Element Nodes Research Articles

Exact conservation in transient electromagnetics using edge elements

In time domain to solve electromagnetic radiation and scattering problems, several numerical methods are available in literature. Among these numerical methods Finite Difference Time Domain (FDTD) method and Time Domain FEM (TDFEM) method are more popular. In the present work, we have proposed a quantity in continuum, which is conserved under certain boundary conditions. Additionally, we have also proposed a time marching scheme within edge element framework, which satisfy this conservation characteristic in algorithmic sense irrespective of the choice of time step. To facilitate this, we have used some conversion algorithm that transform nodal finite element data to edge finite element data.

Nonlinear analysis of laminated composite shells with piezoelectric patches under displacement‐dependent pressure loads

AbstractIn this paper, the three‐dimensional stress field in nonlinear laminated composite shells with piezoelectric patches subjected to displacement‐dependent loads is analyzed. To solve the problem, the geometrically exact (GeX) hybrid‐mixed four‐node solid‐shell element has been developed. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms are taken exactly at the element nodes. The laminated solid‐shell element formulation is based on the choice of an arbitrary number of sampling surfaces (SaS) parallel to the middle surface and located at the Chebyshev polynomial nodes within the layers, in order to introduce the displacements and electric potentials of these surfaces as unknown functions. The outer surfaces and interfaces are also included into a set of SaS. Due to analytical integration utilized to evaluate the tangent stiffness matrix, the proposed GeX piezoelectric solid‐shell element under nonconservative loading exhibits superior performance in the case of coarse meshes and allows the use of only one load step in all considered benchmark problems. Therefore, it can be recommended for the large deformation analysis of doubly curved laminated composite shells with piezoelectric patches, since the SaS formulation allows one to find three‐dimensional solutions of electroelasticity with a prescribed accuracy.

Numerical analysis of downward progressive landslides in long natural slopes with sensitive clay

Landslides occurring in sensitive clay often result in widespread destruction, posing a significant risk to human lives and property due to the substantial decrease in undrained shear strength during deformation. Assessing the consequences of these landslides is challenging and necessitates robust numerical methods to comprehensively investigate their failure mechanisms. While studies have extensively explored upward progressive landslides in sensitive clays, understanding downward progressive cases remains limited. In this study, we utilised the nodal integration-based particle finite element method (N-PFEM) with a nonlinear strain-softening model to analyse downward progressive landslides in sensitive clay on elongated slopes, induced by surcharge loads near the crest. We focused on elucidating the underlying failure mechanisms and evaluating the effects of different soil parameters and strain-softening characteristics. The simulation results revealed the typical pattern for downward landslides, which typically start with a localised failure in proximity to the surcharge loads, followed by a combination of different types of failure mechanisms, including single flow slides, translational progressive landslides, progressive flow slides, and spread failures. Additionally, inclined shear bands occur within spread failures, often adopting distinctive ploughing patterns characterised by triangular shapes. The sensitive clay thickness at the base, the clay strength gradient, the sensitivity, and the softening rate significantly influence the failure mechanisms and the extent of diffused displacement. Remarkably, some of these effects mirror those observed in upward progressive landslides, underscoring the interconnectedness of these phenomena. This study contributes valuable insights into the complex dynamics of sensitive clay landslides, shedding light on the intricate interplay of factors governing their behaviour and progression.

Topological design of microstructured materials using energy-based homogenization theory and proportional topology optimization with the level-set method

