C1 Continuity Research Articles

A Quasi-3D theory for bending, vibration and buckling analysis of FG-CNTRC and GPLRC curved beams

This paper proposes a finite element model based on Quasi-3D theory, which includes both normal and shear effects, to study the bending, vibration, and buckling responses of FG-CNTRC and GPLRC curved beams. A two-node beam element satisfying C1 continuity requirement is utilized to compute displacements, critical buckling loads, and natural frequencies for beams with various boundary conditions. To evaluate the precision of the suggested model, various numerical examples are conducted. Next, a comprehensive parameter study is carried out to study the effects of distribution patterns and fractional volume of reinforced materials, open angles, aspect ratios, and boundary conditions. Numerous numerical results can serve as benchmarks for subsequent investigations.

Accurate free and forced vibration behavior prediction of functionally graded sandwich beams with variable cross-section: A finite element assessment

The current investigation proposes an enhanced finite element beam model to assess the dynamic behavior of functionally graded sandwich beams with variable cross-sections. By integrating the shear effect, this beam model integrates both C0 and C1 continuities to generate elementary matrices using an efficient three-unknown shear deformation beam theory. The model focuses on advanced sandwich beams with functionally graded material face-sheets and metallic cores, exploring various geometric arrangements. The originality of this study addresses variable cross-sections in sandwich beams, evaluating the influence of material composition and geometric configurations on free and forced vibration responses, including different boundary conditions. The methodology uses the eigenvalue method for examining free vibration and applies Newmark’s algorithm for solving forced vibration problems. The current model expands the scope of analysis compared to previous models that primarily focus on uniform cross-sections and conventional material distributions. The developed model incorporates non-uniform geometrics, material gradient parameters, and advanced finite element analysis to enhance precision and efficacy through detailed parametric investigations and comparisons with existing literature. This contribution significantly improves the mechanical modeling of composite sandwich structures, particularly those incorporating functionally graded materials and complex geometrical aspects.

A novel finite element for the vibration analysis of tapered laminated plates

AbstractIn the present study, a new tapered finite element is developed by incorporating the in‐plane effect in our recently introduced element for the vibration analysis of internally tapered laminated plates. Stress and strain fields for a laminated plate with varying thickness along two orthogonal directions are expressed through extending classical laminated plate theory. Shape functions regarding bending and membrane behavior are extracted from the bending and axial basic displacement functions of the Euler‐Bernoulli beam, respectively. Several laminated plates with different taper configurations have been studied, and the results obtained from vibration analysis are compared with those of the literature. Since the element can capture the exact form of variation in thickness, the solution has shown to be competitively accurate with considerably fewer total degrees of freedom and elements in comparison with conventional finite elements.Highlight A new finite element for thin, tapered, laminated plates is introduced. Laminates' stiffness matrix is obtained for arbitrary thickness variation. A novel method is adopted to derive in‐plane shape functions. C1 continuity is guaranteed along the edges of elements. Calculation cost is considerably reduced.

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.

B-ITO: A MATLAB toolbox for isogeometric topology optimization with Bézier extraction of NURBS

The Bézier extraction can decompose the NURBS-based isogeometric analysis (IGA) elements into a series of Bézier elements with C0 continuity, which is considered into developing a higher efficient isogeometric topology optimization (ITO) method. The main focus of the current work is to develop a MATLAB toolbox for isogeometric topology optimization (ITO) with the Bézier extraction of NURBS, termed by the B-ITO. An isogeometric Bézier element data structure is constructed using the extraction operator, which keeps an analogous structure of classic finite element methods (FEM). It can effectively ensure the IGA to be easily integrated into the existing FEM codes with a little modification, which can also promote the development of the ITO in several problems using existing FEMs and topology optimization codes. The B-ITO toolbox mainly consists of the following modules: MODEL, PREIGA, BOUND, EXTRABEZIER2D, SHAPE2D, STIFFBEZ2D, SHEPHARD, ASSEMBLE2D, SOLVE, PLOT and OC. It allows engineers and researchers to easily perform optimization problems in the toolbox with direct modifications of the related modules for design objectives. Moreover, a much higher efficient version of the B-ITO with a module STIFFBER2D is proposed for the rectangular design domains with the help of the local persevering property of the Bernstein basis function space, which can to a great extent reduce the computational cost. Finally, several numerical examples are discussed using the B-ITO to demonstrate its effectiveness and efficiency.

