Download a PDF of the paper titled An explicit predictor/multicorrector time marching with automatic adaptivity for finite-strain elastodynamics, by Nicolas A. Labanda and 2 other authors
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Abstract:We propose a time-adaptive predictor/multi-corrector method to solve hyperbolic partial differential equations, based on the generalized-$\alpha$ scheme that provides user-control on the numerical dissipation and second-order accuracy in time. Our time adaptivity uses an error estimation that exploits the recursive structure of the variable updates. The predictor/multicorrector method explicitly updates the equation system but computes the residual of the system implicitly. We analyze the method's stability and describe how to determine the parameters that ensure high-frequency dissipation and accurate low-frequency approximation. Subsequently, we solve a linear wave equation, followed by non-linear finite strain deformation problems with different boundary conditions. Thus, our method is a straightforward, stable and computationally efficient approach to simulate real-world engineering problems. Finally, to show the performance of our method, we provide several numerical examples in two and three dimensions. These challenging tests demonstrate that our predictor/multicorrector scheme dynamically adapts to sudden energy releases in the system, capturing impacts and boundary shocks. The method efficiently and stably solves dynamic equations with consistent and under-integrated mass matrices conserving the linear and angular momenta as well as the system's energy for long-integration times.
Submission history
From: Nicolas Agustin Labanda Dr. [view email] [v1] Sat, 13 Nov 2021 01:32:55 UTC (40,268 KB) [v2] Mon, 10 Oct 2022 10:52:11 UTC (80,313 KB)
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The idea behind the predictor-corrector methods is to use a suitable combination of an explicit and an implicit technique to obtain a method with better convergence characteristics. The combination of the FE and the AM2 methods is employed often. Here, we use the FE as a predictor equation to get ypn+1 and subsequently use the AM2 as a corrector equation to get the final computed solution yn+1. The method, referred to as the Euler-Trapezoidal method is given below.
Note that in the second (corrector) step, the implicit term for the AM2, f(yn+1,tn+1) is replaced with f(ypn+1,tn+1), i.e., the value of f evaluated at the predicted ypn+1 is used. Hence, the predictor-corrector method described above is an explicit method.
Exercise Problem
Consider the IVP
(25)
Write C programs to compute y(t) in the interval [0,2] using (a). the forward Euler method (b) the AB2 method (c). the Euler-Trapezoidal (predictor-corrector) method and (d). the RK4 method. Plot your numerically computed solutions with h=0.1 along with the exact solution y=1/(1+t). Compare the convergence properties of each one of the above methods by plotting the absolute error for y(2) for h=0.001, 0.01 and 0.1. How do the stability characteristics of these methods compare with one another?