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Part 3: Introduction to Meshless
Methods in Heat Transfer and Fluid
Summary
Introduction
Radial-Basis Function (RBF) Interpolation
Global RBF Meshless Solution of PDE (GRBF)
Localized RBF Meshless Solution of PDE (LRBF)
Blended shape factor
Conclusions
2
Introduction
In traditional numerical methods a Mesh or
Grid is required in order to make
assumptions for the local approximation of
the field variables and/or its derivatives on
the boundary and in the interior of the
domain of interest.
3
Introduction
Meshing constitutes the most time-
consuming and man-power-demanding
aspect of a numerical analysis especially
for Fluid Flow problems where the
numerical solution highly depends on the
quality of the mesh.
4
Introduction
Since the early 1990’s a number of
Meshless Methods have emerged from the
FEM and computational community,
Although these methods are called Mesh-
Free or Element-Free, a background mesh
or shadow elements is often necessary for
integration purposes, rendering the
methods Not Truly Meshless.
5
Introduction
Parallel to the development of these
techniques, a different class of methods
emerged based on interpolation and
collocation of global shape functions:
Method of Fundamental Solutions
Radial-Basis Function Collocation Method***
6
Introduction
These methods offer as their best feature
the ability to globally represent a field
variable in a Truly Meshless way, with no
requirements for background meshes,
point structure, or polygonalization.
7
Introduction
However, as these methods rely on global interpolation
functions, large fully-populated, non-diagonally
dominant, ill-conditioned matrices arise in their
implementation, and therefore, special care must be
taken in the selection and formulation of such
interpolation functions as well as in the solution of the
resulting algebraic systems.
8
Radial-basis Function Interpolation
9
Assume a general field variable may be interpolated in terms of a finite
number of expansion functions as:
Where:
Radial-basis Function Interpolation
10
The expansion functions may be defined to belong to the family of
Radial-Basis functions (RBF). Such functions consist of algebraic
expressions uniquely defined in terms of the Euclidean distance from
an ‘expansion point’ or ‘data center’ to a general field point. Some
examples of these functions are:
i) Polyharmonics Splines RBF:
ii) Multiquadrics RBF:
iii) Gaussian RBF:
n=1 inverse MQ
n=2 MQ
Radial-basis Function Interpolation
11
In all cases, the Euclidean distance is defined as:
With:
Radial-basis Function Interpolation
12
Furthermore, the expansion coefficients, α, may be determined by
least-squares or direct collocation of the known field variable at discrete
locations or ‘data centers’.
For this purpose, assume a finite number points
NB are used as ‘data centers’ on the boundary of a domain o
NI are used as ‘data centers’ inside the domain of interest:
Boundary data center
Internal data center
Global RBF Meshless Solution of PDE
13
Let us now implement the global RBF interpolation to directly
approximate the solution of PDE, this the Kansa Method.
For example, assume 2D steady-state advection-diffusion of energy in
a medium with constant thermophysical properties and no internal
energy generation, governed by:
With general Boundary Conditions:
E.J. Kansa, Multiquadrics – A scattered data approximation scheme with applications to computational
fluid-dynamics (I), Comput. Math.Appl. 19 (1990) 127–145.
Global RBF Meshless Solution of PDE
14
Where:
- T is the temperature field
- u and v are the x and y components of the velocity field
- is the density
- c is the specific heat capacity
- k is the thermal conductivity
- g1, g2, and g3 are coefficients that set the type and value of the
boundary conditions.
