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The development and optimization of particle accelerators are essential for advancing our understanding of the properties of matter, energy, and space-time. Particle accelerators are complex devices whose behavior involves many physical effects and scales. Therefore, advanced computational tools utilizing high-performance computing (HPC) are essential for accurately modeling them. In the past decade, the DOE SciDAC program has produced such accelerator-modeling tools, which have been employed to tackle some of the most difficult accelerator science problems. In this article we discuss the Synergia beam-dynamics framework and its applications to high-intensity particle accelerator physics. Synergia is a hybrid Python/C++ accelerator simulation package capable of handling the entire spectrum of beam dynamics simulations. We present the design principles, key physics and numerical models in Synergia and its performance on HPC platforms. Finally, we present the results of current Synergia applications for the Fermilab proton source chain upgrade, known as the Proton Improvement Plan.