First-order Element Research Articles

Improvement on the axial force accuracy of the ANCF Euler-Bernoulli beam element with the second-order approximate function of the centerline and precise constraint equations

The conventional Euler-Bernoulli beam element of the absolute nodal coordinate formulation (ANCF) yields incorrect axial force due to the existence of the first-order approximate function of the centerline and over-constraint equations. This work aims to improve the axial force accuracy of the element by using the second-order centerline approximate function and precise constraint equations. The second-order function includes the quintic Hermite shape function and the vector of nodal coordinates containing zeroth to second-order derivatives of position. The elastic force vector of the element and its tangent stiffness were then formulated based on the decoupled elastic line method. The exact kinematic requirements of three types of constraint points were fully clarified. On this basis, the corresponding concise constraint equations, constraint Jacobian and Hessian matrices were rigorously derived in the context of ANCF, where G2 continuity was demonstrated to be satisfied at the node acted by an inclined concentrated force, the relative rotation constraint of the rigid joint was described using the dot product of the gradients on both sides of the joint, and the fixing gradient direction at the partially-clamped end was expressed by using the zero cross product of the end gradients in the initial and deformed configurations. Large deformation problems of typical beams under fixed and follower loads were solved using five types of elements, where the results of the ABAQUS B22 elements were taken as a reference for validation. It is found that when using the first- and second-order beam elements having over-constraint equations of the three types of constraint points, obvious axial force error occurs near the corresponding constraint points. In comparison, a few second-order elements containing the derived constraint equations output correct axial force consistent with ABAQUS. However, a large number of the first-order elements containing the correct constraint equations still yield inaccurate axial force in certain beam regions.

Targeting performance and user-friendliness: GPU-accelerated finite element computation with automated code generation in FEniCS

This paper studies the use of automated code generation to provide user-friendly GPU acceleration for solving partial differential equations (PDEs) with finite element methods. By extending the FEniCS framework and its automated compiler, we have achieved that a high-level description of finite element computations written in the Unified Form Language is auto-translated to parallelised CUDA C++ code. The auto-generated code provides GPU offloading for the finite element assembly of linear equation systems which are then solved by a GPU-supported linear algebra backend.Specifically, we explore several auto-generated optimisations of the resulting CUDA C++ code. Numerical experiments show that GPU-based linear system assembly for a typical PDE with first-order elements can benefit from using a lookup table to avoid repeatedly carrying out numerous binary searches, and that further performance gains can be obtained by assembling a sparse matrix row by row. More importantly, the extended FEniCS compiler is able to seamlessly couple the assembly and solution phases for GPU acceleration, so that all unnecessary CPU–GPU data transfers are eliminated. Detailed experiments are used to quantify the negative impact of these data transfers, which can entirely destroy the potential of GPU acceleration if the assembly and solution phases are offloaded to GPU separately. Finally, a complete, auto-generated GPU-based PDE solver for a nonlinear solid mechanics application is used to demonstrate a substantial speedup over running on dual-socket multi-core CPUs, including GPU acceleration of algebraic multigrid as the preconditioner.

Two-Step Identification Method and Experimental Verification of Weld Damage at Joints in Spatial Grid Structures

Welded joints in grid structures are susceptible to damage and destruction when exposed to random excitation. The complexity of the grid structure poses challenges for realizing the damage recognition of welded joints. In this study, a two-step method is proposed specifically for damage identification of welded joints in grid structures, combining wavelet analysis and fuzzy pattern recognition to accurately identify the location and extent of damage in welded joints. Firstly, the structure is divided based on the analysis of the influence range of joint damage. Key joints are selected within the sub-regions where sensors are installed, and the acceleration response of these key joints is measured. Wavelet analysis is then applied to identify the sub-regions where weld damage occurs. Secondly, an equivalent finite element model is established for joints with varying degrees of damage. The damage index, calculated as the ratio of the absolute value of the difference in the first-order element strain mode of the members, increases with the degree of damage during joint weld damage. By monitoring the changes in the damage index of sensitive members, which exhibit significant changes with varying weld damage degrees, a damage pattern database is constructed for each sub-region. The membership degree between joint damage and the patterns in the pattern database is then calculated to determine the location and degree of weld damage. To validate the effectiveness of the proposed method, an experiment was conducted using a grid structure model with replaceable members. Highly sensitive FBG sensors were designed to measure the acceleration response of the joints, resulting in accurate identification of damaged sub-regions solely through the measurement of key joint acceleration responses. Furthermore, within the damaged sub-regions, the fuzzy pattern recognition method precisely determined the location and degree of weld damage in the joints. The experimental results demonstrate that the proposed method effectively reduces the complexity of the structure by dividing the grid structure into sub-regions, and enables the two-step identification method to achieve successful damage identification for the joints in the grid structure with high efficiency and accuracy.