Global RBF Meshless Solution of PDE
15
The temperature T(x) can be globally interpolated over NB boundary
data centers and NI internal data centers:
Introducing this expansion into the generalized boundary conditions:
Which can be reduced to:
Applied over i=1…NB
data centers
Global RBF Meshless Solution of PDE
16
Applying the governing equation to the expansion yields:
Which can be reduced to:
Applied over i=1…NI
data centers
17
2
2
1
1
j
jr
c
Using the inverse multiquadric:
3 22
2 2
3 22
2 2
3 22
2 2
5 22 2
2
2 2 2
1
1
1
12 1
/
/
/
/
( )
(y )
j j j
j
j j j j j
j
j j j j j
j
j j
j
r r
r c c
r x x r
x r x c c
r y r
y r y c c
r r
c c c
The following are needed
And for a boundary condition with a normal derivative
3 22
2 2
11
/
( ) (y )
j j j
x y
j
j x j y
n nn x x
rx x n y n
c c
Global RBF Meshless Solution of PDE
18
Collocating the expanded boundary condition equation at the NB
boundary data centers and the expanded governing equation at the NI
internal data centers leads to a square linear algebraic set for the
expansion coefficients as:
Where:
And:
Global RBF Meshless Solution of PDE
19
Testing this approach in a problem of fully-developed flow between
heated parallel plates at constant temperature. The problem is solved
with a commercial CFD package (Fluent 6.0) using a FVM mesh with
4825 nodes. The Meshless RBF collocation is done over a point
distribution of 110 boundary points and 250 internal points.
Global RBF Meshless Solution of PDE
20
The temperature contour plots for the two solution approaches are
shown below revealing good accuracy on a relatively coarse data
center distribution
T: 0 10 20 30 40 50 60 70 80 90 100
Global RBF Meshless Solution of PDE
21
The temperature profiles after a 1/4, 1/2, 3/4, and full-length are shown
below for the Fluent CFD and Meshless solutions as well as the heat
flux at the bottom plate as a function of position along the channel.
T [K]
y[m
]
270 280 290 300 310 320 330 340 350 360 3700
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
1/4 CFD
1/4 Meshless
1/2 CFD
1/2 Meshless
3/4 CFD
3/4 Meshless
1/1 CFD
1/1 Meshless
Global RBF Meshless Solution of PDE
(0,0) (3,0)
(3,3) (9,3)
(9,6)(3,6)
(3,9)(0,9)
T = 0 q = -1
T = 0
T = 0
T = 0q = -1
T = 0 q = -1
(0,0) (3,0)
(3,3) (9,3)
(9,6)(3,6)
(3,9)(0,9)
T = 0 q = -1
T = 0
T = 0
T = 0q = -1
T = 0 q = -1
22
The Global RBF Meshless approach is tested now for robustness in an
irregular solid region with a regular and irregular point distribution for a
steady heat conduction problem.
Global RBF Meshless Solution of PDE
23
The temperature fields are shown below revealing that the fidelity of the
approximation is not lost by the irregularity of the point distribution.
Global RBF Meshless Solution of PDE
Despite the apparent accuracy and robustness
of the Global RBF Meshless approach the
issues of ill-conditioning and high memory and
CPU power demands become more notorious
as the size of the problem increases and
obvious when dealing with 3D large-scale
problems.
Proper selection of the RBF expansion functions
shape parameter c is an area of current
research.
24
Radial-basis Function Interpolation
25
• Therefore, for numerical reasons, it becomes important to mitigate this
issue by:
• Efficient preconditioning
• Domain decomposition
• Localized expansion
Domain Decomposition RBF Meshless Solution of PDE
One approach that can be followed to mitigate some of these issues consists on domain decomposition.
By properly implementing Domain Decomposition along with an effective iteration scheme that guarantees continuity and smoothness of field variables across interfaces, independent algebraic systems can be formed effectively reducing the number of degrees of freedom, CPU and memory requirements.
