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Views of Treil's Frames

Figure 1. Here are Sagittal A, Frontal B, and Axial C views of the maxillo-facial frame. The maxillo-facial frame is defined by eight anatomical landmarks: both supraorbital foramen (RSO, LSO), both infra orbital foramen (RIO, LIO), both mental foramen (RM, LM), and both head malleus (RHM, LHM). According to Moss theory, all these landmarks are located on the trigeminal neuro-matricial facial growth directions. The axes of inertia of the maxillo-facial frame are in blue the principal, in white the second longest, and in red the shortest.

This Mathematica notebook contains only the numerical issues covered in Treil's keynote lecture. The biological and clinical details can be found in the enclosed references.

The shape function interpolants are the basic objects in any finite element method. In morphometric analysis (i.e., quantifying shape and size changes), a kinematic formulation is adequate without any reference to stress or stress-strain-like constitutive relations. Hence the algebraic expressions of shape functions can be used directly to calculate the non-mechanical growth strains--the local changes in shape and size of the body--in time. In a finite element region with n-number of nodes there are n shape functions. Since the intent of the shape function interpolants is to reproduce m pre-selected algebraic shapes, it is shown here that from given shape functions the remaining m interpolants can be obtained. This new result substantially economizes the computing effort required to deduce the complete set of interpolants.

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