Broyden's Method Research Articles

A note on the equivalence of Broyden and Block ABS methods

A note on the equivalence of Broyden and Block ABS methods

Quadrature based Broyden-like method for systems of nonlinear equations

A new iterative method based on the quasi-Newton approach for solving systems of nonlinear equations, especially large scale is proposed. We used the weighted combination of the Trapezoidal and Simpson quadrature rules. Our goal is to enhance the efficiency of the well known Broyden method by reducing the number of iterations it takes to reach a solution. Local convergence analysis and computational results are given.

Finite Element Framework for Computational Fluid Dynamics in FEBio.

The mechanics of biological fluids is an important topic in biomechanics, often requiring the use of computational tools to analyze problems with realistic geometries and material properties. This study describes the formulation and implementation of a finite element framework for computational fluid dynamics (CFD) in FEBio, a free software designed to meet the computational needs of the biomechanics and biophysics communities. This formulation models nearly incompressible flow with a compressible isothermal formulation that uses a physically realistic value for the fluid bulk modulus. It employs fluid velocity and dilatation as essential variables: The virtual work integral enforces the balance of linear momentum and the kinematic constraint between fluid velocity and dilatation, while fluid density varies with dilatation as prescribed by the axiom of mass balance. Using this approach, equal-order interpolations may be used for both essential variables over each element, contrary to traditional mixed formulations that must explicitly satisfy the inf-sup condition. The formulation accommodates Newtonian and non-Newtonian viscous responses as well as inviscid fluids. The efficiency of numerical solutions is enhanced using Broyden's quasi-Newton method. The results of finite element simulations were verified using well-documented benchmark problems as well as comparisons with other free and commercial codes. These analyses demonstrated that the novel formulation introduced in FEBio could successfully reproduce the results of other codes. The analogy between this CFD formulation and standard finite element formulations for solid mechanics makes it suitable for future extension to fluid-structure interactions (FSIs).

Determination of Gas Pressure Distribution in a Pipeline Network using the Broyden Method

A potential problem in natural gas pipeline networks is bottlenecks occurring in the flow system due to unexpected high pressure at the pipeline network junctions resulting in inaccurate quantity and quality (pressure) at the end user outlets. The gas operator should be able to measure the pressure distribution in its network so the consumers can expect adequate gas quality and quantity obtained at their outlets. In this paper, a new approach to determine the gas pressure distribution in a pipeline network is proposed. A practical and user-friendly software application was developed. The network was modeled as a collection of node pressures and edge flows. The steady state gas flow equations Panhandle A, Panhandle B and Weymouth to represent flow in pipes of different sizes and a valve and regulator equation were considered. The obtained system consists of a set of nonlinear equations of node pressures and edge flowrates. Application in a network in the field involving a large number of outlets will result in a large system of nonlinear equations to be solved. In this study, the Broyden method was used for solving the system of equations. It showed satisfactory performance when implemented with field data.

A numerical model for design and optimization of surface textures for tilting pad thrust bearings

A numerical model based on the Reynolds equation to study textured tilting pad thrust bearings considering mass-conserving cavitation and thermal effects is presented. A non-uniform and adaptive finite volume method is utilized and two methods are compared and selected regarding their efficiency in handling discontinuities; specifically placing additional nodes closely around discontinuities and directly incorporating discontinuities in the discrete system. Multithreading is applied to improve the computational performance and three root-finding methods to evaluate the bearing equilibrium are compared; namely Newton-Raphson method, Broyden's method with Sherman-Morrison formula and a continuation approach with fourth-order Runge-Kutta method. Results from the equivalent untextured bearing are utilized to accelerate the computation of the textured bearing and results are validated by comparison with CFD data.

