Generalization of Lagrange's implicit function theorem to n-dimensions
Lagrange implicit functions generalization to n dimensions, developing inversion as power series
Feagin, T.↗
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Design of an essentially non-oscillatory reconstruction procedure on finite-element type meshes
An essentially non-oscillatory reconstruction for functions defined on finite-element type meshes was designed. Two related problems are studied: the interpolation of possibly unsmooth multivariate functions on arbitrary meshes and the reconstruction of a function from its average in the control volumes surrounding the nodes of the mesh. Concerning the first problem, we have studied the behavior of the highest coefficients of the Lagrange interpolation function which may admit discontinuities of locally regular curves. This enables us to choose the best stencil for the interpolation. The choice of the smallest possible number of stencils is addressed. Concerning the reconstruction problem, because of the very nature of the mesh, the only method that may work is the so called reconstruction via deconvolution method. Unfortunately, it is well suited only for regular meshes as we show, but we also show how to overcome this difficulty. The global method has the expected order of accuracy but is conservative up to a high order quadrature formula only. Some numerical examples are given which demonstrate the efficiency of the method.
Abgrall, R.↗
Evolution of assumed stress hybrid finite element
Early versions of the assumed stress hybrid finite elements were based on the a priori satisifaction of stress equilibrium conditions. In the new version such conditions are relaxed but are introduced through additional internal displacement functions as Lagrange multipliers. A rational procedure is to choose the displacement terms such that the resulting strains are now of complete polynomials up to the same degree as that of the assumed stresses. Several example problems indicate that optimal element properties are resulted by this method.
Pian, T. H. H.↗
A Variational Analysis of Divergence Profiles Based upon Column-Integrated Mass, Moisture and Energetic Constraints with Satellite-Derived Boundary Fluxes
A diagnostic study is made of the mean global divergent circulation based upon a constrained least action principle that minimizes column-integrated divergent kinetic energy subject to constraints on mass, moisture, available potential energy (ape) and total kinetic energy. The concept of gross moist stability was incorporated in the prescription of Lagrange weight function associated with the ape constraint in order to simulate the net effects of cumulus convective heating in the tropics. The variational analyses were validated satisfactorily with the original NCEP/NCAR-reanalysis divergence fields for the Septembers of 1987 and 1988. Further analyses show that in the tropical ascending regions, the analyzed divergences are dominated by the mass and ape constraints; the moisture constraint is implicitly satisfied while the kinetic energy constraint is highly dependent on the ape constraint. In the subtropical descending regions, the analyzed divergences are dominated by the mass, moisture and kinetic energy constraints; the ape constraint is implicitly satisfied. When the constraint integrals were blended with the satellite-derived boundary flux data from GPCP precipitation and ERBE/SRB radiation estimates, the newly analyzed divergences are significantly stronger than the reanalysis divergences in the areas where the estimates of precipitation rates are higher. With few exceptions, the increases in upper-layer divergences are coupled with nearly equal increases in lower-layer convergences.
Lu, Huei-Lin↗
Investigating the Numerical Stability of Using an Impedance Boundary Condition to Model Broadband Noise Scattering With Acoustic Liners
Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for achieving aircraft noise reduction and are characterized by a frequency-dependent impedance value. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners. A Broadband Impedance Model will be discussed in which the liner impedance is specified along a wide range of frequencies. The liner impedance boundary condition will be derived and coupled with a time-domain boundary integral equation to model acoustic scattering by a flat plate consisting of both un-lined and lined surfaces. The scattering solution will be obtained iteratively using both spatial and temporal basis functions and the stability will be demonstrated through eigenvalue analysis. Stability will be assessed for its dependence on time step, spatial discretization, as well as temporal basis function order. Both second- and third-order Lagrange temporal basis functions are considered.
acoustics↗
Investigating the Numerical Stability of Using an Impedance Boundary Condition to Model Broadband Noise Scattering With Acoustic Liners
Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for achieving aircraft noise reduction and are characterized by a frequency-dependent impedance value. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners. A Broadband Impedance Model will be discussed in which the liner impedance is specified along a wide range of frequencies. The liner impedance boundary condition will be derived and coupled with a time-domain boundary integral equation to model acoustic scattering by a flat plate consisting of both un-lined and lined surfaces. The scattering solution will be obtained iteratively using both spatial and temporal basis functions and the stability will be demonstrated through eigenvalue analysis. Stability will be assessed for its dependence on time step, spatial discretization, as well as temporal basis function order. Both second- and third-order Lagrange temporal basis functions are considered.
