The aggregated unfitted finite element method for elliptic problems

Abstract

Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hinders the practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the new method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.

A scalable parallel finite element framework for growing geometries. Application to metal additive manufacturing

“…Even at the rate of 1 [s] per time step, fully resolved high‐fidelity AM simulations, involving, eg, other physics than heat transfer, stronger nonlinearities, historical variables, amongst others, can still take long hours in massively parallel systems. Higher efficiency and parallelism could be attained by resorting to (1) unfitted FE methods to eliminate the mesh body‐ and layer‐fitting requirement and (2) weakly scalable adaptive and nonlinear space‐time solvers that also exploit concurrency in time.…”

“…In this way, inactive cells do not play any role in the numerical approximation; they have no contribution to the global linear system, in contrast with the quiet-element method. 38 Besides, note the similarities of this approach to the one employed in unfitted FE methods, 52,53 distinguishing interior and cut (active) cells from exterior (inactive) cells.…”

“…In AM, for instance, it is frequently required that the mesh of the component conforms to the layers. On the other hand, the method presented later can be adapted with little effort to a more general setting, where the growth‐fitting requirement is dismissed by resorting to unfitted or immersed boundary methods . In this case, is a triangulation of an artificial domain Ω art , such that it includes the final physical domain, ie, Ω f ⊂Ω art , but it is also characterised by a simple geometry, easy to mesh with Cartesian grids.…”

Superconvergent patch recovery with constraints for three‐dimensional contact problems within the Cartesian grid Finite Element Method

“…The arbitrary intersection between the Cartesian grids and the analysis domain may produce some elements with a low volume of material inside, as in the example shown in Figure A. In that case the stiffness associated to the external node colored in red becomes very small, which results in an ill‐conditioning of the FE formulation and also a poor quality of the FE stress field computed at those pathological elements. In the classical SPR procedure, the patch associated to the red node in Figure A would consist only in that pathological element.…”

Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

“…Without dedicated treatments, the severe ill-conditioning of linear systems derived from immersed finite element methods generally forestalls convergence of iterative solution procedures. Multiple resolutions for these conditioning problems have been proposed, the most prominent of which are the ghost penalty, e.g., [4,5,37], constraining, extending, or aggregation of basis functions, e.g., [13,[38][39][40][41][42][43][44][45][46], and preconditioning, which is discussed in detail below.…”