Forced vibration response of axially functionally graded non-uniform plates considering geometric nonlinearity

Nonlinear forced vibration analysis on axially functionally graded (AFG) non-uniform plates has been carried out in the present study. Nonlinear strain displacement relations are used to incorporate geometric nonlinearity in the system. The problem is formulated using energy method and governing equations are derived on the basis of Hamilton's principle. An indirect approach is adopted for solving the problem where, the problem is reduced to a static case by assuming that the dynamic system fulfils the force equilibrium conditions at maximum amplitude of excitation. One of the multidimensional Secant methods, known as Broyden method, is utilised for solving the set of nonlinear equations. Different combinations of clamped and simply supported boundary conditions are considered in the analysis and generated results are furnished in non-dimensional frequency-amplitude plane. Taper parameter and excitation amplitude are observed to significantly affect the forced vibration response of the plates. The deflection shapes for various boundary conditions are presented and the forced vibration response in the proximity of second mode is also investigated. The results of the present analysis are validated with the studies available in the literature.

A Modified Two-Timescale Incremental Harmonic Balance Method for Steady-State Quasi-Periodic Responses of Nonlinear Systems

A modified two-timescale incremental harmonic balance (IHB) method is introduced to obtain quasi-periodic responses of nonlinear dynamic systems with combinations of two incommensurate base frequencies. Truncated Fourier coefficients of residual vectors of nonlinear algebraic equations are obtained by a frequency mapping-fast Fourier transform procedure, and complex two-dimensional (2D) integration is avoided. Jacobian matrices are approximated by Broyden's method and resulting nonlinear algebraic equations are solved. These two modifications lead to a significant reduction of calculation time. To automatically calculate amplitude–frequency response surfaces of quasi-periodic responses and avoid nonconvergent points at peaks, an incremental arc-length method for one timescale is extended for quasi-periodic responses with two timescales. Two examples, Duffing equation and van der Pol equation with quadratic and cubic nonlinear terms, both with two external excitations, are simulated. Results from the modified two-timescale IHB method are in excellent agreement with those from Runge–Kutta method. The total calculation time of the modified two-timescale IHB method can be more than two orders of magnitude less than that of the original quasi-periodic IHB method when complex nonlinearities exist and high-order harmonic terms are considered.

English

We propose a new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR or LU decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires O(n 2) arithmetic operations per iteration in contrast with the Newton method, which requires O(n 3) operations per iteration. Computational experiments confirm the high efficiency of the new method.

Pairing in neutron matter: New uncertainty estimates and three-body forces

We present solutions of the BCS gap equation in the channels ${}^1S_0$ and ${}^3P_2-{}^3F_2$ in neutron matter based on nuclear interactions derived within chiral effective field theory (EFT). Our studies are based on a representative set of nonlocal nucleon-nucleon (NN) plus three-nucleon (3N) interactions up to next-to-next-to-next-to-leading order (N$^3$LO) as well as local and semilocal chiral NN interactions up to N$^2$LO and N$^4$LO, respectively. In particular, we investigate for the first time the impact of subleading 3N forces at N$^3$LO on pairing gaps and also derive uncertainty estimates by taking into account results for pairing gaps at different orders in the chiral expansion. Finally, we discuss different methods for obtaining self-consistent solutions of the gap equation. Besides the widely-used quasi-linear method by Khodel et al. we demonstrate that the modified Broyden method is well applicable and exhibits a robust convergence behavior. In contrast to Khodel's method it is based on a direct iteration of the gap equation without imposing an auxiliary potential and is straightforward to implement.

JFNK method with a physics-based preconditioner for the fully implicit solution of one-dimensional drift-flux model in boiling two-phase flow

The Jacobian-free Newton-Krylov (JFNK) method with an efficient physics-based preconditioner is applied for the numerical solution of the one-dimensional drift-flux model with the closure constitutive equations. Additional closure correlations, including the flow pattern dependent heat transfer correlations and the flow pattern independent kinematic constitutive correlation, are used to close the governing equations of one-dimensional drift-flux model. The governing equations have been discretized using the first-order upwind method for spatial discretization and the fully implicit method for temporal discretization. An efficient physics-based preconditioner derived from the semi-implicit solution of governing equations is used in the JFNK method to improve the efficiency and numerical stability. The numerical verification and code validation have been performed for subcooled boiling two-phase flow in a vertical tube. By comparing with the other methods (JFNK method without preconditioner, the Broyden method and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method), the preconditioning JFNK method shows the robustness and a good computational efficiency. Moreover, by comparing the numerical simulation results with the experimental results, it is found that the JFNK method with a physics-based preconditioner shows the good accuracy for the numerical simulation for one-dimensional boiling two-phase flow.

