Results for “boundary value problems” — Page 28

NASA NTRS2002

Closed-Loop Endoatmospheric Ascent Guidance

This paper will present a complete formulation of the optimal control problem for atmospheric ascent of rocket powered launch vehicles subject to usual load constraints and final condition constraints. We shall demonstrate that the classical finite difference method for two-point-boundary-value-problems (TPBVP) is suited for solving the ascent trajectory optimization problem in real time, therefore closed-loop optimal endoatmospheric ascent guidance becomes feasible. Numerical simulations with a the vehicle data of a reusable launch vehicle will be provided.

Lu, Ping↗

NASA NTRS1972

Generalized characteristics method for elastic wave propagation problems

Characteristic equations are derived in generalized curvilinear coordinates. Linear elastic, isotropic, and homogeneous constitutive equations have been used in the derivation. The generalized characteristic equations readily lend themselves to any requirements of space dimension and geometry. A simple boundary value problem is solved to indicate the applicability of these equations.

Ziv, M.↗

DOE OSTI2018

Oblique water wave damping by two submerged thin vertical porous plates of different heights

Oblique water wave damping by two fully submerged vertical parallel porous plates of different heights is investigated within the framework of linear water wave theory. A channel of infinite horizontal extent but of finite depth is divided into three regions in each of which three different velocity potentials are considered and subsequently the corresponding boundary value problems are formulated. By means of eigenfunction expansion and least square method, complete analytical solution in each of the regions is obtained. With the help of the matching conditions along the vertical boundaries between any two successive regions, the reflection and transmission coefficients, and hence the energy loss (in %), are obtained numerically by applying a matrix method. Numerical study is carried out for various relevant parameters and it is found that with an appropriate choice of the parameters and positioning of the porous plates, both of the reflection and transmission coefficients become significantly low and consequently the porous plates can be utilized as an effecting wave absorber. Comparison of this work with earlier available work is made, and the good agreement in the results suggests the applicability of this method in further engineering purposes. A particular case of two free surface piercing plates is also investigated on similar line and the results are presented in order to observe the difference with the earlier case.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1974

Fracture and contact problems for an elastic wedge

The plane elastostatic contact problem for an infinite elastic wedge of arbitrary angle is discussed. The medium is loaded through a frictionless rigid wedge of a given symmetric profile. Using the Mellin transform formulation the mixed boundary value problem is reduced to a singular integral equation with the contact stress as the unknown function. With the application of the results to the fracture of the medium in mind, the main emphasis in the study has been on the investigation of the singular nature of the stress state around the apex of the wedge and on the determination of the contact pressure.

Erdogan, F.↗

NASA NTRS1979

A fast semi-implicit algorithm for problems of mixed type

Certain physical processes are modeled by partial differential equations which are parabolic over part of the domain and elliptic over the remainder. A family of semi-implicit algorithms which are well suited to initial-boundary value problems of this mixed type is discussed. One important feature of these algorithms is the use of an approximate inverse for the solution of the implicit linear system. A strong error analysis results in an estimate of the total error as a function of approximate inverse error e and time step h.

Frederickson, P. O.↗

NASA NTRS1996

Optimization of Residual Stresses in MMC's through Process Parameter Control and the use of Heterogeneous Compensating/Compliant Interfacial Layers. OPTCOMP2 User's Guide

A user's guide for the computer program OPTCOMP2 is presented in this report. This program provides a capability to optimize the fabrication or service-induced residual stresses in unidirectional metal matrix composites subjected to combined thermomechanical axisymmetric loading by altering the processing history, as well as through the microstructural design of interfacial fiber coatings. The user specifies the initial architecture of the composite and the load history, with the constituent materials being elastic, plastic, viscoplastic, or as defined by the 'user-defined' constitutive model, in addition to the objective function and constraints, through a user-friendly data input interface. The optimization procedure is based on an efficient solution methodology for the inelastic response of a fiber/interface layer(s)/matrix concentric cylinder model where the interface layers can be either homogeneous or heterogeneous. The response of heterogeneous layers is modeled using Aboudi's three-dimensional method of cells micromechanics model. The commercial optimization package DOT is used for the nonlinear optimization problem. The solution methodology for the arbitrarily layered cylinder is based on the local-global stiffness matrix formulation and Mendelson's iterative technique of successive elastic solutions developed for elastoplastic boundary-value problems. The optimization algorithm employed in DOT is based on the method of feasible directions.

