KKT conditions satisﬁed using adaptive neighboring in hybrid cellular automata for topology optimization
Penninger, Charles L.
Watson, Layne T.
Renaud, John E.
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The hybrid cellular automaton (HCA) method is a biologically inspired algorithm capable of topology synthesis that was developed to simulate the behavior of the bone functional adaptation process. In this algorithm, the design domain is divided into cells with some communication property among neighbors. Local evolutionary rules, obtained from classical control theory, iteratively establish the value of the design variables in order to minimize the local error between a ﬁeld variable and a corresponding target value. Karush-Kuhn-Tucker (KKT) optimality conditions have been derived to determine the expression for the ﬁeld variable and its target. While averaging techniques mimicking intercellular communication have been used to mitigate numerical instabilities such as checkerboard patterns and mesh dependency, some questions have been raised whether KKT conditions are fully satisﬁed in the ﬁnal topologies. Furthermore, the averaging procedure might result in cancellation or attenuation of the error between the ﬁeld variable and its target. Several examples are presented showing that HCA converges to different ﬁnal designs for different neighborhood conﬁgurations or averaging schemes. Although it has been claimed that these ﬁnal designs are optimal, this might not be true in a precise mathematical sense—the use of the averaging procedure induces a mathematical incorrectness that has to be addressed. In this work, a new adaptive neighboring scheme will be employed that utilizes a weighting function for the inﬂuence of a cell’s neighbors that decreases to zero over time. When the weighting function reaches zero, the algorithm satisﬁes the aforementioned optimality criterion. Thus, the HCA algorithm will retain the benefits that result from utilizing neighborhood information, as well as obtain an optimal solution.