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Shape and topology optimization using CutFEM
Department of Mathematics, University College London, Gower Street, London, United Kingdom.
Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.ORCID iD: 0000-0001-7352-1550
Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
2017 (English)In: Simulation for Additive Manufacturing 2017, Sinam 2017, International Center for Numerical Methods in Engineering (CIMNE), 2017, p. 208-209Conference paper, Oral presentation with published abstract (Refereed)
##### Abstract [en]

We present a shape and topology optimization method based on the cut finite element method, see [1],[2], and [3], for the optimal compliance problem in linear elasticity and problems involving restrictionson the stresses.The elastic domain is defined by a level-set function, and the evolution of the domain is obtained bymoving the level-set along a velocity field using a transport equation. The velocity field is defined tobe the largest decreasing direction of the shape derivative that resides in a certain Hilbert space and iscomputed by solving an elliptic problem, associated with the bilinear form in the Hilbert space, with theshape derivative as right hand side. The velocity field may thus be viewed as the Riesz representationof the shape derivative on the chosen Hilbert space.We thus obtain a coupled problem involving three partial differential equations: (1) the elasticity problem,(2) the elliptic problem that determines the velocity field, and (3) the transport problem for thelevelset function. The elasticity problem is solved using a cut finite element method on a fixed backgroundmesh, which completely avoids re–meshing when the domain is updated. The levelset functionand the velocity field is approximated by standard conforming elements on the background mesh. Wealso employ higher order cut approximations including isogeometric analysis for the elasticity problem.In this case the levelset function and the velocity field are represented using linear elements on a refinedmesh in order to simplify the geometric and quadrature computations on the cut elements. To obtain astable method, stabilization terms are added in the vicinity of the cut elements at the boundary, whichprovides control of the variation of the solution in the vicinity of the boundary. We present numericalexamples illustrating the performance of the method.We also study an anisotropic material model that accounts for the orientation of the layers in an additivemanufacturing process and by including the orientation in the optimization problem we determine theoptimal choice of orientation.We present numerical results including test problems and engineering applications in additive manufacturing.

References

[1] E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: discretizing geometryand partial differential equations. Internat. J. Numer. Methods Engrg., 104(7):472–501, 2015.

[2] E. Burman, D. Elfverson, P. Hansbo, M. G. Larson, and K. Larsson. Shape optimization using thecut finite element method. Technical report, 2016. arXiv:1611.05673.

[3] E. Burman, D. Elfverson, P. Hansbo, M. G. Larson, and K. Larsson. A cut finite element method forthe Bernoulli free boundary value problem. Comput. Methods Appl. Mech. Engrg., 317:598–618,2017.

##### Place, publisher, year, edition, pages
International Center for Numerical Methods in Engineering (CIMNE), 2017. p. 208-209
##### Keywords [en]
Cut finite element method, Isogeometric analysis, Level set, Optimal build orientation, Shape and topology optimization
##### National Category
Computational Mathematics
##### Identifiers
Scopus ID: 2-s2.0-85045505881OAI: oai:DiVA.org:hj-39385DiVA, id: diva2:1204972
##### Conference
2017 Simulation for Additive Manufacturing, Sinam 2017; Munich; Germany; 11 - 13 October 2017
Available from: 2018-05-09 Created: 2018-05-09 Last updated: 2019-03-22Bibliographically approved

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Hansbo, Peter

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##### By organisation
JTH, Product DevelopmentJTH. Research area Product Development - Simulation and Optimization
##### On the subject
Computational Mathematics

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Cite
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