High-order Continuity Research Articles

A symmetric interior-penalty discontinuous Galerkin isogeometric analysis spatial discretization of the self-adjoint angular flux form of the neutron transport equation

This paper presents the first application of a symmetric interior-penalty discontinuous Galerkin isogeometric analysis (SIP-DG-IGA) spatial discretization to the self-adjoint angular flux (SAAF) form of the multi-group neutron transport equation. The penalty parameters are determined, for general element types, from a mathematically rigorous coercivity analysis of the bilinear form. The proposed scheme produces a compact spatial discretization stencil. It also yields symmetric positive-definite (SPD) matrices, which can be efficiently solved using pre-conditioned conjugate gradient (PCG) solution algorithms. The proposed discretization scheme is verified using the method of manufactured solutions (MMS) and several nuclear reactor physics benchmark verification test cases. For sufficiently smooth elliptic problems, the proposed spatial discretization can exploit higher-order continuity, or k-refinement, of the NURBS basis to consistently yield greater numerical accuracy per degree of freedom (DoF) than standard h-refinement. Since this is a discontinuous scheme, it can also accurately model significant changes in the neutron scalar flux that may occur near the material interfaces of heterogeneous problems.

A scaled boundary shell formulation in isogeometric analysis for static and dynamic analysis

AbstractIn modern applications of computer‐aided design (CAD) for the analysis of shell structures, isogeometric analysis is a powerful tool that integrates both design and analysis. An exact geometry description and a straightforward computation without loss of information are advantageous, especially for shell structures such as roofs, satellite hulls, or car bodies. In addition, the scaled boundary method provides a scale separation with a semi‐analytical solution procedure to consider a three‐dimensional linear elastic constitutive law. The presented approach deals with a scaled boundary solid shell formulation in the framework of isogeometric analysis. The formulation utilizes a normal scaling strategy which scales the shell along its normal vector at each point on the discretized bottom surface. This is fundamentally different from the well‐known radial scaling strategy, where each point on the problem domain is obtained from a fixed scaling center. This results in a separation of the analysis into an in‐plane direction and a scaling (normal) direction. By introducing the scaling, a scaled boundary differential equation is derived that is dependent on the scaling parameter only. Choosing a proper set of conditions, the differential equation can be solved by a Padé expansion. While the in‐plane direction is solved in a weak sense, the thickness direction along the normal vector can be solved analytically resulting in a semi‐analytical procedure. The isogeometric description of the CAD structure inherently yields the exact normal vector, its derivatives and a higher order continuity throughout the structure. Herein, the focus is on the solution technique in the thickness direction and the challenges are addressed. The power of the formulation is outlined in several numerical examples of static and dynamic analysis and a comparison to shell formulations in literature is provided.

An efficient isogeometric buckling optimization framework for multi-patch laminated shells: Leveraging 3D solid elements, lamination parameters and penalty method

Buckling is a common failure mode of laminated shell structures during service, and isogeometric analysis methods are highly suitable for the buckling analysis due to their accurate geometric modeling capability and high-order continuity. However, the isogeometric buckling analysis for a multi-patch structure and its optimal design still face challenges in computational efficiency. To address this issue, this paper proposes an integrated framework for the buckling optimization of multi-patch laminated shells, which leverages 3D solid elements, lamination parameters, and the penalty coupling method. Solid-shell elements have advantages in shell analysis due to their simplicity, absence of rotational degrees of freedom, and adaptive layering in the thickness direction. Lamination parameters simplify the optimization of fiber orientations in laminated shells by condensing the design variable space and improving the convexity of the optimization problem. The penalty method can handle the coupling in multiple patches at a relatively low cost. Nevertheless, lamination parameters are rarely applied to the case of solid-shell elements. In addition, despite using lamination parameters, the number of design variables for multi-patch structures remains significantly large. To tackle these issues, this study derives the solid-shell formulation in lamination parameters and the sensitivity analysis formulae in the reverse mode. Experiment validation is conducted to assess the accuracy of the proposed method based on lamination parameters with solid elements, the feasibility of the multi-patch optimization model, and the computational efficiency of optimization sensitivity based on the reverse mode compared to the forward counterpart common in the literature.