26
Domain Decomposition RBF Meshless Solution of PDE
Boundary
Point
Internal
Point
++ +
++ +
1 2 3 4
1(1) 1(2) 1(3) 1(4)
3(1) 3(2) 3(3) 3(4)
4(1) I4(2) II
4(3) III4(4) I
2(1) 2(4) II2(2) III
2(3)
27
The Domain Decomposition approach can be summarized as:
28
Guaranteeing field variable continuity and smoothness across the
interfaces can be accomplished by imposing the following interface
conditions along an iteration process until convergence is satisfied
through an iterative norm:
Domain Decomposition RBF Meshless Solution of PDE
29
To verify this approach consider the following heat conduction problem
in a long rectangular medium with available exact solution:
T = 0T = 0
q = -1q = -1
Domain Decomposition RBF Meshless Solution of PDE
30
Exact
1-Region
2-Region
4-Region
Domain Decomposition RBF Meshless Solution of PDE
31
8-Region
16-Region
32-Region
64-Region
Domain Decomposition RBF Meshless Solution of PDE
32
Number of Sub-Domains
CP
Utim
e(s
)
0 16 32 48 6410
0
101
102
103
104
Number of Sub-Domains
Sto
rag
e(M
B)
0 16 32 48 640
20
40
60
80
100
120
Number of Sub-Domains
RA
MR
eq
uir
em
en
t(k
B)
0 16 32 48 640
10000
20000
30000
40000
50000
60000
Domain Decomposition RBF Meshless Solution of PDE
The Domain Decomposition scheme adapted to
the Global RBF Meshless method effectively
reduces CPU and memory demands as well as
the size of the resulting algebraic systems.
However, special care most be taken in its
implementation as the effectiveness of the
iteration process may depend on the artificial
decomposition of the domain which in place
requires user intervention.
33
Domain Decomposition RBF Meshless Solution of PDE
3. Localized RBF Meshless Solution of PDE
An alternative approach is proposed consisting on RBF
interpolation over localized topologies of influence
points, pioneered by Prof. Sarler and his group.
This approach allows for optimization of the interpolation
(selection of shape parameter c) as well as CPU and
memory demands.
In addition, no user intervention is necessary as the point
distribution and generation of localized topologies is
completely automated.
This approach can be implemented in an iterative time-
stepping process.
34
Localized RBF Meshless Solution of PDE
35
The localized topology of NF influence points is automatically
generated around each data center xc.
xc
Topology Data Center xc
Topology Influence Points
Localized RBF Meshless Solution of PDE
36
The Localized RBF interpolation is based on the selection of localized
topologies of influence points as follows (in ALMA2D – can be found at
fbm.centecorp.com) :
. Surface is modeled by quadratic sub-parametric (constant) BEM patches
. Each has an outward drawn normal associated with it.
. No data point is at a corner
Localized RBF Meshless Solution of PDE
37
The RBF interpolation of a function (x) is performed over NF influence
points in the topology of a data center xc. In addition, a series of NP
polynomials Pj(x) may be added to the expansion to ensure exact
interpolation of constant and linear fields and solvability of the
equations*
*Karageorghis, A., Chen, C.S., and Smyrlis, Y., (2006) Applied Numerical Mathematics
Localized RBF Meshless Solution of PDE
38
The expansion coefficients may be determined as:
Where:
Note here [C] the is directly the RBF interpolation matrix unlike the global approach.
Localized RBF Meshless Solution of PDE
39
Now, let us assume that we require to compute any derivative of the
test function at the data center xc. The linear derivative operator can be
applied to the expansion as follows:
Introducing the expansion coefficients in this expression:
Where:
Localized RBF Meshless Solution of PDE
40
Therefore, the computation of any linear differential operator (or
integral) applied over the field variable at the data center xc can be
performed by a simple vector-vector multiplication of a pre-generated
and stored vector and the field variable values in the topology of
influence as:
1
1
1
( , )
.
.
( , )
( , )
( , )
.
.
( , )
T
c c
NF c c
c c
c c
N c c
x y
x
x y
x y xC
P x yx
x
P x y
x
• For example,
1
1
1
( , )
.
.
( , )
{ x }( , )
.
.
