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NASA NTRSDate not supplied

Derivatives Across Adaptive-Step Integration Accurately and Efficiently

The numerical integration of ordinary differential equations (ODEs) is common in trajectory optimization. Previous studies have shown that the use of adaptive-step Runge-Kutta integration schemes can introduce errors in the derivatives due to non-differentiable behavior within the integration scheme itself. The authors previously demonstrated the use of an implicit adaptation to the Runge-Kutta methods to mitigate this issue, but at a significant computational cost. In this work, the authors demonstrate an explicit adaptation to the Runge-Kutta methods to address the issue that does not incur a significant computational cost.

Kaushik Ponnapalli↗

NASA NTRS1993

Multigrid methods for flow transition in three-dimensional boundary layers with surface roughness

The efficient multilevel adaptive method has been successfully applied to perform direct numerical simulations (DNS) of flow transition in 3-D channels and 3-D boundary layers with 2-D and 3-D isolated and distributed roughness in a curvilinear coordinate system. A fourth-order finite difference technique on stretched and staggered grids, a fully-implicit time marching scheme, a semi-coarsening multigrid method associated with line distributive relaxation scheme, and an improved outflow boundary-condition treatment, which needs only a very short buffer domain to damp all order-one wave reflections, are developed. These approaches make the multigrid DNS code very accurate and efficient. This allows us not only to be able to do spatial DNS for the 3-D channel and flat plate at low computational costs, but also to do spatial DNS for transition in the 3-D boundary layer with 3-D single and multiple roughness elements, which would have extremely high computational costs with conventional methods. Numerical results show good agreement with the linear stability theory, the secondary instability theory, and a number of laboratory experiments. The contribution of isolated and distributed roughness to transition is analyzed.

Liu, Chaoqun↗

NASA NTRS2022

Vision-based Velocimetry over Unknown Terrain with a Low-Noise IMU

This paper presents a novel approach to terrain-relative navigation with a visual camera and Inertial Measurement Unit (IMU). The proposed algorithm uses an Extended Kalman Filter (EKF) to combine an IMU propagated state estimate with batch correction estimates computed over a sliding window of measurements. The batch correction algorithm follows the Maximum Likelihood Estimation (MLE) approach used in other Bundle Adjustment systems. Unlike other systems, the proposed system parameterizes the state over the entire window in terms of the state at a single epoch. By ignoring IMU error over the window duration, we obtain a state epoch MLE that jointly estimates the epoch state and terrain parameters with drastically reduced computationally cost. This paper presents the general architecture which can be adapted for various state parameterizations and measurement inputs. For space applications with high-accuracy IMUs, the reduction in computational cost comes with only a modest increase in estimation errors. The increase in error is quantified via a linear covariance analysis presented in this paper. Furthermore, we present simulation results which show the applicability of this algorithm to planetary landing problems.

San Martin, A. Miguel↗

NASA NTRS2019

Using Graph Coloring to Compute Total Derivatives More Efficiently in OpenMDAO

When they are applicable, gradient based optimization algorithms are the most efficient way to solve design optimization problems. Although gradient based methods are generally efficient, they can be made significantly more so through the usage of analytic techniques to compute the necessary total derivatives. The traditional forward (direct) and reverse (adjoint) analytic techniques have computational costs that scale linearly with the number of design variables and the number of constraints, respectively. In this work, we present an application of a graph coloring algorithm to the analytic techniques for computing total derivative Jacobians in order to achieve much better computational scaling than the pure analytic methods can provide alone. A detailed theoretical explanation of how coloring algorithms interact with analytic derivative methods is presented that illustrates specific types of sparsity patterns that must be present in total derivative Jacobians in order for this coloring technique to be effective. The new technique has been implemented as a feature in the OpenMDAO framework and the implementation is demonstrated on two example problems. The performance on the example problems up to 50% reduction in compute cost for optimizations with bi-directional coloring compared to traditional constraint aggregation. Additionally, the results show how coloring technique alleviates some of the numerical difficulties that constraint aggregation can cause, leading to the ability to solve larger problems. It is expected that the new method will have wide applicability to multidisciplinary optimization problems, and that its availability in OpenMDAO will offer significant computational savings for users without the need for them to implement the coloring algorithm themselves.

