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Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid

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Page 1: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

Part 3: Introduction to Meshless

Methods in Heat Transfer and Fluid

Page 2: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 3: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 4: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 5: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 6: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 7: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 8: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 9: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

Radial-basis Function Interpolation

9

Assume a general field variable may be interpolated in terms of a finite

number of expansion functions as:

Where:

Page 10: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 11: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

Radial-basis Function Interpolation

11

In all cases, the Euclidean distance is defined as:

With:

Page 12: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 13: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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.

Page 14: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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.

Page 15: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 16: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 17: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 18: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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:

Page 19: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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.

Page 20: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 21: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 22: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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.

Page 23: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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.

Page 24: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 25: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

Radial-basis Function Interpolation

25

• Therefore, for numerical reasons, it becomes important to mitigate this

issue by:

• Efficient preconditioning

• Domain decomposition

• Localized expansion

Page 26: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 27: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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:

Page 28: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 29: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 30: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

30

Exact

1-Region

2-Region

4-Region

Domain Decomposition RBF Meshless Solution of PDE

Page 31: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

31

8-Region

16-Region

32-Region

64-Region

Domain Decomposition RBF Meshless Solution of PDE

Page 32: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 33: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 34: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 35: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 36: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 37: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 38: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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.

Page 39: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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:

Page 40: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 41: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 42: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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:

Page 43: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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.

Page 44: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 45: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 46: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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.

Page 47: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 48: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 49: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 50: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 51: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 52: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

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

Page 53: Part 3: Introduction to Meshless Methods in Heat Transfer and Fluid · 2018-09-26 · Meshless Methods have emerged from the FEM and computational community, Although these methods

Solution of NS for Compressible Flow

Meshless Point distribution was very refined

53

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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

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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

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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

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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

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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

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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

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• 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

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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

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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

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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

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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

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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

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Numerical Experiment

1-D Inviscid Burgers Equation

Constant Shape Parameter Variable Shape Parameter

66

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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

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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

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Numerical Experiment

2D Advection Equation Diagonal Wave

Constant High Value

Shape Parameter

Dispersive

Blended ApproachConstant Low Value

Shape Parameter

Diffusive

69

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70

EOL