A Use of Lighthill's Generalized Functions in the Solution of Optimal Control Problems
Lighthill generalized functions applied to solving two-point boundary value problems in optimal control theory
Armstrong, E. S., Jr.↗
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Control optimization of a lifting body entry problem by an improved and a modified method of perturbation function
A study of the solution problem of a complex entry optimization was studied. The problem was transformed into a two-point boundary value problem by using classical calculus of variation methods. Two perturbation methods were devised. These methods attempted to desensitize the contingency of the solution of this type of problem on the required initial co-state estimates. Also numerical results are presented for the optimal solution resulting from a number of different initial co-states estimates. The perturbation methods were compared. It is found that they are an improvement over existing methods.
Garcia, F., Jr.↗
Excitation of instability waves in a two-dimensional shear layer by sound
The excitation of instability waves in a plane compressible shear layer by sound waves is studied. The problem is formulated mathematically as an inhomogeneous boundary-value problem. A general solution for abitrary incident sound wave is found by first constructing the Green's function of the problem. Numerical values of the coupling constants between incident sound waves and excited instability waves for a range of flow Mach number are calculated. The effect of the angle of incidence in the case of a beam of acoustic waves is analyzed. It is found that for moderate subsonic Mach numbers a narrow beam aiming at an angle between 50 to 80 deg to the flow direction is most effective in exciting instability waves.
Tam, C. K. W.↗
Vertical slender jets
The shape of a vertical slender jet of fluid falling steadily under the force of gravity is studied. The problem is formulated as a nonlinear free boundary-value problem for the potential. Surface tension effects are neglected. The use of perturbation expansions results in a system of equations that can be solved by an efficient numerical procedure. Computations were made for jets issuing from orifices in various shapes including an ellipse, a rectangle, and an equilateral triangle. Computational results are presented illustrating the propagation of discontinuities and the formation of thin sheets of fluid.
Geer, J. F.↗
Hybrid near-optimal aeroassisted orbit transfer plane change trajectories
In this paper, a hybrid methodology is used to determine optimal open loop controls for the atmospheric portion of the aeroassisted plane change problem. The method is hybrid in the sense that it combines the features of numerical collocation with the analytically tractable portions of the problem which result when the two-point boundary value problem is cast in the form of a regular perturbation problem. Various levels of approximation are introduced by eliminating particular collocation parameters and their effect upon problem complexity and required number of nodes is discussed. The results include plane changes of 10, 20, and 30 degrees for a given vehicle.
Calise, Anthony J.↗
Approximate mode decomposition in a compressible anisotropic plasma.
Approximate mode decomposition for treating boundary value problems for uniform compressible anisotropic plasma
ANISOTROPIC FLUID↗
Analysis of the weighted shifted boundary method for the Poisson and Stokes problems
The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by properly shifting the location and values of the boundary conditions. This avoids integration over cut cells and the associated implementation issues. Recently, the Weighted SBM (WSBM) was proposed for the Navier-Stokes equations with free surfaces and the Stokes flow with moving boundaries. The attribute “weighted” in the name WSBM stems from the fact that its variational form is weighted with the elemental volume fraction of active fluid. The motivation for the development of the WSBM was the preservation of the volume of active fluid to a higher degree of accuracy, which in turn resulted in improved stability and robustness characteristics in moving-boundary, time-dependent simulations. In this article, we present the numerical analysis of the WSBM formulations for the Poisson and Stokes problems. We give mathematical conditions under which the bilinear forms defining the discrete variational formulations are uniformly coercive (Poisson problem) or inf-sup stable (Stokes problem). By these results, stability and optimal convergence is proven in the natural norm; L2-error estimates can also be derived.
Approximate domain boundaries↗
Learning Only on Boundaries: A Physics-Informed Neural Operator for Solving Parametric Partial Differential Equations in Complex Geometries
Recently, deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we present a novel physics-informed neural operator method to solve parameterized boundary value problems without labeled data. By reformulating the PDEs into boundary integral equations (BIEs), we can train the operator network solely on the boundary of the domain. This approach reduces the number of required sample points from $O(N^d)$ to $O(N^{d-1}$), where $d$ is the domain’s dimension, leading to a significant acceleration of the training process. Additionally, our method can handle unbounded problems, which are unattainable for existing physics-informed neural networks (PINNs) and neural operators. Finally, our numerical experiments show the effectiveness of parameterized complex geometries and unbounded problems.
