Boundary value problems with interior point boundary conditions
Self adjoint boundary value problems with interior point boundary conditions using differential operator in Hilbert space
Krall, A. M.↗
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A superlinear interior points algorithm for engineering design optimization
We present a quasi-Newton interior points algorithm for nonlinear constrained optimization. It is based on a general approach consisting of the iterative solution in the primal and dual spaces of the equalities in Karush-Kuhn-Tucker optimality conditions. This is done in such a way to have primal and dual feasibility at each iteration, which ensures satisfaction of those optimality conditions at the limit points. This approach is very strong and efficient, since at each iteration it only requires the solution of two linear systems with the same matrix, instead of quadratic programming subproblems. It is also particularly appropriate for engineering design optimization inasmuch at each iteration a feasible design is obtained. The present algorithm uses a quasi-Newton approximation of the second derivative of the Lagrangian function in order to have superlinear asymptotic convergence. We discuss theoretical aspects of the algorithm and its computer implementation.
Herskovits, J.↗
Review of LMIs, Interior Point Methods, Complexity Theory, and Robustness Analysis
From end of intro: ...We would like to show that for certain problems in systems and control theory, there exist algorithms for which corresponding (xi) can be viewed as a certain measure of robustness, e.g., stability margin.
control systems control system analysis↗
An Infeasible Interior-Point Arc-Search Algorithm for Spacecraft Trajectory Optimization
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optimization, trajectory design↗
Entropy Splitting for High Order Numerical Simulation of Compressible Turbulence
A stable high-order numerical scheme for direct numerical simulation (DNS) of shock-free compressible turbulence is presented. The method is applicable to general geometries. It contains no upwinding, artificial dissipation, or filtering. Instead the method relies on the stabilizing mechanisms of an appropriate conditioning of the governing equations and the use of compatible spatial difference operators for the interior points (interior scheme) as well as the boundary points (boundary scheme). An entropy-splitting approach splits the inviscid flux derivatives into conservative and nonconservative portions. The spatial difference operators satisfy a summation-by-parts condition, leading to a stable scheme (combined interior and boundary schemes) for the initial boundary value problem using a generalized energy estimate. A Laplacian formulation of the viscous and heat conduction terms on the right hand side of the Navier–Stokes equations is used to ensure that any tendency to odd–even decoupling associated with central schemes can be countered by the fluid viscosity. The resulting methods are able to minimize the spurious high-frequency oscillations associated with pure central schemes, especially for long time integration applications such as DNS. For validation purposes, the methods are tested in a DNS of compressible turbulent plane channel flow at low values of friction Mach number, where reference turbulence data bases exist. It is demonstrated that the methods are robust in terms of grid resolution, and in good agreement with published channel data. Accurate turbulence statistics can be obtained with moderate grid sizes. Stability limits on the range of the splitting parameter are determined from numerical tests.
N.D. Sandham↗
Transonic flow solutions on general 3D regions using composite-block grids
The construction of computational fluid dynamics (CFD) codes for complicated regions is greatly simplified by a composite-block grid structure since, with the use of a surrounding layer of points on each block, a flow code is then only required basically to operate on rectangular computational regions. The necessary correspondence of points on the surrounding layers (image points) with interior points (object points) is set up by the grid code and made available to the CFD solution code.
Thompson, Joe F.↗
Conservative supra-characteristics method for splitting the hyperbolic systems of gasdynamics with computed boundaries for real and perfect gases
Implicit methods developed by Beam and Warming (1978 and Briley and McDonald (1977) make it possible to overcome the hyperbolic stiffness of the conservative compressible Navier-Stokes equations in the fine wall region computational mesh for high Reynold's number flow. Certain difficulties related to the use of these methods could be overcome by employing an approach reported by Roe (1981). In the present investigation Roe's conceptual framework has been adopted for constructing globally conservative finite difference methods. A globally conservative upwind finite difference method (CSCM) consisting of both implicit interior point and boundary point equations is constructed from a new characteristics based flux difference splitting. It is found that the employed upwind eigenvector split scheme which combines fully coupled implicit interior point and boundary point approximations has the desired properties of robust stability and accuracy.
