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At least 433 records · Page 24
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Implicit extensions of an explicit multirate Runge–Kutta scheme

In this work, we propose a new method that extends conservative explicit multirate methods to implicit explicit-multirate methods. We develop extensions of order one and two with different stability properties on the implicit side. The method is suitable for time-stepping adaptive mesh refinement PDE discretizations with different degrees of stiffness. A numerical example with an advection-diffusion problem illustrates the new method's properties.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2016

Parallel Aircraft Trajectory Optimization with Analytic Derivatives

Trajectory optimization is an integral component for the design of aerospace vehicles, but emerging aircraft technologies have introduced new demands on trajectory analysis that current tools are not well suited to address. Designing aircraft with technologies such as hybrid electric propulsion and morphing wings requires consideration of the operational behavior as well as the physical design characteristics of the aircraft. The addition of operational variables can dramatically increase the number of design variables which motivates the use of gradient based optimization with analytic derivatives to solve the larger optimization problems. In this work we develop an aircraft trajectory analysis tool using a Legendre-Gauss-Lobatto based collocation scheme, providing analytic derivatives via the OpenMDAO multidisciplinary optimization framework. This collocation method uses an implicit time integration scheme that provides a high degree of sparsity and thus several potential options for parallelization. The performance of the new implementation was investigated via a series of single and multi-trajectory optimizations using a combination of parallel computing and constraint aggregation. The computational performance results show that in order to take full advantage of the sparsity in the problem it is vital to parallelize both the non-linear analysis evaluations and the derivative computations themselves. The constraint aggregation results showed a significant numerical challenge due to difficulty in achieving tight convergence tolerances. Overall, the results demonstrate the value of applying analytic derivatives to trajectory optimization problems and lay the foundation for future application of this collocation based method to the design of aircraft with where operational scheduling of technologies is key to achieving good performance.

aircraft↗
NASA NTRS1994

Drag reduction strategies

previously a description was given of an active control scheme using wall transpiration that leads to a 15% reduction in surface skin friction beneath a turbulent boundary layer, according to direct numerical simulation. In this research brief further details of that scheme and its variants are given together with some suggestions as to how sensor/actuator arrays could be configured to reduce surface drag. The research which is summarized here was performed during the first half of 1994. This research is motivated by the need to understand better how the dynamics of near-wall turbulent flow can be modified so that skin friction is reduced. The reduction of turbulent skin friction is highly desirable in many engineering applications. Experiments and direct numerical simulations have led to an increased understanding of the cycle of turbulence production and transport in the boundary layer and raised awareness of the possibility of disrupting the process with a subsequent reduction in turbulent skin friction. The implementation of active feedback control in a computational setting is a viable approach for the investigation of the modifications to the flow physics that can be achieved. Bewley et al. and Hill describe how ideas from optimal control theory are employed to give 'sub-optimal' drag reduction schemes. The objectives of the work reported here is to investigate in greater detail the assumptions implicit within such schemes and their limitations. It is also our objective to describe how an array of sensors and actuators could be arranged and interconnected to form a 'smart' surface which has low skin friction.

Hill, D. Christopher↗
NASA NTRS1985

A simulation of rotor-stator interaction using the Euler equations and patched grids

An unsteady Euler code to study rotor-stator interaction problem was developed. The code uses patched grids that move relative to each other to simulate the motion of the rotor airfoils with respect to the stator airfoils. The Osher integration scheme is used in conjunction with an implicit relaxation approach. The scheme is second order accurate in space and time, and is also TVD in each spatial direction. The numerical results were found to be periodic in time, thus demonstrating the capability of the integration and zonal schemes in simulating periodic time dependent flow. The pressure contours obtained are almost oscillation free because of the TVD nature of the scheme. A new procedure was developed to simulate flows about bodies that move relative to each other. This capability should prove to be very useful in the areas of rotor-stator interaction, propeller-nacelle interaction, and helicopter rotor-fuselage interaction.

