Optimum Design of Infinite and Finite Periodic Structures
This ongoing research effort research is performed by Moshe B.
Fuchs and Michael Ryvkin of the
School of Mechanical Engineering
at Tel Aviv University.
The general purpose of the research is to develop a numerical
methodology for the analysis of modular structures to be used in
a general design scheme of structures.
The static analysis of repetitive strutures is based on
the method of the representative cell, a technique developed by
Michael Ryvkin and Boris Nuller:

Nuller B. and Ryvkin M. (1980),
On boundary value problems for elastic domains of a periodic structure
deformed by arbitrary loads,
In: Proc. of the State Hydraulic Institute, 136, 4955,
Energia, Leningrad (in Russian).

Ryvkin M. and Nuller B. (1997),
Solution of quasiperiodic fracture problems by
the representative cell method,
Comp. Mech., 20, 145149.
The method of the representative cell
can be construed as the static equivalent of
Bloch wave decomposition
or simply the discrete Fourier transform (DFT) used
in the analysis of the dynamic behavior of repetitive structures.
Based on the DFT, the method of the representative cell reduces the
static analysis of the initial
infinite modular structure under arbitrary loading
to the analysis of a
representative cell under appropriate transformed loading and boundary
conditions.
An almost typical boundary value problem
is formulated for the repetitive module which
can be solved by regular closed form
or numerical (finite element) methods,
depending on the complexity of the representative cell.
Having obtained the structural response in the transformed variables
the exact displacements and stresses can be computed at any station along
the structure by the inverse transform.
The advantages of this approach for optimal design are evident.
We now have a means of designing very long repetitive structures
with relatively few variables both for the analysis and for the design.
Although we are dealing
with rather large structures the size of the analysis equations and
of the design
space is commensurate with the dimension of the representative cell.
And since the structural distortions have a tendency to decay very rapidly
beyond the loaded zone, the number of design constraints are also of the
order of the design constraints in the representative cell.
All this has
far reaching implications in the context of multiple reanalysis, sensitivity
calculations and many other aspects of optimum structural design.
It is instructive to mention that the present research does not deal with the
design of extremal composites.
Other investigators have used periodic microstructures
for the topological optimization of multiphase composite.
There, the length scale of variation of the loads applied to the structure
is much larger than the scale of the repeating module and consequently
the stresses applied to the module are of the same period
as the structure, and are given.
The boundary conditions on all modules are thus known and constant
throughout the material.
This paves the way for the analysis of the module.
In the present case, however,
the length scales of the loading and of the repeating modules
are of the same order and the stresses vary spatially from module to module.
We are thus designing periodic structures as opposed to periodic
materials.
 Hakim S. and Fuchs M.B. (1994), A New Criterion for Actuator
Placement in QuasiStatic Control of Flexible Structures Under
Arbitrary Distortions,
45th International Astronautical Conference, Jerusalem 914 Oct.
This paper attempts to quantify the performance of
a given configuration of actuators in an environment of a family of
perturbations (of arbitrary magnitudes).
Every instance of perturbations causes a distortion which is then reduced
by the actuators, leaving a residual distortion.
After scanning all possible perturbations one of these residual distortions
will be maximum. We propose to use this worst case distortion,
J, as a
measure of the quality of the given configuration of actuators.
It turns out that
J is equal to the largest singular value of a rectangular matrix
which depends on the disturbance and the control matrices.
 Hakim S. and Fuchs M.B. (1995), Optimal Actuator Placement with
Minimum Worst Case Distortion Criterion,
AIAA/ ASME/ ACSE/ AHS/ ASC 36th Structures, Structural Dynamics,
and Materials Conference , New Orleans, April.
In this paper the worst residual distortion is employed as
an objective function for finding optimal configurations of actuators.
The structures are a 2D truss beam and a 3D antenna truss, both
representing lightweight structures floating in space.
