High-order Element Research Articles

Extended and Generic Higher-Order Elements for MEMS Modeling

State-dependent resistors, capacitors, and inductors are a common part of many smart engineering solutions, e.g., in MEMS (Micro-Electro-Mechanical Systems) sensors and actuators, Micro/NanoMachines, or biomimetic systems. These memory elements are today modeled as generic and extended memristors (MR), memcapacitors (MC), and meminductors (ML), which are more general versions of classical MR, MC, and ML from the infinite set of the fundamental elements of electrical engineering, known as Higher-Order Elements (HOEs). It turns out that models of many complex phenomena in MEMS cannot be constructed only from classical and state-dependent elements such as R, L, and C, but that other HOEs with generalized behavior should also be used. Thus, in this paper, generic and extended versions of HOEs are introduced, overcoming the existing limitation to MR, MC, and ML elements. The relevant circuit theorems are formulated, which generalize the well-known theorems of classical memory elements, and their application to model complex processes of various physical natures in MEMS is shown.

Vibration characteristics of sandwich plates with an auxetic honeycomb core and laminated three-phase skin layers under blast load

In this article, vibration characteristics of sandwich plates with an auxetic honeycomb core and laminated three-phase skin layers subjected to blast load are studied. Higher-order ES-MITC3 element based on higher-order shear deformation theory (HSDT) to achieve the governing equations. The sandwich plates with the ultra-light feature of the auxetic honeycomb core layer (negative Poisson's ratio) and reinforced by two laminated three-phase skin layers. The obtained results in our work are compared with other previously published to confirm accuracy and reliability. In addition, the effects of parameters such as geometrical and material parameters on the vibration characteristics of sandwich plates with an auxetic honeycomb core and laminated three-phase skin layers are fully investigated.

The usability and limitations of the various absolute nodal coordinate beam elements subjected to torsional and bi-moment loading

This paper explores the usability of several beam elements based on the Absolute Nodal Coordinate Formulation (ANCF) subjected to torsional and bi-moment loading scenarios. ANCF beam element solutions are compared against an analytical solution and beam and solid finite element solutions. When subjected to bi-moment loading, most of studied higher-order ANCF beam elements suffer from strong “non-smooth” displacement continuity between elements, and therefore they should be used with caution. The distribution of warping displacement was shown to be unacceptable for four-node higher-order ANCF beam elements. Numerical results revealed that the higher-order elements failed to characterize local loading scenarios such as a bi-moment. This problem can be alleviated using very fine beam discretization, which would be more computationally intensive.

A boundary-penalized isogeometric analysis for second-order hyperbolic equations

Explicit time-marching schemes are popular for solving time-dependent partial differential equations; one of the biggest challenges these methods suffer is increasing the critical time-marching step size that guarantees numerical stability. In general, there are two ways to increase the critical step size. One is to reduce the stiffness of the spatially discretized system, while the other is to design time-marching schemes with larger stability regions. In this paper, we focus on the recently proposed explicit generalized-α method for second-order hyperbolic equations and increase the critical step size by reducing the stiffness of the isogeometric-discretized system. In particular, we apply boundary penalization to lessen the system’s stiffness. For pth order Cp−1 isogeometric elements, we show numerically that the critical step size increases by a factor of p2−3p+64, which indicates the advantages of using the proposed method, especially for high-order elements. Various examples in one and two dimensions validate the performance of the proposed technique.

3D reconstruction of non-structural surface of a stiffened hull plate based on RGB-D images

Deformation measurement of hull structure is of great significance for enhancing the quality of shipbuilding. In this paper, a three-dimensional (3D) reconstruction method based on RGB-D images is presented for non-structural surface of a stiffened hull plate. The algorithm for object surface region nodes (OSRN) is proposed to extract the features of point clouds and generate higher-order fitting surface model. Firstly, random sample consensus (RANSAC) algorithm and least square method are applied to eliminate the noise of point clouds. Spatial position of targets in RGB image and singular value decomposition (SVD) are utilized to accomplish the registration of point clouds from multiple camera views. Then, the point clouds data are clustered and fused to fit the quadric surface. On this basis, the 3D surface model is reconstructed through higher-order panel element. Two comparative experiments, including the measurements via the RGB-D camera and 3D Laser Scanner, are carried out to verify the applicability and accuracy of the proposed method.

