Isogeometric Finite Element Research Articles

Linear Independence of T-Spline Blending Functions of Degree One for Isogeometric Analysis

Linear independence of the blending functions is a necessary requirement for T-spline in isogeometric analysis. The main work in this paper focuses on the analysis about T-splines of degree one, we demonstrate that all the blending functions of such T-spline of degree one are linearly independent. The advantage owned by one degree T-spline is that it can avoid the problem of judging whether the model is analysis-suitable or not, especially for occasions that need a quick response from the analysis results. This may provide a new way of using T-spline for a CAD and CAE integrating scenario, since one degree T-spline still guarantees the topology flexibility and is compatible with the spline-based modeling system. In addition, we compare the numerical approximations of isogeometric analysis and finite element analysis, and the experiment indicates that isogeometric analysis using T-spline of degree one can reach a comparable result with classical method.

A finite deformation isogeometric finite element approach to fibre-reinforced composites with fibre bending stiffness

It is a common technique in many fields of engineering to reinforce materials with certain types of fibres in order to enhance the mechanical properties of the overall material. Specific simulation methods help to predict the behaviour of these composites in advance. In this regard, a widely established approach is the incorporation of the fibre direction vector as an additional argument of the energy function in order to capture the specific material properties in the fibre direction. While this model represents the transverse isotropy of a material, it cannot capture effects that result from a bending of the fibres and does not include any length scale that might allow the simulation of size effects. In this contribution, an enhanced approach is considered which relies on the introduction of higher-gradient contributions of the deformation map in the stored energy density function and which eventually allows accounting for fibre bending stiffness in simulations. The respective gradient fields are approximated by NURBS basis functions within an isogeometric finite element framework by taking advantage of their characteristic continuity properties. The isogeometric finite element approach that is presented in this contribution for fibre-reinforced composites with fibre bending stiffness accounts for finite deformations. It is shown that the proposed method is in accordance with semi-analytical solutions for a representative boundary value problem. In an additional example it is observed that the initial fibre orientation and the particular bending stiffness of the fibres influence the deformation as well as the stress response of the material.

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The cover image is based on the Research Article – Application Isogeometric finite element–based simulation of the aortic heart valve: Integration of neural network structural material model and structural tensor fiber architecture representations by Wenbo Zhang et al., https://doi.org/10.1002/cnm.3438.

Size-dependent nonlinear bending behavior of porous FGM quasi-3D microplates with a central cutout based on nonlocal strain gradient isogeometric finite element modelling

With the aid of the non-uniform rational B-spline (NURBS)-based isogeometric technique, for the first time, the size-dependent geometrically nonlinear bending characteristics of microplates made of porous functionally graded materials (FGMs) having a central cutout with different shapes are studied. The nonlocal strain gradient continuum elasticity within the framework a hybrid higher-order quasi-3D plate theory is adopted to describe the kinematic relations via only four unknowns. To capture the effective material properties, a porosity-dependent rule of mixture is employed. The nonlocal strain gradient nonlinear load–deflection responses are obtained corresponding to various geometrical and material parameters as well as different boundary conditions. It is revealed that the significance of both the nonlocal and strain gradient reduces. This prediction is the same for all values of the material property gradient index as well as the porosity index. Also, it is demonstrated that a central cutout leads to change the trend of load–deflection response, and this change occurs at a specific value for the applied distributed load which depends on several parameters such as the cutout geometry and boundary conditions. In addition, it is displayed that corresponding to different maximum deflections, the significance of the strain gradient size effect in the absence of nonlocality on the nonlinear flexural stiffness of a porous FGM microplate is more than that of the nonlocal size effect in the absence of the strain gradient size dependency.