Trajectory↗

NASA NTRS1993

Generation and exploration of aggregation abstractions for scheduling and resource allocation

This paper presents research on the abstraction of computational theories for scheduling and resource allocation. The paper describes both theory and methods for the automated generation of aggregation abstractions and approximations in which detailed resource allocation constraints are replaced by constraints between aggregate demand and capacity. The interaction of aggregation abstraction generation with the more thoroughly investigated abstractions of weakening operator preconditions is briefly discussed. The purpose of generating abstract theories for aggregated demand and resources includes: answering queries about aggregate properties, such as gross feasibility; reducing computational costs by using the solution of aggregate problems to guide the solution of detailed problems; facilitating reformulating theories to approximate problems for which there are efficient problem-solving methods; and reducing computational costs of scheduling by providing more opportunities for variable and value-ordering heuristics to be effective. Experiments are being developed to characterize the properties of aggregations that make them cost effective. Both abstract and concrete theories are represented in a variant of first-order predicate calculus, which is a parameterized multi-sorted logic that facilitates specification of large problems. A particular problem is conceptually represented as a set of ground sentences that is consistent with a quantified theory.

Lowry, Michael R.↗

NASA NTRS1995

Scalability of Parallel Spatial Direct Numerical Simulations on Intel Hypercube and IBM SP1 and SP2

The implementation and performance of a parallel spatial direct numerical simulation (PSDNS) approach on the Intel iPSC/860 hypercube and IBM SP1 and SP2 parallel computers is documented. Spatially evolving disturbances associated with the laminar-to-turbulent transition in boundary-layer flows are computed with the PSDNS code. The feasibility of using the PSDNS to perform transition studies on these computers is examined. The results indicate that PSDNS approach can effectively be parallelized on a distributed-memory parallel machine by remapping the distributed data structure during the course of the calculation. Scalability information is provided to estimate computational costs to match the actual costs relative to changes in the number of grid points. By increasing the number of processors, slower than linear speedups are achieved with optimized (machine-dependent library) routines. This slower than linear speedup results because the computational cost is dominated by FFT routine, which yields less than ideal speedups. By using appropriate compile options and optimized library routines on the SP1, the serial code achieves 52-56 M ops on a single node of the SP1 (45 percent of theoretical peak performance). The actual performance of the PSDNS code on the SP1 is evaluated with a "real world" simulation that consists of 1.7 million grid points. One time step of this simulation is calculated on eight nodes of the SP1 in the same time as required by a Cray Y/MP supercomputer. For the same simulation, 32-nodes of the SP1 and SP2 are required to reach the performance of a Cray C-90. A 32 node SP1 (SP2) configuration is 2.9 (4.6) times faster than a Cray Y/MP for this simulation, while the hypercube is roughly 2 times slower than the Y/MP for this application. KEY WORDS: Spatial direct numerical simulations; incompressible viscous flows; spectral methods; finite differences; parallel computing.

Joslin, Ronald D.↗

NASA NTRS1997

Simplifying silicon burning: Application of quasi-equilibrium to (alpha) network nucleosynthesis

While the need for accurate calculation of nucleosynthesis and the resulting rate of thermonuclear energy release within hydrodynamic models of stars and supernovae is clear, the computational expense of these nucleosynthesis calculations often force a compromise in accuracy to reduce the computational cost. To redress this trade-off of accuracy for speed, the authors present an improved nuclear network which takes advantage of quasi- equilibrium in order to reduce the number of independent nuclei, and hence the computational cost of nucleosynthesis, without significant reduction in accuracy. In this paper they will discuss the first application of this method, the further reduction in size of the minimal alpha network. The resultant QSE- reduced alpha network is twice as fast as the conventional alpha network it replaces and requires the tracking of half as many abundance variables, while accurately estimating the rate of energy generation. Such reduction in cost is particularly necessary for future generation of multi-dimensional models for supernovae.