Extensional actuators are assumed to be embedded in some bars of the
truss. Finding best locations for N actuators out of a total
of M bars is a formidable problem. Exhaustive search must be ruled
out and the paper describes some heuristic search techniques.
The paper also introduces a concept of ideal
actuator, which is a theoretical actuator capable of deforming the
structure in any desired manner. Ideal actuators are obviously better
than any other physical one. Employing ideal actuators one can predict
the lowest possible residual distortions without bothering with positioning
the actuator. It turns out that the residual deformation of real actuators
is not very different from the theoretical lower bound.
(
Postscript
file, 524K). See also,
Hakim S. and Fuchs M.B. (1996),
QuasiStatic Optimal Actuator Placement with
Minimum Worst Case Criterion,
AIAA J., 34(7).
 Hakim S. and Fuchs M.B. (1995), Simulated Annealing Techniques
for the Optimal Control of Space Structures,
First World Congress of Structural and Multidisciplinary
Optimization, Goslar, May 28  June 2.
In this paper the worst residual distortion is employed as
an objective function for finding optimal configurations of actuators.
The structures are a 2D truss beam and a 3D antenna truss, both
representing lightweight structures floating in space.
Extensional actuators are assumed to be embedded in some bars of the
truss. Finding best locations for N actuators out of a total
of M bars is a formidable problem. Exhaustive search must be ruled
out and Simulated Annealing was used to approach the optimal
solution. The paper discusses various numerical aspects of the method
and assesses its merits. The solutions are compared with lower
bounds based on ideal actuators.
It is shown that when replacing the type of actuators, dramatic
improvements can sometimes be expected.
(
Postscript file, 158K).
 Hakim S. and Fuchs M.B. (1996), Shape Estimation
of Distorted Flexible Structures,
6th AIAA/ NASA/ USAF Multidisciplinary Analysis & Optimization
Symposium, Bellevue, Sep 4  6.
The present work discusses the optimal placement of sensors in truss structures
in order to obtain best possible information regarding the distortions
of the structure. The estimation goal is to reconstruct the deformed
shape of the structure, at the controlled degrees of freedom, from the
sensor readings.
A basic assumption is that the structure is subjected to a parametric
disturbance field.
We distinguish between disturbances which cause uniform
or arbitrary distortions of the structure,
and disturbances which cause structured distortions.
The estimator is based on the
least squares method, hence the estimated shape is the one with
least RMS displacement for the given sensor readings. To evaluate
the performance of each set of sensors
the measure is the largest possible
error between the estimated and the actual displacements, at the CDOF.
Results show that for reasonable shape estimation a relatively large number
of sensors is needed.
It is also shown that when using sensors which measure mainly the
distortions of the controlled degrees of freedom,
significant improvements in the shape estimation
can be obtained.
(
Postscript file, 158K).
 Hakim S. and Fuchs M.B. (1997), Optimal Geometries of Shape
Controlled Structures,
Submitted
This paper deals with the geometry design of trusses which must maintain
a set of nodes, the controlled degrees of freedom, as undeformed as
possible.
In contrast with previous research the structure is subjected to a family
of disturbances whose total magnitude is bounded in an overall sense but
which is only loosely defined at any given point in time.
Moreover, embedded in the truss are N_c<\I> ideal controllers which will
control the structure in order to reduce the distortions.
The actuators are assumed to apply optimal control corrections.
It is shown that a measure of the distortions is the (N_c+1)<\I>th
singular values of the disturbance influence matrix.
The purpose is therefore to modify the geometry of the structure
in order to minimize that singular value.
One of the difficulties encountered during the optimization is that of
repeated singular values. Their derivative is different from that of a
single singular value and requires more attention.
It is shown that the design tends to generate structures composed
of stiff segments
joined at flexible interfaces where the actuators are located.
These rather peculiar designs may harbor
interesting guide lines in future implementations of smart structures.
(shape.ps)
Last modified by
Moshe Fuchs
on 28 September, 1999.