Finite Element Method Approach Research Articles

Error estimates for second-order SAV finite element method to phase field crystal model

<p style='text-indent:20px;'>In this paper, the second-order scalar auxiliary variable approach in time and linear finite element method in space are employed for solving the Cahn-Hilliard type equation of the phase field crystal model. The energy stability of the fully discrete scheme and the boundedness of numerical solution are studied. The rigorous error estimates of order <inline-formula><tex-math id="M1">$ O(\tau^2+h^2) $</tex-math></inline-formula> in the sense of <inline-formula><tex-math id="M2">$ L^2 $</tex-math></inline-formula>-norm is derived. Finally, some numerical results are given to demonstrate the theoretical analysis.</p>

Terahertz refractive index sensor with high sensitivity based on two‐core photonic crystal fiber

AbstractThis article contributes a highly sensitive novel refractive index (RI) sensor based on two‐core photonic crystal fiber (PCF) is proposed in the terahertz (THz) frequency regime. The sensing performance of the proposed RI sensor is based on the mode‐coupling theory. Two vertical cores of the PCF divided by the central air hole, act as two separate waveguides. The analyte whose refractive index being sensed is filled in the central air hole of the PCF. A full‐vector finite element method (FEM) approach is used to investigate the sensor. Numerical data reveal that the sensitivities of the proposed PCF are 672 600, 940480, 1035300, 1332200, and 1 827 300 nm/RIU for the RI sensing range of analyte from 1.30 to 1.33, 1.34 to 1.37, 1.38 to 1.40, 1.41 to 1.43, and 1.44 to 1.46, respectively. This indicates that the proposed RI sensor is highly sensitive to the refractive index of the analyte.

Analytical Modeling of Participation Reduction in Superconducting Coplanar Resonator and Qubit Designs Through Substrate Trenching

A strategy aimed at decreasing dielectric loss in coplanar waveguides (CPW) and qubits involves the creation of trenches in the underlying substrate within the gaps of the overlying metallization. Participation of contamination layers residing on surfaces and interfaces in these designs can be reduced due to the change in the effective dielectric properties between the groundplane and conductor metallization. Although finite element method approaches have been previously applied to quantify this decrease, an analytical method is presented that can uniquely address geometries possessing small to intermediate substrate trench depths. Conformal mapping techniques produce transformed CPW and qubit geometries without substrate trenching but a non-uniform contamination layer thickness. By parametrizing this variation, one can calculate surface participation through the use of a two-dimensional, analytical approximation that properly captures singularities in the electric field intensity near the metallization corners and edges. Examples demonstrate two regimes with respect to substrate trench depth that capture an initial increase in substrate-to-air surface participation due to the trench sidewalls and an overall decrease in surface participation due to the reduction in the effective dielectric constant, and are compared to experimental measurements to extract loss tangents on this surface.

Voltage Variations and Composite Solid Insulation Effects on Electrical Tree Propagation Based on Finite Element Method

This paper investigates the effect of the voltage variations and the change of solid insulation material on the initiation and propagation of Electrical Tree (ET) through solid dielectrics based on the application of Finite Element Method (FEM) approach using the environmental domain of [COMSOL Multiphysics 𝟓. 𝟑𝒂] program as an innovative way to be used in the complicated study of Electrical Treeing Phenomenon (ETP). This research provides an innovative solution to suppress the initiation point of ET and suppress its growth rates using a composite solid insulation material of Poly – Vinyl Chloride (PVC) and Cross – Linked Polyethylene (XLPE) regardless to the mechanical features of this composition. In this research, the study cases are conducted for High Voltage (HV) and Extra High Voltage (EHV) power system networks as the effect of ETP is to be observed clearly. Simulation model is used in this research to imitate the proposed hyperbolic needle – plane system. Case studies are compared between each others to validate the results

Three-dimensional free vibration analyses of functionally graded laminated shells under thermal environment by a hierarchical quadrature element method

This paper carries out three-dimensional (3D) free vibration analyses of laminated shells made of functionally graded materials (FGMs) in both thermal and non-thermal environments by a hierarchical quadrature element method (HQEM). Two configurations of FGM laminated shells were considered, the first with FGM face sheets and homogeneous core, while the second with homogeneous face sheets and FGM core. Effective material properties of the FGMs were estimated in terms of two micromechanical models called Voigt’s rule of mixture (ROM) and Mori-Tanaka (MT) scheme. For shells in thermal environment, a nonlinear temperature distribution along thickness direction was considered while the elasticity properties were assumed to be temperature-dependent. Natural frequencies obtained by the presented formulation for FGM laminated shells under non-thermal conditions but varied geometric and boundary conditions were compared with those obtained by two-dimensional (2D) finite element method (FEM) and semi-analytical approaches in literatures first to assess the correctness and accuracy. Then the influence of volume fraction index, core thickness, and temperature gradient on natural frequencies of FGM laminated shells under thermal environment were investigated. The three-dimensional formulations of this work were shown to need similar computational cost as 2D ones but with the accuracy of 3D theory.