Two improved classes of Broyden's methods for solving nonlinear systems of equations

Two improved classes of Broyden's methods for solving nonlinear systems of equations

Dynamic Analysis of an Automotive Belt-Drive System With a Noncircular Sprocket by a Modified Incremental Harmonic Balance Method

A dynamic model of an automotive belt-drive system with a noncircular sprocket instead of a round sprocket is developed in this work to study the effect of reducing the angular variation of camshafts. There are two submodels in the belt-drive system, which are an engine model and a belt-drive model, and they are decoupled to simplify the analysis. When the belt-drive system operates at a steady-state, it is described by a nonlinear model with forced excitation, which can be approximated by a linear model with combined parametric and forced excitations. Steady-state responses of the engine and belt-drive models are calculated by a modified incremental harmonic balance method that incorporates fast Fourier transform and Broyden's method, which is efficient and accurate to obtain a periodic response of a multi-degree-of-freedom system. Steady-state responses of the angular variation of camshafts with different values of sprocket parameters are compared to investigate their optimal values to reduce the angular variation of camshafts. The optimal eccentricity and installation angle are larger and smaller than those from the kinematic model, respectively, which is consistent with published experimental results. This study first shows from a dynamic point of view why use of a noncircular sprocket can reduce the angular variation of camshafts in the operating speed of an engine. Simulation of a speed-up procedure for different sprocket parameters shows results that are consistent with steady-state responses. The belt-drive model developed in this work can be used to select sprocket parameters to minimize the angular variation of camshafts and numerically evaluate the dynamic performance of a belt-drive system with a given design of sprocket parameters.

Partitioned quasi-Newton methods for sparse nonlinear equations

In this paper, we present two partitioned quasi-Newton methods for solving partially separable nonlinear equations. When the Jacobian is not available, we propose a partitioned Broyden's rank one method and show that the full step partitioned Broyden's rank one method is locally and superlinearly convergent. By using a well-defined derivative-free line search, we globalize the method and establish its global and superlinear convergence. In the case where the Jacobian is available, we propose a partitioned adjoint Broyden method and show its global and superlinear convergence. We also present some preliminary numerical results. The results show that the two partitioned quasi-Newton methods are effective and competitive for solving large-scale partially separable nonlinear equations.

Inverse heat transfer prediction of the state of the brick wall of a melting furnace

An inverse heat transfer method for predicting the state of the lateral refractory brick wall of a melting furnace is presented. By collecting temperature data with a thermocouple embedded into the brick wall, the inverse method is able to predict (1) the time-varying thickness of the protective bank that covers the inner lining of the furnace wall; (2) the thermal contact resistance between the inner lining and the protective bank; and (3) the possible erosion of the refractory brick wall. The inverse procedure rests on the Levenberg Marquardt algorithm combined with the Broyden method. The effect (1) of the noise on the collected temperature data; (2) of the thermal diffusivity of the brick wall; (3) of the location of the embedded temperature sensor; and (4) of the Biot number on the inverse predictions is investigated. Recommendations are made for the optimum position of the embedded sensor and its operation.

An inverse heat transfer method for predicting the thermal characteristics of a molten material reactor

An inverse heat transfer procedure is presented for predicting the time-varying thickness of the protective bank that covers the lining of the refractory brick walls of a molten material reactor. The inverse method predicts simultaneously influential thermo-physical properties of the reactor such as the thermal conductivity of the refractory brick wall, the thermal conductivity of the solid and of the liquid layers of the phase change material, and the time-varying reactor heat load. The inverse method rests on the Levenberg-Marquardt Method (LMM) combined with the Broyden method (BM). The effect (1) of the initial guesses for the unknown LMM polynomial parameters, (2) of the noise on the recorded temperature data, (3) of the location of the temperature sensors embedded into the brick wall and (4) of the number of recorded temperature data on the inverse predictions is investigated. Recommendations are then made concerning the installation and the operation of the temperature sensors.

