Exact and efficient interpolation using finite elements shape functions
Résumé
The increasing use of finite elements (FE) and optical full-field measurement methods have contributed to the growing need to compare mesh-based degrees of freedom with experimental fields of discrete data points. Applying generic interpolation algorithms (e.g. using linear, cubic and B-spline weight functions) creates a dependency of the interpolated field to non-physical parameters that may affect high-frequency information through implicit filtering and introduces fundamental assumptions to the shape of experimental data field. The alternative is to use the existing FE mesh and shape functions to determine mesh degrees of freedom at each experimental data point. This comparison technique makes no assumptions beyond those already made in the FE model. In this sense, interpolation using element shape functions is exact. However, this approach requires calculating the local FE coordinates of the experimental points (inverse mapping), which is non-trivial and not a standard part of FE software. Basic implementations of FE interpolation can be time consuming and impractical for applications requiring the comparison of large fields to be repeated several times. This article analyzes a two-step process for FE interpolation. First the element containing a given data point is determined, then the interpolation inside the element using reference coordinates. An efficient strategy is proposed, which relies on cross-products, element bounding-boxes and a multi-dimensional storage array (virtual mesh). The strategy yields a linear computation cost with respects to the number of elements in the FE mesh and number of data points in the experimental field, contrary to a quadratic cost from standard approaches. A sample application is given using a plate with a hole.
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