Free vibration analysis of rectangular thin plates with corner and inner cutouts using C1 Chebyshev spectral element method

This paper presents a novel higher-order C1 continuous Chebyshev spectral element method for the free vibration analysis of rectangular thin plates with corner and inner cutouts. To achieve C1 continuity at the interfaces of adjacent elements, the Hermite interpolation functions through nonuniformly distributed Chebyshev nodes are employed to construct the element displacement field. A Gauss–Lobatto type quadrature is used to efficiently evaluate the element matrices. Comparison with other well-established methods show that the proposed element has good convergence and accuracy. The influence of cutout size, location and eccentric ratio on the natural vibration characteristics of plates are investigated.

A numerical approach for hydrodynamic performance evaluation of multi-degree-of-freedom floating oscillating water column (OWC) devices

A numerical approach has been developed in this study for hydrodynamic analysis of complex-shaped oscillating water column (OWC) converters. Physically, in the circumstance when the device is free to move or moored in waves, the airflow movement inside the chamber is coupled tightly with the changing of the internal fluid surface and the hydrodynamic response of the device. By taking these coupling effects into account, appropriate boundary integral equations are formulated with supplemental theoretical relations. The boundary value problem is then solved using a higher-order boundary element method (HOBEM). After obtaining the wave force and dynamic air force, the coupled motions and hydrodynamic efficiency are evaluated by integration. In particular, unlike conventional methods limited to fixed OWCs, the proposed method is proved applicable to those floating with coupled rigid-body motions. An optimal turbine parameter is hence mathematically derived to maximise the wave power absorption. Besides the linear quantities, nonlinear wave drift loads are evaluated via a newly derived formulation that accounts for the far-field contribution and the oscillating air pressure over the internal fluid surface. Based on the above methodology and the resulting tool, numerical studies are carried out for three cylindrical OWC scenarios that are floating in waves, attached by a reflector and connected to a submerged caisson, respectively. Different modes of body motion and typical free-surface oscillation modes in the chamber are analysed and discussed. It is found that, for an OWC with an attached arc-shaped reflector or connected to a submerged caisson, its wave power absorption can be characterised by a series of apparent peaks which are associated with the resonant internal fluid and in close proximity to the resonance frequencies of body motions involving surge, heave or pitch. Numerical results also indicate the benefits of floating OWCs from the coupled rigid-body motion modes and OWC’s resonance, expanding the frequency range of efficient conversion and improving the device’s adaptability to variable oceanic environments.

Experimental and numerical investigation of the deformation behaviour of cables and thin beam-like structures under multi-axial loading

In various applications, it is common to use thin beam-like structures, made of plastic or fibre-reinforced materials, as well as components such as cables. They are flexible, and the most common form of deformation is bending, but they can also be stretched or torqued. Due to their structural composition, a coupling between the different loading directions exists. This is especially pronounced for cables, where the different components interact with each other and the kinematics of each component are different. Thus, to characterise these materials, it is necessary to consider tension, torsion, bending, and a coupling of the three load cases. In this work, such characterisations are performed for a polyvinyl chloride rod, a carbon fibre-reinforced rod, and a coaxial cable. The three materials represent the isotropic and anisotropic material classes and include homogeneous and non-homogeneous cross-sections. An anisotropic elasto-plastic material model is implemented in the finite element method to model the behaviour of such structures. The material model includes anisotropic plasticity so that the structural effects can also be modelled for large deformations. Thin structures are discretised with higher-order elements, and a comparison of the experimental and the simulation results is presented.

