Space–time interface-tracking with topology change (ST-TC)

Abstract

To address the computational challenges associated with contact between moving interfaces, such as those in cardiovascular fluid–structure interaction (FSI), parachute FSI, and flapping-wing aerodynamics, we introduce a space–time (ST) interface-tracking method that can deal with topology change (TC). In cardiovascular FSI, our primary target is heart valves. The method is a new version of the deforming-spatial-domain/stabilized space–time (DSD/SST) method, and we call it ST-TC. It includes a master–slave system that maintains the connectivity of the “parent” mesh when there is contact between the moving interfaces. It is an efficient, practical alternative to using unstructured ST meshes, but without giving up on the accurate representation of the interface or consistent representation of the interface motion. We explain the method with conceptual examples and present 2D test computations with models representative of the classes of problems we are targeting.

Appendix A: Element degeneration

1.1 A. 1 Degeneration identification

We identify the degeneration by utilizing an index set $\mathbf {a} = (a_0, a_1,\ldots ,a_{n_{\mathrm {en}}-1})$, obtained by mapping from the nodal indices of the parent element domain, $k = 0,1,\ldots ,n_{\mathrm {en}}-1$. In this mapping, when there is a degeneration, the lowest nodal index maps to itself as the master index, and the nodal indices of the slaves map to that master index. For example, for a quadrilateral element $(n_{\mathrm {en}}=4)$ with degeneration involving nodal indices 0 and 3, the mapping becomes $\mathbf {a} = (0, 1, 2, 0)$. As another example, we note that the representation $\mathbf {a} = (0, 0, 1, 3)$ would not be considered a valid one in our implementation, because the third mapped index 1 belongs to a slave that has already been mapped to 0. The valid representation in that case would be $\mathbf {a} = (0, 0, 0, 3)$.

We serialize the index set $\mathbf {a}$ with the serial ID number “$s$”, as given by the following expression:

$$\begin{aligned} s\left( \mathbf {a}\right) = \sum \limits _{k=0}^{n_{\mathrm {en}}-1} a_k k!. \end{aligned}$$

(2)

We note that in our implementation strategy, initially the kind of invalid index sets mentioned above are included in the serialization, but eventually they are excluded from the list of acceptable index sets. We also note that, if needed, the index set can be recovered from the ID number with the following modulo operation:

$$\begin{aligned} a_k = \frac{s}{k!} \mod (k+1). \end{aligned}$$

(3)

1.2 A.2 Hexahedral element

Figures 35, 36, 37, 38 and 39 show the degeneration modes of the hexahedral element, excluding those that lead to zero volume. Table 2 shows, for the degeneration modes that do not lead to zero volume, the sets of serial IDs that can be obtained by rotating the parent element to different sets of nodal indices.

Fig. 35

Tetrahedral type

Fig. 36

Pyramid type

Fig. 37

Wedge type

Fig. 38

7-noded type

Fig. 39

Hexahedral type

Table 2 Serial ID numbers for the degeneration modes of the hexahedral element, excluding those that lead to zero volume, and the alternative ID numbers for the modes obtained by rotating the parent element to different sets of nodal indices

We note that an element with its ID number listed in Table 2 could still be invalid, not because of degeneration, but because it could possibly have a negative Jacobian.