An Efficient Second-Order Adaptive Procedure for Inserting CAD Geometries into Hexahedral Meshes using Volume Fractions

TL;DR

The paper presents a spatially accelerated procedure to insert CAD geometries into background hexahedral meshes using volume fractions. It combines a $k$-d tree for fast spatial acceleration with an AMR-like refinement that, at the finest level, uses a plane-based approximation of the CAD surface to compute subvolume fractions via plane–polyhedron intersection, relying only two CAD kernel queries per point. The authors prove a second-order accuracy bound $|V_{exact}-V_h|\le C h^{2}$ for smooth geometries with sufficiently refined meshes and validate this with verification tests across various geometries and mesh types, showing favorable performance relative to uniform sampling and robustness to poorly shaped elements. They also demonstrate HPC applicability via MPI parallelism and discuss extensions to tetrahedral or mixed-element meshes and more sophisticated volume representations as future work.

Abstract

This paper is concerned with inserting three-dimensional computer-aided design (CAD) geometries into meshes composed of hexahedral elements using a volume fraction representation. An adaptive procedure for doing so is presented. The procedure consists of two steps. The first step performs spatial acceleration using a k-d tree. The second step involves subdividing individual hexahedra in an adaptive mesh refinement (AMR)-like fashion and approximating the CAD geometry linearly (as a plane) at the finest subdivision. The procedure requires only two geometric queries from a CAD kernel: determining whether or not a queried spatial coordinate is inside or outside the CAD geometry and determining the closest point on the CAD geometry's surface from a given spatial coordinate. We prove that the procedure is second-order accurate for sufficiently smooth geometries and sufficiently refined background meshes. We demonstrate the expected order of accuracy is achieved with several verification tests and illustrate the procedure's effectiveness for several exemplar CAD geometries.

Figures (15)

• Figure 1: Left: A mesh with four quadrilateral elements (black) and bounding circles associated with nodes in the $k$-d tree (red, blue, green). Right: The $k$-d tree structure associated with the mesh on the left, where the root node is associated with all mesh elements and leaf nodes are associated with a single mesh element.
• Figure 2: Four potential bounding sphere classifications when comparing the sphere's radius to the closest distance to the geometry's surface. (A) The bounding sphere is fully outside of $\Omega$. (B) The bounding sphere is fully inside of $\Omega$. (C) The bounding sphere intersects $\Omega$ and its center $\boldsymbol{x}^C_c \in \Omega$. (D) The bounding sphere intersects $\Omega$ and its center $\boldsymbol{x}^D_c \notin \Omega$.
• Figure 3: Left: A two-dimensional representation of a hexahedron (solid black) subdivided to two AMR levels. Right: A two-dimensional representation of the intersection of a subhex (dashed blue) with a CAD geometry (solid black line), the planar approximation of the geometry's surface within the subhex (solid red line), and the plane's unit normal (solid red line with arrow). The shaded subvolume corresponds to the approximated CAD geometry within the subhex.
• Figure 4: Representations of the verification geometries inserted into a sinusoidally perturbed mesh where the left portion of the domain is rendered with cell volume fractions and the right portion of the domain corresponds to a recovered isosurface.
• Figure 5: An illustration of the AMR subdivision procedure for a chosen element in the case of a sphere inserted into the sinusoidally perturbed mesh with $n_{\text{sub}}=5$ subdivisions. Note that the element has been rotated to more clearly illustrate the AMR structure.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Theorem 1
• proof