Isogeometric topology optimization for maximizing band gap of two-dimensional phononic crystal structures

A smooth and efficient isogeometric topology optimization (ITO) method for maximizing band gaps in two-dimensional phononic crystals is developed in the paper. The band gaps of phononic crystals are computed by the NURBS-based isogeometric analysis, and the material distributions of two-dimensional phononic crystals are represented by the smooth, high-order continuity of the NURBS surface. The densities defined at each control point of the NURBS surface are employed as optimized design variables. The isogeometric analysis optimization using the same spline technique for both geometry and numerical analysis can benefit the optimization procedure and obtain smooth optimized structures, which can be easily used in 3D printing. The maximizing band gaps of phononic crystals for both out-of-plane and in-plane wave modes are obtained here using the present ITO approach. Numerical examples validated the effectiveness and reliability of the ITO method in broadening the band gaps and finding the optimal phononic crystals. And the numerical results show that the smooth optimized structures are obtained with fewer iterations.

A statistical high-order reduced model for nonlinear random heterogeneous materials with three-scale micro-configurations

An effective statistical higher-order three-scale reduced homogenization (SHTRH) method is established to analyze the nonlinear random heterogeneous materials with multiple micro-configurations. Firstly, the various unit cell functions based on the microscale and mescoscale regions are given, and two expected homogenization coefficients are computed through Kolmogorov's strong laws of large number. Further, the nonlinear homogenized equations are formulated, and the corresponding reduced-order multiscale systems for displacement and stress solutions are derived by using the high-order unit cell solutions and homogenized solutions. The key features of the new statistical multiscale methods are (i) the novel reduced models established to solve the inelastic problems of random composites at a fraction of cost, (ii) the high-order homogenized solutions which do not need high-order continuity for the macro solutions of the random problems and (iii) the statistical high-order multiscale algorithms developed for analyzing the nonlinear random composites with three-scale structures. Finally, the effectiveness and correctness of the algorithm are confirmed according to several hyperelastic, plasticity and damage periodic/random composites with multiple-scale configurations. The computation shows that the proposed SHTRH methods are useful for analyzing the macroscopic nonlinear performance, and can efficiently catch the microscopic and mesoscopic information for the random heterogeneous composites.

Wideband Vibro-Acoustic Coupling Investigation in Three Dimensions Using Order-Reduced Isogeometric Finite Element/Boundary Element Method

This study introduces an innovative model-order reduction (MOR) technique that integrates boundary element and finite element methodologies, streamlining the analysis of wideband vibro-acoustic interactions within aquatic and aerial environments. The external acoustic phenomena are efficiently simulated via the boundary element method (BEM), while the finite element method (FEM) adeptly captures the dynamics of vibrating thin-walled structures. Furthermore, the integration of isogeometric analysis within the finite element/boundary element framework ensures geometric integrity and maintains high-order continuity for Kirchhoff–Love shell models, all without the intermediary step of meshing. Foundational to our reduced-order model is the application of the second-order Arnoldi method coupled with Taylor expansions, effectively eliminating the frequency dependence of system matrices. The proposed technique significantly enhances the computational efficiency of wideband vibro-acoustic coupling analyses, as demonstrated through numerical simulations.

