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International audience; Cellular materials, in particular metal or polymer foams and beam lattices, are characterized by a complex architecture where the material is concentrated in slender struts. This study is concerned with the construction of numerically efficient samplespecific digital twin for the analysis and characterization of the mechanical behavior of such meta-materials. A method to build an explicit, light, and analysis-suitable isogeometric beam model directly from digital images is proposed. More precisely, as a main contribution, the Virtual Image Correlation method is extended to the case of multiple interconnected beams with varying cross-sections. Special care is also given to the initialization for better efficiency and robustness. Synthetic and real examples of increasing complexity, i.e. curved beams, lattices, and foams, are presented to account for the performance of the developed workflow from digital images to an isogeometric twin for mechanical modelling and simulation.

This work undertakes to combine the interests of IsoGeometric Analysis (IGA) and standard Finite Element Methods (FEM) for the global/local simulation of structures. The idea is to adopt a hybrid global-IGA/local-FEM modeling, thereby benefiting from: (i) the superior geometric description and per-Degree-Of-Freedom accuracy of IGA for capturing global, regular responses, and (ii) the ability of FEM to compute local, strongly non-linear or even singular behaviors. For the sake of minimizing the implementation effort, we develop a coupling scheme that is fully non-invasive in the sense that the initial global spline model to be enriched is never modified and the construction of the coupling operators can be performed using conventional FE packages. The key ingredient is to express the FEM-to-IGA bridge, based on B{\'e}zier extraction, to transform the initial global spline interface into a FE one on which the local FE mesh can be constructed. This allows to resort to classic FE trace operators to implement the coupling. It results in a strategy that offers the opportunity to simply couple an isogeometric code with any robust FE code suitable for the modelling of complex local behaviors. The method also easily extends in case the users only have at their disposal FE codes. This is the situation that is considered for the numerical illustrations. More precisely, we only make use of the FE industrial software Code Aster to perform efficiently and accurately the hybrid global-IGA/local-FEM simulation of structures subjected locally to cracks, contact, friction and delamination., Comment: Computational Mechanics, Springer Verlag, 2022

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.

In the present study, the stress distribution and first ply failure strength of hybrid laminated composites subjected to uniaxial tensile and compression loads are predicted using Isogeometric analysis (IGA) based on non-uniform rational B-splines technology, commonly referred to as non-uniform rational B-splines (NURBS) technology. The obtained results were compared to finite element method results (ANSYS) published in the literature and analytical results using Autodesk Helius Composite software. Graphite-glass/epoxy hybrid composite plates are analyzed for different lamination orientations and used to calculate first ply failure strength. Non-interactive theories such as maximum stress, maximum strain, and fully interactive criteria such as Tsai-Wu, Tsai-Hill and Hoffman are used to predict the failure strength of different laminates. The failure envelopes of the hybrid composites are developed using the same failure criteria used for the graphite-glass/epoxy hybrid composite. The effects of IGA refinement and plate’s dimension on the present analysis were also established. The IGA results showed good accuracy, robustness and good convergence speed with the analytical software results and the finite element method (FEM) reference value.

In this paper we construct Ritz-type projectors with boundary interpolation properties in finite dimensional subspaces of the usual Sobolev space and we provide a priori error estimates for them. The abstract analysis is exemplified by considering spline spaces and we equip the corresponding error estimates with explicit constants. This complements our results recently obtained for explicit spline error estimates based on the classical Ritz projectors in [Numer. Math. 144(4):889--929, 2020]., Comment: 19 pages, 3 figures. Improved the presentation

This paper focuses on a design problem of branching microchannel heat sink based on solving the thermal-fluid problem in isogeometric analysis approach. A brand new topology optimization framework is proposed to determine the distribution of the microchannels, which comprises two coupled computational layers: the upper layer for channel geometry representation and the lower layer for cooling analysis. In the upper layer, several flexible components fitting two Bezier curves as the central line describe the geometry explicitly, which is quite different from the implicit and conventional level-set description ways of topology optimization, fully releasing the deformation ability of the components. By moving, rotating, deforming, bending, overlapping and merging the level-set-based components, the channel geometry gene-rates and is ready to be projected onto the lower layer for the physical field analysis purpose. The lower layer is discretized using a NURBS patch, and NURBS-based isogeometric analysis is adopted to solve the coupled thermal-fluid problem in the structure and calculate the structural sensitivity so as to drive the upper layer to iterate. Comparing with the conventional topology optimization methods, the proposed meth-od has obvious advantages in terms of computational effectiveness and geometry description. A design of multi-layer cold plate is constructed under this framework, aiming at obtaining the best heat dissipation effect under the constraints of volume dissipation and pressure drop. In the final, simulations and experiments are performed to demonstrate these advantages of this framework.

