Paper modeling results in complex geometries that lead to enormous numerical problems. The complexity lies in the material’s microstructure. Individual paper fibers must be considered for useful material simulations in paper development, where wood composition and other fiber-based parameters are essential. Multiple time-dependent and nonlinear modeling techniques have been proposed in the literature. In this work, a simplified approach to paper modeling is proposed. A simple but effective model can be created by seeing the paper as a network of one-dimensional beams and using linearized one-dimensional beam theory. Working in the industrial collaboration Innovative Simulation Of Paper (ISOP), the model was constructed to be relevant for product developers in papermaking industry, which means fast evaluations and representative results. The model was validated against experimental data for tensile stiffness, tensile strength, and bending resistance in both cross and machine direction for several low-density sheets. These simulations are fast, taking no more than a couple of minutes to generate and evaluate randomly generated paper samples.
For larger simulations, a multiscale approach is proposed. The multiscale method is the Localized Orthogonal Decomposition (LOD) method, a generalized finite element method. In this method, the heterogeneities (fibers) in the paper model are resolved using special local orthogonal projection operators. This work presents the theoretical foundation of using the LOD method on discrete network models, which builds up to an a priori error bound for the multiscale approximations. The theoretical a priori error results are confirmed with numerical examples. Both structural problems and scalar-valued discrete network problems are presented in these examples. This work ends with numerical results showing the successful use of the LOD method on one of the structural simulations in the validation of the paper model, showing that the LOD method can be used for practical simulations.
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