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This paper attempts to merge the concept of hierarchical finite element analysis (FEA) into isogeometric analysis (IGA). The proposed methodology replaces the traditional grid refinement of IGA with custom enrichment functions. The enrichment functions are properly designed B-Spline wavelets tailored to eliminate scale-coupling terms in the stiffness matrix. In this way, the refined solution is synthesized from contributions of smaller independent problems. The proposed approach has two obvious benefits: (1) the calculations performed at each resolution are not discarded when proceeding to a finer one, and (2) it has less computational requirements since the solution is divided into smaller systems. Numerical results on an elasticity problem demonstrate superior performance and accuracy compared to traditional FEA and IGA schemes.

We investigate the nonlinear bending behavior of nanoporous metal foam plates within the framework of isogeometric analysis (IGA) and higher-order plate theory. The nonlocal strain gradient theory (NSGT) taking into account the length scale and nonlocal parameters has been adopted to establish a scale dependent model of metal foam nanoscale plates. Von Karman nonlinear strains are then used to take up the geometric nonlinearity. Different pore dispersions, namely uniform, symmetric and asymmetric, are confirmed. By using the principle of virtual work, nonlinear governing equations are derived and then solved by using an isogeometric analysis and iterative Newton-Raphson method. Influences of the length scale parameter, porosity distributions, nonlocal parameter and nanoporous coefficient on the nonlinear deflection of the plate are numerically experimented in detail. Some findings would play an important role for designing metal foam structures.

In this work, we propose a topology optimization (TO) method using fully adaptive truncated hierarchical B-splines (ATHB-TO), where the adaptive isogeometric analysis mesh and design mesh of ATHB-TO are locally refined and coarsened simultaneously. To meet the demands of full-mesh adaptivity of ATHB-TO, a marking strategy is established based on density variances between adjacent active elements. Then, an adaptive formulation for the mark strategy is proposed to avoid the overrefinement phenomenon that occurred in the adaptive TO method using a fixed mark strategy. Moreover, a density inheritance strategy is devised to project original design variables into design variables concerning a fully adaptive hierarchical mesh. Afterward, ATHB-TO with an adaptive mark strategy is successfully applied to two-dimensional and three-dimensional benchmarks. Numerical results show that the ATHB-TO method can obtain identical optimized designs at a lower computational burden, and improve the convergence rate compared with THB-TO with only hierarchical local refinement considered. Adopting the adaptive mark strategy releases the full potential of ATHB-TO in improving the computational efficiency compared with the fixed mark strategy. Additionally, the adaptive TO schemes outperform TO performed on uniformly refined meshes due to the enhanced numerical efficiency and improved structural performance with a largely accelerated convergence rate. Therefore, ATHB-TO is an effective way of solving structural TO problems.

Overlapping Additive Schwarz (OAS) preconditioners are here constructed for isogeometric collocation discretizations of the system of linear elasticity in both two and three space dimensions. Isogeometric collocation methods are recent variants of isogeometric analysis based on the numerical approximation of the strong form of partial differential equations at appropriate collocation points. Numerical results in two and three dimensions show that two-level OAS preconditioners are scalable in the number of subdomains N, quasi-optimal with respect to the mesh size h and optimal with respect to the spline polynomial degree p. Moreover, two-level OAS preconditioners are more robust than one-level OAS and non-preconditioned GMRES solvers when the material tends to the incompressible limit, as well as in the presence of strong deformation of the NURBS geometry.

This paper proposes a unified solution for in-plane vibration analysis of laminated arbitrary polygonal plates based on the Chebyshev polynomial method. As an innovative point of present work, the ...

Purpose This study aims to develop a new analysis approach devised by incorporating a gradient-enhanced microplane damage model (GeMpDM) into isogeometric analysis (IGA), which shows computational stability and capability in accurately predicting crack propagations in structures with complex geometries. Design/methodology/approach For the non-local microplane damage modeling, the maximum modified von-Mises equivalent strain among all microplanes is regularized as a representative quantity. This characterization implies that only one additional governing equation is considered, which improves computational efficiency dramatically. By combined use of GeMpDM and IGA, quasi-static and dynamic numerical analyses are conducted to demonstrate the capability in predicting crack paths of complex geometries in comparison to FEM and experimental results. Findings The implicit scheme with the adopted damage model shows favorable numerical stability and the numerical results exhibit appropriate convergence characteristics concerning the mesh size. The damage evolution is successfully controlled by a tension-compression damage factor. Thanks to the advanced geometric design capability of IGA, the details of crack patterns can be predicted reliably, which are somewhat difficult to be acquired by FEM. Additionally, the damage distribution obtained in the dynamic analysis is in close agreement with experimental results. Originality/value The paper originally incorporates GeMpDM into IGA. Especially, only one non-local variable is considered besides the displacement field, which improves the computational efficiency and favorable convergence characteristics within the IGA framework. Also, enjoying the geometric design ability of IGA, the proposed analysis method is capable of accurately predicting crack paths reflecting the complex geometries of target structures.

This paper presents a non-classical refined shear deformation theory model in conjunction with the isogeometric analysis for the static bending, free vibration, and buckling behaviors of multilayered microplates. The modified couple stress theory is used to account for the small-scale effect. Taking a five-layer (Al, P3HT: PCBM, PEDOT: PSS, ITO, and Glass) organic solar cell as an example, it is found that the small-scale effects lead to a decrease in deflection, but an increase in the natural frequency and buckling load. With consideration of the size effect (l/h = 1), the stresses are almost 5 times as much as that without the size effect (l/h = 0). This is why the size effect should be taken into account. Besides, the maximum tensile stress occurs in the ITO layer, which is the dangerous layer. In addition, the normalized deflections increase with increasing aspect ratio, but the normalized natural frequencies and normalized buckling loads decrease with increasing aspect ratio.

