When geometric uncertainties arising from manufacturing errors are comparable with the characteristic length or the product responses are sensitive to such uncertainties, the products of deterministic design cannot perform robustly. This paper presents a new level set-based framework for robust shape and topology optimization against geometric uncertainties. We first propose a stochastic level set perturbation model of uncertain topology/shape to characterize manufacturing errors in conjunction with Karhunen-Loève (K-L) expansion. We then utilize polynomial chaos expansion to implement the stochastic response analysis. In this context, the mathematical formulation of the considered robust shape and topology optimization problem is developed, and the adjoint-variable shape sensitivity scheme is derived. An advantage of this method is that relatively large shape variations and even topological changes can be accounted for with desired accuracy and efficiency. Numerical examples are given to demonstrate the validity of the present formulation and numerical techniques. In particular, this method is justified by the observations in minimum compliance problems, where slender bars vanish when the manufacturing errors become comparable with the characteristic length of the structures.

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The functional complexity of native and replacement aortic heart valves (AVs) is well known, incorporating such physical phenomenons as time‐varying non‐linear anisotropic soft tissue mechanical behavior, geometric non‐linearity, complex multi‐surface time varying contact, and fluid–structure interactions to name a few. It is thus clear that computational simulations are critical in understanding AV function and for the rational basis for design of their replacements. However, such approaches continued to be limited by ad‐hoc approaches for incorporating tissue fibrous structure, high‐fidelity material models, and valve geometry. To this end, we developed an integrated tri‐leaflet valve pipeline built upon an isogeometric analysis framework. A high‐order structural tensor (HOST)‐based method was developed for efficient storage and mapping the two‐dimensional fiber structural data onto the valvular 3D geometry. We then developed a neural network (NN) material model that learned the responses of a detailed meso‐structural model for exogenously cross‐linked planar soft tissues. The NN material model not only reproduced the full anisotropic mechanical responses but also demonstrated a considerable efficiency improvement, as it was trained over a range of realizable fibrous structures. Results of parametric simulations were then performed, as well as population‐based bicuspid AV fiber structure, that demonstrated the efficiency and robustness of the present approach. In summary, the present approach that integrates HOST and NN material model provides an efficient computational analysis framework with increased physical and functional realism for the simulation of native and replacement tri‐leaflet heart valves.

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