Hix, W. R.↗

NASA NTRS2001

A Rapid Aerodynamic Design Procedure Based on Artificial Neural Networks

An aerodynamic design procedure that uses neural networks to model the functional behavior of the objective function in design space has been developed. This method incorporates several improvements to an earlier method that employed a strategy called parameter-based partitioning of the design space in order to reduce the computational costs associated with design optimization. As with the earlier method, the current method uses a sequence of response surfaces to traverse the design space in search of the optimal solution. The new method yields significant reductions in computational costs by using composite response surfaces with better generalization capabilities and by exploiting synergies between the optimization method and the simulation codes used to generate the training data. These reductions in design optimization costs are demonstrated for a turbine airfoil design study where a generic shape is evolved into an optimal airfoil.

Rai, Man Mohan↗

NASA NTRS2018

Hybrid Reduced Order Model for N2-N2 Interactions for Application to Dissociation and Energy Transfer Processes

Recent work in the aerothermodynamics community has focused on the development of reduced-order models for thermo-chemical non-equilibrium which avoid the restrictive assumptions of multi-temperature models and the prohibitive cost associated with State-to-State (StS) models. In the present work, this is accomplished by lumping energy states together and assuming groups of states are roughly in equilibrium. As a result, the non-equilibrium behavior of a gas can be captured at a reduced computational cost from a full StS simulation. In this work, we present a hybrid grouping model for studying energy transfer and dissociation in a mixture of nitrogen molecules due to N2-N2 reactions. This is accomplished by making use of a grouping strategy informed by data from the N2-N StS kinetic data. However, due to the massive computational cost associated with constructing StS data for the N2-N2 system, the kinetic data for the hybrid grouping model are calculated using the quasi-classical trajectory (QCT) method by sampling states for trajectory within the groups. This general framework is called the Maximum-Entropy Quasi-Classical Trajectory (ME-QCT) method. The primary challenge associated with this method is that rates for reverse grouped reactions cannot be obtained through detailed balance at a group level, due to the variation of group internal temperatures. To construct the full model for N2-N2 grouped kinetics using the ME-QCT method, detailed balance is invoked at the microscopic level, allowing for the calculation of the full kinetic data from QCT. Results will be presented using the full ME-QCT model for the N2-N2 system in an isothermal and isochoric reactor simulation. In addition, simple CFD test cases for a one-dimensional standing shock and for a quasi-one-dimensional nozzle will be used for demonstration of the ME-QCT method. This method allows for the calculation of non-equilibrium behavior for the N2-N2 system without the prohibitive cost of a full StS simulation. Moreover, it enables the construction of a unified model for the dissociating and recombining non-equilibrium flows.

nonequilibrium modeling↗

NASA NTRSDate not supplied

Improving Multi-Model Trajectory Simulation Estimators using Model Selection and Tuning

Multi-model Monte Carlo methods have been demonstrated to be an efficient and accurate alternative to standard Monte Carlo (MC) in the model-based propagation of uncertainty in entry, descent, and landing (EDL) applications. These multi-model MC methods fuse predictions from low-fidelity models with the high-fidelity EDL model of interest to produce unbiased statistics with a fraction of the computational cost. The accuracy and efficiency of the multi-model MC methods are dependent upon the magnitude of correlations of the low-fidelity models with the high-fidelity model, but also upon the correlation amongst the low-fidelity models, and their relative computational cost. Because of this layer of complexity, the question of how to optimally select the set of low-fidelity models has remained open. In this work, methods for optimal model construction and tuning are investigated as a means to increase the speed and precision of trajectory simulation for EDL. Specifically, the focus is on the inclusion of low-fidelity model tuning within the sample allocation optimization that accompanies multi-model MC methods. Preliminary results indicate that low-fidelity model tuning can significantly improve efficiency and precision of trajectory simulations and provide an increased edge to multi-model MC methods when compared to standard MC. The challenges and potential benefits to exploring a fully iterative and comprehensive optimization strategy in future work are highlighted.

