Efficient hp Error Analysis of a Hybridizable Discontinuous Galerkin Method for Linear Fluid-Structure Interaction

Grunnleggende konsepter

This paper presents a velocity-stress based variational formulation for linear fluid-structure interaction (FSI) problems and analyzes the convergence properties of a hybridizable discontinuous Galerkin (HDG) discretization method. The proposed HDG scheme utilizes symmetric tensors with piecewise polynomial entries of arbitrary degree to approximate the stress components in both 2D and 3D. The stability and quasi-optimal hp error estimates for the semi-discrete and fully discrete schemes are established.

Sammendrag

The paper introduces a velocity-stress based variational formulation for linear fluid-structure interaction (FSI) problems. This formulation employs global energy spaces throughout the entire domain, which simplifies the treatment of interface conditions on the discrete level.

To discretize the problem, a hybridizable discontinuous Galerkin (HDG) method is employed. The key aspects of the HDG discretization are:

1. Symmetric tensors with piecewise polynomial entries of arbitrary degree k ≥ 0 are used to approximate each stress component in both 2D and 3D.
2. The discrete velocity field and the discrete trace variable defined on the mesh skeleton are piecewise polynomials of degree k + 1.

The stability and convergence of the semi-discrete scheme are proven, and quasi-optimal hp error estimates are obtained for stress and velocity in the corresponding L2-norms. The convergence rates are shown to be quasi-optimal with respect to mesh size, while only suboptimal by half a power concerning polynomial degree.

Moreover, the fully discrete scheme based on the Crank-Nicolson method is demonstrated to be stable and convergent.

Numerical results in both two and three dimensions validate the expected convergence rates.

Statistikk

The mass density of the fluid is denoted by ρf > 0.
The dynamic viscosity of the fluid is denoted by μf > 0.
The mass density of the solid is denoted by ρs > 0.
The symmetric and positive definite constant tensor of order 4 relating the stress and strain in the solid is denoted by Cs.

Sitater

"To manage the potential increased computational demand of this monolithic, tensorial-based variational formulation, we apply a hybridizable discontinuous Galerkin (HDG) method [13, 18]. The hybridization technique reduces global degrees of freedom, facilitating efficient implementation through static condensation and enabling effective parallel processing, which makes it attractive for computationally demanding problems."
"The HDG method is also amenable to hp-adaptivity and flexible mesh designs. It has shown promise in various applications such as elastodynamics and Stokes flow [31, 14, 19, 30, 15], and its adaptation to the FSI velocity-pressure-displacement formulation has seen recent developments [38, 21]."

Viktige innsikter hentet fra

An $hp$ Error Analysis of HDG for Linear Fluid-Structure Interaction

by Salim Meddah... klokken arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.13578.pdf

Dypere Spørsmål

How can the proposed HDG method be extended to handle nonlinear FSI models

To extend the proposed HDG method to handle nonlinear FSI models, several modifications and enhancements would be necessary. One approach would be to incorporate nonlinear terms into the governing equations, such as convective terms in the fluid equations or nonlinear constitutive relations in the solid equations. This would require adapting the variational formulation to account for these additional terms and nonlinearities. Additionally, the numerical implementation of the method would need to be adjusted to handle the increased complexity and computational demands of nonlinear models. Techniques such as adaptive mesh refinement, higher-order elements, and advanced solvers may be employed to improve the accuracy and efficiency of the solution for nonlinear FSI problems.

What are the potential limitations or challenges in applying the velocity-stress formulation to more complex FSI problems involving, for example, compressible fluids or nonlinear solid behavior

The velocity-stress formulation, while effective for linear FSI problems, may face limitations or challenges when applied to more complex scenarios involving compressible fluids or nonlinear solid behavior. In the case of compressible fluids, the assumption of incompressibility in the fluid equations would no longer hold, requiring the formulation to be modified to accommodate variations in fluid density. This could introduce additional coupling terms and nonlinearities, complicating the solution process. Similarly, nonlinear solid behavior, such as large deformations or material nonlinearity, would necessitate the inclusion of more sophisticated constitutive models and solution techniques to accurately capture the behavior of the solid domain. These complexities could impact the convergence properties and stability of the numerical method, requiring careful consideration and potentially more advanced numerical strategies.

What are the implications of the suboptimal convergence rates with respect to polynomial degree, and how could this be addressed in future work

The suboptimal convergence rates with respect to polynomial degree in the velocity-stress formulation could have implications for the overall accuracy and efficiency of the method. While the convergence rates are quasi-optimal with respect to mesh size, the suboptimal convergence with respect to polynomial degree may lead to slower convergence and reduced accuracy for higher-order approximations. To address this issue in future work, one possible approach could be to explore alternative discretization schemes or basis functions that offer improved convergence properties for higher polynomial degrees. Additionally, incorporating adaptive strategies that dynamically adjust the polynomial degree based on the local solution behavior could help mitigate the impact of suboptimal convergence rates and enhance the overall performance of the method.