97 MATHEMATICS AND COMPUTING↗
Unsteady, combined radiation and conduction in an absorbing, scattering, and emitting medium.
The transient cooldown of a gray, absorbing, isotropic scattering, emitting, and conducting medium bounded by gray, diffusely emitting and reflecting parallel plates is considered. Numerical solutions are obtained for the initial boundary-value problem with a discontinuous decrease in temperature at one boundary. The quasi-steady equation of radiative transfer is solved using Gaussian quadrature and a matrix eigenvector technique together with explicit numerical solution of the unsteady energy equation. Temperature and energy flux distributions are presented for variations of optical thickness, boundary emissivity, albedo, and conduction-radiation parameter.
Weston, K. C.↗
Hybrid finite element methods
The purpose of this paper is to show how Lagrange multipliers can be used with finite elements to achieve a number of desirable properties in the underlying approximation. For elliptic boundary value problems, variational principles can be developed in which all boundary conditions are natural. In fluid flow problems, one can endow the approximations with physically essential conservation laws.
Fix, G. M.↗
Transformation of Three-Dimensional Regions onto Rectangular Regions by Elliptic Systems
A transformation method is developed which may be used to solve various types of boundary value problems on three-dimensional regions with an arbitrary boundary. The implementation of the method is illustrated in the solution of a potential flow problem. All computations are performed on a cubic mesh in a rectangular region.
Mastin, C. Wayne↗
Parallel architectures for iterative methods on adaptive, block structured grids
A parallel computer architecture well suited to the solution of partial differential equations in complicated geometries is proposed. Algorithms for partial differential equations contain a great deal of parallelism. But this parallelism can be difficult to exploit, particularly on complex problems. One approach to extraction of this parallelism is the use of special purpose architectures tuned to a given problem class. The architecture proposed here is tuned to boundary value problems on complex domains. An adaptive elliptic algorithm which maps effectively onto the proposed architecture is considered in detail. Two levels of parallelism are exploited by the proposed architecture. First, by making use of the freedom one has in grid generation, one can construct grids which are locally regular, permitting a one to one mapping of grids to systolic style processor arrays, at least over small regions. All local parallelism can be extracted by this approach. Second, though there may be a regular global structure to the grids constructed, there will be parallelism at this level. One approach to finding and exploiting this parallelism is to use an architecture having a number of processor clusters connected by a switching network. The use of such a network creates a highly flexible architecture which automatically configures to the problem being solved.
Gannon, D.↗
Robust neighboring extremal guidance for the advanced launch system
With the availability of modern flight computers, realtime neighboring extremal guidance seems feasible. To overcome sensitivity to unknown system parameters and environmental uncertainties, a robust neighboring extremal guidance scheme is proposed. About the optimal trajectory, the accessory problem in the calculus of variations is formed, generating a quadratic cost criterion in the perturbed states and controls. By formulating a disturbance attenuation problem based upon the second variation cost criterion, a differential game is formulated. The game theoretic cost criterion is minimized with respect to the perturbed control but maximized with respect to the unknown parameters in the linearized dynamics. The resulting differential game problem gives rise to a two-point boundary-value problem solved using the sweep method. The sweep method solution provides a linear robust neighboring extremal guidance scheme that is applied to the Advanced Launch System.
Bain, John↗
Low-Thrust Control of a Lunar Mapping Orbit
A method is presented for generating and maintaining a lunar mapping orbit using continuous low-thrust hardware. Optimal control theory is used to maintain a lunar orbit that is low-altitude, near-polar, and Sun-synchronous; three typical requirements for a successful lunar mapping mission. The analysis of the optimal control problem leads to the commonly seen two-point boundary value problem, which is solved using a simple indirect shooting algorithm. Simulations are presented for a 50-day mapping duration, in which it is shown that a very tight control is achieved with thrust levels below 1 N for a 1000 kg spacecraft. A straightforward approach for using the method presented to compute missions of any duration is also discussed.
