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In this paper, a novel isogeometric method for Biot’s consolidation model is constructed and analyzed, using a four-field formulation where the unknown variables are the solid displacement, solid pressure, fluid flux, and fluid pressure. Mixed isogeometric spaces based on B-splines basis functions are employed in the space discretization, allowing a smooth representation of the problem geometry and solution fields. The main result of the paper is the proof of optimal error estimates that are robust with respect to material parameters for all solution fields, particularly in the case of nearly incompressible materials. The analysis does not require a uniformly positive storage coefficient. The results of numerical experiments in two and three dimensions confirm the theoretical error estimates and high-order convergence rates attained by the proposed isogeometric Biot discretization and assess its performance with respect to the mesh size, spline polynomial degree, spline regularity, and material parameters.

Space deployable mechanisms typically employ modular designs, where each unit that can be independently deployed is defined as a generalized deployable unit (GDU), typically consisting of rigid components, flexible components and generalized kinematic pairs. Given the high-performance demands of spatial transmission mechanisms, generalized kinematic pairs frequently integrate flexible factors like torsion springs and flexible hinges, which control the motion according to the attitude and position of the component. In this investigation, a new modeling and solution strategy for rigid-flexible coupled dynamics is established by combining NCF and IGA. It can not only accurately describe the large deformation and large rotation of flexible components, but also consider the influence of flexible factors in the kinematic pairs. The rigid components are modeled by the Natural Coordinate Formula (NCF), and the flexible components are discretized in the inertial coordinate system using spline curves in Isogeometric Analysis (IGA). The intermediate reference coordinates were integrated into the NCF-IGA framework, overcoming the boundary constraints between B-spline and NCF elements, and the geometric constraint equation of the kinematic pair was established. Subsequently, this paper proposes the concept of force constraint in controllable deployable mechanisms. Based on the force constraint equation, the mechanical model of flexible joints is established. The dynamic equations of GDUs are derived considering both geometric and force constraints, and generalized-$\alpha$ method is introduced to solve the equations. Finally, three numerical examples are provided to illustrate the applicability and superiority of the proposed method.

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

This paper aims to develop a size-dependent numerical solution for a comprehensive study on the buckling responses of laminated functionally graded (FG) carbon nanotubes (CNTs)-reinforced composite microplates under linear and nonlinear in-plane compressive loading. According to the new modified couple stress theory and isogeometric analysis (IGA), in conjunction with the higher order plate theory, the size-dependent governing motion equations including material length scale parameter for buckling analysis of CNTs-reinforced laminated composite microplates is established. The research considers five types of loading shapes: uniform, triangular, trapezoidal, sinusoidal, and parabolic. The distribution of CNTs varies across the layer’s thickness, either uniformly or functionally patterns. The material properties of the composite layer are derived by using the New rule of mixture. The accuracy and reliability of the present solution are verified by comparing them with existing models. Different numerical examples illustrate the influences of various parameters on buckling responses of laminated FG-CNTs-reinforced composite microplates. These parameters include size-dependent effects, aspect ratio, volume fraction and distribution of CNTs, fiber orientation, lamination scheme, and boundary conditions. Additionally, the study considers a complex geometry of laminated microplates with flower cutouts to provide a comprehensive analysis.

In this paper, we propose a class of high-order time integration schemes combined with high-order IsoGeometric Analysis (IGA) in three space dimensions. The combined methods offer robust solutions of nonlinear heat diffusion in three-dimensional composites that pose numerical challenges. This tailored strategy significantly enhances computational efficiency, especially crucial when addressing nonlinear heat transfer in three-dimensional enclosures. Leveraging precise geometry representation and seamless high-order element continuity of the IGA, this method effectively exploits these advantages. It emphasizes the vital synergy between high-order spatial discretization and an equivalent high-order time integration scheme. This study also highlights the risks of overlooking this pairing, which can lead to a degradation of the overall high-order accuracy and increased computational demands due to the complexity of high-order nonuniform rational B-splines. Numerical examples, such as applications involving a furnace wall segment and a rail wheel heat transfer, are used to validate the efficiency and accuracy of the combined approach. Consistently surpassing the conventional methods in both aspects, the proposed method notably excels in providing precise solutions for steep heat gradients even on coarse meshes. Consequently, this approach constitutes a substantial advancement in the field of transient heat transfer analysis within composite domains.

