Finite element method for solving geodetic boundary value problems
Finite element method for solving geodetic boundary value problems
复制标题
求解大地边值问题的有限元方法
DOI：
10.1007/s00190-009-0349-7
复制
发表时间：
2010
影响因子：
4.4
通讯作者：
K. Mikula
中科院分区：
文献类型：
--
作者：
Z. Fasková;R. Cunderlík;K. Mikula
本文的目标是提出用于解决地球表面上方 3D 域中的地球潜在问题的有限元方案。为了实现这一目标，我们制定了边值问题 (BVP)，该问题由地球外部的拉普拉斯方程以及诺依曼和狄利克雷边界条件 (BC) 组成。 3D 计算域由地球表面的球形近似或真实三角测量形式的底部边界组成，在该底部边界上给出了表面重力扰动。我们在给出 Dirichlet BC 的地方引入额外的上（球形）和侧（平面和圆锥形）边界。这种椭圆BVP的解是在弱意义上理解的，它始终存在且唯一，并且可以通过有限元法（FEM）有效地找到。我们简要介绍了此类问题的有限元法的推导，包括主要的离散化思想。该方法得出稀疏对称线性系统的解，该解给出了地球在 3D 计算域的每个离散节点中的潜在解。在这一点上，我们的方法不同于其他数值方法，例如边界元法 (BEM)，仅在超曲面上寻找势能。我们在各种情况下应用和测试有限元法。首先，我们将有限元解与均匀球体情况下的已知精确解进行比较。然后，我们利用 DNSC08 数据求解大陆尺度的大地测量 BVP。我们将结果与 EGM2008 位势模型进行比较。最后，我们通过在斯洛伐克进行的 GPS/水准测量测试来研究我们的解决方案的精度，其中我们使用陆地重力测量结果作为输入数据。所有测试均显示与给定解决方案的定性和定量一致性。
The goal of this paper is to present the finite element scheme for solving the Earth potential problems in 3D domains above the Earth surface. To that goal we formulate the boundary-value problem (BVP) consisting of the Laplace equation outside the Earth accompanied by the Neumann as well as the Dirichlet boundary conditions (BC). The 3D computational domain consists of the bottom boundary in the form of a spherical approximation or real triangulation of the Earth’s surface on which surface gravity disturbances are given. We introduce additional upper (spherical) and side (planar and conical) boundaries where the Dirichlet BC is given. Solution of such elliptic BVP is understood in a weak sense, it always exists and is unique and can be efficiently found by the finite element method (FEM). We briefly present derivation of FEM for such type of problems including main discretization ideas. This method leads to a solution of the sparse symmetric linear systems which give the Earth’s potential solution in every discrete node of the 3D computational domain. In this point our method differs from other numerical approaches, e.g. boundary element method (BEM) where the potential is sought on a hypersurface only. We apply and test FEM in various situations. First, we compare the FEM solution with the known exact solution in case of homogeneous sphere. Then, we solve the geodetic BVP in continental scale using the DNSC08 data. We compare the results with the EGM2008 geopotential model. Finally, we study the precision of our solution by the GPS/levelling test in Slovakia where we use terrestrial gravimetric measurements as input data. All tests show qualitative and quantitative agreement with the given solutions.