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indicates an absence of material. Owing to the attainable topological complexity of the design being dependent on the number of elements, a large number is preferred. Large numbers of finite elements increases the attainable topological complexity, but come at a cost. Firstly, solving the FEM system becomes more expensive. Secondly, algorithms that can handle a large number (several thousands of elements is not uncommon) of discrete variables with multiple constraints are unavailable. Moreover, they are impractically sensitive to parameter variations. In literature problems with up to 30000 variables have been reported.
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576:. This indicates 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 optimization are considered as non-design regions.
389:{\displaystyle {\begin{aligned}&{\underset {\rho }{\operatorname {minimize} }}&&F=F(\mathbf {u(\rho ),\rho } )=\int _{\Omega }f(\mathbf {u(\rho ),\rho } )\mathrm {d} V\\&\operatorname {subject\;to} &&G_{0}(\rho )=\int _{\Omega }\rho \mathrm {d} V-V_{0}\leq 0\\&&&G_{j}(\mathbf {u} (\rho ),\rho )\leq 0{\text{ with }}j=1,...,m\end{aligned}}}
1330:
of many engineering applications. Topology optimisation for fluid structure interaction problems has been studied in e.g. references and. Design solutions solved for different
Reynolds numbers are shown below. The design solutions depend on the fluid flow with indicate that the coupling between the fluid and the structure is resolved in the design problems.
1452:
conversion of thermal energy into electric energy and the
Peltier effect concerns the conversion of electric energy into thermal energy. By spatially distributing two thermoelectric materials in a two dimensional design space with a topology optimisation methodology, it is possible to exceed performance of the constitutive thermoelectric materials for
1468:
The current proliferation of 3D printer technology has allowed designers and engineers to use topology optimization techniques when designing new products. Topology optimization combined with 3D printing can result in less weight, improved structural performance and shortened design-to-manufacturing
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is a strongly coupled phenomenon and concerns the interaction between a stationary or moving fluid and an elastic structure. Many engineering applications and natural phenomena are subject to fluid-structure-interaction and to take such effects into consideration is therefore critical in the design
1451:
is a multi-physic problem which concerns the interaction and coupling between electric and thermal energy in semi conducting materials. Thermoelectric energy conversion can be described by two separately identified effects: The
Seebeck effect and the Peltier effect. The Seebeck effect concerns the
1019:
On a broad level, one can visualize that the more the material, the less the deflection as there will be 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
729:
has caused the community to search for other options. One is the modelling of the densities with continuous variables. The material densities can now also attain values between zero and one. Gradient based algorithms that handle large amounts of continuous variables and multiple constraints are
716:
Solving topology optimization problems in a discrete sense is done by discretizing the design domain into finite elements. The material densities inside these elements are then treated as the problem variables. In this case material density of one indicates the presence of material, while zero
904:
There are several commercial topology optimization software on the market. Most of them use topology optimization as a hint how the optimal design should look like, and manual geometry re-construction is required. There are a few solutions which produce optimal designs ready for
Additive
893:, to make sure the derivatives of the objective function are non-zero when the density becomes zero. The higher the penalisation factor, the more SIMP penalises the algorithm in the use of non-binary densities. Unfortunately, the penalisation parameter also introduces non-convexities.
1312:
based on image processing are currently being used to alleviate some of these issues. Although it seemed like this was purely a heuristic approach for a long time, theoretical connections to nonlocal elasticity have been made to support the physical sense of these methods.
1094:
64:. Due to the free forms that naturally occur, the result is often difficult to manufacture. For that reason the result emerging from topology optimization is often fine-tuned for manufacturability. Adding constraints to the formulation in order to
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Design evolution for a fluid-structure-interaction problem from reference. The objective of the design problem is to minimize the structural compliance. The fluid-structure-interaction problem is modelled with Navier-Cauchy and Navier-Stokes
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452:. This function represents the quantity that is being minimized for best performance. The most common objective function is compliance, where minimizing compliance leads to maximizing the stiffness of a structure.
1427:
Design evolution for an off-diagonal thermoelectric generator. The design solution of an optimisation problem solved for electric power output. The performance of the device has been optimised by distributing
1016:) of the structure under the prescribed boundary conditions. The lower the strain energy the higher the stiffness of the structure. So, the objective function of the problem is to minimize the strain energy.
1418:
A sketch of the design problem. The aim of the design problem is to spatially distribute two materials, Material A and
Material B, to maximise a performance measure such as cooling power or electric power
1481:. The third medium contact (TMC) method is an implicit contact formulation that is continuous and differentiable. This makes TMC suitable for use with gradient-based approaches to topology optimization.