Description and implementation of an algebraic multigrid preconditioner for H1-conforming finite element schemes

This paper presents detailed aspects regarding the implementation of the Finite Element Method (FEM) to solve a Poisson’s equation with homogeneous boundary conditions. The aim of this paper is to clarify details of this implementation, such as the construction of algorithms, implementation of numerical experiments, and their results. For such purpose, the continuous problem is described, and a classical FEM approach is used to solve it. In addition, a multilevel technique is implemented for an efficient resolution of the corresponding linear system, describing and including some diagrams to explain the process and presenting the implementation codes in MATLAB®. Finally, codes are validated using several numerical experiments. Results show an adequate behavior of the preconditioner since the number of iterations of the PCG method does not increase, even when the mesh size is reduced.

Probability of serviceability failure in cement-stabilized soft clay excavation

The probabilities of exceedance of the maximum tensile stress in cement-stabilized soft clay, the displacement of retaining structure and bottom upheave are explored with deterministic finite element method and fuzzy finite element approach (FFEA) for a Shanghai pit excavation. Effect of the scale of fluctuation on the probability of exceedance is clearly observed. In the cases with an infinite scale of fluctuation, on account of the asymmetrical configuration of resulting fuzzy number, the probability of exceeding the threshold of the maximum tensile stress is comparatively low and will result in an unsafe design; however, the probability of exceeding the threshold of the displacement of the cement-reinforced soft clay and tunnel are a bit higher, which will consequently lead to a conservative design. For smaller threshold, the scenario with a smaller scale of fluctuation yields a higher probability of exceedance, and the trend reverses when relatively larger threshold is adopted.

Modeling of crack repair using piezoelectric material: XFEM approach

PurposeThe purpose of this paper is to analyze the repair of a straight and angular crack in the structure using a piezoelectric material under thermo-mechanical loading by the extended finite element method (XFEM) approach. This provides a general and simple solution for the modeling of crack in the structure to analyze the repair.Design/methodology/approachThe extended finite element method is used to model crack geometry. The crack surface is modeled by Heaviside enrichment function while the crack front is modeled by branch enrichment functions.FindingsThe effectiveness of the repair is measured in terms of stress intensity factor and J-integral. The critical voltage at which patch repair is most effective is evaluated and presented. Optimal patch shape, location of patch, adhesive thickness and adhesive modulus are obtained for effective repair under thermo-mechanical loading environment.Originality/valueThe presented numerical modeling and simulation by the XFEM approach are of great benefit to analyze crack repair in two-dimensional and three-dimensional structures using piezoelectric patch material under thermo-mechanical loading.

Consideration of maximum impact force design for a rock shed against dry granular flow

This paper focuses on the estimation of the maximum impact force of dry granular flow upon a rock shed using a coupled discrete element method (DEM) and finite element method (FEM). The dry granular flow is modeled as an assembly of discrete particles, and the rock shed is modeled by applying the FEM. The coupled DEM–FEM approach calibrated with a small-scale physical experiment is used to simulate the movement of a dry granular flow impacting the rock shed. Full-scale numerical modeling based on the field model is constructed to estimate the maximum impact force of dry granular flow on a rock shed. Based on the numerical results, three key stages of the impact process are identified: startup streams slippery, impact, and pile-up. The sensitivities of bulk density, impact height, slope angle, cushion thickness, and friction coefficient to the maximum impact force are 1, 0.63, 2.68, 0.09, and 0.73, respectively, in the benchmark model, and the parameters with high sensitivities should be given priority in the design of rock sheds. Moreover, an evaluation formula of maximum impact force is obtained and based on the numerical results and Buckingham’s principle.

Prediction of failure in cancellous bone using extended finite element method.