Optimum material distribution of porous functionally graded plates using Carrera unified formulation based on isogeometric analysis

The current literature deals with the multi-objective optimal material distribution of porous functionally graded plates subjected to free vibration in the framework of Carrera Unified Formulation (CUF) based on isogeometric analysis (IGA). The objective function is to maximize the natural frequencies and minimize the weight of the PFG plate, considering the material properties and porosity through the thickness of the plate. The governing equation of the PFG plate is obtained by using the principle of virtual displacement (PVD) based on the CUF. A non-uniform rational B-splines (NURBS)-based isogeometric finite element model associated with CUF is utilized for the plate free vibration analysis. The Legendre polynomials are used to approximate the thickness direction of plates. The rule of mixture is used to estimate the effective material properties. The Multi-Objective Particle Swarm Optimization (MOPSO) algorithm is employed as an optimizer. Design variables are the power and porosity indices. For verification, several structural design examples are compared with those available in the literature. Several numerical examples for optimizing the PFG plates, including maximization of the first natural frequency and minimization of mass, illustrate the effectiveness and reliability of the proposed method. The results show that the optimum values for the power and porosity indices depend on the objective function and boundary conditions. Furthermore, the clamped boundary condition leads to less mass than the simply-supported boundary condition for each desired first natural frequency.

A Simple-FSDT-Based Isogeometric Method for Piezoelectric Functionally Graded Plates

An efficient isogeometric analysis method (IGA) based on a simple first-order shear deformation theory is presented to study free vibration, static bending response, dynamic response, and active control of functionally graded plates (FGPs) integrated with piezoelectric layers. Based on the neutral surface, isogeometric finite element motion equations of piezoelectric functionally graded plates (PFGPs) are derived using the linear piezoelectric constitutive equation and Hamilton’s principle. The convergence and accuracy of the method for PFGPs with various mechanical and electrical boundary conditions have been investigated via free vibration analysis. In the dynamic analysis, both time-varying mechanical and electrical loads are involved. A closed-loop control method, including displacement feedback control and velocity feedback control, is applied to the static bending control and the dynamic vibration control analysis. The numerical results obtained are accurate and reliable through comparisons with various numerical and analytical examples.

Non-conforming interface coupling and symmetric iterative solution in isogeometric FE–BE​ analysis

In this paper, a three-dimensional (3D) isogeometric finite element (FE) and boundary element (BE) coupling approach based on non-uniform rational B-splines (NURBS) is developed to simulate the deformed body. Based on the geometric exactness and higher smoothness of isogeometric analysis (IGA), NURBS discretization is applied to the approximation of geometric representation, displacement field and traction field simultaneously. An improved power series expansion method (M-PSEM) is used to evaluate the singular integral and is suitable for high order (≥3) NURBS basis functions, which is the limitation of the traditional method. Adaptive integration scheme based on quadtree subdivision technique is adopted to enable the non-singular integration computation more flexible and efficient at optimal computational cost. The relationship between the tractions and nodal external forces on the interface is established by using the virtual work principle. On the other hand, displacement coupling constraints among the non-conforming (or incomplete coincident) surface meshes are constructed by a virtual knot insertion technique. The main advantages of this method are simplicity and robustness, as it is problem-independent and only depends on the NURBS meshes on both sides of the interface. Introducing an enhanced symmetric matrix related to the BE subdomain, the symmetric iterative coupling method (SICM) is extended to solve the final system coupling equation. The final coupling matrix obtained by this iterative method is symmetric, and its dimension is the same as that of the stiffness matrix of IGA finite element method (IGAFEM). Numerical examples are investigated to assess the accuracy and efficiency of the presented method.

Isogeometric symmetric FE-BE coupling method for acoustic-structural interaction

The problems of Acoustic-Structural Interaction (ASI) are commonly encountered in the simulation of the thin-walled structures immersed in the fluid. This interaction can affect the acoustic properties of the fluid and the dynamic characteristics of the structure. In this paper, a Non-Uniform Rational B-Splines (NURBS)-based Isogeometric Finite Element (FE) - Boundary Element (BE) symmetric coupling method is developed to study the ASI problems and the free vibration of the elastic structures submerged in the fluid. The geometry of the thin-walled structures is exactly modelled and discretized by utilizing the Isogeometric Reissner-Mindlin shell elements. This shell theory simplifies the continuity enforcement between NURBS patch boundaries. The Isogeometric Boundary Element Method (IGABEM) is applied to describe the acoustic field. In order to impose coupling constraints, a transformation equation of unknown variables on the interface between sound field and structure field is established, that is, sound pressure versus external force and particle velocity versus displacement. Different from the non-symmetric matrix obtained by the traditional direct BEM, a new variational formulation is introduced to obtain the symmetric coupling coefficient matrix, which maks this coupling approach suitable for a broad class of solvers and improves the robustness of the computation. The reliability and stability of the symmetric coupling method are verified by numerical examples considering acoustic and structural loadings. It is concluded that the symmetric coupling method possesses high accuracy, and can circumvent the limitation of traditional coupling method in solving open boundary problems.