Shape and size optimization framework of stiffened shell using isogeometric analysis

A shape and size optimization framework is developed to improve the structural performance of the stiffened shell using isogeometric analysis. To accurately model the stiffened shell, the Lagrange multiplier method is employed to establish the coupling relationship of isogeometric degenerated shell elements and isogeometric degenerated beam elements. Due to the advantages of the non-uniform rational B-spline (NURBS), the unit normal direction vectors of the shell can be defined analytically, which guarantees stiffeners perpendicular to the skin. Additionally, the stiffeners can be easily generated based on the mapping relationship of the parametric space and the physical space of the shell. Analytical sensitivity is derived in detail, and the gradient-based optimization method can be used to solve the minimum compliance optimization problem owing to the high order continuity of NURBS. Three numerical illustrative examples are presented to verify the effectiveness of the proposed framework for the shape and size optimization of stiffened shells, consisting of a cylindrical shell patch, a hyperbolic shell, and a complex engineering segment. It is worth noting that the proposed framework is especially suitable for optimization problems of the stiffened shell, eliminating the need of complex feature extraction.

Investigating the bending and buckling behaviors of composite porous beams reinforced with carbon nanotubes and graphene platelets using a TRPIM path following mesh-free approach

The aim of the present work consists in investigating the nonlinear behavior of porous beams reinforced with graphene platelets (GPL) and supported carbon nanotubes (CNT), termed functionally graded graphene platelets reinforced composite beam (FG-GPLRC) and functionally graded nanotube carbon reinforced composite beam (FG-CNTRC), respectively. Notably, the distribution of GPL/CNT is explored in both uniform and non-uniform patterns across the beam's thickness. What sets this research apart is its utilization of a refined beam model as enhanced FSDT incorporating nonlinear shear terms which is a crucial advancement in accurately capturing the post-buckling response in certain boundary conditions, a feature lacking in the existing FSDT literature. Innovatively, the post-buckling load-deflection relationship is derived through the solution of governing equations incorporating cubic nonlinearity. This is achieved by employing Galerkin's method alongside a non-iterative high-order continuation technique based on the asymptotic numerical method coupled with the Tchebychev-radial point interpolation method (TRPIM), using a path-following where the solutions are obtained branch-by-branch by eliminating the need for iterative processes. In essence, this research underscores the pivotal role of porosity and GPL/CNT reinforcement in shaping the post-buckling configuration of both perfect and imperfect nanocomposite beams, thereby advancing our understanding of structural behavior in porous nanocomposite materials. The findings of this study illuminate the significant influence of parameters such as porosity coefficient, porosity distribution, GPL/CNT distribution, and GPL-weight/CNT-volume fraction on the nonlinear buckling behavior of porous beams.

Broadband topology optimization of three-dimensional structural-acoustic interaction with reduced order isogeometric FEM/BEM

This paper presents a model order reduction method to accelerate broadband topology optimization of structural-acoustic interaction systems by coupling Finite Element Methods and Boundary Element Methods. The finite element method is used for simulating thin-shell vibration and the boundary element method for exterior acoustic fields. Moreover, the finite element and boundary element methods are implemented in the context of isogeometric analysis, whereby the geometric accuracy and high order continuity of Kirchhoff-Love shells can be guaranteed and meantime no meshing is necessary. The topology optimization method takes continuous material interpolation functions in the density and bulk modulus, and adopts adjoint variable methods for sensitivity analysis. The reduced order model is constructed based on second-order Arnoldi algorithm combined with Taylor's expansions which eliminate the frequency dependence of the system matrices. Numerical results show that the proposed algorithm can significantly improve the efficiency of broadband topology optimization analysis.

A novel quasi-smooth tetrahedral numerical manifold method and its application in topology optimization based on parameterized level-set method