Hydrodynamic analysis of resonant waves within the gap between a vessel and a vertical quay wall

Concerning the side-by-side offloading operations at gravity-based terminals, the wave interaction with a vessel alongside a long vertical wall in both normal and oblique seas was investigated. The imaging principle was applied to convert the original to an equivalent wave-diffraction problem by two symmetrical vessels in open seas exposed to bi-directional incident waves. The boundary integral equation method, in conjunction with a higher-order boundary element method (HOBEM), was used to solve the equivalent problem. Besides the linear free-surface elevation and wave loads, the second-order mean wave drift loads on the vessel was calculated based on either the direct pressure integration or using a semi-cylindrical control surface. Moreover, the solution was also developed for the oblique wave interaction with a two-dimensional barge-wall system using the eigenfunction expansion matching method. Systemic numerical studies were then conducted for the cases of a rectangular barge and a Wigley hull in close proximity to a long wall, respectively. Under oblique incidence, both the odd and even modes of fluid motion within the gap can be induced. In the case of a rectangular barge, the linear wave force can be obviously amplified at the location of the (1, 0) resonant mode, while the linear wave moment at the (2, 0) mode. In addition, the negative mean force along the transverse direction, which tends to push the barge away from the wall, can be apparently enhanced at both the odd and even resonant modes. Besides, in oblique seas, the (1, 0) resonant mode can give rise to the significant amplification of the positive mean yaw moment around the centre of the cross-sectional area of the barge, which tends to pull the stern of the barge (that faces incident waves) closer to the wall, while pushing the bow away from the wall. Such amplification gets less noticeable for higher modes. Numerical results also indicate that the hull geometry imposes a significant impact on the resonant waves in the gap. In the case of a Wigley hull, the amplification of the free-surface response is much less apparent when compared with a rectangular barge.

Optimization of a wave energy converter square array based on the differential evolution algorithm

The aim of optimizing wave energy converters (WECs) is to maximize the power output whilst reducing the cost of energy production. Here, the optimization of a square array composed of four identical cylindrical, heaving-buoy WECs is considered. Four optimization parameters, i.e., the radius and the draft of the buoy, the damping of the power take-off system, and the spacing of the WECs are considered. Two strategies are carried out to optimize the array in regular waves, and the performance of the optimized array in irregular waves is studied. The differential evolution algorithm is selected for the multi-parameter optimization. The higher-order boundary element method is used to simulate the interaction between waves and WECs according to linear potential flow theory. The array optimized by the second strategy extracts the maximum power, but its frequency adaptability is inferior to the array optimized by the first strategy. The optimized array in the first strategy can extract relatively large power from the irregular waves, whilst also exhibiting high energy capture efficiency.

Least-squares finite element method for ordinary differential equations

We consider the least-squares finite element method (lsfem) for systems of nonlinear ordinary differential equations, and establish an optimal error estimate for this method when piecewise linear elements are used. The main assumptions are that the vector field is sufficiently smooth and that the local Lipschitz constant as well as the operator norm of the Jacobian matrix associated with the nonlinearity are sufficiently small, when restricted to a suitable neighborhood of the true solution for the considered initial value problem. This theoretic optimality is further illustrated numerically, along with evidence of possible extension to higher-order basis elements. Examples are also presented to show the advantages of lsfem compared with finite difference methods in various scenarios. Suitable modifications for adaptive time-stepping are discussed as well.

