Annex 1 Laplace function for a regular grid of nodes

Annex 1 Page 199

Annex 1

Laplace function for a regular grid of nodes

Content

A1.1. Laplace function 200

A1.2. Integration on an edge 205

A1.3. Calculation of A^IJ_j 206

Annex 1 Page 200

A1.1. Laplace function

Let us consider the case of a regular grid of nodes with spacing a₁ =a₂ =a(figure A1.1).

Figure A1.1. A regular grid of nodes.

It is easy to show that the Voronoi polygons associated with these nodes are the squares indicated in figures A1.2.

Figure A1.2. Voronoi polygons of a regular grid of nodes.

X1

a

X2

a

Annex 1 Page 201 Let us calculate the value of the Laplace functions at point X located on an edge AD of a Voronoi polygon (figure A1.3).

The position of X is given by the distance:

2 2

u a a ≤ ≤+

− .

Let ξ =2 −1≤ξ ≤+1 a

u

We get the positions of points A,B,C,D:ξ^A =−1; ξ^B =1− 2; ξ^C =−1+ 2; ξ^D =+1

Delaunay triangles Circumcircles Voronoi polygons Figure A1.3. Calculation of Laplace functions on edge AD.

a u

1 2

3 4

5 6

X A B

C D

a

Annex 1 Page 202 case 1: X between A and B

For ξ^A ≤ξ ≤ξ^B, we get from figure A1.4:

5 2 4

1 2

2

1 =h = a ξ + ξ+

h 4 5

) 2 ( 4

) 1 2

( ² ₂

2

1 + +

+

−

− +

=

= ξ ξ

ξ ξ ξ s a

s

2 4

3 1

2

1 +ξ

=

=h a

h

ξ

ξ ξ

ξ

4 1 ) 1 2

( ^{2 2}

4 3

+

−

= +

= a

s s

) 2 ( 2

1

2 2

2

1 +

−

= +

= ξ

ξ α ξ

α ξ

ξ α ξ

α 2

1

2 2

4 3

−

= +

=

∑

= +

−

= +

4 ,

1 2

2

2 2 4 2

I

I ξ ξ

ξ α ξ

2 4

1

−ξ

= Φ

=

Φ 4

2

4 3

ξ

= + Φ

=

Φ Φ₅ =Φ₆ =0

Point X(X₁,X₂) The natural neighbours of point X(X₁,X₂) Figure A1.4. Calculation of Laplace functions on edge AD, zone AB.

a

a

ξ

1 2

3 4

5 6

X A B

C D

h1 h2

h3 h4

s₁ s2

s3 s4

P

Q

R S

Annex 1 Page 203 case 2: X between B and C

Forξ^B ≤ξ ≤ξ^C, we get from figure A1.5:

5 2 4

1 2

2

1 =h = a ξ + ξ+

h 4 5

) 2 ( 4

) 1 2

( ² ₂

2

1 + +

+

−

− +

=

= ξ ξ

ξ ξ ξ s a

s

2 4

3 1

2

1 +ξ

=

=h a

h s₃ =s₄ =a 1+ξ²

5 2 4

1 ₂

6

5 =h = a ξ − ξ+

h 4 5

) 2 ( 4

) 1 2

( ² ₂

6

5 − +

−

−

− −

=

= ξ ξ

ξ ξ ξ s a

s

) 2 ( 2

1

2 2

2

1 +

−

= +

= ξ

ξ α ξ

α α₃ =α₄ =2

) 2 ( 2

1

2 2

6

5 −

−

= −

= ξ

ξ α ξ

α

4 20 4

2 2

6 ,

1 −

= −

∑

I I

5 ) ( 2 8 1

2 2

1 ξ

ξ ⁻

= − Φ

=

Φ )

5 1 1 2( 1

4 2

3 =Φ = − −ξ

Φ )

5 ( 2 8 1

6 2

5 ξ

ξ ⁺

= − Φ

=

Φ

Point X(X₁,X₂) The natural neighbours of point X(X₁,X₂) Figure A1.5. Calculation of Laplace functions on edge AD, zone BC.

1 2

3 4

5 6

X A B

C D

h₁ h₂

h₃ h₄

s1 s₂

s3 s₄

P

Q

R S

T s₅ U s6

h5 h₆

ξ a

a

Annex 1 Page 204 case 3: X between C and D

For ξ^C ≤ξ ≤ξ^D, we get from figure A1.6:

2 4

3 1

2

1 +ξ

=

=h a

h

ξ

ξ ξ

ξ

4 1 ) 1 2

( ^{2 2}

4 3

+

−

= −

= a

s s 5

2 4

1 2

6

5 =^h = ^a ξ − ξ+

h 4 5

) 2 ( 4

) 1 2

( ² ₂

6

5 − +

−

−

− −

=

= ξ ξ

ξ ξ ξ s a

s

ξ ξ α ξ

α 2

1

2 2

4 3

−

− −

=

= 2( 2)

1

2 2

6

5 −

−

= −

= ξ

ξ α ξ

α

∑

= −

−

= −

6 ,

3 2

2

2 2 4 2

I

I ξ ξ

ξ α ξ

2 0

1 =Φ =

Φ

4 2

4 3

ξ

= − Φ

=

Φ ^{5 6} 4

=ξ Φ

=

Φ

Point X(X₁,X₂) The natural neighbours of point X(X₁,X₂) Figure A1.6. Calculation of Laplace functions on edge AD, zone CD.