A new director-based finite element formulation for geometrically exact beams is proposed by weak enforcement of the orthonormality constraints of the directors. In addition to an improved numerical performance, this formulation enables the development of two more beam theories by adding further constraints. Thus, the paper presents a complete intrinsic spatial nonlinear theory of three kinematically different beams which can undergo large displacements and which can have precurved reference configurations. Moreover, the hyperelastic constitutive laws allow for elastic finite strain material behavior of the beams. Furthermore, the numerical discretization using concepts of isogeometric analysis is highlighted in all clarity. Finally, all presented models are numerically validated using exclusive analytical solutions, existing finite element formulations, and a complex dynamical real-world example.

It is well-known that the finite difference discretization of the Laplacian eigenvalue problem $-\Delta u = \lambda u$ leads to a matrix eigenvalue problem (EVP) $A x= \lambda x$ where the matrix $A$ is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in Strang and MacNamara \cite{strang2014functions}. We generalize the results and develop analytical solutions to the generalized matrix eigenvalue problems (GEVPs) $A x= \lambda Bx$ which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities., Comment: 22 pages

In this article, we focus on a design problem of planar compliant mechanisms within the framework of isogeometric topology optimization. An integrated model is developed to identify the optimal deformation transferring path for precise motion output. The model comprises two coupled computational layers: the upper layer for geometry representation and the lower layer for sensitivity calculation. In the upper layer, the structural geometries are described explicitly by parameterized level-set surfaces, which is quite different from the implicit description way of topology optimization, giving great advantages to improve the identification accuracy of elaborate structures (e.g., flexible hinges), which are usually encountered in compliant mechanism systems. By moving, deforming, overlapping and merging these level-set surfaces, the generated shape can be projected onto the lower layer which is discretized using a NURBS patch, and NURBS-based isogeometric analysis is adopted to calculate the structural sensitivity which is then fed back to the upper layer for driving new iteration. The proposed method has the higher ability to search the optimal topology with complex kinematic behavior with respect to the conventional topology optimization methods in terms of computational effectiveness and numerical robustness. Design formulation of compliant mechanisms is constructed under the framework of the proposed method, in which the Jacobian and stiffness matrices of compliant mechanism are optimized simultaneously to achieve kinematic and stiffness requirement respectively. Three typical benchmark problems (e.g., displacement inverter, amplifier, and redirector) are tested to demonstrate these advantages.

This work models spatially uncorrelated (independent) load uncertainty and develops a reduced-order Monte Carlo stochastic isogeometric method to quantify the effect of the load uncertainty on the structural response of thin shells and solid structures. The approach is tested on two demonstrative applications of uncertainty, namely, spatially uncorrelated loading, with (1) Scordelis–Lo Roof shell structure, and (2) a 3D wind turbine blade. This work has three novelties. Firstly, the research models spatially uncorrelated (independent) load uncertainties (including both their magnitude and/or direction) using stochastic analysis. Secondly, the paper advances a reduced-order Monte Carlo stochastic isogeometric method to quantify the spatially uncorrelated load uncertainty. It inherits the merits of isogeometric analysis, which enables the precise representation of geometry and alleviates shell shear locking, thereby reducing the model’s uncertainties. Moreover, the method retains the generality and accuracy of classical Monte Carlo simulation (MCS), with significant efficiency gains. The demonstrative results suggest that there is a cost, which is 3% of the time used by the standard MCS. Furthermore, a significant observation is made from the conducted numerical tests. It is noticed that the standard deviation of the output (i.e., displacement) is strongly influenced when the load uncertainty is spatially uncorrelated. Namely, the standard derivation (SD) of the output is roughly 10 times smaller than the SD for correlated load uncertainties. Nonetheless, the expected values remain consistent between the two cases.

In the context of isogeometric analysis (IGA) of shell structures, the popularity of the solid-shell elements benefit from formulation simplicity and full 3D stress state. However some basic questions remain unresolved when using solid-shell element, especially for large deformation cases with patch coupling, which is a common scene in real-life simulations. In this research, after introduction of the solid-shell nonlinear formulation and a fundamental 3D model construction method, we present a non-symmetric variant of the standard Nitsche’s formulation for multi-patch coupling in association with an empirical formula for its stabilization parameter. An selective and reduced integration scheme is also presented to address the locking syndrome. In addition, the quasi-Newton iteration format is derived as solver, together with a step length control method. The second order derivatives are totally neglected by the adoption of the non-symmetric Nitsche’s formulation and the quasi-Newton solver. The solid-shell elements are numerically studied by a linear elastic plate example, then we demonstrate the performance of the proposed formulation in large deformation, in terms of result verification, iteration history and continuity of displacement across the coupling interface. In the context of isogeometric analysis (IGA), after introduction of the solid-shell nonlinear formulation and a fundamental 3D model construction method, we present a non-symmetric variant of the standard Nitsche’s formulation for multi-patch coupling in association with an empirical formula for its stabilization parameter. An selective and reduced integration scheme is also introduced to address the locking syndrome. In addition, the quasi-Newton iteration format is derived as solver, together with a step length control method. The second order derivatives are totally neglected by the adoption of the non-symmetric Nitsche’s formulation and the quasi-Newton solver. The performance of the proposed formulation in large deformation is demonstrated by several examples.