Pindera, Marek-Jerzy↗

NASA NTRS1986

On similarity solutions of a boundary layer problem with an upstream moving wall

The problem of a boundary layer on a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream is examined. It was previously shown that the solution of this boundary value problem is crucially dependent on the parameter which is the ratio of the velocity of the plate to the velocity of the free stream. In particular, it was proved that a solution exists only if this parameter does not exceed a certain critical value, and numerical evidence was adduced to show that this solution is nonunique. Using Crocco formulation the present work proves this nonuniqueness. Also considered are the analyticity of solutions and the derivation of upper bounds on the critical value of wall velocity parameter.

Hussaini, M. Y.↗

NASA NTRS1987

On similarity solutions of a boundary layer problem with an upstream moving wall

The problem of a boundary layer on a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream is examined. It was previously shown that the solution of this boundary value problem is crucially dependent on the parameter which is the ratio of the velocity of the plate to the velocity of the free stream. In particular, it was proved that a solution exists only if this parameter does not exceed a certain critical value, and numerical evidence was adduced to show that this solution is nonunique. Using Crocco formulation the present work proves this nonuniqueness. Also considered are the analyticity of solutions and the derivation of upper bounds on the critical value of wall velocity parameter.

Hussaini, M. Y.↗

NASA NTRS1972

Calculation of free-fall trajectories using numerical optimization methods.

An important problem in space flight is the calculation of trajectories for nonthrusting vehicles between fixed points in a given time. A new procedure based on Hamilton's principle for solving such two-point boundary-value problems is presented. It employs numerical optimization methods to perform the extremization required by Hamilton's principle. This procedure is applied to the calculation of an Earth-Moon trajectory. The results show that the initial guesses required to obtain an iteration procedure which converges are not critical and that convergence can be obtained to any predetermined degree of accuracy.

Hull, D. G.↗

DOE OSTI2025

A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields

We present a novel numerical method for solving the anisotropic diffusion equation in magnetic fields confined to a periodic box which is accurate and provably stable. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing both boundary conditions and parallel diffusion operator obtained by field line tracing. We prove that the fully-discrete approximation is unconditionally stable. Discrete energy estimates are shown to match the continuous energy estimate given the correct choice of penalty parameters. A nonlinear penalty parameter is shown to provide an effective method for tuning the parallel diffusion penalty and significantly minimises rounding errors. Several numerical experiments, using manufactured solutions, the “NIMROD benchmark” problem and a single island problem, are presented to verify numerical accuracy, convergence, and asymptotic preserving properties of the method. Finally, we present a magnetic field with chaotic regions and islands and show the contours of the anisotropic diffusion equation reproduce key features in the field.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2019

Efficient computation of optimal low thrust gravity perturbed orbit transfers

We have developed a new method for solving low-thrust fuel-optimal orbit transfer problems in the vicinity of a large body (planet or asteroid), considering a high-fidelity spherical harmonic gravity model. The algorithm is formulated via the indirect optimization method, leading to a two-point boundary value problem (TPBVP). We make use of a hyperbolic tangent smoothing law for performing continuation on the thrust magnitude to reduce the sharpness of the control switches in early iterations and thus promote convergence. The TPBVP is solved using the method of particular solutions (MPS) shooting method and Picard-Chebyshev numerical integration. Application of Picard-Chebyshev integration affords an avenue for increased efficiency that is not available with step-by-step integrators. We demonstrate that computing the particular solutions with only a low-fidelity force model greatly increases the efficiency of the algorithm while ultimately achieving near machine precision accuracy. A salient feature of the MPS is that it is parallelizable, and thus further speedups are available. It is also shown that, for near Earth orbits and over a small number of en-route revolutions around the Earth, only the zonal perturbation terms are required in the costate equations to obtain a solution that is accurate to machine precision and optimal to engineering precision. The proposed framework can be used for trajectory design around small asteroids and also for orbit debris rendezvous and removal tasks.

Junkins, John L.↗

NASA NTRS2008

A New Concurrent Multiscale Methodology for Coupling Molecular Dynamics and Finite Element Analyses

The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with FEM nodes at their common interface, necessarily requiring that the finite element mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modeling to two-dimensional material domains due to difficulties in simulating full three-dimensional material processes. In the present work, a new approach to MD-FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and finite element (FE) nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three-dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied.

