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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.

In this paper, we employ the finite element method based on non-uniform rational B-splines function approximation to solve the nonlinear Brinkman-Forcheimer-Darcy equation in a simply connected and bounded Lipschitz domain Ω. We provide both theoretical and numerical studies of the Dirichlet boundary problem. Utilizing a stream function formulation, we demonstrate the well-posedness of the weak form. Furthermore, we approximate the velocity and pressure fields by linearizing the nonlinear terms, resulting in an algebraic system. This Non-uniform rational B-splines method is more effective in terms of the exact representation of the geometry and the good approximation of the solution compared to the virtual element method. To validate the effectiveness of the non-uniform rational B-splines Finite Element Method, we conduct numerical simulations of fluid flow in porous media.

We conducted an isogeometric analysis (IGA) to evaluate the performance of automobile structural member deformation and crash. In automobile crash analysis, ensuring the accuracy of the acceleration, velocity, and load in time series, as well as the structural deformation behavior, is important. To maintain the aforementioned consistency, accurately reproducing the bending and buckling of structural members is indispensable. In this study, we firstly computed the bending and buckling of structural members using IGA and validated its performance by comparing the results with those of a conventional finite element analysis and experiments. In addition, we utilized IGA for the crash analysis of an automobile body.

In this work, free vibration and transient response of porous metal foam magneto-electro-elastic (PMF-MEE) curved nanobeam with elastic boundary conditions (BCs) using an isogeometric analysis (IGA) based on nonlocal elastic and refined higher-order shear beam theories (HSDT) are investigated. The variation of electric and magnetic potentials through the thickness direction of the curved nanobeam is determined by the Maxwell equation and the magneto-electric boundary condition. The new point of this paper is that it does not use traditional boundary conditions, but instead, the first and last positions of the curved nanobeam are linked by an elastic system consisting of transrational springs and rotational springs with stiffness controlled to give arbitrary boundary conditions. Besides, the nonlocal coefficients vary with thickness like other material properties. Hamilton’s principle is applied to establish the governing equations of motion. The Newmark direct integration method is applied to derive the dynamic responses including deflection and phase trajectory. A comprehensive parametric study is carried out to evaluate the influence of input factors on the natural and forced oscillation capabilities of PMF-MEE curved nanobeams. The findings may benefit for the analysis and design of intelligent structures made of magneto-electro-elastic materials.

As the water pipe cooling system is widely applied to controlling temperature in mass concrete structures, the precise simulation of the temperature field in mass concrete with cooling pipes embedded is meaningful. This paper presents an isogeometric analysis (IGA) with NURBS for heat transfer in mass concrete with consideration of the cooling pipe. The proposed method not only achieves the same level of accuracy with fewer nodes but also eliminates the time-consuming process of mesh in the traditional FEM. The coarsest parameter space which depicts small pipe and large concrete precisely is constructed to create an efficient model for numerical computation. In addition, the unique k-refinement in IGA is supposed to be the most appropriate encryption mechanism, and the knot insertion vector for effective refinement is calculated by considering the characteristics of temperature gradient distribution around the cooling pipes. In addition, a different calculation parameter has been discussed to show the stability and flexibility of the IGA. The obtained numerical results demonstrate the accuracy and efficiency of the proposed scheme in the simulation of transient temperature fields in concrete structures with cooling systems.

This paper uses isogeometric analysis (IGA) based on higher-order shear deformation theory (HSDT) to study the dynamic response of bio-inspired helicoid laminated composite (B-iHLC) plates resting on Pasternak foundation (PF) excited by explosive loading. IGA takes advantage of non-uniform rational B-spline (NURBS) basic functions to exactly represent the structure geometry models and the attainment of higher-order approximation conditions. This method also ensures a C1 continuous function in the analysis of transverse shear deformation via HSDT. Furthermore, IGA eliminates the requirement for correction factors and delivers accurate results. Pasternak foundation with two stiffness parameters: springer stiffness (k1) and shear stiffness (k2). The derivation of the governing equations is based on Hamilton's principle. The proposed method is validated through numerical examples. A comprehensive analysis of the impact of geometrical parameters, material properties, boundary conditions (BCs), and foundation stiffness on dynamic response of B-iHLC plates is carried out.

In this paper, we extend our previous work on the dynamic buckling analysis of isogeometric shell structures to the stochastic situation where an isogeometric deterministic dynamic buckling analysis method is combined with spectral-based stochastic modeling of geometric imperfections. To be specific, a modified Generalized-α time integration scheme combined with a nonlinear isogeometric Kirchhoff–Love shell element is used to simulate the buckling and post-buckling problems of cylindrical shell structures. Additionally, geometric imperfections are constructed based on NURBS surface fitting, which can be naturally incorporated into the isogeometric analysis framework due to its seamless CAD/CAE integration feature. For stochastic analysis, the method of separation is adopted to model the stochastic geometric imperfections of cylindrical shells based on a set of measurements. We tested the accuracy and convergence properties of the proposed method with a cylindrical shell example, where measured geometric imperfections were incorporated. The ABAQUS reference solutions are also presented to demonstrate the superiority of the inherited smooth and high-order continuous properties of the isogeometric approach. For stochastic dynamic buckling analysis, we evaluated the buckling load variability and reliability functions of the cylindrical shell with 500 samples generated based on seven nominally identical shells reported in the geometric imperfection data bank. It is noted that the buckling load variability in the cylindrical shell obtained with static nonlinear analysis is also presented to show the differences between dynamic and static buckling analysis.

