approx. Fekete Modes From 1 to 256
GeneralizedFourierAmplitude
(c)
Figure 59: The spectra of u = sin(πx) sin(πy) on different
convex/concave elements; a) using 7th
order orthogonal spectral hull
basis ¯Ψ on a quadrilateral element. b) using 11th ¯Ψ on a hexagonal
element. c) using 16th ¯Ψ on a T-shape element.
92. Application to General Conservation Laws and Fluid
Dynamics
The infinite dimensional solution to general conservation laws
Ui,t + Fij,j = 0, (68)
can be projected in finite fQ, fP spaces (Eqs. (29), (30)) by either
nodal expansion U = ψiUi
modal expansion U = ¯ψkWk
Special Case : 2D compressible Euler equations:
For conservative variables Ui = [ρ, ρui, e]
Fij =
ρuj
ρuiuj
(e + P)uj
. (69)
93. Discontinuous Galerkin (DG) Spectral Hull
The weak form of Eq. (68) is represented by
Ω
ωUi,tdΩ =
Ω
ω,jFijdΩ −
∂Ω
ωFijnj, (70)
We use the Van-Leer flux (Anderson et al., 1985). Other fluxes
and Riemann solvers can be used in the same fashion. See (Bassi
and Rebay, 1997) for the original DG formulation.
Ω
ωUi,tdΩ =
Ω
ω,jFijdΩ −
∂Ω
ω F+
ij nj + F−
ij nj (71)
Boundary integrals computed using Gauss-Legendre rule
(degree of exactness 2p + 1)
Interior integrals computed with the following rules (degree of
exactness 2p)
sub-triangulation of the hull and using conventional
Gauss-Legendre rule (p ≤ 20)
the general polygonal quadrature rule by Sommariva et. al.
(Sommariva and Vianello, 2007). Applicable to any degree of
exactness.
94. Discontinuous Least Squares (DLS) Spectral Hull
Semi-discrete form with Chebyshev differentiation matrix D
D ⊗ Ui + Fij,j = f (72)
Replace flux-vector Jacobians
D ⊗ Ui + Aij(k)Uj,k = f (73)
I(Ui) =
1
2 Ω
D ⊗ Ui + Aij(k)Uj,k − f
2
2
dΩ+
1
2 ∂Ω
α [Ui] 2
2d∂Ω,
(74)
δI(Ui) = 0, for any variational δUi ∈ Rd
. (75)
Following Jiang’s notation (Jiang and Povinelli, 1989),
L = D ⊗ + Aij(k)
∂
∂xk
, (76)
δI(Ui) =
Ω
L(δU)T
(L(U) − f) dΩ +
∂Ω
α (δU[U]) d∂Ω = 0,
(77)
95. Discontinuous Least Squares (DLS) Spectral Hull
Substituting δU = ψk=1...N and U = ψkUk yields the nodal spectral hull
DLS
Ω
L(ψi)
T
L(ψj)dΩ Uj −
∂Ω
α (ψi. ψj in) d∂Ω Uj
+
∂Ωk
αψi. ψjneigh(k).d∂Ωk Uneigh(k) j =
Ω
L(ψi)
T
fdΩ.
For δU = ¯ψk and U = ¯ψkUk, the modal spectral hull DLS is obtained.
Defining diagonal and off-diagonal matrices
¯Adiag =
Ω
L(ψi)
T
L(ψj)dΩ −
∂Ω
α (ψi. ψj in) d∂Ω, (78)
and
¯Boff-diag,k =
∂Ωk
αψi. ψjneigh(k)d∂Ωk, (79)
the final form fits in general discontinuous FEM form:
¯AdiagU+
num. of neighs
k=1
¯Boff-diag,kUneigh k = RHS, RHS =
Ω
L(ψi)
T
fdΩ.
(80)
96. Benchmark Problems
To validate the non-confirming p-refinement for the case of Euler
equations, the method of manufactured solutions (MMS) is used.
Figure 60: The numerical solution converges to the analytical
manufactured solution x-momentum u2 = ρu for non-conforming
p-refinement of compressible Euler equations
97. Benchmark Problems
The next benchmark problem is a cylinder in a low Mach number
flow at M∞ = 0.2. An inaccurate discretization of the interior
fluxes, or geometry, can cause severe asymmetry as shown in
Re. (Bassi and Rebay, 1997). The nodal spectral hull basis
evaluated at approximate Fekete points on triangles are used to
discretize Eq. (71). Various RK and implicit schemes were tested
to yield the same steady-state solution.
100. Benchmark Problems
Linearized Acoustics (zero mean flow)
Aij(1) =
0 ρ0 0
c2
0
ρ0
0 0
0 0 0
, Aij(2) =
0 0 ρ0
0 0 0
c2
0
ρ0
0 0
. (81)
Figure 64: The tessellation of the domain and the interpolation points.
(Left) Lagrange points on triangles. (Right) Approximate Fekete points
on quadrilaterals obtained from the agglomeration of triangles.