( , )
T
c c
NF c c
T
c
c c
N c c
x y
x
x y
xC
P x y
x
P x y
x
Localized RBF Meshless Solution of PDE
41
• Can be applied to a time marching scheme, say for the convection-diffusion
equation
1
{T} {T} {T}T T Tn n n n n n n
i c i c i c
p
kT T t L u x v y
c
2
p
T T Tc u v k T
t x y
Localized RBF Meshless Solution of PDE
42
The automatic point distribution can be accomplished using a quadtree
(octree) scheme to produce the point clustering around high-gradient
areas:
Localized RBF Meshless Solution of PDE
43
Once the point distribution is setup all the localized topologies can be
generated by using a inflating ball scheme to ensure all data centers
are properly surrounded: here the scheme collects 9 points
. Increase constant to r=2.85rmin for ball to collect 25 points.
nj
j
Boundary Point
Internal Shadow Point
Internal Point
• For normal derivative, shadow points are introduced to carry out
finite differencing in the normal direction (layers of shadow points for
higher order) using RBF interpolated values at the boundary point.
( , )b b b Shadow
b Shadow
x y
n r
( , ) Tb bx y
nn
Localized RBF Meshless Solution of PDE
• Shadow points positioned
half the local density.
Point collocation
44
Example
45
Buoyancy-driven flow of liquid aluminum in a 1.25 x 5 cm
rectangular cavity. The left-hand wall is kept at 960 K and the
right-hand wall is kept at 920 K, while the top and bottom walls
are kept insulated. The liquid aluminum has a constant thermal
expansion coefficient β= 0.000117 K-1, yielding a Rayleigh
number Ra= 4,394 based on the width.
Meshless 41X161 points
FVM 101 x 401
46
LCMM for Hemodynamics:
Unsteady Flow
Standard
Optimal
•Unsteady flow with a femoral artery pulsatile flow
waveform inflow and the Carreau non-Newtonian
model:6 (mm)
4 (mm)45º
Outflow (80%)
Inflow
Outflow (20%)
0
100
200
300
400
500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
cycle time (sec)
flo
w r
ate
(m
l/m
in)
FemoralWaveform
t1
t2
t3 t4
0.00 0.02 0.03 0.05 0.07 0.08 0.10 0.12 0.14
0.00 0.02 0.03 0.05 0.07 0.08 0.10 0.12 0.14
0.00 0.05 0.11 0.16 0.21 0.26 0.32 0.37 0.42
0.00 0.05 0.11 0.16 0.21 0.26 0.32 0.37 0.42
t1 t2
LCMM
FVM
0
200
400
600
800
1000
0.064 0.066 0.068 0.070 0.072 0.074
Artery Axial Distance (m)
SW
SS
G (
N/m
3)
Standard ETSDA
Optimal 1 ETSDA
Optimal 2 ETSDA
The averaged spatial wall shear stress gradient
(SWSSG) and temporal wall shear stress gradient
(TWSSG) in conventional model have been reduced by
58% and 35%
El-Zahab, Divo, E., and Kassab, A.J, "Minimization of the Wall Shear Stress Gradients in Bypass Grafts Anastomoses using Meshless CFD and Genetic Algorithms Optimization“ Computer Methods in
Biomechanics and Biomedical Engineering, 2010, Vol. 13, No. 1, pp. 35-47.
El-Zahab, Divo, E., and Kassab, A.J., A Meshless CFD Approach for Evolutionary Shape Optimization of Bypass Grafts Anastomoses, Inverse Problems in Engineering and Science, 2009, Vol. 17, No. 3,
pp. 411–435.
Utilizes RBF interpolations to “fill in the gap”
with finite difference formulations.
Enables working with a relatively more
ordered point distribution.
