Composite Finite Elements for Trabecular Bone Microstructures Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch–Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich–Wilhelms–Universit¨at Bonn vorgelegt von Lars Ole Schwen aus Dusseldorf¨ Bonn, Juli 2010 Angefertigt mit Genehmigung der Mathematisch–Naturwissenschaftlichen Fakultat¨ der Rheinischen Friedrich–Wilhelms–Universitat¨ Bonn am Institut fur¨ Numerische Simulation Diese Dissertation ist auf dem Hochschulschriftenserver der Universitats-¨ und Landes- bibliothek Bonn http://hss.ulb.uni-bonn.de/diss online elektronisch publiziert sowie als Buch mit der ISBN 978-3-938363-78-2 bei HARLAND media erschienen. Erscheinungsjahr: 2010 1. Gutachter: Prof. Dr. Martin Rumpf 2. Gutachter: Prof. Dr. Alexey Chernov Tag der Promotion: 07. Oktober 2010 To my aunt Helga (1947 – 2006) This document was typeset using pdfLATEX, the KOMA-Script scrbook document class, Palladio/Mathpazo and Classico fonts, and (among many others) the microtype package. Cooperations and Previous Publications This thesis was written as part of a joint research project with Prof. Dr. Hans- Joachim Wilke and Dipl.-Ing. Uwe Wolfram (Institute of Orthopaedic Research and Biomechanics, University of Ulm), Prof. Dr. Tobias Preusser (Fraunhofer MEVIS, Bremen), and Prof. Dr. Stefan Sauter (Institute of Mathematics, University of Zurich). Parts of this thesis have been published or submitted for publication in the following journal and proceedings articles: • Florian Liehr, Tobias Preusser, Martin Rumpf, Stefan Sauter, and Lars Ole Schwen, Composite finite elements for 3D image based computing, Computing and Visualization in Science 12 (2009), no. 4, pp. 171–188, reference [217] • Tobias Preusser, Martin Rumpf, and Lars Ole Schwen, Finite element simulation of bone microstructures, Proceedings of the 14th Workshop on the Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields, University of Ulm, July 2007, pp. 52–66, reference [282] • Lars Ole Schwen, Uwe Wolfram, Hans-Joachim Wilke, and Martin Rumpf, Determining effective elasticity parameters of microstructured materials, Proceedings of the 15th Workshop on the Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields, University of Ulm, July 2008, pp. 41–62, reference [311] • Uwe Wolfram, Lars Ole Schwen, Ulrich Simon, Martin Rumpf, and Hans- Joachim Wilke, Statistical osteoporosis models using composite finite elements: A pa- rameter study, Journal of Biomechanics 42 (2009), no. 13, pp. 2205–2209, refer- ence [379] • Lars Ole Schwen, Tobias Preusser, and Martin Rumpf, Composite finite elements for 3D elasticity with discontinuous coefficients, Proceedings of the 16th Workshop on the Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields, University of Ulm, 2009, accepted, reference [310] • Tobias Preusser, Martin Rumpf, Stefan Sauter, and Lars Ole Schwen, 3D composite finite elements for elliptic boundary value problems with discontinuous coefficients, 2010, submitted to SIAM Journal on Scientific Computing, reference [281] • Martin Rumpf, Lars Ole Schwen, Hans-Joachim Wilke, and Uwe Wolfram, Numerical homogenization of trabecular bone specimens using composite finite elements, 1st Conference on Multiphysics Simulation – Advanced Methods for Industrial Engineering, Fraunhofer, 2010, reference [296] Most C++ code developed for this dissertation has been published as part of the QuocMesh software library by AG Rumpf, Institute for Numerical Simulation, University of Bonn. AMS Subject Classifications (MSC2010) 65D05, 65M55, 65M60, 65N30, 65N55, 74B05, 74Q05, 74S05, 80M10, 80M40, 92C10 iii Summary In many medical and technical applications, numerical simulations need to be per- formed for objects with interfaces of geometrically complex shape. We focus on the biomechanical problem of elasticity simulations for trabecular bone microstructures. The goal of this dissertation is to develop and implement an efficient simulation tool for finite element (FE) simulations on such structures, so-called composite FE. We will deal with both the case of material/void interfaces (‘complicated domains’) and the case of interfaces between different materials (‘discontinuous coefficients’). t = 0.0 t = 0.05 t = 0.10 t = 1.0 t = 10.0 t = 20.0 For an aluminum foam embedded in polymethylmethacrylate subject to heating and cooling at the top and bottom, respectively, heat diffusion is simulated and the temperature is visualized. Shearing simulation for a cylindrical specimen Compression simulation for a cuboid specimen of of porcine trabecular bone. Zooms to one corner porcine trabecular bone embedded in polymethyl- of the specimen are shown on the right. All methacrylate. Color in both cases encodes the deformations are scaled