V tomto článku se zabýváme optimální volbou parametru gama pro předpodmínění rozšířrného Lagrangianu metody GMRES pro efektivní řešení lineárních systémů získaných diskretizací nestlačitelných Navier-Stokesových rovnic. Uvažujeme diskretizaci rovnic pomocí přístupu isogeometrické analýzy založené na B-spline. Zajímá nás závislost konvergence na parametru gama pro různé parametry úlohy (Reynoldsovo číslo, zjemnění sítě) a zejména pro různé isogeometrické diskretizace (stupňová a meziprvková spojitost bází B-spline diskretizace). Cílem je umět určit optimální hodnotu gama pro úlohu, která je relativně levná na výpočet, a na základě této hodnoty předpovědět vhodné hodnoty pro jiné úlohy, např. s jemnější sítí, jinou diskretizací atd. Vliv vnitřního řešiče (přímého nebo iteračního založeného na metodě multigridu) je také diskutován. In this paper, we deal with the optimal choice of the parameter gamma for augmented Lagrangian preconditioning of GMRES method for efficient solution of linear systems obtained from discretization of the incompressible Navier-Stokes equations. We consider discretization of the equations using the B-spline based isogeometric analysis approach. We are interested in the dependence of the convergence on the parameter gamma for various problem parameters (Reynolds number, mesh refinement) and especially for various isogeometric discretizations (degree and interelement continuity of the B-spline discretization bases). The idea is to be able to determine the optimal value of gamma for a problem that is relatively cheap to compute and, based on this value, predict suitable values for other problems, e.g., with finer mesh, different discretization, etc. The influence of inner solvers (direct or iterative based on multigrid method) is also discussed.

The adequate coupling of dissimilar materials and structural element types provides effective engineering designs. Particularly, it enables high stiffness and low weight mechanical components, which are largely desired in numerous applications. Besides, Isogeometric Analysis (IGA) has recently stood out as a robust and accurate approach for the mechanical modelling of complex 3D engineering designs directly from Computer-Aided Design (CAD) tools. In this context, this study presents nonlinear Isogeometric Boundary Element Method (IGABEM) formulations for the effective mechanical modelling of nonhomogeneous 3D structural systems reinforced by embedded fibres. The IGABEM based on NURBS surfaces represents the solid whereas the 1D approach of BEM models the reinforcements, which characterises a 1DBEM/IGABEM coupling technique. The sub-region technique enables the nonhomogeneous system modelling whereas connection 1DBEM elements allow for crossing reinforcements along boundary interfaces without the need for surfaces remeshing. Thus, the CAD model information can be straightforwardly used in the mechanical model. Nonlinear time independent formulations are presented accounting for both elastoplastic and bond-slip behaviours. Besides, linear and nonlinear time dependent formulations are presented herein applying the Kelvin-Voigt and Boltzmann viscoelastic rheological models. Five applications illustrate the accuracy and robustness of the proposed formulations. The IGABEM formulations lead to accurate and stable results, in addition to efficient modelling. Besides, it demonstrates excellent performance considering the computational time consuming and the total number of degrees of freedom in each application.

Despite recent advances, there are still challenges in solving higher-order partial differential equations using the finite element method. This thesis focuses on finite element techniques for higher-order problems with applications to isogeometric analysis and statistical inference. In both areas, the efficient solution of higher-order partial differential equations is essential. The construction of isogeometric smooth spline basis functions is crucial for solving higher-order partial differential equations on meshes with arbitrary topologies. We introduce a simple blending approach that produces smooth blended B-splines, referred to as SB-splines, on unstructured quadrilateral and hexahedral meshes. We first define a set of mixed smoothness quadratic B-splines that are $C^0$ continuous in the unstructured regions of the mesh but are $C^1$ continuous everywhere else. Subsequently, the SB-splines are obtained by smoothly blending in the physical space the mixed smoothness B-splines with Bernstein basis functions of equal degree. One of the key novelties of our approach is that the required smooth weight functions are assembled from the available smooth B-splines in the geometric parametrisation. The SB-splines are globally smooth, non-negative, have no breakpoints within the elements, and reduce to conventional B-splines away from the unstructured regions of the mesh. Remarkably, the optimal convergence rates are numerically observed for the Poisson and biharmonic problems in one and two dimensions. The statistical finite element method coherently sythesises data and finite element models using Bayesian inference. However, the Gaussian process inference using covariance matrices is computationally expensive and memory-intensive. Consequently, the size of the finite element model is significantly limited. To this end, we introduce a sparse precision formulation by specifying the Gaussian process priors as Matérn random fields. Instead of kernel parametrisation, the Matérn random fields are parametrised as solutions of a higher-order stochastic partial differential equation. Finite element discretisation of the corresponding weak form results in sparse precision matrices. Specifically, the sparse precision matrices for arbitrary Matérn smoothness are obtained using recursive finite element computation and rational approximation. In addition to achieving improved scalability, we demonstrate the extension of the stochastic partial differential equation approach to non-Euclidean domains and the modelling of non-stationary and anisotropic Matérn random fields., Yang di-Pertuan Agong Scholarship by His Majesty the King of Malaysia