Harl, Nathan↗
Inertial dynamics of an interface with interfacial mass flux: Stability and flow fields’ structure, inertial stabilization mechanism, degeneracy of Landau’s solution, effect of energy fluctuations, and chemistry-induced instabilities
This work focuses on the long-standing problem of inertial dynamics of an interface with interfacial mass flux and reports new mechanisms for the interface stabilization and destabilization. The interface is a phase boundary separating fluids of different densities and having interfacial mass flux. To analyze the interface dynamics from a far field, we develop and apply the general matrix method to rigorously solve the boundary value problem involving the governing equations in the fluid bulk and the boundary conditions at the interface and at the outside boundaries of the domain. We find the fundamental solutions for the linearized system of equations and analyze the interplay of interface stability with flow fields’ structure by directly linking rigorous mathematical attributes to physical observables. We find that the interface is stable when the dynamics conserves the fluxes of mass, momentum, and energy; the stabilization is due to an inertial mechanism causing small oscillations of the interface velocity. In the classic Landau’s dynamics, the postulate of perfect constancy of the interface velocity leads to the development of Landau–Darrieus instability. This destabilization is also linked to the imbalance of the perturbed energy at the interface. The classic Landau’s solution is found to have degeneracy; lifting of the degeneracy may lead to singularity and self-similar dynamics. Our results compare well with traditional theories of combustion and propose new experiments to study the dynamics of the interface and the flow fields in combustible systems. We further conduct reactive molecular dynamics simulations to elucidate the complexity of chemical processes, to study the destabilizing effect of energy fluctuations on the interface stability, and to illustrate the chemistry-induced instabilities. In summary, we identify the extreme sensitivity of the interface dynamics to the interfacial boundary conditions, including the formal properties of fundamental solutions and the qualitative and quantitative properties of the flow fields. This provides new opportunities for studies, diagnostics, and control of multiphase flows in a broad range of processes in nature and technology.
42 ENGINEERING↗
The second-generation Shifted Boundary Method and its numerical analysis
Recently, the Shifted Boundary Method (SBM) was proposed within the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we present enhanced variational SBM formulations for the Poisson and Stokes problems with improved flexibility and robustness. These simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations. First, we show that these new SBM implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive. Second, we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at Dirichlet boundaries, therefore avoiding the calibration of an additional stabilization parameter. Finally, we prove enhanced L 2 -estimates without the cumbersome assumption – of earlier proofs – that the surrogate domain is convex. Instead we rely on a conventional assumption that the boundary of the true domain is smooth, which can also be replaced by requiring convexity of the true domain. The aforementioned improvements open the way to a more general and efficient implementation of the Shifted Boundary Method, particularly in complex three-dimensional geometries. We complement these theoretical developments with numerical experiments in two and three dimensions.
42 ENGINEERING↗
Monotone iterations for nonlinear equations with application to Gauss-Seidel methods.
Monotone iterations for nonlinear elliptic differential equations in boundary-value problems applied to Gauss-Seidel methods
BOUNDARY VALUE↗
The computation of optimal control programmes using a modified successive sweep method.
A second-order method for numerically solving control optimization problems has been developed. The method, referred to as the modified sweep method (MSM), differs from the successive sweep method (SSM) proposed by McReynolds and Bryson (1965) in that the conditions for local control optimality are used to determine the control as an explicit function of the state variables and time. The control is eliminated from the problem and the solution to the resulting two-point boundary value problem can be obtained by linear perturbation methods. The modified sweep method proposed here uncouples the perturbation equations for the state variables and the Lagrange multipliers by using a generalized matrix-Riccati transformation of variables. The resulting algorithm for the numerical iteration process is concerned with determining the initial values of a set of Lagrange multipliers rather than correcting a numerical control programme over the entire time interval of interest.
Colunga, D.↗