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== there are lotus. ==


'''Topology optimisation '''  is a mathematical approach that optimises material layout within a given design space, for a given set of loads and [[boundary conditions]] such that the resulting layout meets a prescribed set of performance targets. Using topology optimisation, engineers can find the best concept design that meets the design requirements.
In a lot of colorful lotus land records.<br><br>ordinary barrel. Also half-human Heights barrel. Half a bucket can be.<br><br>and this pond. Nearly circular shape. About nearly two meters in diameter, so big pond. Inside the one yuan heavy water. Estimated to be sufficient to store ten barrels of heavy water to one yuan.<br><br>both ends of [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 靴 メンズ] the pond in the [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン バッグ] yuan heavy water. Zhang Lin just two colorful, colorful lotus really look the same as [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン セール] described in the reel. Lotus is divided into 288, lotus minimal, there are lotus.<br><br>'Huh?' Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 靴] Yu suddenly surprised again.<br><br>'central pond is arranged a little to hide the magic circle.' Qin Yu force to explore space discovered. This pond central in the end what is being used to hide the magic array block.<br><br>the passage of time, then quickly passed more than three years, more than three years time. [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン バッグ] Liu Yuxi talks nine were sitting cross-legged cry in silence, Qin Yu himself sitting cross-legged in the room heard in silence.<br><br>retreat at the child in the Qin Yu. Suddenly sensed glare to light, when the struggle to open the eyes and yang
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</ul>


Topology optimisation has been implemented through the use of finite element methods for the analysis, and optimisation techniques based on the [[method of moving asymptotes]], [[genetic algorithms]], optimality criteria method, [[level sets]],<ref name="book">[http://books.google.com/books?id=NGmtmMhVe2sC&source=gbs_navlinks_s], a monograph of the subject.</ref> and [[topological derivative]]s.
== leading to a lot of other galaxies Chuansong Zhen. ==


Topology optimisation is used at the concept level of the design process to arrive at a conceptual design proposal that is then fine tuned for performance and manufacturability. This replaces time consuming and costly design iterations and hence reduces design development time and overall cost while improving design performance.
Seeking heart.<br><br>'Yu Huang Majesty broad-minded, will not ......'<br><br>see Emperor Xian Yu also persuade the Vatican, Qin Yu busy hand nonchalantly said: 'Well, I do not have time and クリスチャンルブタン 店舗 you nonsense, leave' Qin Yu finished the whole person speed channeling out, after a great diversion again disappeared.<br><br>Qin Yu next destination 'blue star.'<br><br>blue star, as Angelica galactic core planet, this planet in Star Chuansong Zhen course very much, traffic developed, leading to a lot of other galaxies Chuansong Zhen.<br>'Friends dark astral ......'<br>Emperor Xian Yu Vatican wants to follow to catch up.<br><br>'seems only to let クリスチャンルブタン 偽物 Yu Huang personally resolved, I do not know where now arrived.' Bai star in his own to start a summons to Yu Huang, unfortunately there is no communications success.<br><br>Yu クリスチャンルブタン メンズ 靴 Wong and because he is too far away, only to be summoned クリスチャンルブタン 偽物 within the 'maze Communications' success. Vatican last Emperor Xian Yu Lian cents or so ice will matter to Yu Huang Di reporting.<br><br>Emperor Xian Lian from クリスチャンルブタン通販 the ice to his arraignment
相关的主题文章：
<ul>
 
  <li>?aid=105609</li>
 
  <li>?aid=2</li>
 
  <li>?aid=393439</li>
 
  </ul>


In some cases, proposals from a topology optimisation, although optimal, may be expensive or infeasible to manufacture. These challenges can be overcome through the use of manufacturing constraints in the topology optimisation problem formulation. Using manufacturing constraints, the optimisation yields engineering designs that would satisfy practical manufacturing requirements.
== palace courtyard alone among the Qin Yu ==
In some cases [[Additive manufacturing]] technologies are used to manufacture complex optimized shapes that would otherwise need manufacturing constraints.


