6.2 Remeshing procedure
6.2.1 Remeshing criteria
In order to decide when the analysis has to be aborted and, thus, the remeshing has to be initiated, we consider several remeshing criteria. For this, the most obvious choice is to consider the convergence behavior of the Newton-Raphson procedure, meaning that the analysis is aborted if the Newton-Raphson iteration exceeds its maximum value. In the context of the FCM, this is mainly induced by cut finite cells – since these cells, usually, are susceptible to self-penetration (detF <0) due to large deformations of the fictitious domain. However, in some cases it is of advantage to abort the analysis before the Newton-Raphson method fails. Consequently, specific criteria are needed to evaluate the quality of the deformed finite cells. Keep in mind that the undeformed mesh of each analysis ensures a good mesh quality since Cartesian grids or structured meshes are utilized. In the following, as a means to evaluate the quality of the deformed mesh, we propose three additional criteria to decide whether remeshing has to be initiated or not. Further, all criteria are based on the Jacobi matrix. For an analysis from configurationttot+ 1, the Jacobi matrix of the deformed mesh is consequently given as
Jt=
and the corresponding Jacobi matrix of the deformed mesh is defines as jt+1= ˜Ft+1Jt=
Here,Giandgiare the columns of the Jacobi matrix and denote the tangential vectors.
Keep in mind that, here, we use bold and upright symbols for the vector and matrix quantities in order to distinguish them from tensor ones. Finally, the criteria may be evaluated for different point sets – whereby, in this thesis, we choose the quadrature points xk as the evaluation points. Further, in order to ease the subsequent formulations, we introduce the following abbreviations for the Jacobi matrices
Jk=Jt(xk) and jk=jt+1(xk) , (6.12) the tangential vectors
Gki =Gi(xk) and gki =gi(xk) , (6.13)
6.2 Remeshing procedure
and the deformation gradient
F˜k= ˜Ft+1(xk) . (6.14)
6.2.1.1 Ratio of Jacobians
In the first case we employ a criterion based on the ratio of Jacobians [162–164]. To this end, for each finite cell c, we determine the minimum and maximum values of the determinant of the Jacobi matrix of the deformed mesh and compute the ratio
R=
mink detjk
maxl detjl . (6.15)
Consequently, the ratio of Jacobians evaluates the ratio of volumetric deformations con-sidering its extreme values of a predefined point set. Since the condition detjk>0 has to be satisfied for each point within the cell, the range ofRis given as
0< R≤1 . (6.16)
Thereby, R= 1 defines the optimal case and values close to zero characterize cells of a poor quality. Finally, note that employing Cartesian grids or structured meshes – which is generally the case when applying the FCM – the ratio of Jacobians simplifies to
R= min
k det ˜FkdetJk maxl det ˜FldetJl =
min
k det ˜Fk
maxl det ˜Fl (6.17)
since, with regard to the rectangular cells, the following condition holds for the ratio of Jacobians of the undeformed mesh
detJk
detJl = 1 . (6.18)
6.2.1.2 Orthogonality
In the second criterion, we introduce a measurement evaluating the mesh quality by con-sidering the orthogonality property of each finite cell. Thereby, the cells of the undeformed mesh are optimal in this sense since they include a rectangular shape. Consequently, the criterion has to be developed for the cells of the deformed mesh. Therefore, we proceed as suggested in [165]. In doing so, we introduce the following orthogonality criterion
O= min
i,j,k r
gki·gki gkj·gkj−gki ·gkj2 r
gik·gki gkj·gjk
. (6.19)
which is based on the tangential vectors of the deformed Jacobi matrix, see Eq. (6.11).
Since the condition detjk > 0 holds at each point – which implies that the tangential vectorsgki are linearly independent – the range of the orthogonality criterion is defined as
0< O≤1 . (6.20)
Thereby,O= 1 defines the optimal case and values close to zero indicate cells of a poor quality.
6 A remeshing strategy for the FCM
6.2.1.3 Inverse aspect ratio
As the third criterion, we consider a measurement based on the aspect ratio of the tangen-tial vectors. To this end, we introduce a criterion relating the aspect ratio of the deformed cell with the inverse aspect ratio of the undeformed one. In doing so, the definition of the criterion is given as
A= min
i,j,k gki
2 gkj
2 Gkj
2 Gki
2
. (6.21)
Since the tangential vectors are linearly independent – due to condition detjk>0 – the range ofAis defined as
0< A≤1 . (6.22)
6.2.1.4 Performance of the suggested remeshing criteria
Finally, let us have a look at the performance of the presented remeshing criteria. There-fore, we consider a single cube that is subjected to different load cases: uniaxial pressure, shear, and compression. For the investigations, the cube is discretized by one finite cell.
Further, a hyperelastic and isotropic material behavior based on a polyconvex strain en-ergy density function is assumed. A brief description of the material model is provided in Sec. 2.3.2. Further, the material parameters are listed in Tab. 5.2.
The values of the remeshing criteria of the different cases are plotted in Fig. 6.3, 6.4, and 6.5, respectively. Fig. 6.3 shows the values of the remeshing criteria considering the cube subjected to uniaxial pressure. From the figure it can be seen that due to the deformation of the cell the ratio of Jacobians (R= 1) and the orthogonality (O= 1) do not change during the loading. This is because the deformed cell has a rectangular shape and the determinant of the deformation gradient has the same value at each quadrature point.
However, the inverse aspect ratio changes during the loading. Further, the values of the remeshing criteria for the cube under shear loading are plotted in Fig. 6.4. Here, the values of the orthogonality and the inverse aspect ratio change during loading, while the ratio of the Jacobians (R= 1) does not. Finally, Fig. 6.5 shows the results of the cube under compression. In this special case, all remeshing criteria are not changed during the loading.
6.2 Remeshing procedure
0 0.2 0.4 0.6 0.8 1 1.2
0 2 4 6 8 10 12 14 16
criterion
load step R
O A
Figure 6.3:Remeshing criteria for a single cube under uniaxial pressure.
0 0.2 0.4 0.6 0.8 1 1.2
0 2 4 6 8 10 12 14 16
criterion
load step R
O A
Figure 6.4:Remeshing criteria for a single cube under shear loading.
0 0.2 0.4 0.6 0.8 1 1.2
0 2 4 6 8 10 12 14 16
criterion
load step R
O A
Figure 6.5:Remeshing criteria for a single cube under compression.
6 A remeshing strategy for the FCM