1535:). The selection of the parametrization approach depends mainly on the size of the problem: the CAD approach is preferred for small-to-medium sized models whilst the mesh morphing approach is the best (and sometimes the only feasible one) for large and very large models. The multi-objective Pareto optimization (NSGA II) could be utilized as a powerful approach for shape optimization. In this regard, the Pareto optimization approach displays useful advantages in design method such as the effect of area constraint that other multi-objective optimization cannot declare it. The approach of using a penalty function is an effective technique which could be used in the first stage of optimization. In this method the constrained shape design problem is adapted to an unconstrained problem with utilizing the constraints in the objective function as a penalty factor. Most of the time penalty factor is dependent to the amount of constraint variation rather than constraint number. The GA real-coded technique is applied in the present optimization problem. Therefore, the calculations are based on real value of variables.
25:
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defined on a rectangular box around the shape, which is positive inside of the shape, zero on the boundary of the shape, and negative outside of the shape. One can then evolve this function instead of the shape itself. One can consider a rectangular grid on the box and sample the function at the grid
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Shape optimization can be faced using standard optimization methods if a parametrization of the geometry is defined. Such parametrization is very important in CAE field where goal functions are usually complex functions evaluated using numerical models (CFD, FEA,...). A convenient approach, suitable
489:
One approach is to follow the boundary of the shape. For that, one can sample the shape boundary in a relatively dense and uniform manner, that is, to consider enough points to get a sufficiently accurate outline of the shape. Then, one can evolve the shape by gradually moving the boundary points.
513:
A third approach is to think of the shape evolution as of a flow problem. That is, one can imagine that the shape is made of a plastic material gradually deforming such that any point inside or on the boundary of the shape can be always traced back to a point of the original shape in a one-to-one
110:
is, in addition, concerned with the number of connected components/boundaries belonging to the domain. Such methods are needed since typically shape optimization methods work in a subset of allowable shapes which have fixed topological properties, such as having a fixed number of holes in them.
1201:
273:, which is a way of saying that we would like to find a rather pleasing shape as a solution, not some jumble of rough bits and pieces. Sometimes additional constraints need to be imposed to that end to ensure well-posedness of the problem and uniqueness of the solution.
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In this case the parametrization is defined after the meshing stage acting directly on the numerical model used for calculation that is changed using mesh updating methods. There are several algorithms available for mesh morphing
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Given a known three-dimensional object with a fixed radiation source inside, deduce the shape and size of the source based on measurements done on part of the boundary of the object. A formulation of this
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for a wide class of problems, consists in the parametrization of the CAD model coupled with a full automation of all the process required for function evaluation (meshing, solving and result processing).
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is evolved in the direction of negative shape gradient in order to reduce the value of the cost functional. Higher order derivatives can be similarly defined, leading to
Newtonlike methods.
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Typically, gradient descent is preferred, even if requires a large number of iterations, because, it can be hard to compute the second-order derivative (that is, the
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points. As the shape evolves, the grid points do not change; only the function values at the grid points change. This approach, of using a fixed grid, is called the
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Talebitooti, R.; shojaeefard, M.H.; Yarmohammadisatri, Sadegh (2015). "Shape design optimization of cylindrical tank using b-spline curves".
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1196:{\displaystyle d{\mathcal {F}}(\Omega _{0};V)=\lim _{s\to 0}{\frac {{\mathcal {F}}(\Omega _{s})-{\mathcal {F}}(\Omega _{0})}{s}}}
474:. That is, one starts with an initial guess for a shape, and then gradually evolves it, until it morphs into the optimal shape.
1732:— Simulations and bibliography of the optopo group at Ecole Polytechnique (France). Homogenization method and level set method.
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The idea is again that shapes are difficult entities to be dealt with directly, so manipulate them by means of a function.
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is present, one has to find ways to convert the constrained problem into an unconstrained one. Sometimes ideas based on
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problem. Furthermore, the space of allowable shapes over which the optimization is performed does not admit a
1372:{\displaystyle d{\mathcal {F}}(\Omega _{0};V)=\langle \nabla {\mathcal {F}},V\rangle _{\partial \Omega _{0}}}
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Topological optimization techniques can then help work around the limitations of pure shape optimization.
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Among all three-dimensional shapes of given volume, find the one which has minimal surface area. Here:
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925:{\displaystyle x(0)=x_{0}\in \Omega _{0},\quad x'(s)=V(x(s)),\quad T_{s}(x_{0})=x(s),\quad s\geq 0}
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To solve a shape optimization problem, one needs to find a way to represent a shape in the
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is a valid choice for complex problems that resolves typical issues associated with
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structure, making application of traditional optimization methods more difficult.
659:{\displaystyle f_{t}:\Omega _{0}\to \Omega _{t},{\mbox{ for }}0\leq t\leq t_{0}.}
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413:{\displaystyle {\mathcal {G}}(\Omega )={\mbox{Volume}}(\Omega )={\mbox{const.}}}
100:. In many cases, the functional being solved depends on the solution of a given
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if this limit exists. If in addition the derivative is linear with respect to
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506:. The idea of using a function to represent the shape is at the basis of the
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such as discontinuities in the computed objective and constraint functions.
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If the shape optimization problem has constraints, that is, the functional
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281:
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Find the shape of various mechanical structures, which can resist a given
437:. Here the constraints could be the wing strength, or the wing dimensions.
