Aage, Niels

Christiansen, Asger Nyman

Sigmund, Ole

Tortorelli, Daniel A.

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 20, 2017Combined shape and topology optimization for minimization of von Mises StressChristiansen, Asger Nyman; Tortorelli, Daniel A.; Aage, Niels; Sigmund, OlePublication date:2014Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Christiansen, A. N., Tortorelli, D. A., Aage, N., & Sigmund, O. (2014). Combined shape and topologyoptimization for minimization of von Mises Stress. Abstract from 4th International Conference on EngineeringOptimization, Lisbon, Portugal.4th International Conference on Engineering OptimizationSeptember 8 - 11, 2014, Lisbon, PortugalCombined shape and topology optimizationfor minimization of von Mises StressAsger N. ChristiansenDepartment of Applied Mathematics and Computer Science, Technical University of Denmark,Asmussens Alle´, B.305, 2800 Kgs. Lyngby, Denmark, e-mail: asny@dtu.dkDaniel A. TortorelliDepartment of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign,1206 West Green Street, Urbana, IL 61801, USANiels Aage, Ole SigmundDepartment of Mechanical Engineering, Technical University of Denmark,Nils Koppels Alle´, B.404, 2800 Kgs. Lyngby, DenmarkThe Deformable Simplicial Complex (DSC) method [4] has recently been used in structural topologyoptimization [1] where the objective was to minimize compliance subject to a volume constraint. Wewill apply the same scheme as in [1] but now we minimize the maximal von Mises stress. To generate asmooth cost function, we use the p-norm as an approximation to the max function. This, in combinationwith the finite element discretization, gives rise to a non-optimal jagged surface. To remedy this, weintroduce a filtering of the surface nodes.The DSC method discretizes the design domain into non-overlapping triangular elements (a simplicialcomplex). The elements are either labelled solid (filled with material) or void (filled with air). Con-sequently, the structural surface is represented explicitly by the edges which are sandwiched between avoid and a solid element. The shape and topology of the structure is changed by relabeling elements andmoving the surface nodes. The former is a discrete step which relabels the elements from solid to voidbased on values of their topological derivatives. The latter uses the values of the shape derivatives tofind an improved shape which is within a small perturbation of the current shape. The surface is thendeformed to this improved shape by the DSC method. Advantages of the DSC method include its abil-ity to simultaneously accommodate both topological and shape changes while maintaining well-shapedtriangular elements.In our current application, we want to minimize the maximal von Mises stress subject to a volumeconstraint. Since the max function is not smooth, the differentiable p-normmax(. . . , σvme , . . .) ≈(∑e(σvme )p) 1pis used instead as also proposed by Duysinx and Sigmund [2]. The governing equations are solved withthe finite element method. Furthermore, second order basis functions are used since they solve an issuewith jagged surfaces [1]. However, using the p-norm reintroduces the jagged surfaces and we will thereforeuse a well-known remedy, a filtering scheme. We apply a Gaussian filter to the node coordinates whichis an extension of the ideas proposed by Le et al. [3]. The filtered coordinate vector p˜i of node i isp˜i =∑nj=1 f(dij)pj∑nj=1 f(dij)where fr(dij) is a function of the distance dij between node i and j and r is the filter radius. We havechosen to use a Gaussian function such that f(dij) = e−3·d2ij/r2 if dij < r and f(dij) = 0 otherwise. Notethat the gradients and move limits are also filtered to make a mathematical consistent algorithm.We have successfully minimized the maximal Von Mises stress of several 2D problems such as theNorwegian Sock problem and the sizing problem of a hole in a plate. The results show that stresses areuniformly distributed throughout the structures, wherefore the maximal Von Mises stress is minimized.Before the conference, we plan to apply the approach to more 2D problems and if time permits, extendthe approach to 3D.1References[1] Asger Nyman Christiansen, Morten Nobel-Jørgensen, Niels Aage, Ole Sigmund, and Jakob AndreasBærentzen. Topology optimization using an explicit interface representation. Structural and Multi-disciplinary Optimization, pages 1–13, 2013.[2] Pierre Duysinx and Ole Sigmund. New developments in handling optimal stress constraints in optimalmaterial distributions. In Conference Proceedings of 7th Symposium in Multidisciplinary Analysis andOptimization, AIAA/USAF/NASA/ISSMO, volume 3, pages 1501–1509, St. Louis Missouri, USA,1998.[3] Chau Le, Tyler Bruns, and Daniel Tortorelli. A gradient-based, parameter-free approach to shapeoptimization. Computer Methods in Applied Mechanics and Engineering, 200(9-12):985–996, 2011.[4] Marek Krzysztof Misztal and Jakob Andreas Bærentzen. Topology adaptive interface tracking usingthe deformable simplicial complex. ACM Transactions on Graphics, 31(3):No. 24, 2012.2

Combined shape and topology optimization for minimization of von Mises Stress

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Combined shape and topology optimization for minimization of von Mises Stress

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