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