Recently, isogeometric methodology has been successfully implemented in one-step inverse analysis of sheet metal stamping processes. However, these models are not capable of analyzing forming processes which require severe deformation and/or several forming stages. This paper presents a multi-step inverse isogeometric methodology to enhance the precision of one-step models in predictions of the initial blank, strain distributions, and drawability of the formed parts. This methodology deals with the minimization of potential energy, deformation theory of plasticity, and considering membrane elements. The presented methodology utilizes the NURBS basis functions to create the final, middle, and blank geometries, and also to analyze sheet metal deformation. The characteristics of the applied formulations make it possible to simultaneously observe the effects of changing part parameters on its formability. One advantage of this approach is that the linear system of governing equations is solved without concerning about the convergence. In addition, the presented methodology is able to successfully generate the middle geometry and to restrict the movements of physical nodes along the middle surface, by presenting a new NURBS-based mapping and sliding constraint technique. The performance of the presented model is evaluated under two classical problems including forming of a rectangular box and a two-step drawing of a circular cup. In forming of the rectangular box, the results of the presented model are compared with those of experiment and forward FEM simulation. Also, in the second problem, the presented approach is only compared with forward FEM simulation. Results comparisons indicate the credibility of the presented model in prediction of forming parameters at a low computation time.

The isogeometric analysis boundary element method (IGABEM) has great potential in the simulation of heat conduction problems due to its exact geometric representation and good approximation properties. In this paper, the radial integration IGABEM (RI-IGABEM) is proposed to solve isotropic heat conduction problems in inhomogeneous media with heat source. Similar to traditional BEM, the domain integrals cannot be avoided since the foundational solution for the Laplace equation is used to derive integral equation. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral into an equivalent boundary integral. In addition, using a simple transformation method, the uniform potential method is successfully applied to solve the strongly singular integrals, and the Telles scheme and the element sub-division method are used to solve the weakly singular integrals in RI-IGABEM respectively. In order to validate the accuracy and convergence of the RI-IGABEM in the analysis of the single or multiple boundary heat conduction problems, several 2D and 3D numerical examples are used to discuss the influence of some factors, such as the number of applied points, the order of basis functions, and the position of internal applied points.

This paper presents an enhanced version of our previous work, hybrid non-uniform subdivision surfaces [19], to achieve optimal convergence rates in isogeometric analysis. We introduce a parameter $\lambda$ ($\frac{1}{4}

This study proposes a new plate formulation based on a generalized layerwise theory in the isogeometric analysis (IGA) framework to investigate the bending and free vibration behavior of a laminated composite plate. All degrees of freedom in the isogeometric formulation are only associated with the displacements (i.e., rotation-free). In IGA, NURBS functions are used both to construct the geometry and to approximate the field variables. Merits of utilizing IGA include: the exact geometrical description, high accuracy and efficiancy, higher-order smoothness. The displacement-based layerwise theory assumes an individual displacement field expansion inside each layer, and the transverse displacement component is regarded to be $$C^{0}$$ -continuous at the layer interfaces, thus yielding a more precise description of the stress states. The $$C^{0}$$ continuity enforcement across the thickness in the present formulation is easily implemented by virtue of the knot insertion technique of IGA. The capability of the present isogeometric layerwise formulation in analyzing the static and dynamic responses is evidenced by conducting some numerical tests available in the open literature and comparing the computed outcomes with the reference data.

In this paper, we are interested in high-order algorithms for time discretization and focus upon the high-order implicit/explicit algorithm designs. Five high-order unconditionally stable implicit algorithms, derived by the time continuous Galerkin method, weighting parameter method, collocation method, differential quadrature method, and the modified time-weighted residual method, in first-/second-order transient systems are taken into consideration. The present overview and contributions encompass: (1) The pros and cons of various methodologies for the design of high-order algorithms are first demonstrated. Generally, p unknown variables leads to the optimized $$(2p-1)$$ th-order accurate algorithms with controllable numerical dissipation, and/or 2pth-order accurate algorithms without controllable numerical dissipation. (2) Although it is claimed that the TCG method can achieve 2pth-order accuracy with controllable numerical dissipation, it will be shown in this paper that the conclusion was arrived via an inconsistent analysis for the accuracy and the controllable numerical dissipation. (3) Given the rapid increase on the computational cost for high-order algorithms, the iterative predictor/multi-corrector technique is applied to show a novel design for the high-order explicit algorithms derived from the high-order implicit algorithms for the first-order transient systems. Coupled with the high-order Legendre SEM (likewise isogeometric analysis, DG methods, p-version FEM, etc., can be employed) for the spatial discretization, this newly proposed explicit numerical framework can achieve and preserve high-order accuracy in both space and time. In comparison to the famous explicit Runge-Kutta method, these newly designed explicit algorithms have better solution accuracy with comparable stability region.

We present an efficient isogeometric analysis (IGA) formulation for nonlinear vibration problems of viscoelastic plates. The formulation is based on higher-order shear deformation theory (HSDT), which is guaranteed by p -order nonuniform rational B-splines (NURBS) resulting in C p − 1 continuity in the domain. The governing equations of motion are derived from Hamilton’s principle. The generalized multi-axial Maxwell model is used to express the viscoelastic constitutive equations in global coordinates. The performance of developed approach is demonstrated through several numerical examples. We show that our approach yields more realistic solutions than conventional FEA, especially for coarse discretizations. We find that C 3 -NURBS provide higher accuracy concerning the nonlinear vibration amplitude than C 2 -NURBS. Viscoelastic plates have several active vibration modes adjacent to the fundamental oscillation mode due to the combination of long-time creep and short-time shear moduli. Furthermore, the resultant creep strain leads to large deformations of viscoelastic plates and intensifies the dynamic stress concentration factors. The creep deflection of the plates rapidly increases with damping under sustained loads when the support restrains are released. We observe that the long-time stress-to-thickness ratio is constant for various slenderness ratios, which is relevant during creep.

