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NASA NTRS1996

Solution-Adaptive Cartesian Cell Approach for Viscous and Inviscid Flows

A Cartesian cell-based approach for adaptively refined solutions of the Euler and Navier-Stokes equations in two dimensions is presented. Grids about geometrically complicated bodies are generated automatically, by the recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal cut cells are created using modified polygon-clipping algorithms. The grid is stored in a binary tree data structure that provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite volume formulation. The convective terms are upwinded: A linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The results of a study comparing the accuracy and positivity of two classes of cell-centered, viscous gradient reconstruction procedures is briefly summarized. Adaptively refined solutions of the Navier-Stokes equations are shown using the more robust of these gradient reconstruction procedures, where the results computed by the Cartesian approach are compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.

Coirier, William J.↗

DOE OSTI2020

Demonstration and performance testing of extreme-resolution simulations with static meshes on Summit (CPU & GPU) for a parked-turbine configuration and an actuator-line (mid-fidelity model) wind farm configuration (ECP-Q4 FY2020 Milestone Report)

The goal of the ExaWind project is to enable predictive simulations of wind farms comprised of many megawatt-scale turbines situated in complex terrain. Predictive simulations will require computational fluid dynamics (CFD) simulations for which the mesh resolves the geometry of the turbines and captures the rotation and large deflections of blades. Whereas such simulations for a single turbine are arguably petascale class, multi-turbine wind farm simulations will require exascale-class resources. The primary physics codes in the ExaWind simulation environment are Nalu-Wind, an unstructured-grid solver for the acoustically incompressible Navier-Stokes equations, AMR-Wind, a block-structured-grid solver with adaptive mesh refinement capabilities, and OpenFAST, a wind-turbine structural dynamics solver. The Nalu-Wind model consists of the mass-continuity Poisson-type equation for pressure and Helmholtz-type equations for transport of momentum and other scalars. For such modeling approaches, simulation times are dominated by linear-system setup and solution for the continuity and momentum systems. For the ExaWind challenge problem, the moving meshes greatly affect overall solver costs as reinitialization of matrices and recomputation of preconditioners is required at every time step. The choice of overset-mesh methodology to model the moving and non-moving parts of the computational domain introduces constraint equations in the elliptic pressure-Poisson solver. The presence of constraints greatly affects the performance of algebraic multigrid preconditioners.

17 WIND ENERGY↗

NASA NTRS1994

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells are created using polygon-clipping algorithms. The grid is stored in a binary-tree structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded: a gradient-limited, linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The more robust of a series of viscous flux functions is used to provide the viscous fluxes at the cell interfaces. Adaptively-refined solutions of the Navier-Stokes equations using the Cartesian, cell-based approach are obtained and compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.

Coirier, William J.↗

NASA NTRS1995

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells are created using polygon-clipping algorithms. The grid is stored in a binary-tree data structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded: A gradient-limited, linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The more robust of a series of viscous flux functions is used to provide the viscous fluxes at the cell interfaces. Adaptively-refined solutions of the Navier-Stokes equations using the Cartesian, cell-based approach are obtained and compared to theory, experiment and other accepted computational results for a series of low and moderate Reynolds number flows.

Coirier, William J.↗

DOE OSTI2015

Developing Uintah's Runtime System for Forthcoming Architectures

The need to solve ever-more complex science and engineering problems on computer architectures that are rapidly evolving poses considerable challenges for modern simulation frameworks and packages. While the decomposition of software into a programming model that generates a task-graph for execution by a runtime system make portability and performance possible, it also imposes considerable demands on that runtime system. Here, the challenge of meeting those demands for near-term and longer term systems in the context of the Uintah framework is considered in key areas such as support for data structures, tasks on heterogeneous architectures, performance portability, power management and designing for resilience by the use of replication based upon adaptive mesh refinement processes is addressed. Examples of techniques in these areas are used to illustrate performance improvements obtained for present large-scale systems and the needs of proposed systems considered.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1994

