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A NURBS-enhanced boundary element method for 2D elasticity problems with body forces is proposed in this paper. The non-uniform rational B-spline (NURBS) basis functions are applied to construct the geometry and the model can be reproduced exactly at all stages since the refinement will not change the shape of the boundary. Both open curves and closed curves are considered. The fields are approximated by the traditional Lagrangian basis functions in parameter space, rather than by the same NURBS basis functions for geometry approximation. The parametric boundary elements and collocation nodes are defined from the knot vector of the curve and the refinement of the NURBS curve is easy. Boundary conditions can be imposed directly since the Lagrangian basis functions have the property of delta function. In addition, most methods for the treatment of singular integrals in traditional boundary element method can be applied in the proposed method. To overcome the difficulty for evaluation of the domain integrals in problems with body forces, a line integration method is further applied in this paper to compute the domain integrals without additional volume discretizations. Numerical examples have shown the accuracy of the proposed method.

Isogeometric analysis (IGA) has been widely applied in the finite element method and boundary element method (BEM), in which the geometry discretization error can be avoided. This paper proposes a NURBS-enhanced meshless improved interpolating boundary element-free method for 2D potential problems. In the proposed method, non-uniform rational B-spline (NURBS) basis functions are applied to reproduce the geometry like in IGA, and the boundary integral cells, called isogeometric cells, are defined by the knot vector of NURBS in parameter space. Thus, the geometry can remain the same at all stages because refining a NURBS curve will not change its shape. An improved interpolating moving least-square (IIMLS) method is applied to approximate the field in parameter space, and the boundary nodes can be defined using Greville abscissae definition. Compared with IGA in BEM, the shape functions obtained by IIMLS in the proposed method have the delta function property, and the boundary conditions can be applied directly. In addition, most methods for the treatment of the singular integrals in BEM can be applied easily in the proposed method. Fast multipole method is further coupled with the proposed method for large-scale computation.

The moving least-square (MLS) method has been popular applied in surface construction and meshless methods. However, the moment matrix in MLS method may be singular for ill quality point sets and the computation of the inverse of the singular moment matrix is difficult. To overcome this problem, a regularized moving least-square method with nonsingular moment matrix is proposed. The shape functions obtained from the regularized MLS method still do not have the delta function property and may result in difficulty for imposing boundary conditions in regularized MLS based meshless method. To overcome this problem, a regularized improved interpolating moving least-square (IIMLS) method based on the IIMLS method is also proposed. Compared with the regularized MLS method, the regularized IIMLS not only has nonsingular moment matrices, but also obtains shape functions with delta function property. Shape functions of the proposed methods are compared in 1D and 2D cases, and the methods have been applied in curve fitting, surface fitting and meshless method in numerical examples.

A NURBS-enhanced interpolating boundary element-free method (NE-IIBEFM) is proposed by coupling the meshfree method with the non-uniform rational B-spline (NURBS). The NURBS plays an essential role in computer aided design (CAD) whereby it can be used as the bridge between boundary type meshless method and CAD, which is convenient and is widely utilized in engineering. The NURBS basis functions used in CAD packages for geometry construction are applied to describe the boundary in the numerical procedure. Thus, the geometry of the curve can be reproduced exactly at all stages. The boundary integral cells and nodes are defined from the knot vector in NURBS, and both the open and closed curves are considered. The fields are approximated by an improved interpolating moving least square method, in which the obtained shape functions have the property of delta function. Thus, the boundary conditions can be imposed directly. A line integration method without domain discretization is further coupled with the proposed method for the treatment of domain integrals for problems with body forces.

A numerical method based on the boundary integral equation is proposed for the point location problem. For a bounded domain, the integral value is close to 1.0 if a point is inside the domain, and is close to 0.0 when the point is outside the domain. For convenience of integration, the boundary of the domain can be discretized into boundary integral cells. The idea of isogeometric analysis can be easily coupled with the proposed method, i.e., using the parametric functions in geometric modeling to create the integral cells, which results in a mesh-free procedure for which the geometry can be exactly produced at all stages. Thus, the method can be applied to arbitrary shapes and easily embedded in computer-aided design (CAD) packages. The method is time-consuming if implemented directly; a fast multipole method is coupled with the proposed method to accelerate the integral procedure. Some examples of 2D and 3D cases are tested to show the accuracy and efficiency of the proposed method.

Non-uniform rational B-spline (NURBS) in Isogeometric analysis (IGA) is coupled with the boundary element method (BEM) for 2D potential problems in this paper. The geometry and field are usually approximated by the same basis functions in IGA, such as the B-spline or the NURBS basis functions. In the proposed method, these two kinds of approximation are performed independently, i.e. the geometry is reproduced by the NURBS basis functions while the field is approximated by the traditional Lagrangian basis functions which are used in the conventional BEM. The proposed method has the advantage that the geometry can be reproduced exactly at all stages in IGA methods. Actually, one can use the computer aided design (CAD) software or NURBS library to perform the operations related to the geometry. The field approximation is performed in parameter space and separated from the geometry. Thus, it can be implemented easily as the conventional BEM since most algorithms for BEM can be applied directly, such as the methods for treatment of the singular integrals, and the boundary conditions can be imposed directly. Numerical examples have demonstrated the accuracy of the proposed method.

