Universal meshes, a new paradigm for computing with nonconforming triangulations [electronic resource]
Keywords: Mechanical Engineering

Finite element methods commonly handle evolving domains in one of two ways-- either the changing domain is remeshed at each instant/update, or it is immersed in a background mesh and approximated within it. We introduce a novel approach here that inherits the conceptual simplicity of the former and the computational efficiency of the latter. We describe a method for exactly discretizing planar C2-regular domains immersed in nonconforming triangulations. The key idea in discretizing curved domains is to map triangles in a background mesh to curvilinear ones that conform exactly with the boundary. We construct such mappings using a novel way of parameterizing a curved boundary over a nearby collections of edges with its closest point projection. Then, extending this parameterization to a small neighborhood of the boundary in a piecewise smooth manner yields a discretization for the domain itself. By interpolating the mappings to curvilinear triangles at select points, we recover isoparametric mappings for the immersed domain defined over the background mesh. Indeed, interpolating just at the vertices of the background mesh yields a fast meshing algorithm that involves only perturbing vertices near the boundary. For our method of discretizing of a curved domain to be robust, we have to impose restrictions on the background mesh. Conversely, these restrictions define a family of domains that can be discretized with a given background mesh. We then say that the background mesh is a universal mesh for such a family of domains. The notion of universal meshes is particularly useful in free/moving boundary problems because the same background mesh can serve as the universal mesh for an evolving domain for time intervals independent of the time step. Hence, it facilitates a framework for finite element calculations over evolving domains while using a fixed background mesh. Furthermore, since the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high-order accuracy. The main challenge in our method is determining when a domain can be discretized using a given background mesh. In turn, this depends on when the parameterization that we compute for its boundary is robust. To this end, we identify sufficient conditions under which the restriction of the closest point projection to the selected edges is a homeomorphism onto the boundary. Specifically, we require that the background mesh be sufficiently refined and that certain interior angles of its triangles near the boundary be strictly acute. We provide local computable estimates for the required mesh size and for the Jacobian of the resulting parameterization. We show that the latter is bounded and positive independent of the mesh size. These assumptions, along with a possibly smaller mesh size, also guarantee that the method for discretizing curved domains is robust. Three factors are pivotal to the success of the idea of universal meshes for simulating free/moving boundary problems. Firstly, there are no conformity requirements on the background mesh; none of its vertices need to lie on the boundary. In fact, a sufficiently refined mesh of equilateral triangles suffices to discretize any smooth planar boundary/domain. Secondly, the restrictions we do impose on background meshes can be both easily satisfied and checked. Finally, by using same background mesh to simulate evolving geometries for (reasonably) long times, it is possible to retain the sparsity patterns of data structures involved in the problem. We present numerous examples to demonstrate the high-order of convergence possible with the discretization method. We include simulations of flows over domains with moving boundaries. We are also investigating applications to simulating the propagation of curved cracks and the dynamics of fluid membranes. These examples and applications indicate that universal meshes can be a useful tool in simulating a challenging class of problems in realistic engineering applications.