Isogeometric analysis of integral equations using subdivision

Analysis in computer aided design: Nonlinear isogeometric B-Rep analysis of shell structures

The isogeometric analysis (IGA), which employs non-uniform rational B-splines (NURBS) basis as shape functions for both representation of geometry and approximation of the field variables, owns several inherent advantages to become an effective numerical method such as an exact geometry description with fewer control points, high-order continuity, and high accuracy. Unlike the C0-continuity shape functions in the conventional finite element method (FEM), the high-order basis functions in IGA are not confined to one element, but span on several elements instead. This property makes the programming task difficult, and more importantly they cannot be straightforwardly embedded into the existing FEM framework. In this paper, we provide an effective numerical scheme by further extending the IGA based on the Bezier extraction of NURBS to study mechanical behavior of elasto-plastic problems in three-dimension (3D). The Bezier extraction operator decomposes NURBS functions into a set of Bernstein polynomials and provides C0-continuity Bezier elements for the IGA, which are similar to Lagrange elements structure. Consequently, the implementation of IGA is now similar to that of conventional FEM, and can easily be embedded in most existing FEM codes. The merits of the proposed formulation are also demonstrated by its convergence and validation studies against reference solutions considering both simple and complicated configurations.

Isogeometric analysis (IGA) is a computational technique that integrates computer-aided design (CAD) with finite element analysis (FEA) by employing the same basis functions for both geometry representation and solution approximation. While IGA offers numerous advantages, such as improved accuracy and efficiency, it also presents several challenges related to geometric modeling. Some of these challenges include accurately representing complex geometries with NURBS (Non-Uniform Rational B-Splines) or other basis functions used in IGA and generating high-quality meshes that conform to the complex geometry represented by NURBS curves/surfaces. This paper introduces an analytical framework to provide a more efficient and theoretically grounded method for generating curvilinear configurations and its analytical solution in IGA, bridging the gap between generated data and its physical representations. This innovative approach is distinguished by integrating the NURBS parameterization in curve generation and providing a corresponding framework to achieve a broader and more accurate explanation of meshes and properties, especially constructing new coordinates and calculating the physical displacements under these conditions. Our model enables the analytical understanding of complex curves from the UIUC airfoil and superformula datasets, demonstrating a deeper dive into simulations. This study signifies a pivotal juncture wherein machine-learning-based complex geometrical formulations are synergistically combined with actual isogeometric analysis.

The isogeometric analysis (IGA) that integrates Computer Aided Design(CAD) and Computer Aided Engineering (CAE) is found so far the effectively numerical tool for the analysis of a variety of practical problems. In this paper, we develop further the NURBS based isogeometric analysis framework for piezoelectric structures.The method employs the NURBS basis functions in both geometry representation and analysis. The main advantages of the present method are capable of handling the exactgeometry of conic sections and making the flexibility of refinement and degree elevationwith an arbitrary continuity of basic functions. These features results in high accuracy of approximate solutions for practical applications, especially piezoelectric problems. Three numerical examples are provided to validate excellent performance of the present method.

In this two-part paper we introduce a new formulation for modeling progressive damage in laminated composite structures. We adopt a multi-layer modeling approach, based on Isogeometric Analysis (IGA), where each ply or lamina is represented by a spline surface, and modeled as a Kirchhoff–Love thin shell. Continuum Damage Mechanics is used to model intralaminar damage, and a new zero-thickness cohesive-interface formulation is introduced to model delamination as well as permitting laminate-level transverse shear compliance. In Part I of this series we focus on the presentation of the modeling framework, validation of the framework using standard Mode I and Mode II delamination tests, and assessment of its suitability for modeling thick laminates. In Part II of this series we focus on the application of the proposed framework to modeling and simulation of damage in composite laminates resulting from impact. The proposed approach has significant accuracy and efficiency advantages over existing methods for modeling impact damage. These stem from the use of IGA-based Kirchhoff–Love shells to represent the individual plies of the composite laminate, while the compliant cohesive interfaces enable transverse shear deformation of the laminate. Kirchhoff–Love shells give a faithful representation of the ply deformation behavior, and, unlike solids or traditional shear-deformable shells, do not suffer from transverse-shear locking in the limit of vanishing thickness. This, in combination with higher-order accurate and smooth representation of the shell midsurface displacement field, allows us to adopt relatively coarse in-plane discretizations without sacrificing solution accuracy. Furthermore, the thin-shell formulation employed does not use rotational degrees of freedom, which gives additional efficiency benefits relative to more standard shell formulations.

We propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, includingh-refinement andp-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to theL2norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in theL2norm. Numerical tests are given to validate the theory and gauge the good performance of our method.

Isogeometric analysis (IGA) is a mesh free technique which make use of B-spline basis functions for geometry and field variable representation. Parameterization of B-spline for IGA is the counterpart of meshing as in finite element method (FEM). The objective of parameterization is to find the optimum set of control points for B-spline modelling. The position of control points of a B-spline model has huge effect on IGA results. In this work, the effect of B-spline parameterization on IGA result is presented. One dimensional case of bar with self-weight is solved and compared with exact analytical solution. First fundamental matrix is used as evaluation metric to check the quality of parameterization for 2-D domains. A heat conduction problem of a square domain is presented to study the parameterization effect for 2-D case.

