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This paper presents, for the first time, an effective numerical approach based on the isogeometric analysis (IGA) and the six-variable quasi-three dimensional (3D) higher-order shear deformation theory (HSDT) to study the free vibration characteristics of functionally-graded (FG) graphene origami (GOri)-enabled auxetic metamaterial (GOEAM) plates submerged in a fluid medium. The plate theory incorporates the thickness stretching and the effects of transverse shear deformation without using any shear correction factors. The velocity potential function and Bernoulli's equation are used to derive the hydrodynamic pressure acting on the plate surface. Both horizontally and vertically immersed plate configurations are considered here in the form of inertia effects. The plates are composed of multilayer GOEAMs, with the GOri content varying through the plate's thickness in a layer-wise manner. This design results in graded auxetic growth. The material properties are evaluated by mixing rules and a genetic programming (GP)-assisted micromechanical model. The governing equations of motion for the FG-GOEAM plates immersed in a fluid medium are derived by Hamilton's principle. After validating the convergence and accuracy of the present model, a comprehensive parametric study is carried out to examine the effects of the GOri content, GOri distribution pattern, GOri folding degree, fluid level, immersed depth, and geometric parameter on the natural frequencies of the FG-GOEAM plates. The results show that the natural frequencies for the four GOri distribution patterns increase with the increase in the layer number when the lay number is fewer than 10, and then stabilize after the layer number reaches 10. Besides, in general, the natural frequency of the FG-GOEAM plate in a vacuum or fluid increases when the GOri content increases, while decreases when the GOri folding degree increases. Some additional findings related to the numerical results are presented in the conclusions. It is believed that the present results are useful for the precise design and optimization of FG-GOEAM plates immersed in a fluid medium. [ABSTRACT FROM AUTHOR]

Isogeometric analysis (IGA) has been widely used as a spatial discretization method for phase field models since the seminal work of Gómez et al. (Comput. Methods Appl. Mech. Engrg. 197(49), pp. 4333–4352, 2008), and the first numerical convergence study of IGA for the Cahn-Hilliard equation was presented by Kästner et al. (J. Comput. Phys. 305(15), pp. 360–371, 2016). However, to the best of our knowledge, the theoretical convergence analysis of IGA for the Cahn-Hilliard equation is still missing in the literature. In this paper, we provide the convergence analysis of IGA for the multi-dimensional Cahn-Hilliard equation for the first time. The two important steps to carry out the convergence analysis are (1) we rigorously prove that the L ∞ norm of IGA solution is uniformly bounded for all mesh sizes, and (2) we construct an appropriate Ritz projection operator for the bi-Laplacian term in the Cahn-Hilliard equation. The first- and second-order stabilized semi-implicit schemes are used to obtain the fully discrete schemes. The energy stability analyses are rigorously proved for the resulting fully discrete schemes. Finally, several two- and three-dimensional numerical examples are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This article investigates the nonlinear analysis of hyperelastic planar curved beams based on the total Lagrangian Timoshenko theory. The fully incompressible neo-Hookean model has been selected as the constitutive relation to accurately capture large deformations. The nonlinear equations and boundary conditions derived from the virtual work principle are solved using the Newton–Raphson method. Employing isogeometric analysis with NURBS functions to discretize the unknown kinematics, including displacements and rotation, and the linearized governing equations. The study accounts for conservative and non-conservative forces, validating results against available experimental and numerical results reported in the literature, confirming the proposed formulation's effectiveness and precision. [ABSTRACT FROM AUTHOR]

AbstractThe free vibration analysis of rotating pre-twisted composite beams with arbitrary cross-section geometry is investigated using an isogeometric-based dimensional reduction method. The three-dimensional beam problem is decomposed into a two-dimensional cross-sectional analysis and a one-dimensional beam problem, significantly reducing the numerical cost compared to three-dimensional finite element method. The simultaneous influences of pre-twist and rotational speed are carefully studied for various composite beams. This approach benefits from the numerical advantages of isogeometric analysis, with results showing better agreement with three-dimensional finite element method than other studies in the literature. [ABSTRACT FROM AUTHOR]

Background: The Isogeometric Method (IGM) has been widely explored in the structural analysis of plates with arbitrary and complex shapes, particularly in the context of free vibration analysis. However, a literature survey highlights the scarcity of research on the application of IGM for forced vibration analysis of such structures. This paper aims to address this research gap by conducting a comprehensive forced vibration analysis on plates with complex and arbitrary shapes using the Isogeometric Method. Method: The approach employs Non-Uniform Rational B-Spline (NURBS) interpolation functions and the first-order shear deformation theory of plates. The governing equations are derived through the minimization of potential energy, incorporating Rayleigh damping in the analysis. Representative cases, including an L-shaped plate and a square plate with a heart-shaped cutout, are analyzed. Arbitrary-shaped plates are discretized into patches connected using the bending strip method, obtaining the global equation of motion in matrix form. Transient analysis is performed using the Newmark time-stepping approach, with time history responses extracted. Signal processing tools, such as Fast Fourier Transform (FFT) and Wavelet Transform (WT), are employed to extract natural frequencies from the time history responses. Validation is performed using the finite element method (FEM), considering various loading scenarios, including impulse, harmonic, saw shape, and chirp excitations, as well as resonance and beating cases. Results and Conclusions: Comparative analysis between IGM and FEM reveals that for complicated-shaped plates, IGM provides accurate results in transient analysis with lower degrees of freedom and significantly shorter computation time compared to FEM. Additionally, the frequency spectrum generated by both methods using FFT and WT is found to be fully matched. The results presented in this research contribute significantly to expanding the applicability of IGM in forced vibration analysis. [ABSTRACT FROM AUTHOR]