Second-Order, Single-Band and Dual-Band Bandstop Frequency Selective Surfaces at Millimeter Wave Regime

This communication presents the miniaturized low-profile second-order (SO) bandstop frequency selective surfaces (FSSs) for single- and dual-band resonances. The unit cell comprises two identical first-order (FO) elements in a back-to-back configuration with the metal grid in between. Two thin dielectric substrates separate the three metal layers. The proposed FO unit cell consists of intercell-connected spiral resonators (SRs) with capacitively coupled cross loop dipole. The SO filter response of the proposed FSS exhibits a wide stopband ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{21} &lt; -10$ </tex-math></inline-formula> dB) bandwidth (BW) of 27.6%. The unit cell has the periodicity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.162\lambda _{0}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda _{0}$ </tex-math></inline-formula> is the free space wavelength at the lower passband. The novelty of the proposed work is a miniaturized simple unit cell geometry, at the millimeter-waveband but not limited to it, with polarization insensitive and high angular stable filter response up to 60° with small variations in the BW. An unconventional four-unit cell element FSS is designed for dual-band SO stopband resonances with the BWs of 8.3% and 13.8%. Prototypes of both FSSs were fabricated and their measured results were found consistent with the simulated filter responses.

An innovative and efficient solution for axisymmetric contact problem between structure and half-space

In this paper, with displacement increment as variables, the scaled boundary finite element equations of the axisymmetric plate and half-space are respectively derived based on the principle of virtual work. The dual vector is introduced to solve the boundary stiffness matrix. Linear complementary contact conditions are formulated as B-differentiable equations. With discrete, static assumptions of small displacement, B-differentiable equations system was formed by combining the contact conditions and the stiffness matrix of the plate-foundation. The equations can be iteratively solved by B-differentiable Newton method. In numerical experiments, the contact stress, slip and contact status of each contact node pair are compared with the results of the augmented Lagrangian contact algorithm based on FEM. When adopting different load types, thickness-to-radius ratio, plate-foundation stiffness ratio and friction coefficient, the axisymmetric contact problems between plate and half-space are discretized by 120 one-dimensional first-order elements. The applicability, accuracy and effectiveness of B-SBFEM for axisymmetric contact problems are verified by numerical examples.

Nature-based solutions, mental health, well-being, price fairness, attitude, loyalty, and evangelism for green brands in the hotel context

Given that the polluted environment affects human health, Nature-Based Solutions (NBS) is becoming an emerging issue in the hospitality and tourism industry. This research is designed in order to explore the influence of NBSs on Green Brand Evangelism (GBE) through mental health, emotional well-being, Green Brand (GB) attitude, and Green Brand Loyalty (GBL) in the hotel industry by considering the moderating effect of price fairness. Our theoretical model was built based on the Stimulus-Organism-Response (S-O-R) theory, which successfully embraces the NBS framework, GBL, and GBE. Our findings based on an online survey reveal that NBS as a second-order construct, which embraces the four first-order elements, significantly improves guests' perception of mental health and emotional well-being. The direct and indirect effects of mental health, emotional well-being, GB attitude, and GBL also uncovered and significantly impacted GBE. The hypotheses were generally supported. Furthermore, the moderating impact of price fairness was found.

