Research
Virtual Elements

# Virtual Elements For Engineering Applications

The finite element method (FEM) is a well established tool for solving a wide range of different engineering problems. However in recent years different methods like the virtual element method (VEM) were introduced as tools that brings some new features to the numerical solution of problems in solid and fluid mechanics.

The virtual element method is a competitive discretization scheme for meshes with highly irregular shaped elements and arbitrary number of nodes. VEM can use convex and non-convex polygons/polyhedra to mesh both two and three-dimensional solids. In VEM, the potential density is formulated in terms of suitable polynomial functions, instead of computing the unknown shape functions for complicated element geometries. This results in a rank-deficient structure, therefore it is necessary to add a stabilization term to the formulation. Herein, a robust stabilization technique for VEM will be introduced.

At IKM, latest developments related to the virtual element scheme will be investigated using the software tool AceGen program in the numerical implementation to compute the unknown fields. Of particular interest are problems related to contact mechanics, finite strain elasto-plasticity, thermo-mechanics, crack initiation and propagation and phase-field modeling of brittle and ductile fracture.

## Virtual Elements For Engineering Appications

• Virtual element formulation for trusses and beams
The virtual element method (VEM) was developed not too long ago, starting with the paper Beirao da Veiga et al. (2013) related to elasticity in solid mechanics. The virtual element method allows to revisit the construction of different elements in solid mechanics, however, has so far not been applied to one dimensional structures like trusses and beams. In this project, several VEM elements suitable for trusses and beams are derived. It could be shown that the virtual element methodology produces elements that are equivalent to well know finite elements but also elements that are different, especially for higher order ansatz functions, like 2nd and 3rd order for the truss and 4th order for the beam. It will be shown that these elements can be easily incorporated in classical finite element codes since they have the same nodal degrees of freedom as finite beam elements. Furthermore, the formulation allows to compute nonlinear structural problems undergoing large deflections and rotations.
Year: 2021
• Virtual Kirchhoff-Love plate elements for isotropic and anisotropic materials
The virtual element method allows to revisit the construction of Kirchhoff-Love elements because the C1-continuity condition is much easier to handle in the VEM framework than in the traditional finite element methodology. Here we study the two most simple VEM elements suitable for Kirchhoff-Love plates as stated in (Brezzi and Marini (2013)). The formulation contains new ideas and different approaches for the stabilization needed in a virtual element, including classic and stabilization. An efficient stabilization is crucial in the case of C1-continuous elements because the rank deficiency of the stiffness matrix associated to the projected part of the ansatz function is larger than for C0-continuous elements. This project aims at providing engineering inside in how to construct simple and efficient virtual plate elements for isotropic and anisotropic materials and at comparing different possibilities for the stabilization. Different examples and convergence studies discuss and demonstrate the accuracy of the resulting VEM elements. Finally, reduction of virtual plate elements to triangular and quadrilateral elements with 3 and 4 nodes, respectively, yields finite element like plate elements. These C1-continuous elements can be easily incorporated in legacy codes and demonstrate an efficiency and accuracy that is much higher than provided by traditional finite elements for thin plates.
Team: P. Wriggers, B. Hudobivnik, O. Allix
Year: 2021
• Virtual Element Method for 3D Contact
Contact plays a very important role in engineering problems, where two or more bodies interact with each other through their surfaces. Many techniques were developed in the past, to formulate the contact constraint at the contact interface between two bodies. Nevertheless, VEM provides efficient and robust properties to enforce the contact constraint through the contact interface. Investigations in 2D have been done so far. This work aims an extensions of VEM to 3D contact problems.
Leaders: F. Aldakheel, B. Hudobivnik, P. Wriggers
Team: M. Cihan
Year: 2020
• Virtual Element Method for Dynamic Applications
The Virtual Element Method is a recent developed discretization method, which can be seen as an extension of the classical Galerkin finite element method. It has been applied to various engineering fields, such as elasto-plasticity, multiphysics, damage and fracture mechanics. This project focuses on the extension of VEM towards dynamic applications. In the first part the appropriate computation of the Massmatrix regarding the vitual element ansatzspace will be done. In future works, VEM will be applied to engineering problems, considering the dynamic behavior.
Leaders: F. Aldakheel, B. Hudobivnik, P. Wriggers
Team: M. Cihan
Year: 2019
• 2D VEM for crack-propagation
Leaders: F. Aldakheel, B. Hudobivnik, P. Wriggers
Team: A. Hussein
Year: 2018