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A Lagrange Multiplier Partitioned Scheme for Coupling Reduced Order Models.
Abstract not provided.
Lagrange thermodynamic potential and intrinsic variables for He-3 He-4 dilute solutions
For a two-fluid model of dilute solutions of He-3 in liquid He-4, a thermodynamic potential is constructed that provides a Lagrangian for deriving equations of motion by a variational procedure. This Lagrangian is defined for uniform velocity fields as a (negative) Legendre transform of total internal energy, and its primary independent variables, together with their thermodynamic conjugates, are identified. Here, similarities between relations in classical physics and quantum statistical mechanics serve as a guide for developing an alternate expression for this function that reveals its character as the difference between apparent kinetic energy and intrinsic internal energy. When the He-3 concentration in the mixtures tends to zero, this expression reduces to Zilsel's formula for the Lagrangian for pure liquid He-4. An investigation of properties of the intrinsic internal energy leads to the introduction of intrinsic chemical potentials along with other intrinsic variables for the mixtures. Explicit formulas for these variables are derived for a noninteracting elementary excitation model of the fluid. Using these formulas and others also derived from quantum statistical mechanics, another equivalent expression for the Lagrangian is generated.
Searching for life with the Terrestrial Planet Finder: Lagrange point options for a formation flying interferometer
In this paper, we describe the mission design for TPF assuming a distributed spacecraft concept using formation flight around both a halo orbit around L2 as well as a heliocentric orbit. Although the mission architecture is still under study, the next two years will include study of four design cncepts and a downselect to two concepts around 2005.
Remote Sensing of Daytime Water Leaving Reflectances of Oceans and Large Inland Lakes from EPIC onboard the DSCOVR Spacecraft at Lagrange-1 Point
The NASA’s Earth Polychromatic Imaging Camera (EPIC) on board the Deep Space Climate Observatory (DSCOVR) satellite has been making multiple observations of the entire sunlit Earth in a given day from the Sun-Earth Lagrangian L1 point since the summer of 2015. EPIC contains 10 narrow channels in the 317–780 nm solar spectral range. The data acquired with EPIC have already been used in a variety of scientific investigations, including the study of the global ozone levels, aerosol index and aerosol optical depth, UV reflectivity of clouds over land and ocean, cloud height over land and ocean, and vegetation indices. In this article, we report that EPIC data, particularly for the data measured with narrow channels centered near 443, 551, and 680 nm, can also have important applications in remote sensing of ocean color in different geographical regions. We have modified a version of a multi-channel atmospheric correction algorithm for Moderate Resolution Imaging Spectro Radiometer (MODIS) ocean color applications and adapted the algorithm for processing EPIC data. We present three case studies on water leaving reflectance retrievals from EPIC data acquired over a large turbid river, inland lakes, and oceans. We conclude that a future ocean color instrument on board a satellite at the L1 point, which provides continuous view of the full sunlit disk of the Earth, will complement and extend ocean color observations with the low Earth observing polar orbital and geostationary satellite instruments in both the spatial and time domains.
Space Weather Next Lagrange 5 (L5) Project
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First Observations of Volcanic Eruption Clouds From the L1 Earth‐Sun Lagrange Point by DSCOVR/EPIC
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The discrete Green's function paradigm for two-way coupled Euler-Lagrange simulation
We outline a methodology for the simulation of particle-laden flows whereby the dispersed and fluid phases are two-way coupled. The drag force which couples fluid and particle momentum depends on the undisturbed fluid velocity at the particle location, and this latter quantity requires modelling. We demonstrate that the undisturbed fluid velocity, in the low particle Reynolds number limit, can be related exactly to the discrete Green's function of the discrete Stokes equations. The method is general in that it can be extended to other partial differential equations which may be associated with particle-laden flows, such as the thermal energy equation or Maxwell's equations. In this work, we demonstrate the method of discrete Green's functions by obtaining these functions for the Navier-Stokes equations at low particle Reynolds number in a two-plane channel geometry. We perform verification at low and finite particle Reynolds number for the case of a point-particle settling under gravity parallel to a plane wall, for different wall normal separations. In comparing to other point-particle schemes the discrete Green's function approach is the most robust at low particle Reynolds number, accurate at all wall-normal separations and is the most accurate in the near wall region at finite Reynolds number. We discuss how the accuracy away from the wall at finite Reynolds number could be improved by appealing to Oseen-like discrete Green's functions. Finally we demonstrate that the discrete Green's function approach can have important implications on statistics of particle-laden turbulent channel flow.