1301:
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.
525:
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60:
Topology optimization has a wide range of applications in aerospace, mechanical, bio-chemical and civil engineering. Currently, engineers mostly use topology optimization at the concept level of a
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available. But the material properties have to be modelled in a continuous setting. This is done through interpolation. One of the most implemented interpolation methodologies is the
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1436:(blue) with a density-based topology optimisation methodology. The aim of the optimisation problem is to maximise the electric power output of the thermoelectric generator.
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Picelli, R.; Vicente, W.M.; Pavanello, R. (2017). "Evolutionary topology optimization for structural compliance minimization considering design-dependent FSI loads".
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891:
545:
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1983:
Jenkins, Nicholas; Maute, Kurt (2016). "An immersed boundary approach for shape and topology optimization of stationary fluid-structure interaction problems".
864:
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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
661:
a characteristic that the solution must satisfy. Examples are the maximum amount of material to be distributed (volume constraint) or maximum stress values.
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Mahdavi, A.; Balaji, R.; Frecker, M.; Mockensturm, E. M. (2006). "Topology optimization of 2D continua for minimum compliance using parallel computing".
38:
and sizing optimization in the sense that the design can attain any shape within the design space, instead of dealing with predefined configurations.
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1147:
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Design development and deformation of self-engaging hooks resulting from topology optimization of a contact problem using the TMC method .
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Design evolution for a thermoelectric cooler. The aim of the design problem is to maximise the cooling power of the thermoelectric cooler.
866:. This has been shown to confirm the micro-structure of the materials. In the SIMP method a lower bound on the Young's modulus is added,
2349:
Pedersen, Claus B. W.; Allinger, Peter (2006). "Industrial
Implementation and Applications of Topology Optimization and Future Needs".
1913:
Yoon, Gil Ho (2010). "Topology optimization for stationary fluid-structure interaction problems using a new monolithic formulation".
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But, a straightforward implementation in the finite element framework of such a problem is still infeasible owing to issues such as:
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813:. It interpolates the Young's modulus of the material to the scalar selection field. The value of the penalisation parameter
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Leiva, Juan; Watson, Brian; Kosaka, Iku (1999). "Modern structural optimization concepts applied to topology optimization".
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Critical study of design parameterization in topology optimization; The influence of design parameterization on local minima
737:
605:
1089:{\displaystyle \min _{\rho }\;\int _{\Omega }{\frac {1}{2}}\mathbf {\sigma } :\mathbf {\varepsilon } \,\mathrm {d} \Omega }
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the material. To obtain the total material utilized, an integration of the selection field over the volume can be done.
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Sketch of the well-known wall problem. The objective of the design problem is to minimize the structural compliance.
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is an active field of research. In some cases results from topology optimization can be directly manufactured using
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455:
The material distribution as a problem variable. This is described by the density of the material at each location
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Schramm, Uwe; Zhou, Ming (2006). "Recent
Developments in the Commercial Implementation of Topology Optimization".
409:
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Finally the elasticity governing differential equations are plugged in so as to get the final problem statement.
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cycle. As the designs, while efficient, might not be realisable with more traditional manufacturing techniques.
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There are various implementation methodologies that have been used to solve topology optimization problems.
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Lundgaard, Christian; Alexandersen, Joe; Zhou, Mingdong; Andreasen, Casper
Schousboe; Sigmund, Ole (2018).
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is a mathematical method that optimizes material layout within a given design space, for a given set of
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Design solutions for different
Reynolds number for a wall inserted in a channel with a moving fluid.
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with the goal of maximizing the performance of the system. Topology optimization is different from
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2078:"A density-based topology optimization methodology for thermoelectric energy conversion problems"
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1244:{\displaystyle \mathbf {\nabla } \cdot \mathbf {\sigma } \,+\,\mathbf {F} \;=\;{\mathbf {0} }}
45:(FEM) to evaluate the design performance. The design is optimized using either gradient-based
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2023:"Revisiting density-based topology optimization for fluid-structure-interaction problems"
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IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials
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IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials
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often includes solving a differential equation. This is most commonly done using the
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2380:. Solid Mechanics and Its Applications. Vol. 137. Springer. pp. 239–248.
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2353:. Solid Mechanics and Its Applications. Vol. 137. Springer. pp. 229–238.
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Bendsøe, M. P. (1989). "Optimal shape design as a material distribution problem".
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The earlier stated complexities with solving topology optimization problems using
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is a state field that satisfies a linear or nonlinear state equation depending on
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Frederiksen, Andreas Henrik; Sigmund, Ole; Poulios, Konstantinos (2023-10-07).
