Numerical Analysis of Multi-Domain Systems: Coupled Nonlinear PDEs & DAEs with Noise

We present a numerical modeling and simulation paradigm for multi-domain, multi-physics systems with components modeled both in a lumped and distributed manner. The lumped components are modeled with a system of differential-algebraic equations (DAEs), whereas the possibly nonlinear distributed components that may belong to different physical domains are modeled using partial differential equations (PDEs) with associated boundary conditions (BCs). We address a comprehensive suite of problems for nonlinear coupled DAE-PDE systems including (i) transient simulation, (ii) periodic steady-state (PSS) analysis formulated as a mixed boundary value problem that is solved with a hierarchical spectral collocation technique based on a joint Fourier-Chebyshev representation, for both forced and autonomous systems, (iii) Floquet theory and analysis for coupled linear periodically time-varying (LPTV) DAE-PDE systems, (iv) phase noise analysis for multi-domain oscillators, (v) efficient parameter sweeps for PSS and noise analyses based on first-order and pseudo-arclength continuation schemes. All of these techniques, implemented in a prototype simulator, are applied to a substantial case study: A multi-domain feedback oscillator composed of distributed and lumped components in two physical domains, namely, a nano-mechanical beam resonator operating in the nonlinear regime, an electrical delay line, an electronic amplifier and a sensor-actuator for the transduction between the two physical domains. IEE