This effort targets the original development of CNAPS, an innovative suite of numerical algorithms for continuation analysis of multi-segment trajectories in large-scale, nonlinear dynamical systems with multiple slow and fast timescales, coupled components, and with triggers, resets and switches. Continuation methods have proven very successful for analyzing system behavior of low-dimensional systems. In CNAPS, we aim to dramatically scale continuation methods to complex networked systems with hybrid system trajectories and tens of thousands of states, by developing new multiscale, multisegment, trajectory-discretization algorithms based on asynchronous collocation methods; developing new mesh adaptation algorithms suitable for the asynchronous collocation methods, which accommodate segment-specific discretization error bounds; and constructing domain decomposition methods particular to the network topology and the asynchronous collocation formulation, which enable efficient parallel execution. The core application of CNAPS considered in this multidisciplinary effort is modern power systems that include renewable sources of generation, specifically wind power, and newer forms of load, characterized by multiple coexisting time scales and trigger-induced switching behavior. Analysis of large-disturbance dynamic phenomena in such systems currently relies almost exclusively on forward simulation. While such tools may reveal complex behavior, they offer little help in the design process required to address unacceptable behavior, especially emerging phenomena associated with the increased use of power electronic converters. The development of CNAPS enables intelligent and efficient exploration of transient and steady-state responses of complex power systems, aimed at quantifying design and uncertainty margins for stable, faultless operation.
Problems in the modeling and analysis of a wide variety of physical and biological systems, whether naturally occurring or engineered, pose significant challenges to existing computational infrastructure. This is due to the large numbers of variables that are the typical target of the numerical algorithms, as well as the complexity of the tasks enabled by the analysis. Example tasks investigated in this project include the design of robust power grids; the study of the cloud-formation potential of atmospheric aerosol particles; the shape optimization of microelectromechanical structures; and the sensitivity of chemical reaction networks describing the cell cycle of microorganisms to changes in reaction rates. The outcomes of this project addressing intellectual merit include novel tools of mathematical analysis, original numerical algorithm development, and software implementations, as well as their application to the study of i) threshold-induced instability in models of the life cycle of yeast cells, ii) techniques to increase the range of predictable and linear operating conditions for microbeam resonators, iii) vulnerability of power systems to load uncertainty, and iv) relationships between statistical properties of an initial distribution of aerosol particle sizes and the subsequent fraction of droplet-forming particles. In the latter two applications, the project has made innovative contributions to the analysis of transient inverse problems, characterized by finite-time end-point conditions, in contrast to the more traditional study of steady-state behaviors emphasized in the first two applications. Both intellectual merit and broader impact have been addressed, in collaboration with Frank Schilder of the Technical University of Denmark, through the authoring and dissemination of the Computational Continuation Core, a software development platform for numerical analysis of inverse problems, freely available through a public repository for open-source software. This includes toolboxes for the study of multidimensional families of solutions to mathematical equations with large numbers of unknowns, with particular emphasis on describing changes in time of systems with competing time scales, coupled components, and a combination of continuous and discontinuous dynamics. To this end, the project has proposed original techniques for algorithm parallelization, as well as reduction in computational complexity. In addition, the software development effort has been accompanied by the authoring of extensive documentation, available online and in the textbook Recipes for Continuation, published by the Society for Industrial and Applied Mathematics, as well as of video tutorials and lectures. Further outcomes of the project addressing broader impact include several iterations of an advanced graduate-level course on computational nonlinear analysis, as well as the organization of, and participation in, international research and tutorial workshops on computational methods. Finally, the project has also contributed to the development of an interdisciplinary research infrastructure and work force, by providing multidisciplinary training to project graduate and undergraduate students from mechanical engineering, electrical engineering, and computer science.
