Applications governed by geometric concavity

Examples include analysis of biological entities, and soil-structure interaction. Continuous domains can be modeled using either the finite or boundary element methods. Both require test functions to be used as interpolants. A shape function can be viewed as a specific solution to an elliptic equation on a given polygonal domain. For convex polygons with any number of sides these functions can readily be found in closed form as rational polynomials. However, convexity is too restrictive a requirement for smooth modeling of many continuous domains.

Earthquake Response Behavior Prediction:

The prediction of earthquake's effect on the stability of buildings requires an efficient modeling of the soil structure interaction. Generally the ground is considered to be an semi-infinite domain since the soil is significantly larger than the building. The wave absorbing boundary is necessarily concave. If the boundary is discretized numerical modeling will predict that waves will pass between the arbitrary piece. The spurious waves cloud the solution related to the incoming and outgoing waves which pass through the wave absorbing boundary and are observed in in situ earthquake measurements.

  • Off-shore windmill project: pdf
  • Cloning algorithm and concave elements: pdf

Smooth Computer Generated Graphics:

When an object is rendered it must first be discretized into two dimensional shapes depending on the viewing angle. Any discretization beyond adds more data to the image than is provided in the original shape. Commonly objects are rendered using a triangular domain discretization. The mesh division is not only time consuming, it is unnecessary if convex and concave shape functions can be found consistently in closed form.

  • Fast-smooth graphics generation: Java Implementation
  • Boundary element implementation for generating smooth graphics: pdf

Radiographic Images:

Diagnoses made from MRI, X-ray, CT-scan and other imaging techniques are based on the comparison of medically significant points and boundaries. In order to be able to consistently compare such images coordinate invariant shape analysis is necessary. In the case of the Maxillo-Facial frame nodal points are found from CT scans, these points form the outline of a concave shape. Apparently the concavity is essential to the proper growth and function and thus needs to be considered in comparative analysis. If the domain outlined by the nodes is discretized into convex parts then the concavity information is lost.