Konferencja na temat Optymalizacji Topologicznej

W piątek 19 maja 2006 r. odbyło się pół-dniowe seminarium na temat optymalizacji
konstrukcji:
Konferencja na temat Optymalizacji Topologicznej
Program tej krótkiej konferencji obejmował cztery referaty godzinne:
Matthew Gilbert, Andy Tyas
Department of Civil and Structural Engineering,
University of Sheffield, UK
Identification of optimal truss layouts using an efficient computational procedure
Sławomir Czarnecki
Politechnika Warszawska, Wydział Inżynierii Lądowej
Application of the moving asymptotes method in
compliance optimization of the truss and frame structures
Cezary Graczykowski,
Instytut Podstawowych Problemów Techniki PAN
Tomasz Lewiński
Politechnika Warszawska, Wydział Inżynierii Lądowej
Michell cantilevers constructed within trapezoidal domains
Grzegorz Dzierżanowski
Politechnika Warszawska, Wydział Inżynierii Lądowej
Solution procedures for 2-D continua minimum compliance problem
Referaty dotyczyły tematyki współczesnej: optymalizacji rozkładu prętów w konstrukcjach
dyskretnych i materiałów w konstrukcjach o ciągłym rozkładzie masy. Właśnie ta tematyka
była inspiracją wielu referatów profesorów W. Zalewskiego i W.Zabłockiego.W dyskusji,
prócz wykładowców, wzięli udział: prof. Grzegorz Jemielita, prof. Wacław Szcześniak, dr
hab. Stanisław Jemioło oraz studenci TiKAK.
Udział w obradach był oczywiście bezpłatny. Goście nie obciążyli finansów Wydziału:
sami opłacili pobyt w hotelu na Polnej oraz przeloty samolotem Konferencja ta dowodzi że
możliwe jest organizowanie konferencji bezpłatnych o wysokim poziomie naukowym. Nie
było oczywiście bankietu ani uroczystych obiadów na koszt Wydziału. Konferencję tę na
długo zapamiętają nasi studenci, gdyż wykłady były długie i dokładne. Goście z Anglii
zgodzili się mówić powoli i wyraźnie. Wydaje się że poziom wykładów był wyrównany,
porównywalny z wykładami na kongresach międzynarodowych, z tą różnicą, że na
konferencjach nawet wykłady generalne są o połowę krótsze.
Tomasz Lewiński, 29 maja 2006.
Politechnika Warszawska, Wydział Inżynierii Lądowej
Instytut Mechaniki Konstrukcji Inżynierskich
al. Armii Ludowej 16, PL 00-637 Warszawa
Konferencja na temat
Optymalizacji Topologicznej
19 maja 2006, piątek ,
sala 101, Gmach Wydziału Inżynierii Lądowej PW
9.15
Matthew Gilbert, Andy Tyas
Department of Civil and Structural Engineering,
University of Sheffield, UK
Identification of optimal truss layouts using an efficient
computational procedure
10.15
Sławomir Czarnecki
Politechnika Warszawska, Wydział Inżynierii Lądowej
Application of the moving asymptotes method in
compliance optimization of the truss and frame
structures
11.15
Cezary Graczykowski,
Instytut Podstawowych Problemów Techniki PAN
Tomasz Lewiński
Politechnika Warszawska, Wydział Inżynierii Lądowej
Michell cantilevers constructed within trapezoidal
domains
12.15
Grzegorz Dzierżanowski
Politechnika Warszawska, Wydział Inżynierii Lądowej
Solution procedures for 2-D continua minimum
compliance problem
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Identification of optimal truss layouts using an efficient computational procedure
Matthew Gilbert and Andy Tyas
Department of Civil and Structural Engineering, University of Sheffield, UK
The study of truss layout or topology optimisation has challenged academic researchers for over a century. In
its simplest form, the problem is to find the lightest set of truss bars to transfer a given load (or loads) to
defined supports whilst satisfying equilibrium and stress constraints. However, whilst provably optimal
trusses have been found for many simple problems by researchers such as Michell, Hemp, Chan, Rozvany,
Lewinski and others, identifying optimal layouts using analytical methods can be difficult. Hence a
computational procedure was developed some 40 years ago by Dorn et al., who proposed the ground structure
approach whereby a design space is populated with grid-points, inter-connected by potential truss bars. Linear
programming (LP) may then be used to determine the sub-set of these bars present in the final optimal
structure. Unfortunately, for a problem containing n grid points, there are a total of n(n - 1)/2 potential truss
bars, which means that even using state-of-the-art LP solvers the maximum size of problem that can be solved
this way on a desktop PC is still relatively small (c. 103 nodes and 105-106 potential members). Whilst grids
with this density can be used to give good estimates of simple 2D Michell structures, they cannot be used to
give accurate solutions for more complicated 2D, or even comparatively simple 3D, problems. This has
undoubtedly limited interest in the method.
