A hybrid variational-collocation immersed method for fluid-structure interaction using unstructured T-splines

Abstract

Summary We present a hybrid variational‐collocation, immersed, and fully‐implicit formulation for fluid‐structure interaction (FSI) using unstructured T‐splines. In our immersed methodology, we define an Eulerian mesh on the whole computational domain and a Lagrangian mesh on the solid domain, which moves arbitrarily on top of the Eulerian mesh. Mathematically, the problem reduces to solving three equations, namely, the linear momentum balance, mass conservation, and a condition of kinematic compatibility between the Lagrangian displacement and the Eulerian velocity. We use a weighted residual approach for the linear momentum and mass conservation equations, but we discretize directly the strong form of the kinematic relation, deriving a hybrid variational‐collocation method. We use T‐splines for both the spatial discretization and the information transfer between the Eulerian mesh and the Lagrangian mesh. T‐splines offer us two main advantages against non‐uniform rational B‐splines: they can be locally refined and they are unstructured. The generalized‐α method is used for the time discretization. We validate our formulation with a common FSI benchmark problem achieving excellent agreement with the theoretical solution. An example involving a partially immersed solid is also solved. The numerical examples show how the use of T‐junctions and extraordinary nodes results in an accurate, efficient, and flexible method. Copyright © 2015 John Wiley & Sons, Ltd.

Non-body-fitted fluid–structure interaction: Divergence-conforming B-splines, fully-implicit dynamics, and variational formulation

“…(3) for a given mesh resolution, the DCIB method solves this equation in variational form using the Bubnov-Galerkin method. This is in contrast with our previous works[15,16,25] where Eq. (3) was solved in strong form using isogeometric collocation[84,85].…”

“…The generalization of the IB method to work with rigid solids was done in [8,9], to work with variable densities and viscosities in [10,11], and to work with two-fluid flows in [12,13]. In the last decades, diverse numerical methods have been developed to discretize the mathematical model proposed by the IB method, such as a hybrid finite difference/finite element discretization [14], a NURBS-based discretization [15], a discretization based on T-splines [16], finite volume discretization for the Navier-Stokes equations [17], and lattice Boltzmann discretization for the Navier-Stokes equations [18,19]. These numerical methods have been applied to a variety of problems, such as heart valve analysis and design [20,21,22,23], cell-scale blood flow [24,25,26], aquatic animal locomotion [27,28], tissue cryofreezing [29], capsule dynamics [30], vesicle dynamics [31], particle laden flows [32], and floating structures [33].…”

“…In the last decade, the use of spline functions, such as B-splines, non-uniform rational B-splines (NURBS), analysis-suitable T-splines (ASTS), and hierarchical B-splines, has become widespread in computational mechanics thanks to isogeometric analysis (IGA) [51,54]. In the field of immersed methods for FSI, IGA has already been used to perform NURBS-based and ASTS-based generalizations of the IB method [15,16], develop the immersogeometric method [38,55,39], couple shells with Stokes flows using the boundary integral method [56,57], solve air-blast problems [58], develop a fictitious domain approach [59], and a stabilized cut-cell immersed framework [60].…”

Liquid-vapor transformations with surfactants. Phase-field model and Isogeometric Analysis

“…The contact angle between those interfaces and the boundary will be altered because of the surfactant concentration. 90 Additionally, a three-dimensional numerical simulation is presented. Finally, in Section 6, some concluding remarks are commented.…”

High-Performance Computation of Bézier Surfaces on Parallel and Heterogeneous Platforms

“…Applications such as shape optimization in aerodynamics [14], flow modeling [15], simulation [16] and non-rigid medical image registration [11] require, indeed, high-degree Bézier formulations to cope with the complexity of the underlying data.…”