Unlike single-physics static structural problems, the topology optimization of piezoelectric systems involves three different material coefficient groups, such as the mechanical, electrical, and electromechanical coefficient groups, which correspondingly necessitate three values of penalty exponents.
However, no rule for choosing the values of penalty exponents has yet been studied for piezoelectric systems and in fact, they have been chosen mainly by trial and error.
For each value of N s, we try different values of penalty parameter λ (λ ∈ {2−5,…, 2}) and repeat executing the algorithm 100 times with random initialization.
While reducing the value of penalties is an option, that would entail a host of problems, not least increased indiscipline on the field.
indicates the number of updating penalty parameter, is the penalty parameter, and denotes the value of penalty term.
(d, h, l, p, t, x) Mean value of penalty energy plots ∫ Ω (|∇ϕ| − 1 2dx (closeness measure between LSF and SDF).
In the last column, we draw the plots of mean value of penalty energy ∫ Ω (|∇ϕ| − 1 2dx by the above corresponding methods, which is used to measure the closeness between LSF and SDF.
The same value of penalty is also used for the contact tangent direction p t. Since the direction of slip varies continuously, it is necessary to derive the derivative of T+ΔTt with respect to u: (8) where is the trial friction force assuming that no slip occurs at time T+ΔT and I is a 3 × 3 unit matrix.
Return the value of penalty function to the genetic algorithm.
Thus a smaller value of penalty function means better robustness property of the model.