Adaptive meshfree method for fourth-order phase-field model of fracture using consistent integration schemes

An adaptive fourth-order phase-field model is proposed using consistent element-free Galerkin method (EFG). Convenient construction of high-order interpolation functions in EFG method is fully utilized. Requirement of C1 continuity of shape functions is satisfied in fourth-order phase-field model. Two different consistent integration schemes with high efficiency and high accuracy, namely quadratically consistent one-point integration scheme and three-point integration scheme, are employed to evaluate the fourth-order phase-field equation and displacement equation, respectively. To reduce computational cost, the adaptive strategy of local background mesh refinement is established on basis of strain energy history and phase field. Background mesh is refined via the insertion of nodes at the midpoints of each side of the integration element. Comparing with EFG method, computational accuracy and efficiency are enhanced by the proposed method. Load-displacement response, the smoothness of the resulting stress and crack paths are predicted well.

Adaptive output-constrained finite-time formation control for multiple unmanned surface vessels with directed communication topology

This paper tackles the intricate task of leader-following formation control for multiple unmanned surface vessels (USVs) operating over a directed communication network. Our investigation takes into meticulous consideration several critical variables, including prescribed output constraints, the presence of enigmatic dynamics, and the pervasive influence of external disturbances on each individual USV. Our methodology commences with the introduction of an ingenious output transformation technique, which adeptly converts constrained output variables into their unconstrained counterparts. Subsequently, we craft a novel distributed tracking error metric, artfully harnessing the potential of the transformed output. We build an adaptive, non-singular finite-time formation controller, firmly rooted in the backstepping framework. This controller seamlessly integrates vital components such as virtual velocity control, command filters, neural networks, and parametric adaptive strategies, which is underpinned by our meticulous attention to maintaining C1 continuity within our proposed virtual control law. Furthermore, we introduce the concept of minimum parameter learning, a strategic inclusion aimed at streamlining the computational demands of our controller. Significantly, any potential output overshoot resulting from minimum parameter learning is adeptly managed through our innovative output-constrained control strategy. To substantiate the effectiveness and robustness of our approach, we provide an exhaustive stability analysis, which offers conclusive evidence of our controller’s performance under a diverse array of conditions.

A C0 continuous mixed FE formulation for bending of laminated composite plates based on unified HSDT

AbstractThis study proposes a unified C0 continuous mixed finite element (MFE) formulation for the accurate and efficient prediction of stress components in laminated composite plates relying on Higher Order Shear Deformation Theory (HSDT). This unified form of the MFE accepts any convenient function for the representation of transverse shear deformation. The Hellinger–Reissner variational principle is employed for the derivation of MFE equations within a two‐field formulation involving stress resultants along with kinematical variables. Thus, the displacement and stress resultant fields are obtained directly from the global solution of the system of equations. In this manner, the in‐plane stress components are calculated over constitutive relations at the nodes without any need for error‐prone spatial derivatives. Furthermore, the independent interpolation of kinematic and stress resultant type variables allows the numerical solution to overcome the shear‐locking problem and ensure C0 continuity requirement. Numerical examples include convergence and comparison tests of predicted displacements and stress components under various boundary conditions and material configurations. Various test cases are considered for both the thin and thick plates subjected to sinusoidal and uniformly distributed loads. It is demonstrated that the proposed MFE formulation can capture stress components with high accuracy while being computationally efficient.

