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Additional DOFs \(a_j\) and \(c_k^l\) are usually included with collection \(J\) as well as \(M\) for you to determine discontinuity pertaining to XFEM. The actual collection \(I\) consists of each nodes within arranged \(J\)and hang \(M\) the next: $$\beginaligned L,M\in My spouse and i,J\cap M=\phi \endaligned$$ (Three or more)Whenever factors are totally lower with the break, just about all linked nodes revealed since the reddish hollow circle throughout Fig.?3 from the established \(J\). To be able to signify the particular discontinuous qualities associated with arranged \(J\), Heaviside phase capabilities tend to be included the 2nd expression, that are the next: $$\beginaligned H(a)=\left\{ \beginarrayll 1,(x-x^*)\cdot e_n \ge 0 \\ -1,(x-x^*)\cdot e_n \le 0 \endarray \right. \endaligned$$ (4)where \(x\) is the position of a point in set \(J,\,x^* \) is the position of the point on the crack that is closest to \(x,\,e_n\) is the outward normal to the crack at \(x^* \). Fig. Selleckchem PLX5622 3 The schematic of XFEM. The nodes are shown as the red hollow circle when elements are completely cut by the crack. When elements are completely cut by the crack, all related nodes are shown as the red solid square When elements are partially cut by MLN7243 research buy the crack, the related nodes shown as the red solid square in Fig.?3 form set M. In the third term of Eq.?(2), to represent the discontinuous characteristics of \(M,\,F_l \) combine the radial and angular behavior of the asymptotic linear elastic crack tip displacement, which are as follows: $$\beginaligned \left\ F_l (r,\theta ) \right\&= \left\ \sqrtr\sin \frac\theta 2,\sqrtr\cos \frac\theta 2,\quad \sqrtr\sin \frac\theta 2\sin \theta ,\sqrtr\cos \frac\theta 2\sin \theta \right\ \endaligned$$ (5)where \(r\) and \(\theta \) are the normal and direction vectors of the position of a point \(x\) of set \(M\), respectively. Acquisition of BCs To achieve a unique solution to the XFEM equations (1), two types of BCs need to be determined and imposed. One is the zero displacement BC, which is traction and stress free. The zero displacement area of brain tissue is described as the location of the brain stem and manually selected by experienced doctors. The other type of BC is the displacement of certain crack-related nodes in the operating field where Dabrafenib two retractors are inserted into brain issue. This BC can be calculated by the brain retraction surface-tracking algorithm. These two types of BCs are imposed to the XFEM-based linear elastic model to achieve a unique solution and updated mesh, from which the displacement of all nodes and crack-related information can be obtained. Biomechanical model updating The updated mesh will be exploited by the modified back-interpolation algorithm, which is not only able to warp the pre-retraction images and use ray casting to visualize them?[41, 42], but also able to recognize the crack generated by brain retraction. Regarding crack representation, our modified algorithm can discriminate two level sets?[41, 42].