NURBS distance fields for extremely curved cracks

Abstract

This paper proposes for the first time an intrinsic enrichment for extremely curved cracks in a meshfree framework. The unique property of the proposed method lies in the exact geometric representation of cracks using non-uniform rational B-splines (NURBS). A distance function algorithm for NURBS is presented, resulting in a spatial field which is simultaneously discontinuous over the (finite) curved crack and continuous all around the crack tips. Numerical examples show the potential of the proposed approach and illustrate its advantages with respect to other techniques usually employed to model fracture, including standard finite elements with remeshing and the extended finite element method. This work represents a further step in an ongoing effort in the community to integrate computer aided design with numerical simulations.

Appendices

Appendix 1: The RKPM approximation

This work considers the reproducing kernel particle method (RKPM) for the functional approximation, see [28]. This method can also be viewed as the result of applying a numerical quadrature to the continuous moving least squares approximation that is usually employed in other meshfree methods such as the element-free Galerkin method, see [8, 21] for more details.

The RKPM shape function associated to a particle $\mathbf {x}_I \in \varOmega$ is given by

$$\begin{aligned} \phi _I(\mathbf {x}) = \omega (\mathbf {x}_I,\mathbf {x}) \mathbf {P}^T(\mathbf {x}) \mathbf {M}^{-1}(\mathbf {x}) \mathbf {P}(\mathbf {x}_I) \end{aligned}$$

(33)

where the weighting function is defined as

$$\begin{aligned} \omega (\mathbf {x}_I,\mathbf {x}) = \Delta V_I w \Big ( \frac{\mathbf {x}_I - \mathbf {x}}{\rho } \Big ) \end{aligned}$$

(34)

with $(\Delta V_I, \mathbf {x}_I)$ denoting the quadrature weights and points (particles), $\mathbf {P}(\mathbf {x})$ denotes a complete basis of the subspace of polynomials of degree $k$, $\mathbf {P}(\mathbf {x}) = \{p_0(\mathbf {x}), p_1(\mathbf {x}), \cdots , p_k(\mathbf {x}) \}$, and $\mathbf {M}$ is the so-called moment matrix

$$\begin{aligned} \mathbf {M}(\mathbf {x}) = \sum _{I\in \mathcal {S}_\mathbf {x}^{\rho }}\omega (\mathbf {x}_I,\mathbf {x}) \mathbf {P}(\mathbf {x}_I) \mathbf {P}^T(\mathbf {x}_I) \end{aligned}$$

(35)

where the index set $\mathcal {S}_{\mathbf {x}}^{\rho }$ is defined in Eq. (23).

The moment matrix $\mathbf {M}$ can also be viewed as a Gram matrix defined with a discrete scalar product

$$\begin{aligned} \langle u,v \rangle _\mathbf {x}= \sum _{I\in \mathcal {S}_\mathbf {x}^{\rho }}\omega (\mathbf {x}_I,\mathbf {x}) u(\mathbf {x}_I) v(\mathbf {x}_I) \end{aligned}$$

(36)

and, from a numerical point of view, it is convenient to work with a centred and scaled version to enhance the condition number of the system of normal equations. This correction implies that the following definition of $\mathbf {M}$ is adopted here

$$\begin{aligned} \mathbf {M}(\mathbf {x}) = \sum _{I\in \mathcal {S}_\mathbf {x}^{\rho }}\omega (\mathbf {x}_I,\mathbf {x}) \mathbf {P}\Big ( \frac{\mathbf {x}_I - \mathbf {x}}{\rho } \Big ) \mathbf {P}^T\Big ( \frac{\mathbf {x}_I - \mathbf {x}}{\rho } \Big ) \end{aligned}$$

(37)

where $\rho$ denotes the average of all the compact support radii.

The continuity properties of the RKPM shape functions are clearly linked to the continuity properties of the function $w$ in Eq. (34), see [17], which is usually referred as the kernel of the approximation. This work considers the so-called 2k-th order spline, which is the $C^{k-1}$ function given by

$$\begin{aligned} w(\xi ) = {\left\{ \begin{array}{ll} (1-\xi ^2)^k &{} \qquad 0\le \xi \le 1 \\ 0 &{} \qquad \xi >1 \end{array}\right. } \end{aligned}$$

(38)

Appendix 2: Control data for NURBS objects

This section contains the NURBS description of the two cracks used in Sect. 5.

1.1 Circular arc

There are many options to define a NURBS describing a circle. A commonly used options is to define a quadratic NURBS with four rational segments. The knot vector is

$$\begin{aligned} \varLambda = \{0,0,0,0.25,0.5,0.75,1,1,1 \} \end{aligned}$$

(39)

and the control points and weights are detailed in Table 4 for a circle centred at the origin and with radius $R$.

Table 4 Control points and weights for a circle of radius $R$

In order to define the circular arc depicted in Fig. 15, the radius of the circle is taken as R = $a/\sin (\alpha )$ and the NURBS curve describing a circle is trimmed to the subinterval $\left[ (1-\alpha )/360, 0.5 - (1-\alpha )/360\right]$ where the angle $\alpha$ is given in degrees.

1.2 Parabolic arc

A parabolic arc can be described using a quadratic B-Spline with just three control points. The knot vector is simply

$$\begin{aligned} \varLambda = \{0,0,0,1,1,1 \} \end{aligned}$$

(40)

and the control points for the parabolic arc depicted in Fig. 16 are detailed in Table 5, where $y_a = a^2/(4A)$. The coordinates of the focus of the parabola are $(0,A)$, where $A = a/(2 \tan \alpha )$.

Table 5 Control points for a parabolic arc