A Highly Scalable Matrix-Free Multigrid Solver for μFE Analysis Based on a Pointer-Less Octree

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.