In this study, we propose a novel approach to designing microstructures of porous material with extreme mechanical properties to maximize the bulk or shear modulus. The approach integrates energy-based homogenization theory and proportional topology optimization with the level-set method (HPTO-LSM). A level-set function (LSF) evolution strategy based on the nodal element mutual energies (EME) proportion is proposed and drives the LSF’s evolution. An EME proportion filtering technique is proposed and reduces the redundant branch structure. And a linear scaling strategy is applied to the HPTO-LSM to enhance the topological change ability. The effectiveness of the HPTO-LSM for designing microstructures of cellular materials to maximize the bulk or shear modulus is demonstrated using several 2D and 3D cases. The effects (changes in objective function values or topologies) of the EME proportion filtering scheme and scaling strategy on the HPTO-LSM are discussed. In addition, we compared the HPTO-LSM method with the solid isotropic material with penalization (SIMP) and bi-directional evolutionary structural optimization (BESO) methods through numerical examples. As shown by the results, effective objective function values and 2D and 3D microstructural topologies with clear interfaces and smooth boundaries can be acquired using the new approach. Applying the EME proportion filtering scheme and scaling strategy in the HPTO-LSM not only enhanced the topological variation of microstructures but also facilitated the acquisition of microstructures conducive to engineering applications. Compared to the SIMP and BESO methods, the HPTO-LSM method has significant advantages in obtaining better objective function values and better microstructural topologies.

Equilibrium state of axially symmetric electric solar wind sail at arbitrary sail angles

This paper studies the equilibrium state and trajectory dynamics of an axially symmetric Electric solar wind sail (E-sail) at arbitrary sail angles. The E-sail is assumed operating in a heliocentric-ecliptic orbit at approximately one astronomic unit (au) from the Sun, and experiencing various dynamic disturbances like solar wind pressure, tether tension oscillations, and centrifugal forces. The study derives analytical expressions for the E-sail’s equilibrium state and its maximal coning angle under small coning angle assumption. Subsequently, an improved propulsion model is developed for the E-sail in this equilibrium state. To assess the precision of these formulations, a high-fidelity E-sail dynamic model is constructed using the nodal position finite element method, where the tethers are modeled as two-noded tensile elements and the central spacecraft and remote units are simplified as lumped masses. Through thorough parametric analyses, this paper conclusively demonstrates that the operation of the E-sail at the equilibrium state can be achieved in accordance with the derived analytical prediction of the equilibrium state. Furthermore, the improved propulsion model is employed in trajectory analyses for a mission to reach the solar system’s boundary. The study provides valuable insights and findings and foundation for the practical application and further advancement of the E-sail technology.

Development of Higher Order Stiffened Shell Element (HOST9) for the Static Analysis of Stiffened Laminated Plates

Abstract A new higher order stiffened shell element (HOST9) is developed comprising of a 9 noded flat shell element and a 3 noded stiffener element for analysis of stiffened laminated plates. The shell element is a lagrangian element. The stiffener is a beam element. The stiffness of the stiffener is computed separately and added to that of the shell element at appropriate locations. As a result, the stiffener can be located arbitrarily in the shell element. Static analysis of stiffened laminated plate is carried out using the newly developed element and is validated with the existing literature. It is seen that the proposed element gives comparable results and can be used for the analysis.

A boundary element method of bidomain modeling for predicting cellular responses to electromagnetic fields

Objective. Commonly used cable equation approaches for simulating the effects of electromagnetic fields on excitable cells make several simplifying assumptions that could limit their predictive power. Bidomain or ‘whole’ finite element methods have been developed to fully couple cells and electric fields for more realistic neuron modeling. Here, we introduce a novel bidomain integral equation designed for determining the full electromagnetic coupling between stimulation devices and the intracellular, membrane, and extracellular regions of neurons. Approach. Our proposed boundary element formulation offers a solution to an integral equation that connects the device, tissue inhomogeneity, and cell membrane-induced E-fields. We solve this integral equation using first-order nodal elements and an unconditionally stable Crank–Nicholson time-stepping scheme. To validate and demonstrate our approach, we simulated cylindrical Hodgkin–Huxley axons and spherical cells in multiple brain stimulation scenarios. Main Results. Comparison studies show that a boundary element approach produces accurate results for both electric and magnetic stimulation. Unlike bidomain finite element methods, the bidomain boundary element method does not require volume meshes containing features at multiple scales. As a result, modeling cells, or tightly packed populations of cells, with microscale features embedded in a macroscale head model, is simplified, and the relative placement of devices and cells can be varied without the need to generate a new mesh. Significance. Device-induced electromagnetic fields are commonly used to modulate brain activity for research and therapeutic applications. Bidomain solvers allow for the full incorporation of realistic cell geometries, device E-fields, and neuron populations. Thus, multi-cell studies of advanced neuronal mechanisms would greatly benefit from the development of fast-bidomain solvers to ensure scalability and the practical execution of neural network simulations with realistic neuron morphologies.

Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos (gPC) approach which we consider in the following manuscript. We apply the stochastic Galerkin (SG) method to the stochastic SW equations. Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore. The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations. By transforming the system using Roe variables, the hyperbolicity can be ensured and an entropy-entropy flux pair is known from a recent investigation by Gerster and Herty (Commun. Comput. Phys. 27(3): 639–671, 2020). We use this pair and determine a corresponding entropy flux potential. Then, we construct entropy conservative numerical two-point fluxes for this augmented system. By applying these new numerical fluxes in a nodal DG spectral element method (DGSEM) with flux differencing ansatz, we obtain a provable entropy conservative (dissipative) scheme. In numerical experiments, we validate our theoretical findings.

Deformation state of a graphene sheet within the framework of the continuum moment-membrane theory of elasticity

The paper proposes an approach to finding the stress-strain state (SSS) of structures containing graphene, a novel nanomaterial that has currently found а wide range of practical applications in nanoelectromechanical systems. Graphene is a 2D basic building block for other carbon structures such as membranes, sheets, nanotubes, etc. To describe the SSS of a graphene sheet, the phenomenological continuum moment-membrane theory of plates is used, from which, due to the fact that graphene is an ultrathin material, the concept of thickness is excluded. The physical elasticity relationships of a graphene sheet are expressed through its rigidity characteristics, determined using the harmonic potential of interatomic interactions in carbon. A differential formulation and the corresponding variational formulation are given for the problem of static deformation and determination of the natural frequencies and modes of vibration of a graphene sheet. The variational formulation is based on the Lagrange principle and is implemented numerically using the finite element method. Finite element relations are constructed taking into account moment effects of the behavior of a graphene sheet. For approximation, a 4-node rectangular finite element is used. Numerical solutions to several problems of static deformation of a graphene sheet under conditions of a plane stress state and transverse bending are presented, and the analysis of its natural vibrations is also performed. Good convergence of numerical simulation results in all considered problems is demonstrated. The obtained numerical solutions are essential in designing and calculating resonators in which ultrathin nanostructures are used. The establishment of the fact that a graphene sheet has a high intrinsic frequency falling in the GHz region (for example, quartz resonators are characterized by megahertz frequencies) opens up new prospects for using graphene itself as an ultrasensitive nanomechanical resonator for detecting small masses and ultrasmall displacements.

Biomechanical analysis using finite element analysis of orbital floor fractures reproduced in a realistic experimental environment with an anatomical model.

Introduction: In this study, we attempted to demonstrate the actual process of orbital floor fracture visually and computationally in anatomically reconstructed structures and to investigate them using finite element analysis. Methods: A finite element model of the skull and cervical vertebrae was reconstructed from computed tomography data, and an eyeball surrounded by extraocular adipose was modeled in the orbital cavity. Three-dimensional volume mesh was generated using 173,894 of the 4-node hexahedral solid elements. Results: For the cases where the impactor hit the infraorbital foramen, buckling occurred at the orbital bone as a result of the compressive force, and the von Mises stress exceeded 150MPa. The range of stress components included inferior orbital rim and orbital floor. For the cases where the impactor hit the eyeball first, the orbital bone experienced less stress and the range of stress components limited in orbital floor. The critical speeds for blowout fracture were 4m/s and 6m/s for buckling and hydraulic mechanism. Conclusion: Each mechanism has its own fracture inducing energy and its transmission process, type of force causing the fracture, and fracture pattern. It is possible to determine the mechanism of the fracture based on whether an orbital rim fracture is present.

