Why Isogeometric Analysis ?

Why Isogeometric Analysis ?

Here, I try to present an intuitive understanding about the need, importance and purpose of Isogeometric Analysis (IGA), which was developed in 2005 as an alternate/extension of classical FEM. I have deliberately kept this introductory post math-free. The formulations behind IGA will be dealt in future posts (if any).

With the advancement of technology, modern-day engineering has become heavily dependent on computers. This is especially in the cases of design and analysis stages, which need to be locked before the manufacturing/ construction/ fabrication stage. Both these stages; design and analysis; are completely dependent on each other and are cyclic in nature. Typical design involves developing the structural geometry through a Computer-Aided Design (CAD) environment. This geometry is then passed to the analysis stage where it is converted into an approximate analysis-suitable geometry, which is then meshed and analyzed using Finite Element Methods (FEM). For redesign after analysis, the design stage is again invoked in CAD then followed by the FEM stage. This cyclic nature of communication between the CAD and FEM spaces was studied to take up most of the time in engineering. With the increase in complexity of the structure, this time period will increase exponentially as well.

Hughes et al. (2009) [1]

Despite the fact that both the design and analysis stages deal with the same objects such as engineering geometries, the mathematical foundation for both differs very much. And this is the reason why there is a huge communication gap between both. In the FEM-based analysis stage, we make use of the Lagrange basis function to approximate the geometry and field variable where nodes discretize the structure. While in CAD, we utilize the NURBS basis function which exactly represents a geometry through knots. So an obvious solution to overcome the bottlenecks in CAD and FEM is to use a common basis function, possibly the one that can exactly represent a geometry, so that both design and analysis can be integrated. This is the concept behind IsoGeometric Analysis.

Approximate (Lagrange) geometry vs Exact (NURBS) geometry [1]

Isogeometric Analysis or IGA was proposed by Thomas Hughes and his team in 2005 as an alternate or advancement over conventional FEM. Ever since, the field has received international attention and has been subjected to some extensive research. The basic idea is to use CAD-standard basis functions to model both geometry and field variables in the analysis stage instead of the Lagrange basis function. The origin of IGA was by using Non-Uniform Rational B-splines (NURBS) basis functions. With the advancement in research, so many other types of geometries and basis functions came into existence such as subdivision surfaces, T-Splines, PHT-Splines etc…

Complex geometries modelled using CAD basis function

Despite integrating CAD and FEM, IGA offers many other advantages.

  • Exact modeling of complicated geometries
  • Simplifies mesh refinement
  • Encases higher-order inter-element continuity
  • Computationally efficient
  • A better solution for higher-order functions

There are some areas where IGA scores hugely over the traditional FEM. For the same structure, we can obtain a solution with less degree of freedom (dof) using IGA with the same level of accuracy. This corresponds to a huge computational saving.

IGA requires less degree-of-freedom and hence less computational cost

Another factor is the accuracy and convergence of solutions obtained using IGA. The graph below compares FEM and IGA in the error produced with dof’s. It can be observed that IGA shows convergence with fewer dof’s.

Accuracy of solutions using IGA [2]

An important area where IGA is relevant is the modeling of shell elements. Plates and shells are generally underestimated in FEA because of the inability to provide C1 continuity between elements which causes the shear locking effect that has baffled researchers for decades. IGA dismisses this issue completely as the NURBS basis function allows for C1 inter-element continuity.

References :

[1] Cottrell, J. A., Hughes, T. J. R. & Bazilevs, Y. (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley.

[2] Xu, Kailai & Darve, Eric. (2020). Isogeometric collocation method for the fractional Laplacian in the 2D bounded domain. Computer Methods in Applied Mechanics and Engineering.

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