How is Smoothed Particle Hydrodynamics different from Peridynamics considering both have a concept of horizon or smoothing length? I've been working with Smoothed-Particle Hydrodynamics (SPH) to simulate a ballistic impact problem. While looking into other meshless methods, I came across Peridynamics. It seems the fundamental strength of this method lies in way it reformulates the equation of motion  as an integral equation as opposing to a partial differential equation in continuum mechanics.
However, it seems in SPH the field variable is not being differentiated, so there shouldn't be any issue with the crack or discontinuities.  Right?
$f\left(r_{i}\right) \cong \sum_{j} \frac{m_{j}}{\rho_{j}} f_{j} W\left(\left|r_{i}-r_{j}\right|, h\right)$
$\nabla f\left(r_{i}\right)=\nabla \sum_{j} \frac{m_{j}}{\rho_{j}} f_{j} W_{i j}=\sum_{j} \frac{m}{\rho_{j}} f_{j} \nabla_{j} W_{i j}$
If what I am saying is correct, what advantage does Peridynamics gives over SPH? And what is the difference between the two methods?
Eventually, I have to work on a thermo-mechanical simulation of a ballistic impact with material damage. Which technique out of these two will be suitable for the problem statement?
 A: Ohhhh, welcome to a minefield! This question actually really excites me, because there's so much... animosity (?) towards peridynamics but it's a really useful and powerful method. It also isn't a magic bullet -- nothing is -- so understanding the nuances of it compared to SPH or finite element methods (FEM) is really key.
First, it's really important to note that there are different flavors of both SPH and peridynamics. The field has evolved quite a bit in the past decade and so the understanding in the literature in 2010 may be very different from what it is today.
Under a very specific set of conditions, one can show that SPH and peridynamics are equivalent. In the linked reference, they result in equivalent discrete expressions when:

*

*Peridynamics is formulated using a classical stress tensor;

*SPH is based on a total Lagrangian framework with a linear kernel gradient correction to ensure the forces between two particles are symmetric.

The key takeaway from that equivalency is that SPH requires ad-hoc corrections to the physics to correct for the formulation, while peridynamics does not require any corrections to ensure consistency -- it is automatic in the formulation for a given stress tensor. In other words, peridynamics naturally ensures Newton's third law (called linear admissibility) as well as conservation of angular momentum (called angular admissibility) for all valid force functions. On the other hand, SPH requires corrections to the governing equations to ensure this conservation. In both cases, when linear and angular momentum is conserved, you eliminate the tensile instabilities that can cause SPH to blow up.
The other trouble point, as you noted, is when you have discontinuities. In the limited case where SPH and peridynamics overlap, they will both suffer from rank-deficiency when there are discontinuities and will require ad-hoc fixes to address.
However, peridynamics does not have to use classical stress tensors. One can derive arbitrary constitutive relationships using the fact that peridynamics is an upscaling of molecular dynamics, which requires no derivatives to evaluate properties and relationships. When this is done, there are no gradients of the deformation and so nothing blows up when there are discontinuities across cracks and so on.
I think the last paragraph of the paper I linked above sums it up much better than I can:

It is worthwhile to emphasize that the mathematical foundation of Peridynamics is clear and straightforward. Correct equations of motion emerge from this theory which conserve linear and angular momentum, and approximate linear fields accurately.  All of these desirable features can only introduced into the SPH approximation by ad-hoc procedures.  e.g.  explicit symmetrization.  Thus,  if anything,  the Peridynamic theory provides us with a better route to deriving meshless discretizations than the SPH method.  If the simplest form of meshless discretization, namely nodal integration, is used, the advantages of Peridynamics over SPH vanish and the discrete expressions of both methods become equal.


I came into perdynamics from a fluid dynamics background, so I didn't quite understand what the big deal was. To me, peridynamics was kind of like a finite volume form of the equations, and SPH was kind of like a finite difference form of the equations. We can switch between them easily in fluids, so why is structures so opposed to peridynamics? I never really did get it. And just like FV vs FD in fluids, they are equivalent under a limited set of circumstances, but each has its own strengths where they are not equivalent. For peridynamics, it is much more general than SPH.
In the case of peridynamics, my personal opinion (which I came to during my masters thesis work) is that any problem that will involve cracks, tension, or complex/unknown deformations should use peridynamics. The formulation handles all of that beautifully and naturally. SPH, or FEM for that matter, all require hacks and corrections to bring back physics that peridynamics contains inherently. By not having to take a gradient of the deformation field, you can represent really rich and complex constitutive relations without worrying about cracks and gaps. And it has the added benefit of being an upscaling of molecular dynamics, which enables one to use multi-scale methods to bridge the range of scales between them.
Not all is rosy, of course. While you don't have to take gradients of deformation fields, if you need to account for thermal conductivity in the material, you will still end up having to take a derivative of the temperature field (assuming Fourier diffusion -- if you can come up with another thermal diffusion model that doesn't need a derivative, that would be a novel contribution to the field). And so you'll still have to introduce some ad-hoc regularization to the temperature derivatives around discontinuities. But, I think the fewer places you have to put hacks in, the better and SPH requires hacks in all equations, while peridynamics would only need it in a temperature equation if you go that route.