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Numerical instabilities—The selection of region in the form of a chess board.
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486:. Material is either present, indicated by a 1, or absent, indicated by a 0.
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1287:{\displaystyle \mathbf {\sigma } \;=\;{\mathsf {C}}:\mathbf {\varepsilon } }
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40th Structures, Structural Dynamics, and Materials Conference and Exhibit
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Internal contact can be included in topology optimization by applying the
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A topology optimization problem can be written in the general form of an
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Mathematical method for optimizing material layout under given conditions
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1191:{\displaystyle \int _{\Omega }\rho \,\mathrm {d} \Omega \;\leq \;V^{*}}
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Bluhm, Gore Lukas; Sigmund, Ole; Poulios, Konstantinos (2021-03-04).
1934:
1824:
1754:"Checkerboard Problem in Finite Element Based Topology Optimization"
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2243:
2227:"Internal contact modeling for finite strain topology optimization"
2186:
1547:"Topology optimization using a dual method with discrete variables"
2020:
1758:
International Journal of Advances in Engineering & Technology
1689:. 2nd International Conference on Engineering Optimization, 2010
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since these equations do not have a known analytical solution.
2120:"Topology Optimization of Segmented Thermoelectric Generators"
1862:"Sensitivity filtering from a continuum mechanics perspective"
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Shukla, Avinash; Misra, Anadi; Kumar, Sunil (September 2013).
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method (SIMP). This interpolation is essentially a power law
49:
techniques such as the optimality criteria algorithm and the
520:{\displaystyle \mathbf {u} =\mathbf {u} (\mathbf {\rho } )}
2292:"A finite element method for contact using a third medium"
2118:
Lundgaard, Christian; Sigmund, Ole; Bjørk, Rasmus (2018).
1915:
International Journal for Numerical Methods in Engineering
1805:
International Journal for Numerical Methods in Engineering
41:
The conventional topology optimization formulation uses a
2167:
1635:"Material interpolation schemes in topology optimization"
711:
2289:
1955:
2436:. American Institute of Aeronautics and Astronautics.
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Wriggers, P.; Schröder, J.; Schwarz, A. (2013-03-30).
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806:{\displaystyle E\;=\;E_{0}\,+\,\rho ^{p}(E_{1}-E_{0})}
654:{\displaystyle G_{j}(\mathbf {u} (\rho ),\rho )\leq 0}
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2170:"Topology optimization of self-contacting structures"
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Topology optimization result when filtering is used
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1701:"On some recent advances in shape optimization"
1699:Allaire, Grégoire; Henrot, Antoine (May 2001).
917:Checker Board Patterns are shown in this result
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399:The problem statement includes the following:
72:; topology optimization is thus a key part of
2405:Structural and Multidisciplinary Optimization
2085:Structural and Multidisciplinary Optimization
2030:Structural and Multidisciplinary Optimization
2011:
1985:Structural and Multidisciplinary Optimization
1866:Structural and Multidisciplinary Optimization
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1512:Structural and Multidisciplinary Optimization
933:Topology optimization of a compliance problem
721:Solving the problem with continuous variables
1982:
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1510:(2013). "Topology optimization approaches".
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1380:Design solution and pressure field for Re=40
1368:Design solution and pressure field for Re=10
445:{\displaystyle F(\mathbf {u(\rho ),\rho } )}
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2076:Lundgaard, Christian; Sigmund, Ole (2018).
1682:van Dijk, NP. Langelaar, M. van Keulen, F.
1356:Design solution and velocity field for Re=5
1344:Design solution and velocity field for Re=1
976:. Unsourced material may be challenged and
1860:Sigmund, Ole; Maute, Kurt (October 2012).
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712:Solving with discrete/binary variables
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1633:Bendsøe, M. P.; Sigmund, O. (1999).
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1464:3F3D Form Follows Force 3D Printing
689:{\displaystyle \mathbf {u(\rho )} }
479:{\displaystyle \rho (\mathbf {x} )}
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1725:10.1016/S1620-7742(01)01349-6
1623:, a monograph of the subject.
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1136:{\displaystyle \rho \,\in \,}
1642:Archive of Applied Mechanics
704:Implementation methodologies
7:
1970:10.1016/j.finel.2017.07.005
1479:third medium contact method
1327:Fluid-structure-interaction
1322:Fluid-structure-interaction
908:
833:is generally taken between
51:method of moving asymptotes
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2261:10.1007/s00466-021-01974-x
2204:10.1007/s00466-023-02396-7
1764:(4). CiteSeer: 1769–1774.
1707:. Series IIB - Mechanics.
