Multi-level hp-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes

Abstract

The implementation of hp-adaptivity is challenging as hanging nodes, edges, and faces have to be constrained to ensure compatibility of the shape functions. For this reason, most hp-code frameworks restrict themselves to 1-irregular meshes to ease the implementational effort. This work alleviates these difficulties by introducing a new formulation for high-order mesh adaptivity that provides full local hp-refinement capabilities at a comparably small implementational effort. Its main idea is the extension of the hp-d-method such that it allows for high-order overlay meshes yielding a hierarchical, multi-level hp-formulation of the Finite Element Method. This concept enables intuitive refinement and coarsening procedures, while linear independence and compatibility of the shape functions are guaranteed by construction. The proposed method is demonstrated to achieve exponential rates of convergence-both in terms of degrees of freedom and in run-time-for problems with nonsmooth solutions. Furthermore, the scheme is used alongside the Finite Cell Method to simulate the heat flow around moving objects on a non-conforming background mesh and is combined with an energy-based refinement indicator for automatic hp-adaptivity.

Very high order discontinuous Galerkin method in elliptic problems

“…The mesh refinement techniques are still one of the mainstream of scientific research since it is used for automatic mesh adaptation, e.g. [10,15,38,40,48,49] This paper reveals that it is very easy to combine very high-order finite elements with low-order elements in conforming and nonconforming meshes, so the hp-adaptivity may be very easy to perform in the DGFD method.…”

A high‐order enrichment strategy for the finite cell method

“…In the case of the material interface, the exact solution exhibits a kink that cannot be represented by standard shape functions of the FCM if the interface is located within the element [5]. In order to overcome such a problem, several remedies have been proposed [5,7,13]. In [5], we showed that using the partition of unity method [1,8] together with the high-order representation of the material interface yields very accurate results, both in the displacements and the stresses.…”

A one-field monolithic fictitious domain method for fluid–structure interactions

“…Firstly, our one-field FD method solves the solid and fluid equations together while the clas-415 sical IFEM does not solve the solid equations. Although the implicit form of IFEM can iteratively solve the solid equations, this is different from our onefield FD method which couples the fluid and solid equations monolithically via a direct matrix addition as shown in formulas (34) and (35). Secondly, while both our one-field FD method and DLM/FD solve solid equations, the former 420 solves for just one velocity field in the solid domain using FEM interpolation, while the latter solves one velocity field and one displacement field in the solid domain using Lagrange multipliers.…”

“…stiffness matrices K and K s , matrix B and the force vectors f and f s in equations (33), (34) and (35) are presented. …”

“…Readers can reference [31,32,33,34] for details of the treatment of hanging nodes. The Least-squares method (section 5.1) is used to treat the convection step in all tests unless stated otherwise.…”