The analysis of linear, time-invariant systems by superposition of modes is a longstanding idea, tracing back to the early works by Daniel Bernoulli on the vibrating string and later formalised by Fourier. Time-invariance and linearity allow to describe systems in terms of eigenfunctions and frequencies, called the modes of the system. Such modal shapes and frequencies may be determined experimentally or from a suitable mathematical model. In the latter case, the model is a system of partial differential equations (PDEs), depending on material and geometric properties, type of excitation, and initial and boundary conditions. Modal equations result after an appropriate projection is applied to the system of PDEs, yielding an eigenvalue problem from which the modal frequencies and shapes are determined. The resulting modal equations depend exclusively on time, and output may be extracted as a suitable combination of the time-dependent modal coordinates. Usually, one is interested in computing a physical output at one or more points of the system via a weighted sum resulting from an inverse projection.
Besides being a practical analysis tool, this approach lends itself naturally to the simulation of mechanical vibrations and, thus, to sound synthesis via physics-based modelling. Modal synthesis began in earnest in the 1990s, when frameworks such as Mosaic and Modalys emerged. The early success of modal synthesis was partly due to the ease of implementation, and efficiency, of the modal structure: the orthogonality of the modes yields a bank of parallel damped oscillators. Including complicated loss profiles (necessary for realistic sound synthesis) is also trivial and inexpensive within the modal framework, as is the fine-tuning of the system's resonances. In direct numerical simulation, such as finite differences, distributed nonlinearities can be resolved locally and, in some cases, efficiently via linearly implicit schemes. For the modal approach, the presence of nonlinearities, either lumped or distributed, may become problematic since a coupling occurs between the modes of the associated linear system.
In this seminar, an extension of the modal approach, including nonlinearly coupled subsystems, is presented. The enabling idea is the quadratisation of the potential energy in a fashion analogous to that proposed within the SAV (scalar auxiliary variable) approach. It is then possible to derive discrete-time equations whose update remains explicit while guaranteeing pseudo-energy conservation (necessary for the system's stability). Musical examples, including a real-time music plugin developed within this framework, will be offered.