Curve-induced distortion of polynomial space curves, flat-mapped extension modeling, and their impact on ANCF thin-plate finite elements

Abstract

The Absolute Nodal Coordinate Formulation (ANCF) is a relatively new nonlinear finite element type that uses Hermite splines for shape functions. In this investigation, the ANCF is examined as a possible tool for use in modeling the media in flexible media transport systems, such as printers, copy machines, and roll-to-roll systems. However, it is demonstrated using an example of a thin plate-type ANCF finite element that these elements can suffer from significant membrane locking, which can be problematic for paper or paper-like media. One source of this locking is identified to be a property of all parametric curves that are composed of polynomials. The property is that for parametric polynomial curves, changes in the state of curvature of the curve cause changes in the distribution of points along the curve. This property is labeled Curve-Induced Distortion (CID) by the authors of this paper. CID can cause axial and membrane strain distortion in elements, causing them to be overly stiff. A new solution method is proposed to directly counteract CID in finite elements that use cubic Hermite curves for shape functions, specifically for modeling problems in which bending occurs primarily around one axis, such as paper in printing and media transport machinery. This method is labeled Flat-Mapped Extension Modeling (FMEM). FMEM is a mixed field method that uses a 1D Hermite polynomial kinematically linked to the 3D Hermite curve to represent the axial displacement field. FMEM significantly reduces the effect of CID in the ANCF element tested here. This investigation demonstrates using a single ANCF plate element type that the ANCF’s accuracy can be significantly improved by FMEM with only a small increase in computational cost. It is shown with this plate-element example that without correcting CID, the ANCF element tested is computationally much slower than contemporary methods like the co-rotational formulation for similar accuracy. But with FMEM, the ANCF is significantly faster than the co-rotational formulation for similar accuracy.