First-order VEM for Reissner\u2013Mindlin plates

In this paper, a first-order virtual element method for Reissner–Mindlin plates is presented. A standard displacement-based variational formulation is employed, assuming transverse displacement and rotations as independent variables. In the framework of the first-order virtual element, a piecewise linear approximation is assumed for both displacement and rotations on the boundary of the element. The consistent term of the stiffness matrix is determined assuming uncoupled polynomial approximations for the generalized strains, with different polynomial degrees for bending and shear parts. In order to mitigate shear locking in the thin-plate limit while keeping the element formulation as simple as possible, a selective scheme for the stabilization term of the stiffness matrix is introduced, to indirectly enrich the approximation of the transverse displacement with respect to that of the rotations. Element performance is tested on various numerical examples involving both thin and thick plates and different polygonal meshes.

Efficiency improvement on the ANCF cable element by using the dot product form of curvature

The adoption of the continuum mechanics concept has enabled the absolute nodal coordinate formulation (ANCF) cable element to handle large deformation and large rotation problems. Generally, the generalized elastic forces (GEFs) and tangent stiffness matrices are formulated by differentiating the curvature of the cross product form, which needs labour-intensive construction process, leads to complicated expressions and therefore impairs the computational efficiency of the element. This work aims to improve the computational efficiency of the ANCF cable element by simplifying the GEFs and tangent stiffness matrices based on the dot product form of curvature. The element is interpolated by the curvature constrained interpolation method and defined by following the decoupling elastic line approach, which ensures the second-order accuracy. The material curvature is presented with just dot product terms through two approaches: the original curvature concept approach (OCCA) and the Lagrange's identity approach (LIA). Based on the OCCA and the LIA, concise formulations of GEFs and tangent stiffness matrices are derived through matrix manipulation. In contrast, the GEFs and tangent stiffness matrices derived from the cross product form of curvature are traditionally constructed by deducing every single item individually, which is summarized as the itemized traversal approach (ITA) in this work. To verify the correctness and efficiency of the proposed approaches, some geometrically nonlinear problems of the straight beam and the curved beam are solved and thoroughly analysed. It is found that far fewer number of the second-order elements are needed than that of the first-order element to get the desired accuracy solution. Besides, the convergence rate of the OCCA, LIA and ITA is identical and the solutions of these three approaches coincide with the results obtained by ABAQUS for five-digits because the definitions of the GEFs and tangent stiffness matrices are essentially identical despite different curvature expressions. However, with the same number of second-order elements, the OCCA and LIA are both more than ten times more efficient than the ITA in terms of CPU time, in which the OCCA is the most efficient one. In future, the proposed approaches can be applied in finite element practices for saving computational time without loss of accuracy.

Comparison of second-order serendipity and Lagrange tetrahedral elements for nonlinear explicit methods

This paper evaluates the performances of second-order finite elements for nodal lumped-mass explicit methods in nonlinear solid dynamics, with a particular emphasis on 10-node “serendipity” and 15-node “Lagrange” tetrahedral elements. Historically, many nonlinear explicit finite element codes have exclusively used first-order elements, until a fairly recent flurry of activity that has resulted in higher-order elements becoming available in explicit codes including the authors' in-house one, ParaAble, and the production software EPIC, IMPETUS, LS-DYNA, and Abaqus. A major attractiveness of tetrahedrons is their ease in meshing and higher-order elements can facilitate the avoidance of severe volumetric locking with unstructured C0 meshes, which are generally used with these codes for the discontinuities of inelasticity, contact, etc. They also can improve modeling of flexure and curved shapes as well as eliminate spurious modes without artificial stabilization. The inclusion of face and body centroid nodes with Lagrange interpolants, including the 15-node tetrahedron, has proven to provide robust overall performance with lumped-mass explicit methods and with contact. Nevertheless, versions of the 10-node tetrahedron have also emerged in lumped-mass explicit software. In contrast to hexahedrons, an important observation about tetrahedrons is that the 10-node serendipity version uses about four times fewer quadrature points and a larger time increment than their 15-node Lagrange counterpart, which could result in tremendous computational differences. Serendipity elements, however, notoriously do not nodal mass lump well and tetrahedron versions have not been rigorously evaluated/documented for their effectiveness, as will be done herein with comparisons of those using 15-node tetrahedrons. Using row-summation lumping for Lagrange elements and the ad hoc HRZ scheme for serendipity ones, performances are assessed in common benchmark problems and practical applications using various elastic and inelastic material models and involving large strains/deformations/rotations and severe distortions. Whereas the 10-node tetrahedrons were found to perform much better than their 20-node serendipity hexahedral counterparts, specifically with substantial computational reductions and reasonable predictions, they are not generally quite as accurate or robust as the 15-node tetrahedral elements. The results thus indicate benefits of including both 10- and 15-node tetrahedrons in an explicit code's element library.

