The paper and pulping industries account for a substantial amount of global industrial energy use, which means that small improvements in the process result in significant effects. These improvements could include using more recycled pulp or blending different types of wood from biodiverse forests. However, changing the type of wood used in paper production will directly impact the mechanical properties of the material.
Lab-based testing of new paper materials is expensive and time-consuming, as heavy machinery is used for production. Computer simulations do not have this issue, where efficient digital tools could mitigate unnecessary prototyping by finding optimum configurations before physical production. This may sound straightforward, but because paper-based materials have such a complex microstructure, tools to evaluate commercial-grade paper digitally that consider each individual fiber have only recently become feasible.
The Fraunhofer-Chalmers Research Center for Industrial Mathematics in Gothenburg, in collaboration with the Department of Mathematical Sciences at Chalmers University of Technology and University of Gothenburg, has developed digital tools for micro-scale paper-product evaluation during the last decade. Initial development focused on paper-making, which simulated the machines that formed the paper material using CFD simulations. The results of these simulations are geometries of formed sheets on the fiber scale, with the logical next step being to evaluate the structural properties of the paper using the geometries.
Digital paper models (see Figure 1) that resolve each fiber in the material are numerically challenging, even on small scales. To combat this complexity, we decided to model the fibers as beams in a network model. This means the model is entirely discrete, with each fiber represented as a discretized one-dimensional line. Even with this simplification, the models are challenging, and what’s more, the systems of equations resulting from such discretizations are notoriously ill-conditioned. This means most off-the-shelf iterative solvers are off the table and back on the shelf.
During the last five years, our research group has developed a mathematical framework [1] to develop numerical tools for these discrete network models. This framework proves some fundamental bounds in the discrete setting, entirely based on the network geometry, that enable finite element-type theory. Using this framework, two numerical approaches have been mathematically and numerically shown to work in this setting: an iterative method based on domain decomposition [1] and a multi-scale method based on the localized orthogonal decomposition method [2].
The iterative method in [1] enabled micro-mechanical simulations of paper-based materials on (relatively) huge scales [3] without the need for periodicity or homogenization, which has sparked some interest from people working with paper physics and product development. Simulations that consider each fiber in the material on these scales can give new insights into experiments by re-evaluating them digitally and producing detailed information about the role of each individual fiber. Moving forward, we aim to develop theory for non-linear models and investigate shear and compression experiments within the framework, see Figure 2, which depicts a digital recreation of a compression tester with the paperboard model in Figure 1.
References
[1] Morgan Görtz, Fredrik Hellman, Axel Målqvist, Iterative solution of spatial network models by subspace decomposition, Math. Comp., Vol. 93, pp.233–258, 2024.
[2] Fredrik Edelvik, Morgan Görtz, Fredrik Hellman, Gustav Kettil, Axel Målqvist, Numerical homogenization of spatial network models, CMAME, Vol 418, Part B, 11659, 2024.
[3] Morgan Görtz, Gustav Kettil, Axel Målqvist, Mats Fredlund, Fredrik Edelvik, Iterative method for large- scale Timoshenko beam models, assessed on commercial-grade paperboard, Comput. Mech., 2024.