Defeaturing consists in simplifying geometrical models by removing the geometrical features that are considered not relevant for a given simulation. Feature removal and simplification of computer-aided design models enables faster simulations for engineering analysis problems, and simplifies the meshing problem that is otherwise often unfeasible. The effects of defeaturing on the analysis are then neglected and, as of today, there are basically very few strategies to quantitatively evaluate such an impact. Understanding well the effects of this process is an important step for automatic integration of design and analysis. We formalize the process of defeaturing by understanding its effect on the solution of Poisson equation defined on the geometrical model of interest containing a single feature, with Neumann boundary conditions on the feature itself. We derive an a posteriori estimator of the energy error between the solutions of the exact and the defeatured geometries in $\mathbb{R}^n$, $n\in\{2,3\}$, that is simple, reliable and efficient up to oscillations. The dependence of the estimator upon the size of the features is explicit., 37 pages, 15 figures, 4 tables

We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation .and adaptivity procedure is applied to a series of benchmark problems, demonstrating the suitability of the technique for a range of smooth and non-smooth problems. The adaptivity strategy is also integrated in a scan-based analysis workflow, capable of generating reliable, error-controlled, results from scan data, without the need for extensive user interactions or interventions., Submitted to Journal of Mechanics

Formulations of an extra-degree-of-freedom (DOF) free extended isogeometric analysis (IGA) are presented in this study. The idea is achieved by reconstruction of the coefficients through a mesh-free-based local approximation, based on the framework of a generalized finite-element method. The enrichment functions are embedded in the mesh-free basis implicitly, resulting in an identical number of DOFs. Moreover, the system condition number is of the same level between the enriched IGA and non-enriched version. The approach to handling the blending issues is discussed. Numerical examples with a solution containing sharp features/singularities are designed and studied in terms of accuracy and convergence.

The 2.5D approach is an efficient tool for the dynamic analysis of half-space with longitudinally invariant properties. In this paper, a newly formulated 2.5D version of the NURBS-based isogeometric analysis (IGA) and the scaled boundary IGA (SBIGA) is proposed for analyzing the bounded and unbounded domains, respectively, of a soil-tunnel system subjected to moving railway loads. The bounded domain is discretized by IGA, since the NURBS used can exactly represent the variations in geometry and materials with negligible errors. The unbounded domain is discretized by SBIGA, which automatically satisfies the Sommerfeld radiation condition and can be easily connected with the bounded IGA domain. The 2.5D approach differs from the 2D approach in that both the in-plane and out-of-plane displacements of the profile of the soil-tunnel system are included in analysis. By applying the Fourier transformation, the IGA and SBIGA equations of the 2.5D version in the frequency-wavenumber domain are derived individually and then assembled at the interface of the bounded and unbounded domains. Two examples are prepared for the ground and underground trains moving at subsonic and transonic speeds. Both single and multi moving loads are considered. It was demonstrated that the wave propagation characteristics including the Mach cones at transonic speeds can be precisely predicted. The results in time and frequency domains have been compared with existing ones. It was shown that the proposed method is accurate and reliable for application to a wide range of soil-tunnel systems.

In this study, the application of isogeometric analysis (IGA) is extended to the linear elastodynamics problems. In order to improve the efficiency of time-dependent problems with IGA, a novel effective Krylov-based proper orthogonal decomposition (KPOD) strategy is suggested to establish an efficient extrapolated algorithm by dimension reduction. In this method, a reduced-order model (ROM) is constructed to save the computational cost and the Krylov subspace method is used to achieve effective extrapolation through expanding the solution space formed by proper orthogonal decomposition (POD) basis with the addition of Krylov subspace. To validate the performance of the suggested method, two numerical examples are tested. Moreover, with the increasing of scale of the problem, much more computational cost should be saved. Specifically, the efficiency is increased by more than 20 times with 100,000 degrees of freedom.

