Asymptotic integration algorithms for first-order ODEs with application to viscoplasticity
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When constructing an algorithm for the numerical integration of a differential equation, one must first convert the known ordinary differential equation (ODE), which is defined at a point, into an ordinary difference equation (OΔE), which is defined over an interval. Explicit and implicit, one-step, asymptotic, difference solutions, i.e. OΔE algorithms, are given for a non-linear first-order ODE which is written in the form of a linear ODE. The asymptotic forward integrator is typically underdamped. The asymptotic backward integrator is typically overdamped. The asymptotic midpoint and trapezoidal integrators tend to cancel out this damping by averaging, in some sense, the forward and backward integrations. Viscoplasticity presents itself as a system of non-linear, coupled, first-order ODEs that are mathematically stiff, and therefore difficult to numerically integrate. Considering a general viscoplastic structure, it is demonstrated that one can either integrate the viscoplastic stresses or their associated eigenstrains.