The finite element method for solving the oblique derivative boundary value problems in geodesy

2020 ◽  
Author(s):  
Marek Macák ◽  
Zuzana Minarechová ◽  
Róbert Čunderlík ◽  
Karol Mikula
Keyword(s):  
Finite Element Method ◽  
Boundary Condition ◽  
Finite Element ◽  
Numerical Experiments ◽  
Boundary Value ◽  
Satellite Orbits ◽  
The Finite Element Method ◽  
Hexahedral Elements ◽  
Grid Points ◽  
Element Method

<p>We present a novel approach to the solution of the geodetic boundary value problem with an oblique derivative boundary condition by the finite element method. Namely, we propose and analyse a finite element approximation of a Laplace equation holding on a domain with an oblique derivative boundary condition given on a part of its boundary. The oblique vector in the boundary condition is split into one normal and two tangential components and derivatives in tangential directions are approximated as in the finite difference method. Then we apply the proposed numerical scheme to local gravity field modelling. For our two-dimensional testing numerical experiments, we use four nodes bilinear quadrilateral elements and for a three-dimensional problem, we use hexahedral elements with eight nodes. Practical numerical experiments are located in area of Slovakia that is given by grid points located on the Earth's surface with uniform spacing in horizontal directions. Heights of grid points are interpolated from the SRTM30PLUS topography model. An upper boundary is in the height of 240 km above a reference ellipsoid WGS84 corresponding to an average altitude of the GOCE satellite orbits. Obtained solutions are compared to DVRM05.</p>

scholarly journals The Finite Element Method as a Tool to Solve the Oblique Derivative Boundary Value Problem in Geodesy

2020 ◽  
Vol 75 (1) ◽  
pp. 63-80
Author(s):  
Marek Macák ◽  
Zuzana Minarechová ◽  
Róbert Čunderlík ◽  
Karol Mikula
Keyword(s):  
Finite Element Method ◽  
Boundary Condition ◽  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
The Finite Element Method ◽  
Gravity Field Modelling ◽  
Field Modelling ◽  
Novel Approach ◽  
Element Method

AbstractIn this paper, we propose a novel approach to approximate the solution of the Laplace equation with an oblique derivative boundary condition by the finite element method. We present and analyse diverse testing experiments to study its behaviour and convergence. Finally, the usefulness of this approach is demonstrated by using it to gravity field modelling, namely, to approximate the solution of a geodetic boundary value problem in Himalayas.

Stable finite element method for solving the oblique derivative boundary value problems in geodesy

2021 ◽  
Author(s):  
Marek Macák ◽  
Zuzana Minarechová ◽  
Róbert Čunderlík ◽  
Karol Mikula
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
The Finite Element Method ◽  
Gravity Field Modelling ◽  
Field Modelling ◽  
Local Gravity ◽  
Local Gravity Field Modelling ◽  
Element Method

<p><span>We presents local gravity field modelling in a spatial domain using the finite element method (FEM). FEM as a numerical method is applied for solving the geodetic boundary value problem with oblique derivative boundary conditions (BC). We derive a novel FEM numerical scheme which is the second order accurate and more stable than the previous one published in [1]. A main difference is in applying the oblique derivative BC. While in the previous FEM approach it is considered as an average value on the bottom side of finite elements, the novel FEM approach is based on the oblique derivative BC considered in relevant computational nodes. Such an approach should reduce a loss of accuracy due to averaging. Numerical experiments present </span><span>(i) </span><span>a reconstruction of EGM2008 as a harmonic function over the extremely complicated Earth’s topography in the Himalayas and Tibetan Plateau, and (ii) local gravity field modelling in Slovakia with the high-resolution 100 x 100 m while using terrestrial gravimetric data.</span></p><p><span>[1] </span>Macák, Z. Minarechová, R. Čunderlík, K. Mikula, The finite element method as a tool to solve the oblique derivative boundary value problem in geodesy. Tatra Mountains Mathematical Publications. Vol. 75, no. 1, 63-80, (2020)</p>

Increasing the Solution Accuracy in the Numerical Modeling of Boundary Value Problems Using Finite Element Method Based on Hankel Shape Functions

2019 ◽  
Vol 11 (07) ◽  
pp. 1950062
Author(s):  
S. Farmani ◽  
M. Ghaeini-Hessaroeyeh ◽  
S. Hamzehei-Javaran
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Value Problems ◽  
Numerical Solutions ◽  
Boundary Value ◽  
The Finite Element Method ◽  
Shape Functions ◽  
Element Approach ◽  
Solution Accuracy ◽  
Element Method

A new finite element approach is developed here for the modeling of boundary value problems. In the present model, the finite element method (FEM) is reformulated by new shape functions called spherical Hankel shape functions. The mentioned functions are derived from the first and second kind of Bessel functions that have the properties of both of them. These features provide an improvement in the solution accuracy with number of elements which are equal or lower than the ones used by the classic FEM. The efficiency and accuracy of the suggested model in the potential problems are examined by several numerical examples. Then, the obtained results are compared with the analytical and numerical solutions. The comparisons indicate the high accuracy of the present method.

Transient Stokes Flow in a Wedge

1987 ◽  
Vol 54 (1) ◽  
pp. 203-208 ◽  
Author(s):  
Bohou Xu ◽  
E. B. Hansen
Keyword(s):  
Finite Element Method ◽  
Boundary Condition ◽  
Finite Element ◽  
Boundary Element Method ◽  
Circular Cylinder ◽  
Stokes Flow ◽  
Constant Velocity ◽  
Transient Flow ◽  
The Finite Element Method ◽  
Element Method

The transient flow in the sector region bounded by two intersecting planes and a circular cylinder is determined in the Stokes approximation. The plane boundaries are assumed to be at rest while the cylinder is rotating with a constant velocity starting at t = 0. The problem is solved by means of three different methods, a finite element, a finite difference, and a boundary element method. The corresponding problem in which the constant velocity boundary condition on the cylinder is replaced by a condition of constant stress is also solved by means of the finite element method.

Convergence of the Finite Element Method in the Theory of Elasticity

1968 ◽  
Vol 35 (2) ◽  
pp. 274-278 ◽  
Author(s):  
M. W. Johnson ◽  
R. W. McLay
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Variational Principle ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Careful Consideration ◽  
Stress Singularities ◽  
The Finite Element Method ◽  
Element Method

The foundations of the theory of the finite element method as it applies to linear elasticity are investigated. A particular boundary-value problem in plane stress is considered and the variational principle for the finite element method is shown to be equivalent to it. Mean and uniform convergence of the finite element solution to that of the boundary-value problem is demonstrated with careful consideration given to the stress singularities. A counterexample is presented in which a set of functions, admissible to the variational principle, is shown not to converge.

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