Whip-Bezier: A C1-continuous hardening law for efficient direct and inverse identification

In 3D constitutive models, the hardening law is an essential model ingredient that has a first order effect on the predicted stress–strain response. In most phenomenological plasticity models, the hardening law coincides with the stress–strain curve for monotonic uniaxial loading. When identifying the hardening law for large strains through hybrid experimental–numerical approaches, the numerical properties of the hardening law play an important role. Here, we propose a novel piecewise-defined hardening law based on a composition of quadratic Bezier curve segments. The Whip-Bezier formulation preserves C1 continuity and achieves a parametrization with less parameters than its piecewise-linear counterpart. Both the possibility of identifying the hardening parameters in a decoupled optimization (e.g. from in-plane torsion experiments) and the use of a fully-coupled identification scheme (e.g. from notched tension experiments) are detailed. When contrasted with widely-used analytical expressions, the Whip-Bezier model consistently demonstrates enhanced performance with increasing number of parameters. This improvement comes with minimal statistical variance and reduced dependence on the initial optimization guess. Furthermore, the fitting quality is at least equivalent to that of the Swift-Voce hardening law with the same number of parameters.

Design of Variable Stiffness Trajectories with Cubic Ferguson Curve.

To design a class of full cover trajectories that satisfy curvature limitations and enhance the buckling load of constructed laminates, a variable stiffness laminate is proposed by applying the cubic Ferguson curve. First, the traditional explicit form of the cubic Ferguson curve is redefined as polar coordinates, two connected Ferguson curve segments with three extra parameters are applied to describe full cover trajectories, and the effects on trajectories introduced by these modifications are discussed. Then, the finite element method is used to introduce parameters for analyzing the buckling load of the designed variable stiffness laminates. Numerical experiments show that automatic fiber placement (AFP) trajectories described by the cubic Ferguson curve can automatically reach C1 continuity and can be locally modified by adjusting the introduced parameters. Compared with traditional constant stiffness laminates, the variable stiffness laminates designed using the proposed method exhibit a higher buckling load and better stability.

Overcoming membrane locking in quadratic NURBS-based discretizations of linear Kirchhoff–Love shells: CAS elements

Quadratic NURBS-based discretizations of the Galerkin method suffer from membrane locking when applied to Kirchhoff–Love shell formulations. Membrane locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of the membrane forces. Continuous-assumed-strain (CAS) elements have been recently introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al., CMAME, 2022). In this work, we generalize CAS elements to vanquish membrane locking in quadratic NURBS-based discretizations of linear Kirchhoff–Love shells. CAS elements bilinearly interpolate the membrane strains at the four corners of each element. Thus, the assumed strains have C0 continuity across element boundaries. To the best of the authors’ knowledge, CAS elements are the first assumed-strain treatment to effectively overcome membrane locking in quadratic NURBS-based discretizations of Kirchhoff–Love shells while satisfying the following important characteristics for computational efficiency: (1) No additional degrees of freedom are added, (2) No additional systems of algebraic equations need to be solved, (3) No matrix multiplications or matrix inversions are needed to obtain the stiffness matrix, and (4) The nonzero pattern of the stiffness matrix is preserved. The benchmark problems show that CAS elements, using either 2 × 2 or 3 × 3 Gauss–Legendre quadrature points per element, are an effective locking treatment since this element type results in more accurate displacements for coarse meshes and excises the spurious oscillations of the membrane forces. The benchmark problems also show that CAS elements outperform state-of-the-art element types based on Lagrange polynomials equipped with either assumed-strain or reduced-integration locking treatments.

Higher-order phase field fracture simulation in nearly incompressible viscoelasticity

The phase field modeling of viscoelastic fracture can effectively predict complex crack behavior and offer valuable insights for analyzing time-dependent failure mechanisms. However, its application in nearly incompressible viscoelastic materials presents several numerical challenges, including the volumetric locking and inhibition of crack opening. In addition, the low computational efficiency remains a significant challenge in the development of the phase field models (PFMs). Previous studies have shown that the fourth-order phase field theory can improve the convergence rate of the numerical solution with a smoother phase field. In this study, a fourth-order PFM is proposed, which incorporates the B-bar method and the material penalization to predict the fracture behavior of viscoelastic materials limited by the near compressibility. The additional C1 continuity requirement on the phase field is ensured in the isogeometric framework. No extra degrees of freedom are introduced to resolve the locking problem, which is practical for the PFM, especially in three-dimensional (3D) applications. The loosening of the near incompressibility constraints in the fully damaged zone is achieved by the material penalization, which allows the crack to open freely while preserving the constraints in other regions. The accuracy of the fourth-order PFM is verified through one-dimensional (1D) simulations. Benchmark examples are performed to investigate the numerical behavior of the model. Comparisons between the predicted results and the existing data demonstrate the effectiveness and accuracy of the proposed PFM.

