Free Material Optimization for Shells and Plates [Elektronische Ressource] = Freie Materialoptimierung für Schalen und Platten / Stefanie Gaile. Betreuer: Günter Leugering

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• Publié par: friedrich-alexander-universitat_erlangen-nurnberg
• Publié le: 01 janvier 2011
• Nombre de lectures: 20
• Langue: English
• Poids de l'ouvrage: 4 Mo

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FREE MATERIAL OPTIMIZATION
FOR SHELLS AND PLATES
¨FREIE MATERIALOPTIMIERUNG FUR SCHALEN UND PLATTEN
Der Naturwissenschaftlichen Fakulta¨t
der Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg
zur
Erlangung des Doktorgrades Dr. rer. nat.
vorgelegt von
Stefanie Gaile
aus Mu¨nchenAls Dissertation genehmigt
von der Naturwissenschaftlichen Fakulta¨t
der Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg
Tag der mu¨ndlichen Pru¨fung: 27. Mai 2011
Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink
Erstberichterstatter: Prof. Dr. Gu¨nter Leugering
Zweitberichterstatter: Prof. Dr. Michael StinglAbstract
Within this thesis we develop mathematical models and numerical methods for the Free Ma-
terial Optimization problem for shells and plates. In Chapter 1 we provide a motivation arising
from structural engineering to address this problem and classify the Free Material Optimization
approach within other common methods in structural design optimization.
In Chapter 2 we introduce the foundations of differential geometry and continuum mechanics
necessary for a reliable prediction of the shell’s elastic behavior. The chosen description is based
on the theory of Cosserat continua, a direct approach to the shell as a two-dimensional midsur-
face in physical space endowed with director vectors to provide additional degrees of freedom in
order to model bending and shear deformations. Thus the displacements consist of a translation
of the points on the midsurface and a rotation of the associated director vectors, in which drilling
– the rotation around their own axis – is neglected as the director vectors are considered to be
infinitely thin. We restrict ourselves to displacements that fulfill the Reissner-Mindlin kinemat-
ical assumption: the material lines, that are represented by the director vectors, remain straight
and unstretched during deformation. This requirement leads to a first-order approximation of
the three-dimensional elasticity theory including shear effects, which is known as Naghdi’s shell
model. The material tensors C and D of the shell, whose entries can be depicted as spring
constants connecting all possible directions, are directly derived from the three-dimensional
elasticity tensor by including monoclinic material symmetry as well as the basic assumption of
vanishing normal stresses commonly used in shell theory. Moreover, in the special case of a
planar midsurface the formulas for the membrane, bending and shear strains of this model can
be considerably simplified leading to the Reissner-Mindlin plate model.
Chapter 3 is dedicated to the development of a Free Material Optimization formalism for
Naghdi shells. Free Material Optimization is a subbranch of structural optimization and accord-
ingly deals with the problem of finding the stiffest structure for a given design domain and a
predefined set of loads constructed from a limited amount of material. To this end we consider
the entire elasticity tensors C and D in their most general form as optimization variables and
include only the basic requirements for linear elastic material in the constraints. This freedom in
the design space leads to the ultimately best design, although the optimal material typically does
not preexist in nature and approximations of the optimal structure have to be intricately manu-
factured e.g. by using tapelayering techniques or constructing composites. Moreover, since we
have no information about the density of an arbitrary material, we require another measure for
the amount of used material and employ a summed trace of the elasticity tensors for this purpose.
Based on the minimum potential energy principle specifying the equilibrium of Naghdi shells
we are thus able to state the minimum compliance problem. Its saddlepoint structure allows us
to show existence of at least one optimal solution via a Minimax theorem. Furthermore we prove
that this problem is equivalent to the dual of a nonlinear convex semidefinite program, which we
refer to as the primal problem formulation. To this end we employ Lagrange duality and show a
local maximum principle for matrices that guarantees the existence as well as necessary charac-
teristics of suitable Lagrange multipliers. Within this proof we also obtain information about the
This research was funded by the European Commission within the Sixth Framework Programme STREP 30717
PLATO-N.
iiistructure of the optimal material tensor at the upper and lower material bounds. In contrast to the
original saddlepoint formulation the primal problem is a convex problem, moreover the material
matrices, that tremendously increase the dimension of the discretized problem, are hidden in the
primal problem as Lagrange multipliers. Accordingly the primal problem is considerably better
suited for a numerical approach than the saddlepoint problem and the equivalence proof ensures
that the obtained solutions are also optimal with respect to the original problem formulation.
