Topology Optimization for Incremental Elastoplasticity: A Phase-Field Approach

Abstract

Progresses in additive manufacturing technologies allow the realization of finely graded microstructured materials with tunable mechanical properties. This paves the way to a wealth of innovative applications, calling for the combined design of the macroscopic mechanical piece and its underlying microstructure. In this context, we investigate a topology optimization problem for an elastic medium featuring a periodic microstructure. The optimization problem is variationally formulated as a bilevel minimization of phase-field type. By resorting to Γ -convergence techniques, we characterize the homogenized problem and investigate the corresponding sharp-interface limit. First-order optimality conditions are derived, both at the homogenized phase-field and at the sharp-interface level.

A phase-field approach for detecting cavities via a Kohn-Vogelius type functional

“…Using H 1 −estimates for F δ D (v + ϑ) − F δ D (v) of Proposition 4.9, we have that the first term of the previous equation behaves as ϑ 3 L ∞ (Ω) while the second and the third behave as ϑ 2…”

“…This approach is widely applied in optimization procedures. We refer the reader to [3,11,18,16,17,27,41,42] and references therein for some recent papers on phase-field approaches.…”

“…in (4 21),. we have that the first two terms of the previous equation behaves as ϑ3 L ∞ (Ω) and ϑ 2 L ∞ (Ω) , respectively. Moreover, by means of equation(4.18), and recalling that C δ…”

Overhang penalization in additive manufacturing via phase field structural topology optimization with anisotropic energies

“…Then, given a critical angle threshold ψ (expressed now in radians), the overhang angle constraint in the current context of shape and topology optimization would demand that an optimal geometry for the object Ω 1 has minimal regions where the overhang angles there do not lie in the interval [ψ, 2π−ψ] (i.e., Ω 1 should be self-supporting as much as possible), in addition to other mechanical considerations such as minimal compliance. For structural topology optimization we employ the phase field methodology proposed in [28], later popularized by many authors to other applications such as multi-material structural topology optimization [22,89], compliance optimization [24,78], topology optimization with local stress constraints [29], nonlinear elasticity [73], elastoplasticity [6], eigenfrequency maximization [45,78], graded-material design [32], shape optimization in fluid flow [42,43,44], as well as resolution strategies for some inverse identification problems [21,57]. The key idea is to cast the structural topology optimization problem as a constrained minimization problem for a phase field variable ϕ, where the geometry of an optimal design Ω 1 for the object can be realized as a certain level-set of ϕ.…”

Rate-independent stochastic evolution equations: parametrized solutions

“…Without claiming completeness, we refer the reader to [16] as well as to [7,17,37,38,40] for a collection of abstract results. Applications to plasticity [5,12,13], fracture [1,10,21,28,42], damage [11,22,29], adhesive contact [8,47], delamination [18,46,50], optimal control [58], and topology optimization [2,3] are also available. Recall that, although the artificial viscous term disappears as ε → 0, the choice of the specific form of viscosity actually influences the limit [26,55].…”