101. Benchmark Problems
The exact manufactured solution is shown with blue circles
Figure 65: The discontinuous least squares spectral hull solution of linear
acoustic problem (81) using Lagrange basis on triangle elements.
102. Benchmark Problems
The exact manufactured solution is shown with blue circles
Figure 66: The discontinuous least squares spectral hull solution of linear
acoustic problem (81) using Hull basis on the quad elements
104. Benchmark Problems
M∞ = 0.2, α = 0.0 deg.
Figure 68: The comparison between Continuous Galerkin Spectral
Element solution of Potential Flow using exact Fekete basis functions
(Left) with DG Spectral Hull solution of compressible Euler with
approximate Fekete basis functions
108. Benchmark Problems
Figure 72: Directly in physical space. Pseudocolor plot at step = 19.
∆t = 10−4
.just before blow-up!!!
109. Benchmark Problems
This is not a bug in the code!!!
Issue can be resolved with insights obtained from Lebesgue
constant
Write:
LN max =
Ω (fT f) dΩ
σmin
(82)
For p = 8 hulls in the middle, the value of the Lebesgue constant
is:
LN max =
Ω (1, x, . . . , x8, . . . , y8)T (1, x, . . . , x8, . . . , y8) dΩ
σmin
≈
Ω (x16) dΩ
σmin
≈
Ω
(x16) dΩ
is very large in the physical space far from
origin !!!
110. Benchmark Problems
Figure 73: hp-Spectral Hull solution of subsonic/supersonic vortex
convection in compressible Euler equations. Note the hulls that track the
vortex have order of accuracy p = 8 while the rest of the hulls are
discretized using p = 4 spectral hull basis on approximate Fekete points.
111. Benchmark Problems
Figure 74: hp-Spectral Hull solution of subsonic/supersonic vortex
convection in compressible Euler equations. Note the hulls that track the
vortex have order of accuracy p = 8 while the rest of the hulls are
discretized using p = 4 spectral hull basis on approximate Fekete points.
112. Benchmark Problems
Figure 75: hp-Spectral Hull solution of subsonic/supersonic vortex
convection in compressible Euler equations. Note the hulls that track the
vortex have order of accuracy p = 8 while the rest of the hulls are
discretized using p = 4 spectral hull basis on approximate Fekete points.
113. Benchmark Problems
Figure 76: hp-Spectral Hull solution of subsonic/supersonic vortex
convection in compressible Euler equations. Note the hulls that track the
vortex have order of accuracy p = 8 while the rest of the hulls are
discretized using p = 4 spectral hull basis on approximate Fekete points.
114. Benchmark Problems
M∞ = 0.2, parallel implementation
Figure 77: hp/DG - P2 in the aft-region
115. Benchmark Problems
M∞ = 0.2, parallel implementation
Figure 78: hp/DG - P4 in the aft-region
117. Benchmark Problems
M∞ = 0.2, parallel implementation
Figure 80: hp/Spectral Hull - P5 in the aft-region,
DOF = (5 + 1)(5 + 2)/2 = 21 in P space and DOF = (5 + 1)2
= 36
in Q space
119. Benchmark Problems
M∞ = 0.2, parallel implementation, contours
levels= [0.6879, 0.7750, 0.8621, 0.9492, 1.036]
Figure 82: hp/Spectral Hull - P5 in the
aft-region,DOF = (5 + 1)(5 + 2)/2 = 21 in P space and
DOF = (5 + 1)2
= 36 in Q space
120. Conclusions
Reducing DOF of modern finite element methods is
investigated using a systematic hp-process.
The elements are either agglomerated (h-coarsening) or a hull
grid is directly used and then the polynomial degree of the
hull basis, is increased (p-refinement).
Compared to the conventional continuous/discontinuous
FEM, this mechanism yields more accurate solutions with
smaller DOF.
This methodology is validated using Discontinuous Galerkin
(DG) and Discontinuous Least-Squares.
The spectral properties and convergence is proven by
satisfying the Weierstrass approximation theorem.
121. Conclusions
The approximate Fekete points can be computed once for a
polygonal master element and then tabulated to yield efficient
implementation.
(a) e = 3, p = 9 (b) e = 4, p = 11 (c) e = 8, p = 12 (d) e = 16, p = 20
Figure 83: The approximate Fekete points on a master polygonal hull.
Tabulated in a subroutine hull-basis(d,e,p, XF , U,S,V)
122. Conclusions
Table 3: Comparision of different numerical methods for solving
conservation laws
FD FV Cheby. CG-FEM Spectral Discontinuous FEM Spectral
Fourier (SUPG, Element DG, DLS, Hull
LSFEM, ...) Mortar Element, ...
Spectral Accuracy
Complex Geometry
Easy h-refinement
Easy p-refinement
Imp./Explicit Time
Minimum Degree
of Freedom
Quadrature
Free
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