Book•
The method of weighted residuals and variational principles : with application in fluid mechanics, heat and mass transfer
01 Jan 1972-
About: The article was published on 1972-01-01 and is currently open access. It has received 1451 citations till now. The article focuses on the topics: Fluid mechanics.
Citations
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TL;DR: In this article, a variational iteration method is used to give approximate solutions of the problem of seepage flow in porous media with fractional derivatives, and the results show that the proposed iteration method, requiring no linearization or small perturbation is very effective and convenient.
1,265 citations
TL;DR: An online algorithm based on policy iteration for learning the continuous-time optimal control solution with infinite horizon cost for nonlinear systems with known dynamics, which finds in real-time suitable approximations of both the optimal cost and the optimal control policy, while also guaranteeing closed-loop stability.
1,012 citations
Book•
24 Aug 2001TL;DR: In this paper, the authors introduce the theory of thin plates and thin shells, and apply it to the analysis of shell structures, including the moment theory of circular cylindrical shells.
Abstract: Part 1 Thin plates: introduction the fundamentals of the small-deflection plate bending theory rectangular plates circular plates bending of plates of various shapes plate bending by approximate and numerical methods advanced topics buckling of plates vibration of plates. Part 2 Thin shells: introduction to the general linear shell theory geometry of the middle surface the general linear theory of thin shells the membrane theory of shells applications of the membrane theory to the analysis of shell structures moment theory of circular cylindrical shells the moment theory of shells of revolution approximate theories of shell analysis and their application advanced topics buckling of shells vibration of shells. Appendices: some reference data Fourier series expansion verification of relations of the theory of surfaces derivation of the strain-displacement relations verification of equilibrium equations.
980 citations
01 Aug 2008
TL;DR: It is shown that HDP converges to the optimal control and the optimal value function that solves the Hamilton-Jacobi-Bellman equation appearing in infinite-horizon discrete-time (DT) nonlinear optimal control.
Abstract: Convergence of the value-iteration-based heuristic dynamic programming (HDP) algorithm is proven in the case of general nonlinear systems. That is, it is shown that HDP converges to the optimal control and the optimal value function that solves the Hamilton-Jacobi-Bellman equation appearing in infinite-horizon discrete-time (DT) nonlinear optimal control. It is assumed that, at each iteration, the value and action update equations can be exactly solved. The following two standard neural networks (NN) are used: a critic NN is used to approximate the value function, whereas an action network is used to approximate the optimal control policy. It is stressed that this approach allows the implementation of HDP without knowing the internal dynamics of the system. The exact solution assumption holds for some classes of nonlinear systems and, specifically, in the specific case of the DT linear quadratic regulator (LQR), where the action is linear and the value quadratic in the states and NNs have zero approximation error. It is stressed that, for the LQR, HDP may be implemented without knowing the system A matrix by using two NNs. This fact is not generally appreciated in the folklore of HDP for the DT LQR, where only one critic NN is generally used.
919 citations
Cites methods from "The method of weighted residuals an..."
...Note that the residual error in (29) is explicit, in fact linear, in the tuning parameters W Vi +1. Therefore, to find the least squares solution, the method of weighted residuals may be used [ 11 ]....
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14 Jun 2009
TL;DR: This paper presents an online adaptive algorithm implemented as an actor/critic structure which involves simultaneous continuous-time adaptation of both actor and critic neural networks, and calls this ‘synchronous’ policy iteration.
Abstract: In this paper we discuss an online algorithm based on policy iteration for learning the continuous-time (CT) optimal control solution with infinite horizon cost for nonlinear systems with known dynamics. We present an online adaptive algorithm implemented as an actor/critic structure which involves simultaneous continuous-time adaptation of both actor and critic neural networks. We call this ‘synchronous’ policy iteration. A persistence of excitation condition is shown to guarantee convergence of the critic to the actual optimal value function. Novel tuning algorithms are given for both critic and actor networks, with extra terms in the actor tuning law being required to guarantee closed-loop dynamical stability. The convergence to the optimal controller is proven, and stability of the system is also guaranteed. Simulation examples show the effectiveness of the new algorithm.
648 citations
Cites background from "The method of weighted residuals an..."
...Assumption 1 allows us to bring in informal style of the Weierstrass higher-order approximation Theorem (Abu-Khalaf & Lewis, 2005; Finlayson, 1990) and the results of Hornik, Stinchcombe, andWhite (1990), which state that then there exists a complete independent basis set {φi(x)} such that the solution...
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...Then, as the number of hidden layer neurons N → ∞, the approximation errors ε → 0,∇ε → 0 uniformly (Finlayson, 1990)....
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References
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Book•
01 Jan 1956
TL;DR: Though it incorporates much new material, this new edition preserves the general character of the book in providing a collection of solutions of the equations of diffusion and describing how these solutions may be obtained.
Abstract: Though it incorporates much new material, this new edition preserves the general character of the book in providing a collection of solutions of the equations of diffusion and describing how these solutions may be obtained
20,495 citations
Book•
01 Jan 1955
TL;DR: The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part, denoted as turbulence as discussed by the authors, and the actual flow is very different from that of the Poiseuille flow.
Abstract: The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part. These actual flows show a special characteristic, denoted as turbulence. The character of a turbulent flow is most easily understood the case of the pipe flow. Consider the flow through a straight pipe of circular cross section and with a smooth wall. For laminar flow each fluid particle moves with uniform velocity along a rectilinear path. Because of viscosity, the velocity of the particles near the wall is smaller than that of the particles at the center. i% order to maintain the motion, a pressure decrease is required which, for laminar flow, is proportional to the first power of the mean flow velocity. Actually, however, one ob~erves that, for larger Reynolds numbers, the pressure drop increases almost with the square of the velocity and is very much larger then that given by the Hagen Poiseuille law. One may conclude that the actual flow is very different from that of the Poiseuille flow.
17,321 citations
Book•
01 Jan 1968
TL;DR: This first book to explore the computation of electromagnetic fields by the most popular method for the numerical solution to electromagnetic field problems presents a unified approach to moment methods by employing the concepts of linear spaces and functional analysis.
Abstract: From the Publisher:
"An IEEE reprinting of this classic 1968 edition, FIELD COMPUTATION BY MOMENT METHODS is the first book to explore the computation of electromagnetic fields by the most popular method for the numerical solution to electromagnetic field problems. It presents a unified approach to moment methods by employing the concepts of linear spaces and functional analysis. Written especially for those who have a minimal amount of experience in electromagnetic theory, this book illustrates theoretical and mathematical concepts to prepare all readers with the skills they need to apply the method of moments to new, engineering-related problems.Written especially for those who have a minimal amount of experience in electromagnetic theory, theoretical and mathematical concepts are illustrated by examples that prepare all readers with the skills they need to apply the method of moments to new, engineering-related problems."
6,593 citations
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3,412 citations
2,029 citations