Conventional caustic methods in real or Fourier space produced accelerating optical beams only with convex trajectories. We developed a superposition caustic method capable of winding light beams along nonconvex trajectories. We ascertain this method by constructing a one-dimensional (1D) accelerating beam moving along a sinusoidal trajectory, and subsequently extending to two-dimensional (2D) accelerating beams along arbitrarily elliptical helical trajectories. We experimentally implemented the method with a compact and robust integrated optics approach by fabricating micro-optical structures on quartz glass plates to perform the spatial phase and amplitude modulation to the incident light, generating beam trajectories highly consistent with prediction. The theoretical and implementation methods can in principle be extended to the construction of accelerating beams with a wide variety of nonconvex trajectories, thereby opening up a route of manipulating light beams for fundamental research and practical applications.