== Problem Statement ==
Had lost a war and key central repair, while wind figs, QIN talk about the problem of comprehension aspects. As for the injury itself, Qin Yu, has long been [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン 店舗 東京] a popular tears repaired.<br><br>palace courtyard alone among the Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_10.php クリスチャンルブタン 値段] Yu, Yu Qin, Qin Feng, Qin Zheng, QIN, wind figs five people sitting around the stone table, at the moment on the stone table stood three Cheats, it had left Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン セール] Yu three comprehension Cheats.<br><br>'Kohane, these three cheats you, you and I have seen the wind uncle, you wind uncle say, this three Cheats comprehension of each of the circles are uppermost comprehension Cheats, you wind uncle are now discarded Over the past practice of law, the use of this method of the above first comprehension. 'QIN smile.<br><br>wind figs also exclaimed: 'Kohane is the original Munekado secret pass to [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 東京] practice law, but I also know the secret of Yuan Ying gas only, but obviously not as good as these cheats so deep so proud of mysterious practice of law, it is estimated. Munekado town than we have severe cases of canon. '<br><br>Qin Yu [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_0.php クリスチャンルブタン 店舗] is not talking, but with one hand stretched, spiritual goods is five knives and a black rifles floating
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</ul>


Mathematically one can pose a generic problem as follows:
== 'Nothing ==
:<math>\min_{\rho}\; \int_{\Omega} \phi(\rho) \, \mathrm{d}\Omega</math>


Subject to:
Other tribe still mulberry family palace. '<br><br>'Nothing, in the other direction, I have arranged the tribe.'<br><br>Sang Mo exposed face of evil spirits, while red robe yixian moment, the people turning to red giant octopus, eight Rouzhu startled, of<br><br>towards the south for a streamer will recover the past, Qin Yu faster than the one who is faster on the trace.<br><br>......<br><br>are flying speed towards the south [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン 店舗] Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタンジャパン] Yu suddenly stopped.<br><br>'someone!'<br><br>his spiritual knowledge has clearly found dozens of miles away octopus dozen family tribe, although now found only a dozen, but who knows<br><br>if there are more. Sang Mo bound to have come directly toward their location, Qin Yu had a judge.<br><br>'That Sang ink know where I am, but also the formation of encirclement.'<br><br>stopped for a moment in the Qin Yu Qin Yu spiritual knowledge found behind dozens of miles away [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン パンプス] to have a huge red giant [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタン 東京] octopus came the [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン 銀座] claws.
*<math>\scriptstyle \rho \,\in\, \{0,\, 1\} </math>
相关的主题文章：
*Design Constraints
<ul>
*Governing Differential Equation
 
 
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The problem statement includes the following:
 
 
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'''a.   Objective functional <math>\scriptstyle \left(\int_{\Omega} \phi(\rho) \, \mathrm{d}\Omega\right)</math>:''' This is the goal of the optimization study which is to be minimised over the selection field. For example, one would want to minimise the compliance of a structure to increase structural stiffness.
 
 
  <li>[http://www.yaoyin.net/plus/feedback.php?aid=22 http://www.yaoyin.net/plus/feedback.php?aid=22]</li>
'''b.      Design space <math>\scriptstyle (\Omega)</math>:'''  Design space is the allowable volume within which the design can exist.  Assembly and packaging requirements, human and tool accessibility are some of the factors that need to be considered in identifying this space . With the definition of the design space, regions or components in the model that cannot be modified during the course of the optimisation are considered as non-design regions.
 
 
</ul>
'''c.   The Discrete Selection Field<math>\scriptstyle (\rho)</math>:''' This is the field over which the discrete optimisation is to be performed. It selects or deselects a point on the design space to further the design objective. By selection it has to take the value, 1, and by de-selection it has to take the value, 0.
 
'''d.   Design constraints:''' These are design criteria that need to satisfied. These could include material availability constraints, displacement constraints, etc.
 
'''e.   Governing Differential Equation:''' This is the one that governs the physics of the structure to be built. For example the elasticity equation in the case of stiff structures would be the governing differential equation.
 
== Implementation methodology  ==
The optimisation problem posed above implies the need for a discrete optimisation over every point in domain <math>\scriptstyle \Omega</math>. But, implementation of such a scheme is not yet practicable. So, people adopt continuous optimisation strategies, where they assume that the variable <math>\scriptstyle \rho</math> varies continuously over the domain <math>\scriptstyle (0,\, 1)</math>. This problem is then solved over a set of finite elements obtained from meshing of the domain.
 
=== Topology optimization for stiff structures ===
A stiff structure is one that has the least possible displacement when given certain set of boundary conditions. A global measure of the displacements is the strain energy (also called compliance) of the structure under the prescribed boundary conditions. The lower the strain energy the higher the stiffness of the structure. So, the problem statement involves the objective functional of the strain energy which has to be minimized.  
 
Now the objective functional should be chosen as a function of the selection field <math>\scriptstyle \rho</math>. So in literature, people have interpolated the material properties in terms of the selection field. A widely used interpolation scheme is called the Solid Isotropic Material with Penalization (SIMP). This interpolation is essentially a power law <math>\scriptstyle E \;=\; E_0 \,+\, \rho^n E_1</math> that interpolates the Young's modulus of the material to the scalar selection field. The value of <math>n</math> varies between <math>\scriptstyle [0,\, 3]</math> in general. This has been shown to confirm to micro-structure of the materials.<ref name="Paper Inverse Homogenisation">[http://www.giref.ulaval.ca/~deteix/bois/documents_references/bendsoe1999.pdf], A reference that proved the validity of the interpolation scheme.</ref> So one could view topology optimization to be a process of selection of micro-structure at every point in space so that an objective functional is minimized.
 