Sparse identification of nonlinear unsteady aerodynamics of the oscillating airfoil

Reduced-order models are widely used in aerospace engineering. A model for unsteady aerodynamics is desirable for designing the blades of wind turbines. Recently, sparse identification of nonlinear dynamics with control was introduced to identify the parameters of an input-output dynamical system. In this paper, two models for attached flows and one for separated flows are identified through this technique. For the unsteady lift of the attached flow, Model I is a linear model that presents the dynamic change of an unsteady lift to a static lift. Model II was built based on Model I in order to obtain a more general system with closed-loop control. It has a first-order inert element that delays the overall input of the static lift. The Model II results replicate the training data very well and give an accurate prediction of other oscillating cases with different oscillation amplitudes, reduced frequency or mean angle of attack. For the unsteady lift of the separated flow, Model III is identified as a nonlinear model, which also has a first-order inert element. This model captures the nonlinear aerodynamics of the separated flow and replicates the training cases well. In addition, the prediction of Model III has good agreement with the numerical results.

Discrete Surface Ricci Flow for General Surface Meshing in Computational Electromagnetics Using Iterative Adaptive Refinement

We propose a surface meshing approach for computational electromagnetics (CEM) based on discrete surface Ricci flow (DSRF) with iterative adaptive refinement (AR) in the parametric domain for the automated generation of high-quality surface meshes of arbitrary element type, order, and count. Surfaces are conformally mapped by DSRF to a canonical parametric domain, allowing a canonical seed mesh to be mapped back to an approximation of the original surface. The new DSRF-based meshing technique provides a framework for generation of meshes with high element quality, aimed to greatly enhance the accuracy, conditioning properties, stability, robustness, and efficiency of surface integral equation CEM solutions. We demonstrate the ability of the proposed DSRF technique to produce meshes with near-optimal element corner angles for complicated, highly varied surfaces such as the NASA almond and a fighter jet model, using triangular, quadrilateral, and discontinuous quadrilateral elements. Other element types are also discussed. Where high-fidelity meshing is desired, the technique can capture fine-scale detail using very few high-order elements. Where low-fidelity meshing is desired, DSRF with AR can accurately recreate course-scale detail using standard first-order elements (e.g., flat triangular patches).

Нове кінцеве інтегральне перетворення для рівняння Лапласа в довільній області

Reliability, survivability, as well as the optimal operating mode of operation of the supercomputer will depend on the architecture and efficiency of the cooling system of the hot components of the supercomputer. That is why the number of problems, of great theoretical and practical interest, is the problem of studying the temperature fields arising in elements of arbitrary configuration, cooling a supercomputer. To solve this class of heat conduction problems, the method of finite integral transformations turned out to be the most convenient. This article is the first to construct a new finite integral transformation for the Laplace equation in an arbitrary domain bounded by several closed piecewise-smooth contours. An inverse transformation formula is given. Finding the core of the constructed new finite integral transformation by the finite element method in the Galerkin form for simplex first-order elements reduces to solving a system of algebraic equations. To test the operability of the new integral transformation, calculations were carried out of solutions of the boundary value problem for the Laplace equation obtained using the developed new integral transformation and the well-known analytical solution. The results of comparison the calculations of the solution of the Laplace equation are presented. In the case of a square with a side length equal to one and on one side of the square, the temperature is unity, and on the other, the temperature is zero, with a well-known analytical solution and a solution obtained using the new integral transformation. These results were obtained for 228 simplex first-order elements and 135 nodes. The maximum deviation modulo of these solutions is 0,096, the mathematical expectation of deviations is 0,009, and the variance of the type is 0,001. The developed integral transformation makes it possible to obtain a solution to complex boundary value problems of mathematical physics.