Efficient time-domain spectral element with zigzag kinematics for multilayered strips

Efficient structural elements are widely sought for fast computation of wave propagation response of anisotropic multilayered structures for their design, non-destructive testing, and structural health monitoring purposes. Although zigzag theories are known to combine excellent accuracy with high computational efficiency, no spectral element has been developed so far based on these theories, which would enable fast and accurate simulation of guided waves in such structures. In this article, we develop an efficient time-domain spectral strip element for analyzing ultrasonic wave propagation in multi-material laminated beams and panels while accounting for the zigzag profile of the in-plane displacement across the thickness. The theory superimposes a layerwise linear (zigzag) variation of the in-plane displacement of a layerwise theory onto the global third-order variation of an equivalent single layer (ESL) third-order theory while retaining the computational advantage of the latter. The high-order element has an arbitrary number of unequally spaced nodes with only four kinematic variables at each node, regardless of the number of laminas. The recently proposed generalized Hermite-type C1-continuous Lobatto shape functions interpolate the deflection, while the axial displacement and shear rotation are interpolated using the C0-continuous Lobatto shape functions. A comprehensive numerical study establishes high efficiency, excellent accuracy, and fast convergence of the new element in analyzing free vibration and guided wave propagation in single-material composite and multi-material fiber-metal laminate structures. Its combined performance in terms of these parameters is much superior to its standard FE counterpart, the ESL theory-based elements, and the continuum elements in the considered frequency range.

Time-Domain Simulation of Second-Order Diffracted Forces on Marine Structures in Multidirectional Irregular Seas

Abstract A new method for the accurate discrete technique of directional spectra is proposed for the simulation of the second-order incident potential of directional seas, and a second-order time-domain analysis program is developed for nonlinear wave forces on marine structures in multidirectional irregular seas. The boundary-value problems for the diffracted potential are solved by a higher-order boundary element method (HOBEM). The fourth-order Adams–Bashforth–Moulton method combined with the Newton–Raphson iteration is implemented for time integration. The present method is validated by comparison with the published frequency-domain results for a bottom-mounted vertical cylinder in bichromatic bidirectional waves. The second-order diffracted forces on a truncated cylinder are examined for different combinations of storms and swell seas, and major efforts are focused on investigating the effects of wave directionality, directional spreading, and the spectral peak frequency of mixed seas.

A numerical method for the dynamic stability analysis of supported plates in open and confined axial flow

The interaction with axial flow can significantly alter the mechanical behavior of plates in qualitative and quantitative terms. A numerical method is presented for the dynamic stability analysis of plates subjected to uniform axial flow, which can serve as a general analysis tool for the positively supported plates within linear theory. The plate response is described in the modal space that the corresponding perturbations in the flow field are expressed by means of modal components of the assumed velocity potential field. The modal characteristics of the plate are identified by adopting the heterosis finite element over the Mindlin theory; the fluid action on the plate is computed through a higher-order boundary element solution of the flow problem. The plate-flow interaction is represented in terms of added mass, fluid damping, and fluid stiffness effects. The plate-flow system acts as either gyroscopic conservative or non-conservative, based on the support conditions of the plate leading and trailing edges. The method is applied on two sets of problems that involve rectangular plates in open and confined flow—its performance, accuracy, and adaptability are assessed and the dynamic behavior and stability characteristics of supported plates in axial flow are investigated.

Elements of high order in finite fields specified by binomials

Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to explicitly construct elements of high order in the field $F_q[x]/\langle x^m-a\rangle$. Namely, we find elements with multiplicative order of at least $5^{\sqrt[3]{m/2}}$, which is better than previously obtained bound for such family of extension fields.

Mechanical Deformation Analysis of a Flexible Finger in Terms of an Improved ANCF Plate Element

In recent years, flexible continuum robots have been substantially developed. Absolute nodal coordinates formulation (ANCF) gives a feasible path for simulating the behavior of flexible robots. However, the model of finger-shaped robots is often regarded as a cylinder and characterized by a beam element. Obviously, this is short of characterizing the geometrical feature of fingers in detail, especially under bending conditions. Additionally, for the lower-order plate element, it is hard to characterize the bending behavior of the flexible finger due to fewer nodes; a higher-order plate element often requires an extremely long computing time. In this work, an improved ANCF lower-order plate element is used to increase the accuracy of the Yeoh model and characterize the geometrical structure of silicone rubber fingers, taking into particular consideration the effect of volume locks and multi-body system constraints. Since it is a kind of lower-order plate element, essentially, the computing time is nearly the same as that of conventional lower-order plate elements. The validity of this model was verified by comparing it with the results of the published reference. The flexible finger, manufactured using silicone rubber, is characterized by the novel ANCF lower-order plate element, whereby its mechanical deformation and bending behavior are simulated both efficiently and accurately. Compared to the ANCF beam element, conventional lower-order plate element, and higher-order plate element, the novel plate element in this paper characterizes the external contour of the finger better, reflects bending behavior more realistically, and converges in less computing time.

