Boris Valkov Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 e-mail: bvalkov@alum.mit.edu Chris H. Rycroft School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; Department of Mathematics, Lawrence Berkeley Laboratory, Berkeley, CA 94720 e-mail: chr@seas.harvard.edu Ken Kamrin 1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 e-mail: kkamrin@mit.edu Eulerian Method for Multiphase Interactions of Soft Solid Bodies in Fluids We introduce an Eulerian approach for problems involving one or more soft solids immersed in a fluid, which permits mechanical interactions between all phases. The refer- ence map variable is exploited to simulate finite-deformation constitutive relations in the solid(s) on the same fixed grid as the fluid phase, which greatly simplifies the coupling between phases. Our coupling procedure, a key contribution in the current work, is shown to be computationally faster and more stable than an earlier approach and admits the ability to simulate both fluid–solid and solid–solid interaction between submerged bodies. The interface treatment is demonstrated with multiple examples involving a weakly compressible Navier–Stokes fluid interacting with a neo-Hookean solid, and we verify the method’s convergence. The solid contact method, which exploits distance- measures already existing on the grid, is demonstrated with two examples. A new, gen- eral routine for cross-interface extrapolation is introduced and used as part of the new interfacial treatment. [DOI: 10.1115/1.4029765] 1 Introduction The challenges of simulating fluid–structure interaction (FSI) have been approached from many directions. One set of chal- lenges stem from domain discretization: fluid problems on their own are amenable to solution on an Eulerian computational domain [1–5] and solid deformation is natural to compute within a Lagrangian framework [6–9]. To couple solid and fluid phases in one setting, several approaches have been proposed to mitigate this separation in methodology. One approach is to treat the solid with a standard Lagrangian finite-element framework and to apply an Arbitrary-Lagrange Eulerian method in the fluid [10–12], which remeshes the fluid domain in order to remedy issues of excessive mesh deformation. Other approaches include the family of immersed methods [10,11,13], which keep an ambient station- ary Eulerian grid throughout, on which fluid flow is solved, as well as a moving collection of interacting material points repre- senting the solid structure. Here, discretized delta functions are used to pass information between the grid and the nodes. Some advantages of a fully Eulerian method—fluid and solid both computed on an Eulerian grid—can be directly seen. Since all phases are solved by sweeping through a single fixed mesh, there are computation time advantages. Topological changes are easy to manage on a fixed grid using level sets to track interfaces [4,14]. Furthermore, multiscale and multiphysics coupling can also have advantages when done on an Eulerian grid. One specific example, which will be discussed in more detail later, is the case of multiple solids making contact immersed in a fluid. Finding contact between two solid phases on an Eulerian grid can be achieved using gridwise distance functions or simply identifying grid points which become occupied by multiple solid phases during a trial step. To achieve these goals, one must pose an Eulerian scheme ca- pable of solving finite-deformation solid problems. The recently proposed reference map technique (RMT) is such an Eulerian framework, based on tracking the reference map field [15,16]. Other approaches for solid deformation on a fixed Eulerian grid include hypoelastic implementations [17,18], which may succeed for small elastic strains but lack a thermodynamically consistent form as needed for large deformations, and methods that directly evolve the deformation gradient tensor field as the primitive kine- matic grid variable [19,20,21]. In a previous paper [16], the RMT demonstrated the capability of accurately solving hyperelastic solid deformation problems on a fixed mesh—including shock propagation problems and problems with varied boundary and ini- tial conditions—up to second-order accuracy in space and time. It also provided the first demonstrations of using the method to solve fully coupled problems of FSI. There, the FSI method hinged on a sharp-interface representation, extending on that of the Ghost Fluid Method for fluid–fluid interaction [22]. Sharp methods make a distinct separation between each phase down to the subgrid level. In the current work, our efforts exploit a blurred interface,a simpler and computationally faster implementation, involving fewer numerical extrapolations. A blurred interface method uses a thin transition zone where one phase converts into the other. As the grid size decreases, the corresponding transition zone decreases, and results approach that of a sharp interface method. Here, we show that the blurred interface approach has advantages over the sharp, most important of which is the ability to represent key mechanical behaviors such as submerged solid–solid contact. To satisfy subgrid jump conditions, sharp interface methods require a large number of gridwise extrapolations of kinematic and stress fields across the interface. These produce the “ghost values” for each phase, which represent an extension of each field into the region occupied by the other phase(s). The validity of a sharp interface method is limited by the quality of continued and progressive extrapolation. When used in the FSI method of the previous work [16], accrued extrapolation error can have a desta- bilizing effect—shots of pressure along the interface may errone- ously appear when the interface crosses through grid cell boundaries, in response to the sampling of extrapolated values when new points enter the solid domain. Adding significant solid dissipation or surface tension can penalize these artifacts, but this can alter the physicality of the simulation. We have found that 1 Corresponding author. Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 6, 2015; final manuscript received February 5, 2015; published online March 3, 2015. Editor: Yonggang Huang. The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes. Journal of Applied Mechanics APRIL 2015, Vol. 82 / 041011-1 Copyright V C 2015 by ASME Downloaded from http://asmedigitalcollection.asme.org/appliedmechanics/article-pdf/82/4/041011/6081491/jam_082_04_041011.pdf by guest on 27 November 2021