Основні поняття
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.
Анотація
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:
- Symmetric tensors with piecewise polynomial entries of arbitrary degree k ≥ 0 are used to approximate each stress component in both 2D and 3D.
- 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.
Статистика
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.
Цитати
"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]."