In this paper, we implement the finite detail technique primarily based on T-Splines for approximating solutions to the linear elasticity equations in the connected and bounded Lipschitz domain. Both theoretical and numerical analyses of the Dirichlet and Neumann boundary problems are presented. The Reissner–Mindlin (RM) hypothesis is considered for the investigation of the mechanical performance of a 3D cylindrical shell pipe without and with preformed hole problems under concentrated and compression loading in the linear elastic behavior for trimmed and untrimmed surfaces in structural engineering problems. Bézier extraction from T-Splines is integrated for an isogeometric analysis (IGA) approach. The numerical results obtained, particularly for the displacement and von Mises stress, are compared with and validated against the literature results, particularly with those for Non-Uniform Rational B-Spline (NURBS) IGA and the finite element method (FEM) Abaqus methods. The obtained results show that the computation time of the IGA based on the T-Spline method is shorter than that of the IGA NURBS and FEM Abaqus/CAE (computer-aided engineering) methods. Furthermore, the highlighted results confirm that the IGA approach based on the T-Spline method shows more success than numerical reference methods. We observed that the NURBS IGA method is very limited for studying trimmed surfaces. The T-Spline method shows its power and capability in computing trimmed and untrimmed surfaces.

We propose an asymptotic concentration approach combined with isogeometric analysis (IGA) for the topology optimization of two-dimensional (2D) linear elasticity structures under the commonly-used framework of the solid isotropic materials and penalty (SIMP) model. Based on the SIMP framework, the B-splines are used as basis functions to describe geometric model in structural finite element analysis, which closely combines geometric modeling with structural analysis. Isogeometric analysis is utilized to define the geometric characteristics of the 2D linear elasticity structures, which can greatly improve the calculation accuracy. In addition, to eliminate the gray-scale intervals usually caused by the intermediate density in the SIMP approach, we utilize the asymptotic concentration method to concentrate the intermediate density values on either 0 or 1 gradually. Consequently, the intermediate densities representing gray-scale intervals in topology optimization results are sufficiently eliminated by virtue of the asymptotic concentration method. The effectiveness and applicability of the proposed method are illustrated by several typical examples.

In this paper, a multi-patch weakly singular isogeometric boundary element method (WSIGABEM) for two-dimensional elastostatics is proposed. Since the method is based on the weakly singular boundary integral equation, quadrature techniques, dedicated to the weakly singular and regular integrals, are applied in the method. A new formula for the generation of collocation points is suggested to take full advantage of the multi-patch technique. The generated collocation points are essentially inside the patches without any correction. If the boundary conditions are assumed to be continuous in every patch, no collocation point lies on the discontinuous boundaries, thus simplifying the implementation. The multi-patch WSIGABEM is verified by simple examples with analytical solutions. The features of the present multi-patch WSIGABEM are investigated by comparison with the traditional IGABEM. Furthermore, the combination of the present multi-patch WSIGABEM and the particle swarm optimization algorithm results in a shape optimization method in two-dimensional elastostatics. By changing some specific control points and their weights, the shape optimizations of the fillet corner, the spanner, and the arch bridge are verified to be effective.

In this study, the size-dependent free vibration analysis of a geometrically imperfect single-layer graphene sheet (SLGS) is studied by an isogeometric approach along with the quasi-3D shear and normal deformation theory. Initial geometric imperfections alter the natural frequencies of the graphene sheets that may exist inherently or purposely created by researchers. The initial curvature is modelled by an analytical function in the governing Equations of the plate. A 4-variable quasi-3D theory with a seventh-order distribution function is used to include both shear deformation and thickness stretching influences. A weak form of a nonlocal plate for free vibration analysis is derived that requires the first-order continuity of the displacement fields. Inherent high-order continuity of non-uniform rational B-spline (NURBS) basis functions in isogeometric analysis can meet this condition. A comparison between the present study and other published works reveals the efficiency and accuracy of the proposed method in imperfect SLGS. The results of the present study show a significant effect of initial geometric imperfection on the natural frequency of single-layer graphene sheets.

In this paper, the novel design mode of human-aided design (HAD) is proposed to replace conventional computer-aided design (CAD). In HAD, computers can automatically complete the whole product design via a new isogeometric topology optimization (ITO), while humans just assist to slightly modify the design to meet requirements. An embedded domain ITO is presented to design complex models with irregular design domains, and editable geometric models of optimized results can be automatically generated based on layered ITO results. Three examples are tested to verify the proposed HAD mode, including a 3D cantilever beam with a regular design domain, an automotive part with an irregular design domain, and a Messerschmitt-Bölkow-Blohm (MBB) beam with a multiscale structure. The results demonstrate that the proposed HAD mode can automatically deliver high-quality optimized models; thus, it has great potential as a revolutionary technology to change the current design mode from CAD to HAD.

