A stable added-mass partitioned (AMP) algorithm for elastic solids and incompressible flow

Serino, Daniel A.; Banks, Jeffrey W.; Henshaw, William D.; Schwendeman, Donald W.

doi:10.1016/j.jcp.2019.108923

Cited by 16 publications

(17 citation statements)

References 46 publications

(120 reference statements)

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“…as was done in (3.8). The extensive numerical results in Section 7 and [15] confirm that this is an appropriate choice, and furthermore that the scheme is rather insensitive to the choice of h, C AM and C AD .…”

Section: Velocity Boundary Conditions On ∂ωmentioning

confidence: 74%

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A Stable Added-Mass Partitioned (AMP) Algorithm for Elastic Solids and Incompressible Flow: Model Problem Analysis

Serino¹,

Banks²,

Henshaw³

et al. 2019

SIAM J. Sci. Comput.

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A stable added-mass partitioned (AMP) algorithm is developed for fluid-structure interaction (FSI) problems involving viscous incompressible flow and compressible elastic-solids. The AMP scheme remains stable and second-order accurate even when added-mass and added-damping effects are large. The fluid is updated with an implicit-explicit (IMEX) fractionalstep scheme whereby the velocity is advanced in one step, treating the viscous terms implicitly, and the pressure is computed in a second step. The AMP interface conditions for the fluid arise from the outgoing characteristic variables in the solid and are partitioned into a Robin (mixed) interface condition for the pressure, and interface conditions for the velocity. The latter conditions include an impedance-weighted average between fluid and solid velocities using a fluid impedance of a special form. A similar impedance-weighted average is used to define interface values for the solid. The fluid impedance is defined using material and discretization parameters and follows from a careful analysis of the discretization of the governing equations and coupling conditions near the interface. A normal mode analysis is performed to show that the AMP scheme is stable for a few carefully-selected model problems. Two extensions of the analysis in [1] are considered, including a first-order accurate discretization of a viscous model problem and a second-order accurate discretization of an inviscid model problem. The AMP algorithm is shown to be stable for any ratio of solid and fluid densities, including when added-mass effects are large. On the other hand, the traditional algorithm involving a Dirichlet-Neumann coupling is shown to be unconditionally unstable as added-mass effects become large with grid refinement. The algorithm is verified for accuracy and stability for set of new exact benchmark solutions. These new solutions are elastic piston problems, where finite interface deformations are permitted. The AMP scheme is found to be stable and second-order accurate even for very difficult cases of very light solids.

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Section: Velocity Boundary Conditions On ∂ωmentioning

confidence: 74%

“…The stability and accuracy of the AMP algorithm is verified numerically for new exact solutions. In our companion paper [15], the AMP algorithm is implemented using deforming composite grids for curvilinear geometries.…”

mentioning

confidence: 99%

A Stable Added-Mass Partitioned (AMP) Algorithm for Elastic Solids and Incompressible Flow: Model Problem Analysis

Serino¹,

Banks²,

Henshaw³

et al. 2019

SIAM J. Sci. Comput.

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“…represent the numerical approximation of the exact solution of (24) at discrete points x i on the Cartesian grid Ω h given in (14) and at a fixed time t P r0, T s. Our principal focus is on discretizations of (24) to fourth and sixth-order accuracy, although we also consider second-order accurate approximations as a baseline. A second-order accurate discretization of (24) employs standard centered differences given by…”

Section: Semi-discrete Approximationsmentioning

confidence: 99%

“…CBCs are also useful for problems involving material interfaces, such as conjugate heat transfer [10] and electromagnetics [5,7]. In recent work, we have developed Added-Mass Partitioned (AMP) schemes for a wide range of fluid-structure interaction (FSI) problems, including schemes for incompressible flows coupled to rigid bodies [11][12][13] and elastic solids [14,15]. These strongly-partitioned schemes incorporate AMP interface conditions derived using CBCs and the physical matching conditions at fluid-solid interfaces in order to overcome added-mass instabilities that can occur for the case of light bodies [16,17].…”

Section: Introductionmentioning

confidence: 99%

Local Compatibility Boundary Conditions for High-Order Accurate Finite-Difference Approximations of PDEs

