Stable Dynamical Adaptive Mesh Refinement

We consider accurate and stable interpolation procedures for numerical simulations utilizing time dependent adaptive meshes. The interpolation of numerical solution values between meshes is considered as a transmission problem with respect to the underlying semi-discretized equations, and a theoretical framework using inner product preserving operators is developed, which allows for both explicit and implicit implementations. The theory is supplemented with numerical experiments demonstrating practical benefits of the new stable framework. For this purpose, new interpolation operators have been designed to be used with multi-block finite difference schemes involving non-collocated, moving interfaces.

How tissue fluidity influences brain tumor progression

Mechanical properties of biological tissues and, above all, their solid or fluid behavior influence the spread of malignant tumors. While it is known that solid tumors tend to have higher mechanical rigidity, allowing them to aggressively invade and spread in solid surrounding healthy tissue, it is unknown how softer tumors can grow within a more rigid environment such as the brain. Here, we use in vivo magnetic resonance elastography (MRE) to elucidate the role of anomalous fluidity for the invasive growth of soft brain tumors, showing that aggressive glioblastomas (GBMs) have higher water content while behaving like solids. Conversely, our data show that benign meningiomas (MENs), which contain less water than brain tissue, are characterized by fluid-like behavior. The fact that the 2 tumor entities do not differ in their soft properties suggests that fluidity plays an important role for a tumor’s aggressiveness and infiltrative potential. Using tissue-mimicking phantoms, we show that the anomalous fluidity of neurotumors physically enables GBMs to penetrate surrounding tissue, a phenomenon similar to Saffman−Taylor viscous-fingering instabilities, which occur at moving interfaces between fluids of different viscosity. Thus, targeting tissue fluidity of malignant tumors might open horizons for the diagnosis and treatment of cancer.

Finite Element Method for Fluid Flow in 3D Domains Containing Moving Interfaces

This study presented a three-dimensional (3D) finite element method (FEM) for the numerical analysis of fluid flow in domains containing moving interfaces. This method falls into the general category of Arbitrary Lagrangian Eulerian (ALE) method; based on a fixed mesh that is locally adapted at the moving interfaces and reverts to its original shape once the moving interfaces go over the elements. The 3D domain occupied by the fluid at any time in the simulation is used as the reference domain and is discretized using a mesh of hexahedral tri-linear isoparametric finite elements. The moving interfaces are defined by sets of marker points so that the global mesh is independent of interface movement and eliminates the possibility of mesh entanglement. The mesh never becomes unsuitable due to its continuous deformation, thus eliminating the need for repeated re-meshing and interpolation. A validation is presented via a problem with an analytical solution for the 3D flow between two planes separating at a prescribed speed that shows second order accuracy. The model’s capabilities are illustrated through application to laminar incompressible flows in different geometrical settings that show the flexibility of the technique.