Compliance–Stress Constrained Mass Minimization for Topology Optimization on Anisotropic Meshes

Research Article Compliance–stress constrained mass minimization for topology optimization on anisotropic meshes Nicola Ferro1 · Stefano Micheletti1 · Simona Perotto1 Received: 7 January 2020 / Accepted: 22 May 2020 / Published online: 11 June 2020 © Springer Nature Switzerland AG 2020 Abstract In this paper, we generalize the SIMPATY algorithm, which combines the SIMP method with anisotropic mesh adaptation to solve the minimum compliance problem with a mass constraint. In particular, the mass of the fnal layout is now minimized and both a maximum compliance and a maximum stress can be enforced as either mono- or multi-constraints. The new algorithm, named MSC-SIMPATY, is able to sharply detect the material–void interface, thanks to the anisotropic mesh adaptation. The presented test cases deal with three diferent scenarios, with a focus on the efect of the constraints on the fnal layouts and on the performance of the algorithm. Keywords Topology optimization · Stress constraint · Anisotropic mesh adaptation 1 Introduction in the literature driving topology optimization. Among these, we mention the density-based approaches [8, 10, Topology optimization is of utmost interest in diferent 46], the level-set methods [5, 53], topological derivative branches of industrial design, such as biomedical, space, procedures [49], phase feld techniques [12, 20], evolution- automotive, mechanical, architecture (see, e.g., [2, 13, ary approaches [54], homogenization [4, 9], performance- 19, 23, 57]). Similar formulations can also be adopted for based optimization [39]. We focus on the frst class and, the optimization of structures in diferent contexts, from in particular, on the SIMP (Solid Isotropic Material with fuid–structure interaction to the tailored design of mag- Penalization) method [9, 10, 46] where the material dis- netic or auxetic metamaterials [26, 36, 47, 56]. tribution is modeled via an auxiliary scalar feld, referred One aims at minimizing (or maximizing) a quantity of to as density, taking values between zero (void) and one interest under specifc design constraints, which may be (material) in the design domain. interchanged according to the considered feld of applica- In some recent papers, a new algorithm, called SIM- tion. A standard context leads to minimize the compliance PATY (SIMP with AdaptiviTY), has been proposed to ofer of a structure for a given mass (to optimize the structure an efcient design tool for 3D printing [43, 44]. SIMPATY performance at a given cost) or, vice versa, to minimize algorithm combines SIMP method with an advanced the weight under a prescribed compliance (to produce a mesh adaptation technique based on an a posteriori lightweight structure characterized by a desired stifness). recovery-based error analysis [58, 59]. In particular, the Independently of the number of possible design speci- authors employ anisotropic triangular meshes [27, 28]. fcations, we are led to solve a constrained optimization This choice leads to a very cost-efective procedure, which problem, possibly in an efcient way with a view to the alleviates the end user from most of the post-processing manufacturing process. Several methods are available step, towards the free-form design [25, 26, 44]. The setting * Nicola Ferro, nicola.ferro@polimi.it; Stefano Micheletti, stefano.micheletti@polimi.it; Simona Perotto, simona.perotto@polimi.it | 1MOX – Dipartimento di Matematica, Politecnico di Milano, Piazza L. da Vinci, 32, 20133 Milan, Italy. SN Applied Sciences (2020) 2:1196 | https://doi.org/10.1007/s42452-020-2947-1 Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:1196 | https://doi.org/10.1007/s42452-020-2947-1 considered in these papers addresses the minimization of −∇ ⋅ ()= in the compliance under a mass constraint. = on D In this paper, we move to a more general framework ⋅ ⎧ = 0 on R (1) with a view to more complex applications, by minimiz- ⎪ ( ) = on N ing the structure mass given a maximum stress and/or ⎪ ( ) = on F , a maximum compliance as a mono- or multi-constraint. ⎨ ⎪ The control on the stress takes into account a real require- ⎪where �⊂ℝ2 , is the displacement, ()=2()+ ⎩ ment in many contexts, since it allows us not to overcome I ∶ () is the stress tensor for an isotropic material, with a maximum limit for the mechanical resistance of the ()= ∇ + (∇)T ∕2 the small displacement strain material [14, 37]. The SIMPATY algorithm is here modifed tensor, into the mass-stress-compliance SIMPATY (MSC-SIMPATY) E E procedure to tackle the new optimization setting. The = , = numerical verification, despite preliminary, meets the (1 + )(1 − 2) 2(1 + ) expectation. The designed layouts do depend on the con- the Lamé coefcients, with E the Young