uncertainty quantification↗

NASA NTRSDate not supplied

Improving Multi-Model Trajectory Simulation Estimators using Model Selection and Tuning

Multi-model Monte Carlo methods have been demonstrated to be an efficient and accurate alternative to standard Monte Carlo (MC) in the model-based propagation of uncertainty in entry, descent, and landing (EDL) applications. These multi-model MC methods fuse predictions from low-fidelity models with the high-fidelity EDL model of interest to produce unbiased statistics with a fraction of the computational cost. The accuracy and efficiency of the multi-model MC methods are dependent upon the magnitude of correlations of the low-fidelity models with the high-fidelity model, but also upon the correlation among the low-fidelity models, and their relative computational cost. Because of this layer of complexity, the question of how to optimally select the set of low-fidelity models has remained open. In this work, methods for optimal model construction and tuning are investigated as a means to increase the speed and precision of trajectory simulation for EDL. Specifically, the focus is on the inclusion of low-fidelity model tuning within the sample allocation optimization that accompanies multi-model MC methods. Preliminary results indicate that low-fidelity model tuning can significantly improve efficiency and precision of trajectory simulations and provide an increased edge to multi-model MC methods when compared to standard MC. The challenges and potential benefits to exploring a fully iterative and comprehensive optimization strategy in future work are highlighted.

uncertainty quantification↗

NASA NTRS1974

Numerical orbit integration efficiency of the Delaunay-Similar elements

Orbit equations with a set of conservative and a set of nonconservative perturbing potentials were considered. Scheifele's DS formulation of these equations has dependent variables similar to Delaunay's orbital elements with the true anomaly as the independent variable. Efficiency curves of computing cost v.s. accuracy were constructed for Adams integrators of order of 2 through 15 with several correcting algorithms and for a Runga-Kutta integrator. Considering stability regions, choices were made for the optimally efficient integration modes for the DS elements. Integrating in these modes reduces computing costs for a specified accuracy.

Pierce, S.↗

NASA NTRS2013

Using CFD Surface Solutions to Shape Sonic Boom Signatures Propagated from Off-Body Pressure

The conceptual design of a low-boom and low-drag supersonic aircraft remains a challenge despite significant progress in recent years. Inverse design using reversed equivalent area and adjoint methods have been demonstrated to be effective in shaping the ground signature propagated from computational fluid dynamics (CFD) off-body pressure distributions. However, there is still a need to reduce the computational cost in the early stages of design to obtain a baseline that is feasible for low-boom shaping, and in the search for a robust low-boom design over the entire sonic boom footprint. The proposed design method addresses the need to reduce the computational cost for robust low-boom design by using surface pressure distributions from CFD solutions to shape sonic boom ground signatures propagated from CFD off-body pressure.

Ordaz, Irian↗

NASA NTRS2012

The Pursuit of K: Reflections on the Current State-of-the-Art in Stress Intensity Factor Solutions for Practical Aerospace Applications