Background: Micro structures are gaining prominence in micro-electromechanical systems (MEMS) and nano-electromechanical systems (NEMS). Nonetheless, numerous studies have revealed discrepancies between experimental observations of microstructure mechanical properties and simulation out comes derived from classical continuum theory. Recently, the bi-directional functionally graded microbeams have attracted wide attention from scholars as a new form of material. Purpose: A free vibration of bi-directional functionally graded porous microbeams with geometric imperfections is presented considering the influence of FG power index (nx, nz), boundary condition, geometric imperfection mode, nonlocal parameter and characteristic length. Method: In the present work, an isogeometric analysis formulation in the framework of nonlocal strain gradient theory is developed to investigate the free vibration characteristics of bi-directional FG porous micro-beams with geometrical imperfections. Porosity, FG power index, boundary condition, and size parameters can all be taken into account to determine the vibration behavior of a bi-directional FG porous microbeam with various geometric imperfections. Results: The non-dimensional frequencies of bi-directional functionally graded microbeam are studied based on the variations in crucial parameters such as porosity, FG power index (nx, nz), boundary condition, geometric imperfection mode, nonlocal parameter, and characteristic length. Conclusions: The study finds that strain gradient parameter has a greater impact on vibration than non-local parameter, and the influence of porosity distribution on vibration is more pronounced for even porosity. Geometric imperfections significantly affect vibration, with sine and G1 modes being most sensitive to imperfection amplitude. Applications: These findings demonstrate the importance of considering multiple factors for analyzing the vibration characteristics of bi-directional Functionally graded microbeams with geometrical imperfections in practical engineering applications. [ABSTRACT FROM AUTHOR]

In the moving-morphable-component topology optimization, morphable components are introduced as a geometrical model mapped onto the background finite elements, and their shape parameters are utilized as design variables for topology optimization. Whereas a complex curved geometry ensuring C¹ continuity can be generated using existing curved components, the component curvatures cannot be selected as design variables in the existing methods; thus geometric constraints associated with curvatures cannot also be directly imposed. To address this issue, this study proposes a curvature-based morphable component by introducing the curvilinear geometry representation in the strain-based beam formulation. Since the proposed component is parameterized by curvatures using the curvilinear equation, the component curvatures can be utilized as the design variables. This allows for directly imposing curvature constraints on structural members, thereby accounting for the manufacturability of an optimal topology. It is demonstrated that a symmetric placement of the design variables using the midpoint curvilinear coordinate system is critical in ensuring convergence of the proposed curvature-based component optimization. The symmetric curvature component is further extended to account for multiple curvatures within a single component while ensuring C¹ continuity. Several examples are presented to demonstrate the benefits of the proposed multicurvature component for topology optimization. [ABSTRACT FROM AUTHOR]

The Laplace eigenvalue problem on circular sectors has eigenfunctions with corner singularities. Standard methods may produce suboptimal approximation results. To address this issue, a novel numerical algorithm that enhances standard isogeometric analysis is proposed in this paper by using a single-patch graded mesh refinement scheme. Numerical tests demonstrate optimal convergence rates for both the eigenvalues and eigenfunctions. Furthermore, the results show that smooth splines possess a superior approximation constant compared to their C 0 -continuous counterparts for the lower part of the Laplace spectrum. This is an extension of previous findings about excellent spectral approximation properties of smooth splines on rectangular domains to circular sectors. In addition, graded meshes prove to be particularly advantageous for an accurate approximation of a limited number of eigenvalues. Finally, a hierarchical mesh structure is presented to avoid anisotropic elements in the physical domain and to omit redundant degrees of freedom in the vicinity of the singularity. Numerical results validate the effectiveness of hierarchical mesh grading for simulating eigenfunctions of low and high regularity. [ABSTRACT FROM AUTHOR]