Shape optimization based on isogeometric analysis (IGA) has gained popularity in recent years. Performing shape optimization directly over parameters defining the computer-aided design (CAD) geometry, such as the control points of a spline parametrization, opens up the prospect of seamless integration of a shape optimization step into the CAD workflow. One of the challenges when using IGA for shape optimization is that of maintaining a valid geometry parametrization of the interior of the domain during an optimization process, as the shape of the boundary is altered by an optimization algorithm. Existing methods impose constraints on the Jacobian of the parametrization, to guarantee that the parametrization remains valid. The number of such validity constraints quickly becomes intractably large, especially when 3D shape optimization problems are considered. An alternative, and arguably simpler, approach is to formulate the isogeometric shape optimization problem in terms of both the boundary and the interior control points. To ensure a geometric parametrization of sufficient quality, a regularization term, such as the Winslow functional, is added to the objective function of the shape optimization problem. We illustrate the performance of these methods on the optimal design problem of electromagnetic reflectors and compare their performance. Both methods are implemented for multipatch geometries, using the IGA library G+Smo and the optimization library Ipopt. We find that the second approach performs comparably to a state-of-the-art method with respect to both the quality of the found solutions and computational time, while its performance in our experience is more robust for coarse discretizations.

A new stochastic isogeometric analysis method based on reduced basis vectors (SRBIGA) is proposed for engineering structures with random field material properties and external loads. Based on the Galerkin isogeometric functions, the proposed SRBIGA applies the Karhunen–Loeve expansion to discretize the random field uncertainties. Inspired by the stochastic Krylov subspace theory, the structural responses of linear elasticity structures with random field uncertainties are represented based on the reduced basis vectors. The tremendous advantage of SRBIGA over the spectral stochastic isogeometric analysis (SSIGA) in terms of the computational efficiency is disclosed through the comparison analysis in theoretical aspects. Three illustrative examples demonstrate that the proposed SRBIGA has not only significantly higher efficiency but also higher accuracy and better robustness than the SSIGA and that it can provide a novel and expedient stochastic structural analysis method for practical large-scale complex engineering structures when both material properties and external loads are spatially random.

Isogeometric analysis (IGA) concept is developed as the new numerical approach that reduces the gap between design and analysis by incorporating them both into a unified and single process. In this paper the application of this advanced numerical approach in analyzing composite plates based on different theories have been reviewed. Further, in the review development of different continuum theories based on IGA are also discussed. Firstly, incorporation of different equivalent single layer theories (ESL) with IGA have been discussed. Secondly, due to the consideration of linear displacement throughout the plates in ESL, discussion on IGA based layer wise theories (LWT) have been presented which includes non-linear displacements in each layer of composite plates. The main advantage behind incorporating IGA with these continuum theories is that the higher order and continuity basis function of IGA fulfils need of various continuum theories that results in an accurate and exact analysis of the composite plates. Also, the ease in increasing order of basis function in IGA helps to solve the composite plate problem very efficiently. Based on the advantages the calculated results of various IGA based studies were found to be in good agreement with the already existing numerical methods.

The numerical integration over a planar domain that is cut by an implicitly defined boundary curve is an important problem that arises, for example, in unfitted finite element methods and in isogeo...

The bonded repair techniques seem to be the most frequent procedures in the aviation maintenance. The achieved composite repaired perforated thin-walled plate is a complex geometry with high numerical analysis cost. The NURBS-based Isogeometric Analysis (IGA) proposes a sensible and affordable tool to carry out such geometry analysis. In this context, a well-known technique is to divide the original geometry assembly into number of simple neighbors connected geometries. In the present study the free vibration analysis of the perforated plates repaired on one side with an external bonded composite laminated patch is investigated. A multi-patch geometry modeling approach is implemented in line with the first order shear deformation theory of plates. In order to hold the geometry integrity and uniformity, all the degrees of freedom between adjacent geometry patches are completely tied through implementing a Nitsche method. To show the effectiveness and accuracy of the developed formulation, some representative results are extracted and compared with those from literature. The effects of geometrical as well as material parameters including boundary condition, cutout shape, and repair layup on the dynamic response of the repaired perforated plates are then investigated.

An isogeometric boundary element method (IGABEM) is proposed for solving the steady-state heat transfer problems with concentrated/surface internal heat sources. The isogeometric boundary element method (IGABEM) possesses the advantages of both the isogeometric analysis (IGA) and the boundary element method (BEM), the non-uniform rational B-spline (NURBS) basis functions used in the (computer-aided design) CAD system are flexible and stable in dealing with irregular boundaries, due to which the NURBS basis functions are applied to exactly reconstruct the boundary geometry of the analysis domain, and the physical variables are also approximated by the NURBS as the standard IGA does. Dirac delta function is introduced in the present IGABEM for dealing with the concentrated point heat sources, and a local coordinate system is proposed for the domain integration over line heat source based on the NURBS basis functions. As to the domain integrals caused by the surface heat sources, the radial integration method (RIM) is used to transform the domain integrals to boundary ones without any internal cells for the original version of IGABEM. Several numerical examples involving Dirichlet, Neumann and Robin boundary conditions are analyzed, and circular region with point/line heat source(sources), global and local surface heat sources within multiply connected regions are also considered. Verification of the proposed method has been gained by comparing the present results with those obtained by several other researchers, and solutions achieved by analytical expression together with the fine-meshed ANSYS model are also utilized for the purpose of comparison. Meanwhile, the computational efficiency and convergence of the present method is evaluated by comparing the IGABEM with the finite element method (FEM) as well as the traditional BEM. Good performance of the IGABEM is observed. Finally, heat transfer within a simplified printed circuit board (PCB) with both concentrated (line and point) and surface heat generations is studied, the temperature distribution is achieved, which verifies the applicability of the proposed technoque in practical engineering.