New multigrid approach for three-dimensional unstructured, adaptive grids

A new multigrid method with adaptive unstructured grids is presented. The three-dimensional Euler equations are solved on tetrahedral grids that are adaptively refined or coarsened locally. The multigrid method is employed to propagate the fine grid corrections more rapidly by redistributing the changes-in-time of the solution from the fine grid to the coarser grids to accelerate convergence. A new approach is employed that uses the parent cells of the fine grid cells in an adapted mesh to generate successively coaser levels of multigrid. This obviates the need for the generation of a sequence of independent, nonoverlapping grids as well as the relatively complicated operations that need to be performed to interpolate the solution and the residuals between the independent grids. The solver is an explicit, vertex-based, finite volume scheme that employs edge-based data structures and operations. Spatial discretization is of central-differencing type combined with a special upwind-like smoothing operators. Application cases include adaptive solutions obtained with multigrid acceleration for supersonic and subsonic flow over a bump in a channel, as well as transonic flow around the ONERA M6 wing. Two levels of multigrid resulted in reduction in the number of iterations by a factor of 5.

Parthasarathy, Vijayan↗

NASA NTRS1999

Innovative Language-Based & Object-Oriented Structured AMR Using Fortran 90 and OpenMP

Parallel adaptive mesh refinement (AMR) is an important numerical technique that leads to the efficient solution of many physical and engineering problems. In this paper, we describe how AMR programing can be performed in an object-oreinted way using the modern aspects of Fortran 90 combined with the parallelization features of OpenMP.

software↗

DOE OSTI2022

MFIX-Exa: Performance prediction of multiphase energy conversion devices

MFIX-Exa targets the high-fidelity CFD-DEM model in which particles are unresolved by the fluid grid (typically using a fluid mesh approximately twice the particle diameter) but discrete particle dynamics including collisions are resolved with a simple linear-spring dashpot model. The fluid field is solved with a modern low Mach number formation of a cell-centered, nodal-pressure approximate projection method using a Godunov scheme . Physics capabilities include: complex geometries via embedded boundaries (EBs), open and closed system ideal gas equations of state, species transport and heterogeneous chemi- cal reactions. MFIX-Exa is built on the AMReX software framework, the ECP Block-Structured Adaptive Mesh Refinement (AMR) Co-Design Center, which allows the code to be portable and performant.Further integration into the ECP ecosystem includes HYPRE linear solvers and Ascent for in situ visualization. Current project efforts focus on scaling up realistic simulations to the KPP of the challenge problem: NETs 50kW chemical looping reactor, shakedown and scaling on JSLE and OLCF TDS systems and reaching for any remaining performance improvements.

Musser, Jordan↗

NASA NTRSDate not supplied

Transonic Flutter Dips of the AGARD 445.6 Wing

The AGARD 445.6 configuration is the most popular validation test case for transonic flutter predictions, but the actual extent of truly nonlinear transonic flow for this case is unclear, due to the sparsity of the experimental data, and the thin profile of the wing. This work utilizes a combination of mesh adaptation and the linearized frequency-domain method to obtain high-quality viscous and inviscid flutter predictions; these solutions show a double flutter dip through the transonic Mach range driven by complex shock growth across the wing. A single experimental flutter point lies in this flutter dip area, which is not enough to assess the accuracy of these transonic flutter predictions. Modeling the boundary layer of the wind tunnel wall (as opposed to the commonly-assumed symmetry wall assumption) appears to have a large impact on the predicted flutter boundary, but true mesh convergence of this scenario is a challenge.

Bret K Stanford↗

NASA NTRS2015

Parallel Adaptive High-Order CFD Simulations Characterizing SOFIA Cavitiy Acoustics

This paper presents large-scale MPI-parallel computational uid dynamics simulations for the Stratospheric Observatory for Infrared Astronomy (SOFIA). SOFIA is an airborne, 2.5-meter infrared telescope mounted in an open cavity in the aft fuselage of a Boeing 747SP. These simulations focus on how the unsteady ow eld inside and over the cavity interferes with the optical path and mounting structure of the telescope. A tempo- rally fourth-order accurate Runge-Kutta, and a spatially fth-order accurate WENO-5Z scheme were used to perform implicit large eddy simulations. An immersed boundary method provides automated gridding for complex geometries and natural coupling to a block-structured Cartesian adaptive mesh re nement framework. Strong scaling studies using NASA's Pleiades supercomputer with up to 32k CPU cores and 4 billion compu- tational cells shows excellent scaling. Dynamic load balancing based on execution time on individual AMR blocks addresses irregular numerical cost associated with blocks con- taining boundaries. Limits to scaling beyond 32k cores are identi ed, and targeted code optimizations are discussed.

