Deterministic and Stochastic Fluid-Structure Interaction

Table of Contents

1. Frontmatter

2. Chapter 1. Introduction and Outline

   Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

   Abstract

   This chapter provides and introduction to this book and its outline.

3. Preliminaries

   1. Frontmatter

   2. Chapter 2. Deterministic Preliminaries

      Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

      Abstract

      This chapter reviews essential concepts from real and functional analysis for studying fluid-structure interaction (FSI). It begins with Banach and Hilbert spaces, emphasizing \({\mathbf {L}}^{\mathrm {p}}\)? and Sobolev spaces, and explores the Fourier transform in \({\mathbf {L}}^{2}(\mathbb {R}^{\mathrm {d}}\)). Next, it introduces Bochner spaces (e.g., \({\mathbf {L}}^{\mathrm {p}}(\mathbf {0,T};\ \mathbf {B})\)) for time-dependent PDEs and extends these to moving domains, addressing time-dependent geometries and key inequalities. The chapter then covers compact embeddings and foundational equations of fluid dynamics and elasticity, including a priori estimates and weak solutions. Finally, it introduces constructive methods—Rothe’s method and Lie operator splitting—for approximating solutions to PDEs and FSI problems. These tools are crucial for analyzing evolutionary PDEs and dynamic domain problems.

   3. Chapter 3. Probabilistic Preliminaries

      Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

      Abstract

      This chapter introduces probabilistic concepts for Banach space-valued random variables, essential for analyzing stochastic fluid-structure interaction (FSI) systems. Key topics include:

      • Three modes of convergence for random variables: in probability, weak (distributional), and almost sure, along with their interrelations and relevant classical theorems
      • White noise (both one-dimensional and spacetime) as fundamental random noise in stochastic PDEs
      • Stochastic integration against such noise, crucial for defining solutions to stochastic PDEs and quantifying randomness in dynamics

      While foundational probability theory (e.g., real-valued random variables) is assumed, the chapter extends these ideas to Banach spaces, emphasizing parallels. Advanced generalizations will appear in later chapters. References are provided for deeper dives into measure-theoretic probability, Brownian motion, stochastic integration, and infinite-dimensional stochastic analysis.

4. Deterministic Fluid-Structure Interaction

   1. Frontmatter

   2. Chapter 4. Deterministic FSI with Linear Coupling

      Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

      Abstract

      This chapter examines linearly coupled fluid-structure interaction (FSI) models, where the interaction between fluid and structure is approximated using linear theories, simplifying analysis while retaining key physical insights. The fluid is modeled on a fixed domain, avoiding the complexities of moving boundaries. These models are useful in scenarios with small deformations (e.g., blood flow in certain scenarios, wind effects on some structures) but still pose challenges in well-posedness due to two-way coupling and multiphysics interactions.

      A prototypical linearly coupled FSI problem is introduced, featuring a lower-dimensional elastic structure on the boundary of the fluid domain. The Lie operator splitting method is introduced here to construct weak solutions for the linear case, laying the groundwork for later analysis of more complex models.

   3. Chapter 5. Deterministic FSI with Nonlinear Coupling

      Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

      Abstract

      This chapter examines a benchmark nonlinearly coupled deterministic fluid-structure interaction (FSI) problem, where the fluid domain evolves in time based on the structure’s deformation, introducing geometric nonlinearity. Unlike linearly coupled models, the coupling conditions are evaluated at the current fluid-structure interface, making the problem more complex since the domain is unknown a priori and depends on the solution itself.

      The Lie operator splitting method is employed to construct weak solutions in finite energy spaces, but additional difficulties arise compared to linearly coupled cases. This chapter highlights the key differences between linear and nonlinear FSI coupling.

   4. Chapter 6. Extensions of the Splitting Scheme

      Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

      Abstract

      This chapter highlights the effectiveness of the splitting scheme approach in analyzing fluid-structure interaction (FSI) problems. The Lie operator splitting method, previously demonstrated for both linear and nonlinear FSI models, has proven robust and adaptable, enabling the constructive existence of weak solutions for complex coupled systems. Its versatility extends to physically significant applications, including more intricate FSI scenarios. The discussion sets the stage for further developments in this book by showcasing how this framework can be extended to analyze advanced FSI models of practical relevance.

5. Stochastic Fluid-Structure Interaction (SFSI)

   1. Frontmatter

   2. Chapter 7. Stochastic FSI: A Reduced Model

      Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

      Abstract

      This chapter introduces a reduced stochastic fluid-structure interaction (FSI) model, where the full coupled dynamics are captured by a single stochastic viscous wave equation for the structure displacement. Unlike fully coupled stochastic FSI systems (studied later), this simplified model allows for explicit analysis using fundamental solutions, Fourier methods, and stochastic integration.