An important ingredient of any moving-mesh method for fluid-structure interaction (FSI) problems is the mesh moving technique (MMT) used to adapt the computational mesh in the moving fluid domain. An ideal MMT is computationally inexpensive, can handle large mesh motions without inverting mesh elements and can sustain an FSI simulation for extensive periods of time without irreversibly distorting the mesh. Here we compare several commonly used MMTs which are based on the solution of elliptic partial differential equations, including harmonic extension, bi-harmonic extension and techniques based on the equations of linear elasticity. Moreover, we propose a novel MMT which utilizes ideas from continuation methods to efficiently solve the equations of nonlinear elasticity and proves to be robust even when the mesh undergoes extreme motions. In addition to that, we study how each MMT behaves when combined with the mesh-Jacobian-based stiffening. Finally, we evaluate the performance of different MMTs on a popular two-dimensional FSI benchmark reproduced by using an isogeometric partitioned solver with strong coupling.

In this paper we introduce a C 1 spline space over mixed meshes composed of triangles and quadrilaterals, suitable for FEM-based or isogeometric analysis. In this context, a mesh is considered to be a partition of a planar polygonal domain into triangles and/or quadrilaterals. The proposed space combines the Argyris triangle, cf. Argyris et al. (1968), with the C 1 quadrilateral element introduced in Brenner and Sung (2005), Kapl et al. (2020) for polynomial degrees p ≥ 5 . The space is assumed to be C 2 at all vertices and C 1 across edges, and the splines are uniquely determined by C 2 -data at the vertices, values and normal derivatives at chosen points on the edges, and values at some additional points in the interior of the elements. The motivation for combining the Argyris triangle element with a recent C 1 quadrilateral construction, inspired by isogeometric analysis, is two-fold: on one hand, the ability to connect triangle and quadrilateral finite elements in a C 1 fashion is non-trivial and of theoretical interest. We provide not only approximation error bounds but also numerical tests verifying the results. On the other hand, the construction facilitates the meshing process by allowing more flexibility while remaining C 1 everywhere. This is for instance relevant when trimming of tensor-product B-splines is performed. In the presented construction we assume to have (bi)linear element mappings and piecewise polynomial function spaces of arbitrary degree p ≥ 5 . The basis is simple to implement and the obtained results are optimal with respect to the mesh size for L ∞ , L 2 as well as Sobolev norms H 1 and H 2 .

In this article, the condition number of the stiffness matrix κ ( K ) is compared for three high order finite element methods (FEMs), i.e., the p-version of the FEM, the spectral element method (SEM), and the NURBS-based isogeometric analysis (IGA). Note that only problems in linear elasticity are considered in the analysis. It is well-known that the condition number is one factor strongly influencing the number of significant digits for direct solvers or the required iteration count for iterative solution schemes. Therefore, it is important to investigate the effect of the choice of the shape functions and the element distortion on κ ( K ) . Based on numerous one- and two-, and three-dimensional examples, these influences are comprehensively studied, and the numerical results are compared with condition number estimates extracted from the literature. Overall, a good agreement is observed for p-FEM, SEM and two-dimensional IGA, while discrepancies are noted for three-dimensional isogeometric elements. These are important findings as theoretical results may be only available for very restricted scenarios, where one-element geometries, constant Jacobi matrices of the element maps, etc. are considered.

Adaptive isogeometric methods for the solution of partial differential equations rely on the construction of locally refinable spline spaces. A simple and efficient way to obtain these spaces is to apply the multi-level construction of hierarchical splines, that can be used on single-patch domains or in multi-patch domains with C 0 continuity across the patch interfaces. Due to the benefits of higher continuity in isogeometric methods, recent works investigated the construction of spline spaces with global C 1 continuity on two or more patches. In this paper, we show how these approaches can be combined with the hierarchical construction to obtain global C 1 continuous hierarchical splines on two-patch domains. A selection of numerical examples is presented to highlight the features and effectivity of the construction.

We consider two-point, reaction–diffusion type, singularly perturbed boundary value problems of order 2 ν ∈ Z + , and the approximation of their solution using isogeometric analysis. In particular, we use a Galerkin formulation with B-splines as basis functions, defined using appropriately chosen knot vectors. We prove robust exponential convergence in the energy norm, independently of the singular perturbation parameter. Numerical examples are also presented, which illustrate (and extend) the theory.

The paper is aimed at enhancing computational performance for optimizing the material distribution of tri-directional functionally graded (FG) plates. We exploit advantages of using a non-uniform rational B-spline (NURBS) basis function for describing material distribution varying through all three directions of functionally graded (FG) plates. Two-dimensional free vibration and buckling behaviors of multi-directional (1D, 2D and 3D) FG plates analyzed by using a combination of generalized shear deformation theory (GSDT) and isogeometric analysis (IGA) is first proposed. This approach can help to save a significant amount of computational cost while still ensure the accuracy of the solutions. The effectiveness and reliability of the present method are demonstrated by comparing it to other methods in the literature. The obtained results are in excellent agreement with the reference ones. More importantly, data sets consisting of input-output pairs are randomly generated from the analysis process through iterations for the training process in deep neural networks (DNN). DNN is utilized as an analysis tool to supplant finite element analysis to reduce computational cost. By using DNN, behaviors of the multi-directional FG plates are directly predicted from those material distributions. Optimal material distributions of tri-directional FG plates under free vibration or compression in various volume fraction constraints are found by using modified symbiotic organisms search (mSOS) algorithm for the first time. Moreover, an isogeometric multimesh design technique is also used to diminish a large number of design variables in optimization. Optimal results obtained by DNN are compared with those of IGA to verify the effectiveness of the proposed method.

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.

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.

