On the method of successive approximations <o metode posledovatel'nykh priblizheniy<
Successive approximation method for solution to differential equation system
DIFFERENTIAL EQUATION↗
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An error bound for instantaneous coverage
An error bound is derived for a reliability model approximation method. The approximation method is appropriate for the semi-Markov models of reconfigurable systems that are designed to achieve extremely high reliability. The semi-Markov models of these system are complex, and a significant amount of their complexity arises from the detailed descriptions of the reconfiguration processes. The reliability model approximation method consists of replacing a detailed description of a reconfiguration process with the probabilities of the possible outcomes of the reconfiguration process. These probabilities are included in the model as instantaneous jumps from the fault-occurrence state. Since little time is spent in the reconfiguration states, instantaneous jumps are a close approximation to the original model. This approximation procedure is shown to produce an overestimation for the probability of system failure, and an error bound is derived for this overestimation.
White, Allan L.↗
An investigation of approximate solutions to fluid dynamics and heat transfer problems by variational methods
Approximate solutions to fluid dynamics and heat transfer problems by variational technique based on least squares method
VARIATION METHOD↗
Calculating Flows In Turbomachine Channels
Noniterative integral-entrainment method yields good approximations. Method of approximate calculation of flow in channel of turbomachine based on interaction of viscous flow in boundary layers with inviscid flow in core layer. Faster and more robust than other approximate methods of same type. Suitable for use in preliminary calculations for design and for off-design operation of turbomachinery. Flows in conical diffuser channels represented by two-dimensional boundary-layer and one-dimensional core flows described by equations of new method.
Schumann, Lawrence F.↗
Discrete-continuous variable structural synthesis using dual methods
Approximation concepts and dual methods are extended to solve structural synthesis problems involving a mix of discrete and continuous sizing type of design variables. Pure discrete and pure continuous variable problems can be handled as special cases. The basic mathematical programming statement of the structural synthesis problem is converted into a sequence of explicit approximate primal problems of separable form. These problems are solved by constructing continuous explicit dual functions, which are maximized subject to simple nonnegativity constraints on the dual variables. A newly devised gradient projection type of algorithm called DUAL 1, which includes special features for handling dual function gradient discontinuities that arise from the discrete primal variables, is used to find the solution of each dual problem. Computational implementation is accomplished by incorporating the DUAL 1 algorithm into the ACCESS 3 program as a new optimizer option. The power of the method set forth is demonstrated by presenting numerical results for several example problems, including a pure discrete variable treatment of a metallic swept wing and a mixed discrete-continuous variable solution for a thin delta wing with fiber composite skins.
Schmit, L. A.↗
A method of approximating propellant requirements of low-thrust trajectories
Method of approximating propellant requirements of low thrust trajectories
PROPELLANT↗
A Lyapunov and Sacker–Sell spectral stability theory for one-step methods
Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.
97 MATHEMATICS AND COMPUTING↗
APPROXIMATE SOLUTION TO A NONLINEAR DIFFERENTIAL EQUATION OF THE THIRD ORDER WITH THE N. N. BOGOLYUBOV ASYMPTOTIC METHOD
Approximate solution to nonlinear differential equation of third order with bogolyubov asymptotic method
DIFFERENTIAL EQUATION↗
Dual methods and approximation concepts in structural synthesis
Approximation concepts and dual method algorithms are combined to create a method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit dual functions, which are maximized subject to nonnegativity constraints on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality criteria approach. The dual method is successfully extended to deal with pure discrete and mixed continuous-discrete design variable problems. The power of the method presented is illustrated with numerical results for example problems, including a metallic swept wing and a thin delta wing with fiber composite skins.
Fleury, C.↗
Relaxation and approximate factorization methods for the unsteady full potential equation
The unsteady form of the full potential equation is solved in conservation form, using implicit methods based on approximate factorization and relaxation schemes. A local time linearization for density is introduced to enable solution to the equation in terms of phi, the velocity potential. A novel flux-biasing technique is applied to generate proper forms of the artificial viscosity, to treat hyperbolic regions with shocks and sonic lines present. The wake is properly modeled by accounting not only for jumps in phi, but also for jumps in higher derivatives of phi obtained from requirements of density continuity. The far field is modeled using the Riemann invariants to simulate nonreflecting boundary conditions. Results are presented for flows over airfoils, cylinders, and spheres. Comparisons are made with available Euler and full potential results.
