Assessment of EqP in XFEM for weak discontinuities

Abstract

This work analyzes the use of Equivalent Polynomials (EqP) in the context of the eXtended Finite Element Method (XFEM). Special attention is devoted to the modeling of weak interfaces, here studied as bi-material interfaces, in the framework of the weak form of elastostatic boundary value problems. The well-known pathological oscillations due to the loss of partition of unity in blending elements are also treated by using a local hierarchical enrichment of the approximation space. In order to assess the properties of the studied approximation spaces, some key examples are selected, for which numerical result comparisons are provided by allowing the standard form to also represent the interface. Conventional numerical integration techniques perform the integration of the weak form: the standard Gauss–Legendre quadrature method and the sub-elements (or integration cells, subcells) conformed to the interface. The EqP version of the XFEM investigated here takes advantage of the standard Gauss–Legendre quadrature method and can reproduce the convergence rates of the integration technique with sub-elements, especially for regular meshes. It is an alternative for modeling static or mobile interfaces where the use of the sub-elementation technique presents issues.

Appendices

A Numerical integration using sub-elements

The triangulation of elements crossed by a discontinuity is one of the most used strategies in the XFEM numerical integration. It consists of dividing the regions of a element crossed by interface into triangles and applying a numerical integration procedure, like the Gauss–Legendre procedure, within each sub-element. The triangular sub-elements are easily obtained for convex regions using the Delaunay triangulation.

Given an element with a crossing interface, what is the case of interest, the division into triangular sub-elements was given by the Delaunay triangulation, where each sub-triangle coordinate vertex at the Cartesian plane $\left( x,y\right)$ and the integration points, say the Gauss–Legendre integration quadrature, in the parametric system $\left( r,s\right)$ are known, so the coordinates of the integration points can be easily determined in the Cartesian plane by the use of the shape functions of the triangular element, Eqs. 55 and 56:

$$\begin{aligned}{} & {} x=(x_{2}-x_{1})r+(x_{3}-x_{1})s+x_{1}, \end{aligned}$$

(55)

$$\begin{aligned}{} & {} y=(y_{2}-y_{1})r+(y_{3}-y_{1})s+y_{1}. \end{aligned}$$

(56)

Figure 23 shows an element with triangular sub-elements and the coordinates transformation.

Fig. 23

Coordinate transformation: sub-triangles $\Omega _{\text {sub-e}}$ in a Q4 element $\Omega _{e}$ containing and interface $\Gamma$

Once the integration points in the Cartesian plane were known, it was necessary to transform them for the quadrilateral element’s parametric system $\left( \xi ,\eta \right)$. The mapping from local coordinates $\left( \xi ,\eta \right)$ to the Cartesian plane$\left( x,y\right)$ is performed by

$$\begin{aligned}{} & {} \varvec{x}=\sum \limits _{i}N_{i}\left( \xi ,\eta \right) \varvec{x}_{i}. \end{aligned}$$

(57)

The inverse of this mapping, $\left( x,y\right) \rightarrow \left( \xi ,\eta \right)$, is not simple because it involves a nonlinear system of equations that cannot be solved analytically, so a numerical approximation is necessary. There are several ways to solve this problem. In this paper, we have implemented the method proposed by Murti and Valliappan[17].

In Murti and Valliappan [17], the iteration procedure to determine the inverse mapping has been improved by the intersection of a defined line segment that passes through a point M, defined at the interior of the element, whose Cartesian coordinates are $\varvec{x}_{M}$ and a point P whose the local parametric coordinate, $\xi _{p}$ is known, such that a node of the vertex of the element.

For this characterization, the point M is assumed to be an integration point that needs to be determined in terms of the parametric coordinates, $\xi _{m}$, corresponding to the point $M'$, see Fig. 24.

Fig. 24

Inverse mapping

The segment $\overline{PQ}$ in the Cartesian plane can be easily determined by the linear equation of the line passing by the points P and M. This segment is associated with the curve $P'Q'$ on the local coordinate plane and passing through the point $M'$. Thus, the coordinate of the point $M'$, $\xi _{m}$, can be determined more efficiently when compared to a numerical method without any improvement condition.

