Composite finite elements for 3D image based computing

Abstract

Abstract. We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying image data has been acquired through e. g. computed tomography (CT), magnetic resonance imaging (MRI) or electron microscopy (EM). Based on a level-set description of the computational domain, our approach is capable of automatically providing suitable composite finite element functions that resolve the complicated shapes in the medical/biological data set. It is efficient in the sense that the traversal of the grid (and thus assembling matrices for finite element computations) inherits the efficiency of uniform grids away from complicated structures. The method's efficiency heavily depends on precomputed lookup tables in the vicinity of the domain boundary or interface. A suitable multigrid method is used for an efficient solution of the systems of equations resulting from the composite finite element discretization. The paper focuses on both algorithmical and implementational details. Scalar and vector valued model problems as well as real applications underline the usability of our approach.

3D Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

“…Fig. 4.1 and [50]) in such a way that the subdivision is consistent with neighboring cubes. Let us denote this mesh by G and the set of vertices by N , which by construction coincides with the vertex set of G , i.e.…”

“…The notation and terminology used in this section follow [50], where a similar basic methodology is used and introduced for the construction of CFE on complicated domains with continuous coefficients. Starting from standard affine FE basis functions on a uniform mesh, our aim is to construct CFE basis functions associated with the same nodes such that the corresponding nodal interpolation of a function satisfies the appropriate coupling conditions (3.5) or (3.9) across the interface.…”

“…The resulting matrix is still sparse with a slightly enlarged stencil close to the interface and can be computed in O(card N ) time. For a comprehensive description of the mesh generation and the processing of the interface described by level sets, we refer to [50].…”

“…For the corresponding 2D case and a corresponding derivation in full detail, we refer to [67]. A corresponding CFE approach for complicated single-phase domains is investigated in [50] and applied in a homogenization framework in [68].…”

Shape Optimization under Uncertainty from a Stochastic Programming Point of View

“…They are called composite finite elements, in short CFEs, and have been applied successfully in various applications, e.g. in image based computing [59]. Additionally to domains with complicated boundary, CFEs can be used for problems involving jumping, i.e.…”