A fourth-order finite volume embedded boundary (EB) method is presented for the unsteady Stokes equations. The algorithm represents complex geometries on a Cartesian grid using EB, employing a technique to mitigate the ``small cut-cell"" problem without mesh modifications, cell merging, or state redistribution. Spatial discretizations are based on a weighted least-squares technique that has been extended to fourth-order operators and boundary conditions, including an approximate projection to enforce the divergence-free constraint. Solutions are advanced in time using a fourth-order additive implicit-explicit Runge--Kutta method, with the viscous and source terms treated implicitly and explicitly, respectively. Formal accuracy of the method is demonstrated with several grid convergence studies, and results are shown for an application with a complex bioinspired material. The developed method achieves fourth-order accuracy and is stable despite the pervasive small cells arising from complex geometries.

SummaryThis paper presents a scalable object detection workflow for detecting objects, such as settlements, from remotely sensed (RS) imagery. We have successfully deployed this workflow on Titan supercomputer and utilized it for the task of mapping human settlement at a country scale. The performance of various stages in the workflow was analyzed before making it operational. The workflow implemented various strategies to address issues such as suboptimal resource utilization and long‐tail effects due to unbalanced image workload, data loss due to runtime failures, and maximum wall‐time constraints imposed by Titan's job scheduling policy. A mean shift clustering–based static load balancing strategy was implemented, which partitions the image load such that each partition contained similar‐sized images. Furthermore, a checkpoint‐restart strategy was added in the workflow as a fault‐tolerance mechanism to prevent the data losses due to unforeseen runtime failures. The performance of the above‐mentioned strategies was observed in various scenarios, such as node failure, exceeding wall time, and successful completion. Using this workflow, we have processed an RS data set that has a spatial resolution of 0.31 m and is comprised of 685 675 km2 of area of the Republic of Zambia in under six hours using 5426 nodes of the Titan supercomputer.

AMReX is a C++ software framework that supports the development of block-structured adaptive mesh refinement (AMR) algorithms for solving systems of partial differential equations (PDEs) with complex boundary conditions on current and emerging architectures.

In this paper, we present a fourth-order algorithm to solve Poisson's equation in two and three dimensions. We use a Cartesian grid, embedded boundary method to resolve complex boundaries. We use a weighted least squares algorithm to solve for our stencils. We use convergence tests to demonstrate accuracy and we show the eigenvalues of the operator to demonstrate stability. We compare accuracy and performance with an established second-order algorithm. We also discuss in depth strategies for retaining higher-order accuracy in the presence of nonsmooth geometries.

Supercomputing platforms are expected to have larger failure rates in the future because of scaling and power concerns. The memory and performance impact may vary with error types and failure modes. Therefore, localized recovery schemes will be important for scientific computations, including failure modes where application intervention is suitable for recovery. We present a resiliency methodology for applications using structured adaptive mesh refinement, where failure modes map to granularities within the application for detection and correction. This approach also enables parameterization of cost for differentiated recovery. The cost model is built with tuning parameters that can be used to customize the strategy for different failure rates in different computing environments. We also show that this approach can make recovery cost proportional to the failure rate.

High performance scientific software has many unique and challenging characteristics. These codes typically consist of many different stages of computation with different algorithms and components with diverse requirements. These heterogeneous algorithms, coupled with platform heterogeneity, create serious performance challenges. To retain performance, portability and maintainability of the software on heterogeneous platforms, more abstractions have to be integrated into the software design. Most of these abstractions are still in the research stage and scientific codes have barely started using them. However, it is urgent that we start considering the abstraction interplay in designing the next generation of software architecture. We propose a software architecture for PDE-based scientific codes that combines three abstractions in a code framework suitable for expected heterogeneity in platforms, while retaining separation of concerns, performance and portability of the software. We support our proposal with an example design for an adaptive mesh refinement based framework.

We present an adaptive, finite volume algorithm to solve the incompressible Navier-Stokes equations in complex geometries. The algorithm is based on the embedded boundary method, in which finite volume approximations are used to discretize the solution in cut cells that result from intersecting the irregular boundary with a structured Cartesian grid. This approach is conservative and reduces to a standard finite difference method in grid cells away from the boundary. We solve the incompressible flow equations using a predictor-corrector formulation. Hyperbolic advection terms are obtained by higher-order upwinding without the use of extrapolated data in covered cells. The small-cell stability problem associated with explicit embedded boundary methods for hyperbolic systems is avoided by the use of a volume-weighted scheme in the advection step and is consistent with construction of the right-hand side of the elliptic solvers. The Helmholtz equations resulting from viscous source terms are advanced in time by the Crank-Nicolson method, which reduces solver runtime compared to other second-order time integrators by a half. Incompressibility is enforced by a second-order approximate projection method that makes use of a new conservative cell-centered gradient in cut cells that is consistent with the volume-weighted scheme. The algorithm is also capable of block structured adaptive mesh refinement to increase spatial resolution dynamically in regions of interest. The resulting overall method is second-order accurate for sufficiently smooth problems. In addition, the algorithm is implemented in a high-performance computing framework and can perform structured-grid fluid dynamics calculations at unprecedented scale and resolution, up to 262,144 processor cores. We demonstrate robustness and performance of the algorithm by simulating incompressible flow for a wide range of Reynolds numbers in two and three dimensions: Stokes and low Reynolds number flows in both constructed and image data geometries (Re << 1 to Re = 1), flow past a cylinder (Re = 300), flow past a sphere (Re = 600) and turbulent flow in a contraction (Re = 6300).