Flow diverter implantation has emerged as a highly effective treatment for cerebral aneurysms. However, the variability in patient‐specific vascular anatomy necessitates detailed pre‐operative planning to mitigate potential complications arising from standard, one‐size‐fits‐all devices. To address these challenges, robust predictive simulation tools are essential for optimizing flow diverter design and implantation strategies tailored to the unique characteristics of each patient's anatomy. This study focuses on developing an advanced numerical simulation tool for modeling the mechanical behavior of braided flow diverters during crimping and navigation through patient‐specific vasculature. Using isogeometric analysis (IGA) with NURBS‐based representations in LS‐Dyna, the intricate deformations of the flow diverter wires during catheter crimping are captured with high accuracy. The patient‐specific vessel geometries, derived from imaging data, are integrated into the model to account for variations in vascular structure, ensuring precise alignment and controlled navigation of the device. The navigation process relies on optimizing the central axis of the blood vessel to minimize torsional stress on the flow diverter, reducing the risk of device malfunction or failure during deployment. By incorporating patient‐specific information, such as vessel curvature and tortuosity, the simulation tool enables the prediction of potential issues, thus allowing for intervention planning that is tailored to individual anatomy. This patient‐specific approach enhances the safety and efficacy of flow diverter implantation and serves as a foundation for improved device design prior to prototype development. [ABSTRACT FROM AUTHOR]

The mixed displacement (MD) method was initially developed to mitigate geometrical locking effects in beams, plates, and shells with the intention of having intrinsically locking‐free characteristics while using equal‐order interpolation for all degrees of freedom. In other words, it is an unlocking scheme that works independent of the element shape, polynomial order, and discretization scheme. It includes additional degrees of freedom that adhere to a carefully designed differential relation that can be interpreted as a kinematic law, incorporated in a mixed sense. Certain constraints are to be enforced on these additional degrees of freedom to obtain a well‐posed system of equations. In this work, the MD method is extended for problems in solid mechanics. We present the underlying variational formulation, followed by its application to 2D solid elements. Additionally, we showcase an idea to enforce the additional constraints in a general sense. Various numerical examples, within the framework of the finite element method and isogeometric analysis, are outlined to demonstrate the performance of the MD method in the geometrically linear and geometrically nonlinear cases. [ABSTRACT FROM AUTHOR]

We develop a computational framework to model damage and delamination in laminated polymer composite structures incorporating the effects of temperature and moisture content. The framework is founded on a recently developed comprehensive multi-layer thin-shell formulation based on Isogeometric Analysis, which includes continuum damage, plasticity and cohesive-interface models. To incorporate hygrothermal effects in the modeling, we propose a scaling law that is based on the Arrhenius equation and material glass transition temperature that establishes the dependence of the intra- and interlaminar material properties on the temperature and moisture content. We compute several classical test cases using a combination of environmental conditions and demonstrate that the resulting modeling approach shows a good agreement with the experimental data, both in terms of failure loads reached as well as failure modes predicted. [ABSTRACT FROM AUTHOR]

A general formalism is proposed, based on the definition of a space-time potential, for developing space-time formulations adapted to nonlinear and time dependent behaviors. The focus is given to the case of standard generalized materials that are expressed from the knowledge of two potentials, a strain energy and a dissipation potential in a convex framework with the help of internal variables. Viscoplasticity with isotropic hardening and nonlinear finite viscoelasticity are investigated. Starting from the definition of an appropriate space-time potential, time discontinuous Galerkin forms are developed for use in the case of time singularities (in particular with regard to time integration of internal variables). Furthermore, NURBS approximation are used, such as to propose Space-Time Isogeometric Analysis models. Numerical examples allow to compare the obtained isogeometric space-time models with standard finite-element models (that are based on standard time integration procedures: radial return for viscoplasticity and backward euler for viscosity) and allow to illustrate the new possibilities offered with the proposed space-time formulations. [ABSTRACT FROM AUTHOR]

This paper presents a new mixed finite element model for material and geometric non-linear analysis of composite beams in partial interaction taking into account the non-penetration condition between layers. The Hu–Washizu functional with three independent fields is chosen for the developed mixed formulation. The force fields in the connection are chosen as the redundant forces and approximated using interpolation functions. The remaining force fields are obtained from solving equilibrium equations so that the element equlibrium is verified. Nevertheless, the compatibility as well as the constitutive law is satisfied only in a weak sense. The geometric non-linearity is taken into account by adopting the co-rotational approach. In this paper, the contact condition is imposed at the element level. Augmented Lagrangian method with Uzawa iteration algorithm is used to solve the contact problem. It has been shown that the proposed mixed formulation gives a more accurate result with less elements comparing to classical displacement based model. Besides, the buckling behaviour of delaminated two-layered composite columns has been studied by using the developed mixed formulation model. It has been observed that the buckling strength of the composite column can be overestimated if the uplift is not considered in the model. [ABSTRACT FROM AUTHOR]

We investigate Sobolev regularity of bivariate functions obtained in Isogeometric Analysis when using geometry maps that are degenerate in the sense that the first partial derivatives vanish at isolated points. In particular, we show how the known C 1 -conditions for D-patches have to be tightened to guarantee square integrability of second partial derivatives, as required when computing finite element approximations of elliptic fourth order PDEs like the biharmonic equation. [ABSTRACT FROM AUTHOR]

