Numerical solution of a boundary value problem with effective boundary conditions for calculation of gravity

  • Ivanov Dulus Kh., i.am.djoos@gmail.com M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677000, Russia; Yakutsk Branch of the Regional Scientific and Educational Mathematical Center "Far Eastern Center of Mathematical Research", 48 Kulakovsky Street, Yakutsk 677000, Russia
  • Vabishchevich Petr N., vabishchevich@gmail.com Nuclear Safety Institute of RAS, 52 B. Tulskaya Street, Moscow 115191, Russia; Academy of Science of the Republic of Sakha (Yakutia), 33 Lenin Ave., Yakutsk 677007, Russia
Keywords: gravitational field, Poisson equation, approximate boundary condition, finite element method

Abstract

Forward modeling of gravity field on the base of boundary-value problem solution is a promising technique against traditional summation methods. To calculate gravity of a body with known physical and geometrical properties, one can firstly solve a boundary-value problem for gravitational potential and then calculate its gradient. This approach is more common, but it requires two operations. Another approach is to solve a boundary value problem formulated directly for gravity itself. The main difficulties for methods based on boundary-value problem solution are proper boundary condition, domain size and domain discretization. For the first approach there are plenty of works dealing with these cases in contrast with the second approach. In this paper, authors discuss and compare boundary conditions of two types: the Dirichlet and Robin, in terms of approximation accuracy for the second approach using the finite element method. Calculation results are presented for a test problem, when the gravitational field is produced by a homogeneous body in the shape of a right rectangular prism. A more effective boundary condition is a Robin type condition derived from a simple asymptotic approximation of the gravitational field by the equivalent point mass.

References


[1]
Barnett C. T., “Theoretical modeling of the magnetic and gravitational fields of an arbitrarily shaped three-dimensional body,” Geophys., 41, No. 6, 1353–1364 (1976).

[2]
Blakely R. J., Potential Theory in Gravity and Magnetic Applications, Camb. Univ. Press, Cambridge (1996).

[3]
Brenner S. C. and Scott L. R., TheMathematical Theory of Finite Element Methods, Springer, New York (2008).

[4]
Butler S. L. and Sinha G., “Forward modeling of applied geophysics methods using Comsol and comparison with analytical and laboratory analog models,” Comput. Geosci., 42, 168–176 (2012).

[5]
Cai Y. and Wang C.-Y., “Fast finite-element calculation of gravity anomaly in complex geological regions,” Geophys. J. Int., 162, No. 3, 696–708 (2005).

[6]
Casenave F., M´etivier L., Pajot-M´etivier G., and Panet I., “Fast computation of general forward gravitation problems,” J. Geodesy, 90, No. 7, 655–675, (2016).

[7]
Chakravarthi V., Ramamma B., and Reddy T. V., “Gravity anomaly modeling of sedimentary basins by means of multiple structures and exponential density contrast-depth variations: A space domain approach,” J. Geol. Soc. India, 82, No. 5, 561–569 (2013).

[8]
Cordell L., “Gravity analysis using an exponential density-depth function; San Jacinto Graben, California,” Geophys., 38, No. 4, 684–690 (1973).

[9]
D’Urso M. G., “Analytical computation of gravity effects for polyhedral bodies,” J. Geodesy, 88, No. 1, 13–29 (2014).

[10]
D’Urso M. G. and Trotta S., “Gravity anomaly of polyhedral bodies having a polynomial density contrast,” Surv. Geophys., 38, No. 4, 781–832 (2017).

[11]
D’Urso M. G., “Gravity effects of polyhedral bodies with linearly varying density,” Celest. Mech. Dyn. Astronomy, 120, No. 4, 349–372 (2014).

[12]
Evans L. C., Partial Differential Equations, 2nd ed., Amer. Math. Soc. (2010) (Grad. Stud. Math.; vol. 19).

[13]
Farquharson C. G. and Mosher C. R. W., “Three-dimensional modelling of gravity data using finite differences,” J. Appl. Geophys., 68, No. 3, 417–422 (2009).

[14]
Garcia-Abdeslem J., “The gravitational attraction of a right rectangular prism with density varying with depth following a cubic polynomial,” Geophys., 70, No. 6, J39–J42 (2005).

[15]
Geuzaine C. and Remacle J.-F., “Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities,” Int. J. Numer. Methods Eng., 79, No. 11, 1309–1331 (2009).

[16]
Gharti H. N., Tromp J., and Zampini S., “Spectral-infinite-element simulations of gravity anomalies,” Geophys. J. Int., 215, No. 2, 1098–1117 (2018).

[17]
Gupta H., Encyclopedia of Solid Earth Geophysics, Springer, Dordrecht (2011).

[18]
Guzman S., Forward Modeling and Inversion of Potential Field Data Using Partial Differential Equations, MSc Thes., Colorado School of Mines (2015).

[19]
Haber E., Holtham E., and Davis K., “Large-scale inversion of gravity gradiometry with differential equation,” in: SEG Technical Program Expanded Abstracts 2014, pp. 1302–1307, Soc. Explor. Geophys. (2014).

[20]
Haji T. K., Faramarzi A., Metje N., Chapman D., and Rahimzadeh F., “Challenges associated with finite element methods for forward modelling of unbounded gravity fields,” UK Assoc. Comput. Mech. Conf. Pap., 16, 1–4 (2019).