The false position method was employed, Lim et al. [13], to determine the point $M'$ on the curve $P'Q'$. In Murti and Valliappan [17], the authors used the bisection method, but the number of iterations for this method was higher.

The point P cannot be any vertex of the element. It must be chosen so that the curve $P'Q'$ be defined in the element; that is, it must cover the entire axis $\eta$ at the interval $[-1,1]$. In this paper, when this occurs, the point P is redefined as the vertex opposite to the previous one, while in Murti and Valliappan [17], the authors choose to renumber the element nodes, so there is a transformation of the axis $\eta$ to $\xi$.

Once the integration points are obtained on the parametric system $\left( \xi ,\eta \right)$ the stiffness matrix can be integrated into each triangular sub-element using:

$$\begin{aligned} \textbf{K}_{\text {sub-e}}= & {} \int \limits _{\Omega _{\text {sub-e}}}\det \left[ \textbf{J}_{\text {sub-e}}\right] \textbf{B}^{T}\textbf{EB}\,\textrm{d}\Omega \nonumber \\{} & {} \simeq \sum \limits _{i=1}^{n}w_{i}\det \left[ \textbf{J}_{\text {sub-e}}\right] \textbf{B}^{T}\textbf{EB}, \end{aligned}$$

(58)

where, n is the number of integration points, $w_{i}$ is the weight. The matrices $\textbf{B}$ and $\textbf{E}$ depend on the variables $\eta$ and $\xi$. They are integrated at the points $\left( \xi _{i},\eta _{i}\right)$, the integration points of the triangular sub-element transformed into this coordinate system.

The element stiffness matrix $\textbf{K}_{e}$ is the summation of the sub-elements stiffness matrices $\textbf{K}_{\text {sub-e}}$, that is,

$$\begin{aligned} \textbf{K}_{e}^{SubE}=\sum \limits _{\text {sub-e}}^{nk}\textbf{K}_{\text {sub-e}}, \end{aligned}$$

(59)

where nk is the number of sub-elements in the element $\Omega _{e}$.

B Coefficients for the equivalent polynomials

The coefficients of the polynomial $\tilde{H}$ are the same presented by Ventura [25] and will not be presented in this paper. However, the coefficients for the equivalent polynomial $\tilde{Q}$ are different, see also [24]. When the interface intercepts adjacent sides of the element, the coefficients to the polynomial $\tilde{Q}$, are:

$$\begin{aligned}{} & {} \begin{aligned} q_{1}&=\frac{1}{192}\left( -6a\right. \left( 34\right. -8\xi _{1}-5\xi _{1}^{2}+5\xi _{1}^{3}-5\eta _{2}^{2}\left( 1+\xi _{1}\right) \\&\quad +5\eta _{2}^{3}\left( 1+\xi _{1}\right) +\left. \eta _{2}\left( -8-18\xi _{1}-5\xi _{1}^{2}+5\xi _{1}^{3}\right) \right) \\&\quad -\left( 1+\eta _{2}\right) \left( 1+\xi _{1}\right) \left( 2b(-7+\eta _{2}\left( 21-9\xi _{1}\right) +3\eta _{2}^{2}\left( -4+\xi _{1}\right) \right. \\&\quad +17\xi 1-18\xi _{1}^{2}\\&\quad \left. +9\xi _{1}^{3}\right) +2c\left( -7-18\eta _{2}^{2}+9\eta _{2}^{3}+21\xi _{1}-12\xi _{1}^{2}\right. \\&\quad \left. +\eta _{2}\left( 17-9\xi _{1}+3\xi _{1}^{2}\right) \right) \\&\quad +d\left( 14+3\eta _{2}^{3}\left( -5+\xi _{1}\right) -9\eta _{2}^{2}\left( -3+\xi _{1}\right) \right. \\&\quad \left. -24\xi _{1}+27\xi _{1}^{2}-15\xi _{1}^{3}\right) \\&\quad +\eta _{2}\left. \left. \left( -24+22\xi _{1}-9\xi _{1}^{2}+3\xi _{1}^{3}\right) \right) \right) . \end{aligned} \end{aligned}$$