Parallell↗

DOE OSTI2015

A moving mesh finite difference method for equilibrium radiation diffusion equations

An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

A block-spectral adaptive H-/$p$-refinement strategy for shock-dominated problems

An adaptive H-/p-refinement strategy using a novel sensor is devised and tested in a block-spectral compressible Euler code equipped with adaptive-mesh refinement (AMR) and high-order flux-reconstruction numerics. At each Gauss quadrature point (or solution point) within each spectral block (or mesh element) the discrete velocity jump ΔU = ∂U/∂y 1 Δy 1 + ∂V/∂y 2 Δy 2 + ∂W/∂y 3 Δy 3 is calculated and normalized by the local speed of sound, a. Here, the grid spacing, Δx i , is calculated in each direction as the distance between auxiliary Gauss-Lobatto points, staggered relative to the solution points. The polynomial order is increased from p = 0 to p = p max in regions of weak compression, (ΔU/a) crit < ΔU/a < 0 and kept at p = p max in regions of flow expansion ΔU/a ≥ 0, while staying at the H = 0 base mesh level. Regions experiencing strong compressions, i.e. ΔU/a < (ΔU/a) crit , are H-refined up to H = H max where H max is applied at the location of maximum compression, ΔU/a = min(ΔU/a) in the domain, while keeping p = 0 to guarantee robustness and monotonicity of the solution in the H refined region. The critical value of (ΔU/a) crit = -0.06 is found to effectively separate smooth and non-smooth solution regions, supported by a 1D detonation initiation test case in ideal gas and a shock-to-detonation transition in high explosives. Using this value, the Sod shock tube, Shu-Osher problem, double Mach reflection and a 2D detonation in a high-explosive are simulated with the proposed adaptive H-/p-refinement. In the Sod shock tube case, p-refinement resolves the (weak) contact discontinuity while H-refinement enhances the grid resolution in the shock exploiting the monotonicity of the p = 0 reconstruction. For the Shu-Osher problem, p-refinement captures the small-scale oscillations trailing the shock that would be otherwise attenuated, while H-refinement triggered by the ΔU-sensor appropriately tracks the shock. In the double Mach reflection problem, H-refinement confines the numerical diffusion around the reflected shock while p-refinement recaptures many physical features trailing the shock. Finally, in the 2D high-explosive detonation case, H-refinement follows the leading shock and resolves the curvature of the detonation wave, while p-refinement adds resolution to the trailing reaction zone. Finally, the proposed methodology is tested in a detonation-wave propagation test case in high-explosives with numerical predictions comparing favorably against experiments.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

ZND Verification Tests for Reactive Burn Models in FLAG

The ZND theory, named after Zeldovich, von Neumann, and Doering, provides a simple model for one-dimensional ideal steady-state detonation. It assumes that the detonation wave front starts with a shock that is a discontinuous jump and is followed by a finite-length reaction zone. Reactive burn (also called reactive flow) models are based on ZND theory, as they model the shock initiation and detonation process with a finite reaction rate. The ZND wave propagation test is essentially the only available test case where an analytic solution exists for verification of reactive burn models in numerical codes. However, there are extensions and variants of the ZND test that have been devised for verification of multidimensional flows. The objective of this work is to provide verification of the reactive burn models currently implemented in the Lagrangian hydrocode FLAG and investigate the influence of mesh resolution, artificial viscosity models, and the Arbitrary Lagrangian-Eulerian (ALE) Euler relaxer on the simulation results. The burn models of interest are the Wescott-Stewart-Davis (WSD) model, the Scaled Uniform Reactive Flow (SURF) model (specifically with the SURFplus model extension), and the Arrhenius shock temperature state dependent WSD (AWSD) model. Previously, Ralph Menikoff has used ZND tests for verification of the SURF and SURFplus models in the Eulerian hydrocode xRAGE. The ZND tests here are somewhat different than the approach by Menikoff. In particular, we use a piston-driven ZND detonation wave (via a prescribed constant velocity boundary condition) in a Lagrangian framework whereas Menikoff had a ZND wave followed by a invariant rarefaction wave. The xRAGE simulations were carried out on uniform grids and adaptive mesh refinement (AMR) grids. Although AMR was recently implemented in FLAG and now fully functional for 2D simulations, it will not be evaluated with ZND tests at this time. Some work has been done previously for validating the reactive burn models in FLAG. For example, the validation studies of SURF with shock-to-detonation (SDT) tests, cylinder tests, and gap-stick tests. Further validation of the AWSD, WSD, and SURF models is described in for SDT, multi-shock, cylinder, and corner-turning tests. Recently, a large parameter study with approximately twenty SDT tests was performed to validate the AWSD, WSD, SURF, and SURFplus models while also investigating mesh resolution and artificial viscosity settings. To the best of our knowledge, the current work represents the first documented verification of these burn models in FLAG. The reader should note that many of the tables and figures in this report use units of cm/g/µs, which are the default units for FLAG. However, some lengths and velocities are expressed in µm and mm/µs, respectively, which are typical units for detonation analyses.