We present computational flow analysis of a vertical-axis wind turbine (VAWT) that has been proposed to also serve as a tsunami shelter. In addition to the three-blade rotor, the turbine has four support columns at the periphery. The columns support the turbine rotor and the shelter. Computational challenges encountered in flow analysis of wind turbines in general include accurate representation of the turbine geometry, multiscale unsteady flow, and moving-boundary flow associated with the rotor motion. The tsunami-shelter VAWT, because of its rather high geometric complexity, poses the additional challenge of reaching high accuracy in turbine-geometry representation and flow solution when the geometry is so complex. We address the challenges with a space–time (ST) computational method that integrates three special ST methods around the core, ST Variational Multiscale (ST-VMS) method, and mesh generation and improvement methods. The three special methods are the ST Slip Interface (ST-SI) method, ST Isogeometric Analysis (ST-IGA), and the ST/NURBS Mesh Update Method (STNMUM). The ST-discretization feature of the integrated method provides higher-order accuracy compared to standard discretization methods. The VMS feature addresses the computational challenges associated with the multiscale nature of the unsteady flow. The moving-mesh feature of the ST framework enables high-resolution computation near the blades. The ST-SI enables moving-mesh computation of the spinning rotor. The mesh covering the rotor spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-IGA enables more accurate representation of the blade and other turbine geometries and increased accuracy in the flow solution. The STNMUM enables exact representation of the mesh rotation. A general-purpose NURBS mesh generation method makes it easier to deal with the complex turbine geometry. The quality of the mesh generated with this method is improved with a mesh relaxation method based on fiber-reinforced hyperelasticity and optimized zero-stress state. We present computations for the 2D and 3D cases. The computations show the effectiveness of our ST and mesh generation and relaxation methods in flow analysis of the tsunami-shelter VAWT.

This paper presents an isogeometric analysis with adaptivity using locally refined B-splines (LR B-splines) for steady-state heat conduction simulations in solids. Within this framework, the LR B-splines, which have an efficient and simple local refinement algorithm, are used to represent the geometry, and are also employed for spatial discretization, thus providing a seamless interaction between the CAD models and the numerical analysis. A Zienkiewicz-Zhu a posteriori error estimator in terms of the temperature gradient recovery is used to identify the regions for local mesh refinement. The accuracy and convergence properties of the proposed framework are demonstrated through several two-dimensional isotropic examples. Numerical results indicate good performance of the present method as the adaptive refinement technique yields more accurate solution compared to the uniform global refinement with an improved convergence rate.

This study presents a size dependent model using the higher-order shear deformation theory (HSDT) in conjunction with modified strain gradient theory (MSGT) for free vibration analysis of functionally graded (FG) anisotropic microplates. The FG anisotropic material is made of hexagonal beryllium crystals which can be considered as a hexagonal one. To consider size effects, three material length scale parameters (MLSPs) are added into the elastic constants of the anisotropic material. Based on the principle of virtual work, discretized governing equations of the FG hexagonal microplates are obtained. Subsequently, the natural frequency of the FG anisotropic microplates is determined by using isogeometric analysis (IGA). Numerical results show that the natural frequency of the FG anisotropic microplates is influenced by the geometry, boundary condition, length-to-thickness ratio, exponential factor and material length scale parameter. The results of classical HSDT model can be restored from the present model when three MLSPs equal to zero. Moreover, the differences of the natural frequency predicted by MSGT and classical HSDT can grow up more than 4.5 times.

An isogeometric approach for solving the Laplace–Beltrami equation on a two-dimensional manifold embedded in three-dimensional space using a Galerkin method based on Catmull–Clark subdivision surfaces is presented and assessed. The scalar-valued Laplace–Beltrami equation requires only$$C^0$$C0continuity and is adopted to elucidate key features and properties of the isogeometric method using Catmull–Clark subdivision surfaces. Catmull–Clark subdivision bases are used to discretise both the geometry and the physical field. A fitting method generates control meshes to approximate any given geometry with Catmull–Clark subdivision surfaces. The performance of the Catmull–Clark subdivision method is compared to the conventional finite element method. Subdivision surfaces without extraordinary vertices show the optimal convergence rate. However, extraordinary vertices introduce error, which decreases the convergence rate. A comparative study shows the effect of the number and valences of the extraordinary vertices on accuracy and convergence. An adaptive quadrature scheme is shown to reduce the error.

In this paper, we propose a multigrid method for the solution of the biharmonic problem. Isogeometric Analysis (IGA) is considered in order to easily obtain H 2 -conforming discretizations of the bilaplacian equation, which are difficult to get by means of standard finite element methods. Typically, the design of solvers for isogeometric discretizations that are robust with respect to the polynomial degree is a challenging task. Here, we achieve such robustness by using multiplicative Schwarz methods as smoothers within the multigrid algorithm. The design of the proposed solver is also supported by a local Fourier analysis (LFA), which allows us to choose appropriately the size of the block in the smoother depending on the polynomial degree of the discretization. The robustness and efficiency of the proposed multigrid method is demonstrated through numerical experiments in one- and two-dimensional cases.

This paper concerns a novel isogeometric Timoshenko beam formulation for a geometrically nonlinear analysis of spatial beams using the total Lagrangian description. Constitutive laws for hyperelastic materials, whose behavior varies with their deformations, are widely defined by using strain energy density functions that are written in terms of the Green–Lagrange strain tensor. Many finite element beam formulations for geometrically nonlinear analyses of spatial beams are developed using the Green–Lagrange strain tensor and its energy conjugate, the second Piola–Kirchhoff stress tensor. Unfortunately, there virtually exist no isogeometric Timoshenko beam formulations for this type of analysis that are derived by using this energy conjugate pair. To allow the possibility of considering hyperelastic materials, the present isogeometric beam formulation is developed in the total Lagrangian description using the Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor. The proposed formulation is capable of simulating beam structures that are subjected to large displacements and rotations, without any restriction in magnitude. Three-dimensional beam configurations are reduced into one-dimensional structures using the beam axis and director vectors of the cross sections. The cross-sectional rotation along the beam axis is represented by an orthogonal tensor, which is parameterized by a vector-like parameter. Updating the cross-sectional rotations is performed purely by natural exponentiation and superposition of relevant rotational quantities. To show the accuracy and efficiency of the proposed beam formulation, some benchmark and well-established numerical examples with various types of beam, i.e., straight, curved, pre-twisted beams, are analyzed. The obtained results are compared with those results in the literature, obtained from both analytical and numerical methods.

