We present a proof-of-concept methodology for generating curvilinear polygonal meshes suitable for high-order discretisations by the Virtual Element Method (VEM). A VEM discretisation requires the definition of a set of boundary and internal points used to define basis functions and compute integrals of polynomials. The procedure to locate these points on the boundary borrows ideas from previous work on a posteriori high-order mesh generation in which the geometrical inquiries to a B-rep model of the computational domain are performed via an interface to CAD libraries. Here we describe the steps of the procedure that transforms a straight-sided polygonal mesh, generated using third-party software, into a curvilinear boundary-conforming mesh. We discuss criteria for ensuring and verifying the validity of the mesh. Using an elliptic partial differential equation with Dirichlet boundary conditions as a model problem, we show that VEM discretisations on such meshes achieve the expected rates of convergence as the mesh resolution is increased. This is followed by an illustrative application of the method to the generation of a curvilinear polygonal mesh for an aerofoil geometry. We discuss polygonal curvilinear mesh quality and its enhancement, and use the motion of a cell vertex to appraise three elemental quality metrics, namely convexity, regularity and isotropy, and highlight some of the difficulties associated in their use for mesh quality optimisation. A derivative-free optimisation method is utilised to enhance curvilinear polygonal meshes by maximising a suitable measure of mesh quality. We propose such measure as a combination of the three quality metrics and apply it to optimise a distorted initial mesh for a ring geometry. We show that a suitable version of the convexity metric is effective in untangling invalid meshes. The VEM solution of a model elliptic equation is obtained for a ring geometry where a distorted and an optimised mesh show low errors, indicating that the VEM is robust and relatively insensitive to mesh distortion, and a reduction of the error in the optimised mesh. Finally, we use a more complex geometry, a computational domain for an aerofoil, as a benchmark to further illustrate the ability of the convexity metric to untangle meshes, and also to assess the suitability of two quality measures as optimisation targets to improve the overall quality of curvilinear polygonal meshes.

We aim to tackle the challenge of generating unstructured high-order meshes of complex three-dimensional bodies, which remains a significant bottleneck in the wider adoption of high-order methods. In particular we show that by adopting a variational approach to the generation process, many of the current popular high-order generation methods can be encompassed under a single unifying framework. This allows us to compare the effectiveness of these methods and to assess the quality of the meshes they produce in a systematic fashion. We present a detailed overview of the theory and formulation of the variational framework, and we highlight how such formulation can be effectively exploited to yield a highly-efficient parallel implementation. The effectiveness of this approach is examined by considering a number of two- and three-dimensional examples, where we show how the proposed approach can be used for both mesh quality optimisation and untangling of invalid high-order meshes.

With high-order methods becoming increasingly popular in both academia and industry, generating curvilinear meshes that align with the boundaries of complex geometries continues to present a significant challenge. Whereas traditional low-order methods use planar-faced elements, high-order methods introduce curvature into elements that may, if added naively, cause the element to self-intersect. Over the last few years, several curvilinear mesh generation techniques have been designed to tackle this issue, utilizing mesh deformation to move the interior nodes of the mesh in order to accommodate curvature at the boundary. Many of these are based on elastic models, where the mesh is treated as a solid body and deformed according to a linear or non-linear stress tensor. However, such methods typically have no explicit control over the validity of the elements in the resulting mesh. In this article, we present an extension of this elastic formulation, whereby a thermal stress term is introduced to ‘heat’ or ‘cool’ elements as they deform. We outline a proof-of-concept implementation and show that the adoption of a thermo-elastic analogy leads to an additional degree of robustness, by considering examples in both two and three dimensions.