Fluid-Structure Interactions: Control and Optimization

We focus on fluid-structure interactions (FSI) described by incompressible, viscous fluids coupled with elastic structures that move and deform inside them. Fluid-structure interactions have various medical and engineering applications, ranging from blood flow in stenosed arteries to the design of small-scale unmanned aircrafts. In most of the applications, the goal is the optimization or optimal control of the considered process. Specific examples include: minimizing turbulence, achieving given targets for fluid velocity or pressure, and minimizing the hydro-elastic pressure on the interface between the two environments. Therefore we consider PDE-constrained optimization and optimal control problems governed by moving boundary fluid-elasticity interactions. These interactions are highly nonlinear couplings of parabolic-hyperbolic type, described by a mismatch of regularity of the two solutions at the common interface. We discuss main challenges and describe the results that we obtained so far regarding the existence of optimal controls and the derivation of necessary optimality conditions

The role of structural viscoelasticity in deformable porous media with incompressible constituents: applications in biomechanics

The main goal of this work is to clarify and quantify, by means of mathematical analysis, the role of structural viscoelasticity in the biomechanical response of deformable porous media with incompressible constituents to sudden changes in external applied loads. Models of deformable porous media with incompressible constituents are often utilized to describe the behavior of biological tissues, such as cartilages, bones and engineered tissue scaffolds, where viscoelastic properties may change with age, disease or by design. Here, for the first time, we show that the fluid velocity within the medium could increase tremendously, even up to infinity, should the external applied load experience sudden changes in time and the structural viscoelasticity be too small. In particular, we consider a one-dimensional poro-visco-elastic model for which we derive explicit solutions in the cases where the external applied load is characterized by a step pulse or a trapezoidal pulse in time. By means of dimensional analysis, we identify some dimensionless parameters that can aid the design of structural properties and/or experimental conditions as to ensure that the fluid velocity within the medium remains bounded below a certain given threshold, thereby preventing potential tissue damage. The application to confined compression tests for biological tissues is discussed in detail. Interestingly, the loss of viscoelastic tissue properties has been associated with various disease conditions, such as atherosclerosis, Alzheimer's disease and glaucoma. Thus, the findings of this work may be relevant to many applications in biology and medicine

Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping

AbstractWe consider the wave equation with supercritical interior and boundary sources and damping terms. The main result of the paper is local Hadamard well-posedness of finite energy (weak) solutions. The results obtained: (1) extend the existence results previously obtained in the literature (by allowing more singular sources); (2) show that the corresponding solutions satisfy Hadamard well-posedness conditions during the time of existence. This result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping

Analysis of nonlinear poro-elastic and poro-visco-elastic models

We consider the initial and boundary value problem for a system of partial differential equations describing the motion of a fluid–solid mixture under the assumption of full saturation. The ability of the fluid phase to flow within the solid skeleton is described by the permeability tensor, which is assumed here to be a multiple of the identity and to depend nonlinearly on the volumetric solid strain. In particular, we study the problem of the existence of weak solutions in bounded domains, accounting for non-zero volumetric and boundary forcing terms. We investigate the influence of viscoelasticity on the solution functional setting and on the regularity requirements for the forcing terms. The theoretical analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not viscoelasticity is present. The theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite element discretization. When the data are sufficiently regular, the simulations show that the solutions satisfy the energy estimates predicted by the theoretical analysis. Interestingly, the simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy might become unbounded if indeed the data do not enjoy the time regularity required by the theory

Linearization of a coupled system of nonlinear elasticity and fluid

We model the coupled system formed by an incompressible, irrotational fluid and a nonlinear elastic body. We work with large displacement, small deformation elasticity (or St Venant elasticity), which makes the problem very interesting from the physical point of view. The elastic body is three-dimensional $\Omega \in \mathbb{R}^3$, and thus it can not be reduced to its boundary $\Gamma$ (like in the case of a membrane or a shell). In this paper, we study the static problem, which contrary to common belief, it is more subtle than the dynamical one (since in real life, evolution is more plausible than equilibrium)

Sensitivity Analysis in Poro-Elastic and Poro-Visco-Elastic Models with Respect to Boundary Data

In this article we consider poro-elastic and poro-visco-elastic models inspired by problems in medicine and biology, and we perform sensitivity analysis on the solutions of these fluid-solid mixture problems with respect to the imposed boundary data, which are the main drivers of the system. Moreover, we compare the results obtained in the elastic case vs. visco-elastic case, as it is known that structural viscosity of biological tissues decreases with age and disease. Sensitivity analysis is the first step towards optimization and control problems associated with these models, which is our ultimate goal

Optimization and Control in Free and Moving Boundary Fluid-Structure Interactions

We consider optimization and optimal control problems subject to free and moving boundary nonlinear fluid-elasticity interactions. As the coupled fluid-structure state is the solution of a system of partial differential equations that are coupled through continuity relations on velocities and normal stress tensors, defined on the free and moving interface, the investigation (existence of optimal controls, sensitivity equations, necessary optimality conditions, etc.) is heavily dependent on the geometry of the problem, and falls into moving shape analysis framework.Non UBCUnreviewedAuthor affiliation: NC State UniversityFacult