An Improved Reversed-Flow Formulation of the Galerkin-Kantorovich-Dorodnitsyn Multi-Moment Integral Method

1969 ◽  
Vol 20 (2) ◽  
pp. 191-202 ◽  
Author(s):  
Howard E. Bethel
Keyword(s):  
Integral Method ◽  
Approximate Solutions ◽  
Lower Branch ◽  
Reversed Flow ◽  
New Formulation ◽  
Moment Integral ◽  
Similar Flows

SummaryAn improved reversed-flow formulation of the Galerkin-Kantorovich-Dorodnitsyn multi-moment integral method is presented in this paper. Convergence and accuracy properties of the approximate solutions of the Stewartson lower branch similar flows are given. The approximate solutions obtained with the new formulation for the lower branch similar flows are, in general, more accurate than those obtained with the classical Pohlhausen method or either of the previous formulations used with the GKD method.

Numerical Modeling of Optical Fibers Using the Finite Element Method and an Exact Non-reflecting Boundary Condition

2018 ◽  
Vol 18 (4) ◽  
pp. 581-601
Author(s):  
Rafail Z. Dautov ◽  
Evgenii M. Karchevskii
Keyword(s):  
Finite Element Method ◽  
Boundary Condition ◽  
Finite Element ◽  
Numerical Modeling ◽  
Optical Fibers ◽  
Original Problem ◽  
Approximate Solutions ◽  
The Finite Element Method ◽  
New Formulation ◽  
Adjoint Operators

AbstractThe original problem for eigenwaves of weakly guiding optical fibers formulated on the plane is reduced to a convenient for numerical solution linear parametric eigenvalue problem posed in a disk. The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. Properties of dispersion curves are investigated for the new formulation of the problem. An efficient numerical method based on FEM approximations is developed. Error estimates for approximate solutions are derived. The rate of convergence for the presented algorithm is investigated numerically.

scholarly journals Energetics of collapsible channel flow with a nonlinear fluid-beam model

2021 ◽  
Vol 926 ◽  
Author(s):  
D.Y. Wang ◽  
X.Y. Luo ◽  
P.S. Stewart
Keyword(s):  
Limit Point ◽  
Elastic Beam ◽  
Static Solution ◽  
External Pressure ◽  
Constitutive Law ◽  
Beam Model ◽  
Lower Branch ◽  
Narrow Region ◽  
New Formulation ◽  
Approach Zero

We consider flow along a finite-length collapsible channel driven by a fixed upstream flux, where a section of one wall of a planar rigid channel is replaced by a plane-strain elastic beam subject to uniform external pressure. A modified constitutive law is used to ensure that the elastic beam is energetically conservative. We apply the finite element method to solve the fully nonlinear steady and unsteady systems. In line with previous studies, we show that the system always has at least one static solution and that there is a narrow region of the parameter space where the system simultaneously exhibits two stable static configurations: an (inflated) upper branch and a (collapsed) lower branch, connected by a pair of limit point bifurcations to an unstable intermediate branch. Both upper and lower static configurations can each become unstable to self-excited oscillations, initiating either side of the region with multiple static states. As the Reynolds number increases along the upper branch the oscillatory limit cycle persists into the region with multiple steady states, where interaction with the intermediate static branch suggests a nearby homoclinic orbit. These oscillations approach zero amplitude at the upper branch limit point, resulting in a stable tongue between the upper and lower branch oscillations. Furthermore, this new formulation allows us to calculate a detailed energy budget over a period of oscillation, where we show that both upper and lower branch instabilities require an increase in the work done by the upstream pressure to overcome the increased dissipation.

scholarly journals Efficient Encoding of the Weighted MAX $$k$$-CUT on a Quantum Computer Using QAOA

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Franz G. Fuchs ◽  
Herman Øie Kolden ◽  
Niels Henrik Aase ◽  
Giorgio Sartor
Keyword(s):  
Phase Separation ◽  
Quantum Computer ◽  
Undirected Graph ◽  
Approximate Solutions ◽  
Test Cases ◽  
Weighted Graphs ◽  
Intermediate Scale ◽  
Practical Applications ◽  
Binary Encoding ◽  
New Formulation

AbstractThe weighted MAX $$k$$ k -CUT problem consists of finding a k-partition of a given weighted undirected graph G(V, E), such that the sum of the weights of the crossing edges is maximized. The problem is of particular interest as it has a multitude of practical applications. We present a formulation of the weighted MAX $$k$$ k -CUT suitable for running the quantum approximate optimization algorithm (QAOA) on noisy intermediate scale quantum (NISQ) devices to get approximate solutions. The new formulation uses a binary encoding that requires only $$|V|\log _2k$$ | V | log 2 k qubits. The contributions of this paper are as follows: (i) a novel decomposition of the phase-separation operator based on the binary encoding into basis gates is provided for the MAX $$k$$ k -CUT problem for $$k>2$$ k > 2 . (ii) Numerical simulations on a suite of test cases comparing different encodings are performed. (iii) An analysis of the resources (number of qubits, CX gates) of the different encodings is presented. (iv) Formulations and simulations are extended to the case of weighted graphs. For small k and with further improvements when k is not a power of two, our algorithm is a possible candidate to show quantum advantage on NISQ devices.

Two-Dimensional Flow Losses of a Turbine Cascade With Boundary Layer Injection

1972 ◽  
Author(s):  
W. Tabakoff ◽  
R. Earley
Keyword(s):  
Boundary Layer ◽  
Analytical Method ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Turbine Cascade ◽  
Flow Losses ◽  
Two Dimensional Flow ◽  
Incompressible Boundary ◽  
New Formulation ◽  
Mixing Losses

A method for determining the performance of a two-dimensional turbine cascade with boundary layer injection is developed using existing incompressible boundary layer approximate solutions with a new formulation for the injection. The overall cascade loss includes friction and wake mixing losses. The results of the analysis are compared with experimentally obtained data as a check of the validity of the new analytical method.

Approximate Solutions of Canonical Heat Conduction Equations

1990 ◽  
Vol 112 (4) ◽  
pp. 836-842 ◽  
Author(s):  
B. D. Vujanovic´ ◽  
S. E. Jones
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Heat Conduction ◽  
Integral Method ◽  
Approximate Solutions ◽  
Boundary Layer Theory ◽  
Variational Procedure ◽  
Nonlinear Heat Conduction ◽  
One Dimensional ◽  
The Third

We consider three analytical methods for finding the approximate solutions of one-dimensional, transient, and nonlinear heat conduction problems based upon the canonical equations of heat transfer. The first method can be considered as a generalization or refinement of the integral method. The second is an iterative method similar to that of Targ utilized in boundary layer theory. The third method is a variational procedure introduced in the spirit of Gauss’ variational principle of least constraint.

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