Linear computational cost implicit solver for elliptic problems

In this paper, we use the alternating direction method for isogeometric finite elements to simulate implicit dynamics. Namely, we focus on a parabolic problem and use B-spline basis functions in space and an implicit marching method to fully discretize the problem. We introduce intermediate time steps and separate our differential operator into a summation of the blocks, acting along a particular coordinate axis in the intermediate time steps. We show that the resulting stiffness matrix can be represented as a multiplication of two (in 2D) or three (in 3D) multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. As a result of this algebraic transformations, we get a system of linear equations that can be factorized in linear $O(N)$ computational cost in every time step of the implicit method. We use our method to simulate the heat transfer problem. We demonstrate theoretically and verify numerically that our implicit method is unconditionally stable for heat transfer problems (i.e., parabolic). We conclude our presentation with a discussion on the limitations of the method.

Nonlinear isogeometric multiscale simulation for design and fabrication of functionally graded knitted textiles

We present a nonlinear multiscale modeling and simulation framework for the mechanical design of machine-knitted textiles with functionally graded microstructures. The framework operates on the mesoscale (stitch level), where yarns intermesh into stitch patterns, and the macroscale (fabric level), where these repetitive stitch patterns are composed into a fabric. On the mesoscale, representative unit cells consisting of single interlocked yarn loops, modeled as geometrically exact, nonlinear elastic 3D beams, are homogenized to compute their effective mechanical properties. From this data, a B-Spline response surface model is generated to represent the nonlinear orthotropic constitutive behavior on the macroscale, where the fabric is modeled by a nonlinear Kirchhoff–Love shell formulation and discretized using isogeometric finite elements. These functionally graded textiles with locally varying properties can be designed and analyzed by parameterizing the stitch value, i.e., the loop length of a single jersey stitch, and the knitting direction as mesoscopic design variables of the macroscopic response surface constitutive model. To validate the multiscale simulation and design approach, numerical results are compared against physical experiments of different tensile loading cases for various grading scenarios. Furthermore, the versatility of the method for the design of functionally graded textiles is demonstrated.

An isogeometric finite element approach to fibre-reinforced composites with fibre bending stiffness

In the modelling of fibre-reinforced composites, it is well established to consider the fibre direction in the stored energy in order to account for the transverse isotropy of the overall material, induced by a single family of fibres. However, this approach does not include any length scale and therefore lacks in the prediction of size effects that may occur from the fibre diameter or spacing. By making use of a generalised continuum model including non-symmetric stresses and couple-stresses, the gradient of the fibre direction vector can be taken into account as an additional parameter of the stored energy density function. As a consequence, the enhanced model considers the bending stiffness of the fibres and includes information on the material length scale. Along with additional material parameters, increased continuity requirements on the basis functions follow in the finite element analysis. The isogeometric finite element method provides a framework which can fulfil these requirements of the corresponding weak formulation. In the present contribution, the method is applied to two representative numerical examples. At first, the bending deformation of a cantilever beam is studied in order to analyse the influence of the fibre properties. An increasingly stiff response is observed as the fibre bending stiffness increases and as the fibre orientation aligns with the beam’s axis. Secondly, a fibre-reinforced cylindrical tube under a pure azimuthal shear deformation is considered. The corresponding simulation results are compared against a semi-analytical solution. It is shown that the isogeometric analysis yields highly accurate results for the boundary value problem under consideration.