The paper deals with the isogeometric analysis (IGA) of active composite laminates with piezoelectric layers. IGA is a special formulation of the finite element method (FEM) that aims at seamless integration of geometric and finite element modelling. NURBS basis functions are employed to develop isogeometric shell formulation based on the Reissner-Mindlin kinematics. Piezolayers characterized by electro-mechanical coupled field effects enable active behavior of the considered structures. The electric field acts across the thickness of the piezolayers and is coupled to the in-plane strains. In addition to a number of advantages that NURBS modelling provides, defining the surface normal vector at the points of the control polygon, which are generally not located on the surface, creates certain difficulties. A method of determining the surface normal vectors at the points of the control polygon based on the Greville's points is discussed. In order to demonstrate the applicability of the developed formulation, a benchmark case is computed and the results are compared with those obtained by means of classical FEM formulation, which are available in the literature.

Abstract This publication presents a novel CAD‐integrated avenue for structural analyses of modular constructions, by applying the isogeometric analysis. The approach relies on trimming of segments from an original geometry and allows accordingly to introduce kinematics at the interfaces and to assess the connections of the parametrically defined modules. The proposed research is presented along some exemplary structures.

In electric machine design, efficient methods for the optimization of the geometry and associated parameters are essential. Nowadays, it is necessary to address the uncertainty caused by manufacturing or material tolerances. This work presents a robust optimization strategy to address uncertainty in the design of a three-phase, six-pole permanent magnet synchronous motor (PMSM). The geometry is constructed in a two-dimensional framework within MATLAB®, employing isogeometric analysis (IGA) to enable flexible shape optimization. The main contributions of this research are twofold. First, we integrate shape optimization with parameter optimization to enhance the performance of PMSM designs. Second, we use robust optimization, which creates a min–max problem, to ensure that the motor maintains its performance when facing uncertainties. To solve this bilevel problem, we work with the maximal value functions of the lower-level maximization problems and apply a version of Danskin’s theorem for the computation of generalized derivatives. Additionally, the adjoint method is employed to efficiently solve the lower-level problems with gradient-based optimization. The paper concludes by presenting numerical results showcasing the efficacy of the proposed robust optimization framework. The results indicate that the optimized PMSM designs not only perform competitively compared to their non-robust counterparts but also show resilience to operational and manufacturing uncertainties, making them attractive for industrial applications.

This paper for first time proposes an isogeometric analysis (IGA) for free vibration response of bi-directional functionally graded (BDFG) rectangular plates in the fluid medium. Material properties of the BDFG plate change in both the thickness and length directions via power-law distributions and Mori-Tanaka model. The governing equation of motion of BDFG plate in the fluid-plate system is formulated basing on Hamilton's principle and the refined quasi three-dimensional (3D) plate theory with improved function f(z). The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to determine the added mass. The discrete system of equations is derived from the Galerkin weak form and numerically analyzed by IGA. The accuracy and reliability of the proposed solutions are verified by comparing the obtained results with those published in the literature. Moreover, the effects of the various parameters such as the interaction boundary condition, geometric parameter, submerged depth of plate, fluid density, fluid level, and the material volume control coefficients on the free vibration behavior of BDFG plate in the fluid medium are investigated in detail. Some major findings regarding the numerical results are withdrawn in conclusions.

Creating very detailed finite element models of aero-engines is a very time-consuming process especially if structured meshes have to be generated for thousands of parts. Isogeometric analysis offers the possibility of an accelerated model creation process while achieving higher accuracy by using the non-uniform rational B-spline functions used for the geometry description also as basis functions for the finite element analysis. In this case, no meshing is necessary anymore. Here, we demonstrate this process by applying a self-developed tool which creates a geometry description and a computational model of a part at the same time. By the help of this tool we build up a simplified mechanical aero-engine model, which is used for transient implicit computations simulating the acceleration process of the rotor. We discuss the results as well as the computational time of isogeometric models in comparison to classical finite element models.

This paper extends isogeometric analysis (IGA) based on third-order shear deformation theory (TSDT) for vibration analysis of functionally graded porous (FGP) plates in the thermal environment resting on Kerr foundation. IGA is based on non-uniform rational B-spline (NURBS) basic functions, so it can represent exact geometry and satisfy higher-order approximation (C1 continuous function) conditions of TSDT. This combination ensures high accuracy, including transverse shear deformation, without having to use any correction factors. The temperature field via the plate thickness satisfies the steady‐state heat transfer equation. The governing equations are derived from Hamilton's principle. Then, numerical examples are conducted to validate the effectiveness of the proposed method. Finally, the effects of geometrical parameters, material properties, boundary conditions (BCs), temperature change and foundation stiffness on the thermal vibration of FGP plates are fully considered.