Rai, M. M.↗
NASA NTRS1992

Efficient simulation of incompressible viscous flow over single and multi-element airfoils

Incompressible viscous turbulent flows over single- and multiple-element airfoils are numerically simulated in an efficient manner by solving the incompressible Navier-Stokes equations. The solution algorithm uses the method of pseudocompressibility with an upwind-differencing scheme for the convective fluxes and an implicit line-relaxation scheme to study high-lift take-off and landing configurations and to compute lift and drag at various angles of attack up to stall. Two different turbulence models are tested in computing the flow over an NACA 4412 airfoil. The approach used for multiple-element airfoils involves the use of multiple zones of structured grids fitted to each element. Two different approaches are compared: a patched system of grids and an overlaid Chimera system of grids. Computational results are presented for two-element, three-element, and four-element airfoil configurations. Excellent agreement with experimental surface-pressure coefficients is seen. The code converges in less than 200 iterations, requiring on the order of one minute of CPU time on a CRAY YMP per element in the airfoil configuration.

Rogers, Stuart E.↗
NASA NTRS1993

Efficient simulation of incompressible viscous flow over multi-element airfoils

The incompressible, viscous, turbulent flow over single and multi-element airfoils is numerically simulated in an efficient manner by solving the incompressible Navier-Stokes equations. The solution algorithm employs the method of pseudo compressibility and utilizes an upwind differencing scheme for the convective fluxes, and an implicit line-relaxation scheme. The motivation for this work includes interest in studying high-lift take-off and landing configurations of various aircraft. In particular, accurate computation of lift and drag at various angles of attack up to stall is desired. Two different turbulence models are tested in computing the flow over an NACA 4412 airfoil; an accurate prediction of stall is obtained. The approach used for multi-element airfoils involves the use of multiple zones of structured grids fitted to each element. Two different approaches are compared; a patched system of grids, and an overlaid Chimera system of grids. Computational results are presented for two-element, three-element, and four-element airfoil configurations. Excellent agreement with experimental surface pressure coefficients is seen. The code converges in less than 200 iterations, requiring on the order of one minute of CPU time on a CRAY YMP per element in the airfoil configuration.

Rogers, Stuart E.↗
DOE OSTI2025

Fully implicit crystal plasticity models representing orientations with modified Rodrigues parameters

Here, this work describes a crystal plasticity formulation combining several mathematical, numerical, and implementation choices to produce a highly efficient model. Specifically, the key choices in the implementation are (1) representing orientations with modified Rodrigues parameters, (2) implementing a fully coupled implicit time integration for the elastic stretch, the crystal orientations, and the model internal variables, (3) implementing the model in the NEML2 constitutive modeling framework, based on PyTorch, to vectorize the calculations and port the computation to GPUs and other hardware accelerators, and (4) an exact implementation of the consistent tangent matrix, even for arbitrary coupling to other field variables beyond the displacements, like temperature, neutron fluence, etc. The first two features of the model are, to our knowledge, novel. The paper considers each of these choices individually as well as the final model as a whole. This includes a full description of modified Rodrigues parameters, their advantages over other representations of orientations, the mathematical formulae and tools required to implement a model with modified Rodrigues parameters, and a detailed description of the geometry of the space of modified Rodrigues parameters (in an appendix). It also includes a description of a fully implicit time integration scheme for the orientations and the advantages in representing orientations with modified Rodrigues parameters in implementing such a model. The work then assess, via numerical examples, the advantages of fully coupled implicit time integration versus more common decoupled and explicit time integration schemes. These studies demonstrate the computational advantages of fully coupled integration versus other time integration algorithms, though the performance of the competing models depends on the complexity of the underlying single crystal model. The study concludes by demonstrating that the choice of time integration method affects the sharpness of the predicted texture, with explicit methods for integrating the orientations overestimating texture sharpness and implicit methods underestimating texture sharpness.

Crystal plasticity↗
DOE OSTI2020

Reconciling Work Functions and Adsorption Enthalpies for Implicit Solvent Models: A Pt (111)/Water Interface Case Study

Implicit solvent models are a computationally efficient method of representing solid/liquid interfaces prevalent in electrocatalysis, energy storage, and materials science. However, electronic structure changes induced at the metallic surface by the dielectric continuum are not fully understood. To address this, we perform DFT calculations for the Pt(111)/water interface, in order to compare Poisson–Boltzmann continuum solvation methods with ab initio molecular dynamics (AIMD) simulations of explicit solvent. We show that the implicit solvent cavity can be parametrized in terms of the electric dipole moment change at the equilibrated explicit Pt/water interface to obtain the potential of zero charge (PZC). We also compare the accuracy of aqueous enthalpies of adsorption of phenol on Pt(111) using geometry and charge density based dielectric cavitation methods. The ability to parametrize the cavity according to individual atoms, as afforded in the geometry based approach, is key to obtaining accurate enthalpy changes of adsorption under aqueous conditions. Additionally, we show that the electronic structure changes induced by explicit solvent and our proposed implicit solvent parametrization scheme yield comparable density difference profiles and d-band projected density of states. We therefore demonstrate the capability of implicit solvent approaches to capture both the energetics of adsorption processes and the main electronic effects of aqueous solvent on the metallic surface. Therefore, this work provides a scheme for computationally efficient simulations of interfacial processes for applications in areas such as heterogeneous catalysis and electrochemistry.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2013

Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows

The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multiscale, intermittent and non-homogeneous fluid and ocean flows. The dynamically orthogonal (DO) field equations provide an adaptive methodology to predict the probability density functions of such flows. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier-Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-implicit projection methods are developed for the mean and for the DO modes, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order finite-volumes are employed in physical space with new advection schemes based on total variation diminishing methods. Other results include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic; (ii) symmetric advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. To verify our implementation and study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2008

An Initial Investigation of the Effects of Turbulence Models on the Convergence of the RK/Implicit Scheme

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. This scheme has been applied to both the compressible and essentially incompressible Reynolds-averaged Navier-Stokes (RANS) equations using the algebraic turbulence model of Baldwin and Lomax (BL). In this paper we focus on the convergence of the RK/implicit scheme when the effects of turbulence are represented by either the Spalart-Allmaras model or the Wilcox k-! model, which are frequently used models in practical fluid dynamic applications. Convergence behavior of the scheme with these turbulence models and the BL model are directly compared. For this initial investigation we solve the flow equations and the partial differential equations of the turbulence models indirectly coupled. With this approach we examine the convergence behavior of each system. Both point and line symmetric Gauss-Seidel are considered for approximating the inverse of the implicit operator of the flow solver. To solve the turbulence equations we use a diagonally dominant alternating direction implicit (DDADI) scheme. Computational results are presented for three airfoil flow cases and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and transport-type equations for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses the RK/implicit scheme for the flow equations.

Swanson, R. C.↗
NASA NTRS2019

Effect of Spatial Filtering in Implicit Large-Eddy Simulations of Separated Flows

The relatively high Reynolds number of turbulent flows encountered in various applications puts these problems well beyond the reach of direct numerical simulation (DNS) at present. Meanwhile, lower-fidelity Reynolds-averaged Navier-Stokes (RANS) calculations are known to be not accurate enough in complex problems, such as smooth-body flow separation and other flows involving highly-unsteady phenomena. Hence, given the current infeasibility of DNS and the unsatisfactory performance of RANS, intermediate techniques such as large-eddy simulation (LES) and hybrid RANS-LES, whose fidelity lie between RANS and DNS, have received much attention for application to various problems of practical importance. Modeling of the effect of missing scales on resolved scales, also known as subgrid-scale (SGS) modeling, is an important subject for LES. SGS models can be broadly categorized as explicit or implicit approaches. The explicit approach is based on an SGS model that explicitly appears in the governing equations expressed in the form of so-called “filtered Navier-Stokes equations”, which describe the evolution of the turbulence scales resolved by the LES grid. The effect of the scales unresolved by the grid is represented by the SGS model. The implicit modeling approach, on the other hand, does not employ an explicit model but instead treats the intrinsic dissipation of the numerical discretization scheme as an implicit SGS model. An LES without an explicit SGS model is commonly termed as an implicit LES (ILES). The relative merits of one SGS modeling approach over another is a subject of ongoing debate. We have opted to employ an ILES methodology, based on high-order compact finite-difference and spatial filtering schemes, in our recent investigations of separated flow problems [1, 2]. Further discussion of our preference of ILES over explicit LES is provided in Uzun and Malik [2]. The spatial filtering operation, described in the next section, is treated as an implicit SGS model for the ILES. Some observations made during the course of our recent investigations, which pointed out to excessive numerical dissipation in certain parts of the flowfield, prompted us to take a closer look at the potential effect of the spatial filter on ILES predictions. This technical note is therefore devoted to spatial filter effects in the context of a high Reynolds number, transonic shock-induced separated flow.

