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Results for “variational inequality problems”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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33 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1991

System characterization of positive real conditions

Necessary and sufficient conditions for positive realness in terms of state space matrices are presented under the assumption of complete controllability and complete observability of square systems with independent inputs. As an alternative to the positive real lemma and to the s-domain inequalities, these conditions provide a recursive algorithm for testing positive realness which result in a set of simple algebraic conditions. By relating the positive real property to the associated variational problem, a unified derivation of necessary and sufficient conditions for optimality of both singular and nonsingular problems is derived.

Wang, Q.↗
NASA NTRS1990

System characterization of positive real conditions

Necessary and sufficient conditions for positive realness in terms of state space matrices are presented under the assumption of complete controllability and complete observability of square systems with independent inputs. As an alternative to the positive real lemma and to the s-domain inequalities, these conditions provide a recursive algorithm for testing positive realness which results in a set of simple algebraic conditions. By relating the positive real property to the associated variational problem, a unified derivation of necessary and sufficient conditions for optimality of both singular and nonsingular problems is derived.

Wang, Q.↗
NASA NTRS1973

A result on differential inequalities and its application to higher order trajectory derivatives

A result on differential inequalities is obtained by considering the adjoint differential equation of the variational equation of the right side of the inequality. The main theorem is proved using basic results on differentiability of solutions with respect to initial conditions. The result is then applied to the problem of determining solution behavior using comparison techniques.

Gunderson, R. W.↗
NASA NTRS1993

State-constrained booster trajectory solutions via finite elements and shooting

This paper presents an extension of a FEM formulation based on variational principles. A general formulation for handling internal boundary conditions and discontinuities in the state equations is presented, and the general formulation is modified for optimal control problems subject to state-variable inequality constraints. Solutions which only touch the state constraint and solutions which have a boundary arc of finite length are considered. Suitable shape and test functions are chosen for a FEM discretization. All element quadrature (equivalent to one-point Gaussian quadrature over each element) may be done in closed form. The final form of the algebraic equations is then derived. A simple state-constrained problem is solved. Then, for a practical application of the use of the FEM formulation, a launch vehicle subject to a dynamic pressure constraint (a first-order state inequality constraint) is solved. The results presented for the launch-vehicle trajectory have some interesting features, including a touch-point solution.

Bless, Robert R.↗
NASA NTRS1991

Method and apparatus for configuration control of redundant robots

A method and apparatus to control a robot or manipulator configuration over the entire motion based on augmentation of the manipulator forward kinematics is disclosed. A set of kinematic functions is defined in Cartesian or joint space to reflect the desirable configuration that will be achieved in addition to the specified end-effector motion. The user-defined kinematic functions and the end-effector Cartesian coordinates are combined to form a set of task-related configuration variables as generalized coordinates for the manipulator. A task-based adaptive scheme is then utilized to directly control the configuration variables so as to achieve tracking of some desired reference trajectories throughout the robot motion. This accomplishes the basic task of desired end-effector motion, while utilizing the redundancy to achieve any additional task through the desired time variation of the kinematic functions. The present invention can also be used for optimization of any kinematic objective function, or for satisfaction of a set of kinematic inequality constraints, as in an obstacle avoidance problem. In contrast to pseudoinverse-based methods, the configuration control scheme ensures cyclic motion of the manipulator, which is an essential requirement for repetitive operations. The control law is simple and computationally very fast, and does not require either the complex manipulator dynamic model or the complicated inverse kinematic transformation. The configuration control scheme can alternatively be implemented in joint space.

Seraji, Homayoun↗
DOE OSTI2021

Optimization-based algorithms for nonlinear mechanics and frictional contact

An optimization-based strategy for solving nonlinear mechanics problems is proposed. In contrast to typical nonlinear equation solver algorithms that aim to find zeros in the residual force function, we minimize an energy (or energy-like) function to encourage solutions which are locally stable equilibria. These smooth and potentially non-convex objective functions are minimized using a preconditioned conjugate-gradient trust-region algorithm. Contact is formulated as an inequality constrained minimization problem, and is solved with an augmented Lagrangian algorithm. Friction is included in the approach via a regularized quasi-potential energy, and other dissipative behavior is included through the use of variational constitutive updates. Finally, to accelerate convergence rates for the Lagrange multipliers, we propose a novel multiplier update algorithm utilizing the Fischer-Burmeister function, and demonstrate super-linear solver convergence for some applications.

