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Miniapp: Gaussian Random Fields of Matérn Covariance for Imperfect Materials

This miniapp implements the SPDE method [1] for generating Gaussian random fields of Matérn covariance. We use the resulting random field to model material uncertainties similar to [2,3].

Theory

The mini-app proceeds in three steps. First, we generate a random field $u$ by solving a fractional PDE [1,2] with MFEM. Second, we define a topological support density $v$ [3]. In the third step, we combine $u$ and $v$ to create a topology with random imperfections $w$ similar to [3] $$w = v + (s \cdot T(u) + a),$$ where the scalar parameter s and a can be controlled via the command line parameters --scale and --offset, respectively. Furthermore, you may choose T as the identity transformation (-no-urf) or a pointwise transformation taking the Gaussian random field to a uniform random field (-urf, -umin, -umax). The final geometry is defined as the zero level set $w=0$.

1. Fractional PDE

In the first step, we solve the fractional PDE $$A^\alpha u = \eta W,$$ where $\alpha \in \mathbb{R}^+$, $W$ is Gaussian White noise, and $\eta$ the normalization constant $$\eta = \left( \frac{(2\pi)^{\dim{\Omega}/2} \sqrt{\det{\underline{\underline{\Theta}}}} \Gamma(\nu + \dim{\Omega}/2)} {\nu^{\dim{\Omega}/2} \Gamma (\nu) } \right)^{1/2}.$$ The fractional Operator $A$ is given by $$A = \frac{-1}{2\nu} \nabla \circ \underline{\underline{\Theta}} \nabla + 1,$$ the exponent $\alpha$ is defined as $$\alpha = \frac{2\nu + \dim(\Omega)}{2}.$$

We solve the FPDE with the same approach as in ex33/ex33p. In a nutshell, we compute a rational approximation of the operator A via the triple-A algorithm [4]. Instead of solving a fractional PDE, the rational approximation allows us to solve $N$ integer-order PDEs $$(A + b_i) u_i = c_i \eta W,$$ whose solutions $u_i$ approximate the true solution $u$ of the fractional problem $$u \approx \sum_k u_k.$$ For more details, consider ex33/ex33p and references [4,5,6].

The dimension $\dim (\Omega)$ is implicitly defined via the mesh, but you may specify $\nu$ and $\underline{\underline{\Theta}}$ via the command line arguments -nu and -l1,-l2,-l3,-e1,-e2,-e3. The normalization $\eta$ is computed based on the parameter choice.

2. Topological support

For the topological support, we restrict ourselves to particles and an octet truss with imperfections modeled via the random field. Following [3], a general function for the topological support is $$v (x) = \tau - \rho(x,\mathcal{G}).$$ and one may choose $\rho$ such that the function describes either particles or an octet truss (c.f. [3, Example 2, Example 5]). As a user, You may specify the topology with the command line arguments -top (0 = particles,1 = octet truss).

Sample runs

Generate 5 particles with random imperfections

mpirun -np 4 generate_random_field -o 1 -r 3 -rp 3 -nu 2 \
       -l1 0.015 -l2 0.015 -l3 0.015 -s 0.01 \
       -t 0.08 -n 5 -pl2 3 -top 0 -rs

Generate an Octet-Truss with random imperfections

mpirun -np 4 generate_random_field -o 1 -r 3 -rp 3 -nu 2 \
       -l1 0.02 -l2 0.02 -l3 0.02 -s 0.01 \
       -t 0.08 -top 1 -rs

Generate an Octet-Truss with random imperfections following a uniform distribution

mpirun -np 4 generate_random_field -o 1 -r 3 -rp 3 -nu 2 \
       -l1 0.02 -l2 0.02 -l3 0.02 -umin 0.01 -umax 0.05 \
       -t 0.08 -top 1 -urf -rs

A 2D random field with anisotropy

mpirun -np 4 generate_random_field -o 1 -r 3 -rp 3 -nu 4 \
       -l1 0.09 -l2 0.03 -l3 0.05 -s 0.01 \
       -t 0.08 -top 1 -no-rs -m ../../data/ref-square.mesh

Visualization

The results can be visualized via GLVis or ParaView. GLVis offers quick and lightweight access while ParaView is a more extensive software package. By default, we export four scalar fields to both software packages:

  • The topological support, i.e. the basic geometrical structure
  • The random perturbations
  • The topology with random imperfections
  • The level set

GLVis

To visualize with GLVis, simply have your GLVis server running when you execute the mini-app. Turn on/off with `-gvis/-no-gvis' command line arguments.

ParaView

We export a file <workdir>/ParaView/SurrogateMaterial/SurrogateMaterial.pvd that can be opened and edited with ParaView as you wish. Turn on/off with `-pvis/-no-pvis' command line arguments.

Implementation details

  • The implementation is MPI parallel, if you have built MFEM without MPI, this mini-app will not work.
  • While the mini-app emphasizes [3, Example 2 and 5], users may extend this mini-app to other topologies (e.g. [2, Example 3]) by implementing an appropriate distance metric. We recommend creating a child class of MaterialTopology (see material_metrics.hpp) and swapping the line in generate_random_field.cpp in which we create the respective object.
  • The matrix $\underline{\underline{\Theta}}$ can be specified with length scales -l1,-l2,-l3 and Euler angles -e1,-e2,-e3. We construct a diagonal matrix D = diag([l1,l2,l3]) and rotation matrix R(e1,e2,e3) and compute $\underline{\underline{\Theta}} = R^T D R$.
  • The shape of the particles can be specified with -pl1,-pl2,-pl3. We choose random Euler angles for each particle.

Accompanying presentation

MFEM workshop 2023

References:

[1] Lindgren, F., Rue, H., Lindström, J. (2011). An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(4), 423–498. https://doi.org/10.1111/j.1467-9868.2011.00777.x

[2] Khristenko, U., Constantinescu, A., Tallec, P. L., Oden, J. T., & Wohlmuth, B. (2020). A Statistical Framework for Generating Microstructures of Two-Phase Random Materials: Application to Fatigue Analysis. In Multiscale Modeling & Simulation (Vol. 18, Issue 1, pp. 21–43). Society for Industrial & Applied Mathematics (SIAM). https://doi.org/10.1137/19m1259286

[3] Khristenko, U., Constantinescu, A., Tallec, P. L., & Wohlmuth, B. (2021). Statistically equivalent surrogate material models and the impact of random imperfections on elasto-plastic response (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2112.06655

[4] Nakatsukasa, Y., Sète, O., & Trefethen, L. N. (2018). The AAA algorithm for rational approximation. SIAM Journal on Scientific Computing, 40(3), A1494-A1522. https://doi.org/10.1137/16M1106122

[5] Harizanov, S., Lazarov, R., Margenov, S., Marinov, P., & Pasciak, J. (2020). Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation. Journal of Computational Physics, 408, 109285. https://doi.org/10.1016/j.jcp.2020.109285

[6] Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X., Mao, Z., Cai, W., Meerschaert, M. M., Ainsworth, M., & Karniadakis, G. E. (2020). What is the fractional Laplacian? A comparative review with new results. In Journal of Computational Physics (Vol. 404, p. 109009). Elsevier BV. https://doi.org/10.1016/j.jcp.2019.109009