Computational Mechanics: Geometry and Numerical Simulation

In the Focus Group Computational Mechanics, Hans Fischer Fellow Prof. Alessandro Reali (Mechanics of Solids and Structures, Università di Pavia) collaborates with his host Prof. Ernst Rank (Computation in Engineering, TUM).

Over the last decade, the statement “geometry is the foundation of analysis” has been included among the basic principles of modern Computational Mechanics by an increasing number of researchers. However, from the practical point of view, geometry had so far limited impact on the field, the main reason being that Finite Element Analysis (FEA; i.e., the main engineering numerical analysis tool) was developed in the ‘50/’60's, well before the advent and widespread use of Computer Aided Geometric Design (CAGD; i.e., the main geometric design tool) which occurred only in the ‘70/’80's, and the connection between the two worlds relies on interfaces often far from efficient. As a result, building analysis-suitable geometries is estimated to take up to 80% of the overall analysis time for complex CAGD-based engineering designs. Moreover, since low order FEA methods are typically used, most meshes are made of simple geometrical objects like tetrahedra or hexahedra, which may not be able to represent highly sophisticated geometries with sufficient accuracy. Similar problems are present when the objects to be analyzed are obtained via imaging tools (in particular, in the medical field), and are further amplified when dealing with geometries evolving over time and involving the creation of new material, like in the case of Additive Manufacturing.

The above-mentioned low accuracy of classical FEA typically translates in very expensive simulations (and in some cases even to modeling errors and misleading results), and such a gap definitely has to be dealt with.

Isogeometric Analysis (IGA) was introduced in 2005 specifically aiming at bridging the gap between analysis and geometric design. The idea is to construct isoparametric methods based on splines (which are the basic ingredient of CAGD geometries) in order to make the construction of analysis-suitable geometries much simpler and more efficient. In addition, the higher continuity properties of splines often lead to increased efficiency in terms of approximation power and open the door to the development of new formulations based on higher order partial differential equations. The results have been so far much more than promising and IGA is now regarded as one of the most powerful Computational Mechanics tools, able to attract also industrial interests.

Another relevant and promising simulation framework is the recently developed Finite Cell Method (FCM), which allows to deal with very complex and/or evolving geometries in an incredibly simple and effective way (via the “immersed” concept). FCM can be easily combined with the IGA idea (and with spline functions), giving rise to very powerful and geometrically flexible computational tools.

Given these premises and also considering the fact that the Focus Group Members include some of the pioneers of the above mentioned methodologies, it seems natural to consider the combination of IGA and FCM as the way to go for creating efficient analysis tools for Additive Manufacturing problems, which constitute one of the most interesting modern challenges of Computational Mechanics.