acoustics↗
Safe Tracking Control of an Uncertain Euler-Lagrange System with Full-State Constraints using Barrier Functions
This paper presents a novel, safe tracking control design method that learns the parameters of an uncertain Euler-Lagrange (EL) system online using adaptive learning laws. A barrier function (BF) is first used to transform the full-state constrained EL-dynamics into an equivalent unconstrained dynamics. An adaptive tracking controller is then developed along with the parameter update law in the transformed state space such that the states remain bounded for all time within a prescribed bound. A stability analysis is developed that considers the EL-dynamics’ uncertainty, yielding a semi-globally uniformly ultimately bounded (SGUUB) tracking error and the parameter estimation error. The controller design is validated in simulations using a two-link planar manipulator. The results show the proposed method’s ability to track the reference trajectory while remaining inside each of the predefined state bounds.
Robots↗
New Variational Formulations of Hybrid Stress Elements
In the variational formulations of finite elements by the Hu-Washizu and Hellinger-Reissner principles the stress equilibrium condition is maintained by the inclusion of internal displacements which function as the Lagrange multipliers for the constraints. These versions permit the use of natural coordinates and the relaxation of the equilibrium conditions and render considerable improvements in the assumed stress hybrid elements. These include the derivation of invariant hybrid elements which possess the ideal qualities such as minimum sensitivity to geometric distortions, minimum number of independent stress parameters, rank sufficient, and ability to represent constant strain states and bending moments. Another application is the formulation of semiLoof thin shell elements which can yield excellent results for many severe test cases because the rigid body nodes, the momentless membrane strains, and the inextensional bending modes are all represented.
Pian, T. H. H.↗
Theoretical transfer functions for the Saturn 5 Final report
Laplace transformations and Euler-Lagrange equations for theoretical transfer functions for Saturn 5 launch vehicle
Soifer, M. T.↗
Users manual: Dynamics of two bodies connected by an elastic tether, six degrees of freedom forebody and five degrees of freedom decelerator
The equations of motion and a computer program for the dynamics of a six degree of freedom body joined to a five degree of freedom body by a quasilinear elastic tether are presented. The forebody is assumed to be a completely general rigid body with six degrees of freedom; the decelerator is also assumed to be rigid, but with only five degrees of freedom (symmetric about its longitudinal axis). The tether is represented by a spring and dashpot in parallel, where the spring constant is a function of tether elongation. Lagrange's equation is used to derive the equations of motion with the Lagrange multiplier technique used to express the constraint provided by the tether. A computer program is included which provides a time history of the dynamics of both bodies and the tension in the tether.
Doyle, G. R., Jr.↗
Modeling Principles Using the Relation Between Lagrange Multipliers and Bond Graphs
Final document is attached. Modeling dynamic systems by bond graphs has become state of the art technology since hundreds of researchers around the world have incorporated the technology in many fields of engineering and science. The legacy of its invertor Prof. Henry Paynter at MIT in 1959 is now a fundamental and practical technique to understand reality by building computer models. This paper addresses a particular aspect of this technology when modeling of mechanical systems require relaxation of constraints by means of Lagrange principles. Lagrange's equations are a useful means of describing and solving systems with kinematic constraints. Lagrange multipliers are variables used in equations to find the extremes of multivariate functions. Here we explore the relation of Lagrange multipliers to solve modeling difficulties of a space vehicle with equations with dependent derivatives. Lagrange multipliers were used in conjunction with bond graphs to simulate a system where joints of kinematic linkages produce dependent derivatives. NASA's Morpheus Project lunar lander was used as a case study. The Morpheus Project is a terrestrial test vehicle designed to fly the terminal descent trajectory of a lunar lander to advance the Autonomous Landing Hazard Avoidance Technology (ALHAT). An objective of this study is to apply the modeling approach herein to capture the dynamic movement of the lander as the propellant is sloshed and consumed. This paper expands further the analysis presented by (Granda, J J. Nguyen, L, Carlson, T, Sahragard-Monfared, G., Fornalski, E., Brocker 2016). Using an automated approach bond graph models of state space equations were generated using the Computer Aided Modeling Program (CAMPG). Integral causality models and derivative causality models were considered in order to find the simpler solution for the mathematical dependencies produced in modeling this vehicle.
Granda, Jose J.↗
On stochastic extremum problems - calculus.