Improved force models for Euler - Lagrange computations
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The Newton Modified Barrier Method for QP Problems
The Modified Barrier Functions (MBF) have elements of both Classical Lagrangians (CL) and Classical Barrier Functions (CBF). The MBF methods find an unconstrained minimizer of some smooth barrier function in primal space and then update the Lagrange multipliers, while the barrier parameter either remains fixed or can be updated at each step. The numerical realization of the MBF method leads to the Newton MBF method, where the primal minimizer is found by using Newton's method. This minimizer is then used to update the Lagrange multipliers. In this paper, we examine the Newton MBF method for the Quadratic Programming (QP) problem. It will be shown that under standard second-order optimality conditions, there is a ball around the primal solution and a cut cone in the dual space such that for a set of Lagrange multipliers in this cut cone, the method converges quadratically to the primal minimizer from any point in the aforementioned ball, and continues to do so after each Lagrange multiplier update. The Lagrange multipliers remain within the cut cone and converge linearly to their optimal values. Any point in this ball will be called a "hot start". Starting at such a "hot start", at most Omicron(1n 1n epsilon(exp -1)) Newton steps are sufficient to perform the primal minimization which is necessary for the Lagrange multiplier update. Here, epsilon > 0 is the desired accuracy. Because of the linear convergence of the Lagrange multipliers, this means that only Omicron(1n epsilon(exp -1))omicron(ln 1n epsilon(exp-1)) Newton steps are required to reach an epsilon-approximation to the solution from any "hot start". In order to reach the "hot start", one has to perform Omicron(square root(m) 1n C) Newton steps, where m characterizes the size of the problem and C > 0 is the condition number of the QP problem. This condition number will be characterized explicitly in terms of key parameters of the QP problem, which in turn depend on the input data and the size of the problem.
Modified Interior Distance Functions (Theory and Methods)
In this paper we introduced and developed the theory of Modified Interior Distance Functions (MIDF's). The MIDF is a Classical Lagrangian (CL) for a constrained optimization problem which is equivalent to the initial one and can be obtained from the latter by monotone transformation both the objective function and constraints. In contrast to the Interior Distance Functions (IDF's), which played a fundamental role in Interior Point Methods (IPM's), the MIDF's are defined on an extended feasible set and along with center, have two extra tools, which control the computational process: the barrier parameter and the vector of Lagrange multipliers. The extra tools allow to attach to the MEDF's very important properties of Augmented Lagrangeans. One can consider the MIDFs as Interior Augmented Lagrangeans. It makes MIDF's similar in spirit to Modified Barrier Functions (MBF's), although there is a fundamental difference between them both in theory and methods. Based on MIDF's theory, Modified Center Methods (MCM's) have been developed and analyzed. The MCM's find an unconstrained minimizer in primal space and update the Lagrange multipliers, while both the center and the barrier parameter can be fixed or updated at each step. The MCM's convergence was investigated, and their rate of convergence was estimated. The extension of the feasible set and the special role of the Lagrange multipliers allow to develop MCM's, which produce, in case of nondegenerate constrained optimization, a primal and dual sequences that converge to the primal-dual solutions with linear rate, even when both the center and the barrier parameter are fixed. Moreover, every Lagrange multipliers update shrinks the distance to the primal dual solution by a factor 0 less than gamma less than 1 which can be made as small as one wants by choosing a fixed interior point as a 'center' and a fixed but large enough barrier parameter. The numericai realization of MCM leads to the Newton MCM (NMCM). The approximation for the primal minimizer one finds by Newton Method followed by the Lagrange multipliers update. Due to the MCM convergence, when both the center and the barrier parameter are fixed, the condition of the MDF Hessism and the neighborhood of the primal ninimizer where Newton method is 'well' defined remains stable. It contributes to both the complexity and the numerical stability of the NMCM.
Comparison of Real-Time Pressure Rail Selection Algorithms for the Hybrid Hydraulic Electric Architecture: Case Study on a Track Loader
Abstract The hybrid hydraulic electric architecture (HHEA) seeks to combine the high power/torque/force density of hydraulics with the efficiency of electric machines. A set of common pressure rails is used to provide a majority of the power and this power is modulated by small electric machines to provide precise control for the operator. The HHEA has been studied in previous work using off-line dynamic programming optimization to determine energy efficient pressure rail selections, but this approach requires drive cycle information apriori. A Lagrange multiplier method has also been investigated where a set of gains (Lagrange multipliers) are optimized off-line with the idea the these gains, once determined, could be used for real-time operation. In this work, three new real-time pressure rail selection algorithms that do not require future drive cycle information are investigated; greedy, torque minimizing, and thresholding. The greedy control is found to only use 1% more energy than the globally optimal dynamic programming solution; but a model of energy loss is required.
