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Extending Explicit Guidance Methods to Higher Dimensions, Additional Conditions, and Higher Order Integration

Guidance functions play critical roles in autonomy to steer vehicles and aircraft to the intended target or destination. Explicit guidance (E Guidance) solves the two-point boundary value problem with initial and final conditions for position and velocity. The original formulation of explicit guidance involves translational acceleration commands with a direct relationship to time, and it is possible to modify E Guidance for rotational acceleration. Other extensions for E Guidance include higher dimensions, additional conditions, and higher-order integration of the linearly independent E Guidance functions. The most promising extension involves higher-order integration of the E Guidance functions, but it may be physically impractical by initially moving away from the target. This paper provides a brief overview of some methods that extend E Guidance to higher dimensions, utilize additional conditions, or perform higher-order integration, and if they satisfy the two-point boundary value problem.

explicit guidance↗

NASA NTRS1990

The CFL condition for spectral approximations to hyperbolic initial-boundary value problems

The stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients are studied. Time is discretized by explicit multi-level or Runge-Kutta methods of order less than or equal to 3 (forward Euler time differencing is included), and spatial discretizations are studied by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. It is proved that these fully explicit spectral approximations are stable provided their time-step, delta t, is restricted by the CFL-like condition, delta t less than Const. N(exp-2), where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of approximate L(exp 2)-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. The result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.

Gottlieb, David↗

NASA NTRS1991

The CFL condition for spectral approximations to hyperbolic initial-boundary value problems

Gottlieb, David↗

NASA NTRS1999

Autolanding Trajectory Design for the X-34

An Autolanding I-load Program (ALIP) is developed to design unpowered autolanding trajectories for the X-34 Mach 8 vehicle. The trajectory is comprised of geometric flight segments that are based on the shuttle approach and landing design (steep glideslope, circular flare, exponential flare to shallow glideslope). Enforcing physical constraints such as loads, vertical descent rate, continuity and smoothness reduces the design problem to a two point boundary value problem with conditions on the initial and final dynamic pressure. Finding a solution required the development of trajectory simulation techniques that constrained the flight profile to a prescribed geometry. The design methodology can be extended beyond the autolanding flight regime by repeating the series of geometric segments and solving multiple two-point boundary values problems (one for each series). The techniques described in this paper facilitate rapid design of reference trajectories.

Barton, Gregg H.↗

DOE OSTI2024

TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems

Deep learning (DL), in particular deep neural networks, by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are often not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates for missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, hence promoting better generalization. This paper develops a model-constrained DL approach and its variant TNet—a Tikhonov neural network—which are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solutions with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for one-dimensional (1D) deconvolution, 5 for an inverse 2D heat conductivity problem, 100 for inverse initial conditions for a time-dependent 2D Burgers’s equation, and 50 for inverse initial conditions for 2D Navier–Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. Furthermore, this is possible owing to the model-constrained term, replications, and randomization.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1975

Spherical means of solutions of partial differential equations in a conical region

The spherical means of the solutions of a linear partial differential equation Lu = f in a conical region are studied. The conical region is bounded by a surface generated by curvilinear xi lines and by two truncating xi surfaces. The spherical mean is the average of u over a constant xi surface. Conditions on the linear differential operator, L, and on the orthogonal coordinates xi, eta, and zeta are established so that the problem for the determination of the spherical mean of the solution subjected to the appropriate boundary and initial conditions can be reduced to a problem with only one space variable. Conditions are then established so that the spherical mean of the solution in one conical region will be proportional to that of a known solution in another conical region. Applications to various problems of mathematical physics and their physical interpretations are presented.

Ting, L.↗

NASA NTRS1975

Generalized Du Fort-Frankel methods for parabolic initial boundary value problems

The Du Fort-Frankel difference scheme is generalized to difference operators of arbitrary high order accuracy in space and to arbitrary order of the parabolic differential operator. Spectral methods can also be used to approximate the spatial part of the differential operator. The scheme is explicit, and it is unconditionally stable for the initial value problem. Stable boundary conditions are given for two different fourth order accurate space approximations.