ξ

1 2

3 4

5 6

X A B

C D

h₅

h₆

h₃ h4

s₅ s6

s3 s4

E

F

G a H

1

Annex 1 Page 205 Figure A1.7 illustrates the evolution of the Laplace functions with ξ.

^-0.1

0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Figure A1.7. Laplace functions for a regular grid.

A1.2. Integration on an edge

The integrals of these functions on the interval −1≤ξ ≤+1 are easily computed.

Let

25461073 0.14545853

1 2 5

1 2 ln 5

5 ) 1 2 ( 20 5

1 =

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎠

⎞

⎜⎜⎝

⎛

+

−

− + +

−

c =

C (A1.1)

49077854 0.70908293

1 2 5

1 2 ln 5

5 ) 2 3 ( 10 5

1 =

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎠

⎞

⎜⎜⎝

⎛

− +

+ + −

−

m =

C (A1.2)

Then, we find:

Cc

d

d ≡ Φ =

Φ

∫

∫

− +

−

ξ ξ ¹

1 2 1

1

1 (A1.3)

ξ Φ

4

3 =Φ

Φ

2

1 =Φ

Φ Φ₅ =Φ₆

Annex 1 Page 206 Cm

d

d ≡ Φ =

Φ

∫

∫

− +

−

ξ ξ ¹

1 4 1

1

3 (A1.4)

Cc

d

d ≡ Φ =

Φ

∫

∫

− +

−

ξ ξ ¹

1 6 1

1

5 (A1.5)

A1.3. Calculation of A^IJ_j

Consider a Voronoi cell I and the 8 surrounding nodes (figure A1.8)

Figure A1.8. Cell I and surrounding nodes (here I=5).

By definition,

∑ ∫

∫

∫ ∑

= −

=

=

=

=

1

, 1 , , ,

, ,

2 Φ Φ

Φ _{J t}

s r q p t

t j t

s J

c t pqrs

t j I

J I j IJ

j a n dξ

ds n

ds n A

t I

with the following values for n^t_j

Edge t n₁^t n^t₂

p 0 -1

q 1 0

r 0 1

s -1 0

From section A1.1 above, we see that only the terms A^IJ_j for J =1 to 9 will differ from 0.

The calculation of these non zero terms is summarized in the table below.

a

a

1 2 3

4 I=5 6

7 8

p s q

r X1

X2

9

Annex 1 Page 207 Calculation of A^IJ_j for J =1 to 9 with I =5

J I =5 Contribution of edge Total Formula

p q r s n°

J=1

1 1

AI 0 0 0 -

2

aC_c A₁^I1= - 2

aC_c (A1.6.a)

1 2

AI -

2

aC_c 0 0 0 A₂^I1= -

2

aC_c (A1.6.b)

J=2

2 1

AI 0

2

aCc 0 -

2

aCc A₁^I2= 0 (A1.6.c)

2 2

AI -

2

aCm 0 0 0 A₂^I2= -

2

aCm (A1.6.d)

J=3

3 1

AI 0

2

aCc 0 0 A₁^I3=

2

aCc (A1.6.e)

3 2

AI -

2

aCc 0 0 0 A₂^I3= -

2

aCc (A1.6.f)

J=4

4 1

AI 0 0 0 -

2

aCm A₁^I4= - 2

aCm (A1.6.g)

4 2

AI -

2

aCc 0

2

aCc 0 A₂^I4= 0 (A1.6.h)

J=5

5 1

AI 0

2

aCm 0 -

2

aCm A₁^I5= 0 (A1.6.i)

5 2

AI -

2

aCm 0

2

aCm 0 A₂^I5= 0 (A1.6.j)

J=6

6 1

AI 0

2

aCm 0 0 A₁^I6=

2

aCm (A1.6.k)

6 2

AI -

2

aCc 0

2

aCc 0 A₂^I6= 0 (A1.6.l)

J=7

7 1

AI 0 0 0 -

2

aCc 1⁷

AI = - 2

aCc (A1.6.m)

7 2

AI 0 0

2

aCc 0 A₂^I7= C_c (A1.6.n)

J=8

8 1

AI 0

2

aC_c 0 -

2

aC_c A₁^I8= 0 (A1.6.o)

8 2

AI 0 0

2

aC_m 0 A₂^I8= 2

aC_m (A1.6.p)

J=9

9 1

AI 0

2

aC_c 0 0 A₁^I9=

2

aC_c (A1.6.q)

9 2

AI 0 0

2

aC_c 0 A₂^I9= 2

aC_c (A1.6.r)