Inverse Broyden Research Articles

Comparison of nonlinear solution methods for magnetic equivalent circuits of saturated induction motors

This work compares three nonlinear solution methods for the performance of an induction motor’s magnetic equivalent circuit model with magnetic saturation. The interrelation between magnetic flux density and permeability introduces nonlinearities in the differential system of equations. Three popular nonlinear solution methods are selected for comparison, namely (i) the Gauss–Seidel method, (ii) the Newton–Raphson method and (iii) the inverse Broyden’s method. While all three methods have been applied in this context before, no comparison study has been published to the authors’ best knowledge. The study finds that the inverse Broyden’s method is most performant in terms of the number of required iterations, the computation time per iteration and the resulting total computation time. However, for substantial saturation levels, the authors recommend a hybrid implementation of multiple solution methods to obtain robust and reliable convergence.

Rapid multi-layer method on solving optimal endo-atmospheric trajectory of launch vehicles

ABSTRACTIn order to increase the speed, precision and robustness against the engine failure in solving optimal endo-atmospheric ascent trajectory of a launch vehicle, a rapid multi-layer solving method with improved numerical algorithms was proposed. The proposed method is capable of decomposing a large number of intervals into multiple layers with advantageous convergence property. Firstly, the problem of solving optimal endo-atmospheric ascent trajectory, which was subjected to path constraints and terminal constraints, was transformed into a Hamilton Two Point Boundary Value Problem (TPBVP). Then, through the finite difference method and numerical solving algorithm, the Hamilton TPBVP was iteratively solved with fewer initial discrete intervals. The initial values of higher-layer iterations were obtained by interpolating convergent solutions at sparse nodes into the doubly discrete nodes of high layers. The process was repeatedly performed until the solving precision met the requirements. To decrease the calculation load in solving TPBVPs, two improved solving algorithms without and with fewer Jacobian calculations were studied, respectively the Derivative-free Spectral Algorithm for Nonlinear Equations(DF-SANE) combined with the improved derivative-free nonmonotone line search strategy, and the Modified Newton method with a relaxation factor in combination with the Inverse Broyden Quasi-Newton method, denoted as ‘MN-IBQ’. Simulation verifications showed that the multi-layer method had significantly higher solving speed than the single-layer method. For the improved numerical algorithms, the DF-SANE was trapped in the local convergence problem. While using the proposed MN-IBQ can further increase the solving rate. Typical engine failure simulations showed that the multi-layer method with the MN-IBQ algorithm had not only significantly higher solving speed but also stronger robustness, where the traditional single-layer method could not adapt. In addition, the thrust loss tolerance limits for the multi-layer solving method were given for different engine failure times. The results show promising potential of the proposed approach in trajectory online generation and closed-loop guidance of launch vehicles at the endo-atmospheric ascent stage.

Dynamic Response and Parametric Analysis of Geometrically Nonlinear Functionally Graded Plate With Arbitrary Constraints Under Moving Mass

A comprehensive method is presented to investigate nonlinear dynamic responses of functionally graded rectangular plate with arbitrary boundary conditions under a moving mass, based on Rayleigh–Ritz solutions together with the penalty method (PM) and the Newmark with inverse Broyden quasi-Newton method. Different from the existing methods, all inertial effects associated with the moving mass, which affect the mass, damping, and stiffness matrices of system, are taken into account. The formulations are derived based on classical plate theory and von Karman geometric nonlinearity, which considers large deformation without violating stress failure criteria. Material properties of FG plate vary continuously in the thickness direction according to the power law. The PM is employed to deal with the troublesome arbitrary constraints, which excludes an orthonormalization process for determining these admissible functions, and where only one set of admissible functions is used for arbitrary boundary conditions. The Newmark method together with the rank-one inverse Broyden quasi-Newton method is adopted to solve the nonlinear coupled differential equations of second order in time with more computational efficiency and avoiding numerical instability. Formulation and method of solution are validated by studying their convergence behavior and performing the comparison studies with existing results in the literature in the limit cases. In addition, influences of different material distribution, velocity of moving mass, different conventional, and unconventional boundary conditions on forced and free vibration responses are discussed.

APPLICATION OF INVERSE BROYDEN'S METHOD FOR COMPUTATIONAL ANALYSIS OF HYDROGEN CONTRIBUTION TO THE NEAR-NEUTRAL PH STRESS CORROSION CRACKING

An Inverse Broyden's Method was applied in the context of a molecular statics approach for the analysis of the contribution of hydrogen to the near-neutral pH stress corrosion cracking. A 3D crystal structure was tested numerically. It is shown that the Inverse Broyden's Method provides a computationally efficient technique to evaluate the effect of hydrogen on the material degradation. Simulation results demonstrated that atoms of hydrogen placed near the crack tip produced a strong effect on deformation and crack propagation in bcc iron leading to a 15–20% loss in a residual strength of numerically tested samples.