Introduced in this paper is the complete nonlinear model of the micropolar theory (MPT) for shell-type materials. Considering the three-dimensional kinematic model in a convected curvilinear coordinate system, the Lagrangian description of micromorphic theory (MMT) is formulated first. Due to certain assumptions, i.e. skew-symmetricity of micro-displacement and orthogonality of micro-deformation tensors, the highly nonlinear micropolar continuum theory is obtained. Unlike the conventional knowledge in the literature, it is found that not only MPT is not a simple version of MMT but also the present micropolar formulation can be reduced to the corresponding micromorphic shell elasticity. Also, unlike the Kafadar-Eringen's nonlinear micropolar model which only predicts the large micro-deformations, the developed micropolar theory is able to investigate the physical behavior in case of large elastic macro-deformations, for the first time. The isogeometric analysis (IGA) solution method with standard base functions is used to prevent the locking issues arising in the low-order finite element analysis (FEA) of thin shells. Accordingly, the proposed IGA micropolar shell model possesses 10-DOFs (degrees of freedom) standing for 7 macro-displacements and 3 micro-rotations. Size / inhomogeneity effects of small-scale / micropolar materials are investigated in the benchmark problems of geometrically nonlinear shells. As a result, it is revealed that the paper contributes to literature by developing the nonlinear micropolar shell theory which is able of capturing large macro- and micro-deformations.

This work deals with shape optimization of electric machines using isogeometric analysis. Isogeometric analysis is particularly well suited for shape optimization as it allows to easily modify the geometry without remeshing the domain. A 6-pole interior permanent magnet synchronous machine (IPMSM) is modeled using a multipatch isogeometric approach and rotation of the machine is realized by modeling the stator and rotor domain separately and coupling them at the interface using harmonic basis functions. Shape sensitivity analysis is used to find the shape derivative of the optimization goal functional, i.e., the total harmonic distortion of the electromotive force. This method allows for a freeform shape optimization which is not restricted by a choice of a set of optimization parameters. This freeform shape optimization is applied for the first time to an isogeometric model of an IPMSM minimizing the total harmonic distortion of the electromotive force as a goal functional, where a reduction of 75 % is achieved.

Porosities exist as pores of different sizes within a structure to fabricate lightweight materials and the sintering process. The porous structure gives a lower loading capacity than the perfect design. Crack propagation is also a complicated behavior in this structure. The hybrid phase-field approach is suitable to provide an effective computational tool to model the crack propagation of functionally graded materials with porosity effects. We show the influence of porosity on both the critical force and crack path of the FGM structure. In the framework of isogeometric analysis (IGA), a local refinement multi-patch algorithm based on the Virtual Uncommon-Knot-Inserted Master-Slave (VUKIMS) technique allows us to reduce the computational cost of the phase-field model significantly. The study revealed that cubic NURBS elements with the effective element size of half length-scale parameter could be used to achieve the desired accuracy while maintaining a reasonable computational cost.

Isogeometric analysis (IGA) is an important mesh-free method which utilizes the splines basis functions and control net in the CAD geometry for simulation. Its theoretical basis is continuum mechanics, and it is not suitable for fracture problems. Peridynamics (PD) is a non-local theory based on integral equations, which is suitable for discontinuous problems such as cracks, while the non-local property of PD increases the computational effort. The coupling of IGA and PD promotes the simulation of cracks with the precise CAD geometry and improves the computational efficiency compared with that of pure PD model. In this paper, the coupling approach of IGA and PD (IGA-PD) for plane problems with non-uniform control net is introduced, and the main contribution is a system of algorithms to convert the non-uniform control net into IGA-PD coupling model. A volume division method is proposed to calculate the occupied volumes for the PD nodes. To account for the non-uniform control net, the search range correction method, volume and centroid correlation methods of family nodes and dual-horizon method are applied to improve the computational precision. An exponential gradual constitutive model (EGCM) is applied to calculate the crack propagation of materials. The proposed IGA-PD model with non-uniform control net makes full use of the advantages of the non-local continuum theory, which can solve the fracture problem and alleviate the surface effect of PD. The processing algorithms for the non-uniform control net improve the computational accuracy. Numerical examples of static problems and crack propagations are given. The results prove that the search range, centroid and volume correction methods are important for improving the numerical precision. The plate under stretch and bending, Kalthoff–Winkler impact problem and simulation of concrete material are carried out to show the effectiveness of the proposed IGA-PD model.

Isogeometric analysis is a spline-based discretization method to partial differential equations which show the approximation power of a high-order method. The number of degrees of freedom, however,...

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.