45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗

DOE OSTI2022

Modeling wave propagation in elastic solids via high-order accurate implicit-mesh discontinuous Galerkin methods

Here, a high-order accurate implicit-mesh discontinuous Galerkin framework for wave propagation in single-phase and bi-phase solids is presented. The framework belongs to the embedded-boundary techniques and its novelty regards the spatial discretization, which enables boundary and interface conditions to be enforced with high-order accuracy on curved embedded geometries. High-order accuracy is achieved via high-order quadrature rules for implicitly-defined domains and boundaries, whilst a cell-merging strategy addresses the presence of small cut cells. The framework is used to discretize the governing equations of elastodynamics, written using a first-order hyperbolic momentum-strain formulation, and an exact Riemann solver is employed to compute the numerical flux at the interface between dissimilar materials with general anisotropic properties. The space-discretized equations are then advanced in time using explicit high-order Runge–Kutta algorithms. Several two- and three-dimensional numerical tests including dynamic adaptive mesh refinement are presented to demonstrate the high-order accuracy and the capability of the method in the elastodynamic analysis of single- and bi-phases solids containing complex geometries.

42 ENGINEERING↗

NASA NTRS1998

Using Multi-threading for the Automatic Load Balancing of 2D Adaptive Finite Element Meshes

In this paper, we present a multi-threaded approach for the automatic load balancing of adaptive finite element (FE) meshes The platform of our choice is the EARTH multi-threaded system which offers sufficient capabilities to tackle this problem. We implement the adaption phase of FE applications oil triangular meshes and exploit the EARTH token mechanism to automatically balance the resulting irregular and highly nonuniform workload. We discuss the results of our experiments oil EARTH-SP2, on implementation of EARTH on the IBM SP2 with different load balancing strategies that are built into the runtime system.

Heber, Gerd↗

NASA NTRS1999

High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.

Barth, Timothy↗

DOE OSTI2012

Optimization of multigrid based elliptic solver for large scale simulations in the FLASH code: OPTIMIZATION OF MULTIGRID BASED ELLIPTIC SOLVER IN THE FLASH CODE

FLASH is a multiphysics multiscale adaptive mesh refinement (AMR) code originally designed for simulation of reactive flows often found in Astrophysics. With its wide user base and flexible applications configuration capability, FLASH has a dual task of maintaining scalability and portability in all its solvers. The scalability of fully explicit solvers in the code is tied very closely to that of the underlying mesh. Others such as the Poisson solver based on a multigrid method have more complex scaling behavior. Multigrid methods suffer from processor starvation and dominating communication costs at coarser grids with increase in the number of processors. In this paper, we propose a combination of uniform grid mesh with AMR mesh, and the merger of two different sets of solvers to overcome the scalability limitation of the Poisson solver in FLASH. The principal challenge in the proposed merger is the efficiency of the communication algorithm to map the mesh back and forth between uniform grid and AMR. We present two different parallel mapping algorithms and also discuss results from performance studies of the two implementations. Copyright © 2012 John Wiley & Sons, Ltd.

Computer Science↗

DOE OSTI2020

A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L 2 convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H -1 -type norm, but not L 2 -coercive. Nevertheless, the L 2 convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L 2 norm, or by properly choosing the finite element spaces, providing implicit control of the L 2 norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L 2 convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. Finally, we believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).

97 MATHEMATICS AND COMPUTING↗