Direct structural analysis of domains defined by point clouds

TL;DR

It is shown that oriented point clouds provide sufficient information for these point-membership classifications and a tessellation-free formulation of contour integrals that allows to apply Neumann boundary conditions on point clouds without having to recover the underlying surface is addressed.

Abstract

This contribution presents a method that aims at the numerical analysis of solids represented by oriented point clouds. The proposed approach is based on the Finite Cell Method, a high-order immersed boundary technique that computes on a regular background grid of finite elements and requires only inside-outside information from the geometric model. It is shown that oriented point clouds provide sufficient information for these point-membership classifications. Further, we address a tessellation-free formulation of contour integrals that allows to apply Neumann boundary conditions on point clouds without having to recover the underlying surface. Two-dimensional linear elastic benchmark examples demonstrate that the method is able to provide the same accuracy as those computed with conventional, continuous surface descriptions, because the associated error can be controlled by the density of the cloud. Three-dimensional examples computed on point clouds of historical structures show how the method can be employed to establish seamless connections between digital shape measurement techniques and numerical analyses.

Figures (23)

• Figure 1: From point clouds to simulations: standard pipeline (left) and proposed pipeline (right)
• Figure 2: The core concept of the FCM. The physical domain $\Omega_{\text{phy}}$ is extended by the fictitious domain $\Omega_{\text{fict}}$. Their union, the embedding domain $\Omega_{\cup}$ can be meshed easily. The influence of the fictitious domain is penalized by the scaling factor $\alpha$.
• Figure 3: Spacetree-based integration domains for different values of maximum subdivision depth $k$. Red dots represent integration points that lie in $\Omega_{\text{phy}}$, blue dots are in $\Omega_{\text{fict}}$.
• Figure 4: The process of determining whether a cell is cut by the interface $\partial\Omega_{\text{phy}}$. Seed points are distributed on every cell of the finite cell mesh, and their inside-outside state is evaluated. If there is a pair of points with differing state, the cell is identified as a cut cell. Red and blue dots represent seed points lying in $\Omega_{\text{phy}}$ or $\Omega_{\text{fict}}$, respectively. The cell on the right side is cut, while the one the left side is not.
• Figure 5: Point membership classification on oriented point clouds. The domain is represented by a set of points $\mathbf{p}_i$ and associated normals $\mathbf{n}_i$. Every such pair locally separates the space along a hyperplane into two half-spaces: $\Omega_{i}^-$ and $\Omega_{i}^+$.