# Numerical homogenization

Greetings. My name is Morgan Görtz, and I am a Ph.D. student in numerical analysis working at the Fraunhofer- Chalmers Center for industrial mathematics. My research area is numerical homogenization applied to network models modeling industrial applications. The Swedish Foundation for Strategic Research funds the mathematical side and aims to broaden the mathematical theory and use of homogenization in industrial settings. Currently, the network models analyzed are paper models used to study structural properties before production.

Paper has a weblike structure on the microscopic level. This web consists of individual paper fibers connected through mechanical interlocking and various microscopic forces. Modeling all paper fibers leads to high complexity but is necessary. Without the fiber structure in the model, you lose fiber shape, orientation, composition, and other important parameters used in the paper-making process. Our proposed model sees the fibers as one-dimensional beams, modeled using classical linearized beam theory resulting in a linear network model. There are more advanced approaches analyzed, but those require more processing power and computation time. However, it is easy to lose relevancy with simplifications. Our paper model is developed within an industrial collaborative project called ISOP (Innovative Simulation of Paper) to stay representative, effective, and ultimately relevant. ISOP is performed by a consortium consisting of Albany International, Stora Enso, and Fraunhofer- Chalmers Center, with the end goal being validated and actively used simulation tools in the paper-making process.

Numerical homogenization seeks to find good approximations of heterogeneous problems. One approach is to divide the problem into different scales by using a multiscale method. Consider the paper model. Taking a direct approach by solving the problem with all components at once scales poorly, and models over a couple of square centimeters are affected by resource limitations. Instead, a multiscale method partitions the paper model into parts. Each part forms a smaller problem, less resource-intense, that provide a good structural representation of the fiber structure in that part. These representations are then used to approximate the initial problem by replacing the microscopic, poorly scaling elements.

In my research, I currently work with the Local Orthogonal Decomposition (LOD) method. The method takes a finite element approach. Whereas a classical finite element method needs a sufficiently fine trial space to resolve the heterogeneities, the LOD method aims to find a coarse trial space (representations) that resolves the heterogeneous effects. This coarse trial space is found by taking a standard finite element trial space and adding orthogonal modifications to each nodal function, encoding the microscopic structure into the nodal functions. At this point, the use of the LOD method for network models is experimental. Still, numerical results indicate that the method works optimally for network models and the paper model particularly. Moving forward, we aim to develop the theory, investigate the possibilities, and understand the limitations of the LOD method for network-based models.