The paper presents an isogeometric study for free vibration and buckling behaviors of functionally graded (FG) porous nanocomposite beams reinforced by graphene platelets (GPLs) based on a two-variable shear deformation theory. It is assumed that porosities are dispersed by symmetric and asymmetric models. Besides, GPLs are distributed along the beam thickness both uniformly and nonuniformly. Based on the closed-cell Gaussian random field model and the Halpin–Tsai micromechanical scheme, the mechanical properties of the porous nanocomposite beams are evaluated. In the presented theory, a nonlinear variation for the transverse shear stress across the beam thickness is considered which satisfies shear stress free surface condition. The governing equations are derived using the Hamilton’s principle and then discretized employing the isogeometric analysis (IGA) as an effective numerical method. It is shown that the proposed theory is very simple and efficient to analyze free vibration and buckling of the graphene platelet-reinforced porous composite (GPLRPC) beams.

This study presents a comprehensive investigation of bio-inspired rotating triply periodic minimal surface (RotTPMS) plates under thermo-mechanical loading conditions, advancing sustainable lightweight structural design through computational modeling. Three TPMS architectures including Primitive (P), I-Wrapped Package (IWP), and Gyroid (G) are analyzed using a quasi-3D higher-order shear deformation theory (HSDT) combined with isogeometric analysis (IGA). The research addresses critical thermal stability challenges in lightweight structures by examining thermal buckling and thermo-mechanical buckling behaviors across various crystalline rotations, relative densities, and temperature conditions. Results demonstrate that P-type RotTPMS structures exhibit superior thermal buckling resistance, with critical temperature rises up to 30% higher than other architectures, while maintaining optimal weight-to-performance ratios. The study reveals significant anisotropic characteristics in thermal responses, with crystalline rotations inducing variations of up to 300% in critical buckling loads at elevated temperatures. Temperature-dependent material properties substantially influence structural stability, reducing critical buckling temperatures by $15\%÷30\%$ compared to temperature-independent analyses. These findings provide necessary insights for designing thermally stable lightweight structures in sustainable engineering applications, particularly relevant for aerospace and mechanical systems. The developed computational framework and material design principles contribute to advancing additive manufacturing (AM) capabilities for sustainable structural solutions, aligning with growing demands for energy-efficient and eco-friendly engineering materials.

It is shown that the standard transfinite interpolation in quadrilateral patches may be written in terms of two sets of Bernstein polynomials. The former set is of the same degree with the corresponding blending functions while the latter is of the same degree with the Lagrange polynomials which operate like trial functions along each of the four edges as well as the additional inter-boundaries of the patch. The replacement of the Lagrange polynomials by the Bernstein ones allows the use of control points useful for design. Also it allows the determination of weights that may ensure accurate representation of quadric surfaces including those of revolution (spheres, ellipsoids, hyperboloids, etc). The presentation restricts to the standard Gordon formulation, which refers to structured stencils, similar to rectangular plates reinforced with longitudinal and transverse stiffeners. As an example, the proposed formula is applied to the geometric representation of a cylindrical and a spherical patch. In the latter case a nonlinear programming optimization technique was applied, starting with initial data pertinent to a tensor product surface, from which original weights and control points were obtained for the accurate representation of a spherical cap. In addition, the involved shape functions were successfully applied to the numerical solution of a boundary value problem.

We initiate the study of collocation methods for NURBS-based isogeometric analysis. The idea is to connect the superior accuracy and smoothness of NURBS basis functions with the low computational cost of collocation. We develop a one-dimensional theoretical analysis, and perform numerical tests in one, two and three dimensions. The numerical results obtained confirm theoretical results and illustrate the potential of the methodology.

We provide an overview of the Arbitrary Lagrangian–Eulerian Variational Multiscale (ALE-VMS) and Space–Time Variational Multiscale (ST-VMS) methods we have developed for computer modeling of wind-turbine rotor aerodynamics and fluid–structure interaction (FSI). The related techniques described include weak enforcement of the essential boundary conditions, Kirchhoff–Love shell modeling of the rotor-blade structure, NURBS-based isogeometric analysis, and full FSI coupling. We present results from application of these methods to computer modeling of NREL 5MW and NREL Phase VI wind-turbine rotors at full scale, including comparison with experimental data.

A Balancing Domain Decomposition by Constraints (BDDC) preconditioner for Isogeometric Analysis of scalar elliptic problems is constructed and analyzed by introducing appropriate discrete norms. A main result of this work is the proof that the proposed isogeometric BDDC preconditioner is scalable in the number of subdomains and quasi-optimal in the ratio of subdomain and element sizes. Another main result is the numerical validation of the theoretical convergence rate estimates by carrying out several two- and three-dimensional tests on serial and parallel computers. These numerical experiments also illustrate the preconditioner performance with respect to the polynomial degree and the regularity of the NURBS basis functions, as well as its robustness with respect to discontinuities of the coefficient of the elliptic problem across subdomain boundaries.

