We present a nodal interpolation method to approximate a subdivision model. The main application is to model and represent curved geometry without gaps and preserving the required simulation intent. Accordingly, we devise the technique to maintain the necessary sharp features and smooth the indicated ones. This sharp-to-smooth modeling capability handles unstructured configurations of the simulation points, curves, and surfaces. The surfaces correspond to initial linear triangulations that determine the sharp point and curve features. The method automatically suggests a subset of sharp features to smooth which the user modifies to obtain a limit model preserving the initial points. This model reconstructs the curvature by subdivision of the initial mesh, with no need of an underlying curved geometry model. Finally, given a polynomial degree and a nodal distribution, the method generates a piece-wise polynomial representation interpolating the limit model. We show numerical evidence that this approximation, naturally aligned to the subdivision features, converges to the model geometrically with the polynomial degree for nodal distributions with sub-optimal Lebesgue constant. We also apply the method to prescribe the curved boundary of a high-order volume mesh. We conclude that our sharp-to-smooth modeling capability leads to curved geometry representations with enhanced preservation of the simulation intent.

We propose an optimization method to explore the heuristically best high-order interpolation nodal distributions in the d-dimensional simplex. We consider a twice-differentiable proxy of the Lebesgue constant with multiple local minima that are heuristically explored by means of node relocations and smooth optimizations. For a free node, a proxy of the energy landscape of the Lebesgue constant is obtained through a Delaunay triangulation on the (d+1)-sphere, where the opposite faces of the simplices incident to the node determine an approximation of the uphill energy landscape of the functional around such a node. Accordingly, to explore proximal distributions, we heuristically enforce a tunnel effect by relocating one node to the other side of the uphill of the energy landscape. To heuristically find the best local minima, we explore nodal distributions which always improve the current function value. To exploit the available computational resources, the nodal distributions with better function values are explored first. The results show that the considered heuristics drastically reduce the number of local minima to explore while not having an impact on the best values found. Moreover, in 2D, our nodal distributions present good interpolation properties, and in 3D and 4D, our nodal sets improve the current best interpolative nodal configurations. We conclude that the computed nodal distributions might be suitable for high-order interpolation in the high-dimensional simplex, yet they might be excellent initial approximations for methods optimizing the Lebesgue constant to further improve the interpolation properties.

To estimate the Lebesgue constant, we propose a point refinement method on the d-dimensional simplex. The proposed method features a smooth gradation of the point resolution, neighbor queries based on neighbor-aware coordinates, and a point refinement that algebraically scales as (d+1)d. Remarkably, by using neighbor-aware coordinates, the point refinement method is ready to automatically stop using a Lipschitz criterion. For different polynomial degrees and point distributions, we show that our automatic method efficiently reproduces the literature estimations for the triangle and the tetrahedron. Moreover, we efficiently estimate the Lebesgue constant in higher dimensions. Accordingly, up to six dimensions, we conclude that the point refinement method is well-suited to efficiently estimate the Lebesgue constant on simplices. In perspective, for a given polynomial degree, the proposed point refinement method might be relevant to optimize a set of simplex points that guarantees a small interpolation error.