Convergence proof of the velocity field for a stokes flow immersed boundary method

Mori, Yoichiro

doi:10.1002/cpa.20233

Cited by 59 publications

(93 citation statements)

References 37 publications

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“…As in [5,6], the equations in (2.3) are imposed to ensure unique solutions. The problem is discretized as follows.…”

Section: Model Problem Consider a Stokes Flow Problem On Anmentioning

confidence: 99%

“…As suggested in [8,6,5], the moment order controls the accuracy of the interpolation operation (see equation (2.22)). The smoothing order was introduced in [2] and [5], and has the effect of taming error components with high spatial frequency.…”

Section: Model Problem Consider a Stokes Flow Problem On Anmentioning

confidence: 99%

“…Although it has been justified by numerical results in practice, convergence of the IB method is often unresolved from an analytical point of view. Despite being the topic of many papers [6,8,9,1,3,4], convergence analysis of IB methods is still at a primary stage. As a first step to analyze full dynamic problems, convergence properties of the IB method were studied for a stationary Stokes flow problem in [6].…”

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confidence: 99%

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$L^p$ Convergence of the Immersed Boundary Method for Stationary Stokes Problems

Liu

Mori²

2014

SIAM J. Numer. Anal.

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Abstract.In this paper, we analyze the convergence of the immersed boundary (IB) method as applied to a static Stokes flow problem. Using estimates obtained in [5], we consider a problem in which a d-dimensional structure is immersed in n-dimension, and prove error estimates for both the pressure and the velocity field in the L p (1 ≤ p ≤ ∞) norm. One interesting consequence of our analysis is that the asymptotic error rates in the L 1 norm do not depend on either d or n and in the L p (p > 1) norm they only depend on n − d. The resulting estimates are checked numerically for optimality.Key words. immersed boundary method, L p error estimates, discrete delta functions.1. Introduction. The immersed boundary method has been widely used to solve problems with moving interfaces (fluid-structure interaction, two phase fluid flow, etc) along which variables of interest often possess discontinuities. In IB formulations, these problems are recast as partial differential equations (PDE) over a simpler domain with singular source terms distributed along the interfaces. Dirac delta functions are used to represent the singular source term as a distribution defined on the entire fluid domain. The PDE are often discretized on a Eulerian grid over the fluid domain, while the singular source term is often discretized by integrating over a Lagrangian grid on the interface using regularized Dirac delta functions (often called discrete delta functions) . The discretized equations are then often solved with standard methods.In addition to the IB method, many other methods are often used to solve such free surface problems, with boundary integral methods and level set methods being prime examples. Among these methods, algorithms based on the IB method are often very efficient and easy to implement. Although it has been justified by numerical results in practice, convergence of the IB method is often unresolved from an analytical point of view. Despite being the topic of many papers [6,8,9,1,3,4], convergence analysis of IB methods is still at a primary stage. As a first step to analyze full dynamic problems, convergence properties of the IB method were studied for a stationary Stokes flow problem in [6]. For the velocity field, point-wise and L ∞ error estimates are obtained. Then as an application of the results, L 2 error estimates are studied for a simple dynamic problem. The analysis relies on the geometric structure of the 2D model problem, which is a one-dimensional elastic string immersed in a two-dimensional fluid domain.In [5], the analysis is extended to a more general elliptic model problem. It is worth mentioning that the analysis in [5], despite carried through for an immersed boundary method setup, is developed for more general problems involving discontinuities that are regularized by using discrete delta functions. For an immersed boundary model problem with general choices of n and d, where n is the dimension of the fluid domain and d is the dimension of the immersed structure, the error is decomposed and carefully ...

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“…As in [5,6], the equations in (2.3) are imposed to ensure unique solutions. The problem is discretized as follows.…”

Section: Model Problem Consider a Stokes Flow Problem On Anmentioning

confidence: 99%

Section: Model Problem Consider a Stokes Flow Problem On Anmentioning

confidence: 99%

mentioning

confidence: 99%

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$L^p$ Convergence of the Immersed Boundary Method for Stationary Stokes Problems

Liu

Mori²

2014

SIAM J. Numer. Anal.

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“…This error for the proposed, approximated matrixM n is asymptotically smaller than the O(h) error of the IB Method for points within a distance O(h) of the immersed boundary [12]. Thus, the approximated matrixM n can be use without any deterioration of the overall accuracy of the IB Method.…”

Section: Error Estimate For Approximation Of the Matrix Representatiomentioning

confidence: 86%

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

Ceniceros

Fisher

Roma

2009

Journal of Computational Physics

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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.

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“…No results were included for convergence of the gradient of the velocity. For a volume penalizing IB method applied to Stokes flow, i.e., very viscous, smooth flow, it was shown rigorously in [65] that firstorder convergence of the velocity field can be expected, which was actually achieved in test simulations. In case the solid-fluid interface is allowed to be smoothed, or if it is already sufficiently smooth by itself, a so-called ghost-cell IB method can be shown to yield first order [7] or in selected situations second order [25] converging velocity fields for flow over an undulating channel [89].…”

Section: Introductionmentioning

confidence: 96%

Modeling and simulation of flow in cerebral aneurysms

Mikhal¹

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Convergence proof of the velocity field for a stokes flow immersed boundary method

Cited by 59 publications

References 37 publications

$L^p$ Convergence of the Immersed Boundary Method for Stationary Stokes Problems

$L^p$ Convergence of the Immersed Boundary Method for Stationary Stokes Problems

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

Modeling and simulation of flow in cerebral aneurysms

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