|Simbios Talk by Alex Ten Eyck, Stanford University, March 7, 2007
Title: Discontinuous Gelerkin for nonlinear elasticity with applications to the blood vessel
Abstract: In this talk, we present our analysis of discontinuous Galerkin (DG), a new computational method for tissue mechanics, and propose a new approach to modeling the macroscopic behavior of blood vessel using DG and homogenization. The blood vessel wall is an elastic heterogeneous anisotropic structure for which we observe macroscale mechanical behavior that is determined by its microstructure. An accurate description of the material properties and mechanical behavior is essential for researchers who are modeling blood flow in the arteries. Although, modeling the arteries as rigid can still yield qualitative information about the blood flow, quantitative data, such as pressure, require researchers to model the arteries as elastic structures. Two components of the blood vessel, elastin and collagen, which can be visualized only with nanometer resolution, are the principle mechanical load carrying structures in the blood vessel. Their arrangement in the artery is discontinuous and extremely complex. In a hope to accurately model the blood vessel, we have analyzed the discontinuous Galerkin (DG) method for nonlinear elasticity in detail. We have found this to be a very powerful method comparable to the traditional finite element method. Although this method can become unstable, we have developed a robust adaptive stabilization technique. This technique will stabilize DG for a wide variety of nonlinear elasticity problems. Using recently unpublished three dimensional images of the artery using serial block-face scanning electron microscopy (SBFSEM) we plan to model the blood vessel and extract its macroscopic material properties. The SBFSEM images accurately show the arrangement of elastin and collagen fibers in vitro for both a healthy and diseased rat artery. However, a computational model of the blood vessel accounting for every elastin and collagen fiber is presently impossible. In addition, since we are more concerned with the macroscale behavior, such as the blood flow, we really don't need to know the behavior of each fiber. Our approach to obtaining the macroscopic material properties uses the process of homogenization of periodic structures. This technique will allow us to coarse grain the artery into a more tractable model and extract the homogenized material properties.