Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems I: Well-Posedness

In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations.

Fireshape: a shape optimization toolbox for Firedrake

We introduce Fireshape, an open-source and automated shape optimization toolbox for the finite element software Firedrake. Fireshape is based on the moving mesh method and allows users with minimal shape optimization knowledge to tackle with ease challenging shape optimization problems constrained to partial differential equations (PDEs).

Numerical Analysis on the Thermal Performance in an Excavating Roadway with Auxiliary Ventilation System

A steady and proper thermal environment in deep underground is imperative to ensure worker health and production safety. Understanding the thermal performance in the roadway is the premise of temperature prediction; ventilation design; and improvement in cooling efficiency. A full coupled model incorporated with a moving mesh method was adopted; reflecting the dynamic condition of roadway construction. This study revealed the characteristics of the thermal performance and its evolution law in an excavating roadway. Several scenarios were performed to examine the designs of the auxiliary ventilation system on thermal performance in the roadway. The results show that there is a limitation in the cooling effect by continuously increasing the ventilation volume. Reducing the diameter of the air duct or distances between the duct outlet and the working face will aggravate the heat hazard in the roadway. The heat release from the roadway wall increases with the increase of the advance rate of the working face or roadway section size. Furthermore; an orthogonal experiment was conducted to investigate the effect of major factors on the average air temperature and local heat accumulation in the roadway

Solid Particle Erosion Area of Rotor Blades: Application on Small-Size Unmanned Helicopters

Rotor blades play an important role in unmanned helicopters, and it is of great significance to study the erosion of rotor blades. In this study, titanium alloy (Ti-4Al-1.5Mn) was used as the helicopter rotor blades’ surface material. The commercial software Ansys-Fluent 18.0 was mainly used to study the erosion of solid particles on the helicopter rotor blades. The moving mesh method and the discrete phase method (DPM) were used to construct an erosion model of the blades at different speeds (500, 1000, or 2000 rpm), and at different particle mass flow rates (0.5, 1, or 1.5 kg/s). The results show that the erosion of helicopter blades is mainly observed at the leading edge and at the tip of the blades. At different particle mass flow rates, greater particle mass flow rates lead to greater DPM erosion rates. As the blade speed increases, the maximum DPM erosion rate decreases, but the severely eroded area increases. Finally, the values of the severely eroded area of the helicopter rotor blades and the ratios of the severely eroded area growth are obtained through the image processing method.

Mesh moving techniques in fluid-structure interaction: robustness, accumulated distortion and computational efficiency

An important ingredient of any moving-mesh method for fluid-structure interaction (FSI) problems is the mesh moving technique (MMT) used to adapt the computational mesh in the moving fluid domain. An ideal MMT is computationally inexpensive, can handle large mesh motions without inverting mesh elements and can sustain an FSI simulation for extensive periods of time without irreversibly distorting the mesh. Here we compare several commonly used MMTs which are based on the solution of elliptic partial differential equations, including harmonic extension, bi-harmonic extension and techniques based on the equations of linear elasticity. Moreover, we propose a novel MMT which utilizes ideas from continuation methods to efficiently solve the equations of nonlinear elasticity and proves to be robust even when the mesh undergoes extreme motions. In addition to that, we study how each MMT behaves when combined with the mesh-Jacobian-based stiffening. Finally, we evaluate the performance of different MMTs on a popular two-dimensional FSI benchmark reproduced by using an isogeometric partitioned solver with strong coupling.