Adaptive re-parameterization based on arbitrary scalar fields for shape optimization and surface fitting

- Adaptive re-parameterization method based on arbitrary scalar field is developed.
- The method smoothly re-distributes B-spline control-points towards the needed areas.
- A method for fitting single-patch surfaces to complex geometries is developed.
- The method is appropriate for usage in intelligent shape optimization procedure.

This paper presents a method for re-parameterization based on an arbitrary scalar field named the relaxation field. The relaxation field is applied to re-distribute the control-points of a parametric surface towards the desired areas. The proposed method was developed for possible application in an intelligent shape optimization procedure where a sensitivity field with respect to an objective function (or some other physical field) would be used for constructing the relaxation field. It could hence contribute to the concentrating the control-points at areas where significant changes in the geometry are expected. The method can easily be used in shape optimization since it keeps the number of variables constant during the redistribution of control-points as opposed to adaptive insertion of control points when using T-spline and similar methods.The same method can also be used in surface fitting by choosing the relaxation field based on the geometric error. This leads to an adaptive iterative fitting method. The method was validated by fitting a single patch B-spline surface to triangulated point clouds. The point-clouds were obtained by 3D scanning or from a CAD model. Examples include several complex engineering objects. The proposed method uses a parameterization method based on a combination of harmonic mapping and a mapping method based on a spring mesh. By relaxation using a spring mesh, the method allocates more parametric space to the regions of interest, thus assigning them more control points. The combination of these two mapping methods provides for increased local control while keeping the global smoothness of the parameterization.