An enhanced FIVER method for multi-material flow problems with second-order convergence rate
Introduction
FIVER (finite volume method with exact two-material Riemann problems) is a finite volume (FV) method for the solution of multi-material, fluid and fluid–structure interaction (FSI) problems. Its underlying semi-discretization procedure is based on the solution of local, one-dimensional, two-material Riemann problems. It was originally developed in [1], in the context of explicit time-discretizations, for the solution of compressible, inviscid, two-phase flow problems characterized by simple equations of state (EOS) but large contact discontinuities (density jumps). However, it is equally applicable to the solution of incompressible, multi-material, fluid or FSI problems. In [2], the concept of a one-dimensional fluid–fluid Riemann problem was extended to that of a local fluid–structure Riemann problem, and FIVER was transformed into an Eulerian embedded boundary method (EBM) — or immersed boundary method — for the solution of highly nonlinear fluid and FSI problems. These include those FSI problems which cannot be handled by Arbitrary Lagrangian Eulerian (ALE) methods [3], [4], [5] because of large structural motions, finite deformations, and/or topological changes that challenge the performance of mesh motion schemes [6], [7]. Unlike most other EBMs which operate exclusively on Cartesian grids (for example, see [8], [9], [10] and [11]), FIVER can operate on arbitrary grids. This is noteworthy because even in the pure Eulerian setting, the ability of an EBM to perform on arbitrary, unstructured grids is still particularly advantageous for complex geometries and viscous flows.
For multi-material problems with complex fluid EOS, the analytical solution of a local, one-dimensional, two-material Riemann problem may be either impossible to obtain explicitly, or computationally intensive to evaluate numerically. To address this issue, a computationally efficient sparse grid tabulation technique was developed in [12] for accelerating the numerical solution of arbitrary fluid–fluid and fluid–structure Riemann problems, and thereby enabling the generalization of FIVER to multi-material problems with complex fluid EOS. In [13], the generalized FIVER method was validated for the solution of failure-induced FSI problems. Specifically, it was applied to the simulation of two experiments on the dynamic underwater implosion of cylindrical shells. In both cases, it reproduced with high-fidelity the large deformations of the collapsing structure and the compression waves emanating from it.
FIVER was also extended to implicit time-discretizations in [14], and to viscous, multi-material fluid and FSI problems in [15].
For turbulent viscous flows, Eulerian EBMs typically suffer from the fact that they do not track the boundary layers around dynamic rigid or flexible bodies. Consequently, the application of these methods to such problems requires either high mesh resolutions in a large part of the computational fluid domain, or adaptive mesh refinement. Unfortunately, the first option is computationally inefficient, and the second one is labor intensive to implement. To address these issues, an ALE variant of FIVER which maintains all moving boundary layers resolved during turbulent FSI computations was presented in [16].
Most recently, a generic, comprehensive, and yet effective approach for representing a fractured fluid–structure interface was developed in [17] and incorporated in FIVER. Specifically, this approach enables the coupling of FIVER with many finite element based fracture methods for the solution of multi-material FSI problems with dynamic fracture. These methods include, among others, the interelement fracture [18] and remeshing [19] techniques, the extended finite element method (XFEM) [20], and the element deletion method [21]. Equipped with this fractured fluid–structure interface representation, FIVER was further validated for highly nonlinear FSI problems with crack propagation, flow seepage through narrow cracks, and structural fragmentation [17].
Because of the large scope of applications outlined above, FIVER is today more than a computational method. It is a genuine computational framework for multi-material, inviscid or viscous, laminar or turbulent, fluid and FSI problems. This framework is driven by two simple ideas:
- •
Keeping the underlying semi-discretization scheme as close as possible to a Godunov-type scheme in order to facilitate its implementation in a large body of existing flow solvers.
- •
Solving at each material interface local, one-dimensional, two-material Riemann problems in order to avoid traversing explicitly this interface during the semi-discretization process and therefore achieve robustness with respect to strong contact discontinuities.