On a broad level, one can visualize that more the material, lesser will be the deflection as there is more material to resist the loads. So, the optimization requires an opposing constraint, the volume constraint . This is in reality a cost factor, as we would not want to spend a lot of money on the material. To obtain the total material utilized, an integration of the selection field over the volume can be done.
 
Finally the elasticity governing differential equations are plugged in so as to get the final problem statement.
:<math>\min_{\rho}\; \int_{\Omega} \frac{1}{2} \mathbf{\sigma}:\mathbf{\varepsilon} \,\mathrm{d}\Omega</math>
 
subject to:
*<math>\scriptstyle \rho \,\in\, [0,\, 1] </math>
*<math>\scriptstyle \int_{\Omega} \rho\, \mathrm{d}\Omega \;\leq\; V^*</math>
*<math>\scriptstyle \mathbf{\nabla}\cdot\mathbf{\sigma} \,+\, \mathbf{F} \;=\; {\mathbf{0}} </math>
*<math>\scriptstyle \mathbf{\sigma} \;=\; \mathsf{C}:\mathbf{\varepsilon}</math>
 
But, a straight forward implementation in the Finite Element Framework of such a problem is still infeasible owing to issues such as:
# Mesh dependency—Mesh Dependency means that the design obtained on one mesh is not the one that will be obtained on another mesh. The features of the design become more intricate as the mesh gets refined.
# Numerical instabilities—The selection of region in the form of a chess board.
 
[[File:Checkerboards in Topology Optimization.tif|thumb|Checker Board Patterns are shown in this result.]]
 
Some techniques such as Filtering based on Image Processing are currently being used to alleviate some of these issues.
 
[[File:Topology Optimization with filtereing.tif|thumb|Topology Optimization result when filtering is used]]
 
== Distinction ==
Topology optimization is distinct from [[shape optimization]] since typically shape optimisation methods work in a subset of allowable shapes which have fixed topological properties, such as having a fixed number of holes in them. Therefore topology optimisation is used to generate concepts and shape optimisation is used to fine-tune a chosen design topology.
 
There are various methods used to perform topology optimisation:
* Solid Isotropic Material with Penalisation (SIMP),
* Evolutionary Structural Optimisation (ESO),
* [[Topological derivative]]s.
 
==References==
{{Reflist}}
* Investigation into Structural Topology Optimization Problem Formulations, William Renold, lulu.com, 2007.
* [http://www.springerlink.com/content/t5732j08vk143274/ Recent Developments in the Commercial Implementation of Topology Optimization]; Uwe Schramm, Ming Zhou; IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials: Status and Perspectives, 239–248; 2006 Springer.
*[http://www.springerlink.com/content/l317544685kr4263/?p=86d69fac23964a0aaccc6c10e08701aa&pi=3/ Industrial Implementation and Applications of Topology Optimization and Future Needs]; Claus B.W. Pedersen; Peter Allinger; IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, 229-238; 2006 Springer.
*[http://www.springerlink.com/content/ge6gcby65c0gfeg6/ A 99 line topology optimization code written in Matlab] Ole Sigmund; Int Journal of Structural and Multidisciplinary Optimization, Volume 21, 120-127, 2001 Springer
*[http://www.springerlink.com/content/a7653p3h70116666/ Topology optimization of 2D continua for minimum compliance using parallel computing] Arash Mahdavi; Balaji Raghavan; Mary Frecker; Int Journal of Structural and Multidisciplinary Optimization, Volume 32, 121-132, 2006 Springer
*[http://vrand.com/node/48 Modern Structural Optimization Concepts Applied to Topology Optimization] Juan Pablo Leiva; Brian C. Watson and Iku Kosaka ; 40th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Material Conference. St. Louis, MO, pp.&nbsp;1589–1596, 1999
 
==External links==
* [http://www.topopt.dtu.dk] - page containing information on the SIMP optimization method.
* [http://www.caess.eu/site/Animations.html Topology optimization animations]
 
{{DEFAULTSORT:Topology Optimization}}
[[Category:Mathematical optimization]]
[[Category:Topology]]

Latest revision as of 06:19, 12 January 2015

there are lotus.