This paper aims to study the buckling and bending analysis of curvilinearly stiffened plates by using an isogeometric approach. First-order shear deformation theory is adopted to model the behaviors of stiffened plates. Based on the employed approach, it is easy to model stiffeners with different sizes and layouts, since the degrees of freedom for the whole structure are integrated into those of the original plate leading to a relatively low computational cost. The performance of the proposed approach is validated via a couple of benchmark problems available in the literature, and the effect of different parameters including bending stiffness, relative mass, cross section, and shape of stiffener on the buckling and bending behavior of curvilinearly stiffened plates is investigated. The results show the relative high accuracy of this method using fewer degrees of freedom for stiffeners, which is beneficial in optimization procedures.

In this study, a three-dimensional discontinuous Galerkin isogeometric analysis framework is presented for the analysis of composite laminates. Non-uniform rational B-splines are employed as basis functions for both geometric and computational implementations. From a practical point of view, modeling with multiple non-uniform rational B-spline patches is required in many different applications due to the complexity of computational domains. Then, a special numerical technique is necessary to couple different non-uniform rational B-spline patches to carry out the isogeometric analysis. In this study, therefore, one of the discontinuous Galerkin methods, namely, symmetric interior penalty Galerkin formulation is utilized to deal with multi-patch isogeometric analysis applications. In order to show the applicability of the proposed framework, composite laminates under sinusoidally distributed load with different stacking sequences are studied in the numerical examples. The predicted results are compared with those obtained by the three-dimensional elasticity solutions and various numerical models available in the literature.

For wind turbines operating in cold weather conditions, ice accretion is an established issue that remains an obstacle in effective turbine operation. While the aerodynamic performance of wind turbine blades with ice accretion has received considerable research attention, few studies have investigated the structural impact of blade ice accretion. This work proposes an adaptable projection-based method to superimpose complex ice configurations onto a baseline structure. The proposed approach provides an efficient methodology to include ice accretion in the high-fidelity isogeometric shell analysis of a realistic wind turbine blade. Linear vibration and nonlinear deflection analyses of the blade are performed for various ice configurations to demonstrate the impact of different ice accretion distributions on structural performance. These analyses indicate decreases in the blade natural frequencies and deflection under icing conditions. Such ice-induced changes clearly reveal the need for structural design consideration for turbines operating under icing conditions.

In the present study, the acoustic attenuation performances of mufflers are predicted by employing three-dimensional isogeometric analysis (IGA) and multi-patch technique for the first time. Circular, elliptical and conical reactive expansion chamber mufflers are exactly parameterized by non-uniform rational B-splines (NURBS), and the Helmholtz equation governing the interior acoustic field is also solved by NURBS. Compared with traditional finite element method (FEM) which is rooted in Lagrange and Hermite functions, IGA can effectively avoid the time-consuming meshing and totally preserve exact geometrical information during the modelling-analysis process. The computational effectiveness of multi-patch IGA for calculating transmission loss is verified by comparing with experimental approach, boundary element method and other numerical methods from available research articles. The effects of geometry, such as shapes of the expansion cavity and lengths of the extended inlet/outlet are discussed in detail.

Injective parametrizations are widely used both in theory and in applications. The injectivity of parameteric curves and surfaces means that there are no self-intersections. Toric surface patch is defined by a set of integer lattice points A and corresponding control points and weights, which includes rational tensor product and triangle Bezier patches as special cases. In 2011, Sottile and Zhu presented a geometric method to check the injectivity of 2D toric surface patches. In this paper, we present an improved algorithm of their method. The complexity of the improved algorithm is reduced from O ( n 3 ) to O ( n 2 ) , where n = # ( A ) . Some examples are shown to illustrate the effectiveness of our algorithm. Moreover, the algorithm is also applied to check the injectivity of parameterization in isogeometric analysis.

This paper presents a new total Lagrangian Timoshenko beam formulation with the isogeometric analysis approach for a geometrically nonlinear analysis of planar curved beams. The proposed formulation is derived from the two-dimensional continuum theory, and a beam configuration is characterized by the beam axis and the director vectors of cross sections. Non-uniform rational B-spline (NURBS) curves are used for the geometric representation of the beam axis and the discretization of the unknown kinematics, i.e., the translational displacements of the beam axis and the cross-sectional rotation angle. Several well-established examples covering the analysis of many types of beam, i.e., straight, curved, and free-form beams with varying curvature, are considered. The obtained results by the proposed formulation are compared with those in the literature, and the accuracy and efficiency of the proposed formulation are assessed. The prominent properties of using NURBS curves in analysis, i.e., effective reduction in locking effects, higher accuracy per degree-of-freedom, and better geometric representations, are also verified for the proposed beam formulation.

