On solving nonlinear equations with a one- parameter operator imbedding.
One parameter operator imbedding to modify Newton method for solution of nonlinear equations
Meyer, G. H.↗
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Nonlinear parameter identification: Ballistic range experience applicable to flight testing
The parameter identification scheme being used is a differential correction least squares procedure (Gauss-Newton method). The position, orientation, and derivatives of these quantities with respect to the parameters of interest (i.e., sensitivity coefficients) are determined by digital integration of the equations of motion and the parametric differential equations. The application of this technique to three vastly different sets of data is used to illustrate the versatility of the method and to indicate some of the problems that still remain.
Chapman, G.↗
Mathematical and computational studies of equilibrium capillary free surfaces
The results of several independent studies are presented. The general question is considered of whether a wetting liquid always rises higher in a small capillary tube than in a larger one, when both are dipped vertically into an infinite reservoir. An analytical investigation is initiated to determine the qualitative behavior of the family of solutions of the equilibrium capillary free-surface equation that correspond to rotationally symmetric pendent liquid drops and the relationship of these solutions to the singular solution, which corresponds to an infinite spike of liquid extending downward to infinity. The block successive overrelaxation-Newton method and the generalized conjugate gradient method are investigated for solving the capillary equation on a uniform square mesh in a square domain, including the case for which the solution is unbounded at the corners. Capillary surfaces are calculated on the ellipse, on a circle with reentrant notches, and on other irregularly shaped domains using JASON, a general purpose program for solving nonlinear elliptic equations on a nonuniform quadrilaterial mesh. Analytical estimates for the nonexistence of solutions of the equilibrium capillary free-surface equation on the ellipse in zero gravity are evaluated.
Albright, N.↗
Nonlinear finite element analysis: An alternative formulation
A geometrical nonlinear analysis based on an alternative definition of strain is presented. Expressions for strain are obtained by computing the change in length of the base vectors in the curvilinear element coordinate system. The isoparametric element formulation is assumed in the global Cartesian coordinate system. The approach is based on the minimization of the strain energy, and the resulting nonlinear equations are solved by the modified Newton method. Integration of the first and second variation of the strain energy is performed numerically in the case of two and three dimensional elements. Application is made to a simple long cantilever beam.
Merazzi, S.↗
Finite element analysis of moisture effects in graphite-epoxy composites
A numerical analysis procedure is developed for the micro-mechanical investigation of moisture effects on graphite-epoxy composites. The procedure employs a nonlinear constitutive relationship in a finite element formulation. A quasi-Newton method is used to solve the resulting nonlinear algebraic equations. The results presented indicate that the stresses at the graphite-epoxy interface have a strong dependence on moisture content, fiber spacing, and applied load.
Tobing, F. E. L.↗
Numerical optimization design of advanced transonic wing configurations
A computationally efficient and versatile technique for use in the design of advanced transonic wing configurations has been developed. A reliable and fast transonic wing flow-field analysis program, TWING, has been coupled with a modified quasi-Newton method, unconstrained optimization algorithm, QNMDIF, to create a new design tool. Fully three-dimensional wing designs utilizing both specified wing pressure distributions and drag-to-lift ration minimization as design objectives are demonstrated. Because of the high computational efficiency of each of the components of the design code, in particular the vectorization of TWING and the high speed of the Cray X-MP vector computer, the computer time required for a typical wing design is reduced by approximately an order of magnitude over previous methods. In the results presented here, this computed wave drag has been used as the quantity to be optimized (minimized) with great success, yielding wing designs with nearly shock-free (zero wave drag) pressure distributions and very reasonable wing section shapes.
Cosentino, G. B.↗
Numerical optimization design of advanced transonic wing configurations
A computationally efficient and versatile technique for use in the design of advanced transonic wing configurations has been developed. A reliable and fast transonic wing flow-field analysis program, TWING, has been coupled with a modified quasi-Newton method, unconstrained optimization algorithm, QNMDIF, to create a new design tool. Fully three-dimensional wing designs utilizing both specified wing pressure disributions and drag-to-lift ratio minimization as design objectives are demonstrated. Because of the high computational efficiency of each of the components of the design code, in particular the vectorization of TWING and the high speed of the Cray X-MP vector computer, the computer time required for a typical wing design is reduced by approximately an order of magnitude over previous methods. In the results presented here, this computed wave drag has been used as the quantity to be optimized (minimized) with great success, yielding wing designs with nearly shock-free (zero wave drag) pressure distributions and very reasonable wing section shapes.
