Collocation Software for Boundary-Value ODEs

Ascher, Uri M.; Christiansen, J.; Russell, Robert D.

doi:10.1145/355945.355950

Cited by 627 publications

(296 citation statements)

References 9 publications

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“…Details on COLSYS can be referred to in [37,38]. After obtaining element dynamic stiffness matrix [k i ] , global dynamic stiffness matrix K is assembled in a direct way along the meridian with regular element location vectors.…”

Section: Dynamic Stiffnessmentioning

confidence: 99%

An Exact Dynamic Stiffness Formulation for Predicting Natural Frequencies of Moderately Thick Shells of Revolution

Chen

2018

Mathematical Problems in Engineering

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An exact dynamic stiffness formulation is proposed for calculating the natural frequencies of shells of revolution based on the firstorder Reissner-Mindlin theory. Equations of motion are reduced to be one-dimensional, should the circumferential wave number is specified, and then are rewritten in Hamilton form. Therefore, a shell of revolution with a moderate thickness is analysed as a special skeletal structure with five degrees of freedom at element ends. Dynamic stiffnesses are constructed directly from the one-dimensional governing vibration equations using classic skeletal theory. Number of natural frequencies below a specific value is counted by applying the Wittrick-Williams (W-W) algorithm, and exact eigenvalues can be simply obtained using bisection method. A solution to the number of clamped-end frequencies J 0 in the W-W algorithm is also proposed and proven to be reliable. Numerical examples on a variety of shells with different boundary conditions are investigated, and results are well compared and validated. Influences of a variety of shell parameters on natural frequencies of both cylindrical and spherical shells are discussed in detail, demonstrating that the proposed dynamic stiffness formulation is applicable to analyse natural frequencies of moderately thick shells of revolution with high accuracy.

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Section: Dynamic Stiffnessmentioning

confidence: 99%

An Exact Dynamic Stiffness Formulation for Predicting Natural Frequencies of Moderately Thick Shells of Revolution

Chen

2018

Mathematical Problems in Engineering

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“…The two point boundary value problem (11), (10) can be numerically treated with the Scilab implementation bvode of the Fortran package colnew (see [3,5]) for the numerical solution of multi-point boundary value problems for mixed order systems of ordinary differential equations. This routine has the possibility to directly address higher order differential equation.…”

Section: Numerical Investigationsmentioning

confidence: 99%

“…)dt where T and the extremities X 0 , X 1 are fixed, and 3 We restrict ourselves to these normal extremals. Computations of abnormal extremals is known to be a very difficult task and those may even be not optimal.…”

Section: Proposition 1 (Weak Minimum Principle [8]) Consider the Sysmentioning

confidence: 99%

Inversion in indirect optimal control of multivariable systems

Chaplais

Petit

2008

ESAIM: COCV

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Abstract. This paper presents the role of vector relative degree in the formulation of stationarity conditions of optimal control problems for affine control systems. After translating the dynamics into a normal form, we study the Hamiltonian structure. Stationarity conditions are rewritten with a limited number of variables. The approach is demonstrated on two and three inputs systems, then, we prove a formal result in the general case. A mechanical system example serves as illustration.Mathematics Subject Classification. 34C20, 34H05, 49K15, 93C10, 93C35.

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“…[38] This package solves a general class of mixedorder systems of boundary value ordinary differential equations and is a modification of the COLSYS package developed by Ascher et al [39,40] Each section of the TMB unit is defined by four ordinary differential equations (ODEs): for each component, there is an ODE resulting from the mass balance in a volume element of the bed and another resulting from the mass balance in the particle. Since the TMB unit is composed by four sections, and considering a binary separation, the steady-state TMB model is defined by a set of 16 ODEs.…”

Section: Steady-state Tmb Modelmentioning

confidence: 99%

Design of SMB Chiral Separations Using the Concept of Separation Volume

Rodrigues

Pais

2005

Separation Science and Technology

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The article deals with chiral separation by simulated moving bed (SMB) chromatography. When mass transfer resistances are negligible, equilibrium theory provides explicit criteria for the choice of the SMB operating conditions. However, in the presence of mass transfer resistances, the SMB operating conditions should be evaluated through simulation. Using a package based on the analogy with the true moving bed operation, this work shows how mass transfer resistance can affect the conditions for enantiomers separation, as well as the critical values stated by equilibrium theory. The concept of separation volume is applied to show how the flow-rate constraints, in presence of mass transfer

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Collocation Software for Boundary-Value ODEs

Cited by 627 publications

References 9 publications

An Exact Dynamic Stiffness Formulation for Predicting Natural Frequencies of Moderately Thick Shells of Revolution

An Exact Dynamic Stiffness Formulation for Predicting Natural Frequencies of Moderately Thick Shells of Revolution

Inversion in indirect optimal control of multivariable systems

Design of SMB Chiral Separations Using the Concept of Separation Volume

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