Traditional explicit partitioned schemes exchange boundary conditions between subdomains and can be related to iterative solution methods for the coupled problem. As a result, these schemes may require multiple subdomain solves, acceleration techniques, or optimized transmission conditions to achieve sufficient accuracy and/or stability. We present a new synchronous partitioned method derived from a well-posed mixed finite element formulation of the coupled problem. We transform the resulting Differential Algebraic Equation (DAE) to a Hessenberg index-1 form in which the algebraic equation defines the Lagrange multiplier as an implicit function of the states. Using this fact we eliminate the multiplier and reduce the DAE to a system of explicit ODEs for the states. Explicit time integration both discretizes this system in time and decouples its equations. As a result, the temporal accuracy and stability of our formulation are governed solely by the accuracy and stability of the explicit scheme employed and are not subject to additional stability considerations as in traditional partitioned schemes. We establish sufficient conditions for the formulation to be well-posed and prove that classical mortar finite elements on the interface are a stable choice for the Lagrange multiplier. We show that in this case the condition number of the Schur complement involved in the elimination of the multiplier is bounded by a constant. The paper concludes with numerical examples illustrating the approach for two different interface problems.
Independent meshing of subdomains separated by an interface can lead to spatially non-coincident discrete interfaces. We present an optimization-based coupling method for such problems, which does not require a common mesh refinement of the interface, has optimal H1 convergence rates, and passes a patch test. The method minimizes the mismatch of the state and normal stress extensions on discrete interfaces subject to the subdomain equations, while interface “fluxes” provide virtual Neumann controls.
We consider the problem of finding a mesh such that 1) it is the closest, with respect to a suitable metric, to a given source mesh having the same connectivity, and 2) the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh. This volume correction problem arises in important simulation contexts, such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi-Lagrangian transport schemes. In this paper we formulate volume correction as a constrained optimization problem in which the distance to the source mesh defines an optimization objective, while the prescribed cell volumes, mesh validity and/or cell convexity specify the constraints. We solve this problem numerically using a sequential quadratic programming (SQP) method whose performance scales with the mesh size. To achieve scalable performance we develop a specialized multigrid-based preconditioner for optimality systems that arise in the application of the SQP method to the volume correction problem. Numerical examples illustrate the importance of volume correction, and showcase the accuracy, robustness and scalability of our approach.
We present a new optimization-based, conservative, and quasi-monotone method for passive tracer transport. The scheme combines high-order spectral element discretization in space with semi-Lagrangian time stepping. Solution of a singly linearly constrained quadratic program with simple bounds enforces conservation and physically motivated solution bounds. The scheme can handle efficiently a large number of passive tracers because the semi-Lagrangian time stepping only needs to evolve the grid points where the primitive variables are stored and allows for larger time steps than a conventional explicit spectral element method. Numerical examples show that the use of optimization to enforce physical properties does not affect significantly the spectral accuracy for smooth solutions. Performance studies reveal the benefits of high-order approximations, including for discontinuous solutions.
This study began with a challenge from program area managers at Sandia National Laboratories to technical staff in the energy, climate, and infrastructure security areas: apply a systems-level perspective to existing science and technology program areas in order to determine technology gaps, identify new technical capabilities at Sandia that could be applied to these areas, and identify opportunities for innovation. The Arctic was selected as one of these areas for systems level analyses, and this report documents the results. In this study, an emphasis was placed on the arctic atmosphere since Sandia has been active in atmospheric research in the Arctic since 1997. This study begins with a discussion of the challenges and benefits of analyzing the Arctic as a system. It goes on to discuss current and future needs of the defense, scientific, energy, and intelligence communities for more comprehensive data products related to the Arctic; assess the current state of atmospheric measurement resources available for the Arctic; and explain how the capabilities at Sandia National Laboratories can be used to address the identified technological, data, and modeling needs of the defense, scientific, energy, and intelligence communities for Arctic support.
