Exact and efficient interpolation using finite elements shape functions

The comparison of finite elements (FE) and experimental data fields have become ever more prevalent in numerical simulations. Since FE and experimental data fields rarely match, the interpolation of one field into the other is a fundamental step of the procedure. When one of the fields comes from FE, using the existing FE mesh and shape functions is a natural choice to determine mesh degrees of freedom at data point coordinates. This makes no assumptions beyond those already made in the FE model. In this sense, interpolation using element shape functions is exact. However, crude implementations of this technique generally display a quadratic computation complexity with respect to mesh size and number of data points, which is impractical when large data fields must be compared repeatedly. This document aims at assembling existing numerical procedures to improve the interpolation efficiency. With a combination of cross-products, bounding-boxes and indexing methods, the resulting algorithm shows linear computation cost, providing significant improvement in efficiency.

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