RBF-Enhanced Finite Differencing
Upstream Point
V w1c
w2e
c
w1
w2
e
47
If a required stencil node is missing, build a local topology
about that point and interpolate using RBFP or MLS
Interpolations may be rolled into shape function
formulation to minimize overhead
Allows for utilization of existing node data, as well as
straightforward upwinding
RBF Enhanced Finite Differencing
48
Influence Points
Virtual Points
Data Point
Freedom to use first, second, third order upwinding finite difference formulations or other upwind techniques
One dimensional Stencil
Model integrated meshless method• Overlapping domain
decomposition
• Point adaption on native geometry
• RBF-enhanced RBF
Gerace, S., Erhart, K., Kassab, A., and Divo, E., "A Model-Integrated Localized Collocation Meshless Method for Large Scale Three
Dimensional Heat Transfer Problems," Engineering Analysis, 2014, Vol. 45, pp. 2–19.
Temperature Along Midline of
Bottom Surface
isocontour T = 600K
Commercial 1 code
Structured Mesh
476,889 cells 499,350 nodes
Commercial 2 code
Unstructered Mesh
594,804 cells 113,593 nodes
Meshless
4 Refinement Stages
98,903 nodes
49
Application to the Solution of NS for Compressible Flow
RBF enhanced Finite Difference Meshless Method facilitates implementation for compressible flow of
the Advection Upstream Splitting Method (AUSM) scheme:
Convective terms computed by splitting in two components and manipulating to produce a “Mach-
number weighted average” value, which are evaluated at half-node locations
In this manner, the convected quantities can be upwinded appropriately:
i+1i-1 ii-½ i+½
Upstream Point
V w1c
w2e
c
w1
w2
e50
Solution of NS for Compressible Flow
_________________________________________________________________________________________________________________
NACA-0012 Airfoil
Angle of attack: α = 10o
Mach= 0.8
Transonic case
Mach contours Pressure coefficient
(cp=p/0.5𝛒V∞2)
Gerace, S. , Erhart, K., Divo, E., and Kassab, A.,"Adaptively Refined Hybrid FDM/Meshless Scheme with Applications to
Laminar and Turbulent Flows," CMES: Computer Modeling in Engineering and Science, 2011, Vol. 81, no.1, pp. 35-68. 51
Solution of NS for Compressible Flow
NACA-0012 Airfoil
Angle of attack: α = 10o
Mach= 2.0
Supersonic case
(shock is present)
Mach contoursPressure coefficient
(cp=p/0.5𝛒V∞2)
shock is somewhat smeared52
Solution of NS for Compressible Flow
Meshless Point distribution was very refined
53
Choosing c parameter: c-adaptation RBF interpolation accuracy depends on the shape parameter
For interpolation of smooth functions
Large value shape parameter
High conditioning number interpolation matrix. On the verge of ill-conditioned
leads to exponential convergence of interpolation (Cheng, A. H.-D., Golberg, M.
A., Kansa, E. J., and Zammito, G., 2003)
Large value shape parameter provides better accuracy
54
For interpolation of discontinuous functions
Small value shape parameter
High conditioning number interpolation matrix. On the verge of ill-
conditioned.
Small value shape parameter prevents oscillations and provides better
accuracy
55
Example:
• Three pre-computed RBF’s with 20 collocation points equally
distributed:
f1 with c1=0.1 corresponds to conditioning of ~ 102
f2 with c2= 0.5 corresponds to conditioning of ~ 107
f3 with c3=1.0 corresponds to conditioning of ~ 1012
1( ) tan of x A x x
A=1
ω=1
56
A=1 and ω=1 1( ) tan of x A x x
1
2
3
2 2 34.756%
2 2 2.907%
2 2 0.111%
L d fxx
L d fxx
L d fxx
1
2
3
2 2.968%
2 0.188%
2 0.010%
L dfx
L dfx
L dfx
2
1
( ) ( )1
2( ) ( )
max min
Ni k i
i
k
i i
df x df x
N dx dxL dfx
df x df x
dx dx
22 2
2 21
2 2
2 2
( ) ( )1
2 2( ) ( )
max min
Ni k i
i
k
i i
d f x d f x
N dx dxL d fxx
d f x d f x
dx dx
f1 c1=0.1 K~102
f2 c2=0.5 K~107
f3 c3=1.0 K~1012
L2 Error norms
Higher c is best! 57
A=1 and ω=10 1( ) tan of x A x x
2
1
( ) ( )1
2( ) ( )
max min
Ni k i
i
k
i i
df x df x
N dx dxL dfx
df x df x
dx dx
22 2
2 21
2 2
2 2
( ) ( )1
2 2( ) ( )
max min
Ni k i
i
k
i i
d f x d f x
N dx dxL d fxx
d f x d f x
dx dx
f1 c1=0.1 K~102
f2 c2=0.5 K~107
f3 c3=1.0 K~1012
L2 Error norms
1
2
3
2 5.395%
2 3.039%
2 89.968%
L dfx
L dfx
L dfx
1
2
3
2 2 10.913%
2 2 4.423%
2 2 257.198%
L d fxx
L d fxx
L d fxx
Lowest c is best!Lowest c is best! 58
f1 c2=0.707 K~105
f2 c3=1.41 K~1012
Lowest c is best!