In Section 2, the governing equations of interest, the original FIVER framework and its underlying FV semi-discretization, the chosen interface capturing technique, and the time-discretization of the resulting multi-material semi-discrete fluid or FSI problem are overviewed. In Section 3, the undesirable effect of the geometric error associated with FIVER's surrogate material interface is highlighted using a mathematical analysis of a simple one-dimensional FSI problem. Following some of the ideas presented in [23], a simple approach based on extrapolation and interpolation is presented in Section 4 for eliminating this detrimental effect. In Section 5, an extrapolation component of this approach is equipped with a limiter in order to achieve nonlinear stability for non-smooth problems. In this section, it is also shown that in the one-dimensional inviscid setting, the resulting FIVER method is total variation bounded (TVB) and therefore nonlinearly stable. In Section 6, the numerical properties of the enhanced FIVER framework presented in this paper are illustrated and its performance is assessed for smooth and non-smooth model multi-material problems. In particular, it is shown numerically that when the new FIVER framework is equipped with a second-order FV semi-discretization and applied to the solution of smooth multi-material problems, it delivers a second-order global convergence rate. The robustness of this framework and its potential for complex multi-material problems are also demonstrated in this section with the simulation of the collapse of an air bubble submerged in water and the comparison of the obtained numerical results with the corresponding experimental data [24]. Finally, Section 7 concludes this paper.
Section snippets
Background
As stated in the introduction, FIVER is a FV method for the solution of multi-material problems based on the solution of local, one-dimensional, two-material Riemann problems. When all materials are fluid materials, FIVER is essentially a Godunov-type method for multi-phase flow computations. When at least one material is a solid material, it becomes a genuine EBM for CFD. Here, the basic FIVER framework is reviewed in order to set the stage for the main contributions of this paper which are:
- •
Detrimental effect of the surrogate material interface
Using as a backdrop a simple one-dimensional FSI problem, it is shown here for the first time that the surrogate material interface (2.9) that simplifies the FIVER computations also degrades their accuracy. To this effect, attention is focused on the one-dimensional version of Equations (2.2) whose spatial discretization on a uniform grid of cell size h can be written as where here and throughout the remainder of this paper, the dot symbol designates a time derivative
Enhancements of spatial accuracy for smooth problems
Here, two aspects of FIVER are revisited in order to eliminate the detrimental effect of the surrogate material interface highlighted in the previous section and enhance the overall spatial accuracy of this computational framework:
- •
The solution of a two-material Riemann problem is performed at the captured location of the true material interface , in order to avoid introducing in the FIVER solution process a local geometric error of the order .
- •
The constant extrapolation procedures
Nonlinear stability
It is well-known that, in principle, an extrapolation step such as (4.7) or (2.3) should be equipped with a limiter in order to avoid generating, for non-smooth problems, spurious oscillations in the numerical solution. This issue has been addressed in great details in the thesis work [42] underlying this paper. In particular, it was concluded in [42] that to achieve nonlinear numerical stability, the enhanced version of FIVER proposed here requires a limiter only for the extrapolation step
Preliminaries
Both the original FIVER method and its enhanced counterpart presented in this paper were implemented in the massively parallel flow solver AERO-F [43], [44]. In this CFD code, the semi-discretization of the convective fluxes can be performed using a second-order accurate upwind scheme based on any of the Roe [22], HLLE [45], or HLLC [46] approximate Riemann solvers. Time-discretization can be performed using a first- or second-order implicit time-integrator, or a first-, second-, or
Summary and conclusions
In this paper, attention focuses on enhancing the convergence rate of the original finite volume (FV) method with exact two-material Riemann problems (FIVER). This Eulerian computational method for the solution of multi-material flow problems is robust with respect to large density jumps at fluid–fluid interfaces. It is also equally robust with respect to large structural motions, deformations, and even topological changes at fluid–structure interfaces. It is particularly simple to implement
References (48)
- et al.
A higher-order generalized ghost fluid method for the poor for the three-dimensional two-phase flow computation of underwater implosions
J. Comput. Phys.
(2008) - et al.
Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations
Comput. Methods Appl. Mech. Eng.
(1996) - et al.
Torsional springs for two-dimensional dynamic unstructured fluid meshes
Comput. Methods Appl. Mech. Eng.