The geometries constructed using the standard CAD software are always boundary-represented (B-Rep). Due to this, analysis-suitable volumetric parameterization of the geometries constructed using the standard CAD software is necessary for three-dimensional isogeometric analysis (IGA). This paper presents a user-friendly, robust volumetric parametrization tool capable of processing the B-Rep of the geometries constructed in one of the most popular CAD software, i.e. Rhinoceros®. The developed tool traverses multiple stages, starting with reading the standard 3DM CAD files of an arbitrarily shaped geometry. It then extracts surface control points, organizes them to establish an outer structural framework, and employs Coons volume parameterization to generate internal control points based on user-defined specifications. We use the NURBS-Python library for reading 3DM files and the NURBS toolbox to visualize and evaluate NURBS functions in the constructed geometries. We construct several baseline and intricate shape geometries to showcase the robustness and generality of the developed tool. Additionally, to demonstrate the effectiveness and applicability of our tool for IGA of non-linear three-dimensional problems, we simulate frictionless contact between two deformable geometries constructed using the developed tool. [ABSTRACT FROM AUTHOR]

This article performs the buckling and vibration studies of the sandwich plates with a negative Poisson's ratio honeycomb core. Based on the isogeometric analysis in conjunction with the refined shear deformation theory, the discrete equilibrium equations of the sandwich plate are established. Numerical results show that the plate with negative Poisson's ratios of the honeycomb core layer has a higher load-bearing capacity than that with positive Poisson's ratios. In addition, the optimal inclined angles of honeycomb core for negative Poisson's ratios subjected to different loading types are obtained. The normalized natural frequency of the plate almost decreases by 10% when the Poisson's ratio ν12 of honeycomb core decreases from −0.3 to −5.5. [ABSTRACT FROM AUTHOR]

Given the significant potential of multi-directional functionally graded materials (MFGMs) for customizable performance, it is crucial to develop versatile material models to enhance design optimization in engineering applications. This paper introduces a material model for an MFGM plate described by trigonometric functions, equipped with four parameters to control diverse material distributions effectively. The bending and vibration analysis of MFGM rectangular and cutout plates is carried out utilizing isogeometric analysis, which is based on a novel third-order shear deformation theory (TSDT) to account for transverse shear deformation. The present TSDT, founded on rigorous kinematics of displacements, is demonstrated to surpass other preceding theories. It is derived from an elasticity formulation, rather than relying on the hypothesis of displacements. The effectiveness of the proposed method is verified by comparing its numerical results with those of other methods reported in the relevant literature. Numerical results indicate that the structure, boundary conditions, and gradient parameters of the MFGM plate significantly influence its deflection, stress, and vibration frequency. As the periodic parameter exceeds four, the model complexity increases, causing result fluctuations. Additionally, MFGM cutout plates, when clamped on all sides, display almost identical first four vibration frequencies. [ABSTRACT FROM AUTHOR]

This study commences with the application of an efficient artificial neural network (ANN)-balancing composite motion optimization (BCMO) approach for finding the optimal material distribution of bi-directional functional graded nanocomposite (FGN) thin plates considering geometrically nonlinear behaviors. To this regard, an ANN-based surrogate model of the high-fidelity isogeometric analysis (IGA) of the geometrically nonlinear Kirchhoff–Love plates based on the Von Kármán nonlinearity theory is first constructed and then employed to predict the values of objectives and constraints required in the BCMO framework. The multi-mesh design approach is utilized to form two separate nonuniform rational B-spline meshes for optimal material distribution and analysis meshes. The unknown control point values of the design mesh are herein selected as the continuous design variables. This allows a possibility to fully explore the complex distribution of optimal material profiles without requiring a significant number of variables. Selected numerical examples with different plate geometries and loading conditions are presented to illustrate the merit features of the proposed approach. Obtained results reveal not only its high accuracy and significant efficiency compared with the conventional approach coupling the BCMO with IGA direct analysis but also the stable ability of the BCMO in finding global optimum material distributions in comparison with two commonly used metaheuristics algorithms, i.e., particle swarm optimization and genetic algorithm. [ABSTRACT FROM AUTHOR]

An outliers-free isogeometric modeling method for rotating disk-shaft systems is developed. The Timoshenko beam theory and artificial spring technique are employed for the rotating shaft and elastic boundary conditions. The nonlinear parameterization method is employed for the removal of outliers and three different nonlinear mappings are developed for the discussion of the accuracy of low modes. The energy coupling method between disks and shaft under nonlinear mapping is performed by using the Newton Raphson method. The results show that the isoparametric mapping has better performance in the accuracy of low modes than other nonlinear mapping and the outliers can also be removed, besides, the present method has good convergence rate for different boundary conditions. The accuracy of the proposed method shows good consistency with the Finite Element Method. The time cost of modeling is reduced by 71.4% compared to the traditional rotor model for a multiple disks rotor system, which indicates that the present approach has potential to provide more efficient optimization models of disk-shaft systems. The proposed method can provide a new modeling framework and can be easily extended to the prediction and optimization of vibration characteristics of complex rotor systems with multiple disks and supports. [ABSTRACT FROM AUTHOR]

