Fast simulation of elastic bodies is fundamental to computer graphics, yet current leading methods have a key limitation: they require fixed mesh connectivity. Existing methods leverage this assumption to achieve high performance but fail during topological changes such as cutting, fracturing, or merging. We present CAN, a novel optimizer that fundamentally decouples simulation acceleration from mesh topology. CAN introduces two Hessian-free, curvature-aware components: a Curvature-Aware Momentum (CAM) scheme that prevents overshooting by adaptively decaying momentum based on local gradient variations, and a Curvature-Aware Line Search (CALS) that provides high-quality step sizes via efficient directional curvature approximations. Since CAN relies solely on per-vertex, historical information, it is inherently parallel and topology-agnostic. We demonstrate that CAN achieves superior convergence compared to prior works across a wide range of dynamic-topology scenarios without any precomputation tied to connectivity, establishing a new paradigm for robust and efficient physics-based animation.