Projection-based stabilization of interface Lagrange multipliers in immersogeometric fluid–thin structure interaction analysis, with application to heart valve modeling

Abstract

This paper discusses a method of stabilizing Lagrange multiplier fields used to couple thin immersed shell structures and surrounding fluids. The method retains essential conservation properties by stabilizing only the portion of the constraint orthogonal to a coarse multiplier space. This stabilization can easily be applied within iterative methods or semi-implicit time integrators that avoid directly solving a saddle point problem for the Lagrange multiplier field. Heart valve simulations demonstrate applicability of the proposed method to 3D unsteady simulations. An appendix sketches the relation between the proposed method and a high-order-accurate approach for simpler model problems.

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

“…Moreover, the exact solution of the Eulerian pressure is usually discontinuous at the fluid-solid interface, which leads to poor approximation properties of the discrete pressure spaces used in immersed FSI methods. As a result, immersed FSI methods, whether they follow Peskin's idea or not (see [35,36,37,38,39]), often have profound difficulties to accurately impose the incompressibility constraint at both Eulerian and Lagrangian levels [40,41,6,34,38,25,42]. This issue is extremely important since large errors in the incompressibility constraint are able to even alter the qualitative behavior of numerical solutions in challenging FSI applications, such as heart valves [40,38] and cell-scale blood flow [25].…”

“…To the best of our knowledge, the only work that has tried pointwise divergence-free Eulerian discretization in the context of immersed FSI methods is the recent paper [46]. In [46], divergence-conforming B-splines [47,48,49,50] were applied to the nonboundary-fitted FSI method developed in [38,39]. The method used in [38,46] defines a fluid subproblem and a Kirchhoff-Love shell subproblem with no-tailored discretizations.…”

“…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].…”

Error estimates for projection-based dynamic augmented Lagrangian boundary condition enforcement, with application to fluid–structure interaction

“…Recent years have seen great interest in numerical analysis of fluid-structure interaction (FSI) due to its relevance to structural [27,28], biomedical [29], and other engineering applications [30]. In a recent series of articles [31][32][33][34][35][36], we developed a framework for simulating FSI dynamics of thin, flexible shell structures immersed in a viscous, incompressible fluid, where we assume that the thin structure can cut through the fluid meshes and the fluid/structure meshes do not have to match each other on the fluid-structure interface [37][38][39]. The target application was bioprosthetic heart valve [40] analysis.…”

“…This paper focuses on the particular numerical method introduced in [31], and refinements of it developed in subsequent work [36]. This numerical method is specialized for problems in which the structure is modeled geometrically as a surface of co-dimension one to the fluid sub-problem domain.…”