AbstractFunctionally graded triply periodic minimal surface (FG-TPMS) structures are known as bio-inspired structures. They possess some remarkable advantages such as porous structures with high inter-connectivity, mathematically controllable geometry features and smooth surfaces. As another advantage, FG-TPMS structures can be fast and numerously manufactured by 3D printing technology. Nevertheless, modeling these structures is a challenging task. This paper investigates static bending and free vibration behaviors of FG-TPMS shells. The proposed formulation is established upon isogeometric analysis (IGA) and first-order shear deformation shell theory (FSDT). The governing equations are discretized by a Galerkin weak form and numerically solved by using non-uniform rational B-Spline (NURBS) basis functions. Exact geometries of structures are described via NURBS basis functions. A fitting technique is used to compute the mechanical characteristics of FG-TPMS materials. We investigate behaviors of FG-TPMS shells considering three types of cell geometries which are primitive (P), gyroid (G), I-graph and wrapped package-graph (IWP), and six porosity distribution patterns. The present solutions are verified with the reference ones in the literature. Effects of boundary, type of cell geometry, porosity distribution pattern, length-to-radius and thickness ratios on behaviors of FG-TPMS shells are rigorously studied. Especially, many numerical results of FG-TPMS shells are first proposed in this paper. [ABSTRACT FROM AUTHOR]

In this study, the thermo-mechanical dynamic buckling analysis of two-directional functionally graded porous (2D-FGP) shallow microshells resting on elastic foundations with arbitrary boundary conditions is performed using the isogeometric approach. The isogeometric approach takes advantage of non-uniform rational B-spline basic functions to exactly represent the structure geometry models and the attainment of higher order approximation conditions. The characteristics of the material vary in both the thickness and axial x orientation. Especially, a new point in this material length-scale parameter of microshell is analyzed as a function of spatial coordinates and as a function of the material gradient parameters. Using Hamilton's principle, the modified couple stress theory, and Kirchhoff-Love's shell theory, the governing equations of a 2D-FGP microshell lying on a Pasternak medium are obtained. A computer program that employs an isogeometric analysis method to analyze the static and dynamic buckling of a 2D-FGP doubly curved shallow microshell with various boundary conditions. The numerical results for thin cylindrical, spherical, and hyperbolic paraboloidal shallow microshells with various planforms, such as rectangular and circular, are presented. Several special cases are used to demonstrate the accuracy and efficacy of the developed model by comparing the numerical results obtained by the proposed formulations with other published data. In addition, the effects of certain parameters, such as the length scale parameters, the power-law indexes, the thickness-to-sides ratio, and the radius ratio, on the dynamic thermo-mechanical buckling response of the 2D-FGP shallow microshells with double curvature are explored in depth. [ABSTRACT FROM AUTHOR]

In this article, we apply NURBS-based isogeometric method to solve a nonlocal degenerate parabolic problem. The isogeometric method has advantages over classical finite element method in terms of exact geometry representation and higher order smooth basis functions. We consider the NURBS-based semi-discretization of the problem. The second-order Crank–Nicolson method is applied for the complete discretization of the problem. Error estimates of the semi-discrete and fully-discrete problems are derived. Few numerical examples are provided to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

• We present an effective isogeometric shape optimization method to optimize phononic band gaps of periodic plates. • The particle swarm optimization is employed to solve the constrained dynamic maximization problem. • The thickness, i.e., z-component of control point of B-spline surface is defined as optimal design variable. • Several numerical examples for finding optimal phononic band gaps of periodic plates are studied. This work presents an effective isogeometric shape optimization approach for finding and widening the phononic band gaps of single-phase Mindlin plate structures. As the single-phase material phononic plate is easy to be manufactured by additive manufacturing, it has been investigated here by optimized its thickness profile to find and widen the band gaps. The proposed method utilizes a coarse B-spline surface to model the thickness profile of periodic plate and a fine B-spline surface to model the mid-surface of plate structure for simulation. The optimal design variables are the thickness variables, i.e., the z-components of the control points of the coarse B-spline surface. To avoid specifying the initial control point locations manually, the constrained dynamic maximization problem is solved by a particle swarm optimization (PSO) algorithm here. Various numerical examples demonstrate the effectiveness and reliability of the proposed method in finding optimal phononic band gaps of periodic plates. And the numerical results show that there is no band gap for h max = 3 h min , and the band gaps can be found and widen for h max ≥ 5 h min. The obtained band range is 638.9–823.6 Hz with a decreased central frequency of 731.25 Hz for h max = 5 h min and the width and the number of band gaps are increased as the maximum allowable thickness increases. Finally, one optimized design is fabricated through additive manufacturing, and the experimental frequency response is consistent with the results based on isogeometric analysis. [ABSTRACT FROM AUTHOR]