Effect of incident wave angle to gap resonance between two fixed barges

With two barges arranged side-by-side subject to wave action, fluid motion within the narrow gap between the two barges can be significant under specific incoming wave frequencies. The phenomenon is commonly referred to as the gap resonance. A linear potential model using a higher-order boundary element method (hobem) is applied to investigate the effect of the incident wave angle on the resonant fluid motion within the gap for two fixed identical barges with square corners. An adaptive Gaussian method is used with hobem to improve the accuracy for nearly singular integration. The results show that gap resonant peaks at other incident wave angles can be more critical than the beam sea case which is commonly considered the most critical case. Also, the second mode peak value can surpass the first mode peak value at specific conditions and also be considered when during the design phase. Smaller gap width will increase the resonant frequency and peak value but decrease the incident wave angle at which the maximum resonant peak appears.

Prediction approach for wave forces on large-scale offshore structures under linear wave action

This paper presents a numerical mothed that is implemented by applying higher-order boundary elements in the hybrid integral equation method for predicting the wave forces on arbitrarily-shaped offshore structures. In this study, the virtual cylindrical boundary (i.e., matching boundary) is set to divide the fluid region into inner and outer subdomains. The velocity potentials in the outer domain are expressed using the eigenfunction expansion method. In the inner domain, the simple Green's function is adopted as the fundamental solution and Green's second theorem is applied to derive the boundary integral equation (BIE). The velocity potentials in the two subdomains are matched on the matching boundary. The higher-order boundary elements are ultimately employed to discretize BIE and the velocity potentials in the inner domain are then solved. The wave forces on the truncated and stepped cylinders are calculated using the presented method, and the results match well with analytical results, which illustrates the validity of the presented method. The effects of element sizes and modal truncation numbers are further explored, and the interaction between waves and square columns is subsequently analyzed. Eventually, the proposed method is extended to the practical engineering application of wave diffractions from quasi-elliptic pile caps for sea-crossing bridges. The results show that the proposed method can deal with boundaries with complex geometries. In addition, satisfactory results can be obtained by setting relatively few elements and control points on boundaries, and this method can also effectively predict wave forces on the multi-column array.

Stress and Failure Onset Analysis of Thin Composite Deployables by Global/Local Approach

The purpose of this work is to propose an innovative global/local approach for the failure onset analysis of deployable composite thin and ultrathin structures. The adoption of this technique allows for overcoming the weaknesses related to the commercial calculation codes, which make use of three-dimensional finite elements to capture the local stress field arising within the booms. In this framework, the proposed approach leads to a reduction of computational cost while keeping a high level of accuracy. This is possible thanks to the Carrera unified formulation, which allows for the modeling of higher-order plate elements for the analysis of local critical regions of complex composite assemblies. To show the potentialities of the global/local approach, particular attention is given to the problems that mostly characterize composites, among which are those related to the free edge, the failure concept, and the interlaminar continuity of the displacements and shear stresses. The analyzed booms are the triangular, rollable, and collapsible; a tape-spring hinge; and a telescopic tubular mast. The results show a perfect accuracy of the in-plane stress components and an accurate description of the out-of-plane components, which is essential for a reliable design of these components in terms of free-edge debonding and failure phenomena.

Design of multilayered VAT panels by means of higher-order plate elements

Design of multilayered VAT panels by means of higher-order plate elements