In this paper, a novel quasi-smooth tetrahedral numerical manifold method (NMM) and its two-dimensional (2D) counterpart are proposed. A new topology optimization method is established by combining the quasi-smooth manifold element (QSME) with the parameterized level set method (PLSM). The QSME introduces an innovative displacement function characterized by high accuracy and high-order continuity, effectively addressing the “linear dependence” (LD) issue inherent in traditional high-order NMM. To integrate QSME and PLSM, the corresponding optimization formulations and sensitivity analyses are provided. In order to fully utilize advantages of this novel quasi-smooth NMM and the PLSM, an element subdivision technique based on model recognition is proposed to accurately capture the physical boundaries. Additionally, a volume fraction update method based on element refinement is proposed. Taking advantage of the characteristics of the PLSM, a structure visualization method based on the sign distance function is developed to accurately describe curve boundary. This method allows for precise visualization of optimized structures. This study verifies high efficiency of the QSME-based PLSM for minimum compliance topology optimization problems in both 2D and 3D structures. Some representative structural optimization examples are tested to demonstrate effectiveness of the proposed method in both 2D and 3D problems, especially in complex design domain.

Investigation of FGM membrane wrinkling using reduced models based on a multi-scale method

In this article, we show that the reduced model proposed in our recent article to study the wrinkling of homogeneous elastic membranes does not reproduce the wrinkling observed in the case of an FGM-type composite membrane. New reduced models are then proposed to investigate the wrinkling of membranes made from Functionally Graded Materials (FGM). We examine, by this way, the influence of material functional parameters on determining the critical load and post-critical behavior under different loads. These reduced models, created using a multi-scale method based on Fourier series with slowly varying coefficients, enrich the reference model. In addition, these reduced models also allow us to significantly decrease the computation time and the number of finite elements compared to the complete model. The robustness of these reduced models is demonstrated in the case of wrinkling of membranes modeled by very thin shells, which are discretized using DKT18-type finite elements as described in literature. The resulting nonlinear problems are solved using a High-Order Continuation Method (HOCM).

Accelerating multivariate functional approximation computation with domain decomposition techniques

Modeling large datasets through Multivariate Functional Approximations (MFA) provide an elegant way to handle many visualization and scientific analysis workflows. The process necessitates scalable data partitioning methods to compute MFA representations efficiently without compromising the accuracy or continuity of the reconstructed solution. We propose a domain-decomposed method for computing the MFA with B-spline bases, which reduces the total work per task and uses a restricted Additive Schwarz (RAS) method to converge the control point data degrees-of-freedom along subdomain boundaries. We provide an in-depth analysis of the parallel approach with domain decomposition solvers, aiming to minimize local subdomain error residuals and recover high-order continuity at subdomain interfaces with appropriate choices of knot overlaps. The communication cost, determined by the overlap regions in the RAS implementation, is optimized to recover the numerical error profile of the single subdomain case. Our proposed method stands in contrast to previous methods, which typically only recover either C0 or at best C1 continuity for arbitrary B-spline degree expansions, or those that require post-processing to blend discontinuities in the reconstructed data. We demonstrate the effectiveness of our approach using analytical and real-world datasets in 1D, 2D, and 3D through both strong and weak scaling studies. The performance results indicate that the overall cost of computing the approximation is directly proportional to the underlying nearest-neighbor communication implementation, and is only weakly dependent on the overlap region size that determines the size of the messages. This finding underscores the efficiency and scalability of our proposed method, making it a promising solution for handling large datasets in scientific workflows.

A Hermite-type collocation mesh-free approach for simulating incompressible viscous fluid flows

In this work, a novel Hermite-type mesh-free model for incompressible viscous fluid flows is proposed, as a way, to overcome the numerical difficulties experienced by solving under weak or strong formulation the pure stream-function Navier–Stokes equation. In this model a dimensionless virtual domain is utilized, in order to make an accurate dimensionless shape-functions of the recently developed Hermite-type collocation Weighted Least Square approximation, by using dimensionless scale parameters. Then, a Jacobian transformation matrix is built to express the shape-functions spatial derivatives from the virtual to the considered domain. The resolution of the obtained nonlinear algebraic system is performed by the High Order Continuation Method. Numerical tests were investigated on incompressible viscous isothermal fluid flows in lid-driven cavity, backward-facing step, sudden expansions and around various obstacle shapes. They show the advantages of the proposed model compared to the reference Hermite-type collocation models used for pure stream-function equation and the weak Galerkin procedures as Finite Element Method and Element Free Galerkin method, in term of computation accuracy, number of degree of freedom, implementation simplicity and computation time. The numerical and experimental results of literature confirm the validity and capability of the proposed dimensionless model to simulate fluid flows in a expended-lengths inflow–outflow geometry and around obstacles of circular, inclined elliptical and Joukowski airfoil shapes.