for better visualization. von Mises stress at the interface. Construction of Composite FE. In classical FE simulations, geometric complexity is encoded in tetrahedral and typically unstructured meshes. Composite FE, in contrast, encode geometric complexity in specialized basis functions on a uniform mesh of hexahedral structure. Other than alternative approaches (such as e. g. fictitious domain methods, GFEM, immersed interface methods, partition of unity methods, unfitted meshes, and XFEM), the composite FE are tailored to geometry descriptions by 3D voxel image data and use the corresponding voxel grid as computational mesh, without introducing additional degrees of freedom, and thus making use of efficient data structures for uniformly structured meshes. The composite FE method for complicated domains goes back to Hackbusch and Sauter [Numer. Math. 75 (1997), 447–472; Arch. Math. (Brno) 34 (1998), 105–117] and restricts standard affine FE basis functions on the uniformly structured tetrahedral grid (obtained by subdivision of each cube in six tetrahedra) to an approximation of iv the interior. This can be implemented as a composition of standard FE basis functions on a local auxiliary and purely virtual grid by which we approximate the interface. In case of discontinuous coefficients, the same local auxiliary composition approach is used. Composition weights are obtained by solving local interpolation problems for which coupling conditions across the interface need to be determined. These depend both on the local interface geometry and on the (scalar or tensor-valued) material coefficients on both sides of the interface. We consider heat diffusion as a scalar model problem and linear elasticity as a vector-valued model problem to develop and implement the composite FE. Uniform cubic meshes contain a natural hierarchy of coarsened grids, which allows us to implement a multigrid solver for the case of complicated domains. Near an interface (red line) which is not resolved by the regular computational grid, composite FE basis functions are constructed in such a way that they can approximate functions satisfying a coupling condition (depending on the coefficients) across the interface. Homogenization. Besides simulations of single loading cases, we also apply the composite FE method to the problem of determining effective material properties, e. g. for multiscale simulations. For periodic microstructures, this is achieved by solving corrector problems on the fundamental cells using affine-periodic boundary conditions corresponding to uniaxial compression and shearing. For statistically periodic trabecular structures, representative fundamental cells can be identified but do not permit the periodic approach. Instead, macroscopic displacements are imposed using the same set as before of affine-periodic Dirichlet boundary conditions on all faces. The stress response of the material is subsequently computed only on an interior subdomain to prevent artificial stiffening near the boundary. We finally check for orthotropy of the macroscopic elasticity tensor and identify its axes. young human osteoporotic human porcine bovine For specimens of vertebral trabecular bone of bipeds and quadrupeds, effective elasticity tensors are visualized (where elongation indicates directional compressive stiffness). The human tensors are scaled by 4 relative to the animal tensors. v Notation 1 constant-1 function 1(x) = 1 a thermal diffusivity tensor (p. 13) A(z) set of simplices adjacent to virtual node z (p. 34) B (local) matrices for construction of CFE weights (pp. 43 and 48) c mass-specific heat capacity (p. 13) cM characteristic function of a set M (p. 36) Ck space of real-valued, k times continuously differentiable functions (Ck)3 space of R3-valued, k times continuously differentiable functions C elasticity tensor (p. 14) d space dimension, typically 2 or 3 D(r) set of virtual nodes constrained by a regular node r (p. 34) D( f ) ‘descendants’ in multigrid coarsening (p. 90) di,j Kronecker symbol (p. 17) th ei i unit vector E Young’s modulus (p. 16); FE elasticity block matrix (p. 60) e[u] strain (p. 14) G grid/mesh: G regular cubic grid (p. 28), G regular tetrahedral mesh (p. 28), G4 virtual (tetrahedral) mesh (p. 30) g curved interface (p. 28) G (piecewise) planar interface (p. 29) H± halfspaces (subdomains for a planar interface; p. 19) Hm,p Sobolev space (p. 14) Id identity function Id(x) = x or identity matrix I index set for a node set N : I (p. 28), I4 (p. 30); interpolation operators j global index (p. 28, 98) K, L (local) matrices arising in coupling conditions across an interface (p. 22, 23) L FE stiffness matrix (p. 58) l first Lame-Navier´ number (p. 16); thermal conductivity (p. 13)