Data transfer

In order to proceed with the analysis, as the final step of the remeshing procedure, we have to transfer the necessary data from the old to the new mesh, e.g. displacement-based variables or history variables if nonlinear material models are involved. In this thesis, we introduce alocal radial basis function (RBF) interpolation scheme [167–169] to perform the data transfer. Thereby, we consider the quadrature points of the old mesh as the source points and the quadrature points of the new mesh as the target points. Fig. 6.7 shows the graphical interpretation withI denoting the interpolation operator. Note that the fictitious quadrature points of the old mesh are excluded from the set of the source points, while the set of target points include the fictitious quadrature points. In the following, a source point is denoted asx^sand a target point asx^t.

ξ η

Q₀,J₀ Ω

X=x0

O x₁

˜ u1

q₁,j₁=F˜₁J₀

F˜1 ϕ₁(Ω)

Ω=ϕ₀(Ω)

O x₁

ξ η

Ω Q1,J1

H₁^∗=I(H1)

ϕ1(Ω)

Figure 6.7:Data transfer between old and new mesh where the fictitious quadrature points of the old mesh are excluded from the set of the source points.

Next, we briefly describe the basic idea of thelocalRBF interpolation scheme. In order to compute the values of a target pointx^t, we start off by searching for thenⁿ nearest source points ofx^tand deposit their indices in a setN. So as to find the nearest neighbor, we thereby employ ak-d tree [169, 170] in order to reduce the effort during the searching procedure. For the target pointx^t, we then set up an interpolation scheme that is based on the source points of setN. In the same way, we proceed with the remaining target points. Consequently, for each target point an individual interpolation is generated which can be applied in parallel. Further, in order to distinguish this variant from its global one – where only one global interpolation scheme is generated considering all source points at the same time – we call itlocalRBF interpolation.

6.2 Remeshing procedure

Then, given the setN including thenⁿ nearest source points, an individual valuev^tof a target point is computed by a weighted sum as

v^t=^X

i∈N

λiθx^t−x^s_i

2

. (6.24)

In Eq. (6.24)λidenotes the weight andθ(r) defines the related scalar-valued radial basis function where the argument is the Euclidean norm of the distance between the target pointx^tand the source pointx^s_i. Consequently, in order to computev^t, we first have to determine the unknown weightsλi. To this end, thenⁿunknown weightsλiare computed by solving a linear system of equations

v^s_j=^X

i∈N

λiθx^s_i−x^s_j

2

, j∈N (6.25)

that is obtained by incorporating the interpolation conditions of the source points – where v_j^sdenotes the related value of source pointx^s_j. Further, as the choice forθ(r), commonly used radial basis functions, for instance, are

• theGaussian function (GF):

θ(r) =e^−r2 , (6.26)

• themultiquadric (MQ):

θ(r) =√

1 +r² , (6.27)

• theinverse multiquadric(IMQ):

θ(r) = 1

√1 +r² , (6.28)

• or thethin plate spline(TPS):

θ(r) =r²ln(r) . (6.29)

These RBFs are plotted in Fig. 6.8.

The choice of the RBF is problem-dependent. In the presented remeshing strategy, an important variable for the data transfer is given by the deformation gradient. At each remeshing step, we have to transfer the total deformation gradient of the old mesh to the new one in order to proceed with the simulation. In this thesis, the transfer of the defor-mation gradient is carried out by taking advantage of its relation with the displacement gradient (F =H+I). Consequently, for the remeshing step illustrated in Fig. 6.7, the displacement gradient is interpolated as

H₁^∗=I(H1) (6.30)

where the superscript^∗is introduced in order to distinguish the approximated variable from its original one. In doing so, an approximation of the deformation gradient is obtained as F₁^∗=H₁^∗+I=I(H1) +I . (6.31)

6 A remeshing strategy for the FCM

Figure 6.8:Commonly used radial basis functions.

The reason for interpolating the displacement gradient instead of the deformation gradient is that they have different characteristics. In the case of an undeformed body, the defor-mation gradient corresponds to the identity (F = I), while the displacement gradient, on the other hand, equals zero (H =0). Having recalled the characteristics ofF and H, next, let us again consider the case depicted in Fig. 6.7. As already mentioned, the fictitious quadrature points of the old mesh are excluded from the set of source points while the fictitious quadrature points in the new mesh are included in the set of target points. Consequently, the data transfer based on thelocalRBF can be differentiated in an interpolation phase (from physical source to physical target points) and an extrapolation phase (from physical source to fictitious target points). For the interpolation of the physi-cal points, we aim to achieve target values that are close to the source values. During the extrapolation phase, on the other hand, we intend to obtain a smooth transition of the displacement gradient from the physical into the fictitious domain. In doing so, the goal is to achieve a zero displacement gradient (H =0) for fictitious target points that are placed far away from the physical source points. A radial basis function that complies well with the interpolation and extrapolation requirements of the displacement gradient is the inverse multiquadricRBF given in Eq. (6.28). In this thesis, we utilize a modification of theinverse multiquadricRBF in which the input argument is scaled for each source point individually as follows

In Eq. (6.32),β denotes a scaling factor and ¯ridefines the mean distance of source point x^s_i with respect to itsn^rnearest neighbors. These additional parameters allow to further tune the RBF interpolation.

Summarizing, we set up an individual interpolation scheme for each target point x^t, based on a modification of theinverse multiquadricRBF ˜ϕi(r). To this end, we have to define the following parameters:

• the number of source points per target pointnⁿ whose indices are deposited in set N,

6.3 Finite strain problems

• the number of nearest neighborsn^rper source point in order to compute the mean distance ¯ri, and

• the scaling factorβ.

Thereby, practical experience has shown thatnⁿ= 50,n^r= 3, andβ= 1, . . . ,2 are a good choice.

6.3 Finite strain problems

In this section, we investigate the performance of the remeshing strategy considering prob-lems that undergo large deformations. For all examples, we assume a hyperelastic and isotropic material behavior based on a polyconvex strain energy density function. A brief explanation of the underlying equations of the constitutive model is provided in Sec. 2.3.2.

Further, the material parameters used for all examples are listed in Tab. 5.2.