To address this issue, Gilbert and Tyas have developed an iterative solution strategy which has significantly
increased the size of problem which can be solved. The procedure involves starting with an initial minimal
connectivity ground structure and adding further potential bars at successive iterations. The structures
obtained at the end of the process are provably optimal for the grid discretisation selected. Problems with
ground structures comprising >104 nodes and >109 potential members can be solved, allowing accurate
numerical solutions to be obtained for even geometrically complex problems. The validity of the solutions
may be checked against existing analytical solutions and, most importantly, the method can be used to indicate
the form of structures for which no analytical solutions yet exist (perhaps in due course also enabling new
provably optimal structural forms to be deduced).
In the presentation the iterative solution strategy will initially be described, including extension of the basic
method to permit transmissible (rather than fixed position) applied loads to be treated. Various example
problems will then be examined (e.g. see Fig. 1). It will be shown, apparently for the first time, that the
optimal structural form to carry a uniformly distributed load between pinned supports is not a single parabolic
arch rib (or cable) as is usually supposed, but a more complex structural frame with a central parabolic arch
section. Finally, a key reason for developing the new solution strategy was to enable practical problems to be
tackled. Hence plastic design problems involving multiple load cases and simple “joint-cost” penalties will
also be considered. Stability issues will also be briefly touched upon.
(a)
(b)
(c)
Fig.1 – Sample solutions: (a) vertical 3D cantilever with 2 load cases; (b) 2D cantilever with unequal tensile and
compressive stresses & transmissible load; (c) 2 pin arch problem with fixed or transmissible distributed loading
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Application of the moving asymptotes method
in compliance optimization of the truss and frame structures
Sławomir Czarnecki
Warsaw University of Technology
Faculty of Civil Engineering
Division of Applied Computer Science in Civil Engineering
The problem of designing the stiffest truss and frame with a given and fixed number of joints and
element connections is considered. The design variables are the cross sectional areas of the bars
or/and the nodal points locations. In each case a maximal volume of the structure, constituting an
isoperimetric unilateral condition is prescribed. The nodal force vector is assumed to be independent
of the design variables, hence fixed during the optimization process. The equilibrium equations of
the trusses are obtained by the conventional linear as well as nonlinear finite element analyses taking
into account large nodal displacements and small deformations of members. The equilibrium
equations of the frames are obtained only by the conventional linear finite element analyses.
New optimal layouts of plane and space structures are found by using the moving asymptotes method
(MMA). Application of the MMA is strictly connected with the sensitivity analysis of the
optimization problem. The derivation of the analytical sensitivity formulae for the trusses and
frames, is based on the direct method. In the case of the linear and nonlinear truss structures, such
formulae can be relatively easily obtained due to the dyadic form of the stiffness matrix. In the case
of frame structures, the derivation of the sensitivity formulae is a little more complicated, and
symbolic computation system Maple is used to obtain exact analytical partial derivatives of the
stiffness matrix and other fields.
The MMA proved to be efficient and reliable method for the relatively big number of the variables.
The main advantage of this method – the insensibility to the scaling of the variables, has allowed to
perform numerical procedures for one design vector-parameter: element sizes and shape variables
together.
Geometry and topology optimization – initial and optimal layout of the 2D / 3D Michell truss
Geometry and topology optimization – initial and optimal layout of the 3D frame structure
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Michell cantilevers constructed within trapezoidal domains
Cezary Graczykowski * and Tomasz Lewiński **
(*) Institute of Fundamental Technological Research, Polish Academy of Sciences, Warszawa
(**) Warsaw University of Technology, Faculty of Civil Engineering, Warszawa, Poland
The first part of the presentation refers to the original problem of Michell’s cantilever supported on a
circle. We prove that two formulae for the optimal weight give the same exact result, see [1].
The second part concerns Michell cantilevers transmitting a given point load to a given segment
of a straight-line support, the feasible domain being a part of the half-plane contained between two
fixed half-lines. The axial stress in the optimal cantilevers is assumed to be bounded from below and
from the upper, the ratio of the allowable tensile to the compressive stress being a control parameter
of the problem. The work provides generalization of the results of the paper: T.Lewiński, M.Zhou,
G.I.N. Rozvany, Extended exact solutions for least-weight truss layouts-Part I: Cantilever with a
horizontal axis of symmetry [Int.J.Mech.Sci. 36(1994) 375-398] which was restricted to the case
when the allowable stresses in tension and compression are equal, see [2,3].
This part of the presentation concerns:
i)
analytical formation of the Hencky nets (or the lines of fibres) filling up the interior of the
optimal cantilevers corresponding to an arbitrary position of the point of application of
the given concentrated force,
ii)
construction of the corresponding virtual displacement fields,
iii)
distribution of the force fields in the meaning of W.Hemp - the force fields analysis
introduces a new division of the cantilever domain and gives an alternative method for
computing the optimal weights,
iv)
analysis of concrete examples, new benchmarks for topology optimization.