42 ENGINEERING↗
NASA NTRS1984

Global variation of the para hydrogen fraction in Jupiter's atmosphere and implications for dynamics on the outer planets

A detailed analysis of the Voyager infrared spectrometer measurements on Jupiter's atmosphere is presented, and possible implications of para hydrogen disequilibrium for the energetics and dynamics of that atmosphere are examined. The method of data analysis is described, and results for the large scale latitude variation of the para hydrogen fraction are presented. The Jovian results show pronounced latitude variation, and are compared with other parameters including wind fields, thermal structure, and various indicators of atmospheric clouds. The problem of equilibration rate is reexamined, and it is concluded that on Jupiter the equilibration time is longer than the radiative time constant at the level of emission to space, but that this inequality reverses at greater depths. A model for the interaction of fluid motions with the ortho-para conversion process is presented, and a consistent mixing length theory for the reacting ortho-para mixture is developed. Several implications of the Jovian data for atmospheric energetics and stability on the outer planets are presented.

Conrath, B. J.↗
NASA NTRS1972

Optimum configurations for bangless sonic booms.

A number of optimization problems are posed and solved for supersonic aircraft flight subject to the condition that a shock wave appears only incipiently in the sonic boom signal at a given point. The principal result is one giving the maximum effective gross weight of an aircraft of given effective length under given flight conditions. The calculus of variations with inequality constraints is used, with the novel features of a non-local isoperimetric relation and of only an upper bound on a control variable.

Hayes, W. D.↗
NASA NTRS1972

N-burn optimal analytic trajectories.

Derivation of N-burn analytic solutions for propellant-optimal transfer trajectories of a vehicle in a vacuum between arbitrary boundary conditions. Variational changes in the desired boundary conditions are expressed, in general, in terms of variational changes in the control vector and in the initial state vector. All coefficient matrices are computed recursively in terms of the analytic matrices established from the subarcs of the N-burn solution. The solution is applicable to shuttle ascent (exoatmospheric), rendezvous, and deorbit problems. Consideration is also given to state-variable and control-variable inequality constraints.

Jezewski, D. J.↗
DOE OSTI2020

Primal interface debonding formulation for finite strain isotropic plasticity

In this work, a framework is developed for modeling ductile damage of nonlinear materials whose plastic deformation is characterized using rate independent classical plasticity. This method relies on the assumption that the free energy can be decomposed into elastic, plastic and damage parts. A thermodynamically consistent method is derived which satisfies the second law of thermodynamics in the Clausius–Duhem inequality form. The dissipation associated with plasticity takes place in the domain only, while damage dissipation is localized to the interface. The method is developed using Variational Multiscale ideas to obtain definitions of the interface fluxes within a primal formulation analogous to the Discontinuous Galerkin method, which ensures weakly vanishing interface gap prior to reaching a damage initiation criterion. The local nonlinear problem to calculate both plastic deformation gradient and damage variable follows an incremental approach similar to classical plasticity return mapping algorithm. This elastoplastic damage formulation is developed for material undergoing finite strain, and it naturally accommodates a trapezoidal traction separation law (TSL) whose shape can be varied to model either ductile interface behavior or brittle interface behavior. The formulation's performance is assessed through modeling a patch test and a compact tension specimen.

42 ENGINEERING↗
NASA NTRS1977

Trajectory optimization for the Atlas/Centaur launch vehicle

A method for finding solutions to complex trajectory optimization problems with detailed hardware and launch vehicle constraints is presented. The trajectory optimization problem is formulated as a constrained function minimization problem by modeling the control by a function of a finite number of parameters. Although any parametric control model may be selected, the natural choice of parameters is the actual parameters of the parametric guidance equations flown on the flight digital computer. A variation of the method of multipliers is presented for performing the constrained function minimization. A functional is proposed that allows incorporation of state variable inequality constraints. Finally, optimization of the HEAO-A direct-ascent trajectory is discussed. The HEAO-A mission demonstrates profile optimization to meet range safety, tracking, heating, structural integrity, and attitude rate constraints.

Brusch, R. G.↗
NASA NTRS1986

The expanded LaGrangian system for constrained optimization problems

Smooth penalty functions can be combined with numerical continuation/bifurcation techniques to produce a class of robust and fast algorithms for constrainted optimization problems. The key to the development of these algorithms is the Expanded Lagrangian System which is derived and analyzed in this work. This parameterized system of nonlinear equations contains the penalty path as a solution, provides a smooth homotopy into the first-order necessary conditions, and yields a global optimization technique. Furthermore, the inevitable ill-conditioning present in a sequential optimization algorithm is removed for three penalty methods: the quadratic penalty function for equality constraints, and the logarithmic barrier function (an interior method) and the quadratic loss function (an interior method) for inequality constraints. Although these techniques apply to optimization in general and to linear and nonlinear programming, calculus of variations, optimal control and parameter identification in particular, the development is primarily within the context of nonlinear programming.

