Topology optimization of degradable composite structures with time‐changeable stiffness

Abstract

For effective bone healing, the stiffness of the bone plate should be adjusted to different bone‐healing processes. Thus, the design of stiffness‐changeable structures that take into account the time effect is of importance. To this end, this study introduces a novel topological optimization approach for the composite structural layout design considering material degradation to realize structures with changeable stiffness over time. In this approach, two sets of variables are used: a density field that defines the material layout, and a time field that determines the effect of material degradation on mechanical performance. The continuous degradation update formula is proposed by integrating the Heaviside projected function and Kreisselmeier–Steinhauser function to guarantee its derivability. The objective is to minimize the summed compliance in some specified time steps subject to the constraints of volume fraction. The sensitivity of the aforementioned objective with respect to the design variable is deduced by considering the material degradation over time. The proposed design formulation is general and is demonstrated with several design analyses, considering different fix and degradable interface boundary conditions. Moreover, the results are compared with the results of not considering material degradation and demonstrate the effectiveness of the proposed method.

Discrete variable topology optimization for maximizing single/multiple natural frequencies and frequency gaps considering the topological constraint

“…) 𝝋 2 that is generally not equal to 0. As shown in the following numerical Example 2, the design obtained by the KS formulation ( P F,KS ) in Equation ( 27) does not satisfy the necessary optimality conditions in Equation (24). Consequently, the KS formulation in Equation ( 27), which only considers the sensitivity of the single frequency, bypasses the local optimum solution and converges to the solution with a lower objective function value.…”

“…According to Bratus and Seiranian, 39 the necessary optimality conditions in Equation ( 24) must be satisfied in the optimum multiple frequency design for the original formulation (P F ) in Equation (1). Comparing Equation (24) and Equation (30), the KKT conditions of the KS formulation ( P F,KS ) in Equation ( 27) cannot satisfy the necessary optimality conditions in Equation ( 24) for multiple frequencies unless f 12 = 0.…”

“…Meanwhile, the buckling problem can also appear multiple frequencies. 24 Theoretically speaking, failure to address the issue of multiple frequencies may result in a non-optimum design.…”