Ali Uzun↗
NASA NTRSDate not supplied

Explicit Discontinuous Galerkin Methods for Conservation Laws

The two explicit DG methods in this study are based on a ‘predictor-corrector’ formulation, the first introduced by Lörcher, Gassner, and Munz (2007, 2008) called space–time expansion discontinuous Galerkin or STE-DG scheme, and the second, introduced independently by the author (Huynh 2006, 2013) called the upwind moment scheme. The predictor step of the two methods is essentially identical using a Cauchy-Kovalevsky (CK) procedure, which involves no interaction of the data among neighboring cells. The corrector step also shares the same space-time integration formulation and is where interaction of the data among neighboring cells takes place; the difference, however, is in how the resulting space-time volume integral is estimated. As a consequence of the different estimates, for the case of advection in one spatial dimension (1D), the moment scheme has a CFL (Courant-Friedrichs-Lewy) condition of 1 for all p and is accurate to order 2p+1, i.e., it possesses the super accuracy property, whereas the STE-DG method has a more restrictive CFL condition and is accurate to the expected order of p+1. For 1D advection, compared with the CFL conditions of 1/(2p+1) of standard RK-DG (Runge-Kutta) scheme where space and time discretization are of the same order, the moment scheme allows a significantly larger time step size. It also turns out that the scheme yields a result identical to Van Leer’s scheme III (1977), which amounts to shifting the data a distance of advection corresponding to the time step and projecting the result onto the space of polynomial solutions. Contrary to Van Leer’s approach, however, the space-time ‘predictor-corrector’ formulation facilitates extensions to the case of systems of equations. Concerning 2D extensions, in the case of advection, when the flow is along the diagonal direction, the CFL conditions for the moment schemes become restrictive as will be shown by Fourier (Von Neumann) stability and accuracy analyses. Since the moment scheme employs the right Radau points as collocation points in time, the method is closely related to the implicit Radau IIA scheme, which is stable for any time step size. The role of Radau IIA in relieving stability restriction for these explicit DG schemes remains to be explored

Discontinuous Galerkin↗
NASA NTRS1994

Three-Dimensional Navier-Stokes Method with Two-Equation Turbulence Models for Efficient Numerical Simulation of Hypersonic Flows

A new computational efficient 3-D compressible Reynolds-averaged implicit Navier-Stokes method with advanced two equation turbulence models for high speed flows is presented. All convective terms are modeled using an entropy satisfying higher-order Total Variation Diminishing (TVD) scheme based on implicit upwind flux-difference split approximations and arithmetic averaging procedure of primitive variables. This method combines the best features of data management and computational efficiency of space marching procedures with the generality and stability of time dependent Navier-Stokes procedures to solve flows with mixed supersonic and subsonic zones, including streamwise separated flows. Its robust stability derives from a combination of conservative implicit upwind flux-difference splitting with Roe's property U to provide accurate shock capturing capability that non-conservative schemes do not guarantee, alternating symmetric Gauss-Seidel 'method of planes' relaxation procedure coupled with a three-dimensional two-factor diagonal-dominant approximate factorization scheme, TVD flux limiters of higher-order flux differences satisfying realizability, and well-posed characteristic-based implicit boundary-point a'pproximations consistent with the local characteristics domain of dependence. The efficiency of the method is highly increased with Newton Raphson acceleration which allows convergence in essentially one forward sweep for supersonic flows. The method is verified by comparing with experiment and other Navier-Stokes methods. Here, results of adiabatic and cooled flat plate flows, compression corner flow, and 3-D hypersonic shock-wave/turbulent boundary layer interaction flows are presented. The robust 3-D method achieves a better computational efficiency of at least one order of magnitude over the CNS Navier-Stokes code. It provides cost-effective aerodynamic predictions in agreement with experiment, and the capability of predicting complex flow structures in complex geometries with good accuracy.

Bardina, J. E.↗
NASA NTRS1983

A conservative type-dependent full potential method for the treatment of supersonic flows with embedded subsonic regions

A nonlinear method based on the full potential equation in conservation form, cast in an arbitrary coordinate system, has been developed to treat predominantly supersonic flows with embedded subsonic regions. This type of flow field occurs frequently near the fuselage-canopy junction area and wing leading edge regions for a moderately swept fighter configuration. The method uses the theory of characteristics to accurately monitor the type-dependent flow field. A conservative switching scheme is developed to transition from the supersonic marching algorithm to a subsonic relaxation procedure, and vice versa. An implicit approximate factorization scheme is employed to solve the finite-differenced equation. Results are shown for a few configurations, including a wing-body-wake realistic fighter model having embedded subsonic regions.