In this paper, a new approach is developed for applications of shape optimization on the two-dimensional time harmonic wave propagation (Helmholtz equation) in acoustic problems. The particle swarm optimization (PSO) algorithm - a gradient-free optimization method avoiding the sensitivity analysis - is coupled with two boundary element methods (BEM) and isogeometric analysis (IGA). The first method is the conventional isogeometric boundary element method (IGABEM). The second method is the eXtended IGABEM (XIBEM) enriched with the partition-of-unity expansion using a set of plane waves. In both methods, the computational domain is parameterized and the unknown solution is approximated using non-uniform rational B-splines basis functions (NURBS). In the optimization models, the advantage of IGA is the feature of representing the three models; i.e. shape design/analysis/optimization, using a set of control points, which also represent control variables and optimization parameters, making communication between the three models easy and straightforward. A numerical example is considered for the duct problem to validate the presented techniques against the analytical solution. Furthermore, two different applications for various frequencies are studied; the vertical noise barrier and the horn problems, and the obtained results are compared against previously published numerical methods using sensitivity analysis and genetic algorithms to verify the efficiency of the proposed approaches.

In structural design optimization involving transient responses, time integration scheme plays a crucial role in sensitivity analysis because it affects the accuracy and stability of transient analysis. In this work, the influence of time integration scheme is studied numerically for the adjoint shape sensitivity analysis of two benchmark transient heat conduction problems within the framework of isogeometric analysis. It is found that (i) the explicit approach (β = 0) and semi-implicit approach with β < 0.5 impose a strict stability condition of the transient analysis; (ii) the implicit approach (β = 1) and semi-implicit approach with β > 0.5 are generally preferred for their unconditional stability; and (iii) Crank-Nicolson type approach (β= 0.5) may induce a large error for large time-step sizes due to the oscillatory solutions. The numerical results also show that the time-step size does not have to be chosen to satisfy the critical conditions for all of the eigen-frequencies. It is recommended to use β ≈ 0.75 for unconditional stability, such that the oscillation condition is much less critical than the Crank-Nicolson scheme, and the accuracy is higher than a fully implicit approach.

Purpose This study aims to present a new hybrid formulation based on non-uniform rational b-splines functions and enrichment strategies applied to free and forced vibration of straight bars and trusses. Design/methodology/approach Based on the idea of enrichment from generalized finite element method (GFEM)/extended finite element method (XFEM), an extended isogeometric formulation (partition of unity isogeometric analysis [PUIGA]) is conceived. By numerical examples the methods are tested and compared with isogeometric analysis, finite element method and GFEM in terms of convergence, error spectrum, conditioning and adaptivity capacity. Findings The results show a high convergence rate and accuracy for PUIGA and the advantage of input enrichment functions and material parameters on parametric space. Originality/value The enrichment strategies demonstrated considerable improvements in numerical solutions. The applications of computer-aided design mapped enrichments applied to structural dynamics are not known in the literature.

This paper applies the isogeometric analysis (IGA) based on unified one-dimensional (1D) models to study static, free vibration and dynamic responses of metallic and laminated composite straight beam structures. By employing the Carrera Unified Formulation (CUF), 3D displacement fields are expanded as 1D generalized displacement unknowns over the cross-section domain. 2D hierarchical Legendre expansions (HLE) are adopted in the local area for the refinement of cross-section kinematics. In contrast, B-spline functions are used to approximate 1D generalized displacement unknowns, satisfying the requirement of interelement high-order continuity. Consequently, IGA-based weak-form governing equations can be derived using the principle of virtual work and written in terms of fundamental nuclei, which are independent of the class and order of beam theory. Several geometrically linear analyses are conducted to address the enhanced capability of the proposed approach, which is prominent in the detection of shear stresses, higher-order modes and stress wave propagation problems. Besides, 3D-like behaviors can be captured by the present IGA-based CUF-HLE method with reduced computational costs compared with 3D finite element method (FEM) and FEM-based CUF-HLE method.

In this paper, we present a semi-homogenized extended isogeometric analysis approach for assessment of stress intensity factor of center cracked functionally graded domain containing discontinuities such as holes and inclusions. In the semi-homogenized approach, the presence of discontinuities is considered in the 30% region in the vicinity of crack, and the remaining part of the domain is modeled using equivalent homogenous material properties. The Heaviside and singular crack tip enrichment functions are employed to model the center crack, whereas crack topology is tracked using level set functions. The domain form of interaction integral approach is employed to estimate the stress intensity factors at the tips of the center crack. Several problems containing discontinuities in 30% portion of the domain are investigated using the semi-homogenized approach. The efficiency and accuracy of the developed method are examined by comparing the computed results with those attained by modeling discontinuities in the entire domain. The advantages of the semi-homogenized extended isogeometric analysis approach are highlighted in terms of higher computational efficiency.

In this paper, the vibration and buckling analyses of the FGM (functionally graded material) plates with multiple internal cracks and cutouts under thermal and mechanical loads are numerically investigated using the combined XIGA-PHT (extended isogeometric analysis based on PHT-splines) and FCM (finite cell method). Material properties are graded only in the thickness direction. The effective material properties are estimated by using either the rule of mixture or the Mori-Tanaka homogenization technique. The plate displacement field is based on the HSDT (higher-order shear deformation plate theory) without any requirement of the SCF (shear correction factor). The HSDT model can exactly represent the shear stress distribution and improve the accuracy of solutions. The PHT-splines can naturally fulfill the C1-continuous requirement of the HSDT model. The representation of internal defects is mesh-independent. The discontinuous and singular phenomena induced by the cracks are captured using the enrichment pattern in the XIGA, and the influence of cutouts is implemented by the FCM. The geometries of cutouts are captured by means of adaptive quadrature procedure based on a simple unfitted structural mesh, which avoids the need for multiple patches to describe the complex geometry and eliminates the enforcement of C1-continuity patch-coupling across the patch boundaries. The initial mesh density around the cracks and cutouts can be controlled flexibly utilizing the local refinement property of the PHT-splines. After validating the results of the developed approach with those available in the literature, the effects of material gradient index, side to thickness ratio, boundary conditions, cutout size and crack length on the normalized frequency and the critical buckling parameter are investigated. Numerical results illustrate the effectiveness and accuracy of the present approach.