Cosentino, G. B.↗
An optimization model for the US Air-Traffic System
A systematic approach for monitoring U.S. air traffic was developed in the context of system-wide planning and control. Towards this end, a network optimization model with nonlinear objectives was chosen as the central element in the planning/control system. The network representation was selected because: (1) it provides a comprehensive structure for depicting essential aspects of the air traffic system, (2) it can be solved efficiently for large scale problems, and (3) the design can be easily communicated to non-technical users through computer graphics. Briefly, the network planning models consider the flow of traffic through a graph as the basic structure. Nodes depict locations and time periods for either individual planes or for aggregated groups of airplanes. Arcs define variables as actual airplanes flying through space or as delays across time periods. As such, a special case of the network can be used to model the so called flow control problem. Due to the large number of interacting variables and the difficulty in subdividing the problem into relatively independent subproblems, an integrated model was designed which will depict the entire high level (above 29000 feet) jet route system for the 48 contiguous states in the U.S. As a first step in demonstrating the concept's feasibility a nonlinear risk/cost model was developed for the Indianapolis Airspace. The nonlinear network program --NLPNETG-- was employed in solving the resulting test cases. This optimization program uses the Truncated-Newton method (quadratic approximation) for determining the search direction at each iteration in the nonlinear algorithm. It was shown that aircraft could be re-routed in an optimal fashion whenever traffic congestion increased beyond an acceptable level, as measured by the nonlinear risk function.
Mulvey, J. M.↗
Steady flow past sudden expansions at large Reynolds number. I - Boundary layer solutions
A global Newton method is used to obtain accurate finite difference solutions for uniform inflows to several sudden expansion geometries, avoiding the instabilities encountered at large Reynolds numbers. Steady solutions for lambda values greater than those attained previously are obtained only if lambda is below a critical value. The presence of a singularity at this critical value renders invalid the assumptions of the boundary layer model. The results indicate that for uniform inflows and smaller values of the expansion ratio, the eddy length will no longer increase linearly with R when R becomes sufficiently large.
Milos, F. S.↗
Analysis of a parallelized nonlinear elliptic boundary value problem solver with application to reacting flows
A parallelized finite difference code based on the Newton method for systems of nonlinear elliptic boundary value problems in two dimensions is analyzed in terms of computational complexity and parallel efficiency. An approximate cost function depending on 15 dimensionless parameters is derived for algorithms based on stripwise and boxwise decompositions of the domain and a one-to-one assignment of the strip or box subdomains to processors. The sensitivity of the cost functions to the parameters is explored in regions of parameter space corresponding to model small-order systems with inexpensive function evaluations and also a coupled system of nineteen equations with very expensive function evaluations. The algorithm was implemented on the Intel Hypercube, and some experimental results for the model problems with stripwise decompositions are presented and compared with the theory. In the context of computational combustion problems, multiprocessors of either message-passing or shared-memory type may be employed with stripwise decompositions to realize speedup of O(n), where n is mesh resolution in one direction, for reasonable n.
Keyes, David E.↗
Steady flow past sudden expansions at large Reynolds number. II - Navier-Stokes solutions for the cascade expansion
The equations of motion in the steady laminar flow past a sudden expansion at large Reynolds number, R, reduce to the boundary layer equations as R tends to the freestream valve, when the longitudinal length scale of the separated eddy increases linearly and indefinitely with R. A global Newton method is presently used to obtain finite difference solutions to the steady Navier-Stokes equations up to R of 1000 for a uniform inflow past a cascade of sudden expansions. For large expansion ratio values, the eddy length increases linearly with R; for smaller values, however, where boundary layer equation solutions could not be found, the steady solutions to the Navier-Stokes equations approach the limit of an inviscid eddy in length with increasing R.
Milos, Frank S.↗
MHOST: An efficient finite element program for inelastic analysis of solids and structures
An efficient finite element program for 3-D inelastic analysis of gas turbine hot section components was constructed and validated. A novel mixed iterative solution strategy is derived from the augmented Hu-Washizu variational principle in order to nodally interpolate coordinates, displacements, deformation, strains, stresses and material properties. A series of increasingly sophisticated material models incorporated in MHOST include elasticity, secant plasticity, infinitesimal and finite deformation plasticity, creep and unified viscoplastic constitutive model proposed by Walker. A library of high performance elements is built into this computer program utilizing the concepts of selective reduced integrations and independent strain interpolations. A family of efficient solution algorithms is implemented in MHOST for linear and nonlinear equation solution including the classical Newton-Raphson, modified, quasi and secant Newton methods with optional line search and the conjugate gradient method.
