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An application of FEM to the geodetic boundary value problem. Z . F ašková, R. Čunderlík. Faculty of Civil Engineering Slovak University of Technology in Bratislava, Slovakia. Formulation of mixed geodetic BVP Potential theory Geodetic BVP Mixed geodetic BVP Numerical experiments in ANSYS - PowerPoint PPT Presentation
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An application of FEM to the An application of FEM to the geodetic boundary value geodetic boundary value
problemproblem
ZZ. . FFašková, R. Čunderlíkašková, R. Čunderlík
Faculty of Civil EngineeringSlovak University of Technology in Bratislava, Slovakia
Formulation of mixed geodetic BVPFormulation of mixed geodetic BVP Potential theoryPotential theory
Geodetic BVPGeodetic BVP
Mixed geodetic BVPMixed geodetic BVP
Numerical experiments in ANSYSNumerical experiments in ANSYS Global Quasigeoid ModelGlobal Quasigeoid Model
Gravity (acceleration) g(x):Gravity (acceleration) g(x):
Potential theory - Gravity fieldPotential theory - Gravity field
Gravity potential W(x):Gravity potential W(x):
( ) ( ) ( )
( ) ( )
( ) ( ' )
g c
g
c
W x V x V x x Earth
V x gravitational potential Newton formula
V x centrifugal potential Earth s spinvelocity
( ) ( ) ( ) ( )g cg x W x g x g x x Earth
The Earth
Potential theory – Normal fieldPotential theory – Normal field
Normal bodyNormal body (equipotential ellipsoid, spheroid) (equipotential ellipsoid, spheroid) is given by :is given by :
-- major semi axes major semi axes a
- geopotential coefficient - geopotential coefficient J2,0 (flattening)(flattening)
- geocentric gravitational constant - geocentric gravitational constant GM
- spin velocity - spin velocity
Normal potential U(x):Normal potential U(x):
Normal gravity Normal gravity (x)(x)
( ) ( ) ( )g cU x U x U x x Normal body
( ) ( )x U x x Normal body
Ellipsoid
Disturbing potential T(x):Disturbing potential T(x):
Gravity anomaly Gravity anomaly g(x): g(x):
Gravity disturbance Gravity disturbance g(x)g(x)
3( ) ( )( ) ( ) ( ) g gx U xT x W x U x V x R
Potential theory - Disturbing fieldPotential theory - Disturbing field
( ) ( ) ( ) ( ) ( )g x g x x W x U x
0 0( ) ( ) ( ) ( ) ( )g x g x x W x U x
Ellipsoid
Geodetic BVPGeodetic BVP
3( ) 0 ,T x x R
' '0 0
'0'0
( ) ( )
( ) ( )PQP Q
T P T P
Q QN
t n
Real Earth`sSurface
Telluroid
Geoid
EllipsoidQ´
P´
Q
Q
P
0
0
0
N
Quasigeoid
P´´0
Q´0́
elnq
Height anomaly and geoidal heightHeight anomaly and geoidal height
Stokes-Helmert concept (1849)Stokes-Helmert concept (1849)
Molodenskij concept (1960)Molodenskij concept (1960)
( ) 2( ) ( )
T xg x T x
r R
' '0 0
'0( ) ( ) ( )g Pg QP
( )( )) (g Pg QP
Mixed geodetic BVPMixed geodetic BVP
2
3
1 1
2
( ) 0 ,
: ( ) ( ) at | | ,
: ( ) 0 at | | .
T x x R
T x g x x R
T x x R
2
The air
R1
R2 1
2
2
( )
R radiusof the sphere
thenormal body
R radiusof anartificial
boundary
Numerical experiments in ANSYSNumerical experiments in ANSYS
3D elements (15600 elements) with base 5° * 5° – 1221 nodes
1 2
1
2
( 80,80)
(0,180), (180,360)
6371
20000
B
L L
km
km
Surface gravity disturbancesSurface gravity disturbances
generated fromgenerated from EGM-96 EGM-96
geopotentential geopotentential
coefficientscoefficients by using by using
programprogram f477b f477b
Potential solutionPotential solution
Quasigeoidal heightsQuasigeoidal heights
Comparison of solution with BEMComparison of solution with BEM
Differences between solutionsDifferences between solutions
Thanks for your attentionThanks for your attention