Lombard, C. K.↗
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.
Polyak, Roman A.↗
Discussion of the General Electric proposal to develop a computer program to select thermal coatings for passive temperature control of satellites
The General Electric Company, Missiles and Space Division, submitted/a proposal to NASA for the development of an IBM 7094 computer program which would select the exterior surface coatings for passively controlling spacecraft temperatures. G. E. claims that the "trial and error" procedures currently used can be accomplished more rationally and can therefore be programmed for a digital computer. In the ASME paper, 63-HT-41, which was presented at the ASME-AIChE Heat Transfer Conference at Boston in August, 1963, Costello, Harper, and Cline, described the procedures that have been used at G. E. to develop a coating selection program subject to the following restrictions: 1. Steady-state conditions prevail, 2. Heat transfer occurs by radiation only, 3. Temperatures are optimized at only one interior point in the spacecraft, an d 4. Only the solar absorptance of the external coatings is varied to optimize temperature. The emittance must remain constant at initially specified values. General Electric proposes to develop a generalized program in three steps: 1. The program would vary both solar absorptance and hemispherical emittance to obtain the optimum coating patterns; 2. Temperatures would be optimized at more than one interior point; and 3. The equations would be modified to account for both conduction and radiation heat transfer. In the development of the general program, the scope would be restricted to steady-state heat transfer. Since the thermal designs of most spacecraft are based primarily on nearly equilibrium conditions, the proposed program could have wide application. An obvious extension of the proposed program would be to account for transient temperatures.
IBM 7090 COMPUTER↗
A uniformly valid solution for motion about the interior libration point of the perturbed elliptic-restricted problem
Bounded motion about Lagrange collinear libration points is considered for a perturbed elliptic-restricted problem. A practical application is the motion of a satellite near a libration point collinear with the sun and the earth-moon barycenter. Such a system is treated here as an earth-sun-satellite elliptic restricted problem with lunar perturbations. The method of dual time scales is utilized to develop a uniformly valid three-dimensional analytical solution to the satellite's equations of motion. The analytical development applies somewhat generally to that class of four-body problems where the second primary mass is much greater than the first, and the third primary mass is much greater than the second.
Richardson, D. L.↗
TIGGERC: Turbomachinery Interactive Grid Generator for 2-D Grid Applications and Users Guide
A two-dimensional multi-block grid generator has been developed for a new design and analysis system for studying multiple blade-row turbomachinery problems. TIGGERC is a mouse driven, interactive grid generation program which can be used to modify boundary coordinates and grid packing and generates surface grids using a hyperbolic tangent or algebraic distribution of grid points on the block boundaries. The interior points of each block grid are distributed using a transfinite interpolation approach. TIGGERC can generate a blocked axisymmetric H-grid, C-grid, I-grid or O-grid for studying turbomachinery flow problems. TIGGERC was developed for operation on Silicon Graphics workstations. Detailed discussion of the grid generation methodology, menu options, operational features and sample grid geometries are presented.
Miller, David P.↗
Delayed entry of solar protons (E/p/ over 0.3 MeV) into the earth's magnetotail
The delayed entry of solar protons into the earth's magnetotail is studied using simultaneous observations from three earth-orbiting spacecraft of a number of interplanetary proton events which exhibited abrupt and relatively large discontinuities in intensity associated with magnetic field discontinuities. It is found that such protons have immediate access to the magnetosheath and that delay times increase discontinuously from essentially zero in the magnetosheath to tens of minutes as an observing point moves inward across the magnetopause. Well-defined delay times from 30 to 160 min occur for points interior to the magnetotail. The inferred entry points lie at downstream distances of 80-500 R(E). No discernible relationship is found between the direction and sense of the interplanetary magnetic field and the entry times in the northern and southern lobes of the magnetotail or the entry tiems at various distances from its central axis. These results favor some form of an open model of the magnetotail.
