The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad.
This award will support a twenty-four month research fellowship by Dr. Lorena Bociu to work with Dr. Jean-Paul Zolesio at Centre National de la Recherche Scientifique, Institut Non-Lineaire de Nice, Sophia Antipolis, and with Dr. John Cagnol, at Pole Universitaire Leonard de Vinci, in Paris, France.
The proposed research is focused on wellposedness and stability of finite energy solutions to nonlinear structural acoustic models with curved walls. Structural acoustic interactions are described by a system of coupled equations: the wave equation, which models the acoustic medium in a 3-D chamber, and the dynamic shell equation, describing the flexible wall of the chamber. In turn, the motion of a dynamic shell is described by a set of coupled nonlinear partial differential equations, both of hyperbolic type: an elastic wave for the in-plane displacement, and a nonlinear Kirchhoff equation for the scalar normal displacement. Structural acoustic models, due to their large spectrum of engineering applications, have received a lot of attention in engineering and mathematical literature. However, most of the analysis has been performed on linear models with flat walls. The main novelty of the proposed research is that it will account for nonlinear displacements of the curved wall (i.e. fully nonlinear shell) in a coupling with a nonlinear acoustic medium (nonlinear wave equation). Thus, both nonlinear (topological) and geometric aspects will be at the focus of the proposed research, with Euclidean flat geometry being replaced by Riemannian geometry. More specifically, the following issues, recognized as open problems in the literature, will be addressed: First, for a nonlinear shell with nonlinear boundary sources: local and global existence (or blow-up in finite time), uniqueness and regularity of finite energy solutions. Second, for a 3-D structural acoustic nonlinear model with viscous damping and involving shells on the interface between the media: (i) Hadamard wellposedness of finite energy solutions driven by critical and supercritical sources, along with stability of solutions in the presence of boundary (geometrically restricted) damping, and (ii) quantification of the level of nonlinearity of the damping that is sufficient to ensure that finite energy solutions be global. Thus, nonlinearity of the damping is at the heart of the problem. The solution to this problem will not only provide a novel and important contribution to PDEs and their control, but will also have far-reaching potential for transferable research into engineering-based design. The project will use the dynamic shell model based on intrinsic geometry developed by M.Delfour and J.P.Zolesio, which offers great advantages for an analytic formulation of the problem. Host J.P.Zolesio is also an expert in analysis and control of interactive structures - a dominant theme in the proposal. The project will also benefit from strong interaction with J. Cagnol, who is well experienced in shell analysis, including computations with intrinsic geometry-based codes. The proposed research will provide a mathematical solution to a physical problem that is fundamental in application (noise suppression in an acoustic environment). It should also stimulate new approaches in engineering design, eventually impacting society. The proposed methods could be applied to other PDE models sharing common properties: propagation of singularities, finite speed of propagation and supercriticality. Moreover, good wellposedness theory is fundamental for control theory methods to be applied.
Project Outcomes This project has produced several results of significant importance in engineering applications (in particular in control and stability problems) and numerical computations, where uniqueness of solution and continuous dependence of solution on initial data are the properties most desired. Moreover, this research opens up the possibility of dealing with high-order nonlinear acoustics in various coupled Partial Differential Equations problems that may potentially accommodate strong sources on the interface. Our first set of results pertains to nonlinear wave equations with boundary conditions of Neumann type, and with "energy-building" nonlinear sources acting both in the interior of the domain and on its boundary. These are the natural conditions often present in interactive systems where the wave equation is coupled with other dynamics, for example, structural acoustic problems and fluid structure interactions. For these models, we have obtained: Full Hadamard well-posedness of local (in time) flows (i.e., solutions exist locally, are unique, and depend continuously on initial data in the finite-energy topology). Explicit uniform algebraic decay rates of the finite energy associated with the model. Next, we have obtained groundbreaking results for free or moving boundary fluid-elasticity interactions (both static and dynamic). The linearized model for the fluid-elasticity interaction and the associated well-posedness theory for the linearization will be relevant in engineering design and medical research (e.g., the study and simulation of blood flow in compliant vessels), since it incorporates the influence of the free boundary, unlike all previous models. The new linearization method that we introduced can be applied to other free boundary couplings, thus fueling new research in the field. We have provided the first linearized model that does not ignore the geometry of the problem and takes into account the curvature and the boundary acceleration (unaccounted for in current linear models) of the free/moving common interface. Modeling of this geometrical aspect is critical for a correct physical interpretation of the fluid-structure interaction. Our conclusion is that the boundary curvature and the boundary acceleration can not be neglected when performing sensitivity analysis on the nonlinear coupling of fluid and elasticity. While the total linearization method recovers the linear equations on their respective domains, the coupling on the boundary is much more complicated than just the matching of the respective normal stress tensors, and contains terms involving the matrix of curvatures and the boundary acceleration. We proved existence and uniqueness of solution for the new linearized steady-state coupled system of incompressible fluid and elasticity. The challenge came from the new terms involving the matrix of curvatures in a Robin-Type condition on the boundary of the linearized coupling. Finally, we have obtained several results with immediate applications in shape optimization and control problems for the linear wave equation and coupled systems where the hyperbolic equation is coupled with other dynamics, and the matching conditions at the boundary are of Neumann type. We have proved existence of weak and strong shape derivatives for the solution. This provides the first positive answer on the shape differentiability analysis for hyperbolic problems with non-smooth Neumann boundary conditions, which is an important question in shape optimization. We introduced a new "pseudo-extractor technique" which provides a neat alternative to showing hidden boundary regularity for the solution to the wave equation with mixed (Dirichlet-Neumann) boundary conditions.
