Numerical modeling of the timpani

We propose a time-domain fictitious domain method for the numerical simulation of a kettle drum (or timpani). Such an instrument is made of a circular elastic membrane stretched over an air cavity enclosed by a rigid shell and set into vibrations by the impact of a mallet. The mathematical model couples a 3D linear wave equation for the outside and inside fluids, a 2D wave equation with viscous damping terms on the membrane and a nonlinear differential equation for the mallet. The originality of the so-called fictitious domain approach is to be based on a formulation which does not distinguish explicitly the external fluid from the internal one. This formulation rests upon a velocity-pressure formulation of the acoustic wave equation in the air and a variational mixed formulation of the full problem. The boundary and fluid-structure interface conditions are taken into account in a weak way via a Lagrange multiplier which concides with the pressure jump across the membrane and the shell. The numerical approximation is based on standard and mixed finite elements for the spacial discretization and centered finite differences for time discretization.