CNT/Graphene reinforced composites due to their exceptional mechanical, electrical, thermal, and optical characteristics that have attracted the scientific community working in the area of material development. In protraction, this review work on fracture analysis of CNT/Graphene reinforced composites is summarized from an exponentially growing literature inclusive of fabrication methods and present and potential applications. Owing to these composites' enormous structural applications, the number of literature works on the fracture of the structures using various models is summarized here. The literature contains several techniques to model the CNT/graphene composites that include: Halpin Tsai model, modified Halpin Tsai model, Mori–Tanaka model, homogenization approach, etc. which are discussed in detail. Various combined analysis for fracture toughness and crack growth analysis based on fracture mechanics are cited here using different computational formulations like finite element method (FEM), element free Galerkin method, reproducing kernel particle method, meshless local Petrov–Galerkin method, extended finite element method (XFEM), isogeometric analysis (IGA), etc. In particular, the authors have reported the work done in the area of fracture analysis of these novel composites since its inception that is inclusive of its structure, fabrication, properties, mathematical modeling, applications, and analysis.

In isogeometric analysis (IGA), the boundary representation of computer-aided design (CAD) and the tensor-product non-uniform rational B-spline structure make the analysis of three-dimensional (3D) problems with irregular geometries difficult. In this paper, an IGA method for complex models is presented by reconstructing analysis-suitable models. The CAD model is represented by boundary polygons or point cloud and is embedded into a regular background grid, and a model reconstruction method is proposed to obtain the level set function of the approximate model, which can be directly used in IGA. Three 3D examples are used to test the proposed method, and the results demonstrate that the proposed method can deal with complex engineering parts reconstructed by boundary polygons or point clouds.

In this work, we propose a modified phase-field model for simulating the evolution of mixed mode fractures and compressive driven fractures in porous artificial rocks. For the purpose of validation, the behaviour of artificial rock samples, with either a single or double saw cuts, under uniaxial plane strain compression has been numerically simulated. The simulated results are compared to experimental data, both qualitatively and quantitatively. It is shown that the proposed model is able to capture the commonly observed propagation pattern of wing cracks emergence followed by secondary cracks driven by compressive stresses. Additionally, the typical types of complex crack patterns observed in experimental tests are successfully reproduced, as well as the critical loads.

Topology optimization as a computational approach is used to find an optimal structure with specific objectives such as an ultimately lightweight design. The optimization is normally performed under a series of constraint functions to ensure expected structural performance for safety and resilience. To promote the use of topology optimization in structural engineering, it is crucially important to identify and apply suitable constraints for realistic construction. In this paper, a conceptual attempt aiming for a construction-oriented topology optimization framework is presented. An isogeometric analysis module using NURBS curves for geometric description and discretization is employed in stress analyses, which incorporates the density-based SIMP optimization approach to generate the optimized topology. A Drucker–Prager criterion is adopted to constrain the stresses within the unequal limits of tension and compression for the use of concrete type material. To prevent slim components of potentially high complexity in construction, a minimum-width control is applied as an additional geometric control. In this paper, the conceptual framework is demonstrated using a prestressed concrete beam, while the optimization is performed to find suitable density distribution of concrete and a NURBS-described tendon profile after imposing the mapped prestressed action. The iterative optimization processes are presented and demonstrated to investigate the effect of tensile-compressive strength ratio and the minimum width, which reflects the construction-oriented vision pursuing a more regularized topology after applying the constraints in optimization.

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.

This paper presents a unique solution to the problem of planar compliant mechanism design by means of geometric morphing technology and isogeometric analysis (IGA). A new transformable triangular mesh (TTM) component is developed based on geometric morphing technology, which can generate the required topology with different feature sets from a surface that has zero boundary component (interior hole) under the control of Laplace energy and mesh operations. Such flexible TTM component is helpful in overcoming the initial dependency of conventional topology optimization methods in which the layout of parameterized components often affects final optimized results. As the high-order continuity between the grids of IGA can improve calculation accuracy and numerical stability, IGA is combined with the presented TTM algorithm to establish a two-layer computational model so as to identify the optimal compliant mechanism topology within a given design domain and given displacements of input and output ports. In the upper layer of the model, the compliant limbs are characterized explicitly by triangular grids. By moving, splitting, and refining these triangular grids, the generated shape will then be projected onto the lower layer which is discretized using NURBS elements so as to calculate structural sensitivity for driving new iteration. To demonstrate the benefits provided by such method for compliant mechanism design, several numerical studies are tested, in which the geometry freely evolves along the optimization procedure, resulting in more efficient non-trivial topologies with desired kinematic behavior.

