Direct structural analysis of domains defined by point clouds

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.

Introduction

It is well known in the computational mechanics community that transferring a CAD model into an analysis-suitable finite element mesh may account for as much as 80% of the entire analysis time [1]. Recent years’ research efforts aiming at circumventing this bottleneck have resulted in numerous alternative approaches, leading to significant progress in the quest of establishing seamless connections between geometric modeling and finite element analysis.

However, in some applications of the FEM, the geometries of interest are not directly available in form of CAD models. Typically, this situation arises in the context of biomechanical simulations where models are recorded by means of medical imaging techniques, such as CT scans. As standard finite elements require a CAD model to start with, these methods require special algorithms to recover a geometric model and eventually a finite element mesh from the imaging data — see e.g. [2] for a conceptual overview of these multi-step pipelines.

Volumetric imaging is not always the most feasible approach to record the shape of physical structures. It is especially large objects that do not allow for a cost-effective application of CT scanning. Nonetheless, it can be of importance to be able to compute the structural behavior of large objects — e.g. in the field of cultural heritage preservation, as there are often no digital CAD models available for historical structures. Moreover, even if there are schematic drawings, the shape of the object may differ from them, especially if the structure is exposed to damaging effects such as erosion, floods, earthquakes, or wars. In these cases, other shape measurement techniques need to be employed. The two most popular methods for this purpose are terrestrial laser scanning and close range photogrammetry-based reconstructions. Especially photogrammetry has gained a lot of attention recently, due to the inexpensiveness of the required equipment and because of the rapid development of the computational resources as well as the associated algorithms that allow for efficient, almost real-time reconstructions [3].

The methods of laser scanning and photogrammetry both reproduce the shape of the geometry of interest in the form of point clouds: a set of points representing the surface of the object. Such point clouds are not directly suited for numerical analysis. In order to transform the recorded data into an analysis-suitable model, it needs to pass through several stages, similar to the necessary procedure for models stemming from volumetric imaging.

Usually, these measurement-to-analysis procedures are characterized by the following main steps (Fig. 1):

• Shape acquisition

  A 3D shape measurement technique is employed to capture the shape of the domain of interest, resulting in a point cloud representing the surface of the object.

• Surface reconstruction

  A geometric model is derived from the point cloud information using geometric segmentation and surface fitting methods. The resulting model is stored using standardized geometric representation techniques, such as STL, STEP, or IGES files.

• Mesh generation

  The CAD model from the previous step is discretized into a finite element mesh.

• Finite Element Analysis

  The mesh is handed over to a finite element solver together with the corresponding material properties and structural constraints.

Numerous applications implement the steps above — see e.g. [4], [5], [6], [7], [8] for examples in the preservation of historical structures, or [9] for an application in the context of biomechanical experiments.

Research in different fields of computational science and engineering has resulted in well-established approaches that allow to perform these steps one-by-one. Still, their deep integration into a seamless chain is not trivial, as it requires the interplay of various algorithms. Other than the problems inherent to the data transfer between different implementations, an even bigger challenge is posed by generating a finite element mesh from the geometric model reconstructed in the second step. The fine details recovered by modern surface reconstruction algorithms (e.g.[10], [11], [12]) are not necessarily the details that need to be carried over to a finite element mesh, where the process of refinement is usually governed by the physics of the problem rather than aesthetic aspects. While a geometric defeaturing step may be applied to remove physically uninteresting details, manipulating the geometry carries the danger of introducing flaws in the geometric model, resulting in an invalid, “dirty” geometry that cannot be meshed directly [13], [14].

One method that aims to avoid the difficult task of mesh generation is the Finite Cell Method (FCM), introduced in [15]. The FCM is based on the combination of immersed boundary methods and high-order finite element basis functions used in p-FEM [16] or Isogeometric Analysis [1].

Instead of generating a boundary-conforming discretization, the FCM extends the physical domain of interest by a so-called fictitious domain in such a way that their union forms a simple bounding box that can be meshed easily. To stay consistent with respect to the original problem, the material parameters in the fictitious domain are penalized by a small factor $\alpha$. The introduction of $\alpha$ shifts the analysis effort from mesh generation to numerical integration. The most notable advantages of the FCM are the drastically reduced engineering efforts for preprocessing, the almost costless meshing, and the high accuracy and efficiency of the computation.

In its simplest implementation, the only information that the FCM needs from a geometric model is the inside–outside state — the question whether a given point in space lies in the physical or the fictitious part of the domain. There are numerous geometric representations that are suitable to provide such point membership tests and that have been shown to work well in combination with the FCM, ranging from simple shapes provided by constructive solid geometry [17] to models as complex as metal foams [18].

In this paper, we will demonstrate that a direct analysis of geometries described by oriented point clouds is possible. To this end, we will combine the finite cell method with geometries that are represented by oriented point clouds. The members of the point cloud and the vectors associated to them provide enough information for point membership tests, allowing for structural analyses of objects directly on their cloud representation. This way, the tedious tasks of recovering a geometric model and generating a boundary conforming mesh can be avoided, allowing for significant simplifications in the measurement-to-analysis pipeline.