[Application of the finite element method to craniofacial growth analysis. 4. Three-dimensional application of tensor analysis].

Abstract

The purpose of this study is the 3-dimensional application of finite element method to analyze the craniofacial growth. In our previous studies, 2-dimensional finite element method was applied to analyze and to predict the craniofacial growth. In case of 2-dimensional application, the normal and shear strains were given by the displacements u, v that were linear functions of coordinates x, y. In the 3-dimensional case, six strain components are required to obtain the extension ratio and the directions of three-principal axes. In this study, for the 3-dimensional application, four-noded, pyramidal elements were used and twelve 3-dimensional elements were constructed. Materials were longitudinal frontal and lateral cephalometric X-rays of 4 males and 4 females from 7 to 10 years with normal occlusions. 3-dimensional coordinates of each nodal points were calculated by transforming 2-dimensional coordinates on the frontal and lateral cephalometric X-ray films. Coordinates of the nodal points at 7 years were standards against that results of the finite element method from 8 to 10 years were obtained. They were subjects of the analysis that elements equivalent to the cranial base, the maxillary portion, the maxillary alveolar portion and the posterior pharyngeal portion. Summarized results were as follows. 1. It was observed that the similar figured extension of the anterior cranial base element. 2. The posterior cranial base element extended to right, anterior and upwards direction. 3. On the maxillaly elements, especially, lateral extensions appeared. 4. The element of the posterior pharyngeal portion extended to left and downwards direction. 5. The elements of the maxillary alveolar portion extended to downwards directions. From the above, the directions of transformation of the elements that are selected with voluntary nodal points can be observed. Therefore, 3-dimensional tensor analysis is a method of great significance for obtaining new findings of the craniofacial growth.