Numerical integration over implicitly defined domains with topological guarantee

TL;DR

The paper tackles numerical integration over domains defined implicitly by $f(x,y)$, where topology varies and can be misrepresented by naive discretization. It proposes a hierarchical framework that uses interval arithmetic to automatically locate and preserve the exact boundary of the integration domain, achieving a topological guarantee for the domain $\Omega=\{(x,y)\mid f(x,y)\ge 0\}$. A geometry-based local error estimate guides adaptive subdivision, particularly for boundary cells, by bounding missed regions with a pessimistic, banded area estimate and Bézier-based boundary approximations. Experiments on multiple domains show accurate results with fewer integration points and demonstrated robustness across smooth and singular boundaries, illustrating potential for efficient, topology-correct isogeometric analysis and CAD/CAE integration.

Abstract

Numerical integration over the implicitly defined domains is challenging due to topological variances of implicit functions. In this paper, we use interval arithmetic to identify the boundary of the integration domain exactly, thus getting the correct topology of the domain. Furthermore, a geometry-based local error estimate is explored to guide the hierarchical subdivision and save the computation cost. Numerical experiments are presented to demonstrate the accuracy and the potential of the proposed method.

Figures (8)

• Figure 1: Incorrect classifications of cells: Both cells (with blue frame) should be boundary cells. Only with function values at corner points lead to wrong types: exterior (left) and interior (right), respectively. The interior domains are blue painted.
• Figure 2: Different approximations to the boundary curve of the implicitly-defined domain, where $P_0$ and $P_2$ are intersections of the boundary curve over the cell, $P_1$ is the intersection of tangents at the two intersections, and $P_3$ is a point on the boundary curve.
• Figure 3: Process of obtaining the area of $D_2$ (green). Left: Three sample points with their corresponding geometric distances $r_i,i=0,1,2$. Right: Area of $D_2$ is approximated by a narrow band. Its width is the longest geometric distance ($r_0$ on the left). The boundary of integration domain (blue), bézier approximation to the boundary curve (magenta).
• Figure 4: Annulus area test. (a): the integration domain (blue). Lines in different colors are plotted to show the signs of function values at different levels. (b-d): the cell partitions and integration points of L, Q and our method, respectively. The integration points in interior (boundary) cells are marked in orange (blue).
• Figure 5: Integratoin over annulus with integrand: $x^3y-xy+2.5$. (a-c): the cell partitions and integration points of L, Q and our method, respectively.

Theorems & Definitions (4)

• Example 1
• Example 2
• Example 3
• Example 4