Abstract
The state of the art method to predict bone stiffness is micro finite element (μFE) analysis based on high-resolution computed tomography (CT). Modern parallel solvers enable simulations with billions of degrees of freedom. In this paper we present a conjugate gradient solver that works directly on the CT image and exploits the geometric properties of the regular grid and the basic element shapes given by the 3D pixel. The data is stored in a pointer-less octree. The tree data structure provides different resolutions of the image that are used to construct a geometric multigrid preconditioner. It enables the use of matrix-free representation of all matrices on all levels. The new solver reduces the memory footprint by more than a factor of 10 compared to our previous solver ParFE. It allows to solve much bigger problems than before and scales excellently on a Cray XT-5 supercomputer.
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Adams, M., Brezina, M., Hu, J., Tuminaro, R.: Parallel multigrid smoothing: polynomial versus Gauss–Seidel. J. Comput. Phys. 188(2), 593–610 (2003)
Arbenz, P., van Lenthe, G.H., Mennel, U., Müller, R., Sala, M.: A scalable multi-level preconditioner for matrix-free μ-finite element analysis of human bone structures. Internat. J. Numer. Methods Engrg. 73(7), 927–947 (2008)
Bekas, C., Curioni, A., Arbenz, P., Flaig, C., van Lenthe, G., Müller, R., Wirth, A.: Extreme scalability challenges in micro-finite element simulations of human bone. Concurrency Computat.: Pract. Exper. 22(16), 2282–2296 (2010)
Bielak, J., Ghattas, O., Kim, E.J.: Parallel octree-based finite element method for large-scale earthquake ground simulation. Comp. Model. in Eng. & Sci. 10(2), 99–112 (2005)
Braess, D.: Finite Elements: Theory, fast solvers and applications in solid mechanics, 2nd edn. Cambridge University Press, Cambridge (2001)
Burstedde, C., Wilcox, L.C., Ghattas, O.: p4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees. accepted for publication in SIAM J. Sci. Comput.
Castro, R., Lewiner, T., Lopes, H., Tavares, G., Bordignon, A.: Statistical optimization of octree searches. Computer Graphics Forum 27(6), 1557–1566 (2008)
Swiss National Supercomputing Centre (CSCS), http://www.cscs.ch/
Flaig, C., Arbenz, P.: A Scalable Memory Efficient Multigrid Solver for Micro-Finite Element Analyses Based on CT Images. Parallel Computing 37(12), 846–854 (2011)
Margenov, S., Vutov, Y.: Comparative analysis of PCG solvers for voxel FEM systems. In: Proceedings of the International Multiconference on Computer Science and Information Technology, pp. 591–598 (2006)
The ParFE Project Home Page (2010), http://parfe.sourceforge.net/
van Rietbergen, B., Weinans, H., Huiskes, R., Polman, B.J.W.: Computational strategies for iterative solutions of large FEM applications employing voxel data. Internat. J. Numer. Methods Engrg. 39(16), 2743–2767 (1996)
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)
Samet, H.: The quadtree and related hierarchical data structures. ACM Comput. Surv. 16, 187–260 (1984)
Sampath, R.S., Biros, G.: A parallel geometric multigrid method for finite elements on octree meshes. SIAM J. Sci. Comput. 32(3), 1361–1392 (2010)
Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, London (2000)
Wirth, A., Mueller, T., Vereecken, W., Flaig, C., Arbenz, P., Müller, R., van Lenthe, G.H.: Mechanical competence of bone-implant systems can accurately be determined by image-based micro-finite element analyses. Arch. Appl. Mech. 80(5), 513–525 (2010)
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Flaig, C., Arbenz, P. (2012). A Highly Scalable Matrix-Free Multigrid Solver for μFE Analysis Based on a Pointer-Less Octree. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2011. Lecture Notes in Computer Science, vol 7116. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29843-1_56
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DOI: https://doi.org/10.1007/978-3-642-29843-1_56
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