دانلود رایگان مقاله پارامترسازی دامنه مسطح با اسپلاین - THB

Abstract

Isogeometric analysis uses spline parameterizations to describe the geometry of the physical domain. If such a parameterization is not available, it has to be generated from the domain boundaries. The construction of a good parameterization is crucial since it strongly influences the accuracy of the subsequent analysis. It is of interest to use adaptive techniques such as hierarchical splines for the parameterization, since this facilitates an accurate representation of detailed geometries and potentially improves the efficiency of the subsequent numerical simulation. We use truncated hierarchical (TH) B-splines to generate hierarchical spline spaces, since these functions possess many useful properties, such as linear independence and the fact that they form a non-negative partition of unity. In order to address the trade-off between computational effort and level of difficulty of a specific instance of the problem, we propose three levels of domain parameterization techniques that are based on THB-splines.

7. Conclusion

We proposed several techniques for solving the planar domain parameterization problem in isogeometric analysis. These techniques, which allow adaptive refinement to be performed, are based on the mathematical technology of THB-splines. Consequently we were able to obtain parameterizations with a lower number of degrees of freedom than traditional tensorproduct approaches. THB-splines possess good mathematical properties, such as the fact that they form a non-negative partition of unity, which makes them well suited for applications in geometric design and numerical simulation. The parameterization techniques that we described in this paper were assigned to three levels with increasing computational effort. The simpler methods work for relatively simple domains, while more complicated shapes require additional efforts. The most sophisticated method, which is based on the approximation of the inverse of a harmonic mapping, has two main advantages: Firstly, it does not need any user-selected parameters, except for the polynomial degrees and for the initial (tensor-product) knot configuration. Secondly, it provides error estimators that can guide the adaptive refinement of the THB-splines in the entire domain. In contrast to this, the simpler techniques do not provide such estimators and lead to an adaptive refinement only along the domain boundary.