Convergence of a two-level ideal algorithm for a parametric shape inverse model problem

Similar to the discretization of ordinary or partial differential equations, the numerical approximation of the solution of an optimization problem is possibly subject to numerical stiffness. In the framework of parametric shape optimization, hierarchical representations of the shape can be used for preconditioning, following the idea of Multigrid (MG) methods. In this article, by analogy with the Poisson equation, which is the typical example for linear MG methods, we address a parametric shape inverse problem. We describe the ideal cycle of a two-level algorithm adapted to shape optimization problems that require appropriate transfer operators. With the help of a symbolic calculus software we show that the efficiency of an optimization MG-like strategy is ensured by a small dimension-independent convergence rate. Numerical examples are worked out and corroborate the theoretical results. Applications to antenna design are realized. Finally, some connections with the direct and inverse Broyden–Fletcher–Goldfarb–Shanno preconditioning methods are shown.

The application of inverse Broyden’s algorithm for modeling of crack growth in iron crystals

In the present paper we demonstrate the use of inverse Broyden's algorithm (IBA) in the simulation of fracture in single iron crystals. The iron crystal structure is treated as a truss system, while the forces between the atoms situated at the nodes are defined by modified Morse inter-atomic potentials. The evolution of lattice structure is interpreted as a sequence of equilibrium states corresponding to the history of applied load/deformation, where each equilibrium state is found using an iterative procedure based on IBA. The results presented demonstrate the success of applying the IBA technique for modeling the mechanisms of elastic, plastic and fracture behavior of single iron crystals.

Modeling of hydrogen-assisted cracking in iron crystal using a quasi-Newton method

A Quasi-Newton method was applied in the context of a molecular statics approach to simulate the phenomenon of hydrogen embrittlement of an iron lattice. The atomic system is treated as a truss-type structure. The interatomic forces between the hydrogen-iron and the iron-iron atoms are defined by Morse and modified Morse potential functions, respectively. Two-dimensional hexagonal and 3D bcc crystal structures were subjected to tensile numerical tests. It was shown that the Inverse Broyden's Algorithm-a quasi-Newton method-provides a computationally efficient technique for modeling of the hydrogen-assisted cracking in iron crystal. Simulation results demonstrate that atoms of hydrogen placed near the crack tip produce a strong deformation and crack propagation effect in iron lattice, leading to a decrease in the residual strength of numerically tested samples.

Rapidly convergent procedure to solve the density profile equation in the classical density functional theory

An efficient recursive procedure to solve the density profile equation in the classical density functional theory (DFT) using an inverse Broyden method is described. The present iterative procedure is free of calculation of the Jacobian matrix, and its inversion unavoidable for the well-known Newton-Raphson (NR) method and its variants. Numerical calculation indicates that only the approximate solution and iterative matrix of the lower bulk density case are employed as the corresponding initial guesses of the higher bulk density case, the present recursive procedure can converge quickly to the physical solution with an accuracy of epsilon = 10(-14); therefore, the procedure provides an efficient numerical algorithm for the theory in which acquirement of a density profile of high accuracy is a key step. Extensive numerical calculation shows the advantage of the present inverse Broyden method over Broyles' mixing procedure and a modified Powell hybrid algorithm (a variation of the NR method).

A METHOD FOR QUASI-STATIC ANALYSIS OF TOPOLOGICALLY VARIABLE LATTICE STRUCTURES

The proposed time-independent quasi-static approach for simulations of lattice structures with imperfections is based on integration of the Inverse Broyden's Method suitable for finding the equilibrium state for a large system of atoms interacting through strongly nonlinear potentials and the Recursive Inverse Matrix Algorithm (RIMA) capable of updating the inverse matrix when topological changes (broken or new bonds between the atoms) take place. In this approach, the crystal structure is treated as a truss system while the forces between the atoms situated at the nodes are defined by the inter-atomic potentials. Since both the Broyden's and the RIMA algorithms deal with the inverse matrices of the structure their coupling makes the procedure computationally efficient. In addition, the method allows analysis of lattices subjected to mixed boundary conditions. The developed code was verified by the comparison with an alternative numerical procedure based on energy minimization technique. The model and the code developed were applied to the case of a 2D hexagonal lattice with the mode I crack embedded into the structure. For the cases considered, it was observed that the crack nucleation and growth were accompanied by the dislocation emission.