A Non-Intrusive Stochastic Isogeometric Analysis of Functionally Graded Plates with Material Uncertainty

A non-intrusive approach coupled with non-uniform rational B-splines based isogeometric finite element method is proposed here. The developed methodology was employed to study the stochastic static bending and free vibration characteristics of functionally graded material plates with inhered material randomness. A first order shear deformation theory with an artificial shear correction factor was used for spatial discretization. The output randomness is represented by polynomial chaos expansion. The robustness and accuracy of the framework were demonstrated by comparing the results with Monte Carlo simulations. A systematic parametric study was carried out to bring out the sensitivity of the input randomness on the stochastic output response using Sobol’ indices. Functionally graded plates made up of Aluminium (Al) and Zirconium Oxide (ZrO2) were considered in all the numerical examples.

Shape optimization of heterogeneous materials based on isogeometric boundary element method

In this paper, the isogeometric boundary element method (IGABEM) is used to optimize the shape of heterogeneous materials. In contrast to the isogeometric finite element method (IGAFEM), the iterative optimization algorithm based on IGABEM can be implemented directly from Computer-Aided Design (CAD) without returning the optimization results to CAD designers. The discontinuous element method is extended to IGABEM which deals with corner point problems of inhomogeneous materials. After the singularity of sensitivity analysis of boundary integral equations is demonstrated, the power series expansion method (PSEM) is applied to IGABEM to evaluate various degrees of singularity about sensitivity analysis, which shows more accuracy and efficiency than the element sub-division method (ESDM). A set of control points on the geometric boundary are chosen as design variables, which can be passed from the design model to the analysis model, and the objective function is the elastic energy increment. Finally, several numerical examples in 2D and 3D problems are presented to demonstrate the validity and robustness of the present method.

An isogeometric FE-BE method and experimental investigation for the hydroelastic analysis of a horizontal circular cylindrical shell partially filled with fluid

Abstract In this study, the dynamic characteristics (i.e. natural frequencies and associated mode shapes) of a partially filled horizontal cylindrical shell are investigated experimentally and by an isogeometric finite element-boundary element method. The proposed numerical procedure is divided into two parts. In the first part, the dynamic characteristics of the cylindrical shell under in-vacuo conditions are obtained by the isogeometric finite element method (IGAFEM) based on a linear Kirchhoff-Love shell formulation. In the second part, the fluid-structure interaction effects are calculated in terms of generalized added mass coefficients by using the isogeometric boundary element method (IGABEM), assuming that the structure vibrates in its in-vacuo principle mode shapes. By adopting the linear hydroelasticity theory, it is assumed that the fluid flow is ideal, i.e., an incompressible flow and inviscid fluid. In order to show the versatility of the numerical method, the results are compared with those obtained by the conducted experiments. Relevant numerical challenges in the hydroelastic vibration analysis are highlighted and it is shown that the numerical predictions and experimental results are in good agreement.

Thermal buckling of curvilinearly stiffened laminated composite plates with cutouts using isogeometric analysis

This research presents the Non-Uniform Rational B-Splines (NURBS) based isogeometric finite element analysis of stiffened laminated composite plates. A first-order shear deformation theory is used to derive the governing equations by employing a variational formulation. Isotropic, orthotropic and laminated composite plates stiffened with multiple curvilinear stiffeners of different profiles are investigated. A novel way to achieve displacement compatibility between the panel and stiffeners interfaces is introduced. An easy way of modeling plates with complicated cutouts by using edge curves and generating a ruled NURBS surface between them is described. Influence on the critical thermal buckling load due to the presence of circular and elliptical cutouts is also investigated. Results of parametric studies are presented which show the influence of ply orientation, size and orientation of the cutout, and the position and profile of the curvilinear stiffener. The numerical examples show high reliability and efficiency of the proposed formulation when compared to other published solutions and those obtained using ABAQUS, a commercial software.

Dynamic analysis of carbon nanotube reinforced composite plates by using Bézier extraction based isogeometric finite element combined with higher-order shear deformation theory