1 1( , ) tan tanx o y of x y A x x y y
f
fa
2 10.137%
2 10.138%
2 72.861%
L dfx
L dfy
L Lf
2 1.855%
2 1.855%
2 7.408%
L dfx
L dfy
L Lf
fa
f2
f1
L2 Error norms
L2 Error norms
01 , 5 , 0x y oA x y
Example:
59
• Selection of an appropriate RBF shape parameter, c, not
only depends on the distribution of the collocation
points but it also depends on the function being
interpolated.
• As quality of the expansion becomes field-dependent,
RBF collocation loses the advantage of pre-building
optimized interpolating operators based exclusively on
the point distribution.
• One approach to mitigate this issue is to formulate the
interpolation operators as a blend between multiple
expansions.
Blended Interpolation
60
61
We can create (pre-compute) RBF interpolation functions each
with preset shape parameter values and blend each by a factor 𝝓
Introducing two interpolation functions and the blended function.
𝒇𝒂 𝒙 =
𝒋=𝟏
𝑵
𝜶𝒂𝒋 𝝍𝒂𝒋 𝒙
𝒇𝒃 𝒙 =
𝒋=𝟏
𝑵
𝜶𝒃𝒋 𝝍𝒃𝒋 𝒙
𝒇𝒄 𝒙 = 𝟏 − 𝝓 𝒇𝒂 𝒙 + 𝝓𝒇𝒃 𝒙
We need a form of 𝒇𝒂 𝒙 and 𝒇𝒃 𝒙 in terms of 𝒇 𝒙
Blended Interpolation
We can form a system of equations using the collocation points 𝒙𝒊
𝒋=𝟏
𝑵
𝜶𝒂𝒋 𝝍𝒂𝒋 𝒙𝒊 = 𝒇(𝒙𝒊)
𝒋=𝟏
𝑵
𝜶𝒃𝒋 𝝍𝒃𝒋 𝒙𝒊 = 𝒇 𝒙𝒊
and solve for the weights 𝜶𝒂 and 𝜶𝒃
𝝍𝒂 𝜶𝒂 = 𝒇 ⟹ 𝜶𝒂 = 𝝍𝒂−𝟏 𝒇
𝝍𝒃 𝜶𝒃 = 𝒇 ⟹ 𝜶𝒃 = 𝝍𝒃−𝟏 𝒇
62
Blended Interpolation
We can blend between the two
functions𝒇𝒄 𝒙 = 𝟏 − 𝝓 𝒇𝒂 𝒙 + 𝝓𝒇𝒃 𝒙
𝒇𝒄 𝒙𝒊 = 𝟏 − 𝝓 𝝍𝒂(𝒙𝒊)𝑻 𝝍𝒂
−𝟏 𝒇 + 𝝓 𝝍𝒃(𝒙𝒊)𝑻 𝝍𝒃
−𝟏 𝒇
𝒇𝒄 𝒙𝒊 = 𝟏 − 𝝓 𝝍𝒂(𝒙𝒊)𝑻 𝒇 + 𝝓 𝝍𝒃(𝒙𝒊)
𝑻 𝒇
Discontinuous
Function
Smooth Function
63
Localized Collocation Meshless Method
• Applies also to derivatives for which we can compute
interpolation vectors for low and high shape parameters