The elastic stress intensity factor (SIF, commonly denoted as K) is the foundation of practical fracture mechanics (FM) analysis for aircraft structures. This single parameter describes the first-order effects of stress magnitude and distribution as well as the geometry of both structure/component and crack. Hence, the calculation of K is often the most significant step in fatigue analysis based on FM. This presentation will provide several reflections on the current state-of-the-art in SIF solution methods used for practical aerospace applications, including a brief historical perspective, descriptions of some recent and ongoing advances, and comments on some remaining challenges. Newman and Raju made significant early contributions to practical structural analysis by developing closed-form SIF equations for surface and corner cracks in simplified geometries, often based on empirical fits of finite element (FE) solutions. Those solutions (and others like them) were sometimes revised as new analyses were conducted or limitations discovered. The foundational solutions have exhibited striking longevity, despite the relatively "coarse" FE models employed many decades ago. However, in recent years, the accumulation of different generations of solutions for the same nominal geometry has led to some confusion (which solution is correct?), and steady increases in computational capabilities have facilitated the discovery of inaccuracies in some (not all!) of the legacy solutions. Some examples of problems and solutions are presented and discussed, including the challenge of maintaining consistency with legacy design applications. As computational power has increased, the prospect of calculating large numbers of SIF solutions for specific complex geometries with advanced numerical methods has grown more attractive. Fawaz and Andersson, for example, have been generating literally millions of new SIF solutions for different combinations of multiple cracks under simplified loading schemes using p-version FE methods. These data are invaluable, but questions remain about their practical use, because the tabular databases of key results needed to support practical life analysis can occupy gigabytes of storage for only a few classes of geometries. The prospect of using such advanced numerical methods to calculate in real time only those K solutions actually needed to support a specific crack growth analysis is also tempting, but the stark reality is that the computational cost is still so high that the approach is not practical except for specific, critical application problems. Some thoughts are offered about alternative paradigms. Compounding approaches are some of the earliest building blocks of SIF development for more complex geometries. These approaches are especially attractive because of their very low computational cost and their conceptual robustness; they are, in some ways, an intriguing contrast and complement to the brute-force numerical methods. In recent years, researchers at NRC-Canada have published remarkable results showing how compounding approaches can be used to generate accurate solutions for very difficult problems. Examples are provided of some successes--and some limitations--using this approach. These closed-form, tabulated numerical, and compounding approaches have typically been used for simple remote loading with simple load paths to the crack. However, many significant cracks occur in complex stress gradient fields. This is a job for weight function (WF) methods, where the arbitrary stress distribution on the crack plane in the corresponding uncracked body (typically determined using FE methods) is used to determine K. Several significant recent advances in WF methods and solutions are highlighted here. Fueled by advanced 3D numerical methods, many new solutions have been generated for classic geometries such as surface and corner cracks with wide ranges of geometrical validity. A new WF formulation has also be developed for part-through cracks considering the arbitrary stress gradients in all directions in the crack plane (so-called bivariant solutions). Basic WF methods have recently been combined with analytical expressions for crack plane stresses to develop a large family of accurate SIF solutions for corner, surface, and through cracks at internal or external notches with very wide ranges of shapes, sizes, acuities, and offsets. Finally, WF solutions are much faster than FE or boundary element solutions, but can still be much slower than simple closed-form solutions, especially for bivariant solutions that can require 2D numerical integration. Novel pre-integration and dynamic tabular methods have been developed that substantially increase the speed of these advanced WF solutions. The practical utility of advanced SIF methods, including both WF and direct numerical methods, is greatly enhanced if the FM life analysis can be directly and efficiently linked with digital models of the actual structure or component (e.g., FE models for stress analysis). Two recent advances of this type will be described. One approach directly interfaces the FM life analysis with the FE model of the uncracked component (including stress results). Through a powerful graphical user interface, simplified FM life models can be constructed (and visualized) directly on the component model, with the computer collecting the geometry and stress gradient information needed for the life calculation. An even more powerful paradigm uses expert logic to automatically build an optimum simple fracture model at any and every desired location in the component model, perform the life calculation, and even generate fatigue crack growth life contour maps, all with minimal user intervention. This paradigm has also been extended to the automatic calculation of fracture risk, considering uncertainty or variability in key input parameters such as initial crack size or location. Another new integrated approach links the engineering life analysis, the component model, and a 3D numerical fracture analysis built with the same component model to generate a table of SIF values at a specific location that can then be employed efficiently to perform the life calculation. Some attention must be given to verification and validation (V&V) issues and challenges: how good are these SIF solutions, how good is good enough, and does anyone believe the life answer? It is important to think critically about the different sources of error or uncertainty and to perform V&V in a hierarchal, building-block manner. Some accuracy issues for SIF solutions, for example, may actually involve independent material behavior issues, such as constraint loss effects for crack fronts near component surfaces, and can be a source of confusion. Recommendations are proposed for improved V&V approaches. This presentation will briefly but critically survey the range of issues and advances mentioned above, with a particular view towards assembling an integrated approach that combines different methods to create practical tools for real-world design and analysis problems. Examples will be selectively drawn from the recent literature, from recent enhancements in the NASGRO and DARWIN computer codes, and from previously unpublished research

