Bottom Line Upfront

I found Peridynamics to be unsuitable:

1. It does not represent constant strain modes exactly.
2. The only acceptable formulation is unstable, especially around fracture tips.
3. We can fix the instability, but that doesn't change #1.
4. It is more computationally costly than the Finite Element Method.

Checkout my implementation with PeriFlakes: https://github.com/afqueiruga/PeriFlakes
The Jupyter notebook replicates the graphs in this presentation.

Today, I'm going to talk about the path I took to make those determinations, and discuss the stabilization around fracture tips.

Outline

• The problem I was trying to solve
• The simulation I managed to program
• PeriFlakes
• Briefly: constant strain problems
• Thin crack benchmark problem
• Smoothing methods
• Comparisons including FEM

Motivating Problem

Shale natural gas, Enhanced geothermal, Escape of gasses from nuclear waste, Bubbling of methane through oceanic sediments

Peridynamic Representation

Computational Framework

Videos are not results

It looks good, but not under actual scrutiny:

Queiruga, A. F. and G. J. Moridis, "Numerical experiments on the convergence properties of state-based peridynamic laws and influence functions in two-dimensional problems." Computer Methods in Applied Mechanics and Engineering 322 (2017): 97-122.

It is critical to verify and validate numerical methods on known solutions and expected discretization refinement behavior.

• Can't jump to dynamics and fracturing simulations immediately.

Variety of Peridynamic Formulations

It is well known that bond-based does not work due to $\nu$ problem.

The are two classes of state-based models:

1. The dilation-state methods
   1. Silling's
   2. Oterkus's
2. The deformation gradient (F) correspondence methods
   1. Silling's original formulation
   2. Littlewood 2010's stabilization
   3. Silling 2017's stabilization (back-to-back in CMAME to Queiruga, 2017!)
   4. I tried post-processing smoothing

Hyperparameters

Our goal is to obtain a computer program that solves the problems we want accurately and efficiently.

In the Peridynamics scope, we have the following hyperparameters:

1. Formulation $\mathbf{t}$- discrete choice
2. Influence function $w$ - discrete choice
3. Relative support radius, $m=\delta/h$ - continuous
4. Stabilizer parameter $A$ - continuous
5. Smoothing function $w^s$ - discrete choice (including None)

These are arbitrary. We need to determine which combinations are acceptable, and which of those are the best.

Thin Pressurized Crack

We need to check for convergence to analytical solutions.

\begin{equation} \hat{Z}\left(z = x+iy\right)=P\left(\sqrt{z^{2}-L^{2}}-z\right) \end{equation}

\begin{equation} u_{x} = \frac{1}{2G}\left(\frac{2-4\nu}{2}\mathrm{Re}\left[\hat{Z}\right]-y\mathrm{Im}\left[Z\right]\right) \end{equation} \begin{equation} u_{y} = \frac{1}{2G}\left(\frac{4-4\nu}{2}\mathrm{Im}\left[\hat{Z}\right]-y\mathrm{Re}\left[Z\right]\right) \end{equation}

Calculate an error to judge numerical solution $u^N$: \begin{equation} e = \frac{ \int \left|u^N-u^a \right|^2 \mathrm{d}x }{ \int \left|u^a \right|^2 \mathrm{d}x } \end{equation}

Peridynamic Displacement Field

The F-based method's instability effects bond stretches, making the fracture branch and jump nonphysically.

Smoothing

My ad hoc approach was to post process the solution: \begin{equation} \mathbf{u}^s = \frac{\int_\mathcal{H} w^s \alpha \mathbf{u} \mathrm{d}x'}{\int_\mathcal{H} w^s \alpha \mathrm{d}x'} \end{equation}

The smoothing kernel $w^s$ doesn't have to be the same $w$.

This worked to stabilize the fracture path in my original hydraulic fracturing simulations, but the method never passed any other validations!

The Stabilizers

We can add a stabilizer to the force: \begin{equation} \mathbf{T}\left<\xi\right> = w\alpha \sigma \mathbf{K}^{-1} \mathbf{\xi} + \mathbf{T}^s \end{equation}

Silling's 2017 paper: \begin{equation} \mathbf{T}^s = w\alpha A \frac{18 K}{ w_0 \pi \delta^5} \left( \mathbf{Y} - \mathbf{F}\xi \right), \quad w_0 = \int_\mathcal{H} w\alpha \mathrm{d}x' \end{equation}

Littlewood's 2010 paper: \begin{equation} \mathbf{T}^s = w\alpha A \frac{18 K}{\pi \delta^4} \left( \frac{ \left|\mathbf{Y}\right| - \left|\mathbf{F}\xi\right| }{ \left|\xi\right| } \right) \frac{\mathbf{Y}}{\left|\mathbf{Y}\right|} \end{equation}

$A$ is a new penalty hyperparameter. It would be nice if it were insensitive to the grid size.

Grid Convergence on Fracture

• The smoothing step is the best correction
• Silling's dilation is the best, but recall that it doesn't pass constant strain!

What is our ultimate goal?

We're searching for a numerical method that:

• Passes all standard tests (continuum mechanics solutions)
• Solves the fracture problem efficiently
• Efficiently represents small changes in fractures
• Good tradeoff between accuracy and computational cost

Compare to a discretely meshed fracture solved by FEM:

• Meshed with GMSH, implemented with FEniCS
• Included in PeriFlakes/fem/fem_crack.py

Wall-clock time for matrix/vector assemble + solve.

Error vs. Compute time

• FEM is an order of magnitude more accurate and faster.

Minimal Representable Length

• Bond breaks are discrete events at fixed locations
• Peridynamic grid spacing is the minimal length a fracture can jump
• Same as a the mesh size in node-splitting or element-deleting FEM

Minimal Representable Length

FEM solves a smaller length scale faster!

Conclusions

• Need to do rigorous verifications of our numerical programs.
• Evaluate programs on a static crack problem with a known solution.
• Perform hyperparameter sweeps---every choice is arbitrary.
• Can fix the stability and improve the accuracy of the F-based Peridynamics model.
• Silling's dilation model is the best on the crack, but it doesn't pass all constant-strain tests.
• FEM is faster and more accurate.
• Peridynamics discretization isn't more efficient for a given fracture resolution.

We can churn out and test more Peridynamics models in PeriFlakes!

Thank you

Email: afqueiruga@lbl.gov

Checkout my codes at
https://github.com/afqueiruga/PeriFlakes
and look at the Jupyter notebook and Zenodo database.

I have another upcoming talk at WCCM in July:

#2018172, Numerical Experiments on the Convergence Properties of State-based Peridynamic Laws and Influence Functions in Two-Dimensional Problems