Purpose Domain integrals, known as volume potentials in 3D elasticity problems, exist in many boundary-type methods, such as the boundary element method (BEM) for inhomogeneous partial differential equations. The purpose of this paper is to develop an accurate and reliable technique to effectively evaluate the volume potentials in 3D elasticity problems. Design/methodology/approach An adaptive background cell-based domain integration method is proposed for treatment of volume potentials in 3D elasticity problems. The background cells are constructed from the information of the boundary elements based on an oct-tree structure, and the domain integrals are evaluated over the cells rather than volume elements. The cells that contain the boundary elements can be subdivided into smaller sub-cells adaptively according to the sizes and levels of the boundary elements. The fast multipole method (FMM) is further applied in the proposed method to reduce the time complexity of large-scale computation. Findings The method is a boundary-only discretization method, and it can be applied in the BEM easily. Much computational time is saved by coupling with the FMM. Numerical examples demonstrate the accuracy and efficiency of the proposed method.. Originality/value Boundary elements are used to create adaptive background cells, and domain integrals are evaluated over the cells rather than volume elements. Large-scale computation is made possible by coupling with the FMM.

A line integration method is presented in this paper for evaluation of domain integrals in 3D problems. The method is a boundary-only discretization method and the domain integrals can be computed by sum of integrals on one-dimensional straight lines. Divergence theorem is used to transform the domain integrals into boundary integrals with one-dimensional integrals. The boundary integrals can be evaluated by boundary elements with integral points. Each integral point can be used to construct an integral line, and the domain integrals can be finally computed by line integrals on integral lines. Only the boundary discretization is needed and background cells are used to cut the integral lines into sub-lines to obtain the desired accuracy. The method is proved and applied in boundary element method for 3D potential and elasticity problems. Numerical examples have demonstrated the accuracy of the proposed method.

An adaptive cell-based domain integration method (CDIM) is proposed for the treatment of domain integrals in 3D boundary element method (BEM). The domain integrals are computed in background cells rather than volume elements. The cells are created from the boundary elements based on an adaptive oct-tree structure and no other discretization is needed. Cells containing the boundary elements are subdivided into smaller sub-cells adaptively according to the sizes and levels of the boundary elements; and the sub-cells outside the domain are deleted to obtain the desired accuracy. The method is applied in the 3D potential and elasticity problems in this paper.

A line integration boundary element method (LIBEM) is proposed for three-dimensional elastostatic problems with body forces. The method is a boundary-only discretization method like the traditional boundary element method (BEM), and the boundary elements created in BEM can be used directly in the proposed method for constructing the integral lines. Finally, the body forces are computed by summing one-dimensional integrals on straight lines. Background cells can be used to cut the lines into sub-lines to compute the integrals more easily and efficiently. To further reduce the computational time of LIBEM, the fast multipole method is applied to accelerate the method for large-scale computations and the details of the fast multipole line integration method for 3D elastostatic problems are given. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A line integration method (LIM) is proposed to calculate the domain integrals for 3D problems. In the proposed method, the domain integrals are transformed into boundary integrals and only line integrals on straight lines are needed to be computed. A background cell structure is applied to further simplify the line integrals and improve the accuracy. The method creates elements only on the boundary, and the integral lines are created from the boundary elements. The procedure is quite suitable for the boundary element method, and we have applied it to 3D situations. Directly applying the method is time-consuming since the complexity of the computational time is O(NM), where N and M are the numbers of nodes and lines, respectively. To overcome this problem, the fast multipole method is used with the LIM for large-scale computation. The numerical results show that the proposed method is efficient and accurate.

A background cell-based domain integration method is proposed in this paper for evaluating domain integrals in 3D problems. The cells are created by an adaptive oct-tree structure based on the information of boundary elements, and no other discretization is needed. Cells that contain boundary elements can be subdivided into smaller sub-cells adaptively to obtain the desired accuracy according to the sizes and levels of the boundary elements. Applying the method directly in the boundary element method is time-consuming since the time complexity is O ( NM ), where N and M are the numbers of nodes and cells, respectively. The fast multipole method is coupled with the cell-based domain integration method to further accelerate the computational efficiency, and the main formulations are introduced in this paper. Numerical examples have demonstrated the accuracy and efficiency of the proposed method.

A novel microporous metal-organic framework (MOF) based on Zn(II) ion as node with the chemical formula of {[Zn(H2O)(HL)]·(DMF)2(H2O)2}n (1, H3L ​= ​2-(4-carboxyphenyl)-1H-benzo[d]imidazole-5-carboxylic acid, DMF ​= ​N,N-dimethylformamide), which has rich water-coordinated metal centers and high density of free N-donor groups has been successfully prepared via H3L ligand and Zn(NO3)2·6H2O reaction within H2O, EtOH along with DMF mixed solvent. The presence of open metal sites along with the basic N-donor sites in the resulting activated 1a induces a moderate high sorption capacity of CO2. In addition, because of basic N-donor groups along with Lewis acidic Zn2+ ions, 1a has become an efficient heterogeneous catalyst, which can chemically fix CO2 in the carbonate of cyclic annular at mild room temperature. Furthermore, the influence of compound on the activation of PI3K/AKT signaling pathway was studied by Western blot. The ELISA was carried out the evaluate the IL-18 and IL-6 content in synovial cells after compound treatment.