A universal algorithm for an improved finite element mesh generation: Mesh quality assessment in comparison to former automated mesh-generators and an analytic model

https://doi.org/10.1016/j.medengphy.2004.10.004Get rights and content

Abstract

The FE-modeling of complex anatomical structures is not solved satisfyingly so far. Voxel-based as opposed to contour-based algorithms allow an automated mesh generation based on the image data. Nonetheless their geometric precision is limited.

We developed an automated mesh-generator that combines the advantages of voxel-based generation with improved representation of the geometry by displacement of nodes on the object-surface. Models of an artificial 3D-pipe-section and a skullbase were generated with different mesh-densities using the newly developed geometric, unsmoothed and smoothed voxel generators.

Compared to the analytic calculation of the 3D-pipe-section model the normalized RMS error of the surface stress was 0.173–0.647 for the unsmoothed voxel models, 0.111–0.616 for the smoothed voxel models with small volume error and 0.126–0.273 for the geometric models. The highest element-energy error as a criterion for the mesh quality was 2.61 × 10−2 N mm, 2.46 × 10−2 N mm and 1.81 × 10−2 N mm for unsmoothed, smoothed and geometric voxel models, respectively. The geometric model of the 3D-skullbase resulted in the lowest element-energy error and volume error. This algorithm also allowed the best representation of anatomical details.

The presented geometric mesh-generator is universally applicable and allows an automated and accurate modeling by combining the advantages of the voxel-technique and of improved surface-modeling.

Introduction

The finite element method has been shown to be useful for biomechanic applications in medicine [1], [2]. The complex geometry and material characteristics of biological tissue complicates the finite element (FE)-modeling for biomechanical analysis as compared to pure technical analysis [3], [4], [5]. Therefore, most studies on finite elements are limited to examination of exemplary and simplified models that often have to be created with significant time effort [6], [7]. This way general statements can be derived. However, analysis of selected patients in order to optimize the individual therapeutic strategy is not possible [3], [4], [8], [9]. These limitations impair the application of the FE-method even though precise geometric data and even information on material characteristics are available by cross-sectional imaging techniques.

Two strategies for implementation of geometric information from cross-sectional data to generate FE-meshes were described.

The first strategy is based on modeling of the geometric boundaries of the objects. Typically, surface contours are generated manually or semiautomated first. Applying different techniques the contours are then filled with a finite element mesh [6], [7], [10], [11], [12], [13]. Even for less complex structures that contain branches or cavities, this process can no longer be automated and time-consuming interactive refinement is necessary [14]. The use of brick elements, that are favorable for finite element analysis, often requires geometric simplification as well as manual or partially manual mapped-mesh techniques [4], [8], [9], [15]. Alternatively, FE-models can easily be generated automatically from triangulated surfaces. These surfaces can be rendered using the marching cube technique, that is typically used for 3D-visualization [16], [17], [18], [19]. These models contain a high number of tetrahedral elements as compared to models generated using bricks. Analysis of these models requires more hardware-resources, and subsequently, results in an increased time effort. Furthermore, the calculation-characteristics of tetrahedral elements are not as beneficial as the characteristics of brick-elements due to over stiffening of the models [5], [6], [14].

The second strategy is based on voxel-techniques transforming the voxel information of the cross-sectional data into a FE-mesh. The voxel-technique can be implemented as direct generation of elements for each object defining voxel, typically used for high-resolution modeling of trabecular bone [20], [21], [22], or as integration of a constant number of voxels [3], [14], [23]. For latter implementation, the image data is superimposed with a 3D-grid. Each grid-cell represents a potential FE-element, whereas the grid-nodes determine the coordinates of the corresponding FE-nodes. A voxel-ratio determines which grid-cells are used to generate the FE-mesh. The voxel-ratio is the quotient of object defining voxels and all voxels within a grid-cell [14]. Another method is to generate elements for all grid-cells, containing any object defining voxel and to adjust the material properties according to the voxel-ratio [3].