modulus, the sidered constraints, and only a bound on both compliance Poisson ratio and I the identity tensor, ∶ → ℝ2 , is and stress delivers a structure that is sufciently stif as N the load applied to a portion of the boundary, is the well as failure-free under the applied loads. Moreover, the N unit outward normal vector to , is the portion of stress constraint turns out to be more hard to deal with, D the boundary where the structure is clamped, is the since the nonlinearities involved in the corresponding R segment where a roller boundary condition is assigned, defnition make the convergence of SIMPATY slower. In all and is the normal stress-free boundary, such that cases, however, the predicted layouts are characterized by F ∪ ∪ ∪ = . sharp interfaces between void and material, with no jag- D R N F The weak form of problem (1) is: find ged boundaries, thanks to the employment of anisotropic ∈ U ={ ∈[H1()]2 ∶ = on } , s.t. meshes, which are tailored to the density profle. This also D makes the fnal layouts almost ready for 3D printing, as a(, )=G()∀ ∈ U, (2) shown by the .STL fles in Fig. 12. The layout of the paper is the following. Section 2 with details the SIMP-based topology optimization procedure, ⋅ whereas Sect. 3 deals with the corresponding numerical a( , )= ( )∶( ) d, G( )= d, discretization. Section 4 focuses on the MSC-SIMPATY algo- N rithm, frst by introducing the optimization algorithm and where R =�. The SIMP method modifes the elasticity then the mesh adaptation procedure. Section 5 gathers equation by weighting (2) via the density function , i.e., some numerical tests, which alternate mono- with multi- a continuous design variable, modeling the distribution constrained problems. Finally, some conclusions are drawn of material in the design domain. A priori, the density is in the last section. assumed to be a function in L∞() , taking values in the interval [0, 1], with the understanding that = 0 repre- sents the void and = 1 the full material. To avoid non- 2 The mathematical model physical intermediate densities, the SIMP formulation actually resorts to a suitable power-law penalization on Several methods are available in the literature to identify , namely p with p ≥ max {2∕(1 − ),4∕(1 + )} , which the topologically optimal structure, contained in the initial promotes the extreme values, 0 and 1 [6, 10]. domain and subject to design constraints. Among these, The SIMP linear elasticity equation is thus: find ∈ U we cite approaches based on phase-feld [12, 17, 20], level s.t. set [3, 7, 15, 16], density-based methods [8, 10, 46], and , G , more recent techniques [30, 31, 45, 51]. In this paper, we a( )= ( )∀∈ U (3) adopt the SIMP (Solid Isotropic Material with Penalization) with formulation [10]. a( , )= ( )∶( ) d, (4) 2.1 The SIMP linear elasticity equation p In a structure optimization setting, the standard mathe- and ( )= [2( )+I ∶ ( )] . Notice that the penal- matical model is represented by the linear elasticity prob- ization modifes the material properties weighting the lem [32], i.e., Lamé coefcients. Vol:.(1234567890) SN Applied Sciences (2020) 2:1196 | https://doi.org/10.1007/s42452-020-2947-1 Research Article 2.2 The topology optimization formalize the dependence of the stress feld on the design variable, essentially adopting diferent penalizations [14, To formalize the topology optimization, we have to intro- 17, 18]. The choice in (8) is the most straightforward one in duce a cost functional, J , to be minimized, as well as pre- view of the employment of the SIMP method. Thus, prob- scribed constraints. A standard choice for J is the static lem (5) becomes: fnd ∈ L∞() s.t. compliance of the structure, or the total mass of the fnal a ((), )=G()∀ ∈ U layout. Concerning the constraints, (3) can be used to take S((), ) ≤ Smax into account the physical model at hand, whereas other min M()∶ (9) ∈L∞() C(()) ≤ C constraints are added to include design specifcations, ⎧ max ≤ ≤ 1, such as an upper bound on the local stresses or on the ⎪ min ⎨ fundamental frequency of vibration. According to the with C and S ⎪ a maximum upper bound on the com- max max⎩ application of interest, cost functional and constraints can pliance and on the von Mises stress defned in (8), respec- become interchangeable. tively. Problem (9) amounts to minimizing the mass of the Thus, a general topology optimization problem can be structure while ensuring a certain stifness and stress. formulated as: fnd ∈ L∞() s.t. a( (), )=G( )∀∈ U 3 The numerical discretization min J( (), )∶ Ci( (), ) ≤ ci, i = 1, … , nC (5) ∈L∞() ⎧ min ≤ ≤ 1, We resort to a standard fnite element approximation in ⎪ ⎨ order to numerically solve problem (9). This yields the dis- where C ((), ) ≤⎪c enforces a generic inequality

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