In a lot of colorful lotus land records.

ordinary barrel. Also half-human Heights barrel. Half a bucket can be.

and this pond. Nearly circular shape. About nearly two meters in diameter, so big pond. Inside the one yuan heavy water. Estimated to be sufficient to store ten barrels of heavy water to one yuan.

both ends of クリスチャンルブタン 靴 メンズ the pond in the クリスチャンルブタン バッグ yuan heavy water. Zhang Lin just two colorful, colorful lotus really look the same as クリスチャンルブタン セール described in the reel. Lotus is divided into 288, lotus minimal, there are lotus.

'Huh?' Qin クリスチャンルブタン 靴 Yu suddenly surprised again.

'central pond is arranged a little to hide the magic circle.' Qin Yu force to explore space discovered. This pond central in the end what is being used to hide the magic array block.

the passage of time, then quickly passed more than three years, more than three years time. クリスチャンルブタン バッグ Liu Yuxi talks nine were sitting cross-legged cry in silence, Qin Yu himself sitting cross-legged in the room heard in silence.

retreat at the child in the Qin Yu. Suddenly sensed glare to light, when the struggle to open the eyes and yang 相关的主题文章：

leading to a lot of other galaxies Chuansong Zhen.

Seeking heart.

'Yu Huang Majesty broad-minded, will not ......'

see Emperor Xian Yu also persuade the Vatican, Qin Yu busy hand nonchalantly said: 'Well, I do not have time and クリスチャンルブタン 店舗 you nonsense, leave' Qin Yu finished the whole person speed channeling out, after a great diversion again disappeared.

Qin Yu next destination 'blue star.'

blue star, as Angelica galactic core planet, this planet in Star Chuansong Zhen course very much, traffic developed, leading to a lot of other galaxies Chuansong Zhen.
'Friends dark astral ......'
Emperor Xian Yu Vatican wants to follow to catch up.

'seems only to let クリスチャンルブタン 偽物 Yu Huang personally resolved, I do not know where now arrived.' Bai star in his own to start a summons to Yu Huang, unfortunately there is no communications success.

Yu クリスチャンルブタン メンズ 靴 Wong and because he is too far away, only to be summoned クリスチャンルブタン 偽物 within the 'maze Communications' success. Vatican last Emperor Xian Yu Lian cents or so ice will matter to Yu Huang Di reporting.

Emperor Xian Lian from クリスチャンルブタン通販 the ice to his arraignment 相关的主题文章：

  • ?aid=105609
  • ?aid=2
  • ?aid=393439

palace courtyard alone among the Qin Yu

Had lost a war and key central repair, while wind figs, QIN talk about the problem of comprehension aspects. As for the injury itself, Qin Yu, has long been クリスチャンルブタン 店舗 東京 a popular tears repaired.

palace courtyard alone among the Qin クリスチャンルブタン 値段 Yu, Yu Qin, Qin Feng, Qin Zheng, QIN, wind figs five people sitting around the stone table, at the moment on the stone table stood three Cheats, it had left Qin クリスチャンルブタン セール Yu three comprehension Cheats.

'Kohane, these three cheats you, you and I have seen the wind uncle, you wind uncle say, this three Cheats comprehension of each of the circles are uppermost comprehension Cheats, you wind uncle are now discarded Over the past practice of law, the use of this method of the above first comprehension. 'QIN smile.

wind figs also exclaimed: 'Kohane is the original Munekado secret pass to クリスチャンルブタン 東京 practice law, but I also know the secret of Yuan Ying gas only, but obviously not as good as these cheats so deep so proud of mysterious practice of law, it is estimated. Munekado town than we have severe cases of canon. '

Qin Yu クリスチャンルブタン 店舗 is not talking, but with one hand stretched, spiritual goods is five knives and a black rifles floating 相关的主题文章：

'Nothing

Other tribe still mulberry family palace. '

'Nothing, in the other direction, I have arranged the tribe.'

Sang Mo exposed face of evil spirits, while red robe yixian moment, the people turning to red giant octopus, eight Rouzhu startled, of

towards the south for a streamer will recover the past, Qin Yu faster than the one who is faster on the trace.

......

are flying speed towards the south クリスチャンルブタン 店舗 Qin クリスチャンルブタンジャパン Yu suddenly stopped.

'someone!'

his spiritual knowledge has clearly found dozens of miles away octopus dozen family tribe, although now found only a dozen, but who knows

if there are more. Sang Mo bound to have come directly toward their location, Qin Yu had a judge.

'That Sang ink know where I am, but also the formation of encirclement.'

stopped for a moment in the Qin Yu Qin Yu spiritual knowledge found behind dozens of miles away クリスチャンルブタン パンプス to have a huge red giant クリスチャンルブタン 東京 octopus came the クリスチャンルブタン 銀座 claws. 相关的主题文章：

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