State-of-the-art review on meshless methods in the application of crack problems

Meshless or meshfree methods (MMs) are practical and excellent numerical techniques for solving crack problems. With the inherent limitations of mesh-based methods such as the finite element method (FEM) in modeling cracks, MMs have captivated the attention of researchers worldwide because of their conspicuous merits, including mesh independence and higher-order continuity. However, the most recent review literature on MMs in crack problems is only available until 2018, and no relevant analyses have been reported in detail since then. To fill this gap, this state-of-the-art review explores the novel potential and advancements of MMs in crack problems. Generally, eight approaches are implemented with MMs to reproduce crack behaviors. To improve efficiency and stability, the existing strategies are explored in terms of numerical integration, method coupling, and adaptive analysis. Based on these techniques, 82 highly cited and influential articles are analyzed to determine the specific MMs, simulation techniques, and the enhanced strategies that have been adopted. According to existing studies, various novel MMs have been proposed to analyze cracks. Furthermore, flexible coupling of multiple techniques to simulate cracks can be applied to different forms of cracks in various materials. However, adaptive analysis still needs to be widely performed in the MMs for crack problems due to the absence of a solid mathematical foundation for error estimators. This paper presents the research gaps and the future prospect of MMs in the application of crack problems. It should be stated that significant works on numerical methods for modeling cracks predating 1994 are not cited in this paper, a decision considered by the authors to be non-essential to its scope and objectives.

Optimization of multi-target continuous dynamic trajectory for unmanned aerial vehicles

In the problem of trajectory optimization for unmanned aerial vehicles (UAVs) passing through multiple target points, continuous velocity and acceleration in the process of optimization are essential for maintaining a feasible trajectory. This paper proposes a multi-segment holistic Bezier shaping method (MHBSM) for dynamic trajectory optimization. The MHBSM not only achieves holistic optimization of UAVs passing through a multi-target trajectory but also ensures high-order continuity at the junctions of trajectory segments. Unlike methods that generate non-dynamic trajectories by connecting arcs and straight lines (such as Dubins curves) without considering speed changes, MHBSM optimizes the trajectory while taking into account the dynamics of UAVs. The method optimizes the differential flat output space of a fixed-wing UAV dynamic model and transforms the dynamic optimization problem into a three-dimensional position optimization problem. The problem is then further transformed into an optimization of control points using the Bezier curve. The resulting trajectory satisfies the dynamic constraints. In comparison to the Gaussian pseudospectral method (GPM), MHBSM achieves similar optimization results with only a 2.58% difference, while requiring approximately 1.36% of the computational time on average. The MHBSM demonstrates excellent convergence performance while satisfying the constraints of continuous acceleration and obstacle avoidance.

Higher-Order Continuity of Pullback Random Attractors for Random Quasilinear Equations with Nonlinear Colored Noise

Higher-Order Continuity of Pullback Random Attractors for Random Quasilinear Equations with Nonlinear Colored Noise