Article Highlights The use of a p-multigrid method significantly reduces the CPU timings for higher values of the spline degree. The Multigrid Reduction in Time method shows both strong and weak scalability up to 2048 cores. Iteration numbers are independent of the number of time steps, mesh width and spline degree.

In this paper, an isogeometric analysis method is proposed to obtain the notch tip stress intensity factor for the plane fracture problem with V-notch. Based on the isogeometric analysis, which discretizes the overall structure and constructs the stiffness matrix of the structure, obtains the equilibrium equation, and simultaneously obtains the coordinate information of the nodes. Combined with the displacement expression of the plane with sharp notch in polar coordinate system, the node coordinate information of the whole field is substituted into the analytical expression after coordinate conversion to obtain a transformation matrix, and the isogeometric equilibrium equation is converted into a smaller-dimensional equation. The coefficients of the analytical solution of the displacement field can be calculated, and the stress field and the stress intensity factor of the notch are obtained at the same time. This method unifies numerical modeling and numerical calculation into one platform, which improves the efficiency of numerical analysis; avoids the grid-sensitive problem of iso-geometric analysis fracture problems, and the stress intensity factors can be directly solved without any post-processing program, and avoids the problem of inconsistency in the calculation of isogeometric equations.

This paper develops the new logarithmic higher-order shear deformation theory (LHSDT) incorporating isogeometric method for free and forced vibration analyses of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) plates. In this theory, a logarithmic function is employed to approximate the distribution of shear strains along the plate thickness which satisfies the condition of zero tractions on the top and bottom surfaces of the plate. The plate is assumed to be fabricated from a mixture of carbon nanotubes (CNTs) and a polymeric matrix. The CNTs are either uniformly distributed or functionally graded (FG) along the thickness direction of the plate. The modified rule of mixture scheme is applied to estimate the effective mechanical properties of FG-CNTRC plates. The governing equations are derived from Hamilton’s principle. Furthermore, the Newmark approach is utilized to predict the temporal response of FG-CNTRC plates under different transverse dynamical loadings. The applicability and efficiency of the present formulation in predicting vibrational characteristics of FG-CNTRC plates are investigated through an extensive set of numerical examples considering different configurations of the plate. It is revealed that the computed results are in excellent agreement with other solution methods extracted by the 3D model and other plate theories. Eventually, a detailed parametric study is conducted to explore the influence of related parameters on the natural frequencies and temporal response of FG-CNTRC plates.

Non-negative intensity (NNI) is a quantity which avoids near-field cancellation effects in sound intensity and provides direct visualization of the surface contributions to the radiated sound power. Hence, minimizing the integration of Non-negative intensity on predefined surfaces is implemented to be the design objective of topology optimization for the constrained-layer damping design on plates in this work. Non-negative intensity can be easily computed based on the radiation modes and the particle velocity on the surfaces of interest. Regarding the radiation modes, an eigenvalue analysis for the acoustic impedance matrix is required. After evaluating the objective function, the gradients of the objective function are computed using an adjoint variable method (AVM). These gradients enable the optimization to be solved by the method of moving asymptotes (MMA). Finally, some numerical examples are presented to validate the proposed optimization approach. Numerical results show that the corner radiation properties of the plate can be suppressed by the optimization, minimizing the integration of Non-negative intensity.

In this paper proposes a sensitivity analysis method based on a Polynomial Chaos Expansion (PCE) surrogate model for flexoelectric materials. The non-uniform rational B-splines (NURBS) basis functions to discretize the fourth-order partial differential equation for flexoelectricity and obtains a deterministic solution (electric potential). The mathematical expressions of surrogate model for the flexoelectric materials are established by considering uncertain parameters such as independent Young’s modulus, concentrated load and flexoelectric constants. The sensitivity expression is found by derivation the mathematical expression for the surrogate model. Moreover the finite difference method (FDM) are conducted in numerical examples to demonstrate the validity and correctness of the proposed algorithm.

In this paper, a domain integral method to evaluate the J-integral along with a singular patch method for three-dimensional linear elastic fracture mechanics analyses using Isogeometric analysis (IGA) are presented. The analyses are carried out using a singular patch method that can realize the square-root singularity in the stress distribution in the vicinity of the crack front. The sizes of IGA elements adjacent to the crack front are generally much larger than those of finite elements in the crack analysis. The proposed domain integral method enables a pointwise evaluation of the J-integral at a point on the crack front curve. To achieve it, integral domain with a very small thickness is set at the crack front. To examine the accuracy, the proposed methods are applied to the problems of semi-elliptical crack problems with high aspect ratios. Stress intensity factors are computed by the J-integral method. The proposed methods are found to evaluate the stress intensity factors accurately.

A systematic literature review on numerical strategies suitable to simulate defects and discontinuities in layered and sandwich structures is presented here. Particular care is given to studies whose scope concerns the so-called Moving Mesh (MM) or Isogeometric Analysis (IGA) methods. A general overview of the peculiarities of each approach is also provided. A total of fifteen and twenty-six journal/conference articles were summarised and categorised for MM and IGA methods, respectively. Based on the available literature, it can be stated that MM approaches generally allow a lower computational burden due to the reduced number of re-meshing steps required when compared to standard finite element approaches. Conversely, IGA approaches bring strong advantages in the geometric description of curved shell structures and, due to a non-uniform rational b-spline interpolation function, ensure higher numerical accuracy in stress analysis problems.

