On the use of polynomial series with the Rayleigh-Ritz method

Brown, Ross E.; Stone, M.

doi:10.1016/s0263-8223(97)00113-x

Cited by 29 publications

(15 citation statements)

References 16 publications

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“…This outcome is in agreement with the findings of Ref. [14], where different sets of polynomial expressions were compared and demonstrated to lead to equal results, and it can be explained by recalling that the method of Ritz operates a projection of the exact solution onto the vector space spanned by the trial functions. In this case, the vector space spanned by Legendre and Chebyshev polynomials is the same, thus the solution is approximated with the same level of accuracy.…”

Section: Remarks On the Numerical Efficiency Of Different Basissupporting

confidence: 91%

“…[11][12][13], where different boundary conditions can be handled by proper modification of the trial functions. While different kinds of polynomial expansions were found to have similar convergence properties, they are generally characterized by dissimilar stability properties [14]. Oosterhout et al [15] observed that simple polynomials can be adopted up to 11 trial functions before becoming unstable.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the application of the Ritz method to free vibration and buckling analysis of highly anisotropic plates

Vescovini

Dozio

D′Ottavio

et al. 2018

Composite Structures

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Section: Remarks On the Numerical Efficiency Of Different Basissupporting

confidence: 91%

Section: Introductionmentioning

confidence: 99%

On the application of the Ritz method to free vibration and buckling analysis of highly anisotropic plates

Vescovini

Dozio

D′Ottavio

et al. 2018

Composite Structures

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“…Solutions converge to the exact solution as more terms in this series are retained. Brown and Stone [4] have shown that the particular set of polynomials used does not strictly affect results. Different sets of polynomials of the same degree are expected to influence numerical stability of the solution, with respect to inversion and the extraction of eigenvalues and the conditioning of the Jacobian matrix of the equilibrium system.…”

Section: Potential Energy Functionalmentioning

confidence: 99%

“…Brown and Stone [4] have shown that this argument can be erroneous. For numerical stability of the relevant matrices obtained by the Rayleigh-Ritz method they show that orthogonality may have to be sought at the level of the first or second derivatives of the functions and not of the functions themselves.…”

Section: Potential Energy Functionalmentioning

confidence: 99%

Influence of splices on the buckling of columns

Simão

Coelho

Bijlaard

2012

International Journal of Non-Linear Mechanics

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“…The methods and results are all applicable to higher-order polynomials, there, the undetermined coefficients still enter linearly and they might inherit numerical advantages (power series are numerically stable only up to degree 12..15), but the notation would be less comprehensible. In [14] it was shown that only the polynomial degree but not the type of series are important for convergence. The choice is a simple approximate, but yet, it turns out to be quite accurate.…”

Section: Output Parameterization With a Polynomial Basismentioning

confidence: 99%

Computationally efficient trajectory optimization for linear control systems with input and state constraints

Stumper

Kennel

2011

Proceedings of the 2011 American Control Conference

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This paper presents a trajectory generation method that optimizes a quadratic cost functional with respect to linear system dynamics and to linear input and state constraints. The method is based on continuous-time flatness-based trajectory generation, and the outputs are parameterized using a polynomial basis. A method to parameterize the constraints is introduced using a result on polynomial nonpositivity. The resulting parameterized problem remains linear-quadratic and can be solved using quadratic programming. The problem can be further simplified to a linear programming problem by linearization around the unconstrained optimum. The method promises to be computationally efficient for constrained systems with a high optimization horizon. As application, a predictive torque controller for a permanent magnet synchronous motor which is based on real-time optimization is presented. I. INTRODUCTIONTrajectory optimization in real-time is an important part of many modern control systems, for instance, predictive control. The trajectory optimization problem must be solved in the sampling interval, determined by the plant dynamics. A computationally efficient solution, however, encounters two major obstacles: first, the optimization horizon has a minimum length, either to avoid suboptimality or as stability criterion, and second, the trajectory must satisfy input and state constraints to be feasible. For fast systems, many of the existing constrained predictive control schemes, which are based on numerical iterations, are not applicable.Concerning the horizon length problem, the continuous approach to predictive control is of interest [1]. Using differential flatness, the optimal control problem can be rewritten as output optimization problem. The basis function approach is applied to obtain a finite-parameter optimization problem [2] [3] [4] [5], analogeously to discrete-time optimization. A long optimization horizon is, however, obtained with comparably few optimization parameters.A remaining issue regarding computational efficiency is the inclusion of constraints. The classical method is the application of penalty functions [3]. As an alternative, a coordinate transformation was proposed to modify the constrained problem to a nonlinear unconstrained problem [6], solvable with nonlinear calculus of variations methods. The existing optimization methods with penalty functions or nonlinear coordinate transformations lead to a nonlinear or non-linear-quadratic problem, increasing the computational burden. If only feasibility is of interest, numerical procedures applying time scaling to slow down some variables can be used [1]. This paper treats the special case of the linearly constrained linear-quadratic problem. The linear structure of the system is exploited in the transformation of the problem to a finite-parameter optimization problem. The linear-quadratic structure of the problem is maintained. The unconstrained problem is solved algebraically as it is convex. A quadratic

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On the use of polynomial series with the Rayleigh-Ritz method

Cited by 29 publications

References 16 publications

On the application of the Ritz method to free vibration and buckling analysis of highly anisotropic plates

On the application of the Ritz method to free vibration and buckling analysis of highly anisotropic plates

Influence of splices on the buckling of columns

Computationally efficient trajectory optimization for linear control systems with input and state constraints

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