Gottlieb, D.↗

NASA NTRS1976

Generalized Du Fort-Frankel methods for parabolic initial-boundary value problems

Gottlieb, D.↗

NASA NTRS1990

Minimum fuel trajectory for the aerospace-plane

An overall trajectory for a single-stage-to-orbit vehicle with an initial weight of 234 tons is calculated, and four different propulsion models including turbojet, ramjet, scramjet, and rocket are considered. First, the atmospheric flight in the thicker atmosphere is discussed with emphasis on trajectory optimization, optimization problem, aerodynamic problem, propulsion model, and initial conditions. The performance of turbojet and ramjet-scramjet engines is analyzed; and then the flight to orbit is assessed from the optimization point of view. It is shown that roll modulation saves little during the trajectory, and the combined application of airbreathing propulsion and aerodynamic lift is suggested.

Breakwell, John V.↗

NASA NTRS1972

Numerical method for the solution of large systems of differential equations of the boundary layer type

A numerical method for the solution of large systems of nonlinear differential equations of the boundary-layer type is described. The method is a modification of the technique for satisfying asymptotic boundary conditions. The present method employs inverse interpolation instead of the Newton method to adjust the initial conditions of the related initial-value problem. This eliminates the so-called perturbation equations. The elimination of the perturbation equations not only reduces the user's preliminary work in the application of the method, but also reduces the number of time-consuming initial-value problems to be numerically solved at each iteration. For further ease of application, the solution of the overdetermined system for the unknown initial conditions is obtained automatically by applying Golub's linear least-squares algorithm. The relative ease of application of the proposed numerical method increases directly as the order of the differential-equation system increases. Hence, the method is especially attractive for the solution of large-order systems. After the method is described, it is applied to a fifth-order problem from boundary-layer theory.

Green, M. J.↗

NASA NTRS1976

First order impulsive solutions

A mathematically rigorous derivation is given of first order corrections to multi-impulse approximations to the solutions to space flight optimization problems with bang-bang control. The rocket was subjected to an inverse square gravitational force and to a thrust force with constant magnitude. The mass decreased linearly with time. An optimal impulsive solution was obtained for a problem with given initial and final conditions. The method was then used to obtain first-order corrections to the initial values of the costate variables. Indications are given on how the theory may be extended to higher order corrections. The theory was applied to intercept and rendezvous problems.

Andrus, J. F.↗

NASA NTRS1999

Inversion of Gravity Fields From the Spacecraft Orbital Data Using an Adjoint Operator Approach

In perturbation approximation, the forward problem of orbital dynamics (equations with initial conditions) is linear with respect to variations of coordinates and/or velocities of the spacecraft and to corresponding variations of the gravity field in the models used. The linear operator adjoint to the linear operator of such forward problem turns out to be instrumental in inversion of differences between observed and predicted coordinates/velocities in terms of the updates of harmonics in the initial gravity field model. Based on this approach, the solution of resulting adjoint problem of orbital dynamics can be used to directly evaluate the matrix of partial derivatives of observable differences with respect to the gravity field harmonics. General discussion of the adjoint problem of orbital dynamics is given and an example of a mathematical formalism for the practical retrieval algorithm is presented.

Ustinov, E. A.↗

DOE OSTI2024

Robust wind farm layout optimization

Wake interactions in wind farms cause losses in annual energy production (AEP) on the order of 10%. Wind farm designers optimize the layout of the farm to mitigate wake losses, especially in the dominant site-specific wind directions. As wind turbines and wind farms grow in scale, optimization becomes more complex. Offshore wind farms regularly comprise more than 100 wind turbines and are characterized by complex boundaries due to shipping lanes, neighboring wind farms, and other constraints. Layout optimization methods are broadly split between gradient-based and gradient-free approaches. Gradient-based approaches can converge quickly and perform well for smaller, academic problems but are often sensitive to initial conditions and tuning parameters and require expert knowledge to use. On the other hand, gradient-free approaches can be more robust to problem complexities. We present a robust layout optimization approach based on a random search algorithm. The algorithm is intended for those who are not optimization experts and has few tuning parameters that need specification to achieve satisfactory results. Unlike off-the-shelf methods, which use generally available, non-domain-specific optimization routines that accept as inputs an optimization function and constraint definitions, this approach takes advantage of the relative computational costs of the different evaluations by evaluating cheaper computations first (boundary and minimum distance constraints) and running expensive AEP evaluations only if all other checks pass. Moreover, an outer genetic algorithm allows multiple solutions to evolve in parallel, enabling rapid solution development on high-performance computers. We discuss the relative ease of selecting necessary tuning parameters and demonstrate the efficacy of the genetic random search on a complex layout problem consisting of placing 70 turbines in a nonconvex and unconnected boundary region.