2471:Mathematical optimization
2417:10.1007/s00158-006-0006-1
2316:10.1007/s00466-013-0848-5
2144:10.1007/s11664-018-6606-x
2097:10.1007/s00158-018-1919-1
2042:10.1007/s00158-018-1940-4
1997:10.1007/s00158-016-1467-5
1878:10.1007/s00158-012-0814-4
1524:10.1007/s00158-013-0978-6
1458:thermoelectric generators
569:{\displaystyle (\Omega )}
2386:10.1007/1-4020-4752-5_24
2359:10.1007/1-4020-4752-5_23
1872:(4). Springer: 471–475.
1711:(5). Elsevier: 383–396.
1308:Some techniques such as
47:mathematical programming
2296:Computational Mechanics
2231:Computational Mechanics
2174:Computational Mechanics
1811:(9). Wiley: 2143–2158.
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2136:2018JEMat..47.6959L
1927:2010IJNME..82..591Y
1817:2001IJNME..50.2143B
1717:2001CRASB.329..383A
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900:Commercial software
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2066:. CRC press, 2005.
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862:
859:{\displaystyle }
857:
832:
830:
829:
824:
812:
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799:
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727:binary variables
695:
693:
692:
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685:
660:
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626:
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354:
352: with
351:
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248:
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147:
114:
112:
101:
2506:
2505:
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2500:
2499:
2497:
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2461:
2460:
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2396:
2369:
2345:
2343:Further reading
2340:
2339:
2288:
2284:
2223:
2219:
2166:
2159:
2116:
2112:
2080:
2074:
2070:
2061:
2057:
2025:
2019:
2012:
1981:
1977:
1954:
1950:
1911:
1907:
1898:
1896:
1858:
1854:
1845:
1843:
1825:10.1002/nme.116
1797:
1793:
1784:
1782:
1771:10.1.1.670.6771
1750:
1746:
1737:
1735:
1697:
1693:
1681:
1677:
1637:
1631:
1627:
1620:
1616:
1585:
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1549:
1543:
1539:
1504:
1500:
1495:
1475:
1466:
1412:
1390:
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1383:
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1381:
1378:
1370:
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1366:
1358:
1357:
1354:
1350:
1346:
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1342:
1338:
1324:
1319:
1279:
1270:
1269:
1259:
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1254:
1253:
1235:
1234:
1224:
1214:
1206:
1204:
1201:
1200:
1182:
1178:
1165:
1155:
1151:
1149:
1146:
1145:
1107:
1104:
1103:
1078:
1072:
1064:
1054:
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1044:
1037:
1031:
1028:
1027:
1002:
991:
985:
982:
967:
951:
940:
911:
905:Manufacturing.
902:
896:
877:
873:
871:
868:
867:
838:
835:
834:
818:
815:
814:
794:
790:
781:
777:
768:
764:
753:
749:
739:
736:
735:
723:
714:
706:
672:
670:
667:
666:
622:
613:
609:
607:
604:
603:
583:
580:
579:
555:
552:
551:
532:
529:
528:
509:
501:
493:
491:
488:
487:
468:
460:
457:
456:
419:
411:
408:
407:
383:
382:
350:
321:
312:
308:
302:
301:
289:
285:
274:
265:
261:
243:
239:
236:
206:
202:
201:
193:
170:
158:
154:
128:
113:
103:
98:
96:
93:
92:
82:
17:
12:
11:
5:
2504:
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2483:
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2473:
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2451:
2450:External links
2448:
2447:
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2411:(2): 121–132.
2400:
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2373:
2367:
2344:
2341:
2338:
2337:
2302:(4): 837–847.
2282:
2217:
2180:(4): 967–981.
2157:
2110:
2068:
2055:
2036:(3): 969–995.
2010:
1975:
1948:
1921:(5): 591–616.
1905:
1852:
1791:
1744:
1691:
1675:
1625:
1614:
1595:(4): 193–202.
