Measuring and improving the geometric accuracy of piece-wise polynomial boundary meshes

We present a new disparity functional to measure and improve the geometric accuracy of a curved high-order mesh that approximates a target geometry model. We have devised the disparity to account for compound models, be independent of the entity parameterization, and allow trimmed entities. The disparity depends on the physical mesh and the auxiliary parametric meshes. Since it is two times differentiable on all these variables, we can minimize it with a second-order method. Its minimization with the parametric meshes as design variables measures the geometric accuracy of a given mesh. Furthermore, the minimization with both the physical and parametric meshes as design variables improves the geometric accuracy of an initial mesh. We have numerical evidence that the obtained meshes converge to the target geometry (unitary normal) algebraically, in terms of the element size, with order 2p (2p-1, respectively), where p is the polynomial degree of the mesh. Although we obtain meshes with non-interpolative boundary nodes, we propose a post-process to enforce, if required by the application, meshes with interpolative boundary nodes and featuring the same order of geometric accuracy. In conclusion, we can obtain super-convergent orders, at least for sufficiently smooth parametric curve (surface) entities, for meshes of polynomial degrees up to 4 (3, respectively). In perspective, this super-convergence might enable using a lower polynomial degree to approximate the geometry than to approximate the solution without hampering the required geometric accuracy for high-order analysis.This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 715546. This work has also received funding from the Generalitat de Catalunya under grant number 2017 SGR 1731. This work has been supported by FEDER and the Spanish Government, Ministerio de Ciencia, Innovación y Universidades grant project contract PGC2018-097257-B-C33. The work of Xevi Roca has been partially supported by the Spanish Ministerio de Economía y Competitividad under the personal grant agreement RYC-2015-01633.Peer ReviewedPostprint (published version