(60a)

$$\begin{aligned}{} & {} \begin{aligned} q_{2}&=\frac{1}{2240}\left( -210a\right. \left( 1+\eta _{2}\right) \left( 3+\eta _{2}\left( 7-3\xi _{1}\right) +\eta _{2}^{2}\left( -4+\xi _{1}\right) \right) \\&\quad \left( 1+\xi _{1}\right) \\&\quad - 70b\left( 2\right. 4-6\xi _{1}+3\xi _{1}^{2}+\xi _{1}^{3}-3\eta _{2}\left( 8+4\xi _{1}-3\xi _{1}^{2}+\xi _{1}^{3}\right) +\eta _{2}^{3}\\&\quad \left( 10+6\xi _{1}-3\xi _{1}^{2}+\xi _{1}^{3}\right) -3\left. \eta _{2}^{2}\left( 2-\xi _{1}^{2}+\xi _{1}^{3}\right) \right) \\&\quad -3\left( 1+\eta _{2}\right) \left( 1+\xi _{1}\right) \\&\quad \left( 35\right. c\left( -1+\eta _{2}\right) ^{2}\left( -1+\eta _{2}\left( -5+\xi _{1}\right) -\xi _{1}\right) \\&\quad +2d(19+3\xi _{1}-16\xi _{1}^{2}\\&\quad + 5\eta _{2}^{3}\left( 15-5\xi _{1}+\xi _{1}^{2}\right) -5\eta _{2}^{2}\left( 25-13\xi _{1}+4\xi _{1}^{2}\right) \\&\quad +\left. \eta _{2}\left( 29-47\xi _{1}+29\xi _{1}^{2}\right) \right) . \end{aligned} \end{aligned}$$

(60b)

$$\begin{aligned}{} & {} \begin{aligned} q_{3}&=\frac{1}{2240}\left( -\right. 105b\left( 1+\eta _{2}\right) \left( -1+\eta _{2}\left( -1+\xi _{1}\right) -5\xi _{1}\right) \\&\quad \left( -1+\xi _{1}\right) ^{2}\left( 1+\xi _{1}\right) \\&\quad -210a\left( 1+\eta _{2}\right) \left( 1+\xi _{1}\right) \left( 3+\left( 7-3\eta _{2}\right) \xi _{1}+\left( -4+\eta _{2}\right) \xi _{1}^{2}\right) \\&\quad - 2\left( 35c\right. \left( 2\right. 4-24\xi _{1}-6\xi _{1}^{2}+10\xi _{1}^{3}+\eta _{2}^{2}\left( 3+9\xi _{1}+3\xi _{1}^{2}\right. \\&\quad \left. -3\xi _{1}^{3}\right) \\&\quad +6\eta _{2}\left( -1-2\xi _{1}+\xi _{1}^{3}\right) +\eta _{2}^{3}\left( 1-3\xi 1-3\xi _{1}^{2}+\xi _{1}^{3}\right) \\&\quad + 3d\left( 1+\eta _{2}\right) \left( 1+\xi _{1}\right) \left( 19+29\xi _{1}-125\xi _{1}^{2}+75\xi _{1}^{3}+\right. \\&\quad \eta _{2}\left( 3-47\xi _{1}+65\xi _{1}^{2}-25\xi _{1}^{3}\right) +\eta _{2}^{2}\left( -16+29\xi _{1}\right. \\&\quad \left. -20\xi _{1}^{2}+5\xi _{1}^{3}\right) . \end{aligned} \end{aligned}$$

(60c)

$$\begin{aligned}{} & {} \begin{aligned} q_{4}&=\frac{1}{80}\left( 15a\right. \left( 1+\eta _{2}\right) \left( 5+\eta _{2}\left( -3+\xi _{1}\right) -3\xi _{1}\right) \left( 1+\xi _{1}\right) \\&\quad + 2\left( 3c\left( 1+\eta _{2}\right) \left( 1+\xi _{1}\right) \left( -9+\eta _{2}\left( 7-3\xi _{1}\right) +\eta 2^{2}\left( -4+\xi _{1}\right) \right. \right. \\&\quad \left. +6\xi _{1}\right) \\&\quad + 3b\left( 1+\eta _{2}\right) \left( 1+\xi _{1}\right) \left( -9+7\xi _{1}-4\xi _{1}^{2}\right. \\&\quad \left. +\eta _{2}\left( 6-3\xi _{1}+\xi _{1}^{2}\right) \right) \\&\quad +d\left( -\right. 21+3\xi _{1}-6\xi _{1}^{2}+10\xi _{1}^{3}+\eta _{2}^{3}\left( 10+6\xi _{1}-3\xi _{1}^{2}+\xi _{1}^{3}\right) \\&\quad - 3\eta _{2}^{2}\left( 2-\xi _{1}^{2}+\xi _{1}^{3}\right) +\eta _{2}\left( 3-3\xi _{1}+6\xi _{1}^{3}\right) . \end{aligned} \end{aligned}$$