In this paper, we develop a convergence theory for Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for isogeometric multi-patch discretizations of the Poisson problem, where the patches are coupled using discontinuous Galerkin. The presented theory provides condition number bounds that are explicit in the grid sizes ℎ and in the spline degrees 𝑝. We give an analysis that holds for various choices for the primal degrees of freedom: vertex values, edge averages, and a combination of both. If only the vertex values or both vertex values and edge averages are taken as primal degrees of freedom, the condition number bound is the same as for the conforming case. If only the edge averages are taken, both the convergence theory and the experiments show that the condition number of the preconditioned system grows with the ratio of the grid sizes on neighboring patches.

Numerical methods studied the functionally graded piezoelectric porous (FGPP) microplates are still somewhat restricted. In this study, a size-dependent approach in the framework of isogeometric analysis (IGA) via the modified strain gradient theory (MSGT) and the higher-order shear deformation theory (HSDT) for electro-mechanical vibration analysis of FGPPo microplates is presented for the first time. A FGPP microplate is created from a mixture of two materials which their properties vary continuously through the thickness plate obeying a modified power-law expression. Two porosity distribution types including even and uneven are considered. An electric potential field which satisfies the Maxwell’s equation in the quasi-static approximation is adopted. To study size effects, MSGT is proposed with the addition of three material length scale parameters (MLSPs) into the elastic constants. It is general due to when two or three MLSPs are set equal to zeros, MSGT can be recovered into the modified couple stress theory model (MCST) or classical higher-order shear deformation theory, respectively. Governing equations deduced from the weak form of FGPP microplates are obtained by utilizing the principle of virtual work, subsequently, natural frequencies of FGPP microplates are determined by IGA. The influence of external electric voltages, power-law index, porosity coefficient, porosity distribution, various boundary conditions, and material length scale parameters on the natural frequencies of FGPP microplates is investigated. To demonstrate the reliability and effectiveness of the proposed method, the present results are compared with various available publications in the literature. Moreover, although there are no analytical solutions, we strongly furthermore present answers of some FGPP microplates being considered as reference results for upcoming studies.

An extended approach is developed by blending isogeometric analysis and the first-order local maximum entropy for the standard and the enhanced fields, respectively. Isogeometric analysis facilitates the accurate parametrization of the geometry in general, particularly the exact geometric parametrization of the conic curves and quadratic surfaces using NURBS. On the other hand, the local maximum entropy leads to an improved estimate for the enhanced part due to its infinite continuity. Moreover, local maximum entropy paves the way to a nonelementwise crack propagation owing to its meshfree characteristic. To enforce compatibility, the shifting technique is amended for the meshfree enhanced part to localize the effect of the Heaviside function to a narrow region around the crack. Next, a blending technique is exploited to remove the effect of the discontinuity in front of the crack tip. The viability of the approach is illustrated at the hand of several examples comprising straight and curved crack propagation.

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.

In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree $p$, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, also in this case the given matrices are ill-conditioned both in the low and high frequencies for large $p$. More precisely, in the fractional scenario the symbol has a single zero at $0$ of order $\alpha$, with $\alpha$ the fractional derivative order that ranges from $1$ to $2$, and it presents an exponential decay to zero at $\pi$ for increasing $p$ that becomes faster as $\alpha$ approaches $1$. This translates in a mitigated conditioning in the low frequencies and in a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B-spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B-splines. Finally, we perform a numerical study of the approximation behavior of polynomial B-spline collocation. This study suggests that, in line with non-fractional diffusion problems, the approximation order for smooth solutions in the fractional case is $p+2-\alpha$ for even $p$, and $p+1-\alpha$ for odd $p$., Comment: 23 pages, 13 figures

Modern electronic devices are usually subjected to thermo-mechanical loads and prone to buckle during their operation. Thermo-mechanical stability is a crucial standard for their reliable applications. This paper explores the size effect on the thermo-mechanical behavior of the organic solar cells. An effective isogeometric analysis method combined with modified couple stress theory is presented. The thermo-mechanical buckling load-bearing capacities of the organic solar cells subjected to various in-plane loadings, temperatures, and geometrical parameters are discussed. Numerical results show that the size effect has significant effect on the thermo-mechanical load-bearing capacity. The stability region changes minimally when the material length scale parameter $$l$$ to cell thickness $$h$$ ratio is less than 0.2, while the stability region increases remarkably when it is larger than 0.2. Notably, if the material length scale parameter increases to its thickness, the stability region increased almost 25 times than that without size effect. Furthermore, the stability region is narrowest if the temperature is uniform across the thickness direction of the cell.