Nakazawa, S.↗
Efficient numerical simulation of electron states in quantum wires
A new algorithm is presented for the numerical simulation of electrons in a quantum wire as described by a two-dimensional eigenvalue problem for Schroedinger's equation coupled with Poisson's equation. Initially, the algorithm employs an underrelaxed fixed point iteration to generate an approximation which is reasonably close to the solution. Subsequently, this approximate solution is employed as an initial guess for a Jacobian-free implementation of an approximate Newton method. In this manner the nonlinearity in the model is dealt with effectively. The effectiveness of this approach is demonstrated in a set of numerical experiments which study the electron states on the cross section of a quantum wire structure based on III-V semiconductors at 4.2 and 77 K.
Kerkhoven, Thomas↗
Rapid Design of Gravity Assist Trajectories
Several International Solar Terrestrial Physics (ISTP) missions require the design of complex gravity assisted trajectories in order to investigate the interaction of the solar wind with the Earth's magnetic field. These trajectories present a formidable trajectory design and optimization problem. The philosophy and methodology that enable an analyst to design and analyse such trajectories are discussed. The so called 'floating end point' targeting, which allows the inherently nonlinear multiple body problem to be solved with simple linear techniques, is described. The combination of floating end point targeting with analytic approximations with a Newton method targeter to achieve trajectory design goals quickly, even for the very sensitive double lunar swingby trajectories used by the ISTP missions, is demonstrated. A multiconic orbit integration scheme allows fast and accurate orbit propagation. A prototype software tool, Swingby, built for trajectory design and launch window analysis, is described.
Carrico, J.↗
Rapid design of gravity assist trajectories
Several International Solar-Terrestrial Physics (ISTP) missions require the design of complex gravity-assisted trajectories in order to investigate the interaction of the solar wind with the earth's magnetic field. These trajectories present a formidable trajectory design and optimization problem. This paper discusses the philosophy and methodology that enable an analyst to design and analyze such trajectories. The authors describe what is called 'floating end-point' targeting, which allows the inherently nonlinear multiple body problem to be solved with simple linear techniques. This paper demonstrates how floating end-point targeting combines analytic approximations with a Newton method targeter to achieve trajectory design goals quickly, even for the very sensitive double lunar swingby trajectories used by the ISTP missions. A Multiconic orbit integration scheme allows fast and accurate orbit propagation. This paper also describes a prototype software tool, Swingby, which has been built for trajectory design and launch window analysis.
Carrico, J.↗
Solution of free-boundary problems using finite-element/Newton methods and locally refined grids - Application to analysis of solidification microstructure
A new method is presented for the solution of free-boundary problems using Lagrangian finite element approximations defined on locally refined grids. The formulation allows for direct transition from coarse to fine grids without introducing non-conforming basis functions. The calculation of elemental stiffness matrices and residual vectors are unaffected by changes in the refinement level, which are accounted for in the loading of elemental data to the global stiffness matrix and residual vector. This technique for local mesh refinement is combined with recently developed mapping methods and Newton's method to form an efficient algorithm for the solution of free-boundary problems, as demonstrated here by sample calculations of cellular interfacial microstructure during directional solidification of a binary alloy.
Tsiveriotis, K.↗
Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation
We study parallel two-level overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, Newton-Krylov-Schwarz (NKS), employs an inexact finite-difference Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and, economical for this class of mixed elliptic-hyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fill-in in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributed-memory parallel computer.
Cai, Xiao-Chuan↗
Nonlinear Transient Problems Using Structure Compatible Heat Transfer Code
The report documents the recent effort to enhance a transient linear heat transfer code so as to solve nonlinear problems. The linear heat transfer code was originally developed by Dr. Kim Bey of NASA Largely and called the Structure-Compatible Heat Transfer (SCHT) code. The report includes four parts. The first part outlines the formulation of the heat transfer problem of concern. The second and the third parts give detailed procedures to construct the nonlinear finite element equations and the required Jacobian matrices for the nonlinear iterative method, Newton-Raphson method. The final part summarizes the results of the numerical experiments on the newly enhanced SCHT code.
Hou, Gene↗