Nonlinear motion modeling and analysis of underwater cables under complex loads based on global nodal position finite element method in Hamiltonian formulation

The nonlinear dynamic response of underwater cable is essential for underwater equipment system. This paper presents an effective global nodal position finite element method in Hamiltonian formulation to investigate the nonlinear motion of underwater cable under complex loads. The method fully considers the axial, bending, torsional and shear deformation of underwater cable. The non-linear Green-Lagrange strain-displacement relationship is applied to obtain the strain energy. Furthermore, the fourth-order circular cross-section quadrature is introduced to integrate the volume of underwater cable element after deformation. The dynamic governing equations in Hamiltonian formulation are derived and converted into the corresponding canonical equations, which are calculated by the symplectic difference algorithm. Then an initial curved pendulum system, a 180° turn towed underwater cable system and the spiral turning motion of ROV are designed to verify the presented method. The results show that compared with the existing Hamiltonian global nodal position finite element method which ignores torsional and shear deformation, the proposed method is more accurate in describing the nonlinear motion of the underwater cable with axial, bending, torsional and shear deformation.

Numerical modelling of downward progressive landslides in sensitive clay

Landslides in sensitive clays are catastrophic events that threaten life and property. When plastic strain occurs in sensitive clays, there is a significant reduction in the undrained shear strength. Consequently, even minor triggers can lead to multiple progressive landslides, causing extensive devastation. Investigating landslides in sensitive clay presents challenges, necessitating robust numerical methods to comprehend failure modes and collapse behaviors, particularly in the post-failure phase. Although several numerical methods have been utilized for simulations, most have primarily focused on upward progressive landslides. In this study, we adopted the nodal integration-based particle finite element method to simulate downward progressive landslides triggered by surcharge loads from an embankment near the crest in sensitive clay on a long slope. We also considered nonlinear strain softening in sensitive clays. The effects of the strain softening rate and strength gradient of the sensitive clay layer on the failure mechanisms and destruction extent were investigated. The simulation results reveal that a downward progressive landslide initiates with a localized flow slide surrounding the embankment and usually involves a combination of various failure modes.

Three-dimensional nonlinear stress analysis of functionally graded shells subjected to displacement-dependent loads

The nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading, developed in the companion paper, is here extended to the stress analysis of functionally graded material (FGM) shells. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms can be taken exactly at element nodes. The FGM solid-shell element formulation is based on the choice of N sampling surfaces (SaS) parallel to the middle surface and located at Chebyshev polynomial nodes within the shell as well as on the outer surfaces, to introduce the displacements of these surfaces as basic unknowns. The use of Lagrange polynomials of degree N–1 in the distributions of displacements, strains, stresses, and elastic coefficients through the thickness of the shell leads to an efficient 3N-parameter shell formulation. To develop the FGM four-node solid-shell element based on the hybrid-mixed method, in which strains and stresses are utilized as primary variables, the Hu-Washizu variational principle is applied. Due to the analytical integration used to evaluate the tangent stiffness matrix, the developed FGM solid-shell element subjected to displacement-dependent loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered benchmark problems. It has been established that the difference between the second Piola-Kirchhoff and Cauchy stress tensors in non-conservative problems for FGM shells undergoing moderately large strains can be significant.

A compressible layerwise third-order shear deformation theory with transverse shear stress continuity for laminated sandwich plates

Sandwich structures with a soft core typically exhibit multiple deformation modes such as bending, twisting and compression when subjected to transverse loads. Compressive behavior of soft core significantly affects the structural response, which is usually neglected in conventional plate theories. Additionally, conventional plate theories do not take into account transverse shear stress continuity. In order to address this concern and precisely characterize the characteristics of arbitrary multilayer laminated structures, a novel electromechanical coupled finite element model is developed. This model is based on a compressible layerwise third-order shear deformation theory with a soft core. Notably, the proposed model theory requires only twelve degrees of freedom for each quadratic element node. The accuracy of the model is validated first. Afterwards, several numerical investigations are carried out for piezolaminated sandwich structures with different width-to-thickness ratios, reinforcement angle and skew angle. The results demonstrate the applicability of the current model in predicting the behavior of laminated sandwich plates.