On C0 and C1 continuity of envelopes of rotational solids and its application to 5-axis CNC machining

We study the smoothness of envelopes generated by motions of rotational rigid bodies in the context of 5-axis Computer Numerically Controlled (CNC) machining. A moving cutting tool, conceptualized as a rotational solid, forms a surface, called envelope, that delimits a part of 3D space where the tool engages the material block. The smoothness of the resulting envelope depends both on the smoothness of the motion and smoothness of the tool. While the motions of the tool are typically required to be at least C2, the tools are frequently only C0 continuous, which results in discontinuous envelopes. In this work, we classify a family of instantaneous motions that, in spite of only C0 continuous shape of the tool, result in C0 continuous envelopes. We show that such motions are flexible enough to follow a free-form surface, preserving tangential contact between the tool and surface along two points, therefore having applications in shape slot milling or in a semi-finishing stage of 5-axis flank machining. We also show that C1 tools and motions still can generate smooth envelopes.

A Surface Subdivision Scheme Based on Four‐Directional S13 Non‐Box Splines

AbstractIn this paper, we propose a novel surface subdivision scheme called non‐box subdivision, which is generalized from four‐directional S13 non‐box splines. The resulting subdivision surfaces achieve C1 continuity with the convex hull property. This scheme can be regarded as either a four‐directional subdivision or a special quadrilateral subdivision. When used as a quadrilateral subdivision, the proposed scheme can control the shape of the limit surface more flexibly than traditional schemes due to the natural introduction of auxiliary face control vertices.

A Galerkin approach for analysing coupling effects in the piezoelectric semiconducting beams

The piezoelectric semiconductor combines the characteristics of both piezoelectric and semiconducting materials, making it highly promising for a variety of engineering applications. The present work adopts a Galerkin approach to derive a general weak form of the coupled governing equations for the analysis of piezoelectric semiconductors. The coupling effect between the strain, the electric field, and the gradients of the concentrations of holes and electrons has been considered. Piezoelectric semiconductors are often deployed in the shape of rods and beams and therefore commonly experience axial and transverse load. Thus, the Euler–Bernoulli beam theory is employed to formulate a numerical approach to analyse the coupling behaviour of such piezoelectric semiconductor beams. In order to satisfy the C1 continuity requirement of Euler–Bernoulli beam theory without introducing additional degrees of freedom, a discretisation with B-Splines is used. Non-mechanical variables, including electrical potential, holes and electrons concentrations, are decomposed into constant, linear and quadratic terms through series expansion along the thickness, which ensures compatibility with beam formulation. To validate the proposed method, an N-type piezoelectric semiconducting beam under a sinusoidal distributed load, for which an analytical solution is available, is initially examined. Subsequently, complex examples including a cantilever beam with generalised loading conditions and a beam with PN-junction are analysed to demonstrate the capabilities of the proposed method in handling more challenging scenarios.

The interior penalty virtual element method for the fourth-order elliptic hemivariational inequality

We develop an interior penalty virtual element method (IPVEM) for solving a Kirchhoff plate contact problem, which can be described by a fourth-order elliptic hemivariational inequality (HVI). The virtual element space in IPVEM is constructed by modifying the H2-conforming virtual element space, and the number of degrees of freedom is greatly reduced. To force the C1 continuity, the interior penalty scheme is adopted, that is, we introduce a penalty term for the jump of normal derivatives on mesh edges and two additional terms to guarantee the consistency and symmetry of the scheme. With certain assumptions, the well-posedness of the discrete problem is proved. Furthermore, a priori error estimation is established for the IPVEM for the fourth-order elliptic HVI, and we show that the lowest-order VEM achieves optimal convergence order. Finally, some numerical examples are presented to support the theoretical results.

