Figure 1: Two ships in stormy seas near Longfellow island. We refine the domain near the ships by placing grids in their object spaces to add detail and allow them to propel themselves using their two-way solid-fluid coupled propellers. AbstractWe introduce a new method for large scale water simulation using Chimera grid embedding, which discretizes space with overlapping Cartesian grids that translate and rotate in order to decompose the domain into different regions of interest with varying spatial resolutions. Grids can track both fluid features and solid objects, allowing for dynamic spatial adaptivity without remeshing or repartitioning the domain. We solve the inviscid incompressible NavierStokes equations with an arbitrary-Lagrangian-Eulerian style semiLagrangian advection scheme and a monolithic SPD Poisson solver. We modify the particle level set method in order to adapt it to Chimera grids including particle treatment across grid boundaries with disparate cell sizes, and strategies to deal with locality in the implementation of the level set and fast marching algorithms. We use a local Voronoi mesh construction to solve for pressure and address a number of issues that arise with the treatment of the velocity near the interface. The resulting method is highly scalable on distributed parallel architectures with minimal communication costs.

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Two-way solid fluid coupling techniques typically calculate fluid pressure forces that in turn drive the solid motion. However, when solids are in close proximity (e.g. touching or in contact), the fluid in the thin gap region between the solids is difficult to resolve with a background fluid grid. Although one might attempt to address this difficulty using an adaptive, body-fitted, or ALE fluid grid, the size of the fluid cells can shrink to zero as the bodies collide. The inability to apply pressure forces in a thin lubricated gap tends to make the solids stick together, since collision forces stop interpenetration but vanish when the solids are separating leaving the fluid pressure forces on the surface of the solid unbalanced in regards to the gap region. We address this problem by adding pressure degrees of freedom onto surfaces of rigid bodies, and subsequently using the resulting pressure forces to provide solid fluid coupling in the thin gap region. These pressure degrees of freedom readily resolve the tangential flow along the solid surface inside the gap and are two-way coupled to the pressure degrees of freedom on the grid allowing the fluid to freely flow into and out of the gap region. The two-way coupled system is formulated as a symmetric positive-definite matrix which is solved using the preconditioned conjugate gradient method. Additionally, we provide a mechanism for advecting tangential velocities on solid surfaces in the gap region by extending semi-Lagrangian advection onto a curved surface mesh where a codimension-one velocity field tangential to the surface is defined. We demonstrate the convergence of our method on a number of examples, such as underwater rigid body separation and collision in both two and three spatial dimensions where typical methods do not converge. Finally, we demonstrate that our method not only works for the aforementioned "wet" contact, but also works in conjunction with "dry" contact where there is no fluid in the gap between the solids.

We present a novel method for simulating compressible flow on a multitude of Cartesian grids that can rotate and translate. Following previous work, we split the time integration into an explicit step for advection followed by an implicit solve for the pressure. A second order accurate flux based scheme is devised to handle advection on each moving Cartesian grid using an effective characteristic velocity that accounts for the grid motion. In order to avoid the stringent time step restriction imposed by very fine grids, we propose strategies that allow for a fluid velocity CFL number larger than 1. The stringent time step restriction related to the sound speed is alleviated by formulating an implicit linear system in order to find a pressure consistent with the equation of state. This implicit linear system crosses overlapping Cartesian grid boundaries by utilizing local Voronoi meshes to connect the various degrees of freedom obtaining a symmetric positive-definite system. Since a straightforward application of this technique contains an inherent central differencing which can result in spurious oscillations, we introduce a new high order diffusion term similar in spirit to ENO-LLF but solved for implicitly in order to avoid any associated time step restrictions. The method is conservative on each grid, as well as globally conservative on the background grid that contains all other grids. Moreover, a conservative interpolation operator is devised for conservatively remapping values in order to keep them consistent across different overlapping grids. Additionally, the method is extended to handle two-way solid fluid coupling in a monolithic fashion including cases (in the appendix) where solids in close proximity do not properly allow for grid based degrees of freedom in between them.

In large-scale optimization problems, distributed asynchronous stochastic gradient descent (DASGD) is a commonly used algorithm. In most applications, there are often a large number of computing nodes asynchronously computing gradient information. As such, the gradient information received at a given iteration is often stale. In the presence of such delays, which can be unbounded, the convergence of DASGD is uncertain. The contribution of this paper is twofold. First, we propose a delay-adaptive variant of DASGD where we adjust each iteration's step-size based on the size of the delay, and prove asymptotic convergence of the algorithm on variationally coherent stochastic problems, a class of functions which properly includes convex, quasi-convex and star-convex functions. Second, we extend the convergence results of standard DASGD, used usually for problems with bounded domains, to problems with unbounded domains. In this way, we extend the frontier of theoretical guarantees for distributed asynchronous optimization, and provide new insights for practitioners working on large-scale optimization problems.