For the first time, the isogeometric analysis (IGA) approach is used to model and analyze free and forced vibrations of doubly-curved magneto-electro-elastic (MEE) composite shallow shell resting on the visco-Pasternak foundation in a hygro-temperature environment. The doubly-curved MEE shallow shell types include spherical shallow shell, cylindrical shallow shell, saddle shallow shell, and elliptical shallow shell subjected to blast load are investigated. The Maxwell equation and electromagnetic boundary conditions are used to determine the vary of the electric and magnetic potentials. The MEE shallow shell's equations of motion are derived from Hamilton's principle and refined higher-order shear theory. Then, the IGA method is used to derive the laws of natural frequencies and dynamic responses of the shell under various boundary conditions. The accuracy of the model and method is verified through reliable numerical comparisons. Aside from this, the impact of the input parameters on the free and forced vibration of the doubly-curved MEE shallow shell is examined in detail. These results may be useful in the design and manufacture of military structures such as warships, fighter aircraft, drones and missiles. [ABSTRACT FROM AUTHOR]

High-performance finite element research has always been a major focus of finite element method studies. This article introduces isogeometric analysis into the finite element method and proposes a new isogeometric finite element method. Firstly, the physical field is approximated by uniform B-spline interpolation, while geometry is represented by non-uniform rational B-spline interpolation. By introducing a transformation matrix, elements of types C0 and C1 are constructed in the isogeometric finite element method. Subsequently, the corresponding calculation formats for one-dimensional bars, beams, and two-dimensional linear elasticity in the isogeometric finite element method are derived through variational principles and parameter mapping. The proposed method combines element construction techniques of the finite element method with geometric construction techniques of isogeometric analysis, eliminating the need for mesh generation and maintaining flexibility in element construction. Two elements with interpolation characteristics are constructed in the method so that boundary conditions and connections between elements can be processed like the finite element method. Finally, the test results of several examples show that: (1) Under the same degree and element node numbers, the constructed elements are almost consistent with the results obtained by traditional finite element method; (2) For bar problems with large local field variations and beam problems with variable cross-sections, high-degree and multi-nodes elements constructed can achieve high computational accuracy with fewer degrees of freedom than finite element method; (3) The computational efficiency of isogeometric finite element method is higher than finite element method under similar degrees of freedom, while as degrees of freedom increase, the computational efficiency between the two is similar. [ABSTRACT FROM AUTHOR]

Isogeometric analysis (IGA) has emerged as a promising approach in the field of structural optimization, benefiting from the seamless integration between the computer-aided design (CAD) geometry and the analysis model by employing non-uniform rational B-splines (NURBS) as basis functions. However, structural optimization for real-world CAD geometries consisting of multiple non-matching NURBS patches remains a challenging task. In this work, we propose a unified formulation for shape and thickness optimization of separately parametrized shell structures by adopting the free-form deformation (FFD) technique, so that continuity with respect to design variables is preserved at patch intersections during optimization. Shell patches are modeled with isogeometric Kirchhoff-Love theory and coupled using a penalty-based method in the analysis. We use Lagrange extraction to link the control points associated with the B-spline FFD block and shell patches, and we perform IGA using the same extraction matrices by taking advantage of existing finite element assembly procedures in the FEniCS partial differential equation (PDE) solution library. Moreover, we enable automated analytical derivative computation by leveraging advanced code generation in FEniCS, thereby facilitating efficient gradient-based optimization algorithms. The framework is validated using a collection of benchmark problems, demonstrating its applications to shape and thickness optimization of aircraft wings with complex shell layouts.

Thin walled structures such as plates and shells are widely used in many engineering fields. To Predict its fracture behavior is of great significance for integrity design and strength evaluation of engineering structures. Numerical simulation of the fracture behavior of hyperelastic plates and shells is a challenge due to complex kinematic description, hyperelastic constitutive relationship, geometric nonlinearity and the degradation on elastic parameter caused by fracture damage. Combining Kirchhoff Love (K-L) shell theory with the fracture phase field method, and numerically discretizing the first and second order partial derivatives of displacement field and phase field by using T-splines and meeting the requirements of K-L plate and shell theory for the C1 continuity of the shape function, a model for the isogeometric analysis numerical formulation of the phase field fracture in hyperelastic K-L plates and shells is established. The fracture failure behavior of hyperelastic K-L plates and shells under the uniform load and displacement load is simulated, and the effect of the Gaussian curvature on the fracture behavior of hyperelastic K-L shells is studied. The simulation results show that the present numerical scheme can effectively capture the complex crack propagation path of plates and shells under the uniform load, and the displacement field can effectively reflect the crack distribution of materials. The thin shell with negative Gaussian curvature shows the excellent fracture performance under the internal pressure, and can withstand the greater internal pressure.