A mathematical model is proposed to simulate the fully hydro-mechanical coupling of two-phase cement-based slurry migration in saturated deformable geomaterials from microscopic to macroscopic scale. The mass conservation equations and the momentum balance equations for cement-based slurry and geomaterials are derived based on the thermodynamically constrained averaging theory (TCAT). The Galerkin discretization of the governing equations of hydro-mechanical coupling are developed by the isogeometric analysis (IGA) integrated with the Bézier extraction operator, and the numerical calculation is implemented with the generalized backward Euler method. The presented modeling is verified by comparison of the numerical calculation with the experimental tests, and the pore fluid pressure of the stratum and the slurry concentration of cement-based slurry migration in saturated deformable geomaterials are discussed. The modeling presented provides an effective alternative method to simulate cement-based slurry migration and explore isothermal multiphase coupled problems.

This paper presents a novel quasi-three-dimensional shear deformation theory called the spectral displacement formulation (SDF) for analyzing the free vibration of functionally graded plates. The SDF expresses the unknown displacement field as a unique form of the Chebyshev series in the thickness direction. By increasing the truncation number in the Chebyshev series, the bending analysis results can approach the three-dimensional elasticity solution and satisfy the traction-free boundary conditions without requiring a shear correction factor. The SDF is an extension of the classical plate theory, thereby naturally avoiding the shear-locking phenomenon. These characteristics enable the SDF to apply to plates of arbitrary thickness while maintaining accuracy. The nonuniform rational B-spline-based isogeometric approach is employed to enhance the applicability of this theory to free vibration analysis of functionally graded plates with complex geometries and different boundary conditions. Numerical examples are presented to demonstrate the accuracy and reliability of the proposed method in analyzing the free vibration of functionally graded plates.

This study focuses on numerical solution to the Brinkman equation with mixed Dirichlet–Neumann boundary conditions utilizing isogeometric analysis (IGA) based on non-uniform rational B-splines (NURBS) within the Galerkin method framework. The authors suggest using different choices of compatible NURBS spaces, which may be considered a generalization of traditional finite element spaces for velocity and pressure approximation. In order to investigate the numerical properties of the suggested elements, two numerical experiments based on a square and a quarter of an annulus are discussed. The preliminary results for the Stokes problem are presented in References.

A new quasi-three-dimensional (3D) shear deformation theory, called the spectral displacement formulation (SDF), is proposed for high-precision static bending analyses of functionally graded plates. The main idea is to expand unknown displacement fields into Chebyshev series of a unique form in the thickness direction; the truncation numbers are set to be adjustable to meet various application requirements. Specifically, 3D elasticity solutions and traction-free boundary conditions can be approached by increasing the number of Chebyshev bases. The SDF is also an extension of the classical plate theory and naturally avoids the shear locking problem, making it versatile for functionally graded material (FGM) plates of arbitrary thicknesses. The C1 continuity requirement for the discretization of the generalized displacements is conveniently fulfilled by the nonuniform rational B-splines (NURBS)-based isogeometric method. Numerical examples demonstrate the excellent performance of the proposed method for the displacement and stress analyses of functionally graded plates. The high precision and versatility of the present method have manifested its great potential applications in strain-based or stress-based reliability analysis, optimization design, fatigue analysis, and fracture analysis of FGM plates, and other related fields.

We construct an overlapping additive Schwarz preconditioner for the biharmonic Dirichlet problems discretized by isogeometric analysis based on generalized B-splines (GB-splines) and analyze its optimal convergence rate bound that is cubic in the ratio between subdomains and overlap sizes. Our analysis is validated through a set of numerical experiments that illustrate good behavior of the proposed preconditioner with respect to the model parameters.

Application of isogeometric analysis (IGA) for curved beams is very convenient for its ability of exact representation of curved geometries. Several beam formulation has been presented since the introduction of IGA. In this paper, two different beam formulations are presented: Bernoulli beam formulation of A. M. Bauer et al. [1], and Timoshenko beam element introduced by G. Zhang et al. [2]. Both beam elements are implemented and their performance is documented on the fully threedimensional example of helicoidal spring.

Flexoelectricity is a symmetry independent electromechanical coupling phenomenon that outperforms piezoelectricity at micro and nanoscales due to its size-dependent behavior arising from gradient terms in its constitutive relations. However, due to this gradient term flexoelectricity, to exhibit itself, requires specially designed geometry or material composition of the dielectric material. First of its kind, the present study put forward a novel strategy of achieving electric field gradient and thereby converse flexoelectricity, independent of geometry and material composition of the material. The spatial variation of the electric field is established inside the dielectric material, Ba0.67Sr0.33TiO3 (BST), by manipulating electrical boundary conditions. Three unique patterns of electrode placement are suggested to achieve this spatial variation. This varying direction of electric field gives rise to electric field gradient, the prerequisite of converse flexoelectricity. A multi-physics coupling based theoretical framework is established to solve the flexoelectric actuation by employing isogeometric analysis (IGA). Electromechanically coupled equations of flexoelectricity are solved to obtain the electric field distribution and the resulting displacements thereby. The maximum displacements of 0.2 nm and 2.36 nm are obtained with patterns I and II, respectively, while pattern III can yield up to 85 nm of maximum displacement.