(1998) - et al.
A three-dimensional torsional spring analogy method for unstructured dynamic meshes
Comput. Struct.
(2002) - et al.
A Cartesian grid embedded boundary method for Poisson's equation on irregular domains
J. Comput. Phys.
(1998) - et al.
Large-scale fluid–structure interaction simulation of viscoplastic and fracturing thin-shells subjected to shocks and detonations
Comput. Struct.
(2007) - et al.
FIVER: a finite volume method based on exact two-phase Riemann problems and sparse grids for multi-material flows with large density jumps
J. Comput. Phys.
(2012) - et al.
Dynamic implosion of underwater cylindrical shells: experiments and computations
Int. J. Solids Struct.
(2013) - et al.
A second-order time-accurate implicit finite volume method with exact two-phase Riemann problems for compressible multi-phase fluid and fluid–structure problems
J. Comput. Phys.
(2014) - et al.
An ALE formulation of embedded boundary methods for tracking boundary layers in turbulent fluid–structure interaction problems
J. Comput. Phys.
(2014)
Numerical simulations of fast crack growth in brittle solids
J. Mech. Phys. Solids
Approximate Riemann solver – parameter vectors and difference schemes
J. Comput. Phys.
A systematic approach for constructing higher-order immersed boundary and ghost fluid methods for fluid and fluid–structure interaction problems
J. Comput. Phys.
Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method
J. Comput. Phys.
Partitioned analysis of coupled mechanical systems
Comput. Methods Appl. Mech. Eng.
Partitioned procedures for the transient solution of coupled aeroelastic problems – part I: model problem, theory, and two-dimensional application
Comput. Methods Appl. Mech. Eng.
Partitioned procedures for the transient solution of coupled aeroelastic problems – part II: energy transfer analysis and three-dimensional applications
Comput. Methods Appl. Mech. Eng.
Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity
Comput. Methods Appl. Mech. Eng.
Coupling an Eulerian fluid calculation to a Lagrangian solid calculation with the ghost fluid method
J. Comput. Phys.
Application of a three-field nonlinear fluid–structure formulation to the prediction of the aeroelastic parameters of an F-16 fighter
Comput. Fluids
TOP/DOMDEC, a software tool for mesh partitioning and parallel processing
J. Comput. Syst. Eng.
An analysis of a new stable partitioned algorithm for FSI problems. Part II: incompressible flow and structural shells
J. Comput. Phys.
Algorithms for interface treatment and load computation in embedded boundary methods for fluid and fluid–structure interaction problems
Int. J. Numer. Methods Fluids
Mixed explicit/implicit time integration of coupled aeroelastic problems: three-field formulation, geometric conservation and distributed solution
Int. J. Numer. Methods Fluids
Cited by (42)
Efficient solution of bimaterial Riemann problems for compressible multi-material flow simulations
2023, Journal of Computational PhysicsFluid–solid coupled simulation of hypervelocity impact and plasma formation
2023, International Journal of Impact EngineeringAerodynamic optimization with large shape and topology changes using a differentiable embedded boundary method
2023, Journal of Computational PhysicsPlasma formation in ambient fluid from hypervelocity impacts
2023, Extreme Mechanics LettersSimulating laser-fluid coupling and laser-induced cavitation using embedded boundary and level set methods
2023, Journal of Computational PhysicsCitation Excerpt :The localized domains of Eqs. (13) and (17) are both updated at every time step. The level set equation (13) can be solved using finite difference and finite volume methods on structured and unstructured meshes (e.g., [29,30,33,48,49]). In the numerical tests presented in this paper, the equation is solved using a third-order upwind finite difference method with a second-order explicit Runge-Kutta time-integrator.
Numerical simulation data of bubble-structure interactions in near-field underwater explosion
2022, Data in BriefCitation Excerpt :The structural dynamics is simulated using finite element method implemented in the AERO-S solver [3]. The two solvers are coupled using an embedded boundary method and the FInite Volume method with Exact two-material Riemann problems (FIVER) [4–6]. Fig. 1 (a) illustrates the problem investigated in this work.
- 1
Vivian Church Hoff Professor of Aircraft Structures.