Bi-cubic analysis-suitable++ T-splines (AS++ T-splines) (Li in Comput Methods Appl Mech Eng 333:462–474, 2018) are T-splines defined on less restricted T-meshes than analysis-suitable T-splines (AS T-splines), which are both important tools in isogeometric analysis (IGA). In this paper, we generalize the bi-cubic AS++ T-splines to arbitrary degrees and describe some important mathematical properties. Specifically, we develop the conditions under which an AS++ T-spline space belongs to another AS++ T-spline space. This result provides the foundation for the optimized local refinement (Zhang in Comput Methods Appl Mech Eng 342:32–45, 2018) and also is one of the keys for AS++ T-spline approximation. In the end, the optimal approximation properties of the associated T-spline space are developed for arbitrary AS++ T-spline space with the assumption of existence of dual basis, which is automatically satisfied for bi-cubic AS++ T-spline spaces.

Isogeometric analysis (IGA) has become an alternative to standard finite element analysis (FEA) in many areas of engineering. Its powerful tools for model generation and flexibility of basis functions make this relatively new approach attractive for tire analysis and its computational challenges.This contribution summarizes the benefits of IGA for complex tire simulations starting from model generation and the subsequent transition to the environment of numerical analysis without losing accuracy at the parametrizing stage. It presents results of further development work on earlier pioneering examples of the application of IGA in pneumatic tire analysis. In addition to the analysis of vertical stiffness, for the first time, velocity and acceleration fields are addressed and compared with experimental results and standard FEA simulations, with a focus on benefits of the continuity of basis functions within the contact patch. The numerical issues that arise in IGA at the enforcement of contact and the application of inelastic materials with inclusions of reinforcing layers are studied. Moreover, the important advantages of the possibility to use higher order functions for simulations of tire maneuvers are addressed within the steady-state framework. Numerical examples are provided to illustrate the capabilities of IGA. Concluding remarks on the results close the publication.

This article aims to utilize IsoGeometric analysis (IGA) and Level set method for topology optimization of elastoplastic plane stress problems. The IGA is employed to model geometry of the problem and calculate unknown displacements by satisfying equilibrium for materially-nonlinear problems. The von Mises elastoplastic material behavior model with and without isotropic strain hardening is utilized. The normal velocity is derived by the pointwise gradient based sensitivity analysis of the Level set function and the control net is updated at each optimization iteration. The reaction–diffusion equation is also employed to update the design variables, which eliminates the signed distance function dependency, and the algorithm provides capability of hole nucleation. The objective is to maximize the toughness of structure, which is defined as the total external work within a specified displacement, while certain amount of material in the design domain is used. In order to demonstrate the ability and efficiency of the proposed method, several numerical examples are presented.

The purpose of this study is to present a quasi-three-dimensional (quasi-3D) shear deformation theory for static bending and free vibration analyses of porous sandwich functionally graded (FG) plates with graphene nanoplatelets (GPLs) reinforcement. In addition, we propose a novel sandwich plate model with various outstanding features in terms of structural performance. The quasi-3D theory-based isogeometric analysis (IGA) in conjunction with refined plate theory (RPT) is first exploited to capture adequately the thickness stretching effect for porous sandwich FG plate structures reinforced with GPLs. The Non-Uniform Rational B-Splines (NURBS)-based IGA is employed in order to describe exactly the geometry models as well as approximate the unknown field with higher-order derivatives and continuity requirements while the RPT model includes only four essential variables. The sandwich FG plates consist of a core layer containing internal pores reinforced by GPLs and two functionally graded materials (FGMs) skin layers. Effective mechanical properties can be evaluated by employing the Halpin-Tsai model along with the rule of mixture. Various combinations of two porosity distributions and three GPL dispersions in the core layer are thoroughly investigated. Several numerical investigations are conducted to examine the effects of several key parameters on the static bending and free vibration behaviors of sandwich FG plate structures.

We expose an efficient strategy to deal with shape optimization of dynamical systems exhibiting flutter-type instability induced by friction, such as the considered disc-pad system. The stability of such systems can be analyzed through complex eigenvalue analysis, through which we present a squeal noise criterion to be minimized as a computationally expensive black-box function. The computational domain is discretized through isogeometric formulation for its advantages in optimization and superior approximation properties which are well studied in structural dynamics. To be computationally efficient with the expensive black-box function, we defined the optimization based on efficient global optimization scheme in the context of multi-objective optimization, with the integration of Isogeometric design-through-analysis methodology. As gradient information is hard to access for such black-box functions, in addition to the presence of constraints, we relied on meta-heuristic approach as a more generic strategy for realizing optimization of such functions in multi-objective context. As one such scheme with its own advantages was observed to provide lack of resolution to define expected improvement (EI) with a single reference value, we propose a multi-reference acquisition strategy which can be defined through a fast and efficient algorithm with fewer adaptation to the existing scheme. Results show the efficiency of this approach for our applicative example, which can be extended to other such applications as well.

