Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

Abstract

The Immersed Boundary Method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 70's which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged, leads to a linear system of equations for the interface configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin's lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system's matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix-vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve.

Partially implicit motion of a sharp interface in Navier–Stokes flow

“…For our accuracy tests we use the familiar test problem in which the initial interface is an ellipse stretched from its natural circular configuration, as in [5,10,11,17,18,24,25,33]. Usually the initial velocity is set to zero and the interface markers are equally spaced in the parameter θ as in (71) below.…”

“…This choice is comparable to that in [24,5]. We place the initial interface markers equally spaced in θ and assume the initial pressure and velocity are those determined by the Stokes equations.…”

“…It could be simplified by lagging the update of the interface location in the first integral, as has been done with implicit methods for the immersed boundary method, giving a more refined approximation than the one used here. For recent work on implicit methods for the IBM applied to Navier-Stokes flow, see [4,5,11,24,25].…”

“…Considerable effort has led to the design of implicit and semiimplicit methods to alleviate the difficulty [5,16,23,24,25,33]. Another approach, however, is to identify the source of greatest stiffness arising from small scales or large wavenumbers in the interface motion and to modify the velocity to approximate an implicit step in the high wavenumbers, thereby avoiding the iterative solves needed for a fully implicit method.…”

“…We verify convergence, test the validity of the partially implicit approximation, and estimate time steps with the force (4). We illustrate the application to a nonlinear force, as in [24,5]. We also present examples with a bending force which can be compared with calculations of inextensible vesicles [13,12,34].…”