Kirchhoff–Love shell formulation based on triangular isogeometric analysis

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.

A hierarchical approach to the a posteriori error estimation of isogeometric Kirchhoff plates and Kirchhoff–Love shells

Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach

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) 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.

An efficient model reduction method for buckling analyses of thin shells based on IGA

We consider the wavelet-based estimators of mean regression function with long memory moving average errors and investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, in the presence of long memory non-Gaussian moving average noise, wavelet estimators still achieve nearly optimal convergence rates and provide explicitly the extraordinary local adaptability. The theory is illustrated with some numerical examples.

Isogeometric analysis (IGA) connects CAE with standard CAD design tools and has been utilized for a decade in structural optimization. FEA has a number of drawbacks, including a lack of linkage between analytic and geometric models, a shortage of continuity between components, and a lack of accuracy for high-order elements. Isogeometric analysis effectively overcomes these obstacles. This paper proposes the isogeometric topological optimization mathematical model depending on the asymmetric constitutive relationship of tension and compression microscopic materials. We provide a new density-based topology optimization formulation in which the design domain is limited to the B-spline space. A rectangular domain with a tensor-product is embedded in a design domain that is arbitrarily shaped. The density field is represented using B-splines. The intrinsic filter provided by the B-spline representation for topology optimization is governed by the degrees of B-spline and the knot spans. This B-spline effectively removes numerical artifacts and controls the minimum feature-length in; when the B-spline basis functions span many analysis elements. Benchmark problems in structural topology optimization are computed through Evolutionary Structural Optimization (ESO) algorithm. Comparisons of the optimal structure topologies using finite element analysis and isogeometric collocation methods are analyzed. The influence of different tension-compression stiffness ratios on the final structure topologies is listed and analyzed. The numerical results show the reliability and performance of the proposed mathematical formulation and asymmetry of natural structure.

Local refinement is crucial in all computer methods for flow problems, e.g., to resolve boundary layers. Locally Refinable (LR) B-splines are a recent approach to local refinement within isogeometric analysis based on piecewise polynomials. In this work we present numerical investigations of the use of LR B-splines in isogeometric analysis of flow problems. The problems considered are based on the steady-state, incompressible Navier-Stokes equations in two dimensions. We give a brief introduction to LR B-splines and to isogeometric analysis in fluid mechanics in general. We propose two families of LR B-spline discretizations of the pressure and velocity fields for solving the mixed formulation of the governing equations using isogeometric analysis. And through representative examples, we investigate the stability of the discretizations, we study their ability to reproduce an analytical solution, and we examine their performance on a standard benchmark problem.

In this paper, we apply semi-Lagrangian discontinuous Galerkin (SLDG) methods for linear hyperbolic equations in one space dimension and analyze the error between the numerical and exact solutions under the L 2 L^2 -norm. In all the previous works, the theoretical analysis of the SLDG method would suggest a suboptimal convergence rate due to the error accumulation over time steps. However, numerical experiments demonstrate an optimal convergence rate and, if the terminal time is large, a superconvergence rate. In this paper, we will prove optimal convergence and optimal superconvergence rates. There are three main difficulties: 1. The error analysis on overlapping meshes. Due to the nature of the semi-Lagrangian time discretization, we need to introduce the background Eulerian mesh and the shifted mesh. The two meshes are staggered, and it is not easy to construct local projections and to handle the error accumulation during time evolution. 2. The superconvergence of time-dependent terms under the L 2 L^2 -norm. The error of the numerical and exact solutions can be divided into two parts, the projection error and the time-dependent superconvergence term. The projection strongly depends on the superconvergence rates. Therefore, we need to construct a sequence of projections and gradually improve the superconvergence rates. 3. The stopping criterion of the sequence of projections. The sequence of projections are basically of the same form. We need to show that the projections exist up to some certain order since the superconvergence rate cannot be infinity. Hence, we will seek some “hidden” condition for the existence of the projections. In this paper, we will solve all the three difficulties and construct several local projections to prove the optimal convergence and superconvergence rates. Numerical experiments verify the theoretical findings.

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

Based on our proposed adaptivity strategy for the vibration of Reissner–Mindlin plate, we develop it to apply for the vibration of Kirchhoff plate. The adaptive algorithm is based on the Geometry-Independent Field approximaTion (GIFT), generalized from Iso-Geometric Analysis (IGA), and it can characterize the geometry of the structure with NURBS (Non-Uniform Rational B-Splines), and independently apply PHT-splines (Polynomial splines over Hierarchical T-meshes) to achieve local refinement in the solution field. The MAC (Modal Assurance Criterion) is improved to locate unique, as well as multiple, modal correspondence between different meshes, in order to deal with error estimation. Local adaptivity is carried out by sweeping modes from low to high frequency. Numerical examples show that a proper choice of the spline space in solution field (with GIFT) can deliver better accuracy than using NURBS solution field. In addition, for vibration of heterogeneous Kirchhoff plates, our proposed method indicates that the adaptive local h-refinement achieves a better solution accuracy than the uniform h-refinement.

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