In this paper, we present numerical results obtained by an in-house incompressible fluid flow solver based on isogeometric analysis (IgA) for the standard benchmark problem for incompressible fluid flow simulation – lid-driven cavity flow. The steady Navier-Stokes equations are solved in their velocity-pressure formulation and we consider only inf-sup stable pairs of B-spline discretization spaces. The main aim of the paper is to compare the results from our IgA-based flow solver with the results obtained by a standard package based on finite element method with respect to degrees of freedom and stability of the solution. Further, the effectiveness of the recently introduced rIgA method for the steady Navier-Stokes equations is studied. The authors dedicate the paper to Professor K. Kozel on the occasion of his 80th birthday.

This paper presents a comparison of different time- and frequency-domain solvers for the steady-state simulation of the eddy current phenomena, due to the motion of a permanent magnet array, occurring in the soft-magnetic stator core of electrical machines that exhibits nonlinear material characteristics. Three different dynamic solvers are implemented in the framework of the isogeometric analysis, namely the traditional time-stepping backward-Euler technique, the space-time Galerkin approach, and the harmonic balance method, which operates in the frequency domain. Two-dimensional electrical machine benchmarks, consisting of both slotless and slotted stator core, are considered to establish the accuracy, convergence, and computational efficiency of the presented solvers.

Abstract Topology Optimization (TO) is a powerful numerical technique to determine the optimal material layout in a design domain, which has accepted considerable developments in recent years. The classic Finite Element Method (FEM) is applied to compute the unknown structural responses in TO. However, several numerical deficiencies of the FEM significantly influence the effectiveness and efficiency of TO. In order to eliminate the negative influence of the FEM on TO, IsoGeometric Analysis (IGA) has become a promising alternative due to its unique feature that the Computer-Aided Design (CAD) model and Computer-Aided Engineering (CAE) model can be unified into a same mathematical model. In the paper, the main intention is to provide a comprehensive overview for the developments of Isogeometric Topology Optimization (ITO) in methods and applications. Finally, some prospects for the developments of ITO in the future are also presented.

The isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressive-iterative approximation (IG-LSPIA), to solve the fitting problem in the collocation method. IG-LSPIA starts with an initial blending function, where the control coefficients are combined with the B-spline basis functions and their derivatives. A new blending function is generated by constructing the differences for collocation points (DCP) and control coefficients (DCC), and then adding the DCC to the corresponding control coefficients. The procedure is performed iteratively until the stop criterion is reached. We prove the convergence of IG-LSPIA and show that the computation complexity in each iteration of IG-LSPIA is related only to the number of collocation points and unrelated to the number of control coefficients. Moreover, an incremental algorithm is designed; it alternates with knot refinement until the desired precision is achieved. After each knot refinement, the result of the last round of IG-LSPIA iterations is used to generate the initial blending function of the new round of iteration, thereby saving great computation. Experiments show that the proposed method is stable and efficient. In the three-dimensional case, the total computation time is saved twice compared to the traditional method.

In this paper, we develop a T-spline-based isogeometric method for the large deformation of Kirchhoff–Love shells considering highly nonlinear elastoplastic materials. The adaptive refinement is implemented, and some relatively complex models are considered by utilizing the superiorities of T-splines. A classical finite strain plastic model combining von Mises yield criteria and the principle of maximum plastic dissipation is carefully explored in the derivation of discrete isogeometric formulations under the total Lagrangian framework. The Bézier extraction scheme is embedded into a unified framework converting T-spline or NURBS models into Bézier meshes for isogeometric analysis. An a posteriori error estimator is established and used to guide the local refinement of T-spline models. Both standard T-splines with T-junctions and unstructured T-splines with extraordinary points are investigated in the examples. The obtained results are compared with existing solutions and those of ABAQUS. The numerical results confirm that the adaptive refinement strategy with T-splines could improve the convergence behaviors when compared with the uniform refinement strategy.

Isogeometric analysis (IGA) is an effective numerical method for connecting computer-aided design and engineering, which has been widely applied in various aspects of computational mechanics. IGA involves Galerkin and collocation formulations. Exploiting the same high-order non-uniform rational B-spline (NURBS) bases that span the physical domain and the solution space leads to increased accuracy and fast computation. Although IGA Galerkin provides optimal convergence, IGA collocation performs better in terms of the ratio of accuracy to computational time. Without numerical integration, by working directly with the strong form of the partial differential equation over the physical domain defined by NURBS geometry, the derivatives of the NURBS-expressed numerical solution at some chosen collocation points can be calculated. In this study, we survey the methodological framework and the research prospects of IGA. The collocation schemes in the IGA collocation method that affect the convergence performance are addressed in this paper. Recent studies and application developments are reviewed as well.

