# Numerical homogenization of network models and micro-mechanical simulation of paperboard Doctoral thesis, 2023

Micro-mechanical simulations of paper products involve complex geometries and challenging numerical problems. This work considers micro-mechanical simulations where individual paper fibers are modeled and resolved. This level of detail is useful in paper-product development, where wood composition and other fiber-based parameters are essential. Several time-dependent and nonlinear micro-mechanical models have been proposed in the literature for accuracy, but these models are limited to small problems.

This work evaluates linear network models as a possible effective tool for paper development, as they permit full commercial-grade paper to be modeled on consumer hardware. This evaluation was performed in the industrial collaboration Innovative Simulation Of Paper (ISOP), with the goal of developing numerically efficient micro-mechanical simulations that are useful for paper product developers. In this work, the linear model was shown to produce accurate results for tensile stiffness, bending stiffness, and tensile strength for paper products with low surface weight. Moreover, accurate tensile stiffness and bending stiffness simulations were possible with commercial-grade three-ply paperboards. Bending stiffness simulations using micro-mechanical models are not well studied, but from this evaluation, it is clear they are now possible on consumer-grade hardware.

Increasing the size of these micro-mechanical models requires specialized numerical techniques that are less resource-intensive. This work developed the theoretical foundation for a finite element-inspired mathematical theory on models based on networks. With this foundation, two resource-efficient methods, an iterative solver and a multiscale method, were mathematically motivated for the discrete network setting. These methods were also validated numerically for the mentioned micro-mechanical paper models. For the iterative approach, bending resistance simulations of models larger than the computational limit of a direct approach were possible.

domain decomposition

paper model

local orthogonal decomposition

Bending

paper simulation

network model

multiscale

Pascal, Hörsalsvägen 1

## Author

### Morgan Görtz

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

### ITERATIVE SOLUTION OF SPATIAL NETWORK MODELS BY SUBSPACE DECOMPOSITION

Mathematics of Computation,; Vol. 93(2024)p. 233-258

Journal article

### Numerical homogenization of spatial network models

Computer Methods in Applied Mechanics and Engineering,; Vol. 418(2024)

Journal article

### Multiscale methods for solving wave equations on spatial networks

Computer Methods in Applied Mechanics and Engineering,; Vol. 410(2023)

Journal article

### Network model for predicting structural properties of paper

Nordic Pulp and Paper Research Journal,; Vol. 37(2022)p. 712-724

Journal article

### A numerical multiscale method for fiber networks

World Congress in Computational Mechanics and ECCOMAS Congress,; Vol. 300(2021)

Paper in proceeding

### Görtz, M., Kettil, G., Målqvist, A., Fredlund, M., Edelvik, F. Iterative method for large-scale Timoshenko beam models, assessed on commercial-grade paperboard

Paperboard is a layered material typically composed of different types of fibers. These fibers have various characteristics; some add good surface properties, and others bulk to the board. Understanding the effects of different fiber characteristics is essential in paper-product development. In this work, paperboard properties are evaluated with a micro-mechanical model where each fiber in the material is considered. These fibers are modeled using one-dimensional linear Timoshenko beams in a linear network model. Here, tensile stiffness and bending stiffness are simulated and validated against experiments and multi-laminar theory.

Novel numerical techniques and a mathematical framework are presented to evaluate these micro-mechanical models. With this framework, two methods are mathematically motivated and numerically validated for linear network models on elliptic-type problems. The first method is a multi-scale method that creates an accurate coarse-scale representation of the model well suited for periodic models and geometrically linear time-dependent problems. The second method is a domain decomposition method, an iterative method that can enable large-scale simulations due to memory efficiency and parallelizability.

### Subject Categories

Applied Mechanics

Basic sciences

### Infrastructure

C3SE (Chalmers Centre for Computational Science and Engineering)

Materials Science

### ISBN

978-91-7905-976-7

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 5442

### Publisher

Chalmers

Pascal, Hörsalsvägen 1