[21]
Haji T. K., Faramarzi A., Metje N., Chapman D., and Rahimzadeh F., “Development of an infinite element boundary to model gravity for subsurface civil engineering applications,” Int. J. Numer. Anal. Methods Geomech., 44, No. 3, 418–431 (2020).

[22]
Höschl V. and Burda M., “Computation of gravity anomalies caused by three-dimensional bodies of arbitrary shape and arbitrary varying density,” Stud. Geophys. Geodaet., 25, No. 4, 315–320 (1981).

[23]
Howell L. E., Forward Modeling the Gravitational Field Using a Direct Solution of Poisson’s Equation, MSc Thes., Colorado School of Mines (2013).

[24]
Jahandari H. and Farquharson C. G., “Forward modeling of gravity data using finite-volume and finite-element methods on unstructured grids,” Geophys., 78, No. 3, G69–G80 (2013).

[25]
Jiang L., Liu J., Zhang J., and Feng Z., “Analytic expressions for the gravity gradient tensor of 3D prisms with depth-dependent density,” Surv. Geophys., 39, No. 3, 337–363 (2018).

[26]
Kwok Y.-K., “Gravity gradient tensors due to a polyhedron with polygonal facets,” Geophys. Prospect., 39, No. 3, 435–443 (1991).

[27]
Logg A., Mardal K. A., and Wells G., Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Springer-Verl., Berlin; Heidelberg (2012) (Lect. Notes Comput. Sci. Eng.; vol. 84).

[28]
Long J. and Farquharson C. G., “Three-dimensional forward modelling of gravity data using mesh-free methods with radial basis functions and unstructured nodes,” Geophys. J. Int., 217, No. 3, 1577–1601 (2019).

[29]
Maag E., Capriotti J., and Li Y., “3D gravity inversion using the finite element method,” in: SEG Technical Program Expanded Abstracts 2017, pp. 1713–1717, Soc. Explor. Geophys. (2017).

[30]
Martin R., Chevrot S., Komatitsch D., Seoane L., Spangenberg H., Wang Y., Dufr´echou G., Bonvalot S., and Bruinsma S., “A high-order 3-D spectral-element method for the forward modelling and inversion of gravimetric data-application to the western Pyrenees,” Geophys. J. Int., 209, No. 1, 406–424 (2017).

[31]
May D. A. and Knepley M. G., “Optimal, scalable forward models for computing gravity anomalies,” Geophys. J. Int., 187, No. 1, 161–177 (2011).

[32]
Nagy D., “The gravitational attraction of a right rectangular prism,” Geophys., 31, No. 2, 362–371 (1966).

[33]
Nagy D., Papp G., and Benedek J., “The gravitational potential and its derivatives for the prism,” J. Geodesy, 74, No. 7-8, 552–560 (2000).

[34]
Okabe M., “Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies,” Geophys., 44, No. 4, 730–741 (1979).

[35]
Paul M. K., “The gravity effect of a homogeneous polyhedron for three-dimensional interpretation,” Pure Appl. Geophys., 112, No. 3, 553–561 (1974).

[36]
Plouff D., “Gravity and magnetic fields of polygonal prisms and application to magnetic terrain corrections,” Geophys., 41, No. 4, 727–741 (1976).

[37]
Poh´anka V., “Optimum expression for computation of the gravity field of a homogeneous polyhedral body,” Geophys. Prospect., 36, No. 7, 733–751 (1988).

[38]
Hamayun and Prutkin, I. and Tenzer, R., “The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution, J. Geodesy, 83, No. 12, 1163 (2009).

[39]
Rao, C. V. and Raju, M. L. and Chakravarthi, V., “Gravity modelling of an interface above which the density contrast decreases hyperbolically with depth,” J. Appl. Geophys., 34, No. 1, 63–67 (1995).

[40]
Ren Z., Chen C., Zhong Y., Chen H., Kalscheuer T., Maurer H., Tang J., and Hu X., Exact gravity field for polyhedrons with polynomial density contrasts of arbitrary orders,

[41]
arXiv:1810.11768 (2018).

[42]
Ren Z., Chen C., Zhong Y., Chen H., Kalscheuer T.,Maurer H., Tang J., and Hu X., “Recursive analytical formulae of gravitational fields and gradient tensors for polyhedral bodies with polynomial density contrasts of arbitrary non-negative integer orders,” Surv. Geophys., 41, 695–722 (2020).

[43]
Ren Z., Tang J., Kalscheuer T., and Maurer H., “Fast 3-D large-scale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method,” J. Geophys. Res., Solid Earth, 122, No. 1, 79–109 (2017).

[44]
Saad Y., Iterative Methods for Sparse Linear Systems, SIAM (2003).

[45]
Talwani M.,Worzel J. L., and Landisman M., “Rapid gravity computations for two-dimensional bodies with application to the Mendocino submarine fracture zone,” J. Geophys. Res., 64, No. 1, 49–59 (1959).

[46]
Tsoulis D., “Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals,” Geophys., 77, No. 2, F1–F11 (2012).

[47]
Zhang J. and Jiang L., “Analytical expressions for the gravitational vector field of a 3-D rectangular prism with density varying as an arbitrary-order polynomial function,” Geophys. J. Int., 210, No. 2, 1176–1190 (2017).
How to Cite
Ivanov, D. and Vabishchevich, P. (2021) “Numerical solution of a boundary value problem with effective boundary conditions for calculation of gravity”, Mathematical notes of NEFU, 28(1), pp. 93-113. doi: https://doi.org/10.25587/SVFU.2021.74.56.008.
Section
Mathematical Modeling