Van Allen, J. A.↗
TIGGERC: Turbomachinery interactive grid generator energy distributor and restart code
A two dimensional multi-block grid generator was developed for a new design and analysis system for studying multi-blade row turbomachinery problems with an axisymmetric viscous/inviscid 'average passage' through flow code. TIGGERC is a mouse driven, fully interactive grid generation program which can be used to modify boundary coordinates and grid packing. TIGGERC generates grids using a hyperbolic tangent or algebraic distribution of grid points on the block boundaries and the interior points of each block grid are distributed using a transfinite interpolation approach. TIGGERC generates a blocked axisymmetric H grid, C grid, I grid, or O grid for studying turbomachinery flow problems. TIGGERC was developed for operation on small high speed graphic workstations.
Miller, David P.↗
Sufficient conditions for a local minimum of the Bolza problem with a variable initial point
Sufficient conditions for a weak relative minimum of a form of the Bolza problem of variational calculus are derived. The variational problem considered includes arbitrary numbers of constraints on the initial and terminal points, but assumes no control constraints or interior point constraints. Testing of the second-order conditions requires the backward integration of fewer matrix elements than in the case of previously published sets of conditions. The derivation, though lengthy, is felt to be simpler in concept than previous derivations. The application of these conditions is demonstrated in the context of a simple geometric problem.
Wood, Lincoln J.↗
Implementation of a Point Algorithm for Real-Time Convex Optimization
The primal-dual interior-point algorithm implemented in G-OPT is a relatively new and efficient way of solving convex optimization problems. Given a prescribed level of accuracy, the convergence to the optimal solution is guaranteed in a predetermined, finite number of iterations. G-OPT Version 1.0 is a flight software implementation written in C. Onboard application of the software enables autonomous, real-time guidance and control that explicitly incorporates mission constraints such as control authority (e.g. maximum thrust limits), hazard avoidance, and fuel limitations. This software can be used in planetary landing missions (Mars pinpoint landing and lunar landing), as well as in proximity operations around small celestial bodies (moons, asteroids, and comets). It also can be used in any spacecraft mission for thrust allocation in six-degrees-of-freedom control.
Acikmese, Behcet↗
The Split Coefficient Matrix method for hyperbolic systems of gasdynamic equations
The Split Coefficient Matrix (SCM) finite difference method for solving hyperbolic systems of equations is presented. This new method is based on the mathematical theory of characteristics. The development of the method from characteristic theory is presented. Boundary point calculation procedures consistent with the SCM method used at interior points are explained. The split coefficient matrices that define the method for steady supersonic and unsteady inviscid flows are given for several examples. The SCM method is used to compute several flow fields to demonstrate its accuracy and versatility. The similarities and differences between the SCM method and the lambda-scheme are discussed.
Chakravarthy, S. R.↗
Examples Of Synthesis Of Dual-Shaped Reflectors
Report presents examples to demonstrate validity and utility of method of synthesis of offset dual-shape reflectors. Method of synthesis described by the authors in previous journal article. Current report reviews derivation of partial differential equations and iterative method of numerical solution. Discusses significance of starting point of numerical integration on each reflector surface; this point could be at center, on outer rim, or at interior point. Emphasizes that one of notable attributes of partial differential equations is speed with which they can be solved.
Galindo, Victor↗
A New Method for the Accurate Calculation of Derivatives at Structured Multiblock Grid Singularities
The new Structured Mapping At Reduced dimension Topology elements (SMART) method for the accurate calculation of derivatives at multiblock structured grid block interfaces is presented. In the SMART method, a ’local’ curvilinear mapping is defined at each interface point, giving a consistent multidimensional definition of the derivatives. The resulting system of equations is overdetermined, and a least squares method is used to determine the value of the derivatives in the local coordinate system. In the special case of a nonsingular interface point, the SMART method automatically recovers the spatial differencing stencils used at the interior points in the neighboring grid blocks.
Computational Aeroacoustics↗