Poore, A. B.↗
NASA NTRS1988

The expanded Lagrangian system for constrained optimization problems

Smooth penalty functions can be combined with numerical continuation/bifurcation techniques to produce a class of robust and fast algorithms for constrained optimization problems. The key to the development of these algorithms is the Expanded Lagrangian System which is derived and analyzed in this work. This parameterized system of nonlinear equations contains the penalty path as a solution, provides a smooth homotopy into the first-order necessary conditions, and yields a global optimization technique. Furthermore, the inevitable ill-conditioning present in a sequential optimization algorithm is removed for three penalty methods: the quadratic penalty function for equality constraints, and the logarithmic barrier function (an interior method) and the quadratic loss function (an interior method) for inequality constraints. Although these techniques apply to optimization in general and to linear and nonlinear programming, calculus of variations, optimal control and parameter identification in particular, the development is primarily within the context of nonlinear programming.

Poore, A. B.↗
DOE OSTI2025

McCormick envelopes in mixed-integer PDE-constrained optimization

McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. This lack of attention may be due to the distributed nature of such problems, which on the one hand leads to infinitely many linear constraints (generally state constraints that may be difficult to handle) in addition to the state equation for a pointwise formulation of the McCormick envelopes and renders bound-tightening procedures that successively improve the resulting convex relaxations computationally intractable. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. This yields convex relaxations that underestimate the original problem up to an a priori error estimate that depends on the mesh size of the discretization. These approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We show that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero. We provide a computational example, for which we certify all of our imposed assumptions. The results point to both the potential of the methodology and the gaps in the research that need to be closed. Our methodology provides a framework first for obtaining pointwise underestimators for nonconvexities and second for approximating them with finitely many linear inequalities in an infinite-dimensional setting.

Approximations and Expansions↗
NASA NTRS2023

Chemical Thermodynamics and the Mathematical Integration of Reaction Kinetics

Key advances in the development of numerical methods for non-reacting compressible flows have been enabled by translating physical requirements into concrete numerical guidelines, such as the satisfaction of entropy inequalities for shock-capturing techniques [Lax, Contributions to Nonlinear Functional Analysis (1971) 603-634]. In the present work, we present nonlinear numerical analysis tools that draw from Chemical Thermodynamics , the branch of Nonequilibrium Thermodynamics that deals with chemical reactions. Through Gibbs formalism, chemical thermodynamics provides a well-known theoretical expression for the chemical equilibrium constant of a reaction in terms of reduced chemical potentials. A less-known, yet extremely valuable result, due to [Krambeck, Arch. Ration. Mech. Anal. , 38 (1970) 317], states that when this expression is implemented, mass-action kinetic models are consistent with the dynamical prescriptions of the 2nd law of thermodynamics. For fixed-temperature ordinary differential equations modeling constant-volume reacting gas mixtures, this leads to a decreasing Helmholtz free energy. If the temperature is allowed to vary in accordance with conservation of energy (1st law), this leads to the statement of increasing entropy. These nonlinear prescriptions can, and should be, used to further develop temporal integration techniques for reaction kinetics. We demonstrate that Krambeck's result holds even when the equilibrium constants are approximated from data. We prove this result by constructing the implicit free energy and the implicit entropy inherent to a given approximation. This is first done for a 5-species, 17-reaction model problem for air. With this structure established, elements of discrete entropy-stability theory [Tadmor, Acta Numer. , 12 (2003) 451] are leveraged to examine the consistency of time-integration schemes with these prescriptions. Using chemical potentials, one can compute the respective contributions of the kinetics model and of the temporal scheme to free energy/entropy variations. We introduce a nonlinear-stable version of the Discontinuous-Galerkin (DG) scheme in time which shows robustness improvements over the standard linearly-stable version. Most notably, the maximum timestep that can be resolved with the nonlinearly-stable variant tends to grow with polynomial order, in contrast to the linearly-stable variant. We generalize our constructions to arbitrary systems of reversible chemical reactions, ultimately showing that the compressible reacting Euler system admits the opposite of the implicitly constructed thermodynamic entropy as a mathematical entropy . This lays important theoretical foundations towards robust scheme development [Harten, J. Comput. Phys. 49 (1983) 151-164].

STMD↗