Lagrange multiplier technique for determining stationary points /of functions/ or stationary functions /of integrals/ of expected value of random functions with certain random constraints
LAGRANGE MULTIPLIER↗
Aerodynamic design and optimization in one shot
This paper describes an efficient numerical approach for the design and optimization of aerodynamic bodies. As in classical optimal control methods, the present approach introduces a cost function and a costate variable (Lagrange multiplier) in order to achieve a minimum. High efficiency is achieved by using a multigrid technique to solve for all the unknowns simultaneously, but restricting work on a design variable only to grids on which their changes produce nonsmooth perturbations. Thus, the effort required to evaluate design variables that have nonlocal effects on the solution is confined to the coarse grids. However, if a variable has a nonsmooth local effect on the solution in some neighborhood, it is relaxed in that neighborhood on finer grids. The cost of solving the optimal control problem is shown to be approximately two to three times the cost of the equivalent analysis problem. Examples are presented to illustrate the application of the method to aerodynamic design and constraint optimization.
Ta'asan, Shlomo↗
Shape optimization governed by the Euler equations using an adjoint method
A numerical approach for the treatment of optimal shape problems governed by the Euler equations is discussed. Focus is on flows with embedded shocks. A very simple problem is considered: the design of a quasi-one-dimensional Laval nozzle. A cost function and a set of Lagrange multipliers are introduced to achieve the minimum. The nature of the resulting costate equations is discussed. A theoretical difficulty that arises for cases with embedded shocks is pointed out and solved. Finally, some results are given to illustrate the effectiveness of the method.
Iollo, Angelo↗
Nonlinear programming extensions to rational function approximations of unsteady aerodynamics
This paper deals with approximating unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft. Two methods of formulating these approximations are extended to include both the same flexibility in constraining them and the same methodology in optimizing nonlinear parameters as another currently used 'extended least-squares' method. Optimal selection of 'nonlinear' parameters is made in each of the three methods by use of the same nonlinear (nongradient) optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is of lower order than that required when no optimization of the nonlinear terms is performed. The free 'linear' parameters are determined using least-squares matrix techniques on a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from the different approaches are described, and results are presented which show comparative evaluations from application of each of the extended methods to a numerical example. The results obtained for the example problem show a significant (up to 63 percent) reduction in the number of differential equations used to represent the unsteady aerodynamic forces in linear time-invariant equations of motion as compared to a conventional method in which nonlinear terms are not optimized.
Tiffany, Sherwood H.↗
Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamic forces
The approximation of unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft are discussed. Two methods of formulating these approximations are extended to include the same flexibility in constraining the approximations and the same methodology in optimizing nonlinear parameters as another currently used extended least-squares method. Optimal selection of nonlinear parameters is made in each of the three methods by use of the same nonlinear, nongradient optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is lower order than that required when no optimization of the nonlinear terms is performed. The free linear parameters are determined using the least-squares matrix techniques of a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from different approaches are described and results are presented that show comparative evaluations from application of each of the extended methods to a numerical example.
Tiffany, Sherwood H.↗
Comment of fitting ab initio intermolecular potentials for scattering calculations
The angular dependence of the interaction between two rigid linear rotors is considered, and one of the two expansions in current use is shown to be inappropriate. Interpolation of tabulated radial interactions is also discussed, and numerical tests are presented which demonstrate the accuracy of Lagrange and spline algorithms as a function of the spacing between points.
Green, S.↗
Contact solution algorithms
Two algorithms for obtaining static contact solutions are described in this presentation. Although they were derived for contact problems involving specific structures (a tire and a solid rubber cylinder), they are sufficiently general to be applied to other shell-of-revolution and solid-body contact problems. The shell-of-revolution contact algorithm is a method of obtaining a point load influence coefficient matrix for the portion of shell surface that is expected to carry a contact load. If the shell is sufficiently linear with respect to contact loading, a single influence coefficient matrix can be used to obtain a good approximation of the contact pressure distribution. Otherwise, the matrix will be updated to reflect nonlinear load-deflection behavior. The solid-body contact algorithm utilizes a Lagrange multiplier to include the contact constraint in a potential energy functional. The solution is found by applying the principle of minimum potential energy. The Lagrange multiplier is identified as the contact load resultant for a specific deflection. At present, only frictionless contact solutions have been obtained with these algorithms. A sliding tread element has been developed to calculate friction shear force in the contact region of the rolling shell-of-revolution tire model.
Tielking, John T.↗