Development of a nonlinear Finite Element beam model for dynamic contact problems

E. Svenning, Master thesis, Chalmers, supervisor B. Andersson, A. Mark, examiner K. Runesson, June 2011.

A Finite Element model for simulation of paper forming has been developed and validated. Paper forming is the fi rst step in the paper machine where a fiber suspension leaves the headbox and flows through a forming fabric. The fi bers land on the fabric and start to form the fi ber web. Understanding this process is important for the development of better paper products, because the orientation and distribution of the individual fibers during this step have a large influence on the fin al quality. Simulation of paper forming o ffers great challenges since it involves structures with large displacements and large rotations, flow with complex boundaries, fluid-structure interaction with strong coupling and dynamic collisions.

The fi ber model, which is based on a dynamic co-rotational formulation of the Euler-Bernoulli beam equation, accounts for geometric nonlinearities under the assumption of small strains. Two contact models have been implemented, a penalty method and the impulse based method Decomposition Contact Response. These models can handle fi ber-fi ber collisions as well as collisions between fibers and the forming fabric, which may have arbitrary geometry. Friction is included in the models and elastic/inelastic collisions are accounted for with the coefficient of restitution. The fiber model was implemented in C++ and the nonlinear system of equations was solved with Newton’s method. The flow around the fibers was simulated with the CFD software IBOFlow developed at FCC. IBOFlow is based on a finite volume discretization on a Cartesian octree grid that can be dynamically refi ned and coarsened. The flow around the moving fibers is resolved and the Hybrid Immersed Boundary Method is used to model the presence of fibers in the flow.

Extensive validation of the implementation has been performed against several demanding test cases from the literature. These cases include static instability with postbuckling, large amplitude oscillation of slender structures and dynamic impacts. Large e ffort was dedicated to making the code robust and efficient. The code was used to study two fluid-structure interaction problems. First, a single fi ber oscillating in a cross flow was studied and the numerical results were compared to an analytical solution obtained from a Fourier series expansion of the Euler-Bernoulli beam equation. Paper forming with two forming fabrics of di fferent geometry was also studied. A qualitative comparison of the resulting distribution and orientation of paper fibers was made.

Photo credits: Nic McPhee