Differential Mueller matrix imaging of partially depolarizing optically anisotropic biological tissues

Since recently, a number of innovative polarization-based optical imaging modalities have been introduced and extensively used in various biomedical applications, with an ultimate aim to attain the practical tool for the optical biopsy and functional characterization of biological tissues. The techniques utilize polarization properties of light and Mueller matrix mapping of microscopic images of histological sections of biological tissues or polycrystalline films of biological fluids. The main drawback of currently developed laser polarimetry approaches and Mueller matrix mapping techniques is poor reproducibility of experimental data. This is due to azimuthal dependence of polarization and ellipticity values of most matrix elements to sample orientation in respect to incidence light polarization. Current study aims to generalize the methods of laser polarimetry for diagnosis of partially depolarizing optically anisotropic biological tissues. A method of differential Mueller matrix mapping for reconstruction of linear and circular birefringence and dichroism parameter distributions of partially depolarizing layers of biological tissues of different morphological structure is introduced and practically implemented. The coordinate distributions of the value of the first-order differential matrix elements of histological sections of brain tissue with spatially structured, optically anisotropic fibrillar network, as well as of parenchymatous tissue of the rectum wall with an “islet” polycrystalline structure are determined. Within the statistical analysis of polarization reproduced distributions of the averaged parameters of phase and amplitude anisotropy, the significant sensitivity of the statistical moments of the third and fourth orders to changes in the polycrystalline structure of partially depolarizing layers of biological tissue is observed. The differentiation of female reproductive sphere connective tissue is realized with excellent accuracy. The differential Mueller matrix mapping method for reconstruction of distributions of linear and circular birefringence and dichroism parameters of partially depolarizing layers of biological tissues of different morphological structures is proposed and substantiated. Differential diagnostics of changes in the phase (good balanced accuracy) and amplitude (excellent balanced accuracy) of the anisotropy of the partially depolarizing layers of the vagina wall tissue with prolapse of the genitals is realized. The maximum diagnostic efficiency of the first-order differential matrix method was demonstrated in comparison with the traditional methods of polarization and Mueller matrix mapping of histological sections of light-scattering biological tissues.

A Concept of the Non-Stationary Filtering Network with Reduced Transient Response

This paper presents a concept of the non-stationary filtering network with reduced transient response consisting of the first-order digital elements with time-varying parameters. The digital filter section is based on the analog system. In order to design the filtering network, the analog prototype was subjected to the discretization process. The time constant and the gain factor were then temporarily varied in time in order to suppress the transient response of the designed filtering structure. The optimization method, based on the Particle Swarm Optimization (PSO) algorithm which is aimed at reducing the settling time by a proper parameter manipulation, is presented. Simulation results proving the usefulness of the proposed concept are also shown and discussed.

Quadrilateral 2D linked-interpolation finite elements for micropolar continuum

Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation. In order to satisfy convergence criteria, the newly presented finite elements are modified using the Petrov–Galerkin method in which different interpolation is used for the test and trial functions. The elements are tested through four numerical examples consisting of a set of patch tests, a cantilever beam in pure bending and a stress concentration problem and compared with the analytical solution and quadrilateral micropolar finite elements with standard Lagrangian interpolation. In the higher-order patch test, the performance of the first-order element is significantly improved. However, since the problems analysed are already describable with quadratic polynomials, the enhancement due to linked interpolation for higher-order elements could not be highlighted. All the presented elements also faithfully reproduce the micropolar effects in the stress concentration analysis, but the enhancement here is negligible with respect to standard Lagrangian elements, since the higher-order polynomials in this example are not needed.

Fluid dynamics analysis method for MRF of first order discontinuous optical elements

Magnetorheological finishing of first-order discontinuous optical elements is one of the difficult problems that restrict the development of high-precision and high-efficiency optical manufacturing in China. In this paper, a fluid dynamics analysis method for magnetorheological finishing concerning the first order discontinuous optical elements is presented. Firstly, reasonable assumptions are made for the flow field under the practical condition of magnetorheological finishing. Secondly, based on the micro-element hydrodynamic equation, a flow field analysis method suitable for the first order discontinuous surface shape is established. Finally, a numerical calculation method for the flow field governing equation is established based on the finite difference method and numerical iteration method. Through numerical simulation for cutting distance ranging over 1~18 mm, it is found that the pressure distribution pattern of the first-order discontinuous surface obtained by the proposed method is correct, and the discontinuous pressure drop generated by the proposed method is consistent with that of the experimental observation.

