Mathematical and Computational Modelling of Human Physiology 

Last October marked the tenth anniversary of the appearance of the first paper [1] describing a vision of how to address a major problem in Computer Aided Engineering (CAE). The motivation was as follows: Designs are encapsulated in Computer Aided Design (CAD) systems. Simulation is performed in Finite Element Analysis (FEA) programs. FEA requires the conversions of CAD descriptions to analysis-suitable formats from which finite element meshes can be developed. The conversion process involves many steps, is tedious and labor intensive, and is the major bottleneck in the engineering design-through-analysis process, accounting for more than 80% of overall analysis time, which remains an enormous impediment to the efficiency of the overall engineering product development cycle.

The approach taken in [1] was given the name Isogeometric Analysis. Since its inception it has become a focus of research within both the fields of FEA and CAD and is rapidly becoming a mainstream analysis methodology and a new paradigm for geometric design [2]. The key concept utilized in the technical approach is the development of a new foundation for FEA, based on rich geometric descriptions originating in CAD, resulting in a single geometric model that serves as a basis for both design and analysis.

In this short course I will introduce Isogeometric Analysis, describe some of the basic tools and methods, identify a few areas of current intense activity, and areas where problems remain open, representing opportunities for future research [3].

References

1. T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs, Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement, Computer Methods in Applied Mechanics and Engineering, 194, (1 October 2005), 4135-4195.
2. J. A. Cottrell, T. J. R. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, Chichester, U.K., 2009.
3. Isogeometric Analysis Special Issue (eds. T. J. R. Hughes, J. T. Oden and M. Papadrakakis), Computer Methods in Applied Mechanics and Engineering, 284, (1 February 2015), 1-1182.

Outline

1. Isogeometric Analysis
a. Background and brief history
2. B-splines, NURBS
a. Linear elasticity
i. Approximation theory
b. Spectral approximation
i. Vibrations
ii. Eigenvalue problems
c. Nearly-incompressible solids
d. Nonlinear solids
e. Shells (w/wo rotations)
f. Contact
g. Collocation
h. Reduced quadrature
i. Phase-field methods
j. Fluids and fluid-structure interaction
3. Analysis-suitable IGA Technologies
4. T-splines and Trimmed NURBS
a. Extraordinary points
b. Design-through-analysis
i. Surfaces
ii. Volumes
c. Boundary element methods
d. Hierarchical B-splines and immersed boundary methods
i. Solids
e. Phase-field modeling of crack propagation
i. Brittle fracture
5. Conclusions and Future Prospects