A comparison of different methods for calculating tangent-stifess matrices in a massively parallel computational peridynamics code




Brothers, Michael D.

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In order to maintain the quadratic convergence properties of the first-order Newton's method in quasi-static nonlinear analysis of solid structures it is crucial to obtain accurate, algorithmically consistent tangent-stiffness matrices. For an extremely small class of nonlinear material models, these consistent tangent-stiffness operators can be derived analytically; however, most often in practice, they are found through numerical approximation of derivatives. A goal of the study described in this thesis was to establish the suitability of an under-explored method for computing tangent-stiffness operators, referred to here as 'complex-step'. Compared are four methods of numerical derivative calculation: automatic differentiation, complex-step, forward finite difference, and central finite difference in the context of tangent-stiffness matrix calculation in a massively parallel computational peridynamics code. The complex-step method was newly implemented in the peridynamics code for the purpose of this comparison. The methods were compared through in situ profiling of the code for Jacobian accuracy, solution accuracy, speed, efficiency, Newton's method convergence rate and parallel scalability. The performance data was intended to serve as practical guide for code developers and analysts faced with choosing which method best suit the needs of their application code. The results indicated that complex-step produces Jacobians very similar, as measured by a low l 2 norm of element wise difference, to automatic differentiation. The values for this accuracy metric computed for forward finite difference and central finite difference indicated orders of magnitude worse Jacobian accuracy than complex-step, but convergence vstudy results showed that convergence rate and solution was not strongly affected. Ultimately it was speculated that further studies on the effect of Jacobian accuracy may better accompany experiments conducted on plastic material models or towards the evaluation of approximate and Quasi-Newton's methods


The author has granted permission for their work to be available to the general public.


Jacobian, newton's method, nonlinear systems, parallel computing, peridynamics, tangent-stiffness



Mechanical Engineering