Isogeometric analysis: an overview and computer implementation aspects (1205.2129v2)

Abstract: Isogeometric analysis (IGA) represents a recently developed technology in computational mechanics that offers the possibility of integrating methods for analysis and Computer Aided Design (CAD) into a single, unified process. The implications to practical engineering design scenarios are profound, since the time taken from design to analysis is greatly reduced, leading to dramatic gains in efficiency. The tight coupling of CAD and analysis within IGA requires knowledge from both fields and it is one of the goals of the present paper to outline much of the commonly used notation. In this manuscript, through a clear and simple Matlab implementation, we present an introduction to IGA applied to the Finite Element (FE) method and related computer implementation aspects. Furthermore, implemen- tation of the extended IGA which incorporates enrichment functions through the partition of unity method (PUM) is also presented, where several examples for both two-dimensional and three-dimensional fracture are illustrated. The open source Matlab code which accompanies the present paper can be applied to one, two and three-dimensional problems for linear elasticity, linear elastic fracture mechanics, structural mechanics (beams/plates/shells including large displacements and rotations) and Poisson problems with or without enrichment. The Bezier extraction concept that allows FE analysis to be performed efficiently on T-spline geometries is also incorporated. The article includes a summary of recent trends and developments within the field of IGA.

• The paper introduces isogeometric analysis by demonstrating the use of NURBS to precisely represent geometry in computational mechanics.
• It details computational techniques such as k-refinement and Bézier extraction that improve the efficiency and accuracy of the analysis.
• Applications in fracture mechanics and structural analysis are validated through benchmarks like edge-cracked plates and pinched cylinders.

An Overview of Isogeometric Analysis and Its Implementation Aspects

The paper "Isogeometric Analysis: An Overview and Computer Implementation Aspects" provides a comprehensive examination of Isogeometric Analysis (IGA), focusing on its application to the integration of computational mechanics and Computer-Aided Design (CAD). The authors, Vinh Phu Nguyen, Stéphane P.A. Bordas, and Timon Rabczuk, illuminate the potential of IGA to transform engineering design by integrating design and analysis processes, thus enhancing efficiency and accuracy. This essay summarizes key concepts, findings, and implications from the paper, emphasizing implementation details and theoretical advancements.

Core Concepts

Isogeometric Analysis is a numerical approach that proposes the use of Non-Uniform Rational B-Splines (NURBS)—the basis of many CAD software—as functions for both geometric representation and computational analysis. This dual application ensures that the geometry is exactly represented at all computational stages, a departure from traditional finite element methods (FEM) where geometry is often approximated.

Computational Aspects and Implementation

The paper meticulously outlines the computational workflow of IGA, emphasizing its basis on NURBS. The implementation is facilitated using Matlab\textsuperscript{\textregistered{} with various routines developed for one, two, and three-dimensional problems. The isogeometric formulation is leveraged for elasticity problems, with specific attention to boundary condition application, numerical integration, and the assembly of system equations.

Several computational methods are detailed:

• k-refinement: A method unique to IGA that involves a combination of p-refinement (increasing polynomial order) and h-refinement (subdivision of elements) for more accurate solutions.
• Bézier Extraction: Simplifies the integration of NURBS into existing FEM codes by expressing NURBS basis functions in terms of Bernstein polynomials.

Applications in Fracture Mechanics and Structural Analysis

The paper reviews the application of IGA in fracture mechanics, particularly in modeling discontinuities like cracks using extended IGA (XIGA). XIGA enhances IGA with enrichment functions, allowing accurate simulation of crack propagation without mesh regeneration. Implementation aspects of XIGA in both 2D and 3D contexts are provided.

For structural mechanics, the paper illustrates the efficiency of IGA in modeling thin structures like plates and shells. The intrinsic smoothness of NURBS offers a distinct advantage in achieving rotation-free plate and shell formulations.

Benchmarks and Examples

The validation of the described methodologies is supported by benchmarks, including problems like the pinched cylinder and edge-cracked plates. These examples illustrate the applicability of IGA in analyzing complex engineering problems effectively.

Future Directions and Implications

The potential of IGA to streamline CAD and FEA into a singular robust framework suggests significant implications for industrial applications, facilitating iterative design processes and reducing errors associated with geometry approximation. The authors note remaining challenges, particularly in developing error estimators, adaptive refinement strategies, and efficient integration schemes.

Conclusion

This paper underscores the transformative capabilities of isogeometric analysis in computational mechanics, offering detailed implementation strategies and demonstrating the significance of tightly integrating design with analysis. As the field evolves, IGA is poised to become a critical tool in engineering analysis, driving innovations in design and analysis integration. The detailed Matlab\textsuperscript{\textregistered{} source code and discussions present a valuable resource for researchers aiming to implement or extend IGA-related computational strategies.