Constructing Trivariate B-splines with Positive Jacobian by Pillow Operation and Geometric Iterative Fitting

TL;DR

This paper introduces a novel method for constructing trivariate B-spline solids with guaranteed positive Jacobian by segmenting tetrahedral meshes with pillow operations and applying geometric iterative fitting, ensuring smoothness and validity.

Contribution

It proposes a new approach combining pillow operations and iterative fitting to generate positive Jacobian TBSs, addressing a complex geometric constraint in isogeometric analysis.

Findings

Effective generation of TBS with positive Jacobian demonstrated

Improved smoothness between adjacent TBS segments achieved

Algorithm shown to be efficient and practical in experiments

Abstract

The advent of isogeometric analysis has prompted a need for methods to generate Trivariate B-spline Solids (TBS) with positive Jacobian. However, it is difficult to guarantee a positive Jacobian of a TBS since the geometric pre-condition for ensuring the positive Jacobian is very complicated. In this paper, we propose a method for generating TBSs with guaranteed positive Jacobian. For the study, we used a tetrahedral (tet) mesh model and segmented it into sub-volumes using the pillow operation. Then, to reduce the difficulty in ensuring a positive Jacobian, we separately fitted the boundary curves and surfaces and the sub-volumes using a geometric iterative fitting algorithm. Finally, the smoothness between adjacent TBSs is improved. The experimental examples presented in this paper demonstrate the effectiveness and efficiency of the developed algorithm.