mfem/miniapps/spde

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