arXiv subjects

Pierre-Frédéric Villard

Publications and source records attributed to Pierre-Frédéric Villard.

3 recordsLinked to original sources

arXiv2026

The least squares RBF-PU method for linear elasticity in the diaphragm geometry

The human diaphragm is vital for respiration and its simulation can aid in further understanding complications arising in individuals that have been subjected to mechanical ventilation. The diaphragm geometry is very thin and hence its numerical simulation poses a challenge for most numerical methods. In this paper, we compute the deformation of real diaphragm geometries which are modelled as linearly elastic and are subject to realistic boundary conditions. The method used is the unfitted least-squares radial basis function partition of unity method, which is well suited to solve problems on thin geometries given its unfitted nature and the shape and refinement of the cylindrical patches that are used. We derive a theoretical proof showing that the method converges with high order to the solution of this problem.

math.NA↗

arXiv2024

Cosserat Rods for Modeling Tendon-Driven Robotic Catheter Systems

Tendon-driven robotic catheters are capable of precise execution of minimally invasive cardiac procedures including ablations and imaging. These procedures require accurate mathematical models of not only the catheter and tendons but also their interactions with surrounding tissue and vasculature in order to control the robot path and interaction. This paper presents a mechanical model of a tendon-driven robotic catheter system based on Cosserat rods and integrated with a stable, implicit Euler scheme. We implement the Cosserat rod as a model for a simple catheter centerline and validate its physical accuracy against a large deformation analytical model and experimental data. The catheter model is then supplemented by adding a second Cosserat rod to model a single tendon, using penalty forces to define the constraints of the tendon-catheter system. All the model parameters are defined by the catheter properties established by the design. The combined model is validated against experimental data to confirm its physical accuracy. This model represents a new contribution to the field of robotic catheter modeling in which both the tendons and catheter are modeled by mechanical Cosserat rods and fully-validated against experimental data in the case of the single rod system.

cs.RO↗

arXiv2024

An RBF partition of unity method for geometry reconstruction and PDE solution in thin structures

The main respiratory muscle, the diaphragm, is an example of a thin structure. We aim to perform detailed numerical simulations of the muscle mechanics based on individual patient data. This requires a representation of the diaphragm geometry extracted from medical image data. We design an adaptive reconstruction method based on a least-squares radial basis function partition of unity method. The method is adapted to thin structures by subdividing the structure rather than the surrounding space, and by introducing an anisotropic scaling of local subproblems. The resulting representation is an infinitely smooth level set function, which is stabilized such that there are no spurious zero level sets. We show reconstruction results for 2D cross sections of the diaphragm geometry as well as for the full 3D geometry. We also show solutions to basic PDE test problems in the reconstructed geometries.

math.NA↗