Geometric Morphometrics and Finite Element Analysis: First Results from a Joint Formalism for Modeling Strain

Finite element analysis (FEA) is a robust and widely exploited 20th-century approach to the computational approximation of partial differential equations, particularly for continuum mechanics. Geometric morphometrics (GMM), a more recent methodology, blends tools from computer vision and multivariate statistics in a toolkit for pattern analysis of shape variation that has been applied extensively in conjunction with deformable-template methods since 1990, especially within organismal biology and medical imaging. In the absence of any canonical statistical method for finite-element studies, the possibility of a bridge between GMM and FEA is becoming a focus of active concern in several contemporary biosciences. The two existing traditions agree on the centrality of the biharmonic equation but differ in representations of the geometry of space, in many details of boundary condition specification, in the role of the uniform (affine) strains, and in the constraints linking changes of multiple coordinates. This speculative essay reviews some preliminary issues arising in the course of attempts to construct a preliminary synthesis along these lines for application to data sets comprising multiple instances of the same design or Bauplan that can be characterized by variation of a shorter or longer vector of scalar parameters. An approach to this goal is demonstrated according to which the GMM toolkit is represented by a nonstandard version of its sufficient scalar statistic, Procrustes distance, while the FEA domain is represented by a less-familiar summary statistic, the net strain energy stored in the deformed configuration. The approaches are compared by recourse to an ancient example, the Euler–Bernoulli analysis of a cantilevered beam, for which the net strain energy can be computed by a simple geometric analysis of the same parameters that specify the strain being simulated. The finding is rather surprising: for simulations involving constant load at the end of the beam, strain energy and Procrustes distance are exactly proportional. This simplicity argues for the centrality of this pair of scalar summaries over any more extensive multivariate vector representation of the deformations being studied. A concluding comment sketches extensions to other biomathematically relevant scenarios for which analytic solutions can be approximated from the technical literature.