### Isogeometric Analysis

Isogeometric Analysis Introduce Accurate Shape and Analysis Models
Advanced analysis of the properties of natural and human made objects and structures most often combines a description of the physics involved by system of partial differential equations, and a geometric description of the objects and structures addressed. From the early days of computers there has been a non-stop emphasis on the evolution of better and more efficient methods for solving advanced mathematical problems on computers. While early uses addressed single steps in advance analysis processes, there has during the last decades been introduced an every increasing interoperability between computer systems utilized at different steps in such processes. As the systems become more and more integrated discrepancies between representations and approaches in different systems become evident, and show up as bottleneck in the total information process. Such discrepancies have often in the time when electronic information exchange between computers systems was not on the agenda, and the representations and approaches of each system was just optimized with respect to the challenge addressed, and the best possible use of the very limited computational power. Computer Aided Design (CAD) and Finite Element Analysis (FEA) are two such classes of systems where interoperability .is severely hampered by the discrepancies of the approaches and representations of CAD and FEA. Isogeometric analysis has the potential to introduce efficient interoperability of CAD and FEA.

### Discrepancies of CAD and FEA representation of shape

FEA as we know it today was already well established in the 1960s, while current CAD-technology was developed after 1970.  While the focus of CAD is to make 3D models that are sufficiently accurate for production purposes, FEA needs compact geometric descriptions where unnecessary detail is removed to focus the analysis on essential properties of the objects.. Consequently CAD-models provide accurate descriptions of the outer and shells of objects, while FEA provides a 3-variate volumetric model with a crude outer shape description.

• CAD: Exact descriptions of the inner and outer shells of the object. The shells are described by a patchwork of elementary surfaces (plan, cylinder, cone, sphere, torus,..)  and (trimmed NURBS-surface. The relation between the surface patches is kept in a data structured denoted B-representation. Adjacent surface will not necessarily match exactly, small gaps within predefined  tolerances are allowed.
• FEA: The object has a complete 3-variate parametric description composed of structures of Finite Elements. Adjacent Finite Elements match exactly along the shared face. However, the Finite Elements are usually of low degree (1 or 2), and will consequently be a crude representation of the shape of inner and outer hulls.

The above discrepancies make information exchange between CAD and FEA  time consuming, complex and expensive.

### Isogeometric representation combines the best of CAD and FEFA

Interoperability of CAD and FEA requires that the shape accuracy of CAD has to be reflected in the Finte Elements, while the water-tight connectivity of FEA has to be reflected in CAD. This will:

• Allow FEA to exactly represent all shapes from CAD
• Allow FEA to use higher level elements
• Introduce water-tight 3-variate volumetric description in CAD

Published October 19, 2010