A Class of Analytic Solutions for Verification and Convergence Analysis of Linear and Nonlinear Fluid-Structure Interaction Algorithms

TL;DR

This work introduces a comprehensive class of analytic fluid–structure interaction (FSI) solutions that span 2D linear and 3D nonlinear regimes, including quasi-static and transient cases with both linear elastic and hyperelastic (neo-Hookean) solids. Building on a harmonic time dependence and axisymmetric geometry, the authors derive closed-form expressions for fluid velocity, solid displacement, and pressures, with constants fixed via coupling conditions and, in some cases, resonance criteria such as $\omega_n=(2n+1)\pi/(2(H_o-H_i))\sqrt{\mu_s/\rho_s}$. The analytic solutions are integrated into a monolithic finite-element framework using a Lagrange multiplier to enforce kinematic and dynamic coupling, and are validated through space–time convergence studies and physiologically relevant parameter sets, demonstrating first-order temporal and cubic spatial convergence under appropriate refinement. The open-source MATLAB implementations provided in the Supplementary Material enable rigorous verification and convergence analysis across multiple FSI algorithms, making this work a valuable benchmark for method development and comparison in computational biomechanics and engineering.

Abstract

Fluid-structure interaction (FSI) problems are pervasive in the computational engineering community. The need to address challenging FSI problems has led to the development of a broad range of numerical methods addressing a variety of application-specific demands. While a range of numerical and experimental benchmarks are present in the literature, few solutions are available that enable both verification and spatiotemporal convergence analysis. In this paper, we introduce a class of analytic solutions to FSI problems involving shear in channels and pipes. Comprised of 16 separate analytic solutions, our approach is permuted to enable progressive verification and analysis of FSI methods and implementations, in two and three dimensions, for static and transient scenarios as well as for linear and hyperelastic solid materials. Results are shown for a range of analytic models exhibiting progressively complex behavior. The utility of these solutions for analysis of convergence behavior is further demonstrated using a previously published monolithic FSI technique. The resulting class of analytic solutions addresses a core challenge in the development of novel FSI algorithms and implementations, providing a progressive testbed for verification and detailed convergence analysis.

Figures (11)

• Figure 1: Fluid and solid reference domains, $\Omega_f^0$ and $\Omega_s^0$, in two dimensions with respective boundaries at the inlet ($\Gamma_f^I$ and $\Gamma_s^I$), the outlet ($\Gamma_f^O$ and $\Gamma_s^O$) and the wall ($\Gamma_f^W$ and $\Gamma_s^W$). The common interface boundary is denoted as $\Gamma^\lambda$. Further, the domain length is given as $L$, the fluid domain height as $H_i$ and the fluid / solid domain height as $H_o$.
• Figure 2: Fluid and solid reference domains in three dimensions: The fluid reference domain $\Omega_f^0$ is shown in red for $x, y > 0$, the solid reference domain $\Omega_s^0$ is shown in blue for $y > 0$, and the the common interface boundary is indicated in opaque gray. The respective boundaries are denoted as $\Gamma_f^I$ and $\Gamma_s^I$ at the inlet, $\Gamma_f^O$ and $\Gamma_s^O$ at the outlet, and $\Gamma_f^W$ and $\Gamma_s^W$ at the wall. Furthermore, the domain length is given as $L$, the fluid domain radius as $H_i$ and the fluid / solid domain radius as $H_o$.
• Figure 3: Coarse, medium and fine mesh refinement levels for simulating the nonlinear transient FSI case in three dimensions.
• Figure 4: The analytic solution for the transient linear FSI case in two dimensions with density $\rho_f = \rho_s = 1$, fluid viscosity $\mu_f = 0.01$, solid stiffness $\mu_s = 0.1$ and cycle length $T = 1.024$: Fluid and solid velocity, $v_f$ (left) and $v_s$ (top left), and solid displacement, $u_s$ (top right), along the $y$-axis, and fluid / solid pressure, $p_f$ and $p_s$ (bottom), over time $t$ at three positions along the $x$-axis.
• Figure 5: The analytic solution for the transient nonlinear FSI case in three dimensions with fluid and solid density $\rho_f = 2.1$ and $\rho_s = 1$, fluid viscosity $\mu_f = 0.03$, solid stiffness $\mu_s = 0.1$ and cycle length $T = 1.024$: Fluid and solid velocity, $v_f$ (left) and $v_s$ (top left), and solid displacement, $u_s$ (top right), along the $y$-axis. Further, fluid and solid pressure, $p_f$ and $p_s$ (bottom) over time $t$ at three positions along the $z$-axis.