Is your industrial marketing work working? Developing a composite index of market change

We define market work as purposeful efforts by a focal actor to perform and transform markets and focus on the connection between market work and market change. To enable an evaluation of the effectiveness of market work, we delineate the domain of market change and provide an operational definition of its elements and develop a composite index of market change. Our research process consisted of four steps: (1) Specification of the elements of market change; (2) Specification of indicators for the identified elements; (3) Reliability and validity assessments; and (4) Validating the market change index. We identified six elements of market change. Markets can be changed by changes in Products & Price, Customers & Use, Channels, Supply-side Network, Representations and Norms. In the qualitative phase of the study we identified at total of 22 facets of these elements, which were later developed into 25 indicators. The overall model was operationalized as a formative first-order, formative second-order model, where first-order elements are formatively measured latent constructs that form a more abstract general (second-order) latent construct - market change. The research process involved testing the reliability and validity of the model and the final market change index.

New Type of Second-Order Tetrahedral Edge Elements by Reducing Edge Variables for Quasi-Static Field Analysis

Second-order tetrahedral edge elements afford much better accuracy in magnetic field analysis compared with first-order elements. However, they often entail higher computational cost due to the higher number of unknowns, many of them being redundant. The partial tree-gauging approach eliminates only a portion of the redundant edge variables, often reducing the overall computational cost. In this paper, we propose a new partial tree-gauging approach in second-order tetrahedral edge elements by unifying the two edges on the sides of elements into one, resulting in a new type of second-order tetrahedral edge element. Using the proposed elements in the analysis of two linear quasi-static magnetic field problems reveal the superiority of the edge unification over the conventional partial tree gauging in term of computational load.

P-multigrid expansion of hybrid multilevel solvers for discontinuous Galerkin finite element discrete ordinate (DG-FEM-SN) diffusion synthetic acceleration (DSA) of radiation transport algorithms

Effective preconditioning of neutron diffusion problems is necessary for the development of efficient DSA schemes for neutron transport problems. This paper uses P-multigrid techniques to expand two preconditioners designed to solve the MIP diffusion neutron diffusion equation with a discontinuous Galerkin (DG-FEM) framework using first-order elements. These preconditioners are based on projecting the first-order DG-FEM formulation to either a linear continuous or a constant discontinuous FEM system. The P-multigrid expansion allows the preconditioners to be applied to problems discretised with second and higher-order elements. The preconditioning algorithms are defined in the form of both a V-cycle and W-cycle and applied to solve challenging neutron diffusion problems. In addition a hybrid preconditioner using P-multigrid and AMG without a constant or continuous coarsening is used. Their performance is measured against a computationally efficient standard algebraic multigrid preconditioner. The results obtained demonstrate that all preconditioners studied in this paper provide good convergence with the continuous method generally being the most computationally efficient. In terms of memory requirements the preconditioners studied significantly outperform the AMG.

Experimental Identification and Vibration Control of A Piezoelectric Flexible Manipulator Using Optimal Multi-Poles Placement Control

This paper presents experimental identification and vibration suppression of a flexible manipulator with piezoelectric actuators and strain sensors using optimal multi-poles placement control. To precisely identify the system model, a reduced order transfer function with relocated zeros is proposed, and a first-order inertia element is added to the model. Comparisons show the identified model match closely with the experimental results both in the time and frequency domains, and a fit of 97.2% is achieved. Based on the identified model, a full-state multi-poles placement controller is designed, and the optimal locations of the closed loop poles are determined where the move distance of the closed loop poles is the shortest. The feasibility of the proposed controller is validated by simulations. Moreover, the controller is tested for different locations of the closed loop poles, and an excellent performance of the optimal locations of the closed loop poles is shown. Finally, the effectiveness of the proposed controller is demonstrated by experiments. Results show that the vibrations of the expected modes are significantly diminished. Accordingly, multi-mode vibrations of the manipulator are well attenuated.