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

Abstract

In this work, we analyze the convergence of the recent numerical method for enforcing fluid–structure interaction (FSI) kinematic constraints in the immersogeometric framework for cardiovascular FSI. In the immersogeometric framework, the structure is modeled as a thin shell, and its influence on the fluid subproblem is imposed as a forcing term. This force has the interpretation of a Lagrange multiplier field supplemented by penalty forces, in an augmented Lagrangian formulation of the FSI kinematic constraints. Because of the non-matching fluid and structure discretizations used, no discrete inf-sup condition can be assumed. To avoid solving (potentially unstable) discrete saddle point problems, the penalty forces are treated implicitly and the multiplier field is updated explicitly. In the present contribution, we introduce the term dynamic augmented Lagrangian (DAL) to describe this time integration scheme. Moreover, to improve the stability and conservation of the DAL method, in a recently-proposed extension we projected the multiplier onto a coarser space and introduced the projection-based DAL method. In this paper, we formulate this projection-based DAL algorithm for a linearized parabolic model problem in a domain with an immersed Lipschitz surface, analyze the regularity of solutions to this problem, and provide error estimates for the projection-based DAL method in both the [Formula: see text] and [Formula: see text] norms. Numerical experiments indicate that the derived estimates are sharp and that the results of the model problem analysis can be extrapolated to the setting of nonlinear FSI, for which the numerical method was originally proposed.

A low-distortion mesh moving method based on fiber-reinforced hyperelasticity and optimized zero-stress state

“…When this method was first introduced in [137][138][139], it consisted of simply dropping J M from Eq. (68).…”

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

“…This example is used in order to numerically measure the convergence rates of the proposed method. As explained in [4,87,88,14], immersed FSI formulations often lead to exact solutions with low regularity at the fluid-solid interface. More specifically, the velocity has C 0 continuity and the pressure is discontinuous at the interface for co-dimension one solids [4,88].…”

“…As explained in [4,87,88,14], immersed FSI formulations often lead to exact solutions with low regularity at the fluid-solid interface. More specifically, the velocity has C 0 continuity and the pressure is discontinuous at the interface for co-dimension one solids [4,88]. This is also the case for co-dimension zero solids as long as the elastic traction of the solid at the interface is nonzero [4].…”

An immersogeometric formulation for free-surface flows with application to marine engineering problems

“…Since that, the research on immersed methods has been growing significantly. Some recent developments using immersed approach can be found in [28][29][30][31][32][33][34].…”