Model
Digital Document
Publisher
Florida Atlantic University
Description
The rotorcraft trim solution involves a search for control inputs for
required flight conditions as well as for corresponding initial conditions for
periodic response or orbit. The control inputs are specified indirectly to
satisfy flight conditions of prescribed thrust levels, rolling and pitching
moments etc. In addition to the nonlinearity of the equations of motion and
control inputs, the control inputs appear not only in damping and stiffness
matrices but also in the forcing-function or input matrix; they must be found
concomitantly with the periodic response from external constraints on the
flight conditions. The Floquet Transition Matrix (FTM) is generated for
perturbations about that periodic response; usually, a byproduct of the trim
analysis. The damping levels or stability margins are computed from an
eigenanalysis of the FTM. The Floquet analysis comprises the trim analysis
and eigenanalysis and is routinely used for small order systems (order N <
100). However, it is practical for neither design applications nor
comprehensive analysis models that lead to large systems (N > 100); the execution time on a sequential computer is prohibitive. The trim analysis
takes the bulk of this execution time.
Accordingly, this thesis develops concepts and methods of parallelism
toward Floquet analysis of large systems with computational reliability
comparable to that of sequential computations. A parallel shooting scheme
with damped Newton iteration is developed for the trim analysis. The scheme
uses parallel algorithms of Runge-Kutta integration and linear equations
solution. A parallel QR algorithm is used for the eigenanalysis of the FTM.
Additional parallelism in each iteration cycle is achieved by concurrent
operations such as perturbations of initial conditions and control inputs,
follow-up integrations and formations of the columns of the Jacobian matrix.
These parallel shooting and eigenanalysis schemes are applied to the
nonlinear flap-lag stability with a three-dimensional dynamic wake (N ~
150). The stability also is investigated by widely used sequential schemes of
shooting with damped Newton iteration and QR eigenanalysis. The
computational reliability is quantified by the maximum condition number of
the Jacobian matrices in the Newton iteration, the eigenvalue condition
numbers and the residual errors of the eigenpairs. The saving in computer
time is quantified by the speedup, which is the ratio of the execution times of
Floquet analysis by sequential and parallel schemes. The work is carried out
on massively parallel MasPar MP-1, a distributed-memory, single-instruction
multiple-data or SIMD computer. A major finding is that with increasing
system order, while the parallel execution time remains nearly constant, the
sequential execution time increases nearly cubically with N. Thus,
parallelism promises to make large-scale Floquet analysis practical.
required flight conditions as well as for corresponding initial conditions for
periodic response or orbit. The control inputs are specified indirectly to
satisfy flight conditions of prescribed thrust levels, rolling and pitching
moments etc. In addition to the nonlinearity of the equations of motion and
control inputs, the control inputs appear not only in damping and stiffness
matrices but also in the forcing-function or input matrix; they must be found
concomitantly with the periodic response from external constraints on the
flight conditions. The Floquet Transition Matrix (FTM) is generated for
perturbations about that periodic response; usually, a byproduct of the trim
analysis. The damping levels or stability margins are computed from an
eigenanalysis of the FTM. The Floquet analysis comprises the trim analysis
and eigenanalysis and is routinely used for small order systems (order N <
100). However, it is practical for neither design applications nor
comprehensive analysis models that lead to large systems (N > 100); the execution time on a sequential computer is prohibitive. The trim analysis
takes the bulk of this execution time.
Accordingly, this thesis develops concepts and methods of parallelism
toward Floquet analysis of large systems with computational reliability
comparable to that of sequential computations. A parallel shooting scheme
with damped Newton iteration is developed for the trim analysis. The scheme
uses parallel algorithms of Runge-Kutta integration and linear equations
solution. A parallel QR algorithm is used for the eigenanalysis of the FTM.
Additional parallelism in each iteration cycle is achieved by concurrent
operations such as perturbations of initial conditions and control inputs,
follow-up integrations and formations of the columns of the Jacobian matrix.
These parallel shooting and eigenanalysis schemes are applied to the
nonlinear flap-lag stability with a three-dimensional dynamic wake (N ~
150). The stability also is investigated by widely used sequential schemes of
shooting with damped Newton iteration and QR eigenanalysis. The
computational reliability is quantified by the maximum condition number of
the Jacobian matrices in the Newton iteration, the eigenvalue condition
numbers and the residual errors of the eigenpairs. The saving in computer
time is quantified by the speedup, which is the ratio of the execution times of
Floquet analysis by sequential and parallel schemes. The work is carried out
on massively parallel MasPar MP-1, a distributed-memory, single-instruction
multiple-data or SIMD computer. A major finding is that with increasing
system order, while the parallel execution time remains nearly constant, the
sequential execution time increases nearly cubically with N. Thus,
parallelism promises to make large-scale Floquet analysis practical.
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