We present a unified reproducing kernel linkage for the meshfree and isogeometric methods, two classes of representative computational methods which have gained significant attentions over the past years. Now it is well understood that in the reproducing kernel meshfree shape functions, the consistency conditions are formulated via using the nodal points as the reproducing locations. However, we show that the monomial reproducing points for different order B-spline basis functions are distinct and consequently we propose a rational method to compute reproducing points for B-spline basis functions. We further prove that after properly defining meshfree nodes, nodal supports and consistency conditions, the reproducing kernel meshfree shape functions are capable of exactly representing the isogeometric B-spline and NURBS basis functions. As a result, a unified reproducing kernel linkage or correspondence is established between meshfree methods and isogeometric analysis. Thus the meshfree shape functions and isogeometric basis functions can be constructed on the same reproducing kernel platform. The outcomes of this linkage are two-folds. On the one hand, this linkage can enhance the meshfree methods through the geometry exact isogeometric formulation. On the other hand, this linkage enables a meshfree isogeometric analysis, i.e., the local model refinement, one notable difficulty in isogeometric analysis, can be realized in a meshfree manner. Several examples are presented to illustrate the proposed methodologies.
Key words: Meshfree method, isogeometric analysis, consistency condition, reproducing kernel formalism, local refinement
Dongdong Wang received his Ph.D. in civil engineering with an Outstanding Ph.D. Award from University of California, Los Angeles in 2003. In 2004 he joined the faculty of civil engineering department of Xiamen University, where presently he is a professor in the field of structural engineering and mechanics. His research interests center around computational solid and structural mechanics, in particular, the development of efficient and accurate meshfree and isogeometric methods for solid, structural, and geotechnical problems. He has received several awards for his contributions in computational mechanics, including the APACM Young Investigator Award from Asian-Pacific Association for Computational Mechanics (2007), the ICACM Young Investigator Award (2011), the ICACM Fellow Award (2013) and the ICACM Computational Mechanics Award (2016) from International Chinese Association for Computational Mechanics, and the Qianlingxi Young Investigator Award of Computational Mechanics (2012) from Qianlingxi Mechanics Award Foundation and Chinese Association for Computational Mechanics, respectively. Currently he is an executive council member of ICACM and a general council member of IACM.