Object oriented implementation of the T-spline based isogeometric analysis

THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis

Dynamic analysis of two-dimensional elasticity problems using the Isogeometric analysis method and its comparison with the finite element method is the subject of this research. For this purpose, formulation of the governing differential equation is obtained by using the B-spline basis functions. Some numerical examples employing consistent and lumped mass matrices are presented in order to compare the results of Isogeometric analysis and finite elements. The effect of increasing the number of control points and the degree of B-spline basis functions on the accuracy of the solution are also investigated. Furthermore, since the Newmark method is employed for transient dynamic analysis, the effects of parameters α and β on the solution accuracy are investigated. The obtained results demonstrate higher accuracy of the isogeometric analysis method in comparison with the FEM, despite employing fewer numbers of degrees of freedom and consequently reduction in the dimensions of system of equations and computational effort.

Multivariate basis splines (B‐splines) and Non‐uniform rational B‐splines (NURBS) lack adaptivity due to their tensor product structure. Truncated hierarchical B‐splines (THB‐splines) provide a solution for this. THB‐splines organize the parameter space into a hierarchical structure, which enables efficient approximation and representation of functions with different levels of detail. The truncation mechanism ensures the partition of unity property of B‐splines and defines a more scattered set of basis functions without overlapping on the multi‐level spline space. Transferring these multi‐level splines into Bézier elements representation facilitates straightforward incorporation into existing finite element (FE) codes. By separating the multi‐level extraction of the THB‐splines from the standard Bézier extraction, a more general independent framework applicable to any sequence of nested spaces is created. The operators for the multi‐level structure of THB‐splines and the operators of Bézier extraction are constructed in a local approach. Adjusting the operators for the multi‐level structure from an element point of view and multiplying with the Bézier extraction operators of those elements, a direct map between Bézier elements and a hierarchical structure is obtained. The presented implementation involves the use of an open‐source Octave/MATLAB isogeometric analysis (IGA) code called GeoPDEs. A basic Poisson problem is presented to investigate the performance of multi‐level Bézier extraction compared to a standard THB‐spline approach.

Multi-material problems often exhibit complex geometries along with physical responses presenting large spatial gradients or discontinuities. In these cases, providing high-quality body-fitted finite element analysis meshes and obtaining accurate solutions remain challenging. Immersed boundary techniques provide elegant solutions for such problems. Enrichment methods alleviate the need for generating conforming analysis grids by capturing discontinuities within mesh elements. Additionally, increased accuracy of physical responses and geometry description can be achieved with higher-order approximation bases. In particular, using B-splines has become popular with the development of IsoGeometric Analysis. In this work, an eXtended IsoGeometric Analysis (XIGA) approach is proposed for multi-material problems. The computational domain geometry is described implicitly by level set functions. A novel generalized Heaviside enrichment strategy is employed to accommodate an arbitrary number of materials without artificially stiffening the physical response. Higher-order B-spline functions are used for both geometry representation and analysis. Boundary and interface conditions are enforced weakly via Nitsche’s method, and a new face-oriented ghost stabilization methodology is used to mitigate numerical instabilities arising from small material integration subdomains. Two- and three-dimensional heat transfer and elasticity problems are solved to validate the approach. Numerical studies provide insight into the ability to handle multiple materials considering sharp-edged and curved interfaces, as well as the impact of higher-order bases and stabilization on the solution accuracy and conditioning.

Precise simulation of realistic physical phenomena and maintaining high efficiency will be the ultimate goal of computational mechanics. As an effort of such endeavor, an isogeometric analysis (IGA) was proposed by integrating finite element analysis (FEA) and computer-aided design. IGA efficiently predicts a physical behavior with higher fidelity than the original FEA. However, an application to various geometries will be cumbersome owing to the absence of an appropriate pre-processor. To alleviate such limitation, various reinforcements have been attempted, including a finite element (FE)-based IGA. Derived from those initiatives, alternatives will be suggested herein by using a Bernstein–Bézier FE. To obtain approximate C1 continuity for a general tetrahedral FE, an approximated Worsey–Piper element split will be presented. Also, Bézier mesh generation will be adopted for an enhanced geometric representation. The present attempt will be validated by comparing the results for various curved geometry. Furthermore, the present method will be applied to a much more complicated configuration to demonstrate the geometric applicability.

Before isogeometric analysis can be applied to solving a partial differential equation posed over some physical domain, one needs to construct a valid parametrization of the geometry. The accuracy of the analysis is affected by the quality of the parametrization. The challenge of computing and maintaining a valid geometry parametrization is particularly relevant in applications of isogemetric analysis to shape optimization, where the geometry varies from one optimization iteration to another. We propose a general framework for handling the geometry parametrization in isogeometric analysis and shape optimization. It utilizes an expensive non-linear method for constructing/updating a high quality reference parametrization, and an inexpensive linear method for maintaining the parametrization in the vicinity of the reference one. We describe several linear and non-linear parametrization methods, which are suitable for our framework. The non-linear methods we consider are based on solving a constrained optimization problem numerically, and are divided into two classes, geometry-oriented methods and analysis-oriented methods. Their performance is illustrated through a few numerical examples.

formula omitted] regularity properties of singular parameterizations in isogeometric analysis