This study intends to explore the free vibration and dynamic transient response of carbon nanotube reinforced composite (CNTRC) plates by using the Bézier extraction based isogeometric analysis (IGA). Bézier extraction decomposes non-uniform rational B-spline (NURBS) basis functions into the linear combinations of Bernstein polynomial basis through the Bézier extraction operator, hence providing piecewise C0-continuous Bézier elements. The IGA can thus be implemented in the familiar finite element method (FEM) framework. Higher-order shear deformation theory (HSDT) able to remedy the shear locking problem with no use of the shear correction factor is coupled with the variant IGA. The HSDT adopts a hybrid type shape function, and the governing equilibrium equations for the dynamic behavior are formulated by virtue of the Hamilton's principle. The validity of the Bézier extraction based isogeometric approach coupled with the HSDT in obtaining the solutions for the free vibration and dynamic transient response is first evidenced. Compared with the classical FEM based on the first-order shear deformation theory, the proposed IGA method combined with the HSDT is proved to be more accurate, effective and efficient. Illustrative parametric studies are also carried out to further scrutinize the dynamic behavior of CNTRC plates with various reinforcement schemes. The sinusoidally distributed transverse loads with different time-history patterns and harmonic excitation loading are applied for the dynamic transient analysis, and the damping effect on the temporal response is also examined through the Rayleigh damping model.

Isogeometric analysis of FG-CNTRC plates in combination with hybrid type higher-order shear deformation theory

Abstract Bezier extraction based isogeometric finite element analysis method coupled with a new hybrid type higher-order shear deformation theory (HSDT) is presented for the static bending and buckling analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. By mapping non-uniform rational B-spline basis functions to Bezier element in the forms of Bernstein polynomial basis via the Bezier extraction operator, the isogeometric analysis (IGA) can be implemented in the conventional finite element method (FEM) framework. This study focuses on the introduction of the Bezier extraction based IGA combined with the HSDT for the static analysis through the formulation and implementation of the second-order derivatives of the Bezier element basis functions, which has hardly been attempted in the open literature. Compared to the conventional FEM based on first-order shear deformation theory, the present IGA method based on the HSDT surmounts the shear locking without exploiting the shear correction factor. The proposed isogeometric approach is thus believed to be more accurate and effective and achieves higher convergence as the polynomial order increases. Performance accuracy of the Bezier extraction based isogeometric approach is first evidenced by comparing the computed results with the reference solutions, followed by the illustrative examples to further explore the flexural and buckling behavior of FG-CNTRC plates with various reinforcement patterns.

Isogeometric free vibration of sector cylindrical shells with carbon nanotubes reinforced and functionally graded materials

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.

Comparing quantum, molecular and continuum models for graphene at large deformations

In this paper, the validity and accuracy of three interatomic potentials and the continuum shell model of Ghaffari and Sauer [1] are investigated. The mechanical behavior of single-layered graphene sheets (SLGSs) under uniaxial stretching, biaxial stretching and pure bending is studied for this comparison. The validity of the molecular and continuum models is assessed by direct comparison with density functional theory (DFT) data available in the literature. The molecular simulations are carried out employing the MM3, Tersoff and REBO + LJ potentials. The continuum formulation uses an anisotropic hyperelastic material model in the framework of the geometrically exact Kirchhoff-Love shell theory and isogeometric finite elements. Results from the continuum model are in good agreement with those from DFT. The results from the MM3 potential agree well up to the point of material instability, whereas those from the REBO + LJ and Tersoff potentials agree only for small deformations. Only the Tersoff potential is found to yield auxetic response in SLGSs under uniaxial stretch. Additionally, the transverse vibration frequencies of a pre-stretched graphene sheet and a carbon nanocone are obtained using the continuum model and molecular simulations with the MM3 potential. The variations of the frequencies from these approaches agree within an error of ≈5%.

NURBS-based thermal buckling analysis of graphene platelet reinforced composite laminated skew plates

A nonuniform rational B-spline isogeometric finite element formulation is presented in this research to analyze the thermal buckling behavior of composite laminated skew plates reinforced by graphene platelets. Formulation is based on the first-order shear deformation plate theory. It is assumed that each layer of the composite laminated plate may have different volume fraction of graphene platelets leading to a through-the-thickness piecewise functionally graded medium. The equivalent properties of the plate are obtained by means of the Halpin–Tsai rule. The developed solution method may be used for arbitrary combinations of boundary conditions. The accuracy of the developed formulation is depicted via comparison studies with respect to the available data in the open literature. Novel numerical results are also given to show the effects of volume fraction of graphene platelets, distributed patterns of graphene platelets, and geometric characteristics of the skew plate.