1

Icosahedral mesh simulatorTSU 07. 08. 2012.

K. Natobaidze N. Qozashvili T. Aslanishvili

N. Chumbadze

B. Tsutskhvashvili G. Bakradze

5th Georgian-German School and Workshop inBasic Science, 2012

5th Georgian-German School and Workshop in Basic Science

2

Task

icosahedron

5th Georgian-German School and Workshop in Basic Science

𝑢′ (𝑡 )= 𝑓 ( 𝑡 ,𝑢 (𝑡 ) )

3

Tools5th Georgian-German School and

Workshop in Basic Science

4

Userswhere is ‘start’

button ?

5th Georgian-German School and Workshop in Basic Science

5

MethodEulerClass

Fields

data1data2data3data4data5ficomaxValuevEndvStart

Methods

~MethodEulerdrawPointdrawPointsdrawSolutionPointsgetNeighbourSuminitinitMatrixMethodEulerminimizeForDrawsolvesolveMatrix

5th Georgian-German School and Workshop in Basic Science

6

Features5th Georgian-German School and

Workshop in Basic Science

7

Features

min area max area min value max value

5th Georgian-German School and Workshop in Basic Science

8

Features5th Georgian-German School and

Workshop in Basic Science

9

Effects5th Georgian-German School and

Workshop in Basic Science

10

Effects5th Georgian-German School and

Workshop in Basic Science

11

Effects5th Georgian-German School and

Workshop in Basic Science

12

OpenGL

glutCreateWindow()

glutInitWindowSize(x,y)

glutInitWindowPosition(x,y)

5th Georgian-German School and Workshop in Basic Science

13

OpenGL

glutDisplayFunc(display)• glutReshapeFunc(reshape)•

5th Georgian-German School and Workshop in Basic Science

14

OpenGL

mouse ‘listener’ keyboard ‘listener’

5th Georgian-German School and Workshop in Basic Science

15

Structure

start point

end point

5th Georgian-German School and Workshop in Basic Science

16

Structure5th Georgian-German School and

Workshop in Basic Science

17

Subdivision1) divide container

2) add row

3) add column

5th Georgian-German School and Workshop in Basic Science

18

Subdivision5th Georgian-German School and

Workshop in Basic Science

19

Neighbours5th Georgian-German School and

Workshop in Basic Science

20

ProblemDifferential Equation on the Sphere• tє(0,T] •u u0 is known

now later

?!

5th Georgian-German School and Workshop in Basic Science

21

•

stencil

Differential equation

𝑢𝑖𝑗

𝑢𝑖𝑗(1) 𝑢𝑖𝑗 (2)

𝑢𝑖𝑗(3)𝑢𝑖𝑗 (6 )

𝑢𝑖𝑗 (5) 𝑢𝑖𝑗(4 )

5th Georgian-German School and Workshop in Basic Science

22

Discretized analog –Explicit Euler scheme

• Where , , is known.• is time step.

5th Georgian-German School and Workshop in Basic Science

23

Other Numerical Methods•explicit Runge–Kutta methods

5th Georgian-German School and Workshop in Basic Science

24

•Other explicit Runge–Kutta methods

SecondOrderapproximation

ThirdOrderapproximation

),( 001 yxfk

1002 3

1,31 hkyhxfk

2003 3

2,32 hkyhxfk

32101 4

3041 kkkhyy

3101 4

341 kkhyy

FourthOrderapproximation

432101

3004

2003

1002

001

226

,21,

21

21,

21),(

kkkkhyy

hkyhxfk

hkyhxfk

hkyhxfk

yxfk

5th Georgian-German School and Workshop in Basic Science

25

Future plans • Local mesh refinement• Introduce different stencils• Extend developed software to concrete equations • Solve concrete problems

5th Georgian-German School and Workshop in Basic Science

26

Thank you გმადლობთ

Special thanks to Ramaz Botchorishvili , Tinatin Davitashvili and Nino

Tkheshelashvili for their support

5th Georgian-German School and Workshop in Basic Science

27

waiting for Batumi5th Georgian-German School and

Workshop in Basic Science