We develop finite element data structures for T-splines based on Bezier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bezier elements are defined in terms of a fixed set of polynomial basis functions, the so-called Bernstein basis. The Bezier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T-junctions, referred to as ‘hanging nodes’ in finite element analysis and a fundamental feature of T-splines. A detailed example is presented to illustrate the ideas. Copyright © 2011 John Wiley & Sons, Ltd.

Isogeometric finite element data structures based on Bézier extraction of T-splines

Local refinement of analysis-suitable T-splines

https://orbilu.uni.lu/bitstream/10993/11850/1/2013_IGABEM_Tsplines3D_CMAME_Scott.pdf

Isogeometric boundary element analysis using unstructured T-splines

The finite cell method is an embedded domain method, which combines the fictitious domain approach with higher-order finite elements, adaptive integration, and weak enforcement of unfitted essential boundary conditions. Its core idea is to use a simple unfitted structured mesh of higher-order basis functions for the approximation of the solution fields, while the geometry is captured by means of adaptive quadrature points. This eliminates the need for boundary conforming meshes that require time-consuming and error-prone mesh generation procedures, and opens the door for a seamless integration of very complex geometric models into finite element analysis. At the same time, the finite cell method achieves full accuracy, i.e. optimal rates of convergence, when the mesh is refined, and exponential rates of convergence, when the polynomial degree is increased. Due to the flexibility of the quadrature based geometry approximation, the finite cell method can operate with almost any geometric model, ranging from boundary representations in computer aided geometric design to voxel representations obtained from medical imaging technologies. In this review article, we first provide a concise introduction to the basics of the finite cell method. We then summarize recent developments of the technology, with particular emphasis on the research topics in which we have been actively involved. These include the finite cell method with B-spline and NURBS basis functions, the treatment of geometric nonlinearities for large deformation analysis, the weak enforcement of boundary and coupling conditions, and local refinement schemes. We illustrate the capabilities and advantages of the finite cell method with several challenging examples, e.g. the image-based analysis of foam-like structures, the patient-specific analysis of a human femur bone, the analysis of volumetric structures based on CAD boundary representations, and the isogeometric treatment of trimmed NURBS surfaces. We conclude our review by briefly discussing some key aspects for the efficient implementation of the finite cell method.

The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models

Contact treatment in isogeometric analysis with NURBS

Isogeometric sizing and shape optimisation of beam structures

On the numerical integration of trimmed isogeometric elements

Isogeometric design of anisotropic shells: Optimal form and material distribution

A layerwise theory for laminated composites in the framework of isogeometric analysis

One of the major goals of isogeometric analysis is direct design-to-analysis, i.e., using computer-aided design (CAD) files for analysis without the need for mesh generation. One of the primary obstacles to achieving this goal is CAD models are based on surfaces, and not volumes. The boundary element method (BEM) circumvents this difficulty by directly working with the surfaces. The standard basis functions in CAD are trimmed nonuniform rational B-spline (NURBS). NURBS patches are the tensor product of one-dimensional NURBS, making the construction of arbitrary surfaces difficult. Trimmed NURBS use curves to trim away regions of the patch to obtain the desired shape. By coupling trimmed NURBS with a nonsingular BEM, the formulation proposed here comes close achieving the goal of direct design to analysis. Example calculations demonstrate its efficiency and accuracy.

Attila P. Nagy

Papers

Isogeometric sizing and shape optimisation of beam structures

On the numerical integration of trimmed isogeometric elements

Isogeometric design of anisotropic shells: Optimal form and material distribution

A layerwise theory for laminated composites in the framework of isogeometric analysis

A multi-patch nonsingular isogeometric boundary element method using trimmed elements