Adaptive isogeometric phase-field modeling of the Cahn–Hilliard equation: Suitably graded hierarchical refinement and coarsening on multi-patch geometries

We present an adaptive scheme for isogeometric phase-field modeling, to perform suitably graded hierarchical refinement and coarsening on both single- and multi-patch geometries by considering truncated hierarchical spline constructions which ensure C1 continuity between patches. We apply the proposed algorithms to the Cahn–Hilliard equation, describing the time-evolving phase separation processes of immiscible fluids. We first verify the accuracy of the hierarchical spline scheme by comparing two classical indicators usually considered in phase-field modeling, for then demonstrating the effectiveness of the grading strategy in terms of accuracy per degree of freedom. A selection of numerical examples confirms the performance of the proposed scheme to simulate standard modes of phase separation using adaptive isogeometric analysis with smooth hierarchical spline constructions.

Kirchhoff–Love shell representation and analysis using triangle configuration B-splines

This paper presents the application of triangle configuration B-splines (TCB-splines) for representing and analyzing the Kirchhoff–Love shell in the context of isogeometric analysis (IGA). TCB-splines offer flexibility in modeling complex geometries with C1 continuity, making them naturally fit into the Kirchhoff–Love shell formulation with complex geometries. We first propose a linear least-squares-based framework to reparametrize the mid-surface of a thin shell, which consists of multiple (trimmed) NURBS patches and is topologically equivalent to an open disk with a finite number of holes, into a single TCB-surface defined over a carefully computed parametric domain. We then utilize TCB-splines for geometric representation and solution approximation in shell analysis. We verify the accuracy and robustness of our method by applying it to linear and nonlinear benchmark shell problems. The applicability of the proposed approach to shell analysis is further exemplified by performing geometrically nonlinear Kirchhoff–Love shell simulations of a pipe junction and a front bumper represented by a single patch of TCB-splines.

Combined SGC-Ball Interpolation Curves: Construction and IGEO-Based Shape Optimization

With the swift advancement of the geometric modeling industry and computer technology, traditional generalized Ball curves and surfaces are challenging to achieve the geometric modeling of various complex curves and surfaces. Constructing an interpolation curve for the given discrete data points and optimizing its shape have important research value in engineering applications. This article uses an improved golden eagle optimizer to design the shape-adjustable combined generalized cubic Ball interpolation curves with ideal shape. Firstly, the combined generalized cubic Ball interpolation curves are constructed, which have global and local shape parameters. Secondly, an improved golden eagle optimizer is presented by integrating Lévy flight, sine cosine algorithm, and differential evolution into the original golden eagle optimizer; the three mechanisms work together to increase the precision and convergence rate of the original golden eagle optimizer. Finally, in view of the criterion of minimizing curve energy, the shape optimization models of combined generalized cubic Ball interpolation curves that meet the C1 and C2 smooth continuity are instituted. The improved golden eagle optimizer is employed to deal with the shape optimization models, and the combined generalized cubic Ball interpolation curves with minimum energy are attained. The superiority and competitiveness of improved golden eagle optimizer in solving the optimization models are verified through three representative numerical experiments.

Adaptive grid deformation method for CFD application to hull optimization

This work introduces a new grid deformation method for an efficient CFD-based optimization of ship hull forms. The method uses a two-level point transformation technique to manipulate grid points with a small number of design points. At the first level, generic B-spline is employed to transfer grid points based on movements of control points sampled inside a control box, ensuring accuracy and smoothness of surface modification. At the second level, Radial Basis Function with Wendland’s C2 continuity is adopted to interpolate movements of control points based on relatively few design points. It is shown to be effective in preserving good mesh quality and efficiency. The method is applied to the deformation of hull surfaces for ship models KVLCC2 and KCS, and to investigate the effects of bulbous bow on calm-water resistance with fixed Lpp. A regression model is proposed for ship length, location of buoyancy, wet surface area, and displacement. Numerical results show that the present method is well-suited for CFD-based hull form optimization with better efficiency.