(60d)

$$\begin{aligned}{} & {} \begin{aligned} q_{5}&=\frac{1}{64}\left( 1+\eta _{2}\right) \left( 1+\xi _{1}\right) (-14c+17d+16c\eta _{2}\\&\quad -15d\eta _{2}+30a\left( -1+\xi _{1}\right) ^{2}\\&\quad +42c\xi _{1}-21d\xi _{1}-18c\eta _{2}\xi _{1}+13d\eta _{2}\xi _{1}-24c\xi _{1}^{2}\\&\quad +27d\xi _{1}^{2}+6c\eta _{2}\xi _{1}^{2}\\&\quad - 9d\eta _{2}\xi _{1}^{2}-15d\xi _{1}^{3}+3d\eta _{2}\xi _{1}^{3}\\&\quad +2b\left( -16+17\xi _{1}-18\xi _{1}^{2}+9\xi _{1}^{3}\right) . \end{aligned} \end{aligned}$$

(60e)

$$\begin{aligned}{} & {} \begin{aligned} q_{6}&=\frac{1}{64}\left( 1+\eta _{2}\right) \left( 1+\xi _{1}\right) (-32c+17d+30a\left( -1+\eta _{2}\right) ^{2}\\&\quad +34c\eta _{2}-21d\eta _{2}-36c\eta _{2}^{2}\\&\quad +27d\eta _{2}^{2}+18c\eta _{2}^{3}-15d\eta _{2}^{3}-15d\xi _{1}+13d\eta 2\xi _{1}\\&\quad -9d\eta _{2}^{2}\xi _{1}+3d\eta _{2}^{3}\xi _{1}\\&\quad + 2b\left. \left( -7+\eta _{2}\left( 21-9\xi _{1}\right) +3\eta _{2}^{2}\left( -4+\xi _{1}\right) +8\xi _{1}\right) \right) . \end{aligned} \end{aligned}$$

(60f)

$$\begin{aligned}{} & {} \begin{aligned} q_{7}&=\frac{3}{448}\left( 1+\eta _{2}\right) \left( 1+\xi _{1}\right) (14a\left( -7+21\xi _{1}-12\xi _{1}^{2}\right. \\&\quad \left. +\eta _{2}\left( 8-9\xi _{1}+3\xi _{1}^{2}\right) \right) \\&\quad +7b\left( 17-21\xi _{1}+27\xi _{1}^{2}-15\xi _{1}^{3}\right. \\&\quad \left. +\eta _{2}\left( -15+13\xi _{1}-9\xi _{1}^{2}+3\xi _{1}^{3}\right) \right) \\&\quad +2\left( 7\right. c\left( \eta _{2}^{2}\left( 5-4\xi _{1}+\xi _{1}^{2}\right) -2\eta _{2}\left( 3-5\xi _{1}+2\xi _{1}^{2}\right) \right. \\&\quad \left. +2\left( 2-8\xi _{1}+5\xi _{1}^{2}\right) \right) \\&\quad + d\left( -3\right. 9+51\xi _{1}-75\xi _{1}^{2}+45\xi _{1}^{3}\\&\quad +\eta _{2}\left( 41-45\xi _{1}+39\xi _{1}^{2}-15\xi _{1}^{3}\right) \\&\quad \left. \left. \left. +\eta _{2}^{2}\left( -32+23\xi _{1}-12\xi _{1}^{2}+3\xi _{1}^{3}\right) \right) \right) \right) . \end{aligned} \end{aligned}$$

