An Explicit Nite Element Scheme for Time-dependent Kirchhoo-love Equations

1 Motivation 1.1 A goal : the numerical modeling of the guitar This work is concerned with sound synthesis, the goal of which is to reproduce numerically the sound of musical instruments. We are interested in the study of the guitar. In our model the soundboard is coupled to one or more strings by means of a bridge. The soundboard is itself coupled with an open air cavity and the outer air. The system is excited by the plucking of a string. We obtain a system of partial diierential equations which we seek to solve numerically in the time domain. One of the main diiculties of this work, from the physical point of view as well as from the numerical point of view, is the modeling of the soundboard, a thin anisotropic wooden layer (in the shape of a guitar) clamped on its outer boundary and pierced by a hole. We consider here the Kirchhoo-Love's dynamic plate equations with xed boundary conditions on the clamped outer boundary and free boundary conditions along the hole. 1.2 The Kirchhoo-Love dynamic plate model Let ! be a C 1 bounded domain in R 2 and a partition of its boundary : ! = 0 l , where 0 is the xed boundary ((0) 6 = 0) and l the free one. The plate occupies the domain = !] ? =2; =22 in R 3 , where the thickness, is supposed small with respect to the dimensions of !. The unknown u is the vertical displacement of the plate. It is sought as a 2D function in the space: 1