Faculty of Civil Engineering, Slovak University of Technology in Bratislava

Dept. of Mathematics and Descriptive Geometry (1) &

Dept. of Theoretical Geodesy (2)

R. Čunderlík (2), Z. Fašková (1), J. Janák (2), K. Mikula (1), M. Šprlák (2)

An Application of Finite Element

Method to the Geodetic

Boundary Value Problem

Formulation of mixed geodetic BVP

Motivation for the quasigeoid determination

Potential theory

Mixed geodetic boundary value problem

Numerical experiments in ANSYS

Global Quasigeoid Model

Local Quasigeoid Model

Leveling – the main idea

Global Positioning System

Global Positioning System

Gravity (acceleration) g(x):

Potential theory - Gravity field

Gravity potential W(x):

The Earth

g c

g

c

W(x) V (x) V (x) x Earth

V (x) gravitationalpotential (Newton formula)

V (x) centrifugalpotential (Earth spin velocity)

g cg(x) W(x) g (x) g (x) x Earth

Potential theory - Normal field

Normal body (equipotential ellipsoid, spheroid) is given by :

Normal potential U(x):

Normal gravity g(x) Ellipsoid

2,0

semimajor axis a

geopotentialcoefficient J

geocentricgravitationalconst. GM

spin velocity

g cU(x) U (x) U (x) x Normalbody

(x) U(x) x Normalbodyg

Disturbing potential T(x):

Gravity disturbance dg(x):

Gravity anomaly Dg(x):

Potential theory - Disturbing field

Ellipsoid

g g

3(x) U (x)T(x) W(x) U(x) V x R

0g(x) g(x) (x )gD

g(x) g(x) (x)d g

Mixed geodetic BVP

i

EGM 96 1

r

2

3, ,6

EGM 96

T(x) 0 x

Tg (x) on

n

T(x) 0 on

T0

n

or on

T(x) T (x)

d

D

ANSYS - Global Quasigeoid Model

3D elements with base 5° x 5°

1 2

1

2

( 80,80)

(0,180), (180,360)

6371km

20000km

Input data - Surface gravity disturbances

generated from

EGM-96 geopotential

coefficients by using

program f477b

Disturbing potential solution

Quasigeoidal heights

Comparison of solution

Differences between solutions

Convergence of solution

Surface gravity disturbances in Europe

generated from

EGM-96 geopotential

coefficients by using

program GRAFIM

European Quasigeoid Model

Disturbing potential solution

Quasigeoidal heights, step 0.5°

FEM

EGM-96

BEM

Res = (EGM-96)-(FEM)

E

U

R

O

P

A

Quasigeoidal heights, step 1°

FEM

EGM-96

BEM

Res = (EGM-96)-(FEM)

A

S

I

A

FEM BEM

EGM-96 Res = (EGM-96)-(FEM)

I

N

D

O

N

E

S

I

A

FEM BEM

EGM-96 Res = (EGM-96)-(FEM)

NORTH AMERIKA

FEM BEM

EGM-96 Res = (EGM-96)-(FEM)

S O U T H

A M E R I K A

THANK YOU FOR ATTENTION