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.
The methods described above usually generate a trivariate solid to fill a given B-rep model. However, the generation of a TBS by fitting a tet mesh model is much easier than by filling a B-rep model because it is very easy to produce a tet mesh using popular software such as, TetGen @cite_4 , and NetGen @cite_6 . Hence, it is feasible to generate a TBS by fitting a tet mesh model. In Ref. , a tet mesh model is fitted by the geometric iterative method to generate a TBS. However, there are some regions with negative Jacobian values close to the boundary. In this experiment, a tet mesh is first segmented into seven sub-volumes, each of which is fitted with a TBS by a geometric iterative fitting method. In this way, the generated TBSs are ensured to be valid, i.e. the Jacobian value at any point of the TBSs is positive.
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In this paper, we study the progressive iteration approximation property of a curve (tensor product surface) generated by blending a given data point set and a set of basis functions. The curve (tensor product surface) has the progressive iteration approximation property as long as the basis is totally positive and the corresponding collocation matrix is nonsingular. Thus, the B-spline and NURBS curve (surface) have the progressive iteration approximation property, and Bezier curve (surface) also has the property if the corresponding collocation matrix is nonsingular. A dual channel time domain reflectometer includes a pair of input lines connected at respective nodes to a reference flat current pulse generator and a traveling wave sampling gate, respectively. The sampling gates are actuated by a balanced strobe generator which includes a waveguide coupler for coupling a high amplitude fast rise time pulse to each of the gates simultaneously. Pulses of requisite amplitude and shape are generated by a circuit responsive to a strobe trigger input which drives a step recovery diode.