A method for local approximation of a planar deformation field

Abstract

We present a method of approximation of a deformation eld based on the local a ne transformations constructed based on n nearest neighbors with respect to points of adopted grid. The local a ne transformations are weighted by means of inverse distance squared between each grid point and observed points (nearest neighbors). This work uses a deformation gradient, although it is possible to use a displacement gradient instead -the two approaches are equivalent. To decompose the deformation gradient into components related to rigid motions (rotations, translations are excluded from the deformation gradient through di erentiation process) and deformations, we used a polar decomposition and decomposition into a sum of symmetric and an anti-symmetric matrices (tensors). We discuss the results from both decompositions. Calibration of a local a ne transformations model (i.e., number of nearest neighbors) is performed on observed points and is carried out in a cross-validation procedure. Veri cation of the method was conducted on simulated data-grids subjected to known (functionally generated) deformations, hence, known in every point of a study area.

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“…The deformation gradient tensor F is given by T . [ 57 ] The right Cauchy–Green deformation tensor ( C ) is then calculated by multiplying F with its transpose ( F′ ): $$\[ \begin{array}{*{20}{c}}{{\bm{C}} = {\bm{F'}}\;\cdot\;{\bm{F}}}\end{array} \]$$…”

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“…From a mechanics perspective, a more robust definition of strain can be calculated in the Lagrangian reference frame to account for multi-directional components of deformation. The 3-dimensional Green-Lagrange strain tensor ( E ) depends on the 3D deformation gradient tensor ( F ) and the identity matrix ( I ) ( 28 , 29 ) shown below in Equation (1).…”

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“…The rigid "stretching-and-rotation" geometric transformation can be generalized in the spirit of velocity fields by introducing locally varying scaling and rotation factors, that is, using deformation fields based on local affine transformations (Ligas et al, 2019). This where = √( − 0 ) 2 + ( − 0 ) 2 is the distance of a point ( , ) from the center of the deformation ( 0 , 0 ), is a rotation angle, and is a scaling parameter controlling the swirl extension.…”

“…The rigid "stretching-and-rotation" geometric transformation can be generalized in the spirit of velocity fields by introducing locally varying scaling and rotation factors, that is, using deformation fields based on local affine transformations (Ligas et al, 2019). This method enables the simulation of fields with convenient anisotropy mimicking for instance cyclonic shapes via a swirl-like coordinate transformation given by…”

“…The rigid “stretching‐and‐rotation” geometric transformation can be generalized in the spirit of velocity fields by introducing locally varying scaling and rotation factors, that is, using deformation fields based on local affine transformations (Ligas et al., 2019). This method enables the simulation of fields with convenient anisotropy mimicking for instance cyclonic shapes via a swirl–like coordinate transformation given by where is the distance of a point from the center of the deformation , is a rotation angle, and is a scaling parameter controlling the swirl extension.…”