Computational design is encountering barriers due to chaotic dynamics. The characteristic unpredictability, aperiodicity, and sensitivity to initial conditions in chaotic dynamics requires computationally costly time-dependent analysis. Computationally costly analysis precludes sampling-based optimization, indicating that chaotic systems would significantly benefit from gradient-based optimization. Unfortunately, these same chaotic characteristics lead to truncation error and numerical overflow in the standard discretely exact adjoint method for computing gradients. Even finite difference methods are inaccurate unless very long simulations are computed. Current state of the art approaches are unnecessarily mathematically complex, significantly more expensive to compute compared to the standard adjoint method, and have reported questionable robustness. This Small Business Technology Transfer (STTR) proposal introduces a new method to approximate gradients of long-time-averaged objective functions for chaotic dynamic systems. In contrast to current state of the art approaches, the proposed method has a computational cost similar to the standard time-dependent adjoint method. In certain cases, computational cost is a small fraction of the standard time-dependent adjoint method. Preliminary results are shown to demonstrate viability. The university research plan will increase mathematical understanding and improve robustness of the proposed method. The small business will demonstrate shape optimization of tens to thousands of parameters for 2D flow.
