author = {Schwen, Lars Ole},
  title = {Composite Finite Elements for Trabecular Bone Microstructures},
  school = {University of Bonn},
  year = {2010},
  type = {Dissertation},
  abstract = {In many medical and technical applications, numerical simulations
	need to be performed 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 simulations on such structures, so-called
	composite finite elements. We will deal with both the case of material/void
	interfaces (complicated domains) and the case of interfaces between
	different materials (discontinuous coefficients). In classical finite
	element simulations, geometric complexity is encoded in tetrahedral
	and typically unstructured meshes. Composite finite elements, 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, generalized finite
	element methods, immersed interface methods, partition of unity methods,
	unfitted meshes, and extended finite element methods), the composite
	finite elements 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 finite element method for complicated domains
	goes back to Wolfgang Hackbusch and Stefan Sauter and restricts standard
	affine finite element basis functions on the uniformly structured
	tetrahedral grid (obtained by subdivision of each cube in six tetrahedra)
	to an approximation of the interior. This can be implemented as a
	composition of standard finite element 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 finite elements. Uniform cubic
	meshes contain a natural hierarchy of coarsened grids, which allows
	us to implement a multigrid solver for the case of complicated domains.
	Besides simulations of single loading cases, we also apply the composite
	finite element 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.},
  pdf = { 1},
  isbn = {978-3-938363-78-2},
  publisher = {HARLAND media}