An effective computational approach that can simultaneously deal with several complexities in geometries, material properties, and size effects is of non-trivial yet important tasks for the design and optimization of device components in micro- and nano-electromechanical systems. This study is devoted to the development of such an effective computational approach, which is formed by integrating the modified couple stress theory into isogeometric analysis framework. The proposed approach is applied to mechanical problems of laminated Timoshenko curved microbeams, i.e., anazlying static bending, natural frequency and transient response. The developed method enables to describe and examine nontrivial features in both material and geometrical properties of laminated curved microbeams. Geometries of curved microbeams and displacement approximation are accurately described by the non-uniform rational B-spline basis functions. For laminated composites, the material coupling is taken into account with the use of equivalent elastic modulus; and the deepness term is properly included in laminate stiffness parameters. In addition, the small-scale size effects are captured with the use of modified couple stress theory. The accuracy and performance of proposed method are demonstrated through several numerical examples, in which the static bending, free vibration and transient response of laminated curved microbeams are considered under effects of some factors such as aspect ratios, lay-ups, boundary conditions and size dependence.

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 ...

This work presents an efficient methodology for the optimum design of functionally graded structures using a Kriging-based approach. The method combines an adaptive Kriging framework with a hybrid particle swarm optimization (PSO) algorithm to improve the computational efficiency of the optimization process. In this approach, the surrogate model is used to replace the high-fidelity structural responses obtained by a NURBS-based isogeometric analysis. In addition, the impact of key factors on surrogate modelling, as the correlation function, the infill criterion used to update the surrogate model, and the constraint handling is assessed for accuracy, efficiency, and robustness. The design variables are related to the volume fraction distribution and the thickness. Displacement, fundamental frequency, buckling load, mass, and ceramic volume fraction are used as objective functions or constraints. The effectiveness and accuracy of the proposed algorithm are illustrated through a set of numerical examples. Results show a significant reduction in the computational effort over the conventional approach.

Flexoelectricity refers to a phenomenon which involves a coupling of the mechanical strain gradient and electric polarization. In this study, a meshless Fragile Points Method (FPM), is presented for analyzing flexoelectric effects in dielectric solids. Local, simple, polynomial and discontinuous trial and test functions are generated with the help of a local meshless Differential Quadrature approximation of derivatives. Both primal and mixed FPM are developed, based on two alternate flexoelectric theories, with or without the electric gradient effect and Maxwell stress. In the present primal as well as mixed FPM, only the displacements and electric potential are retained as explicit unknown variables at each internal Fragile Point in the final algebraic equations. Thus the number of unknowns in the final system of algebraic equations is kept to be absolutely minimal. An algorithm for simulating crack initiation and propagation using the present FPM is presented, with classic stress-based criterion as well as a Bonding-Energy-Rate(BER)-based criterion for crack development. The present primal and mixed FPM approaches represent clear advantages as compared to the current methods for computational flexoelectric analyses, using primal as well as mixed Finite Element Methods, Element Free Galerkin (EFG) Methods, Meshless Local Petrov Galerkin (MLPG) Methods, and Isogeometric Analysis (IGA) Methods, because of the following new features: they are simpler Galerkin meshless methods using polynomial trial and test functions; minimal DoFs per Point make it very user-friendly; arbitrary polygonal subdomains make it flexible for modeling complex geometries; the numerical integration of the primal as well as mixed FPM weak forms is trivially simple; and FPM can be easily employed in crack development simulations without remeshing or trial function enhancement.

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.

Parameterizations, which map parametric domains into certain domains, are widely used in computer aided design, computer aided geometric design, computer graphics, isogeometric analysis, and related fields. The parameterizations of curves, surfaces, and volumes are injective means that they do not have self-intersections. A 3D toric volume is defined via a set of 3D control points with weights that correspond to a set of finite 3D lattice points. Rational tensor product or tetrahedral Bezier volumes are special cases of toric volumes. In this paper, we proved that a toric volume is injective for any positive weights if and only if the lattice points set and control points set are compatible. An algorithm is also presented for checking the compatibility of the two sets by the mixed product of three vectors. Some examples illustrate the effectiveness of the proposed method. Moreover, we improve the algorithm based on the properties and results of clean and empty tetrahedrons in combinatorics.