Absorbing boundary layer control for linear one-dimensional wave propagation problems

Abstract

In this work, a boundary layer control scheme for one-dimensional wave propagation problems is presented that provides reflection-less absorption of incident waves in numerical simulations. The desired absorption properties are formulated as an optimization problem. By using the dispersion relation to predict the unbounded wave propagation as a reference trajectory, a constant-gain state-feedback boundary layer controller is found. Since no additional auxiliary variables are introduced, this approach is highly computationally efficient, making it suitable for simulations under real-time requirements. The performance of the boundary layer controller is first evaluated and demonstrated on the scalar wave equation (vibrating string), for which reference absorbing boundary conditions are well established. Afterward, the moving Euler–Bernoulli beam under axial tension is considered, for which a good absorbing performance is achieved.