Al¹,

Banks²,

Henshaw³

et al. 2021

Preprint

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We describe a new approach to derive numerical approximations of boundary conditions for highorder accurate finite-difference approximations. The approach, called the Local Compatibility Boundary Condition (LCBC) method, uses boundary conditions and compatibility boundary conditions derived from the governing equations, as well as interior and boundary grid values, to construct a local polynomial, whose degree matches the order of accuracy of the interior scheme, centered at each boundary point. The local polynomial is then used to derive a discrete formula for each ghost point in terms of the data. This approach leads to centered approximations that are generally more accurate and stable than one-sided approximations. Moreover, the stencil approximations are local since they do not couple to neighboring ghost-point values which can occur with traditional compatibility conditions. The local polynomial is derived using continuous operators and derivatives which enables the automatic construction of stencil approximations at different orders of accuracy. The LCBC method is developed here for problems governed by second-order partial differential equations, and it is verified for a wide range of sample problems, both time-dependent and time-independent, in two space dimensions and for schemes up to sixth-order accuracy.

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“…The AMP condition, which is a non-standard Robin-type boundary condition involving the fluid stress tensor, requires no adjustable parameters and in principle is applicable at the discrete level to couple the fluid and structure solvers of any accuracy and of any approximation methods (finite difference, finite element, finite volume, spectral element methods, etc). Within the finite-difference framework, the AMP algorithms have been developed and implemented to solve FSI problems involving the interaction of incompressible flows with a wide range of structures, such as elastic beams/shells [25,26], bulk solids [27][28][29] and rigid bodies [30][31][32]. It has been shown in these works that the AMP schemes are second-order accurate and stable without sub-time-step iterations, even for very light structures when added-mass effects are strong.…”

Section: Introductionmentioning

confidence: 99%

A split-step finite-element method for incompressible Navier-Stokes equations with high-order accuracy up-to the boundary

2020

Journal of Computational Physics

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An efficient and accurate finite-element algorithm is described for the numerical solution of the incompressible Navier-Stokes (INS) equations. The new algorithm that solves the INS equations in a velocity-pressure reformulation is based on a split-step scheme in conjunction with the standard finite-element method. The split-step scheme employed for the temporal discretization of our algorithm completely separates the pressure updates from the solution of velocity variables. When the pressure equation is formed explicitly, the algorithm avoids solving a saddle-point problem; therefore, our algorithm has more flexibility in choosing finite-element spaces. In contrast, popular mixed finite-element methods that solve the INS equations in the primitive variables (or velocity-divergence formulation) lead to discrete saddle-point problems whose solution depends on the choice of finite-element spaces for velocity and pressure that is subject to the wellknown Ladyzenskaja-Babuška-Brezzi (LBB) (or inf-sup) condition. For efficiency and robustness, Lagrange (piecewise-polynomial) finite elements of equal order for both velocity and pressure are used. Accurate numerical boundary condition for the pressure equation is also investigated. Motivated by a post-processing technique that calculates derivatives of a finite element solution with super-convergent error estimates, an alternative numerical boundary condition is proposed for the pressure equation at the discrete level. The new numerical pressure boundary condition that can be regarded as a better implementation of the compatibility boundary condition improves the boundary-layer errors of the pressure solution. A normal-mode analysis is performed using a simplified model problem on a uniform mesh to demonstrate the numerical properties of our methods. Convergence studies using P 1 elements support the analytical results and demonstrates that our algorithm with the new numerical boundary condition achieves the optimal second-order accuracy for both velocity and pressure up-to the boundary. Benchmark problems are also computed and carefully compared with existing studies. Finally, as an example to illustrate that our approach can be easily adapted for higher-order finite elements, we solve the classical flow-past-a-cylinder problem using P n finite elements with n ≥ 1.

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A stable added-mass partitioned (AMP) algorithm for elastic solids and incompressible flow

Cited by 16 publications

References 46 publications

A Stable Added-Mass Partitioned (AMP) Algorithm for Elastic Solids and Incompressible Flow: Model Problem Analysis

A Stable Added-Mass Partitioned (AMP) Algorithm for Elastic Solids and Incompressible Flow: Model Problem Analysis

Local Compatibility Boundary Conditions for High-Order Accurate Finite-Difference Approximations of PDEs

A split-step finite-element method for incompressible Navier-Stokes equations with high-order accuracy up-to the boundary

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