We develop divergence-conforming B-spline discretizations for the numerical solution of the steady Navier–Stokes equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. They are (at least) patchwise C⁰ and can be directly utilized in the Galerkin solution of steady Navier–Stokes flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. Consequently, discrete variational formulations employing the new discretization scheme are automatically momentum-conservative and energy-stable. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Euler flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions subject to a smallness condition, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. These numerical experiments also suggest our discretization methodology is robust with respect to Reynolds number and more accurate than classical numerical methods for the steady Navier–Stokes equations.

T-splines are an important tool in IGA since they allow local refinement. In this paper we define analysis-suitable T-splines of arbitrary degree and prove fundamental properties: Linear independence of the blending functions and optimal approximation properties of the associated T-spline space. These are corollaries of our main result: A T-mesh is analysis-suitable if and only if it is dual-compatible. Indeed, dual compatibility is a concept already defined and used in L. Beirão da Veiga et al.⁵ Analysis-suitable T-splines are dual-compatible which allows for a straightforward construction of a dual basis.

We establish several fundamental properties of analysis-suitable T-splines which are important for design and analysis. First, we characterize T-spline spaces and prove that the space of smooth bicubic polynomials, defined over the extended T-mesh of an analysis-suitable T-spline, is contained in the corresponding analysis-suitable T-spline space. This is accomplished through the theory of perturbed analysis-suitable T-spline spaces and a simple topological dimension formula. Second, we establish the theory of analysis-suitable local refinement and describe the conditions under which two analysis-suitable T-spline spaces are nested. Last, we demonstrate that these results can be used to establish basic approximation results which are critical for analysis.

We study a reformulated version of Reissner–Mindlin plate theory in which rotation variables are eliminated in favor of transverse shear strains. Upon discretization, this theory has the advantage that the "shear locking" phenomenon is completely precluded, independent of the basis functions used for displacement and shear strains. Any combination works, but due to the appearance of second derivatives in the strain energy expression, smooth basis functions are required. These are provided by Isogeometric Analysis, in particular, NURBS of various degrees and quadratic triangular NURPS. We present a mathematical analysis of the formulation proving convergence and error estimates for all physically interesting quantities, and provide numerical results that corroborate the theory.

The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The adaptivity analysis holds in any space dimensions. We consider a simple residual-type error estimator for which we provide a posteriori upper and lower bound in terms of local error indicators, taking also into account the critical role of oscillations as in a standard adaptive finite element setting. The error estimates are properly combined with a simple marking strategy to define a sequence of admissible locally refined meshes and corresponding approximate solutions. The design of a refine module that preserves the admissibility of the hierarchical mesh configuration between two consecutive steps of the adaptive loop is presented. The contraction property of the quasi-error, given by the sum of the energy error and the scaled error estimator, leads to the convergence proof of the AIGM.

In this paper, we develop approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard Sobolev norms and semi-norms. The presented approximation error estimates do not depend on the polynomial degree of the splines but only on the grid size. We will see that the approximation lives in a subspace of the classical B-spline space. We show that for this subspace, there is an inverse inequality which is also independent of the polynomial degree. As the approximation error estimate and the inverse inequality show complementary behavior, the results shown in this paper can be used to construct fast iterative methods for solving problems arising from isogeometric discretizations of partial differential equations.

Isogeometric analysis is applied to boundary integral equations corresponding to boundary-value problems governed by Laplace’s equation. It is shown that the smoothness of geometric parametrizations central to computer-aided design can be exploited for regularizing integral operators to obtain high-order collocation methods involving superior approximation and numerical integration schemes. The regularization is applicable to both singular and hyper-singular integral equations, and as a result one can formulate the governing integral equations so that the corresponding linear algebraic equations are well-conditioned. It is demonstrated that the proposed approach allows one to compute accurate approximate solutions which optimally converge to the exact ones.

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension $d\ge 2$. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (respectively, the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.

We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and continue the study of its numerical properties. We prove that our AIGM is optimal in the sense that delivers optimal convergence rates as soon as the solution of the underlying partial differential equation belongs to a suitable approximation class. The main tool we use is the theory of adaptive methods, together with a local upper bound for the residual error indicators based on suitable properties of a well selected quasi-interpolation operator on hierarchical spline spaces.