(60g)

$$\begin{aligned}{} & {} \begin{aligned} q_{8}&=\frac{3}{448}\left( 1+\eta _{2}\right) \left( 1+\xi _{1}\right) (119c-78d-147c\eta _{2}\\&\quad +102d\eta _{2}+189c\eta _{2}^{2}\\&\quad -150d\eta _{2}^{2}-105c\eta _{2}^{3}+90d\eta _{2}^{3}-105c\xi _{1}+82d\xi _{1}\\&\quad +91c\eta _{2}\xi _{1}-90d\eta _{2}\xi _{1}\\&\quad -63c\eta _{2}^{2}\xi _{1}+78d\eta _{2}^{2}\xi _{1}+21c\eta _{2}^{3}\xi _{1}-30d\eta _{2}^{3}\xi _{1}\\&\quad -64d\xi _{1}^{2}+46d\eta _{2}\xi _{1}^{2}\\&\quad -24d\eta _{2}^{2}\xi _{1}^{2}+6d\eta _{2}^{3}\xi _{1}^{2}+14a\left( -7+\eta _{2}\left( 21-9\xi _{1}\right) \right. \\&\quad \left. +3\eta 2^{2}\left( -4+\xi _{1}\right) +8\xi _{1}\right) \\&\quad +14b\left( 4-6\xi _{1}+5\xi _{1}^{2}+\eta _{2}^{2}\left( 10-4\xi _{1}+\xi _{1}^{2}\right) \right. \\&\quad \left. \left. -2\eta _{2}\left( 8-5\xi _{1}+2\xi _{1}^{2}\right) \right) \right) . \end{aligned} \end{aligned}$$

(60h)

And when the interface intercepted opposite sides of the element, the coefficients to the polynomial $\tilde{Q}$, are

$$\begin{aligned}{} & {} \begin{aligned} q_{1}&=\frac{1}{48}\left( -\right. 3a\left( \xi _{1}+\xi _{2}\right) \left( -18+5\xi _{1}^{2}+5\xi _{2}^{2}\right) -b(9+9\xi _{1}^{4}\\&\quad +9\xi _{1}^{3}\xi _{2}-16\xi _{2}^{2}+9\xi _{2}^{4}+\xi _{1}\xi _{2}\left( -22+9\xi _{2}^{2}\right) \\&\quad +\xi _{1}^{2}\left( -16+9\xi _{2}^{2}\right) )+\left( \xi _{1}-\xi _{2}\right) \left( d\right. \left( \xi _{1}+\xi _{2}\right) \\&\quad \left( -5+6\xi _{1}^{2}+3\xi _{1}\xi _{2}+6\xi _{2}^{2}\right) \\&\quad \left. \left. +c\left( -10+9\xi _{1}^{2}+12\xi 1\xi _{2}+9\xi _{2}^{2}\right) \right) \right) . \end{aligned} \end{aligned}$$

(61a)

$$\begin{aligned}{} & {} q_{2}=\frac{1}{280}\left( 3d\left( \xi _{1}-\xi _{2}\right) ^{3}+35b\left( \xi _{1}+\xi _{2}\right) ^{3}\right. \nonumber \\{} & {} \left. +105a\left( -2+\xi _{1}+\xi _{2}\right) \left( 2+\xi _{1}+\xi _{2}\right) \right) . \end{aligned}$$

(61b)

$$\begin{aligned}{} & {} \begin{aligned} q_{3}&=\frac{1}{560}\left( -\right. 3d\left( 35+\right. 55\xi _{1}^{4}+25\xi _{1}^{3}\xi _{2}-84\xi _{2}^{2}+55\xi _{2}^{4}\\&\quad +3\xi _{1}^{2}\left( -28+5\xi _{2}^{2}\right) +\xi _{1}\xi _{2}\left. \left( -42+25\xi _{2}^{2}\right) \right) \\&\quad +35\left( 3\right. a\left( \xi _{1}-\xi _{2}\right) \\&\quad \left( -6+3\xi _{1}^{2}+4\xi _{1}\xi _{2}+3\xi _{2}^{2}\right) +\left( \xi _{1}+\xi _{2}\right) \\&\quad \left( c\left( 18-7\xi _{1}^{2}+4\xi _{1}\xi _{2}-7\xi _{2}^{2}\right) +3b\left( \xi _{1}-\xi _{2}\right) \right. \\&\quad \left. \left. \left( -3+2\xi _{1}^{2}+\xi _{1}\xi _{2}+2\xi _{2}^{2}\right) \right) \right) . \end{aligned} \end{aligned}$$