Isogeometric boundary element method for axisymmetric steady-state heat transfer

Isogeometric analysis (IGA) utilizes the Non-Uniform Rational B-Splines (NURBS) basis functions, which are commonly employed in Computer Aided Design (CAD), to both construct the problem’s geometry and approximate the field variables. In axisymmetric scenarios, the 3D problem can be simplified to 2D, requiring only the discretization of the 1D curve boundary for the boundary element method (BEM). This study proposes an isogeometric boundary element method (IGABEM) to simulate steady-state heat transfer in axisymmetric domains. Both the precise geometry of the boundary and the physical variables are approximated with the same NURBS basis functions, yielding higher-order continuity, superior accuracy per degree of freedom (DOF), and continuous nodal gradients with fewer DOFs than the traditional BEM. Additionally, a zoning method is introduced to handle heterogeneities. Five cases including a solid cylinder, a revolution hyperboloid with a coaxial cylindrical tube, an ellipsoid with a spherical cavity, a bimaterial ellipsoid and a complex geometry with a toroidal tube validate the accuracy, convergence, computational efficiency of IGABEM, and the applicability in addressing multiply-connected domains. The method outperforms Finite Element Method (FEM) and conventional BEM in accurately handling steady-state axisymmetric heat transfer issues under Dirichlet, Neumann, and Robin boundary conditions.

Vibration Bandgap Analysis of Periodic Composite Microplates using Isogeometric Approach

For periodic composite microplates, the microstructure dependent effect has great influence on the elastic wave vibration band gaps. In this paper, an effective isogeometric analysis method based on the modified couple stress theory (MCST) is proposed to calculate the vibration band gaps in periodic composite microplates. The MCST is employed to capture the microstructure effect of vibration band gaps in periodic composite microplates. The high order continuity requirement of MCST can be easily satisfied by applying the NURBS based isogeometric analysis approach. The elasto-dynamics and discrete equations of isogeometric analysis are obtained under the Mindlin kinematics assumptions. The present approach is validated by comparison with the analytical solution. And the influence of microstructure effect, volume fraction and unit cell length on the band gaps are investigated in details.

Analytical and isogeometric solutions of flexoelectric microbeams based on a layerwise beam theory

The theory of layerwise beam, which considers both microstructure and flexoelectric effects, is used in the curvature-based flexoelectricity theory. The displacement field function of the layerwise beam model is given. The governing equations and boundary conditions are obtained using a variational formulation based on Hamilton's principle. Analytical solutions for static bending and free vibration of a simply supported layerwise beam are derived. Isogeometric analysis (IGA) with higher-order continuity considering the flexoelectric effect is presented to numerically solve the static bending and free vibration problems. The results obtained from the analytical and IGA approaches are compared through several examples in which the microstructure and flexoelectric effects on the mechanical responses of layerwise nanobeams are analyzed.

Nonlinear thermo-mechanical stress-driven modeling of nano arches augmented by higher order double-scaled kernel

Modeling the nonlocal behavior of small-scale structures is almost controversial. The recently developed approach, known as the stress-driven nonlocal model, in which the stress is the input of nonlocal integral, has been proved to pose as a remedy for the ill-posing behavior of strain-driven solutions. Though it is still in its early stages, stress-driven modeling of the nonlinear behavior of nano-scaled problems needs further thought. The coupled displacement field, in particular, makes nano-arches more challenging. The stress-driven formulation, enhanced by constitutive boundary conditions, reveals the thermo-mechanical buckling and post-buckling features of shallow nano-arches for the first time in this research. Furthermore, the single-scaled Helmholtz kernel is replaced with the more appropriate double-scaled kernel known as the bi-Helmholtz averaging nonlocal kernel. Higher-order continuity of the bi-Helmholtz kernel and, hence, constitutive boundary conditions and higher-order differential equations corresponding to the stress-driven scheme hold the key to efficiency. The arch is subjected to uniform temperature and radial pressure at the same time. To demonstrate the response of the nano-arch during buckling and post-buckling, the nonlinear equilibrium path, whether for mechanical or thermal loading, is given for the first time by prevailing on the challenges of the higher-order problems. Moreover, it would be beneficial if one could have the chance to recognize how and when the arch buckles. These valuable data are available in this study to give the limiting parameters at the moment of buckling, including limiting temperature, nonlocal parameter, and geometry.