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rate of convergence for Peridynamics

wallstedt's picture

Hello everyone - long time reader, first time poster.  I have worked with meshfree methods for a while but have recently read a few papers on Peridynamics.  The theory seems very interesting, although everything feels new, especially constitutive modeling.  But my question relates to something that I have not heard discussions of: the rate of convergence of a typical Peridynamic implementation.  What I mean is: can a Peridynamic solver offer a guarantee that error is reduced as the mesh size or particle spacing is refined?  If any demonstrations of first or second order accuracy have been published in any Peridynamic paper, I would appreciate a reference to them.  Thanks,

Phil

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Rich Lehoucq's picture

Phil,

There are several published papers that consider the rate of convergence as the mesh size is refined. Please see chapter 1 of Peridynamic Theory of Solid Mechanics by Stewart Silling and I, to appear in Advances in Applied Mechanics, and available as Technical report SAND 2010-1233J, for pointers to the literature. Keep in mind that before one can talk about reducing error, it must be demonstrated that there is a solution and that it's unique. These sort of mathematical questions are being addressed and chapter 1 cites the relevant papers and work in progress. 

Note that first or second order convergence rates depend upon the norm used and the regularity, i.e. smoothness, of the solution and the quality of the approximation. One of the interesting aspects of peridynamics is that the equations are valid even if the deformation is discontinuous and the location of the discontunuities are not required to be known in advance. The resulting mathematical theory and numerical analysis then explain that the best possible convergence rate may only be square root. However, away from the discontunity, the convergence rate will be first or second order, depending how the error is measured and the quality of the approximation.

Regards,

Rich Lehoucq

wallstedt's picture

Rich, thank you for the pointer to the SAND report; the refs from chapter one seem to go in the right direction and several of them were new to me.  I wanted to follow up on a few points you made, and make sure I'm heading in the right direction.  To start things off we make the necessary assumptions of existence, uniqueness, and smoothness (no cracks) in space and time.  So then we can make comparisons to typical FE behavior and to manufactured solutions.  In most methods the details can get a bit tricky: the way boundary conditions are applied, the way time stepping is initialized, mass lumping, order of reproducibility for elements or MLS approximations, the definition of error, etc. etc.  What I ultimately hope to learn is the specifics of all those picky details that are necessary to get second order convergence with, say, a typical L-2 norm.  If Peridynamics can demonstrate second order accuracy for smooth problems that would be fantastic, and lower order for discontinuous problems is to be expected.  Cheers,

Phil

Rich Lehoucq's picture

Phil,

Yes, second order accuracy on smooth problems does occur for linear peridynamics.

My colleague Max Gunzburger at FSU and his student have a paper under review where a 1d finite peridynamic bar is discretized with continuous and discontinuous FE and convergence rates verified to both smooth and discontinuous manufactured solutions. Another colleague Qiang Du at PSU and his student show the approximation error for both a 1d finite bar and 2d problem on a square; this paper has been accepted for publication in the SIAM J. of Num. Anal. You might contact both Max and Qiang for copies.

Any implementation has details, such as those you list, but mass lumping is not an issue if discontinuous FE are used. This may (or maybe not) seem obvious but is a point that deserves a bit of discussion. Discontinuous Galerkin methods for conventional weak formulations of linear elasticity are nonconforming. Asymptoptic convergence to discontinuous problems only occurs if the element boundaries are aligned with the discontinuities. On the other hand, a discontinuous FE for peridynamics is conforming and any discontunuities in the solutiuon only affect the rate of convergence.

Thanks for your interest.

Rich

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