(61c)

$$\begin{aligned}{} & {} \begin{aligned} q_{4}&=\frac{1}{20}\left( \xi _{1}\right. ^{2}\left( -15a-9b\xi _{1}+7d\xi _{1}\right) +3\left( -b+d\right) \xi _{1}^{2}\xi _{2}\\&\quad + 3\left( 5a+\left( b+d\right) \xi _{1}\right) \xi _{2}^{2}+\left( 9b+7d\right) \xi _{2}^{3}\\&\quad +\left. 6c\left( -5+2\xi _{1}^{2}+\xi _{1}\xi _{2}+2\xi _{2}^{2}\right) \right) . \end{aligned} \end{aligned}$$

(61d)

$$\begin{aligned}{} & {} \begin{aligned} q_{5}&=\frac{1}{16}\left( 1\right. 5a\left( \xi _{1}+\xi _{2}\right) \left( -2+\xi _{1}^{2}+\xi _{2}^{2}\right) \\&\quad + b\left( -15+9\xi _{1}^{4}+9\xi _{1}^{3}\xi _{2}-10\xi _{2}^{2}+9\xi _{2}^{4}+\xi _{1}^{2}\left( -10+9\xi _{2}^{2}\right) \right. \\&\quad \left. +\xi _{1}\xi _{2}\left( -10+9\xi _{2}^{2}\right) \right) -\left( \xi _{1}-\xi _{2}\right) (d\left( \xi _{1}+\xi _{2}\right) \\&\quad \left. \left. +c\left( -10+9\xi _{1}^{2}+12\xi _{1}\xi _{2}+9\xi _{2}^{2}\right) \right) \right) . \end{aligned} \end{aligned}$$

(61e)

$$\begin{aligned}{} & {} q_{6}=-\frac{1}{8}\left( \xi _{1}-\xi _{2}\right) \left( 4c+b\left( -\xi _{1}+\xi _{2}\right) +2d\left( \xi _{1}+\xi _{2}\right) \right) . \end{aligned}$$

(61f)

$$\begin{aligned}{} & {} \begin{aligned} q_{7}&=-\frac{3}{112}\left( d\right. \left( 35\right. -33\xi _{1}^{4}-15\xi _{1}^{3}\xi _{2}+28\xi _{2}^{2}-33\xi _{2}^{4}+\xi _{1}\xi _{2}\\&\quad \left( 14-15\xi _{2}^{2}\right) +\xi _{1}^{2}\left. \left( 28-9\xi _{2}^{2}\right) \right) +7\left( a\right. \left( \xi _{1}-\xi _{2}\right) \\&\quad \left( -10+9\xi _{1}^{2}+12\xi _{1}\xi _{2}+9\xi _{2}^{2}\right) +\left( \xi _{1}+\xi _{2}\right) \\&\quad \left( c\left( 10-7\xi _{1}^{2}+4\xi _{1}\xi _{2}-7\xi _{2}^{2}\right) +\right. \\&\quad \left. \left. b\left( \xi _{1}-\xi _{2}\right) \left( -5+6\xi _{1}^{2}+3\xi _{1}\xi _{2}+6\xi _{2}^{2}\right) \right) \right) . \end{aligned} \end{aligned}$$

(61g)

$$\begin{aligned}{} & {} \begin{aligned} q_{8}&=\frac{3}{56}\left( \xi _{1}-\xi _{2}\right) (\left( 7b-9d\right) \xi _{1}^{2}\\&\quad +7a\left( \xi _{1}-\xi _{2}\right) -10d\xi _{1}\xi _{2}-\left( 7b+9d\right) \xi _{2}^{2}-14c\left( \xi _{1}+\xi _{2}\right) . \end{aligned} \end{aligned}$$

(61h)