Yamakov, Vesselin↗

NASA NTRS2008

An Embedded Statistical Method for Coupling Molecular Dynamics and Finite Element Analyses

The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with FEM nodes at their common interface, necessarily requiring that the finite element mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modeling to two-dimensional material domains due to difficulties in simulating full three-dimensional material processes. In the present work, a new approach to MD-FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and finite element (FE) nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three-dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied.

Saether, E.↗

NASA NTRS1983

Eigenmode analysis of unsteady one-dimensional Euler equations

The initial boundary value problem describing the evolution of unsteady linearized perturbations of a steady, uniform subsonic flow is analyzed. The eigenmodes and eigenfrequencies of the system are derived and several examples are presented to illustrate the effect of different boundary conditions on the exponential decay rate of the eigenmodes. The resultant implications for the stability and convergence rates of finite difference computations are discussed.

Giles, M.↗

NASA NTRS1985

Topics in spectral methods

After detailing the construction of spectral approximations to time-dependent mixed initial boundary value problems, a study is conducted of differential equations of the form 'partial derivative of u/partial derivative of t = Lu + f', where for each t, u(t) belongs to a Hilbert space such that u satisfies homogeneous boundary conditions. For the sake of simplicity, it is assumed that L is an unbounded, time-independent linear operator. Attention is given to Fourier methods of both Galerkin and pseudospectral method types, the Galerkin method, the pseudospectral Chebyshev and Legendre methods, the error equation, hyperbolic partial differentiation equations, and time discretization and iterative methods.

Gottlieb, D.↗

NASA NTRS1986

Kalman filtering, smoothing and recursive robot arm forward and inverse dynamics

The inverse and forward dynamics problems for multi-link serial manipulators are solved by using recursive techniques from linear filtering and smoothing theory. The pivotal step is to cast the system dynamics and kinematics as a two-point boundary-value problem. Solution of this problem leads to filtering and smoothing techniques identical to the equations of Kalman filtering and Bryson-Frazier fixed time-interval smoothing. The solutions prescribe an inward filtering recursion to compute a sequence of constraint moments and forces followed by an outward recursion to determine a corresponding sequence of angular and linear accelerations. In addition to providing techniques to compute joint accelerations from applied joint moments (and vice versa), the report provides an approach to evaluate recursively the composite multi-link system inertia matrix and its inverse. The report lays the foundation for the potential use of filtering and smoothing techniques in robot inverse and forward dynamics and in robot control design.

Rodriguez, G.↗

DOE OSTI2020

τ-SIMPLE Algorithm for the closure problem in homogenization of stokes flows

Estimation of the permeability tensor is crucial for many natural and industrial porous media applications. Homogenization provides a rigorous framework to calculate effective parameters from pore-scale images of a representative unit cell of the porous medium by solving a boundary value problem, also known as closure problem, subject to global constraints. However, the latter are hard to satisfy for arbitrarily complex geometries. We have developed a novel computational algorithm to calculate rigorously the permeability tensor. The approach, here referred to as τ-SIMPLE, is based on introducing an artificial time scale τ to satisfy the global constraint within the SIMPLE iteration. Here, we show that the proposed algorithm has high accuracy for both two-dimensional and three-dimensional periodically arranged geometries.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1981

Development of a linear unsteady aerodynamic analysis for finite-deflection subsonic cascades

A linear unsteady potential flow analysis, which accounts for the effects of blade geometry and steady turning, is being developed to predict the aerodynamic response to blade vibrations in the fan, compressor or turbine stages of modern jet engines. In previous work numerical solutions were restricted to cascades of sharp-edged blades aligned with the mean flow. Under the present effort the solution procedure has been extended to treat blades with rounded or blunt edges. As part of this effort an analytical model-problem study has been conducted to clarify the behavior of first-order perturbation solutions in the vicinity of airfoil edges. Further, a numerical approximation using concepts from singular perturbation theory has been developed to resolve the unsteady boundary value problem for cascades of blunt leading-edged blades. Numerical results for NACA 0012 cascades, including detailed results in the vicinity of a blade leading edge, are presented and evaluated.

Verdon, J. M.↗