Enhancing Isogeometric Analysis With NURBS-Based Synthesis

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

Isogeometric Analysis (IGA) has emerged as a powerful computational method that seamlessly integrates computer-aided design (CAD) and finite element analysis (FEA). It provides an innovative approach for solving complex engineering problems by employing the same basis functions used in CAD representations, such as Non-Uniform Rational B-Splines (NURBS), to discretize the computational domain. However, one of the challenges in IGA is the occurrence of crosspoints, which are regions where multiple NURBS patches meet. This presents a novel development of a crosspoint modification technique for Isogeometric Analysis, leveraging the capabilities of Artificial Intelligence (AI) algorithms. The aim is to overcome the limitations posed by crosspoints and enhance the accuracy and efficiency of IGA simulations. It utilizes AI-based methods, such as deep learning and pattern recognition, to identify and modify the crosspoint regions in the NURBS-based representation. The AI model is trained on a large dataset of known crosspoint configurations and their corresponding modifications, enabling it to learn the underlying patterns and relationships between the geometry and the required modifications. This trained model is then employed to automatically identify and modify crosspoints in new IGA problems, streamlining the pre-processing stage and reducing human intervention. The crosspoint modification technique incorporates various strategies to ensure geometric continuity and smooth transition across the modified crosspoint regions. These strategies include local refinement of the NURBS control points, adaptively adjusting the degrees of the adjacent patches, and optimizing the blending functions in the crosspoint vicinity. Several numerical experiments and comparative studies are conducted. The results demonstrate that the crosspoint modification using AI significantly improves the accuracy of the IGA simulations, particularly in regions affected by crosspoints. Moreover, the computational efficiency is enhanced as the automated modification reduces the manual effort required for crosspoint handling.
 This research contributes to the advancement of Isogeometric Analysis by introducing a novel AI-based crosspoint modification technique. By effectively addressing the challenges associated with crosspoints, the proposed method enhances the accuracy and efficiency of IGA simulations. The integration of AI algorithms with IGA holds great potential for future applications in various engineering disciplines, including structural analysis, fluid dynamics, and electromagnetics.

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) offers significant improvement in geometric accuracy over finite element analysis (FEA) by utilizing non-uniform rational B-spline (NURBS) basis functions, which allow a precise representation of complex geometries. Within the IGA framework, the robustness and efficiency of contact formulations can be further improved by refining contact surfaces using higher-order NURBS functions. This paper presents a contact surface enrichment strategy based on a varying-order NURBS discretization for three-dimensional (3D) frictional large deformation contact problems. The strategy enhances contact surface resolution, keeping the overall computational cost low by avoiding unnecessary refinement in non-contact regions. It also achieves better accuracy and efficiency in capturing complex contact behavior, making it a promising framework for solving general 3D frictional large deformation contact problems. A detailed algorithm is provided to facilitate user implementation. A comprehensive set of numerical examples is also presented to show the effectiveness of the strategy.

NURBS-enhanced boundary element method based on independent geometry and field approximation for 2D potential problems

We present an algorithm for fast generation of quasi-uniform and variable-spacing nodes on domains whose boundaries are represented as computer-aided design (CAD) models, more specifically non-uniform rational B-splines (NURBS). This new algorithm enables the solution of partial differential equations within the volumes enclosed by these CAD models using (collocation-based) meshless numerical discretizations. Our hierarchical algorithm first generates quasi-uniform node sets directly on the NURBS surfaces representing the domain boundary, then uses the NURBS representation in conjunction with the surface nodes to generate nodes within the volume enclosed by the NURBS surface. We provide evidence for the quality of these node sets by analyzing them in terms of local regularity and separation distances. Finally, we demonstrate that these node sets are well-suited (both in terms of accuracy and numerical stability) for meshless radial basis function generated finite differences discretizations of the Poisson, Navier-Cauchy, and heat equations. Our algorithm constitutes an important step in bridging the field of node generation for meshless discretizations with isogeometric analysis.

High-performance bending and buckling analyses of cylindrical shells resting on elastic foundation using isogeometric scaled boundary finite element method

Isogeometric size optimization of bi-directional functionally graded beams under static loads

Application of isogeometric analysis in structural shape optimization

An isogeometric analysis (IGA) framework is presented to construct and solve dispersion relations for generating the band structure of periodic materials with complicated geometries representing phononic crystals and elastic metamaterials. As the dispersive properties depend on the microstructural geometry, an accurate representation of microstructural geometrical features is paramount. To this end, the ability of isogeometric analysis to exactly model complex curved geometries is exploited, and wave propagation in infinitely periodic solids is combined with isogeometric analysis. The benefits of IGA are demonstrated by comparing the results to those obtained using standard finite element analysis (FEA). It is shown that the IGA solutions can reach the same level of accuracy as FEA while using significantly fewer degrees of freedom. IGA is applied to phononic crystals and elastic metamaterials and the band structure for a variety of unit cells with complex microstructural geometries is investigated to illustrate the desirable dispersive effects in these metamaterials.

The classic finite difference method (FDM) has been successfully adopted in the simulation of dendritic solidification, which is based on phase-field theory. Nevertheless, special strategies of boundary integral and projection are required for applying a supercooling rate to a droplet surface. In the present study, isogeometric analysis (IGA) is employed to discretize the phase-field equation due to the two advantages of Non-Uniform Rational B-Splines (NURBS) basis functions, namely an arbitrary order of derivatives and exact description of complex geometry. In addition, an improved, easy way to apply the supercooling rate on a melt droplet surface is proposed to avoid the integral and projection of the cellular boundary required in FDM. Firstly, dendrite growth in a square computational domain is simulated to verify the performance of IGA. Then, the influences of latent heat, anisotropic mode and initial angle on the dendrite shapes are studied by the presented IGA, FDM and finite element method (FEM). Finally, dendritic solidification in a droplet under different cooling rates along irregular boundaries is performed by the proposed IGA.

Interrogation of spline surfaces with application to isogeometric design and analysis of lattice-skin structures

This work aims at including nonlinear elastic shell models in a multibody framework. We focus our attention to Kirchhoff–Love shells and explore the benefits of an isogeometric approach, the latest development in finite element methods, within a multibody system. Isogeometric analysis extends isoparameteric finite elements to more general functions such as B-splines and NURBS (Non-Uniform Rational B-Splines) and works on exact geometry representations even at the coarsest level of discretizations. Using NURBS as basis functions, high regularity requirements of the shell model, which are difficult to achieve with standard finite elements, are easily fulfilled. A particular advantage is the promise of simplifying the mesh generation step, and mesh refinement is easily performed by eliminating the need for communication with the geometry representation in a CAD (Computer-Aided Design) tool. Target applications are wind turbine blades and twist beam rear suspensions. First numerical examples demonstrate an impressive convergence behavior of the isogeometric approach even for a coarse mesh, while offering substantial savings with respect to the number of degrees of freedom.

A novel isogeometric scaled boundary finite element method for bending and free vibration analyses of laminated plates with rectilinear and curvilinear fibers constrained or free from elastic foundations

The application of isogeometric analysis to the neutron diffusion equation for a pincell problem with an analytic benchmark