      The model considers a viscous incompressible fluid (stationary Stokes equations) interacting with an elastic membrane subjected to space-time white noise. Due to the geometry and linear coupling, the fluid’s effect on the structure is encoded via a Dirichlet-to-Neumann operator, yielding the stochastic viscous wave equation on \(\mathbb {R}^{2}\).

      Key results include the following:

      • Well-posedness of mild solutions in dimensions 1 and 2, surpassing classical results for stochastic heat/wave equations.
      • Improved Hölder regularity of solutions, attributed to fluid viscosity’s regularization and favorable space-time scaling.

      While distinct from the splitting schemes used earlier, this model’s analysis provides foundational insights into stochastic FSI and showcases techniques for handling space-time white noise.

   3. Chapter 8. Stochastic FSI: A Fully Coupled Model with Linear Coupling

      Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

      Abstract

      This chapter develops a general framework for stochastic FSI on bounded domains, extending beyond the simplified unbounded-domain model of Chap. 7. Unlike the earlier stochastic viscous wave equation (solvable via Fourier analysis), bounded domains require new techniques. Here, a 2D stochastic Stokes flow interacts with a 1D stochastically forced elastic membrane (modeled by a Koiter shell equation).

      Key contributions:

      • Operator splitting method divides the system into fluid/structure subproblems with stochastic effects, constructing weak solutions analytically.
      • Probabilistic compactness arguments (tightness, Skorokhod theorem, Gyöngy-Krylov lemma) replace deterministic tools to handle convergence in law and upgrade it to almost sure convergence for passing to limits.

      This work establishes the first application of operator splitting to stochastic FSI, demonstrating its versatility for bounded domains and paving the way for more realistic applications.

   4. Chapter 9. Stochastic FSI: A Fully Coupled Model with Nonlinear Coupling

      Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

      Abstract

      This chapter addresses nonlinearly coupled stochastic FSI in moving domains where domain motion is significant-unlike Chap. 8’s fixed-interface model. The system couples a 2D viscous incompressible fluid with a linearly elastic shell across a stochastically moving interface, driven by multiplicative noise affecting both fluid (volumetric) and structure (boundary).

      Key Advances

      • Weak martingale solutions (analytically and probabilistically weak) are constructed via a stochastic operator splitting scheme, extending Chap. 5’s deterministic approach.
      • New challenges arise from random domain motion, requiring tailored probabilistic compactness arguments beyond Chap. 8’s fixed-domain setting. This work demonstrates robustness of FSI models to real-world noise, particularly relevant in, e.g., hemodynamics where cardiac motion introduces inherent stochasticity.

6. Deterministic Fluid-Poroelastic Structure Interaction (FPSI)

   1. Frontmatter

   2. Chapter 10. FPSI with Linear Coupling

      Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

      Abstract

      This chapter examines a linearly coupled fluid-poroelastic structure interaction (FPSI) system, where an incompressible viscous fluid (modeled by time-dependent Stokes equations) interacts with a multilayered poroelastic structure comprising a thin plate and a thick Biot medium. The coupling occurs across a fixed interface, with displacements approximated on fixed domains—valid for small deformations. Key coupling conditions include the Beavers-Joseph-Saffman condition, posing challenges for velocity trace regularity.

      Two scenarios are analyzed:

      1.

      A linear dynamic Biot model for the thick poroelastic medium

       

      2.

      A nonlinear quasi-static Biot model, where permeability depends nonlinearly on fluid content (motivated by biological applications)

       

      Existence of weak, finite-energy solutions is proven using Rothe’s method and energy estimates, with compactness tools (e.g., Aubin-Lions lemma) for the nonlinear case. Uniqueness and strong solution criteria are also discussed.

   3. Chapter 11. FPSI with Nonlinear Coupling

      Sunčica Čanić, Jeffrey Kuan, Boris Muha, Krutika Tawri

      This chapter investigates a nonlinearly coupled fluid-poroelastic structure interaction (FPSI) system, where an incompressible fluid (modeled by Navier-Stokes equations) interacts with a multilayered poroelastic structure comprising a thin reticular plate and a thick Biot medium. Unlike previous linearly coupled models, both the fluid and Biot domains are time dependent, with their configurations determined by structural displacements, introducing geometric nonlinearities that complicate analysis.

      Key contributions:

      • A well-posedness theory for weak solutions to a regularized FPSI problem, proven via a splitting scheme and compactness arguments adapted for moving domains

      • A weak-classical consistency result, showing that regularized weak solutions converge to smooth solutions of the original problem (when they exist) as regularization vanishes

      This chapter extends FPSI analysis to nonlinear coupling and moving domains, addressing challenges unique to Navier-Stokes-Biot systems with dynamic interfaces.

7. Backmatter