CraigMcClung, R.↗

NASA NTRS2012

Efficient Computation of Separation-Compliant Speed Advisories for Air Traffic Arriving in Terminal Airspace

A class of problems in air traffic management asks for a scheduling algorithm that supplies the air traffic services authority not only with a schedule of arrivals and departures, but also with speed advisories. Since advisories must be finite, a scheduling algorithm must ultimately produce a finite data set, hence must either start with a purely discrete model or involve a discretization of a continuous one. The former choice, often preferred for intuitive clarity, naturally leads to mixed-integer programs, hindering proofs of correctness and computational cost bounds (crucial for real-time operations). In this paper, a hybrid control system is used to model air traffic scheduling, capturing both the discrete and continuous aspects. This framework is applied to a class of problems, called the Fully Routed Nominal Problem. We prove a number of geometric results on feasible schedules and use these results to formulate an algorithm that attempts to compute a collective speed advisory, effectively finite, and has computational cost polynomial in the number of aircraft. This work is a first step toward optimization and models refined with more realistic detail.

Sadovsky, Alexander V.↗

NASA NTRSDate not supplied

Informing the Space Launch System Booster Separation Initial CFD Run Matrix with Observed Parametric Sensitivity

It currently requires significant computational cost to simulate the flow physics of the booster separation event on the Space Launch System. This comes from the large parametric space in which the event occurs, as pre-separation flight conditions and separated booster core-relative trajectories can vary. Functionally removing the risk of core-booster collision mandates careful assessment of the fluid dynamics in terms of several trajectory parameters. However, simulating the entire trajectory envelope is computationally intractable given the high parametric dimension. In order to reduce the uncertainty in the resulting low-parametric-resolution aerodynamic booster separation database, a data-driven approach was developed to select which breakpoints should be studied by simulations and experiments and which should be relegated to a regression-based interpolation procedure. This technique works by simulating cases where the flow physics are most sensitive to changes in the parameters and leaving the less parametrically sensitive regions for interpolation. The result is a booster separation run matrix whose computational cost is comparable to that of previous database generations but has lower interpolation errors.

SLS↗

NASA NTRSDate not supplied

Informing the Space Launch System Booster Separation Initial CFD Run Matrix with Observed Parametric Sensitivity

SLS↗

NASA NTRS1999

Design Optimization of Stiffened Panels with Postbuckling Constraints

The funding provided by the grant is used to complete the final stages of the development of a geometrically nonlinear analysis and design capability for the static response of compressively loaded prismatic plate structures. The analysis is based on the nonlinear finite strip method and is applicable for structures, such as stiffened panels or box columns, that can be modeled as assemblages of finite length plate strips. In an effort to reduce the computational cost of the nonlinear finite strip method, thus making it suitable for use in the design optimization environment, reduced basis techniques as described in various references by Noor are used in conjunction with the finite strip method. In addition, an efficient scheme for tracing the nonlinear equilibrium paths through highly nonlinear response curves was implemented. The new scheme, which is referred to as the normal flow algorithm, is based on homotopy methods, and is capable of negotiating highly nonlinear limit point instabilities with a reduced computational cost compared to the popular Ricks/Wempner and Chrisfield algorithms.

Guerdal, Zafer↗