Globally convergent homotopy algorithms for nonlinear systems of equations

Watson, Layne T.

doi:10.1007/bf01857785

Cited by 74 publications

(34 citation statements)

References 65 publications

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“…Parameter embedding methods, also known as continuation methods [8,9] are robust and accurate numerical techniques employed to solve nonlinear algebraic equations [1,62,63]. They are used to find multiple solutions of equations that possess multiple solutions [46].…”

Section: Parameter Embedding and Continuation Methodsmentioning

confidence: 99%

“…Increasing λ from 0 to 1 results in "losing" the curve. The difficulty is resolved by making λ a function of a new parameter, the arc length s. This method is known as the arc length continuation [61,63].…”

Section: Homotopy Methods: Backgroundmentioning

confidence: 99%

“…There are several approaches for implementing homotopy methods [63]. One set of algorithms is based on the ordinary differential equations.…”

Section: Numerical Solvermentioning

confidence: 99%

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DC operating points of transistor circuits

Trajković

2012

NOLTA

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Abstract:Finding a circuit's dc operating points is an essential step in its design and involves solving systems of nonlinear algebraic equations. Of particular research and practical interests are dc analysis and simulation of electronic circuits consisting of bipolar junction and fieldeffect transistors (BJTs and FETs), which are building blocks of modern electronic circuits. In this paper, we survey main theoretical results related to dc operating points of transistor circuits and discuss numerical methods for their calculation.

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Section: Parameter Embedding and Continuation Methodsmentioning

confidence: 99%

Section: Homotopy Methods: Backgroundmentioning

confidence: 99%

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DC operating points of transistor circuits

Trajković

2012

NOLTA

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“…The globalization methods we present in this work use parameter continuation and are related to probability-one homotopy 11 and path-following algorithms. 12 Homotopy and path-following methods are applied in fixed-point problems, root finding, bifurcation problems, and interior-point optimization algorithms, to name a few examples.…”

mentioning

confidence: 99%

Globalization Strategies for Inexact-Newton Solvers

Hicken

Zingg

2009

19th AIAA Computational Fluid Dynamics

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Globalization strategies are necessary in practical inexact-Newton flow solvers to ensure convergence when the initial iterate is far from the solution. In this work, we present two novel globalizations based on parameter continuation. The first continuation method parameterizes the boundary conditions while the second parameterizes the numerical dissipation. In both cases, a continuation parameter is used to create a sequence of modified nonlinear equations. The solution of each equation in the sequence provides an initial estimate for the subsequent problem until the desired convergence tolerance is reached. When applied to benign inviscid flows, the proposed globalization methods have similar efficiency compared with the more common pseudo-transient continuation. They are significantly more robust on difficult inviscid problems. Moreover, the continuation-based methods require less tuning.

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“…The parameters that minimize the difference between the response of the Adams model and that of the MapleSim model are identified using homotopy optimization [4], an optimization technique designed to avoid converging to a local minimum. The use of homotopy optimization in this work is related to the work of Abarbanel et al [5], and follows the approach presented by Vyasarayani et al [6].…”

Section: Introductionmentioning

confidence: 99%

Reduction of a Vehicle Multibody Dynamic Model Using Homotopy Optimization

Hall¹,

Loh²,

Schmitke³

et al. 2013

Archive of Mechanical Engineering

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Despite the ever-increasing computational power of modern processors, the reduction of complex multibody dynamic models remains an important topic of investigation, particularly for design optimization, sensitivity analysis, parameter identification, and controller tuning tasks, which can require hundreds or thousands of simulations. In this work, we first develop a high-fidelity model of a production sports utility vehicle in Adams/Car. Single-link equivalent kinematic quarter-car (SLEKQ, pronounced "sleek") models for the front and rear suspensions are then developed in MapleSim. To avoid the computational complexity associated with introducing bushings or kinematic loops, all suspension linkages are lumped into a single unsprung mass at each corner of the vehicle. The SLEKQ models are designed to replicate the kinematic behaviour of a full suspension model using lookup tables or polynomial functions, which are obtained from the high-fidelity Adams model in this work. The predictive capability of each SLEKQ model relies on the use of appropriate parameters for the nonlinear spring and damper, which include the stiffness and damping contributions of the bushings, and the unsprung mass. Homotopy optimization is used to identify the parameters that minimize the difference between the responses of the Adams and MapleSim models. Finally, the SLEKQ models are assembled to construct a reduced 10-degree-of-freedom model of the full vehicle, the dynamic performance of which is validated against that of the high-fidelity Adams model using four-post heave and pitch tests.

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Globally convergent homotopy algorithms for nonlinear systems of equations

Cited by 74 publications

References 65 publications

DC operating points of transistor circuits

DC operating points of transistor circuits

Globalization Strategies for Inexact-Newton Solvers

Reduction of a Vehicle Multibody Dynamic Model Using Homotopy Optimization

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