1672:
Introduction to Shape
Optimization: Theory, Approximation and Computation
1657:
Shapes and
Geometries - Analysis, Differential Calculus, and Optimization
123:
119:
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344:{\displaystyle {\mathcal {F}}(\Omega )={\mbox{Area}}(\partial \Omega )}
1717:
Differentiation with respect to the domain in boundary value problems
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A Survey on Level Set
Methods for Inverse Problems and Optimal Design
1278:{\displaystyle \nabla {\mathcal {F}}\in L^{2}(\partial \Omega _{0})}
1652:. European Journal of Applied Mathematics, vol.16 pp. 263–301.
1544:
1004:{\displaystyle \Omega _{0}\mapsto T_{s}(\Omega _{0})=\Omega _{s}.}
486:, and follow its evolution. Several approaches are usually used.
1687:
Numerical
Methods in Sensitivity Analysis and Shape Optimization
1719:. Numer. Funct. Anal. and Optimiz., 2(7&8), 649-687 (1980).
122:, shape optimization can be posed as the problem of finding a
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is called the shape gradient. This gives a natural idea of
16:
Problem of finding the optimal shape under given conditions
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1618:
Ashok D. Belegundu, Tirupathi R. Chandrupatla. (2003)
1608:. Applied Mathematical Sciences 146, Springer Verlag.
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Optimization
Concepts and applications in Engineering
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which is optimal in that it minimizes a certain cost
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1674:. Society for Industrial and Applied Mathematic.
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1606:Shape optimization by the homogenization method
466:Shape optimization problems are usually solved
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88:theory. The typical problem is to find the
1074:with respect to the shape is the limit of
456:fit leads to a shape optimization problem.
231:{\displaystyle {\mathcal {G}}(\Omega )=0.}
69:Learn how and when to remove this message
1014:Then the Gâteaux or shape derivative of
32:This article includes a list of general
1040:{\displaystyle {\mathcal {F}}(\Omega )}
673:Iterative methods using shape gradients
181:{\displaystyle {\mathcal {F}}(\Omega )}
1738:
1702:Applied Shape Optimization for Fluids
1655:Delfour, M.C.; Zolesio, J.-P. (2001)
1567:
1402:{\displaystyle \nabla {\mathcal {F}}}
1700:Mohammadi, B.; Pironneau, O. (2001)
18:
444:while having a minimal mass/volume.
13:
1685:Laporte, E.; Le Tallec, P. (2003)
1670:Haslinger, J.; Mäkinen, R. (2003)
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697:and the family of transformations
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497:Another approach is to consider a
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241:Usually we are interested in sets
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38:it lacks sufficient corresponding
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1633:Bendsøe M. P.; Sigmund O. (2003)
677:Consider a smooth velocity field
278:infinite-dimensional optimization
1429:{\displaystyle \partial \Omega }
104:defined on the variable domain.
23:
1648:Burger, M.; Osher, S.L. (2005)
1588:10.1016/j.compfluid.2014.12.004
1443:) of the objective functional
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1487:{\displaystyle {\mathcal {G}}}
1460:{\displaystyle {\mathcal {F}}}
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269:and consist of finitely many
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514:fashion. Mathematically, if
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1704:. Oxford University Press.
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1067:{\displaystyle \Omega _{0}}
744:{\displaystyle \Omega _{0}}
561:{\displaystyle \Omega _{t}}
534:{\displaystyle \Omega _{0}}
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541:is the initial shape, and
478:Keeping track of the shape
1751:Mathematical optimization
751:under the velocity field
422:The answer, given by the
276:Shape optimization is an
1506:Geometry parametrization
424:isoperimetric inequality
84:is part of the field of
254:{\displaystyle \Omega }
138:{\displaystyle \Omega }
96:while satisfying given
53:more precise citations.
1550:Topological derivative
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717:{\displaystyle T_{s}}
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504:Eulerian approach
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454:least-squares
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1730:Optopo Group
1716:
1701:
1686:
1671:
1656:
1649:
1637:. Springer.
1634:
1619:
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1579:
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1569:
1529:pseudosolids
1521:
1509:
1502:, can work.
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1013:
935:
676:
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569:
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282:vector space
275:
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196:of the form
191:
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81:
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1582:: 100–112.
1498:, like the
936:and denote
470:, by using
468:numerically
124:bounded set
98:constraints
51:introducing
1740:Categories
1561:References
1517:re-meshing
462:Techniques
271:components
261:which are
194:constraint
151:functional
147:minimizing
115:Definition
94:functional
34:references
1424:Ω
1421:∂
1390:∇
1359:Ω
1355:∂
1351:⟩
1334:∇
1331:⟨
1310:Ω
1264:Ω
1260:∂
1244:∈
1234:∇
1176:Ω
1162:−
1150:Ω
1128:→
1099:Ω
1056:Ω
1032:Ω
990:Ω
974:Ω
957:↦
948:Ω
917:≥
811:Ω
807:∈
733:Ω
641:≤
635:≤
613:Ω
609:→
600:Ω
550:Ω
523:Ω
395:Ω
376:Ω
336:Ω
333:∂
314:Ω
263:Lipschitz
249:Ω
217:Ω
173:Ω
133:Ω
59:June 2020
1659:. SIAM.
1545:SU2 code
1539:See also
828:′
499:function
288:Examples
267:boundary
1599:Sources
1441:Hessian
426:, is a
47:improve
1708:
1693:
1678:
1663:
1641:
1626:
1612:
1382:where
452:using
442:stress
406:const.
387:Volume
355:with
36:, but
265:or C
90:shape
1706:ISBN
1691:ISBN
1676:ISBN
1661:ISBN
1639:ISBN
1624:ISBN
1610:ISBN
1285:and
435:drag
428:ball
325:Area
1584:doi
1580:109
1121:lim
1047:at
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1578:.
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149:a
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