The voxel-technique can be automated easily, and therefore, provides the opportunity to generate individual finite element meshes with limited time effort. Furthermore, based on the voxel information of the image data the material properties can directly be assigned to the corresponding elements [3], [24], [25], [26]. The surface of these meshes, however, is angular as it is derived from the corresponding bricks. Thus, calculation of external and internal surfaces of the model is biased [4], [27]. For biomedical application, these surfaces often represent the region where relevant interaction takes place (joint articulations, bone-implant borders, appositional bone growth) and experimental measurements of strain and stress are performed [14], [27].

To overcome these limitations, different smoothing algorithms have been developed [6], [14], [23]. Camacho et al. [14] introduced a centroid-based algorithm that can be applied to arbitrarily complex structures. The authors could show that the mean bias and element-to-element-oscillation of the calculated surface element stress were significantly reduced in smoothed voxel models compared to unsmoothed voxel models. This effect was found to be independent of mesh refinement. Previously, Viceconti et al. [5] showed that the calculation of surface element stress does not necessarily become more precise with increasing mesh refinement. Furthermore, the precision achievable by increasing mesh refinement is limited as the amount of mesh-data grows exponentially, and thus, quickly exceeds the resources of even modern workstations, although special solvers have been developed to speed up analysis and to reduce storage requirements [5], [22].

Distortion of the surface brick elements is a disadvantage of all voxel-smoothing techniques as the brick shape is ideally suited for finite element analysis [3], [5], [8]. Guldberg et al. [4] reported that the resulting inaccuracies were of minor importance compared to the inaccuracies resulting from the only limited knowledge about the material properties of biological systems. Furthermore, smoothing techniques change the model's geometry according to a formula rather than to the actual surface geometry. Thus, geometric inaccuracy can sometimes be severe in areas where the anatomical structures are of similar or smaller dimension than the finite elements [4]. In models of the skeletal system, this especially occurs in the area of joint spaces, processes and foramina. An increased mesh refinement does not provide a practical solution to this problem as described above.

This study introduces a finite element mesh-generator that applies a geometric voxel-technique. The described generator does not approximate the surface geometry by conventional smoothing techniques but by distortion of the grid-nodes based on voxel information and surface geometry. The resulting mesh quality of the newly developed and the previously described mesh-generators is compared using an analytic 3D-pipe-section model as a reference. The practical feasibility was studied applying a skullbase model according to Camacho et al. [14].

Section snippets

Mesh-generators

Four different methods of finite element mesh generation were compared:

  • Mapped-mesh-generator;

  • Unsmoothed voxel mesh-generator;

  • Smoothed voxel mesh-generator with surface smoothing modified according to an algorithm by Camacho et al. [14];

  • Geometric voxel mesh-generator with precise surface modelling.

The last three methods are fully automated techniques. For each of these methods a C++ program (Microsoft Visual C++ 6.0) was developed that calculates nodes, elements and material data based on the

3D-pipe-section

To compare the mesh-generators and to perform an analytic calculation as a reference a 3D-pipe-section model was used (Fig. 3). Translations were disabled in all directions for nodes with y = 0 within the model-coordinate-system. An equally distributed overall force of 25 N was applied to nodes with x = 0 pointing in the direction of the local-y-axis. Isotropic material properties were set as E = 104 MPa and ν = 0.31 [14].

An analytic result for the elastic deformation and stress was calculated according

Results

The FE data of the generated pipe-section-models is listed in Table 1 for different mesh-densities. The volume error of the mapped-mesh-model (0%  0.11%) was negligible. The volume error of the remaining models was 3.80%  30.81% (unsmoothed voxel, voxel-ratio 0.375), −16.28% to −51.90% (unsmoothed voxel, voxel-ratio 0.5), −2.92% to 5.50% (smoothed voxel, voxel-ratio 0.375), −16.69% to −59.02% (smoothed voxel, voxel-ratio 0.5) and 3.13%–8.09% (geometric voxel).

The normalized RMS error of stress at

Discussion

As the presented geometric voxel mesh-generator is based on a voxel algorithm it provides the benefits of this technique compared to contour-based meshing techniques. Applying the available precise image data, the algorithm allows a fully-automated, and therefore, fast FE-mesh generation of even complex anatomical structures [3], [4], [5], [14], [30]. Interactive contour-segmentation, that is time-consuming and often requires simplifications, is not necessary [4], [8]. The geometric voxel

Acknowledgement

This study was supported by a Research Grant of the Novartis-Foundation.

References (30)

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