A Fast Detection Algorithm for Change Detection in National Forestland “One Map” Based on NLNE Quad-Tree

The National Forestland “One Map” applies the boundaries and attributes of sub-elements to mountain plots by means of spatial data to achieve digital management of forest resources. The change detection and analysis of forest space and property is the key to determining the change characteristics, evolution trend and management effectiveness of forest land. The existing spatial overlay method, rasterization method, object matching method, etc., cannot meet the requirements of high efficiency and high precision at the same time. In this paper, we investigate a fast algorithm for the detection of changes in “One Map”, taking Sichuan Province as an example. The key spatial characteristic extraction method is used to uniquely determine the sub-compartments. We construct an unbalanced quadtree based on the number of maximum leaf node elements (NLNE Quad-Tree) to narrow down the query range of the target sub-compartments and quickly locate the sub-compartments. Based on NLNE Quad-Tree, we establish a change detection model for “One Map” (NQT-FCDM). The results show that the spatial feature combination of barycentric coordinates and area can ensure the spatial uniqueness of 44.45 million sub-compartments in Sichuan Province with 1 m~0.000001 m precision. The NQT-FCDM constructed with 1000–6000 as the maximum number of leaf nodes has the best retrieval efficiency in the range of 100,000–500,000 sub-compartments. The NQT-FCDM shortens the time by about 75% compared with the traditional spatial union analysis method, shortens the time by about 50% compared with the normal quadtree and effectively solves the problem of generating a large amount of intermediate data in the spatial union analysis method. The NQT-FCDM proposed in this paper improves the efficiency of change detection in “One Map” and can be generalized to other industries applying geographic information systems to carry out change detection, providing a basis for the detection of changes in vector spatial data.

Multi-domain wavelet boundary element method for calculating two-dimensional stress intensity factors

In order to improve the accuracy of stress intensity factors (SIFs) calculated by traditional boundary element methods (BEM), the multi-domain wavelet boundary element method (WBEM) is proposed. Firstly, by adjusting the nodes of the B-spline wavelet element on the interval, crack-tip elements are constructed. Since B-spline wavelet on the interval (BSWI) has excellent compact support characteristics and is particularly suitable for describing solution domains with large gradient changes, the constructed crack-tip can reduce the numerical oscillation effect near the crack tip. Secondly, the crack-tip elements are implemented into WBEM. And the combination of WBEM and multi-domain technology can effectively handle interface cracks. Thirdly, the crack problem solving strategy based on multi-domain WBEM can directly evaluate the SIFs of cracks. Finally, several numerical examples involving homogeneous media and bi-material models are given to verify that the proposed method is simple and highly accurate.

WIME Systematic Innovation Method based on 6WHI and Total Factor Matrix Analysis

The involved object of systematic innovation and management is neither a single goal nor a single meth-od. Firstly, the research object is a systematic and organic whole composed of multiple node elements with a certain degree of correlation. Secondly, the research method is multi-dimensional, comprehensive, all round and highly efficient. Innovation should break through the existing thinking framework, change the existing structure, add new components, give new energy and produce new functions. The 5W1H is a questioning approach and a problem-solving method that answers all the basic elements within a problem which are what, who, when, where, why, and how. It aims to view ideas from various perspectives and gain an in-depth understanding of a specific situation. On the basis of 5W1H analysis method, this paper puts forward the 6WHI systematic innovation management method, which increases the query words weather and if of the innovation expansion solution, increases the number of questions from 6 to 8, and expands innovative forward-looking and innovative solutions diversity, with smoother pronunciation, more exciting storm. Furthermore, a systematic innovation method based on 6WHI and total factor analysis is established, and a more general and normative model WIME method is presented. This method can be utilized as a continuous process-improvement technique in an organization. Taking the 17 letters of “if question patents” as the prefix, they are just 17 common key elements in system management and innovation. For each element mentioned above, 8 questions and answers of 6WHI are given to realize systematic management and innovation. The above two types of work are arranged according to the matrix format, and then the weighted evaluation method is introduced to evaluate the above management and innovation work, so as to give a general and standardized systematic management and innovation (WIME) method.

