Progress in computational methods has been stimulated by the widespread availability of cheap computational power leading to the improved precision and efficiency of simulation software. Simulation tools become indispensable tools for engineers who are interested in attacking increasingly larger problems or are interested in searching larger phase space of process and system variables to find the optimal design. In this paper, we show and introduce a new approach to a computational method that involves mixed time stepping scheme and allows to decrease computational cost. Implementation of our algorithm does not require a parallel computing environment. Our strategy splits domains of a dynamically changing physical phenomena and allows to adjust the numerical model to various sub-domains. We are the first (to our best knowledge) to show that it is possible to use a mixed time partitioning method with various combination of schemes during binary alloys solidification. In particular, we use a fixed time step in one domain, and look for much larger time steps in other domains, while maintaining high accuracy. Our method is independent of a number of domains considered, comparing to traditional methods where only two domains were considered. Mixed time partitioning methods are of high importance here, because of natural separation of domain types. Typically all important physical phenomena occur in the casting and are of high computational cost, while in the mold domains less dynamic processes are observed and consequently larger time step can be chosen. Finally, we performed series of numerical experiments and demonstrate that our approach allows reducing computational time by more than three times without losing the significant precision of results and without parallel computing.
Multi-rate Control Design under Input Constraints via Fixed-Time Barrier Functions