We present an algorithm to produce the necessary geometric information for finite volume calculations in the context of Cartesian grids with embedded boundaries. Given an order of accuracy for the overall calculation, we show what accuracy is required for each of the geometric quantities and we demonstrate how to calculate the moments using the divergence theorem. We demonstrate that, for a known flux, these moments can be used to create a flux divergence of the expected order.

We present a hybrid geometric-algebraic multigrid approach for solving Poisson’s equation on domains with complex geometries. The discretization uses a novel fourth-order finite volume cut cell representation to discretize the Laplacian operator on a Cartesian mesh. This representation is based on a weighted least-squares fit to a cell-averaged discretization, which is used to provide a conservative and accurate framework for the multi-resolution discretization, despite the presence of cut cells. We use geometric multigrid coarsening with an algebraic multigrid bottom solver, so that the memory overhead of algebraic coarsening is avoided until the geometry becomes under-resolved. With tuning, the hybrid approach has the simplicity of geometric multigrid while still retaining the robustness of algebraic multigrid. We investigate at what coarse level the transition should occur, and how the order of accuracy of the prolongation operator affects multigrid convergence rates. We also present some converged solutions as examples of how the use of adaptivity and a cell connectivity graph can affect performance in cases with under-resolved geometries.

We present a method for generating higher-order finite volume discretizations for Poisson's equation on Cartesian cut cell grids in two and three dimensions. The discretization is in flux-divergence form, and stencils for the flux are computed by solving small weighted least-squares linear systems. Weights are the key in generating a stable discretization. We apply the method to solve Poisson's equation on a variety of geometries, and we demonstrate that the method can achieve second and fourth order accuracy in both truncation and solution error for these examples. We also show that the Laplacian operator has only stable eigenvalues for each of these examples.

Author(s): Devendran, Dharshi; Graves, Daniel T.; Johansen, Hans | Abstract: We present a hybrid geometric-algebraic multigrid approach for solving Poisson's equation on domains with complex geometries. The discretization uses a novel fourth-order finite volume cut cell representation to discretize the Laplacian operator on a Cartesian mesh. This representation is based on a weighted least-squares fit to a cell-averaged discretization, which is used to provide a conservative and accurate framework for the multi-resolution discretization, despite the presence of cut cells. We use geometric multigrid coarsening with an algebraic multigrid bottom solver, so that the memory overhead of algebraic coarsening is avoided until the geometry becomes under-resolved. With tuning, the hybrid approach has the simplicity of geometric multigrid while still retaining the robustness of algebraic multigrid. We investigate at what coarse level the transition should occur, and how the order of accuracy of the prolongation operator affects multigrid convergence rates. We also present some converged solutions as examples of how the use of adaptivity and a cell connectivity graph can affect performance

We present new prototype tools for optimizing building solar energy impacts in urban regions, to enable better real-time control and policy decisions for energy supply and demand response. The concept is demonstrated with a prototype that estimates the amount of direct sunlight available to building surfaces in complex urban landscapes, taking into consideration local weather predictions (via cloud cover simulation). We also calculate partial shadows from visual obstructions, due to their effect on the availability of solar energy and building energy usage. The prototype has the potential to make better day-ahead predictions that can help balance energy supply and demand during peak load hours. This can lead to better strategies for control of heating, air conditioning and alternatives (such as local energy storage in batteries or co-generation) to offset peak energy demand. However, in addition it can be used as a statistical optimization tool for informing local policy decisions related to solar energy incentives and demand response programs. We apply the approach to a prototype calculation on models of a hypothetical city and a section of downtown San Francisco. We briefly discuss optimization opportunities in response to the variability and uncertainty in solar energy for individual buildings in an urban landscape.

Over the last decade block-structured adaptive mesh refinement (SAMR) has found increasing use in large, publicly available codes and frameworks. SAMR frameworks have evolved along different paths. Some have stayed focused on specific domain areas, others have pursued a more general functionality, providing the building blocks for a larger variety of applications. In this survey paper we examine a representative set of SAMR packages and SAMR-based codes that have been in existence for half a decade or more, have a reasonably sized and active user base outside of their home institutions, and are publicly available. The set consists of a mix of SAMR packages and application codes that cover a broad range of scientific domains. We look at their high-level frameworks, their design trade-offs and their approach to dealing with the advent of radical changes in hardware architecture. The codes included in this survey are BoxLib, Cactus, Chombo, Enzo, FLASH, and Uintah.

We present an unsplit method for the time-dependent compressible Navier-Stokes equations in two and three dimensions.We use a conservative, second-order Godunov algorithm.We use a Cartesian grid, embedded boundary method to resolve complex boundaries.We solve for viscous and conductive terms with a second-order semiimplicit algorithm.We demonstrate second-order accuracy in solutions of smooth problems in smooth geometries and demonstrate robust behavior for strongly discontinuous initial conditions in complex geometries.

We present a second-order accurate projection method to solve the incompressible Navier-Stokes equations on irregular domains in two and three dimensions. We use a finite-volume discretization obtained from intersecting the irregular domain boundary with a Cartesian grid. We address the small-cell stability problem associated with such methods by hybridizing a conservative discretization of the advective terms with a stable, nonconservative discretization at irregular control volumes, and redistributing the difference to nearby cells. Our projection is based upon a finite-volume discretization of Poisson’s equation. We use a second-order, L∞-stable algorithm to advance in time. Block structured local refinement is applied in space. The resulting method is second-order accurate in L1 for smooth problems. We demonstrate the method on benchmark problems for flow past a cylinder in 2D and a sphere in 3D as well as flows in 3D geometries obtained from image data.