We present a novel formulation based on an immersed coupling of Isogeometric Analysis (IGA) and Peridynamics (PD) for the simulation of fluid-structure interaction (FSI) phenomena for air blast. We aim to develop a practical computational framework that is capable of capturing the mechanics of air blast coupled to solids and structures that undergo large, inelastic deformations with extreme damage and fragmentation. An immersed technique is used, which involves an a priori monolithic FSI formulation with the implicit detection of the fluid-structure interface and without limitations on the solid domain motion. The coupled weak forms of the fluid and structural mechanics equations are solved on the background mesh. Correspondence-based PD is used to model the meshfree solid in the foreground domain. We employ the Non-Uniform Rational B-Splines (NURBS) IGA functions in the background and the Reproducing Kernel Particle Method (RKPM) functions for the PD solid in the foreground. We feel that the combination of these numerical tools is particularly attractive for the problem class of interest due to the higher-order accuracy and smoothness of IGA and RKPM, the benefits of using immersed methodology in handling the fluid-structure coupling, and the capabilities of PD in simulating fracture and fragmentation scenarios. Numerical examples are provided to illustrate the performance of the proposed air-blast FSI framework.

This paper proposes a novel generic methodology for kinematics of spatial rigid-multibody systems with and without lubricated joints. In this method CAD surface representations in the form of non-uniform rational B-splines (NURBS) are used to address the interface kinematics. This eliminates the time and effort needed to manually parameterize the interface geometry, by enabling a direct use of the engineering designs encapsulated in CAD systems. Furthermore, the use of NURBS for surface representation allows integration of tribodynamics into an isogeometric analysis (IGA) setting. The kinematic formulation is based on a new implicit matrix approach for implicitization of CAD surfaces in three-dimensional space. The construction of such implicit matrices and their properties are explained, and explicit expressions for the gap height distance, velocity and relative velocities in a general clearance joint are provided.

Topology optimization has been an interesting area of research in recent years. The main focus of this paper is to use an evolutionary swarm intelligence algorithm to perform Isogeometric Topology optimization of continuum structures. A two-dimensional plate is analyzed statically and the nodal displacements are calculated. The nodal displacements using Isogeometric analysis are found to be in good agreement with the nodal displacements acquired by standard finite element analysis. The sizing optimization of the beam is then performed. In order to determine the stress at each point in the beam a formulation is presented. The optimal cross-section dimensions by performing Isogeometric analysis are acquired and verified with the cross-section dimensions achieved by hiring bending stress and shear stress criteria, as well. The topology optimization of a two-dimensional simply supported plate continuum and a problem on three-dimensional continuum are optimized and presented. The results show that the minimum weight which is found by applying Isogeometric topology optimization gives better results compared to the traditional finite element analysis.

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.

This paper proposes an analytical solution and isogeometric analysis numerical approach for buckling analysis of size-dependent beams based on a reformulated strain gradient elasticity theory (RSGET). The superiority of this method is that it has only one material parameter for couple stress and another material parameter for strain gradient effects. Using the RSGET and the principle of minimum potential energy, both non-classical Euler–Bernoulli and Timoshenko beam buckling models are developed. Moreover, the obtained governing equations are solved by an exact solution and isogeometric analysis approach, which conforms to the requirements of higher continuity in gradient elasticity theory. Numerical results are compared with exact solutions to reveal the accuracy of the current isogeometric analysis approach. The influences of length–scale parameter, length-to-thickness ratio, beam thickness and boundary conditions are investigated. Moreover, the difference between the buckling responses obtained by the Timoshenko and Euler–Bernoulli theories shows that the Euler–Bernoulli theory is suitable for slender beams.

In times of the COVID-19, reliable tools to simulate the airborne pathogens causing the infection are extremely important to enable the testing of various preventive methods. Advection-diffusion simulations can model the propagation of pathogens in the air. We can represent the concentration of pathogens in the air by “contamination” propagating from the source, by the mechanisms of advection (representing air movement) and diffusion (representing the spontaneous propagation of pathogen particles in the air). The three-dimensional time-dependent advection-diffusion equation is difficult to simulate due to the high computational cost and instabilities of the numerical methods. In this paper, we present alternating directions implicit isogeometric analysis simulations of the three-dimensional advection-diffusion equations. We introduce three intermediate time steps, where in the differential operator, we separate the derivatives concerning particular spatial directions. We provide a mathematical analysis of the numerical stability of the method. We show well-posedness of each time step formulation, under the assumption of a particular time step size. We utilize the tensor products of one-dimensional B-spline basis functions over the three-dimensional cube shape domain for the spatial discretization. The alternating direction solver is implemented in C++ and parallelized using the GALOIS framework for multi-core processors. We run the simulations within 120 minutes on a laptop equipped with i7 6700 Q processor 2.6 GHz (8 cores with HT) and 16 GB of RAM.