References
[1] C.Graczykowski, T.Lewiński, The lightest plane structures of a bounded stress level transmitting a point load to a
circular support, Control and Cybernetics, 34(2005) no 1, 227-253
[2] Graczykowski, C., Lewiński, T. Michell cantilevers constructed within trapezoidal domains. Part I: Geometry of
Hencky nets. Part II: Virtual displacement fields. Part III: Force fields. Part IV: Complete exact solutions of selected
optimal designs and their approximations by trusses of finite number of joints Structural and Multidisciplinary
Optimization, 2006 in press
[3] C. Graczykowski, T. Lewiński, Force fields within Michell-like cantilevers transmitting a point load to a straight
support, pp 55-65. In: M.Bendsøe, N.Olhoff, O.Sigmund (Eds.). IUTAM Symposium on Topological Design
Optimization of Structures, Machines and Materials. Status and Perspectives. Vol 137 of the Series: Solid Mechanics
and its Applications. [The Conference held at Rungstedgaard, Denmark, October 26 - 29, 2005]. Springer, Dordrecht
2006.
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Solution procedures for 2-D continua minimum compliance problem
Grzegorz Dzierżanowski
Institute of Structural Mechanics, Warsaw University of Technology, Warsaw, Poland
Correct formulation of minimum compliance problem for Kirchhoff plate takes the form
J = min { ∫ [ M : (d : M ) + λθ ] dx θ ∈ L∞ (Ω;[0,1] ), M ∈ S (Ω), d ∈ Lθ } ,
(1)
Ω
where M denotes the tensor of statically admissible moments, d stands for the constitutive
(compliance) tensor, Lθ denotes a space of microstructural laminates obtained in the theory of
homogenization, θ refers to a function of material layout on the plate middle surface (Ω), and λ
stands for the Lagrange multiplier for the isoperimetric condition, which determines the maximal
area of one of the materials on (Ω).
By exchanging the ‘min’ operations over d ∈ Lθ and θ ∈ L∞ (Ω;[0,1]) with the integration,
which is justified by the local character of the G-convergence theory, one is able to obtain the
following modified formula
∗
J = min { ∫ min [ 2Wopt
(M,θ ) + λθ ] dx M ∈ S (Ω) } ,
Ω
0 ≤ θ ( x ) ≤1
(2)
where 2W*opt (M,θ) = min { M : (d : M) | d ∈ Lθ }. On the other hand local minimization over the
constitutive tensors followed by the use of the Castigliano Theorem leads to the displacement-based
form of minimum compliance problem
J = min max { 2 ∫ qv ds + ∫ [ − 2Wopt (κ ,θ ) + λθ ] dx θ ∈ L∞ (Ω;[0,1] ), v ∈ V (Ω)}
Γσ
(3)
Ω
where 2Wopt (κ,θ) = min { κ : (D : κ) | D ∈ Lθ }, κ denotes the curvature change tensor, D stands for
constitutive (stiffness) tensor, q refers to a load function, and v denotes the kinematically admissible
displacement function. Above introduced W*opt and Wopt can be expressed analytically by making use
of the translation method of Gibianski and Cherkaev, (G&C, 1984). Direct application of Castigliano
Theorem to formula (1) leads to the alternative variant of the minimum compliance problem
J = min max min{2 ∫ qv ds + ∫ [−κ : (D : κ ) + λθ ] dx θ ∈ L∞ (Ω, [ 0,1] ), v ∈V (Ω), D ∈ Lθ } (4)
Γσ
Ω
The numerical analysis of optimal design process given by formulae (3), (4) consists of
iterative search of values (θ, D) thus minimizing J, see (B&K, 1988); (C&L, 2001); (A, 2002);
(D, 2004), while thanks to notation (2) one is able to perform this task by one-step, simultaneous
nonlinear static analysis and optimization of material distribution.
References
Allaire G (2002) Shape Optimization by the Homogenization Method, Springer, New York.
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using homogenization method, Comp.
Meth. Appl. Mech. Eng., 71, 197-224.
Czarnecki S, Lewiński T (2001) Optimal layouts of a two-phase isotropic material in thin elastic plates, Proc. 2nd Europ.
Conf. on Computational Mechanics, Kraków, ECCM-2001, CD-ROM.
Dzierżanowski G (2004) Optimization of material distribution in elastic surface girders (in Polish), PhD Thesis, Warsaw
University of Technology.
Gibianski LV, Cherkaev AV (1984) Designing composite plates of extremal rigidity. Preprint no. 914, Fiz.-Tekhn. Inst.
A.F. Joffe AN SSSR, (in Russian), see also: Cherkaev AV and Kohn RV (eds), Topics in the mathematical modelling of
composite materials, Birkhäuser, Boston 1997.
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