and blend as needed by
𝜕𝑓 𝑥, 𝑦
𝜕𝑥=
𝜕𝑓𝐿 𝑥, 𝑦
𝜕𝑥+ 𝜙
𝜕𝑓𝐻 𝑥, 𝑦
𝜕𝑥−𝜕𝑓𝐿 𝑥, 𝑦
𝜕𝑥
𝜕𝑓 𝑥, 𝑦
𝜕𝑦=
𝜕𝑓𝐿 𝑥, 𝑦
𝜕𝑦+ 𝜙
𝜕𝑓𝐻 𝑥, 𝑦
𝜕𝑦−𝜕𝑓𝐿 𝑥, 𝑦
𝜕𝑦
• The blending parameter 𝝓 is calculated using flux
limiter approach
– Successive Gradient
– Minmod Limiter
𝑟𝑖 =𝑢𝑖 − 𝑢𝑖−1𝑢𝑖+1 − 𝑢𝑖
𝜙 = max(0,min 𝑟, 1 )
64
Numerical Experiment
1-D Inviscid Burgers Equation
Use the inviscid Burgers
equation as a model
equation
𝜕𝑢(𝑥, 𝑡)
𝜕𝑡+ 𝑢 𝑥, 𝑡
𝜕𝑢 𝑥, 𝑡
𝜕𝑥= 0
BC: 𝑢 0, 𝑡 = 1
IC: 𝑢 𝑥, 0 =
1 𝑓𝑜𝑟 𝑥 ≤ 0.50 𝑓𝑜𝑟 𝑥 > 0.5
65
Numerical Experiment
1-D Inviscid Burgers Equation
Constant Shape Parameter Variable Shape Parameter
66
Numerical Experiment
1-D advection equation as a model equation
𝜕𝑢(𝑥,𝑡)
𝜕𝑡+ 𝑎
𝜕𝑢 𝑥,𝑡
𝜕𝑥= 0 𝑎 = 1
BC: 𝑢 0, 𝑡 = 0
IC: 𝑢 𝑥, 0 = 1 𝑓𝑜𝑟 0.1 ≤ 𝑥 < 0.40 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
• RBF Interpolation
– Low Shape Parameter• Stable
• Dissipative
• Similar to low order schemes
– High Shape Parameter• Dispersive –Overshoots/Oscillations
• Can lead to instabilities
• Similar to high order schemes
– RBF Blended Interpolation• Blends between low and high shape
parameter RBF interpolation
• Steep gradients are detected using
the successive gradients
67
Numerical Experiment
2D Advection Equation Diagonal Wave
𝜕𝑢
𝜕𝑡+ 𝑈1
𝜕𝑢
𝜕𝑥+ 𝑈2
𝜕𝑢
𝜕𝑦= 0
𝑢 𝑥, 0, 𝑡 = 2 𝑓𝑜𝑟 𝑥 ≤ 0.2
𝑢 𝑥, 0, 𝑡 = 1 𝑓𝑜𝑟 𝑥 > 0.2
𝑢 0, 𝑦, 𝑡 = 2 𝑓𝑜𝑟 𝑦 ≤ 0.2
𝑢 0, 𝑦, 𝑡 = 1 𝑓𝑜𝑟 𝑦 > 0.2
𝑢 𝑥, 0 = 0
Where 𝑈1 =2
2,𝑈2 =
2
2
Inlet: u = 2
Outlet
Inlet: u =1
Inlet: u = 2
Inlet: u =1
Outlet
68
Numerical Experiment
2D Advection Equation Diagonal Wave
Constant High Value
Shape Parameter
Dispersive
Blended ApproachConstant Low Value
Shape Parameter
Diffusive
69
70
EOL