A contact method for B-spline material point method with application in impact and penetration problems

Abstract

A novel contact algorithm for the B-spline material point method (referred to as cBSMPM) is proposed to address impact and penetration problems. The proposed contact algorithm is based on the Lagrangian multiplier method and enables the cBSMPM to accurately predict the contact, friction, and separation of two continuum bodies, where the numerical results are free from the cell-crossing noise of particles presented in the conventional MPM. In cBSMPM, the contact algorithm is implemented on the computational background grid built from the control points associated with the knot vectors of the B-splines. Correspondingly, a comprehensive criterion, including the nodal momentum condition and the physical distance between the bodies, is introduced to detect the contact event accurately. The Greville abscissa is utilized to determine the coordinates of computational grid nodes, facilitating the calculation of the actual distance between the approaching bodies. A comprehensive set of numerical examples is presented, and the numerical results from the proposed method agree well with the analytical solution and the experimental data documented in the literature, where the effectiveness of the proposed criterion is demonstrated in avoiding spurious contact and the corresponding stress oscillations. Moreover, it is demonstrated that increasing the B-spline basis function order can improve solution accuracy in terms of smooth stress/pressure field for impact and penetration problems.

Appendix A. B-spline

A basic component for the construction of B-splines is the knot vector, which is a non-decreasing set of coordinates in the parametric domain. An open 1D knot vector can be written as $\varvec{\Xi }=\left\{ \xi _0, \xi _1, \ldots , \xi _{n+\mathrm{p-1}}, \xi _{n+\textrm{p}}\right\}$, where $\xi _0=\cdots =\xi _{\textrm{p}}<\xi _{\mathrm{p+1}}<\cdots <\xi _{n}=\cdots =\xi _{n+\mathrm p}$, $\xi _i$ is $i$th knot, $i$ is the knot index, $i=0,1, \ldots , n+\textrm{p}$, and n and $\textrm{p}$ are the number and polynomial order of B-spline functions, respectively.

The B-spline basis functions are defined using the Cox-de Boor formula as follows.

$$\begin{aligned} N_i^0(\xi )=\left\{ \begin{array}{lc} 1 &{} \text{ if } \xi _i \le \xi \le \xi _{i+1} \\ 0 &{} \text{ otherwise } \end{array}\right. \end{aligned}$$

(A.1)

and for $\textrm{p} =1,2,3,\ldots ,$ they are defined by

$$\begin{aligned} N_i^{\textrm{p}}(\xi )= & {} \frac{\xi -\xi _i}{\xi _{i+\textrm{p}}-\xi _i} N_i^{\mathrm{p-1}}(\xi )\nonumber \\{} & {} +\frac{\xi _{i+\mathrm{p+1}}-\xi }{\left( \xi _{i+\mathrm{p+1}}-\xi _{i+1}\right) } N_{i+1}^{\mathrm{p-1}}(\xi ) \end{aligned}$$

(A.2)

In this definition, $0/0=0$ is assumed. The derivative of those basis functions is given as

$$\begin{aligned} \frac{d N_i^{\textrm{p}}(\xi )}{d \xi }= & {} \frac{\textrm{p}}{\xi _{i+\textrm{p}}-\xi _i} N_i^{\mathrm{p-1}}(\xi )\nonumber \\{} & {} -\frac{\textrm{p}}{\left( \xi _{i+\mathrm{p+1}}-\xi _{i+1}\right) } N_{i+1}^{\mathrm{p-1}}(\xi ) \end{aligned}$$

(A.3)

The above is the definition of B-spline basis functions, from which it can be noted that the basis functions have the following properties.

• (1) Non-negative

  $$\begin{aligned} N_i^{\textrm{p}}(\xi ) \ge 0, \ \ \ \ \ \ \forall \xi \in \varvec{\Xi }\end{aligned}$$

  (A.4)

• (2) Partition of unity

  $$\begin{aligned} \sum _{i=1}^n N_i^{\textrm{p}}(\xi )=1, \ \ \ \ \ \ \forall \xi \in \varvec{\Xi }\end{aligned}$$

  (A.5)

• (3) Compact support

The support of each basis function $N_i^{\textrm{p}}$ is compact and contained in the interval $\left[ \xi _i, \xi _{i+\mathrm{p+1}}\right]$, and it becomes larger as the order increases.