1579:
1537:
1506:Sigmund, Ole;
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110:
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62:design process
15:
9:
6:
4:
3:
2:
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2422:
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2414:
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2387:
2383:
2379:
2374:
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2364:
2360:
2356:
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2347:
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2325:
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2317:
2313:
2309:
2305:
2301:
2297:
2293:
2286:
2278:
2274:
2270:
2266:
2262:
2258:
2254:
2250:
2245:
2240:
2236:
2232:
2228:
2221:
2213:
2209:
2205:
2201:
2197:
2193:
2188:
2183:
2179:
2175:
2171:
2164:
2162:
2153:
2149:
2145:
2141:
2137:
2133:
2129:
2125:
2121:
2114:
2106:
2102:
2098:
2094:
2090:
2086:
2079:
2072:
2065:
2059:
2051:
2047:
2043:
2039:
2035:
2031:
2024:
2017:
2015:
2006:
2002:
1998:
1994:
1990:
1986:
1979:
1971:
1967:
1963:
1959:
1952:
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1909:
1895:
1891:
1887:
1883:
1879:
1875:
1871:
1867:
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1842:
1838:
1834:
1830:
1826:
1822:
1818:
1814:
1810:
1806:
1802:
1795:
1781:
1777:
1772:
1767:
1763:
1759:
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1748:
1734:
1730:
1726:
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1710:
1706:
1702:
1695:
1688:
1686:
1679:
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1435:
1432:(yellow) and
1431:
1425:
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1331:
1328:
1314:
1311:
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1220:
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1211:
1199:
1183:
1179:
1174:
1161:
1152:
1144:
1127:
1123:
1120:
1113:
1109:
1102:
1101:
1100:
1073:
1069:
1065:
1059:
1056:
1045:
1038:
1026:
1025:
1024:
1021:
1017:
1015:
1012:(also called
1011:
1010:strain energy
1000:
997:
989:
986:December 2018
979:
975:
971:
965:
964:
960:
955:This section
953:
949:
944:
943:
931:
923:
915:
906:
897:
894:
878:
874:
850:
846:
843:
820:
795:
791:
787:
782:
778:
769:
765:
760:
754:
750:
745:
741:
733:
728:
718:
709:
701:
699:
679:
648:
645:
639:
636:
630:
614:
610:
602:
586:
578:
549:
534:
510:
498:
462:
454:
435:
426:
413:
406:
402:
401:
400:
379:
376:
373:
370:
367:
364:
361:
358:
355:
347:
344:
338:
335:
329:
313:
309:
298:
295:
290:
286:
282:
279:
271:
262:
258:
252:
244:
240:
198:
186:
177:
164:
155:
151:
144:
135:
122:
119:
116:
108:
105:
91:
90:
89:
87:
77:
75:
71:
67:
63:
58:
56:
52:
48:
44:
39:
37:
33:
29:
25:
21:
2481:Construction
2433:
2408:
2404:
2377:
2350:
2299:
2295:
2285:
2234:
2230:
2220:
2177:
2173:
2127:
2123:
2113:
2088:
2084:
2071:
2058:
2033:
2029:
1988:
1984:
1978:
1961:
1957:
1951:
1918:
1914:
1908:
1897:. Retrieved
1869:
1865:
1855:
1844:. Retrieved
1808:
1804:
1794:
1783:. Retrieved
1761:
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1747:
1736:. Retrieved
1708:
1704:
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1683:
1678:
1645:
1641:
1628:
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1588:
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1540:
1515:
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1430:Skutterudite
1325:
1307:
1296:
1099:subject to:
1098:
1022:
1018:
1007:
992:
983:
968:Please help
956:
903:
895:
731:
724:
715:
707:
664:
398:
83:
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2491:3D printing
1508:Maute, Kurt
665:Evaluating
601:constraints
32:constraints
2465:Categories
2244:2010.14277
2187:2305.06750
1899:2021-06-17
1846:2020-08-02
1785:2022-02-14
1738:2021-09-12
1493:References
1405:equations.
1014:compliance
2332:254032357
2324:0178-7675
2277:225076340
2269:0178-7675
2212:1432-0924
2152:105113187
2105:126031362
2050:125798826
2005:124632210
1964:: 44–55.
1943:122993997
1894:253680268
1886:1615-1488
1833:1097-0207
1780:2231-1963
1766:CiteSeerX
1733:1620-7742
1574:122845784
1560:: 14–24.
1532:124426387
1310:filtering
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1261:σ
1216:σ
1212:⋅
1208:∇
1184:∗
1175:≤
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1162:ρ
1157:Ω
1153:∫
1114:∈
1110:ρ
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1050:Ω
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1039:ρ
957:does not
788:−
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646:≤
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2476:Topology
2425:61564700
1841:38860291
1670:11368603
1609:18253872
909:Examples
106:minimize
2304:Bibcode
2249:Bibcode
2192:Bibcode
2132:Bibcode
1923:Bibcode
1813:Bibcode
1713:Bibcode
1650:Bibcode
978:removed
963:sources
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2421:S2CID
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2273:S2CID
2239:arXiv
2182:arXiv
2148:S2CID
2101:S2CID
2081:(PDF)
2046:S2CID
2026:(PDF)
2001:S2CID
1939:S2CID
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