AbstractIsogeometric high-order analysis of a sandwich beam is presented in this article. The kinematic and geometric modeling is applied using the higher order modeling based on the stretchable models. The suggested model in this article can be applied for the sport equipment specially for analysis of diving beams boards and beams. The Variational-based approach is extended to derive governing equations. The results are analytically obtained in order to seek impact of significant parameters of the problem on the responses. The results are confirmed through comparison with available ones in the literature. [ABSTRACT FROM AUTHOR]

In the present work the geometrically exact beam model is formulated in terms of unit quaternions. A projection‐based discretization approach is proposed which is based on a normalization of the quaternion approximation. The discretization relies on NURBS shape functions and, alternatively, on Lagrangian interpolation. The redundancy of the quaternions is resolved by applying the method of Lagrange multipliers. In a second step the Lagrange multipliers are eliminated circumventing the need to solve saddle point systems. The resulting finite elements retain the objectivity of the underlying beam formulation. Optimal rates of convergence are observed in representative numerical examples. [ABSTRACT FROM AUTHOR]

AbstractThis study addresses the multi-material optimization problem by the effective and robust isogeometric proportional topology optimization (IGA-PTO) approach. Non-uniform rational B-spline basis functions are used for descriptions of geometry, displacement field, and density fields of material phases. The alternating active-phase algorithm is employed to convert the optimization problem with multiple constraints into a series of sub-problems of single constraint. Several examples are exhibited, and intensive comparisons with the gradient-based optimality criteria (OC) algorithm are presented to highlight the superior performance of the proposed PTO algorithm. Finally, the optimized topologies are prototyped using 3D printing technology to evidence the manufacturing feasibility. [ABSTRACT FROM AUTHOR]

In modern applications of computer‐aided design (CAD) for the analysis of shell structures, isogeometric analysis is a powerful tool that integrates both design and analysis. An exact geometry description and a straightforward computation without loss of information are advantageous, especially for shell structures such as roofs, satellite hulls, or car bodies. In addition, the scaled boundary method provides a scale separation with a semi‐analytical solution procedure to consider a three‐dimensional linear elastic constitutive law. The presented approach deals with a scaled boundary solid shell formulation in the framework of isogeometric analysis. The formulation utilizes a normal scaling strategy which scales the shell along its normal vector at each point on the discretized bottom surface. This is fundamentally different from the well‐known radial scaling strategy, where each point on the problem domain is obtained from a fixed scaling center. This results in a separation of the analysis into an in‐plane direction and a scaling (normal) direction. By introducing the scaling, a scaled boundary differential equation is derived that is dependent on the scaling parameter only. Choosing a proper set of conditions, the differential equation can be solved by a Padé expansion. While the in‐plane direction is solved in a weak sense, the thickness direction along the normal vector can be solved analytically resulting in a semi‐analytical procedure. The isogeometric description of the CAD structure inherently yields the exact normal vector, its derivatives and a higher order continuity throughout the structure. Herein, the focus is on the solution technique in the thickness direction and the challenges are addressed. The power of the formulation is outlined in several numerical examples of static and dynamic analysis and a comparison to shell formulations in literature is provided. [ABSTRACT FROM AUTHOR]

In the present article, the panel flutter behavior of nanocomposite hybrid laminated quadrilateral panel is studied. A three-phase carbon-nanotube (CNT)/polymer/fiber laminated composite is considered where the probable agglomeration of the CNT phase in the nano-matrix mixture is also addressed. Diverse use of fiber phase material in different plies is assumed in the context of hybrid laminated composite. A combination of nanocomposite polymer matrix, tow-steered curvilinear reinforcing fibers within the plies, and hybrid multi-material laminate is considered. The panel flutter behavior of the arbitrary quadrilateral hybrid laminated nanocomposite plate is formulated by a non-uniform rational B-splines-type isogeometric analysis method based on the first-order shear deformation plate theory. The nonideal mixture of the nanocomposite is also concerned about through applying a micromechanics approach. Representative results are obtained and compared with those available in the literature to demonstrate the quality of the proposed formulation. The effects of different parameters including the layup, mixture quality, boundary condition, geometry, and flow direction on the flutter behavior are investigated by performing parametric studies. [ABSTRACT FROM AUTHOR]

This study introduces extended isogeometric analysis method (XIGA) as a tool to investigate fracture behavior in isotropic and orthotropic materials under uniaxial and biaxial mechanical loads. Utilizing isogeometric analysis knot spans to discretize the problem domain and second order perturbation technique within XIGA framework derives mean and coefficient of variance values for mixed-mode stress intensity factors (MMSIFs) by considering variations in input parameters, including crack length, crack angle and material elastic properties. Mixed-mode SIF is evaluated using the proper interaction integral method subjected to different loading conditions showcases versatility of the proposed approach. The alignment between the existing literature and Monte Carlo simulations demonstrates a robust correspondence, validating the accuracy and reliability of the stochastic XIGA technique. This research establishes the efficacy of the approach in comprehensively addressing fracture analysis across different mechanical loading scenarios. [ABSTRACT FROM AUTHOR]

This paper introduces a novel method for optimal shape design of thick shells. We consider shells based on the Reissner-Mindlin theory, with the assumption of linear elastic material behavior. The goal is to find the optimal material distribution within the shell’s mid-surface. This is achieved using a cost function that minimizes the volume while considering stress-based constraints, with the material distribution represented by a level set function. The evolution of the shape is driven by the gradient of the cost function within the framework of a Hamilton–Jacobi equation. Both the level set and the displacement fields are described using computer aided design compatible tools, within the framework of isogeometric analysis. This allows for precise definition of the optimal shape and straightforward export of the resulting design to commercial software for manufacturing. Furthermore, the proposed method handles complex, non-conforming multi-patch geometries thanks to an augmented Lagrangian formulation. The latter guarantees strong compatibility with real-world engineering applications. The effectiveness of the method is demonstrated through its application to various three-dimensional multi-patch geometries under different loading conditions. [ABSTRACT FROM AUTHOR]

Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh. [ABSTRACT FROM AUTHOR]

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Purpose: The article is dedicated to the study of the size-dependent nonlinear vibration characteristics of multilayer functionally graded graphene platelets-reinforced (FG-GPLs) composite tapered microbeams. Methods: The Timoshenko beam theory (TBT) is applied as the fundamental framework of the theoretical model, integrated with the modified couple stress theory (MCST) and the von Kármán geometric nonlinearity hypothesis. The modified Halpin–Tsai micromechanical model and rule of mixture are employed to determine the properties of the composites. Combining Hamilton's principle and isogeometric analysis, the governing equations as well as boundary conditions for the nonlinear characteristics of the tapered microbeam are derived, and the direct iteration technique is employed for the numerical solution. Results: The comparative analysis of the calculated values with the results reported in the public literature verifies the accuracy and efficacy of the proposed model and calculation strategy. Then, a detailed parametric investigation is carried out to explore the nonlinear characteristics of FG-GPLs tapered microbeams, and the effects of various factors, including taper ratio, boundary conditions and GPLs distribution patterns, are examined. Conclusions: The findings showed that the change in the taper ratio along the thickness direction has a greater impact than the change in the taper ratio along the width direction. In addition, the increase of GPLs weight fraction and small-scale parameters will lead to a notable increase in the FG-GPLs tapered microbeam's nonlinear frequency, while the increase in the length-to-thickness ratio has little effect on this value. Meanwhile, regardless of the boundary conditions, the nonlinear frequency ratio corresponds to the X-GPLs distribution pattern is always the largest, followed by the U-GPLs and O-GPLs in order. [ABSTRACT FROM AUTHOR]

Spectral and singular value symbols are valuable tools to analyse the eigenvalue or singular value distributions of matrix-sequences in the Weyl sense. More recently, Generalized Locally Toeplitz (GLT) sequences of matrices have been introduced for the spectral/singular value study of the numerical approximations of differential operators in several contexts. As an example, such matrix-sequences stem from the large linear systems approximating Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), Integro Differential Equations (IDEs), using any discretization on reasonable grids via local methods, such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, Discontinuous Galerkin etc. Studying the asymptotic spectral behaviour of GLT sequences is useful in analysing classical techniques for the solution of the corresponding PDEs/FDEs/IDEs and in designing novel fast and efficient methods for the corresponding large linear systems or related large eigenvalue problems. The theory of GLT sequences, in combination with the results concerning the asymptotic spectral distribution of perturbed sequences of matrices, is one of the most powerful and successful tools for computing the spectral symbol f. In this regard, it would be beneficial to design an automatic procedure to compute the spectral symbols of such matrix-sequences and Ahmed Ratnani partially pursued it. Here, in the case of one-dimensional and two-dimensional differential problems, we continue in this direction by proposing an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming that it is a GLT sequence satisfying mild conditions. [ABSTRACT FROM AUTHOR]

Summary: In this paper, a generalized particle domain method (GPDM) is proposed and developed within the framework of the convected particle domain interpolation method. This new method generates particles directly from non‐uniform rational B‐spline (NURBS)‐based CAD file of a continuum body. The particle domain corresponds to a NURBS element even for trimmed elements of solids with complex geometries. The shape functions and the gradient of shape functions are evaluated using NURBS basis functions to map material properties between particles and grid nodes. It approves that this proposed GPDM can track the domain of particles accurately and avoid the issue of cell‐crossing instability. Several numerical examples are presented to demonstrate the high performance of this proposed new particle domain method. It is shown that the results obtained using the proposed GPDM are consistent with the experimental data reported in the literature. Further development of the generalized particle domain method can provide a link to the material point method and isogeometric analysis. [ABSTRACT FROM AUTHOR]

The present research applies a 2D refined plate theory and isogeometric analysis (IGA) for free vibration analysis of functionally graded (FG) sandwich plates, whose governing equations are treated based on a unified formulation (UF), and nonuniform rational Lagrange (NURL)-based IGA technique. The constitutive model of FG materials is approximated via a Voigt’s rule of mixture based on an equivalent single-layer (ESL) theory. The present framework offers several advantages, including high precision of vibration response by employing higher-order plate theory and the capability of NURL basis functions to capture the exact form of plate geometries. Moreover, higher-order theories postulated by the UF are exempt from the Poisson locking phenomenon and do not require a shear correction factor. Additionally, by employing UF, the effect of thickness stretching on vibration response is considered. Furthermore, higher-order NURL basis functions effectively mitigate shear locking. A large numerical investigation shows the accuracy of results and investigates the effects of several key parameters, such as gradient index, thickness-to-length ratios, layer-to-thickness ratios, and boundary conditions, on the vibration response of FG sandwich plates. [ABSTRACT FROM AUTHOR]