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In vivo visualization of cardiovascular structures is
possible using medical images. However, one has to realize that the resulting 3D
geometries correspond to in vivo conditions. This entails an internal
stress state to be present in the in vivo measured geometry of
e.g. a blood vessel due to the presence of the blood pressure. In order
to correct for this in vivo stress, this paper presents an inverse method
to restore the original zero-pressure geometry of a structure, and to recover the
in vivo stress field of the final, loaded structure. The proposed
backward displacement method is able to solve the inverse problem iteratively using fixed
point iterations, but can be significantly accelerated by a quasi-Newton technique in
which a least-squares model is used to approximate the inverse of the Jacobian. The here
proposed backward displacement method allows for a straightforward implementation of the
algorithm in combination with existing structural solvers, even if the structural solver
is a black box, as only an update of the coordinates of the mesh needs to be
In the second part of the paper, we compare the solutions produced
in the framework of the conference “Mathematical and numerical
aspects of low Mach number flows” organized by INRIA and MAB in
Porquerolles, June 2004, to the reference solutions described in
Part 1. We make some recommendations on how to produce good
quality solutions, and list a number of pitfalls to be avoided.
There are very few reference solutions in the literature on
non-Boussinesq natural convection flows. We propose here a test
case problem which extends the well-known De Vahl Davis
differentially heated square cavity problem to the case of large
temperature differences for which the Boussinesq approximation is
no longer valid. The paper is split in two parts: in this first
part, we propose as yet unpublished reference solutions for cases
characterized by a non-dimensional temperature difference of 0.6,
Ra = 106 (constant property and variable property cases) and
Ra = 107 (variable property case). These reference solutions were
produced after a first international workshop organized by CEA and
LIMSI in January 2000, in which the above authors volunteered to
produce accurate numerical solutions from which the present
reference solutions could be established.
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