Abstract
The numerical approach for solving the fixed gravimetric boundary value problem (FGBVP) based on the finite element method (FEM) with mapped infinite elements is developed and implemented. In this approach, the 3D semi-infinite domain outside the Earth is bounded by the triangular discretization of the whole Earth’s surface and extends to infinity. Then the FGBVP consists of the Laplace equation for unknown disturbing potential which holds in the domain, the oblique derivative boundary condition (BC) given directly at computational nodes on the Earth’s surface, and regularity of the disturbing potential at infinity. In this way, it differs from previous FEM approaches, since the numerical solution is not fixed by the Dirichlet BC on some part of the boundary of the computational domain. As a numerical method, the FEM with finite and mapped infinite triangular prisms has been derived and implemented. In experiments, at first, a convergence of the proposed numerical scheme to the exact solution is tested. Afterwards, a numerical study is focused on a reconstruction of the harmonic function (EGM2008) above the Earth’s topography. Here, a special discretization of the Earth’s surface which is able to fulfil the conditions that arise from correct geometrical properties of finite elements, and it is suitable for parallel computing is implemented. The obtained solutions at nodes on the Earth’s surface as well as nodes that lie approximately at the altitude of the GOCE satellite mission have been tested.
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All datasets generated and/or analysed within the experiments are available from the corresponding author.
References
Becker, J.J., Sandwell, D.T., Smith, W.H.F., Braud, J., Binder, B., Depner, J., Fabre, D., Factor, J., Ingalls, S., Kim, S.H., Ladner, R., Marks, K., Nelson, S., Pharaoh, A., Trimmer, R., Von Rosenberg, J., Wallace, G., Weatherall, P.: Global Bathymetry and Elevation Data at 30 Arc Seconds Resolution: SRTM30 PLUS. Marine Geodesy 32(4), 355–371 (2009)
Bettess, P.: Infinite elements. International Journal for Numerical Methods in Engineering 11(1), 53–64 (1977)
Bettess, P.: More on infinite elements. International Journal for Numerical Methods in Engineering 15(11), 1613–1626 (1983)
Bjerhammar, A., Svensson, L.: On the geodetic boundary value problem for a fixed boundary surface. A satellite approach. Bulletin Géodésique 57(1–4), 382–393 (1983)
Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Springer-Verlag, New York (2002)
B. Bucha, J. Janák, A MATLAB-based graphical user interface program for computing functionals of the geopotential up to ultra-high degrees and orders. Computers & Geosciences 56, 186-196 http://dx.doi.org/10.1016/j.cageo.2013.03.012 (2013)
Čunderlík, R., Mikula, K., Mojžeš, M.: Numerical solution of the linearized fixed gravimetric boundary-value problem. Journal of Geodesy 82, 15–29 (2008)
Čunderlík, R., Mikula, K.: Direct BEM for high-resolution gravity field modelling. Studia Geophysica et Geodaetica 54(2), 219–238 (2010)
Čunderlík, R., Mikula, K., Špir, R.: An oblique derivative in the direct BEM formulation of the fixed gravimetric BVP. International Association of Geodesy Symposia 137, 227–231 (2012)
Damjanic, F., Owen, D.R.J.: Mapped infinite element in transient thermal analysis. Computers & Structures 19, 673–687 (1984)
Droniou, J., Medl’a, M., Mikula, K.: Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modeling. Journal of Computational Physics 398, 108876 (2019). https://doi.org/10.1016/j.jcp.2019.108876
Fašková, Z., Čunderlík, R., Janák, J., Mikula, K., Šprlák, M.: Gravimetric quasigeoid in Slovakia by the finite element method. Kybernetika 43(6), 789–796 (2007)
Z. Fašková, Numerical Methods for Solving Geodetic Boundary Value Problems. PhD Thesis, SvF STU, Bratislava, Slovakia, (2008)
Fašková, Z., Čunderlík, R., Mikula, K.: Finite element method for solving geodetic boundary value problems. Journal of Geodesy 84(2), 135–144 (2010)
Freeden, W., Kersten, H.: A constructive approximation theorem for the oblique derivative problem in potential theory. Mathematical Methods in Applied Sciences 3, 104–114 (1981)
Holota, P.: Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation. Journal of Geodesy 71, 640–651 (1997)
Kaiming, X., Zhijun, Z.: Three-dimensional finite/infinite elements analysis of fluid flow in porous media. Applied Mathematical Modelling 30(9), 904–919 (2006)
W. Keller, Finite differences schemes for elliptic boundary value problems. Section IV Bulletin IAG, No. 1. (1995)
Klees, R.: Boundary value problems and approximation of integral equations by finite elements. Manuscripta Geodaetica 20, 345–361 (1995)
Koch, K.R., Pope, A.J.: Uniqueness and existence for the geodetic boundary value problem using the known surface of the earth. Bulletin Géodésique 46, 467–476 (1972)
Kumar, P.: Infinite elements for numerical analysis of underground excavations. Tunnelling and Underground Space Technology 15(1), 117–124 (2000)
Lehmann, R., Klees, R.: Numerical solution of geodetic boundary value problems using a global reference field. Journal of Geodesy 73, 543–554 (1999)
