Modern mechanics is unthinkable without computations and, more generally, numerical simulations. Analytical or closed-form solutions are seldom available for practically relevant engineering problems. Predicting static deformations and stability of a structure under given loads, its dynamic behaviour or the steady-state vibration response requires modern techniques of computational mechanics, including various kinds of symbolic computations, finite element analysis, time integration, etc.
The Research Group Mechanics of Solids is focusing on efficient simulations of thin and slender structures such as curved rods in space, plates or shells. For that sake, we use not only commercially available software, but we develop our own simulation tools, which can be tailored to specific problem-oriented needs for improved simulation efficiency. Although finite elements play the most prominent role in our work, we also use other schemes like finite differences or the Ritz-Galerkin method, if more appropriate. This enables us to apply the best fitting method to a specific problem to produce accurate and trustworthy solutions to a variety of problems in a relatively short time.
Numerical Modelling with Finite Elements.
The numerical schemes developed by our group are founded on modern analytical techniques, such as the technique of asymptotic splitting and methods of Lagrangian mechanics, which directly provide us with reliable dimensionally reduced models of thin and slender bodies, namely one-dimensional continua for rods or material surfaces for shells. Based on the knowledge of proper degrees of freedom of material particles, we develop consistent numerical models using the Finite Element method.
As our approach to numerical modelling rests on fundamental theoretical results, the developed computational schemes are less prone to the effects of locking and numerical instabilities, especially in geometrically nonlinear problems with essential changes of curvature of rods and shells. We concentrate our numerical modelling on both elastic and inelastic problems, where in particular plasticity and electroelasticity are considered. Special attention is paid to non-material finite element formulations, which allow for the flow of material particles across the boundaries of the finite elements in time and are particularly useful for axially moving structures.
Implementations and Applications
Being thoroughly tested against analytical solutions as well as other results available in the literature, we use the numerical schemes in both research-oriented as well as industrially relevant applications. As examples, we mention the simulation of smart devices or the simulation of belt drives and metal strips in hot roll milling. The algorithms are implemented in the form of stand-alone in-house software, which is capable of solving electromechanically coupled and contact problems of large deformations of rods, plates and shells.
Vetyukov, Yu. "Finite element modeling of Kirchhoff‐Love shells as smooth material surfaces, opens an external URL in a new window." ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 94, no. 1‐2 (2014): 150-163.