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In this paper we introduce a constructive approach to study well-posedness of solutions to stochastic uid-structure interaction with stochastic noise. We focus on a benchmark problem in stochastic uidstructure interaction, and prove the existence of a unique weak solution in the probabilistically strong sense. The benchmark problem consists of the 2D time-dependent Stokes equations describing the ow of an incompressible, viscous uid interacting with a linearly elastic membrane modeled by the 1D linear wave equation. The membrane is stochastically forced by the time-dependent white noise. The uid and the structure are linearly coupled. The constructive existence proof is based on a time-discretization via an operator splitting approach. This introduces a sequence of approximate solutions, which are random variables. We show the existence of a subsequence of approximate solutions which converges, almost surely, to a weak solution in the probabilistically strong sense. The proof is based on uniform energy estimates in terms of the expectation of the energy norms, which are the backbone for a weak compactness argument giving rise to a weakly convergent subsequence of probability measures associated with the approximate solutions. Probabilistic techniques based on the Skorohod representation theorem and the Gyongy-Krylov lemma are then employed to obtain almost sure convergence of a subsequence of the random approximate solutions to a weak solution in the probabilistically strong sense. The result shows that the deterministic benchmark FSI model is robust to stochastic noise, even in the presence of rough white noise in time. To the best of our knowledge, this is the  rst well-posedness result for stochastic uid-structure interaction. 

We develop a novel mathematical and computational framework for geometric optimization of mesh-like devices such as stents, based on modeling mesh-like structures as networks of one-dimensional curved rods. To simplify calculations, the curved rods are approximated by piecewise straight rods. Constrained optimization problems for different cost functionals are stated and mathematically analyzed. The cost functionals considered include: (1) stents' compliance, (2) norm of displacement, (3) norm of contact moment (which is related to fatigue), and (4) multicriteria optimization in which stents are optimized to achieve maximal radial stiffness and minimal bending rigidity. The optimization parameters are stent's vertices, namely, the location of points where the stent struts meet. Existence of solutions to the mathematically posed optimization problems is obtained, and a numerical method based on the gradient descent algorithm is proposed to find the solutions. Three representative stents' geometries are numerically analyzed to show that the optimization algorithms provide tangible solutions. The stent geometries considered are those of Palmaz type stents, single zig-zag stent rings, and Express type stents. Interesting findings are obtained, including several new stent designs. Several optimized stents are presented, including an optimized Palmaz stent with a reduction in contact moment of 30%, and optimized Express and Palmaz stents with a reduction in compliance by more than 70%. 

We give a brief survey of the recent progress in the area of mathematical well-posedness for moving boundary problems describing fluid-structure interaction between incompressible, viscous fluids and elastic, viscoelastic, and rigid solids. 

Abstract. We consider a nonlinear, moving boundary, fluid-structure interaction problem between a time dependent incompressible, viscous fluid flow, and an elastic structure composed of a cylindrical shell supported by a mesh of elastic rods. The fluid flow is modeled by the time-dependent Navier- Stokes equations in a three-dimensional cylindrical domain, while the lateral wall of the cylinder is modeled by the two-dimensional linearly elastic Koiter shell equations coupled to a one-dimensional system of conservation laws defined on a graph domain, describing a mesh of curved rods. The mesh supported shell allows displacements in all three spatial directions. Two-way coupling based on kinematic and dynamic coupling conditions is assumed between the fluid and composite structure, and between the mesh of curved rods and Koiter shell. Problems of this type arise in many ap- plications, including blood flow through arteries treated with vascular prostheses called stents. We prove the existence of a weak solution to this nonlinear, moving boundary problem by using the time discretization via Lie operator splitting method combined with an Arbitrary Lagrangian-Eulerian approach, and a non-trivial extension of the Aubin-Lions-Simon compactness result to problems on moving domains. 

The biological response of a coronary artery can be assessed measuring the radial stress of the arterial wall, which depend on the location, arterial tortuosity, and cardiac cycle. We sought to study the radial stress and investigate which geometric distribution of stent struts is associated with favorable biologic response in tortuous coronary arteries. 

The biological response of a coronary artery can be assessed measuring the radial stress of the arterial wall, which depend on the location, arterial tortuosity, and cardiac cycle. We sought to study the radial stress and investigate which geometric distribution of stent struts is associated with favorable biologic response in tortuous coronary arteries.