An effective free vibration optimization procedure in combination with the isogeometric approach (IGA), particle swarm optimization (PSO) and an integrated global and local parameterization is presented. The natural frequency of functionally graded (FG) plates is calculated by the IGA based on the Bézier extraction of non-uniform rational B-splines (NURBS) with the cubic NURBS basis function. The material composition is assumed to vary only in the thickness direction, and the volumetric fraction is described by the NURBS basis function in light of the superior properties of NURBS curves. The volume fractions of the control points are then optimized by the PSO. In most of the previous work, the control points for the volume fraction are usually equally spaced, which is incapable of identifying the optimal location of the graded zones in most cases. To overcome this bottleneck, a novel local refinement strategy is proposed. The reliability and effectiveness of the proposed approach are demonstrated through several numerical examples. It is interesting to observe that the optimal results are sandwich or laminate plates, and few parameters are involved in the integrated global and local parameterization.

In this study, a Monte Carlo simulation (MCs)-based isogeometric stochastic Finite Element Method (FEM) is proposed for uncertainty quantification in the vibration analysis of piezoelectric materials. In this method, deterministic solutions (natural frequencies) of the coupled eigenvalue problem are obtained via isogeometric analysis (IGA). Moreover, MCs is employed to solve various uncertainty parameters, including separate elastic and piezoelectric constants and their combined cases.

This paper presents a novel non-classical Timoshenko–Ehrenfest beam model based on a reformulated strain gradient elasticity theory. The strain gradient effect, couple stress effect, and velocity gradient effect for vibration are included in the new model by only one material length scale parameter for each. The variational formulation and Hamilton’s principle are applied to derive the governing equations and boundary conditions. Both an analytical solution and an isogeometric analysis approach are proposed for static bending and free vibration of the microbeam. A non-uniform rational B-splines (NURBS) isogeometric analysis with high-order continuity can effectively fulfill the higher derivatives of the displacement variables in the reformulated gradient beam model. Convergence studies and comparisons to the corresponding analytical solutions verify the model’s performance and accuracy. Finally, different boundary conditions, material length scale parameters, and beam thicknesses are investigated in order to certify the applicability of the proposed approach.

In Isogeometric boundary element method（IGA-BEM）, Non-Uniform Rational B-Splines（NURBS） which are used for describing the geometric boundary in a computer-aided geometric design（CAGD） are employed as shape functions of Boundary Element Method. IGA-BEM not only can accurately describe the geometric boundary, but also unify the design model, analysis model and optimization model, so it can avoid the repeated meshing and mesh distortion problems in the optimization process and it also can improve the efficiency of the design, calculation and optimization. In this article IGA-BEM is combined with particle swarm optimization（PSO） to study the shape optimization of two-dimensional linear elastic structure. Compared with the gradient-based optimization algorithms, gradient-free PSO algorithm can avoid the complicated sensitivity analysis process and make the optimization process easier. The efficiency and accuracy of the combined isogeometric boundary element method-PSO are demonstrated through two shape optimization examples.

The complex geometries of lightweight structures can easily be modeled with modern CAD systems that use NURBS. Isogeometric B-Rep Analysis (IBRA) adopts the NURBS geometries along with their topology for the analysis model, which can thus fulfill both design purposes and solve structural problems in one software environment. Especially in lightweight design, which is characterized by a strong interaction between shape and structural behavior, this presents a major advantage. Results can be used as a new input to consecutive analyses within this novel approach. This is especially useful for the simulation of construction stages or in the design cycle of membrane structures, where mechanically driven optimizations, such as formfinding and cutting pattern generation, as well as structural analysis, are dependent on each other. This paper focusses on parametric design with IBRA and the advantages of the uniform model description and workflow. Not only geometrical but also mechanical properties can be parameterized and thus offer a large variety of options to explore for designers as well as engineers. In addition to the possibility of optimization through formfinding methods, the provided framework can conveniently be combined with other tools and thus be extended by external optimization loops where geometry and structural parameters are modified simultaneously. The successful integration of parametric IBRA in a CAD environment, such as Rhino and Grasshopper is presented. Selected examples demonstrate the significance of the CAD-integrated unified workflow for the interactive analysis loops of lightweight structures.

An isogeometric numerical procedure based on non-uniform rational B-splines is applied to solve the free vibration of carbon nanotubes reinforced and functionally graded material sector cylindrical shells. A typical isogeometric analysis cylindrical shell element is introduced to transform curvilinear domain to parametric domain. The effects of the shear deformation, rotary inertia and axis extensibility are taken into consideration. The material properties are estimated by a mixture rule and effective carbon nanotube parameters. The weak form is derived in detail by applying virtual principle. The novelty of this paper is to use the isogeometric finite element method to analyze the dynamic behaviors of sector cylindrical shells with functionally graded material and four types of carbon nanotubes distributed material. By numerical examples, the convergence and accuracy of the current method are validated. Then, a series of natural frequencies and mode shapes are presented to serve as benchmark solutions in future researches. Moreover, the effects of boundary conditions, geometric proprieties as well as material parameters on the frequencies of the carbon nanotubes reinforced and functionally graded material sector cylindrical shells are examined.