Speaker
Description
The position-based finite element formulation (PFEF) assumes the nodal positions rather than the displacements as the primary unknowns. This formulation enables the derivation of simple analytical expressions of the secant and tangent stiffness matrices of isoparametric elements with any hyperelastic constitutive law [1].
Accordingly, the nonlinear governing equations are expressed as:
\begin{equation}
\mathbf{M} \ddot{\mathbf{x}} \left( t \right) + \mathbf{D} \dot{\mathbf{x}} \left( t \right) + \mathbf{S} \left[ \mathbf{x} \left( t \right) \right] \mathbf{x} \left( t \right) = \mathbf{p} \left( t \right) + \mathbf{r} \left( t \right),
\end{equation}
where $\mathbf{M}$, $\mathbf{D}$, and $\mathbf{S} \left(\mathbf{x}\right)$ are the global mass, damping, and secant stiffness matrices, respectively; $\mathbf{x}$ is the nodal position vector, $\mathbf{p}$ and $\mathbf{r}$ are the vectors of nodal loads and restraint reactions, respectively; an upper dot represents differentiation w.r.t. time, $t$.
The governing equations are solved by using an incremental-iterative arc-length method in statics and Newmark's method in dynamics.
In this talk, we present some examples concerning cable nets [2], wrinkling membranes [3], and curved rods [4]. Furthermore, we discuss the application of the PFEF for structural optimisation problems [5].
References
[1] Valvo PS (2025) Symmetric stiffness matrices for isoparametric finite elements in nonlinear elasticity. Comput Mech 79:919-943. https://doi.org/10.1007/s00466-024-02539-4
[2] Fisicaro P, Pasini A, Valvo PS (2022) Simulation of Deployable Cable Nets for Active Debris Removal in Space. J Phys Conf Ser 2412:012010. https://doi.org/10.1088/1742-6596/2412/1/012010
[3] Valvo PS (2025) Position-based finite element formulation for the analysis of wrinkled membranes. In: Trovalusci P, Sadowski T, Ibrahimbegovic A (eds.) Multiscale and Multiphysics Modelling for Advanced and Sustainable Materials, Adv Struct Mater 231:417-429, Springer. https://doi.org/10.1007/978-3-031-84379-2_31
[4] Lottici L, Fisicaro P, Scheid SP, Valvo PS (submitted) A position-based formulation of the Hermite finite element for planar Kirchhoff rods: linear static and dynamic analysis. Comput Mech.
[5] Boyd S, Vandenberghe L (2004) Convex Optimization. Cambridge University Press. https://doi.org/10.1017/CBO9780511804441