Assessment of methods for computing the closest point projection, penetration, and gap functions in contact searching problems

Summary In computational contact mechanics problems, local searching requires calculation of the closest point projection of a contactor point onto a given target segment. It is generally supposed that the contact boundary is locally described by a convex region. However, because this assumption is not valid for a general curved segment of a three-dimensional quadratic serendipity element, an iterative numerical procedure may not converge to the nearest local minimum. To this end, several unconstrained optimization methods are tested: the Newton–Raphson method, the least square projection, the sphere and torus approximation method, the steepest descent method, the Broyden method, the Broyden–Fletcher–Goldfarb–Shanno method, and the simplex method. The effectiveness and robustness of these methods are tested by means of a proposed benchmark problem. It is concluded that the Newton–Raphson method in conjunction with the simplex method significantly increases the robustness of the local contact search procedure of pure penalty contact methods, whereas the torus approximation method can be recommended for contact searching algorithms, which employ the Lagrange method or the augmented Lagrangian method. Copyright © 2015 John Wiley & Sons, Ltd.

BROYDEN'S METHOD FOR OPERATORS WITH REGULARLY CONTINUOUS DIVIDED DIFFERENCES

We present a new convergence analysis of popular Broyden's method in the Banach/Hilbert space setting which is applicable to non- smooth operators. Moreover, we do not assume a priori solvability of the equation under consideration. Nevertheless, without these simplifying as- sumptions our convergence theorem implies existence of a solution and superlinear convergence of Broyden's iterations. To demonstrate practi- cal merits of Broyden's method, we use it for numerical solution of three nontrivial infinite-dimensional problems.

NUMERICAL STUDY OF A TIME-DOMAIN FINITE ELEMENT METHOD FOR NONLINEAR MAGNETIC PROBLEMS IN THREE DIMENSIONS (Invited Paper)

In this work, numerical analysis of nonlinear ferromagnetic problems is presented using the three-dimensional time-domain flnite element method (TDFEM). Formulated with the second- order nonlinear partial difierential equation (PDE) combined with the inverse Jiles-Atherton (J-A) vector hysteresis model, the nonlinear problems are solved in the time domain with the Newton- Raphson method. To solve the ordinary difierential equation (ODE) representing the magnetic hysteresis accurately and e-ciently, several ODE solvers are speciflcally designed and investigated. To improve the computational e-ciency of the Newton-Raphson method, the multi-dimensional secant methods, aka Broyden's methods, are incorporated in the nonlinear TDFEM solver. A nonuniform time-stepping scheme is also developed using the weighted residual approach to remove the requirement of a uniform time-step size during the simulation. The capability and the performance of the proposed methods are demonstrated by various numerical examples.

On Broyden-like update via some quadratures for solving nonlinear systems of equations

In this work, we propose a new alternative approximation based on the quasi-Newton approach for solving systems of nonlinear equations using the average of midpoint and Simpson's quadrature. Our goal is to enhance the efficiency of the method (Broyden's method) by reducing the number of iterations it takes to reach a solution. Local convergence analysis and computational results showing the relative efficiency of the proposed method are given.

116 分離型連成解法による複数のき裂を有する固体-固体連成解析の収束性能の向上

We have investigated the computational performance of Broyden method for plural cracks problems using a solid-solid coupling analysis approach. In this investigation, we found that the convergence rate of Broyden method deteriorates when the location of cracks in the model is asymmetry. In this paper, we introduced improved Broyden method with the aim of accelerating the convergence rate of the analysis for plural cracks problem. As a result, the convergence rate was not improved even with the improved method. Here, we illustrate the current status of research and development.