17 WIND ENERGY↗

DOE OSTI2021

Phase field benchmark problems for nucleation

In this work, we present nucleation phase field model benchmark problems, expanding on our previous benchmark problems on diffusion, precipitation, dendritic growth, linear elasticity, fluid flow and electrochemistry. Nucleation is the process in which either a new thermodynamic phase or a new structure is created, such as solidification from the melt, or self-assembly of particulates. Based on where the nucleation occurs, it can be divided into two main categories: homogeneous nucleation and heterogeneous nucleation. In the first nucleation benchmark problem, we focus on homogeneous nucleation for both single seed under different initial conditions and multiple seeds. The second nucleation benchmark problem focuses on athermal heterogeneous nucleation and nucleation behavior near the free growth limit with different undercooling driving forces.

36 MATERIALS SCIENCE↗

NASA NTRS1976

Thermal instability in post-flare plasmas

The cooling of post-flare plasmas is discussed and the formation of loop prominences is explained as due to a thermal instability. A one-dimensional model was developed for active loop prominences. Only the motion and heat fluxes parallel to the existing magnetic fields are considered. The relevant size scales and time scales are such that single-fluid MHD equations are valid. The effects of gravity, the geometry of the field and conduction losses to the chromosphere are included. A computer code was constructed to solve the model equations. Basically, the system is treated as an initial value problem (with certain boundary conditions at the chromosphere-corona transition region), and a two-step time differencing scheme is used.

Antiochos, S. K.↗

NASA NTRS1989

Heating and acceleration of heavy ions during solar flares

The heating of the various ion species during solar flares is examined in an attempt to develop a unified model incorporating the enrichments in abundances seen by ISEE 3 and some of the characteristics of the soft X-ray line emissions. Problems associated with identifying the initial conditions and the driving mechanisms for the particle acceleration are addressed. The acceleration of heated plasma ions as they propagate upward into the corona is studied via particle simulations.

Winglee, R. M.↗

NASA NTRS2007

Simulations of Boundary-Layer Transition

For incompressible benchmark flows, we have demonstrated the capability of the parabolized stability equations (PSE) to simulate the transition process in excellent agreement with microscopic experiments and direct Navier-Stokes simulations at modest computational cost. Encouraged by these results, we have developed the PSE methodology of three-dimensional boundary-layers in general curvilinear coordinates for the range from low to hypersonic speeds, and for both linear and nonlinear problems. For given initial and boundary conditions, the approach permits simulations from receptivity through linear and secondary instabilities into the late stages of transition where significant changes in skin friction and heat transfer coefficients occur. We have performed transition simulations for a variety of two- and three-dimensional similarity solutions and for realistic flows over swept wings at subsonic and supersonic speeds, the pressure ans suction side of turbine blades at low and medium turbulence levels, and over a blunt cone at Mach number Ma = 8. We present selected results for different transition mechanisms with emphasis on the late stage of transition and the evolution of wall-shear stress and heat transfer.

Herbert, Thorwald↗

NASA NTRS1979

Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems

Finite difference methods for the numerical solution of mixed initial boundary value problems for hyperbolic equations are studied. The reported investigation has the objective to develop a technique for the total error analysis of a finite difference scheme, taking into account initial approximations, boundary conditions, and interior approximation. Attention is given to the Cauchy problem and the initial approximation, the homogeneous problem in an infinite strip with inhomogeneous boundary data, the reflection of errors in the boundaries, and two different boundary approximations for the leapfrog scheme with a fourth order accurate difference operator in space.

Skollermo, G.↗