Modeling/simulation of transient linear heat conduction problems via integrating a wide variety of space/time methods and choices

Abstract

We propose an implementation of a general purpose multi-spatial method, multi-time scheme subdomain Differential Algebraic Equations (DAE) framework allowing a mix of different space discretization methods while interfacing altogether different time integration algorithms on a single body analysis for the linear first order transient systems. With the concept of subdomains and DAE framework which allows constraint in time as well as space, we can divide a body into multiple subdomains and implement different spatial methods (finite elements, particles, etc.) in different regions of the body including altogether different time integrators and enable targeting an area with a specific method which can fully utilize its best features. The robust Generalized Single Step Single Solve (GS4) family of Linear Multistep (LMS) framework with second order time accuracy encompasses most of the developments over the last 50 years or so including new and optimal designs, and provides a wide variety of choices available to the analyst in a single analysis setting. We also advance, analyze and present the GS4 parameter studies regarding parameter selection. The GS4 readily allows flexibility in matching different time integration schemes, both existing traditional time schemes and other new and optimal competitive designs while maintaining second order time accuracy. Such implementation of coupling a wide variety of different spatial and time integration algorithms is not possible (especially implicit-implicit algorithm couplings) in the current state of the technology. Existing traditional practices of coupling multiple spatial and time algorithms show limitations in reduced order of accuracy, and consistency and the like in various field variables typical of, and including, temperature, temperature rate, and Lagrange multipliers. Various combinations of spatial methods and time algorithms between subdomains are tested with simple linear first order system problems for readers to readily mimic the results and demonstrate the overall robustness and efficacy for numerical computations.