Shankar, V.↗
Core Combination Method of Approximation (CCMA) for Neutron Fluence Calculation
Neutron fluence, or accumulative neutron fluence rate, is a key parameter of the neutron irradiation resistance of materials (Wagner, Bergner et al. 2016). So we must try to obtain it during irradiation experiments. Neutron fluence is defined as the time integral of neutron fluence rate. To get the quantity, or to do this integral, we need to know how neutron fluence rate varied with time. The integral might not be difficult when we conduct a material irradiation experiment, because we can put certain kind of neutron detectors in the location of species, to measure neutron fluence rate (Tanaka, Endo et al. 2011). At this circumstance, the reactor power is usually maintained to be constant (Ferroukhi, Hollard et al. 2009; Dupre, Vasiliev et al. 2015). So the neutron fluence is equal to the product of neutron fluence rate and irradiation time. However, it's not the same case when we want to get the neutron fluence of structure materials used in the SPRR- 300 (Swimming Pool Research Reactor -300). This reactor is a light-water-moderated, enriched-uranium-fueled research reactor developed by China. Structural materials, such as graphite, are of great value for research (Asamoah, Gyampo et al. 2012). Graphite was used as reflector in reactor. It is a waste of material and a waste of money to dismantle it simply. However, to reuse post-irradiation graphite assemblies faces some challenges, such as neutron fluence calculation, radiation protection, etc. The former will be discussed in this paper, whereas the latter will remain to be studied for the following time. The problem is how to acquire neutron fluence distribution in structure materials, since specific measurement of neutron fluence rate were not conducted intentionally during historical reactor operation. So we need to figure it out by a method of calculation. Considering a large number of reactor refueling, we propose an approximate method, Core Combination Method of Approximation (CCMA), to calculate accumulative neutron fluence rate of key structure materials during a long period of irradiation time.
21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
Comparing Capacity Credit Calculations for Wind: A Case Study in Texas
The degree to which wind energy can contribute to the capacity needed to meet resource adequacy requirements, also known as capacity credit (CC), varies regionally with wind resource and correlation to net load. CC is an important metric widely used for resource planning and resource adequacy assessments. However, there are multiple methods for computing and estimating CC, depending on specific needs, access to data, and computational burden. It is unclear the extent to which the CC computation method may influence the result. To address this, we use a probabilistic resource adequacy tool and multiple approximation methods to systematically assess the CC of wind for near-term wind deployment under a case study in Texas. We find that proper consideration of transmission constraints is important; some approximation methods may overestimate the CC of wind due to a lack of consideration of transmission constraints, while other approximation methods may underestimate the CC by not capturing the ability of wind to be shipped to neighboring regions. In this case study, we find that several approximation methods do come close to the CC calculated by more robust probabilistic methods. However, the best approximation method may vary on a case-by-case basis, depending on system-specific considerations.
17 WIND ENERGY↗
Response of joint dominated space structures
An approximate method is developed for estimating the transient response of nonlinear systems in terms of linearized modes of response. Its advantages are that it is computationally more efficient than the time integration method and that it is possible to view the design problem in the more traditional physical terms of modal response. The major drawback of the approximate method is loss of accuracy. It seems that both approximate methods and time integration have their roles in design. Approximate methods provide efficient tools for performing parametric studies and they supply physical insights into how to optimize system performance that are not easily inferred from strictly numerical methods. Time integration provides a method for assessing the accuracy of the approximate solution for key simulations and for fine tuning the final design. In the procedure presented the nonlinear system is approximated by an equivalent linear system in which the system parameters are constant over the range of transient response.
Source record↗
Boundary integral equation method calculations of surface regression effects in flame spreading
A solid-phase conduction problem that is a modified version of one that has been treated previously in the literature and is applicable to flame spreading over a pyrolyzing fuel is solved using a boundary integral equation (BIE) method. Results are compared to surface temperature measurements that can be found in the literature. In addition, the heat conducted through the solid forward of the flame, the heat transfer responsible for sustaining the flame, is also computed in terms of the Peclet number based on a heated layer depth using the BIE method and approximate methods based on asymptotic expansions. Agreement between computed and experimental results is quite good as is agreement between the BIE and the approximate results.
Altenkirch, R. A.↗
A stochastic biomass blending problem in decentralized supply chains
Blending biomass materials of different physical or chemical properties provides an opportunity to adjust the quality of the feedstock to meet the specifications of the conversion platform. We propose a model which identifies the right mix of biomass to optimize the performance of the thermochemical conversion process at the mini-mum cost. This is a chance-constraint programming (CCP) model which takes into account the stochastic nature of biomass quality. The proposed CCP model ensures that process requirements, which are impacted by physical and chemical properties of biomass, are met most of the time. We consider two problem settings, a centralized and a decentralized supply chain. We propose a mixed-integer linear program to model the blending problem in the centralized setting and a bilevel program to model the blending problem in the decentralized setting. We use the sample average approximation method to approximate the chance constraints, and propose solution algorithms to solve this approximation. We develop a case study for South Carolina using data provided by the Billion Ton Study. Based on our results, the blends identified consist mainly of pine and softwood residues. The blends identified and the suppliers selected by both models are different. The cost of the centralized supply chain is 2%–6% lower. The implications of these results are twofold. First, these results could lead to improved collaborations in the supply chain. Second, these results provide an estimate of the approximation error from assuming centralized decision making in the supply chain.
09 BIOMASS FUELS↗
Uniform waveguides with arbitrary cross-section considered by the point-matching method.
Point-matching method, approximate technique for wave equations of uniform waveguides with very general cross sections
WAVEGUIDE↗