Interactive Computation, Parameter Continuation, and Visualization

Maddocks, John H.; Manning, Robert S.; Paffenroth, Randy; Rogers, Kathleen A.; Warner, J.

doi:10.1142/s0218127497001333

Cited by 5 publications

(2 citation statements)

References 10 publications

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“…However, by using a judicious itinerary of parameter switching, one may compute an approximation of the two-dimensional surface using a collection of one-dimensional curves [18,22].…”

Section: The Sheets Of Buckled Equilibriamentioning

confidence: 99%

Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

Hoffman¹,

Manning²,

Paffenroth³

2002

SIAM J. Appl. Dyn. Syst.

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Abstract. We consider the problem of minimizing the energy of an inextensible elastic strut with length 1 subject to an imposed twist angle and force. In a standard calculus of variations approach, one first locates equilibria by solving the Euler-Lagrange ODE with boundary conditions at arclength values 0 and 1. Then one classifies each equilibrium by counting conjugate points, with local minima corresponding to equilibria with no conjugate points. These conjugate points are arclength values σ ≤ 1 at which a second ODE (the Jacobi equation) has a solution vanishing at 0 and σ. Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter λ, such as the force or twist angle in the elastic strut, this computation must be repeated for every value of λ of interest.Here we present an alternative approach that takes advantage of the presence of a parameter λ. Rather than search for conjugate points σ ≤ 1 at a fixed value of λ, we search for a set of special parameter values λm (with corresponding Jacobi solution ζ m ) for which σ = 1 is a conjugate point. We show that, under appropriate assumptions, the index of an equilibrium at any λ equals the number of these ζ m for which ζ m , Sζ m < 0, where S is the Jacobi differential operator at λ. This computation is particularly simple when λ appears linearly in S.We apply this approach to the elastic strut, in which the force appears linearly in S, and, as a result, we locate the conjugate points for any twisted unbuckled rod configuration without resorting to numerical solution of differential equations. In addition, we numerically compute two-dimensional sheets of buckled equilibria (as the two parameters of force and twist are varied) via a coordinated family of one-dimensional parameter continuation computations. Conjugate points for these buckled equilibria are determined by numerical solution of the Jacobi ODE.

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“…However, by using a judicious itinerary of parameter switching, one may compute an approximation of the two-dimensional surface using a collection of one-dimensional curves [18,22].…”

Section: The Sheets Of Buckled Equilibriamentioning

confidence: 99%

Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

Hoffman¹,

Manning²,

Paffenroth³

2002

SIAM J. Appl. Dyn. Syst.

Self Cite

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“…Furthermore, several solutions of the problem may exist, but only one is usually obtained. To characterize the nonlinear behaviour that produces these different solutions, continuation techniques are particularly efficient [20][21][22][23][24][25][26][27]. Yet, these methods compute multidimensional manifolds, thus continuous solution spaces to systems with more variables than equations.…”

Section: Introductionmentioning

confidence: 99%

Analytical expression of elastic rods at equilibrium under 3D strong anchoring boundary conditions

Ameline

Haliyo

Huang

et al. 2018

Journal of Computational Physics

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A general-purpose method is presented and implemented to express analytically one stationary configuration of an ideal 3D elastic rod when the end-to-end relative position and orientation are imposed. The mechanical equilibrium of such a rod is described by ordinary differential equations and parametrized by six scalar quantities. When one end of the rod is anchored, the analytical integration of these equations lead to one unique solution for given values of these six parameters. When the second end is also anchored, six additional nonlinear equations must be resolved to obtain parameter values that fit the targeted boundary conditions. We find one solution of these equations with a zero-finding algorithm, by taking initial guesses from a grid of potential candidates. We exhibit the symmetries of the problem, which reduces drastically the size of this grid and shortens the time of selection of an initial guess. The six variables used in the search algorithm, forces and moments at one end of the rod, are particularly adapted due to their unbounded definition domain. More than 850 000 tests are performed in a large region of configurational space, and in 99.9% of cases the targeted boundary conditions are reached with short computation time and a precision better than 10 −5. We propose extensions of the method to obtain many solutions instead of only one, using numerical continuation or starting from different initial guesses.

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Higher-Dimensional Continuation

Henderson¹

2007

Understanding Complex Systems

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Interactive Computation, Parameter Continuation, and Visualization

Cited by 5 publications

References 10 publications

Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

Analytical expression of elastic rods at equilibrium under 3D strong anchoring boundary conditions

Higher-Dimensional Continuation

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