Publications

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The Next Generation Global Atmosphere Model LDRD project developed a suite of atmosphere models: a shallow water model, an x - z hydrostatic model, and a 3D hydrostatic model, by using Albany, a finite element code. Albany provides access to a large suite of leading-edge Sandia high- performance computing technologies enabled by Trilinos, Dakota, and Sierra. The next-generation capabilities most relevant to a global atmosphere model are performance portability and embedded uncertainty quantification (UQ). Performance portability is the capability for a single code base to run efficiently on diverse set of advanced computing architectures, such as multi-core threading or GPUs. Embedded UQ refers to simulation algorithms that have been modified to aid in the quantifying of uncertainties. In our case, this means running multiple samples for an ensemble concurrently, and reaping certain performance benefits. We demonstrate the effectiveness of these approaches here as a prelude to introducing them into ACME.

This report summarizes FY16 progress towards enabling uncertainty quantification for compress- ible cavity simulations using model order reduction (MOR). The targeted application is the quan- tification of the captive-carry environment for the design and qualification of nuclear weapons systems. To accurately simulate this scenario, Large Eddy Simulations (LES) require very fine meshes and long run times, which lead to week -long runs even on parallel state-of-the-art super- computers. MOR can reduce substantially the CPU-time requirement for these simulations. We describe two approaches for model order reduction for nonlinear systems, which can yield sig- nificant speed-ups when combined with hyper-reduction: the Proper Orthogonal Decomposition (POD)/Galerkin approach and the POD/Least-Squares Petrov Galerkin (LSPG) approach. The im- plementation of these methods within the in-house compressible flow solver SPARC is discussed. Next, a method for stabilizing and enhancing low-dimensional reduced bases that was developed as a part of this project is detailed. This approach is based on a premise termed "minimal sub- space rotation", and has the advantage of yielding ROMs that are more stable and accurate for long-time compressible cavity simulations. Numerical results for some laminar cavity problems aimed at gauging the viability of the proposed model reduction methodologies are presented and discussed.

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This report describes work performed from June 2012 through May 2014 as a part of a Sandia Early Career Laboratory Directed Research and Development (LDRD) project led by the first author. The objective of the project is to investigate methods for building stable and efficient proper orthogonal decomposition (POD)/Galerkin reduced order models (ROMs): models derived from a sequence of high-fidelity simulations but having a much lower computational cost. Since they are, by construction, small and fast, ROMs can enable real-time simulations of complex systems for onthe- spot analysis, control and decision-making in the presence of uncertainty. Of particular interest to Sandia is the use of ROMs for the quantification of the compressible captive-carry environment, simulated for the design and qualification of nuclear weapons systems. It is an unfortunate reality that many ROM techniques are computationally intractable or lack an a priori stability guarantee for compressible flows. For this reason, this LDRD project focuses on the development of techniques for building provably stable projection-based ROMs. Model reduction approaches based on continuous as well as discrete projection are considered. In the first part of this report, an approach for building energy-stable Galerkin ROMs for linear hyperbolic or incompletely parabolic systems of partial differential equations (PDEs) using continuous projection is developed. The key idea is to apply a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables. It is shown that, for many PDE systems including the linearized compressible Euler and linearized compressible Navier-Stokes equations, the desired transformation is induced by a special inner product, termed the “symmetry inner product”. Attention is then turned to nonlinear conservation laws. A new transformation and corresponding energy-based inner product for the full nonlinear compressible Navier-Stokes equations is derived, and it is demonstrated that if a Galerkin ROM is constructed in this inner product, the ROM system energy will be bounded in a way that is consistent with the behavior of the exact solution to these PDEs, i.e., the ROM will be energy-stable. The viability of the linear as well as nonlinear continuous projection model reduction approaches developed as a part of this project is evaluated on several test cases, including the cavity configuration of interest in the targeted application area. In the second part of this report, some POD/Galerkin approaches for building stable ROMs using discrete projection are explored. It is shown that, for generic linear time-invariant (LTI) systems, a discrete counterpart of the continuous symmetry inner product is a weighted L2 inner product obtained by solving a Lyapunov equation. This inner product was first proposed by Rowley et al., and is termed herein the “Lyapunov inner product“. Comparisons between the symmetry inner product and the Lyapunov inner product are made, and the performance of ROMs constructed using these inner products is evaluated on several benchmark test cases. Also in the second part of this report, a new ROM stabilization approach, termed “ROM stabilization via optimization-based eigenvalue reassignment“, is developed for generic LTI systems. At the heart of this method is a constrained nonlinear least-squares optimization problem that is formulated and solved numerically to ensure accuracy of the stabilized ROM. Numerical studies reveal that the optimization problem is computationally inexpensive to solve, and that the new stabilization approach delivers ROMs that are stable as well as accurate. Summaries of “lessons learned“ and perspectives for future work motivated by this LDRD project are provided at the end of each of the two main chapters.

The construction of stable reduced order models using Galerkin projection for the Euler or Navier-Stokes equations requires a suitable choice for the inner product. The standard L2 inner product is expected to produce unstable ROMs. For the non-linear Navier-Stokes equations this means the use of an energy inner product. In this report, Galerkin projection for the non-linear Navier-Stokes equations using the L2 inner product is implemented as a first step toward constructing stable ROMs for this set of physics.