A Convenient Smoothly-Continuous Basis Function for Interpolation on Unstructured Triangular Grids: Application to Inversion of Geodetic Data for Forces, Strains and Velocities
Abstract
Finite element methods are used ubiquitously for geodynamical modeling of stress and strain and interpreting geodetic data. GPS station distribution necessitates the use of unstructured grids. Basis functions need to have at most one degree of freedom per GPS station, and we would like them to be smoothly continuous so they are differentiable. Here we examine several elements, including Lagrange, Quartic and Argyris, and the properties of the resultant basis functions. In particular, a Lagrangian element specifies the values of the function at the nodes and the midpoints on the side of the triangular gird and has 6 degrees of freedom, a Quartic element defines the function values and first derivatives at the nodes as well as the function values and normal derivative at the midpoints on the sides of the triangular grid and has 15 degrees of freedom, and an Argyris element defines the function values, first and second derivatives at the nodes and normal derivatives at the midpoints of the sides for a total of 21 degrees of freedom. We reduce these degrees of freedom to one per node in the following manner. For the Lagrange and Quartic elements we simply minimize integrals of the square of the first derivatives of the basis functions. For Argyris elements the same procedure alone is inadequate because it is invalid to calculate second derivatives by minimizing integrals of the square of the first derivatives. We circumvent this problem by minimizing integrals of the square of the second derivatives to obtain the second derivative values and only then minimize the integrals of the square of the first derivatives to determine the remaining parameters. All three elements result in basis functions that are effectively scale invariant, by which we mean that they have uniform values of the integral of the square of the first derivatives, irrespective of the spatial spans of the basis functions. Of these element types, the Argyris is clearly the most versatile because unlike Lagrange and Quartic elements, it provides continuous first derivatives everywhere. We illustrate the use of the elements by inverting GPS velocity data for force terms in the force balance equations with the force terms expanded in terms of each of the basis functions. We compare the variations in resulting force terms and implied strain rates that are due to the different properties of the elements.
- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2013
- Bibcode:
- 2013AGUFMNG41A1656H
- Keywords:
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- 3200 MATHEMATICAL GEOPHYSICS;
- 3225 MATHEMATICAL GEOPHYSICS Numerical approximations and analysis;
- 3252 MATHEMATICAL GEOPHYSICS Spatial analysis;
- 3294 MATHEMATICAL GEOPHYSICS Instruments and techniques