The finite element method (FEM) has very broad applications in a lot of research areas, and isogeometric analysis (IGA) is a new advancement based on FEM to integrate design with analysis. This chapter reviews the basic algorithm of finite element analysis (FEA) and its new developments, including IGA, extended FEM and immersed FEM. As a popular and powerful numerical method to solve partial differential equations over complex domains, FEM has been developed rapidly and used in many research areas including computational medicine, biology and engineering. The FEM is a general technique to solve boundary value problems with uniformly and non-uniformly spaced grids or meshes. In the implementation, the element stiffness matrix and element load vector are computed element by element, and then assembled together into the global stiffness matrix and global load vector. FEA has been applied …

We explore T-splines, a generalization of NURBS enabling local refinement, as a basis for isogeometric analysis. We review T-splines as a surface design methodology and then develop it for engineering analysis applications. We test T-splines on some elementary two-dimensional and three-dimensional fluid and structural analysis problems and attain good results in all cases. We summarize the current status of T-splines, their limitations, and future possibilities.

The authors are the originators of isogeometric analysis, are excellent scientists and good educators. It is very original. There is no other book on this topic. Ren de Borst, Eindhoven University of Technology Written by leading experts in the field and featuring fully integrated colour throughout, Isogeometric Analysis provides a groundbreaking solution for the integration of CAD and FEA technologies. Tom Hughes and his researchers, Austin Cottrell and Yuri Bazilevs, present their pioneering isogeometric approach, which aims to integrate the two techniques of CAD and FEA using precise NURBS geometry in the FEA application. This technology offers the potential to revolutionise automobile, ship and airplane design and analysis by allowing models to be designed, tested and adjusted in one integrative stage. Providing a systematic approach to the topic, the authors begin with a tutorial introducing the foundations of Isogeometric Analysis, before advancing to a comprehensive coverage of the most recent developments in the technique. The authors offer a clear explanation as to how to add isogeometric capabilities to existing finite element computer programs, demonstrating how to implement and use the technology. Detailed programming examples and datasets are included to impart a thorough knowledge and understanding of the material. Provides examples of different applications, showing the reader how to implement isogeometric models Addresses readers on both sides of the CAD/FEA divide Describes Non-Uniform Rational B-Splines (NURBS) basis functions

This chapter contains sections titled: The isoparametric concept Boundary value problems (BVPs) Numerical methods Boundary conditions Multiple patches revisited Comparing isogeometric analysis with classical finite element analysis Appendix 3.A: Shape function routine Appendix 3.B: Error estimates Notes

This chapter contains sections titled: Elastodynamics Semi-discrete methods Space–time finite elements

This chapter contains sections titled: Introduction The evolution of FEA basis functions The evolution of CAD representations Things you need to get used to in order to understand NURBS-based isogeometric analysis Notes

This chapter contains sections titled: The polar form of polynomials The polar form of B-splines Note

This chapter contains sections titled: Longitudinal vibrations of an elastic rod Rotation-free analysis of the transverse vibrations of a Bernoulli–Euler beam Transverse vibrations of an elastic membrane Rotation-free analysis of the transverse vibrations of a Poisson–Kirchhoff plate Vibrations of a clamped thin circular plate using three-dimensional solid elements The NASA aluminum testbed cylinder Wave propagation Appendix 5.A: Kolmogorov n-widths Notes

This chapter discusses the major challenges faced in engineering design and analysis. Going from an initial geometric design to a finite element mesh constitutes the most time-consuming step of the overall design through analysis process. The concept of isogeometric analysis has thus far addressed this issue by utilizing the NURBS functions commonly found in CAD packages as an isoparametric basis for both the geometry and the solution space. We extend the methodology to include T-splines, a technology rapidly gaining favor in the geometry community due to its flexibility and efficiency in representing complex objects. This efficiency stems from its locally refinable basis, a feature ideally suited to finite element analysis where local refinement is frequently desirable. Examples are presented.

We present an LES-type variational multiscale theory of turbulence. Our approach derives completely from the incompressible Navier–Stokes equations and does not employ any ad hoc devices, such as eddy viscosities. We tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows. In the calculations, we employed linear, quadratic and cubic NURBS. A dispersion analysis of simple model problems revealed NURBS elements to be superior to classical finite elements in approximating advective and diffusive processes, which play a significant role in turbulence computations. The numerical results are very good and confirm the viability of the theoretical framework.