Applications of Mesh Simplification Techniques from Computer Graphics in Computational Mechanics

Nathan Wilson

Stanford University

CS 448 class contribution, March 14, 2000

Introduction

In the fields of engineering and computer science numerous examples of crossover technology exist. For example, in the field of computer graphics much current effort has revolved around numerically simulating the behavior and interaction of objects obeying physical laws. These current efforts in computer graphics on simulating deformable bodies and collision detection can draw on a rich history of over 30 years (and over 100,000 technical publications) of work done in the field of computational mechanics on numerical techniques such as the finite-element, finite-difference, boundary-element, and finite-volume methods.

I propose interesting examples exist where the fields of engineering and computational mechanics can benefit from work done in computer graphics. Probably the most obvious example is in the rendering of complex datasets resulting from detailed engineering analysis. Techniques of mesh compression and mesh simplification will have a profound influence on the field engineering analysis when used as enabling technology for internet-based design tools. The engineering design and analysis software market is a multi-billion dollar industry, and in the authors opinion the dominant companies of the future will be those that harness and succeed in delivering internet-based design tools.

The ability and interest to simulate more complex problems from mathematical physics has increased due to the tremendous growth in computational power. Historically, the use of simulation tools has been fragmented into distinct areas of the design process. Distinct software was used in drafting (CAD), manufacturing (CAM), and simulated based-design (CAE). In recent years, great interest has been expressed in unifying the software so one can design based on simulating the manufacture and then the behavior of a product.

The simulation of a given manufacturing process to produce a device can involve drastically different physical phenomenon than the physics governing the operation of the device. For example, integrated-circuit fabrication techniques involve numerous chemically driven steps of addition (deposition) and removal (etching) of material from a silicon wafer. Figure 1 shows a corner of a micro-mechanical structure with a simulated isotropic deposition of material. The deposition simulation was performed using the level-set method [1] and requires a highly dense uniformly spaced grid (since the simulation was performed using an explicit finite-difference scheme). Figure 2 shows the resulting surface mesh extracted using a marching-cubes algorithm. While this mesh density was required to simulate the manufacture of the device, it is unnecessarily large for the mechanical simulation of the device using the finite-element method.

Mesh Simplification

Mesh simplification, also known as decimation, takes a given faceted surface representation and reduces the number of triangles or polygons used in the representation. Mesh simplification accelerates graphical rendering and reduces the transmission costs of geometric models. Numerous techniques exist for decimation [2-5]. For example, a mesh reduction technique known as "progressive simplicial meshes" was developed by Hoppe et. al. [3] and is shown in Figure 3. In this paper, we compare the results of two decimation algorithms. The first algorithm was developed by Schroeder et. al. [4] and is an example of an algorithm derived in the computer graphics community. A modified version of the algorithm detailed in [4] is part of the Visualization Toolkit [6]. The second algorithm was developed by Shephard et. al. [5] and can be found in the meshing tool MEGA from RPI [7].

Example 1: MEMS (micro-electromechanical systems) fabrication [8, 9, 10]

Figure 4 shows a shaded decimation of the corner given in Figure 2 using Vtk. Several different levels of detail are shown. Visually, the results are appealing even for the significantly smaller mesh. In the computer graphics community, the visual results are what matter and Vtk produces acceptable results. Figure 5 compares the surface mesh obtained from Vtk with that generated by MEGA. The resulting surface meshes are drastically different in that MEGA maintains the quality of the surface mesh for subsequent analysis.

Example 2: Bio-engineering (geometric model construction from MRI data) [11,12]

The level-set method has also been applied to extract models of the vascular system [11]. Figure 6 shows an aortic bifurcation extracted from a Magnetic Resonance Imaging (MRI) dataset using the level set method. Figure 7 shows the results of decimating the models using Vtk and MEGA. Again, the Vtk model has triangles of poor quality that will adversely affect the accuracy of the hemodynamic simulation results.

Conclusions and Remarks

The examples shown indicate how very similar end goals, such as representing a surface using fewer triangles, can take on very different manifestations in different fields. The criteria in computer graphics of visual similarity is different from that required in computational mechanics of desirable mesh properties (such as small aspect ratios). As other examples in the introduction indicate, both fields can learn a great deal from each other and very likely additional technology transfer between the two disciplines will continue to occur in the future.

References

[1] J.A. Sethian and D. Adalsteinsson, "An overview of level set methods for etching, deposition, and lithography development," IEEE Trans Semiconductor Manufacturing, 10(1), 1997, pp. 167-184.

[2] H. Hoppe, "Progressive Meshes", Proc. Siggraph 1996, pp. 99-108.

[3] J. Popovic and H. Hoppe, "Progressive Simplicial Complexes", Proc. Siggraph 1997, pp. 217-224.

[4] W. J. Schroeder, J. A. Zarge, and W. E. Lorensen, "Decimation of triangle meshes," Computer Graphics (SIGGRAPH '92 Proc.), 26(2):65-70, July 1992.

[5] H. L. de Cougny, "Refinement and Coarsening of Surface Meshes", Engineering with Computers, vol. 14, 1998, pp. 214-222.

[6] Vtk, http://www.kitware.com/

[7] MEGA, http://scorec.rpi.edu/software/Software.html

[8] N. M. Wilson, K. Wang, D. Yergeau, and R. W. Dutton, "GEODESIC: A New and Extensible Geometry Tool and Framework with Application to MEMS", being presented at Modeling and Simulation of Microsystems, March 27-29, 2000.

[9] N. M. Wilson, R. W. Dutton, and P. M. Pinsky, "Utilizing Existing TCAD Simulation Tools to Create Solid Models for the Simulation Based Design of MEMS Devices", Proceedings of International Mechanical Engineering Conference and Exposition, November 15-20, 1998, pp. 565-570.

[10] N. M. Wilson, S. Liang, P. M. Pinsky, and R. W. Dutton, "A Novel Method to Utilize Existing TCAD Tools to Build Accurate Geometry Required for MEMS simulation", Proceedings of Modeling and Simulation of Microsystems, April 19-21, 1999, pp. 120-123.

[11] K. Wang, C.A. Taylor, Z. Hsiau, D. Parker, and R.W. Dutton, "Level Set Methods and MR Image Segmentation for Geometric Modeling in Computational Hemodynamics," Proc. of the 20th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 1998, 20(6), p. 3079-3082.

[12] K.C. Wang, D. Parker, R.W. Dutton, and C.A. Taylor, "Level Sets for Vascular Model Construction in Computational Hemodynamics," Proc. of the 1999 ASME Summer Bioengineering Meeting, Big Sky, MT.

Acknowledgements

Many thanks to Ken Wang (Stanford University) for the original level set simulations.