G.M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, ISBN: 978-981-4452-32-8 (2013)
M. Macák, R. Čunderlík, K. Mikula, Z. Minarechová, An upwind-based scheme for solving the oblique derivative boundary-value problem related to the physical geodesy. Journal of Geodetic Science, Volume 5, Issue 1, (2015)
Macák, M., Čunderlík, R., Mikula, K., Minarechová, Z.: Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S 14(3), 987–999 (2021)
M. Macák, K. Mikula, Z. Minarechová, Solving the oblique derivative boundary-value problem by the finite volume method, ALGORITMY 2012, 19th Conference on Scientific Computing, Podbanske, Slovakia, September 9-14, 2012, Proceedings of contributed papers and posters, Publishing House of STU, 75-84 (2012)
M. Macák, Z. Minarechová K. Mikula, A novel scheme for solving the oblique derivative boundary-value problem, Studia Geophysica et Geodaetica 58(4): 556-570, (2014)
Macák, M., Minarechová, Z., Čunderlík, R., Mikula, K.: The finite element method as a tool to solve the oblique derivative boundary value problem in geodesy. Tatra Mountains Mathematical Publications. 75(1), 63–80 (2020)
Marques, J.M.M.C., Owen, D.R.J.: Infinite elements in quasi-static materially nonlinear problems. Computers & Structures 18(4), 739–751 (1984)
MATLAB version 2018 b, Natick, Massachusetts: The MathWorks Inc., 2018
Medl’a, M., Mikula, K., Čunderlík, R., Macák, M.: Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography. Journal of Geodesy 92, 1–19 (2018)
P. Meissl, The use of finite elements in physical geodesy. Report 313, Geodetic Science and Surveying, The Ohio State University, (1981)
Minarechová, Z., Macák, M., Čunderlík, R., Mikula, K.: High-resolution global gravity field modelling by the finite volume method. Studia Geophysica et Geodaetica 59, 1–20 (2015)
Minarechová, Z., Macák, M., Čunderlík, R., Mikula, K.: On the finite element method for solving the oblique derivative boundary value problems and its application in local gravity field modelling. Journal of Geodesy 95, 70 (2021)
D. Mráz, M. Bořík, J. Novotný, On the Convergence of the h-p Finite Element Method for Solving Boundary Value Problems in Physical Geodesy, Freymueller J.T., Sánchez L. (eds) International Symposium on Earth and Environmental Sciences for Future Generations. International Association of Geodesy Symposia, vol 147. Springer, Cham (2016)
Pavlis, N.K., Holmes, S.A., Kenyon, S.C., Factor, J.K.: The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). Journal of Geophysical Research 117, B04406 (2012). https://doi.org/10.1029/2011JB008916
J.N. Reddy, An Introduction to the Finite Element Method, 3rd Edition, McGraw-Hill Education, New York, ISBN: 9780072466850 (2006)
Shaofeng, B., Dingbo, C.: The finite element method for the geodetic boundary value problem. Manuscripta geodaetica 16, 353–359 (1991)
Simoni, L., Schrefler, B.A.: Mapped infinite elements in soil consolidation. International Journal for Numerical Methods in Engineering 24, 513–527 (1987)
G.L.G. Sleijpen, D.R. Fokkema, Bicgstab (l) for Linear Equations Involving Unsymmetric Matrices with Complex Spectrum, http://dspace.library.uu.nl/handle/1874/16827, (1993)
Šprlák, M., Fašková, Z., Mikula, K.: On the application of the coupled finite-infinite element method to the geodetic boundary value problem. Studia Geophysica et Geodaetica 55, 479–487 (2011)
Xia, K., Bai, M.: Modeling fluid flow and solid deformation in naturally fractured reservoirs. The University of Oklahoma, Norman, Rock Mechanics Institute (1998)
Yin, Z., Sneeuw, N.: Modeling the gravitational field by using CFD techniques. Journal of Geodesy 95, 68 (2021). https://doi.org/10.1007/s00190-021-01504-w
Zienkiewicz, O.C., Emson, C., Bettess, P.: A novel boundary infinite element. International Journal for Numerical Methods in Engineering 19, 340–393 (1983)
Zienkiewicz, O.C., Bando, K., Bettess, P., Emson, C., Chiam, T.C.: Mapped infinite elements for exterior wave problems. International Journal for Numerical Methods in Engineering 21, 1229–1251 (1985)
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This work was supported by Grants APVV-19-0460, VEGA 1/0486/20 and VEGA 1/0436/20.
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Open access funding provided by The Ministry of Education, Science, Research and Sport of the Slovak Republic in cooperation with Centre for Scientific and Technical Information of the Slovak Republic
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Marek Macák developed the theory and performed the numerical simulations. Zuzana Minarechová contributed to develop the theory and wrote the manuscript. Lukáš Tomek developed the meshing of the computational domain. Róbert Čunderlík and Karol Mikula devised the project. All authors discussed the results and contributed to the final manuscript.
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Macák, M., Minarechová, Z., Tomek, L. et al. Solving the fixed gravimetric boundary value problem by the finite element method using mapped infinite elements.. Comput Geosci 27, 649–662 (2023). https://doi.org/10.1007/s10596-023-10224-3
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DOI: https://doi.org/10.1007/s10596-023-10224-3