Analytical Dynamics and Nonrigid Spacecraft Simulation
Application to the simulation of idealized spacecraft are considered both for multiple-rigid-body models and for models consisting of combination of rigid bodies and elastic bodies, with the elastic bodies being defined either as continua, as finite-element systems, or as a collection of given modal data. Several specific examples are developed in detail by alternative methods of analytical mechanics, and results are compared to a Newton-Euler formulation. The following methods are developed from d'Alembert's principle in vector form: (1) Lagrange's form of d'Alembert's principle for independent generalized coordinates; (2) Lagrange's form of d'Alembert's principle for simply constrained systems; (3) Kane's quasi-coordinate formulation of D'Alembert's principle; (4) Lagrange's equations for independent generalized coordinates; (5) Lagrange's equations for simply constrained systems; (6) Lagrangian quasi-coordinate equations (or the Boltzmann-Hamel equations); (7) Hamilton's equations for simply constrained systems; and (8) Hamilton's equations for independent generalized coordinates.
The TPF Mission at L2
The Terrestrial Planet Finder (TPF) is one of the center pieces of NASA's Origins Program. The goal of TPF is to identify terrestrial planets around stars nearby the Sun. For this purpose, a space-based interferometer with a baseline of approximately 100 m is required. To achieve such a large baseline, a distributed system of five spacecraft flying in formation is an efficient approach. Since the TPF instruments need a cold and stable environment, a halo orbit about 4 is ideal. First, we describe formation flight near the Lagrange point is feasible for the TPF mission. Second, we propose a novel approach for human servicing of Lagrange point missions by placing a Lunar service station in an Lunar L1 orbit. The TPF spacecraft can be transferred to a Lunar L1 orbit in a few days and requires relatively little delta-V. This efficient transfer results from the system of low energy pathways connecting the entire Solar System generated by the Lagrange points. The halo orbits are the portals of this Interplanetary . A Lunar Station at the L,portal, in addition to servicing missions from the Sun-Earth Lagrange points, may play an even more important role in the future development of space.
Scalable preconditioning for the stabilized contact mechanics problem
We present a family of preconditioning strategies for the contact problem in fractured and faulted porous media. We combine low-order continuous finite elements to simulate the bulk deformation with piecewise constant Lagrange multipliers to impose the frictional contact constraints. This formulation is not uniformly inf-sup stable and requires stabilization. We improve previous work by Franceschini et al. (2020) by introducing a novel jump stabilization technique that requires only local geometrical and mechanical properties. We then design scalable preconditioning strategies that take advantage of the block structure of the Jacobian matrix using a physics-based partitioning of the unknowns by field type, namely displacement and Lagrange multipliers. The key to the success of the proposed preconditioners is a pseudo-Schur complement obtained by eliminating the Lagrange multiplier degrees of freedom, which can then be efficiently solved using an optimal multigrid method. Numerical results, including complex real-world problems, are presented to illustrate theoretical properties, scalability and robustness of the preconditioner. A comparison with other approaches available in the literature is also provided.
Discovering exact, gauge-invariant, local energy-momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds
Gyrokinetic theory is arguably the most important tool for numerical studies of transport physics in magnetized plasmas. However, exact local energy-momentum conservation laws for the electromagnetic gyrokinetic system have not been found despite the continuous effort. Without such local conservation laws, energy and momentum can be instantaneously transported across spacetime, which is unphysical and casts doubt on the validity of numerical simulations based on the gyrokinetic theory. The standard Noether procedure for deriving conservation laws from corresponding symmetries does not apply to gyrokinetic systems because the gyrocenters and electromagnetic fields reside on different manifolds. To overcome this difficulty, we develop a high-order field theory on heterogeneous manifolds for classical particle-field systems and apply it to derive exact, local conservation laws, in particular the energy-momentum conservation laws, for the electromagnetic gyrokinetic system. A weak Euler-Lagrange equation is established to replace the standard Euler-Lagrange equation for the particles. It is discovered that an induced weak Euler-Lagrange current enters the local conservation laws. And it is the new physics captured by the high-order field theory on heterogeneous manifolds. A recently developed gauge-symmetrization method for high-order electromagnetic field theories using the electromagnetic displacement-potential tensor is applied to render the derived energy-momentum conservation laws electromagnetic gauge-invariant.