Finite element analysis (FEA) of stress distribution in platform-switched short dental implants.

The distribution of stress on short platform switched dental implants is of interest. Hence, the mandibular posterior molar area was modelled using a three-dimensional finite element method (FEM) with a continuous 1.5 mm cortical bone thickness and an inner cancellous bone core. The implants used in the study were 5 mm long, 4.5 mm wide and 3.5 mm wide at the abutments. 120 N of force was applied in both the vertical and oblique (20° and 35°) directions to create a realistic simulation. ANSYS Workbench was generated for each model. Von Mises stress was assessed in the cortical and cancellous bones at varying depths. Ten noded tetrahedron elements with three degrees of freedom per node were employed to interpret translations on the x, y, and z axes. The stress-based biomechanical behaviour of platform switched short osseo-integrated implants varied across all 5 positions in FEM simulations, based on the depth of implant placement, the direction of applied force, and the shape of the bone. Data shows that opposite forces to the vertical forces caused more damage. Thus, the implantation of subcrestal implants resulted in reduced stress on the cortical and cancellous bone.

General formulation for floating body with elastic mooring in irregular waves: A hybrid linear and nonlinear framework and validation

In the present study, an advanced computational platform has been developed for offshore renewable energy converter systems, considering nonlinear wave input, body dynamics, mooring line dynamics, and kinematic and mechanical constraints. In particular, a hybrid linear and nonlinear hydrodynamic model is formulated through coupling with a numerical wave tank generated by OceanWave3D. A novel nodal position finite element model considering geometrical and material nonlinearities is developed to simulate the nonlinear behaviour of the mooring dynamics. Then, an implicit solver with an iteration process solves the dynamics of the rigid body, the dynamics of the mooring lines, and kinematic and mechanical constraints simultaneously. Therefore, the developed platform is capable of studying the wave body interaction with mooring lines in extreme wave conditions. Wave basin model tests of a single float on a nonlinear mooring line are carried out under a series of irregular waves for comparison. The developed model was also validated against commercial software under identical conditions for idealised cases. In steep wave conditions, the hybrid nonlinear/linear hydrodynamics give improved predictions over the linear hydrodynamic method.

3D Numerical Cross-Section Analysis of a Tapered Beam Slice

Cross-section analysis is an important tool used to recover stresses and strains in a structure at specific cross-sections of arbitrary geometries, without the need for a full 3D model. This is particularly essential for large-scale structures such as aircrafts, wind turbine blades, etc. where making a full model can be computationally very expensive or impractical. The majority of currently available cross-section analysis frameworks are based on stepwise prismatic assumptions, which are hardly suited for the analysis of tapered beams. In fact, high-fidelity stress analysis obtained from analytical and full 3D models shows that predictions of stepwise prismatic approximations can significantly deviate from the correct solution of tapered beams. In this work, a prismatic 3D cross-section analysis method is extended to analyze a symmetrically tapered finite cross-section slice. In this study, the cross-section slice is discretized with 8-node and 20-node solid elements. The boundary conditions are applied as six constraint equations via the Lagrange multiplier method. The external nodal forces acting on the cross-section faces are obtained from the equivalent tractions induced by the cross-section forces. The developed numerical model is validated against the exact analytical solutions of a wedge as well as commercial finite element (FE) software COMSOL and it is shown that the numerically predicted displacement and stress fields agree well with those provided by the wedge’s analytical solution and the FE COMSOL results. This work contributes to the advancement of high-fidelity numerical tapered cross-section analysis methods with an application for many engineering structures.