Shankar, V.↗
NASA NTRS1984

Application of a full potential method for computation of three-dimensional supersonic flows

A nonlinear aerodynamic analysis technique based on the full potential equation in conservative form has been modified to permit treatment of supersonic flows with embedded subsonic regions (typically near the fuselage-canopy juncture and the wing leading edge). Solution procedures for the equations do not require any specific form of geometry or physical grid system. This results in the capability to analyze easily very complex geometries provided the posed problem lies within the isentropic restrictions of the full potential theory. Characteristic signal propagation theory is used to monitor the type dependent flow and a conservative switching scheme is employed to transition from the supersonic marching algorithm to a subsonic relaxation procedure and vice versa. An implicit approximate factorization scheme is used to solve the finite-difference equations. These modifications now permit analysis of fully three-dimensional flowfields including the interference effects due to lifting surface wakes. Improved grid generation capability allows analysis of complete complex aircraft geometries (fuselage, wing, tail, wing wake, and tail wake). Results are presented showing very good correlations with experimental surface pressure data and aerodynamic force data at both design and off-design operating points. Configurations examined include several waverider concepts, an arrow wing-body with wake, an advanced tactical fighter concept, and a fighter forebody-canard configuration.

Jones, K. M.↗
NASA NTRS1992

Development of iterative techniques for the solution of unsteady compressible viscous flows

A research effort was initiated at Georgia Tech in February 1991 on the development of efficient techniques for the computation of 2-D and 3-D unsteady compressible flow problems. It was found that in 2-D unsteady viscous flow applications, the generalized minimal residual (GMRES) scheme was able to significantly improve the accuracy and stability characteristics of an existing 2-D ADI (Alternating Direction Implicit) time marching scheme. That is, the GMRES/ADI combination allowed 10 to 20 times larger time steps compared to an ADI scheme. Because the GMRES algorithm requires 5 to 10 times the CPU work compared to the ADI scheme, the combined GMRES/ADI scheme yields a net factor of 2 savings in CPU cost. During the past year, we also experimented with GMRES/multigrid/ADI combination. The purpose of this combination was to compute the low frequency components of the change in the flow properties from one time step to the next on a coarse grid. This strategy reduces the memory requirements of the GMRES method roughly by a factor of 4-8 for steady flow problems.

Sankar, Lakshmi N.↗
NASA NTRS1997

Massively Parallel and Scalable Implicit Time Integration Algorithms for Structural Dynamics

Explicit codes are often used to simulate the nonlinear dynamics of large-scale structural systems, even for low frequency response, because the storage and CPU requirements entailed by the repeated factorizations traditionally found in implicit codes rapidly overwhelm the available computing resources. With the advent of parallel processing, this trend is accelerating because of the following additional facts: (a) explicit schemes are easier to parallelize than implicit ones, and (b) explicit schemes induce short range interprocessor communications that are relatively inexpensive, while the factorization methods used in most implicit schemes induce long range interprocessor communications that often ruin the sought-after speed-up. However, the time step restriction imposed by the Courant stability condition on all explicit schemes cannot yet be offset by the speed of the currently available parallel hardware. Therefore, it is essential to develop efficient alternatives to direct methods that are also amenable to massively parallel processing because implicit codes using unconditionally stable time-integration algorithms are computationally more efficient when simulating the low-frequency dynamics of aerospace structures.

Farhat, Charbel↗
NASA NTRS1993

Time-accurate Navier-Stokes calculations with multigrid acceleration

A numerical scheme to solve the unsteady Navier-Stokes equations is described. The scheme is implemented by modifying the multigrid-multiblock version of the steady Navier-Stokes equations solver, TLNS3D. The scheme is fully implicit in time and uses TLNS3D to iteratively invert the equations at each physical time step. The design objective of the scheme is unconditional stability (at least for first- and second-order discretizations of the physical time derivatives). With unconditional stability, the choice of the time step is based on the physical phenomena to be resolved rather than limited by numerical stability which is especially important for high Reynolds number viscous flows, where the spatial variation of grid cell size can be as much as six orders of magnitude. An analysis of the iterative procedure and the implementation of this procedure in TLNS3D are discussed. Numerical results are presented to show both the capabilities of the scheme and its speed up relative to the use of global minimum time stepping. Reductions in computational times of an order of magnitude are demonstrated.

Melson, N. Duane↗