application of fem to the geodetic boundary value problem · faculty of civil engineering, slovak...
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![Page 1: Application of FEM to the geodetic boundary value problem · Faculty of Civil Engineering, Slovak University of Technology in ... R. Čunderlík (2), Z. Fašková (1), ... Application](https://reader031.vdocuments.us/reader031/viewer/2022022523/5b3886667f8b9a1a678b79d6/html5/thumbnails/1.jpg)
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
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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
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Leveling – the main idea
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Global Positioning System
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Global Positioning System
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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
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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
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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
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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
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ANSYS - Global Quasigeoid Model
3D elements with base 5° x 5°
1 2
1
2
( 80,80)
(0,180), (180,360)
6371km
20000km
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Input data - Surface gravity disturbances
generated from
EGM-96 geopotential
coefficients by using
program f477b
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Disturbing potential solution
Quasigeoidal heights
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Comparison of solution
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Differences between solutions
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Convergence of solution
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Surface gravity disturbances in Europe
generated from
EGM-96 geopotential
coefficients by using
program GRAFIM
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European Quasigeoid Model
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Disturbing potential solution
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Quasigeoidal heights, step 0.5°
FEM
EGM-96
BEM
Res = (EGM-96)-(FEM)
E
U
R
O
P
A
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Quasigeoidal heights, step 1°
FEM
EGM-96
BEM
Res = (EGM-96)-(FEM)
A
S
I
A
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FEM BEM
EGM-96 Res = (EGM-96)-(FEM)
I
N
D
O
N
E
S
I
A
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FEM BEM
EGM-96 Res = (EGM-96)-(FEM)
NORTH AMERIKA
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FEM BEM
EGM-96 Res = (EGM-96)-(FEM)
S O U T H
A M E R I K A
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THANK YOU FOR ATTENTION