Moreover we introduce the minimum weight problem formulation which has recently gained
much attention due to the development of numerical solvers that render this problem’s structure
computationally tractable.
In Chapter 4 we focus on the numerical solution of the preceding problem formulations. To
this end we apply a finite element method to obtain discretized versions of the previously intro-
duced optimization problems. After a sensitivity analysis we are able to compute solutions by
employing the nonlinear semidefinite programming code PENSCP, an efficient solver for prob-
lems arising from Free Material Optimization. We test our software on a collection of numerical
test examples frequently used in the structural optimization of shells and show the validity of
our results by comparing them with solutions originating from other prominent material opti-
mization approaches. Our software does not only provide reliable results including extensive
information about the optimal material properties, but is also designed for realistic applications
as it allows the simultaneous optimization of structures composed of solids as well as shells.
Moreover it is possible to include condensed data structures in order to decrease the problem
dimension by a static condensation of segments that are constructed from fixed material or to
simulate the characteristics of element types which are not comprised in our software package.
In Chapter 5 we discuss the extension of the Free Material Optimization problem for shells
by multidisciplinary optimization constraints. We consider linear displacement constraints in a
discrete context, which can be utilized to manipulate the shape of the deformed structure. In
the case of stress constraints we distinguish between in-plane stresses and out-of-plane stresses
and apply them in order to avoid material damage or even failure due to high stresses. For
the formulation of eigenfrequency constraints we introduce a dynamic model describing the
free vibrations of Naghdi shells. Therefrom we deduce a semidefinite matrix constraint that
can be employed to raise the natural frequency of the structure to prohibit resonance excitation
and ultimately a resonance disaster. The final type of constraints regarded by us are buckling
constraints, which address the susceptibility of shells to geometrical imperfections and load
perturbations. The resulting sudden deformations are captured by a nonlinear strain model,
which is used for the derivation of nonlinear semidefinite matrix constraints in order to avoid
buckling behavior.
The thesis is concluded by a summary emphasizing continuative questions for future research
in Chapter 6.
ivZusammenfassung
Im Rahmen dieser Dissertation entwickeln wir mathematische Modelle und numerische Metho-
den, um das freie Materialoptimierungsproblem fu¨r Schalen und Platten zu lo¨sen. In Kapitel 1
stellen wir die aus der Bautechnik stammende Motivation fu¨r dieses Problem vor und ordnen
den Ansatz der freien Materialoptimierung in Bezug zu anderen gebra¨uchlichen Methoden in
der Strukturoptimierung ein.
Im 2. Kapitel stellen wir die Grundlagen der Differentialgeometrie und der Kontinuumsme-
chanik vor, die fu¨r eine zuverla¨ssige Vorhersage des elastischen Verhaltens der Schale no¨tig
sind. Die gewa¨hlte Beschreibung basiert auf der Theorie der Cosserat-Medien, einem direk-
ten Zugang zur Schale in Form einer zweidimensionalen Mittelfla¨che im physischen Raum, die
mit Direktorvektoren ausgestattet ist. Die dadurch zur Verfu¨gung gestellten zusa¨tzlichen Frei-
heitsgrade dienen dazu, Biege- und Scherdeformationen zu modellieren. Demzufolge bestehen
die Verschiebungen aus der Translation der Punkte auf der Mittelfla¨che und einer Rotation der
zugeho¨rigen Direktorvektoren, wobei das Drillen – die Rotation der Vektoren um ihre eige-
ne Achse – vernachla¨ssigt wird, da man annimmt, dass die Direktorvektoren unendlich du¨nn
sind. Wir beschra¨nken uns auf Verschiebungen, die die kinematischen Annahmen der Reissner-
Mindlinschen Theorie erfu¨llen: die durch die Direktorvektoren repra¨sentierten Materiallinien
bleiben wa¨hrend der Deformation gerade und vera¨ndern dabei auch nicht ihre La¨nge. Diese
Bedingung fu¨hrt zu einer Na¨herung erster Ordnung der dreidimensionalen Elastizita¨tsthe