The objective of our study is to develop extended finite element method models of cancellous bone specimens that are capable of accurately predicting the onset and propagation of cracks under mechanical loading. In order to do so, previously published three-point bending test results of a single trabecula were replicated using two different extended finite element method approaches, namely, elastic-plastic-fracture and elastic-fracture that considered different configurations of the elasto-plastic properties of bone from which the best approach to fit the experimental data was identified. The behavior of a single trabecula was then used in 2D extended finite element method models to quantify the strength of the trabecular tissue of the forearm along three perpendicular anatomical axes. The results revealed that the elastic-plastic-fracture model better represented the experimental data in the model of a single trabecula. Considering the 2D trabecular specimens, the elastic fracture model predicted higher strength than the elastic-plastic-fracture model and there was no difference in stiffness between the two models. In general, the specimens exhibited higher failure strain and more ductile behavior in compression than in tension. In addition, strength and stiffness were found to be higher in tension than compression on average. It can be concluded that with proper parameters, extended finite element method is capable of simulating the ductile behavior of cancellous bone. The models are able to quantify the tensile strength of trabecular tissue in the various anatomical directions reporting an increased strength in the longitudinal direction of forearm cancellous bone tissue. Extended finite element method of cancellous bone proves to be a valuable tool to predict the mechanical characteristics of cancellous bones as a function of the microstructure.

Comparative study of stochastic slope stability analysis based on conditional and unconditional random field

Abstract In this study, by coupling the conditional and unconditional random field with finite element methods, the stability of a real slope is investigated. In the first method, which is used for the case of an Unconditional Random Field (URF), the analysis is carried out similar to the approach of the Random Finite Element Method (RFEM) using the mean and standard deviation (std.) of soil parameters derived from all boreholes. In the second, which is a Conditional Random Field (CRF), the soil parameters are mapped onto the selected analysis section and are then predicted on the section mesh elements based on geostatistical estimations. As outputs of the stochastic analysis, the probability density functions of safety factor and maximum lateral displacement of the slope from the two methods are compared with each other. The results of this case study indicate that the use of the CRF affects not only the std. of the safety factor and maximum lateral displacement distribution but can also cause variations in the mean value. Furthermore, the system reliability index of the slope stability decreases when the analysis is done based on the URF.

Modeling the mechanics of calcium regulation in T lymphocyte: A finite element method approach

Changes in cellular Ca[Formula: see text] concentration control a variety of physiological activities including hormone and neurotransmitter release, muscular contraction, synaptic plasticity, ionic channel permeability, apoptosis, enzyme activity, gene transcription and reproduction process. Spatial–temporal Ca[Formula: see text] dynamics due to Ca[Formula: see text] release, buffering and re-uptaking plays a central role in studying the Ca[Formula: see text] regulation in T lymphocytes. In most cases, Ca[Formula: see text] has its major signaling function when it is elevated in the cytosolic compartment. In this paper, a two-dimensional mathematical model to study spatiotemporal variations of intracellular Ca[Formula: see text] concentration in T lymphocyte cell is proposed and investigated. The cell is assumed to be a circular shaped geometrical domain for the representation of properties of Ca[Formula: see text] dynamics within the cell including important parameters. Ca[Formula: see text] binding proteins for the dynamics of Ca[Formula: see text] are itself buffer and other physiological parameters located in Ca[Formula: see text] stores. The model incorporates the important biophysical processes like diffusion, reaction, voltage-gated Ca[Formula: see text] channel, leak from endoplasmic reticulum (ER), efflux from cytosol to ER via sarco–ER Ca[Formula: see text] adenosine triphosphate (SERCA) pumps, buffers and Na[Formula: see text] exchanger. The proposed mathematical model is solved using a finite difference method and the finite element method. Appropriate initial and boundary conditions are incorporated in the model based on biophysical conditions of the problem. Computer simulations in MATLAB R2019b are employed to investigate mathematical models of reaction–diffusion equation. The effect of source, buffer, Na[Formula: see text]/Ca[Formula: see text] exchanger, etc. on spatial and temporal patterns of Ca[Formula: see text] in T lymphocyte has been studied with the help of numerical results. From the obtained results, it is observed that, the coordinated combination of the incorporated parameters plays a significant role in Ca[Formula: see text] regulation in T lymphocytes. ER leak and voltage-gated Ca[Formula: see text] channel provides the necessary Ca[Formula: see text] to the cell when required for its proper functioning, while on the other side buffers, SERCA pump and Na[Formula: see text]/Ca[Formula: see text] exchanger makes balance in the Ca[Formula: see text] concentration, so as to prevent the cell from death as higher concentration for longer time is harmful for the cell and can cause cell death.

Comparative study of granular solid-structure interaction in a uni-axial compression system

Interaction between granular solids and confining structures is an elementary problem encountered in subsurface structural design and bulk solids storing and handling. A classic scenario is uni-axial compression of granular solids in a deformable cylindrical container. Despite being apparently simple in loading condition, the understanding of this scenario remains limited, mainly due to complex interactive deformation between the two components via frictional interfaces. This paper comparatively examines such a uni-axial compression particulate system by a laboratory experiment and two different numerical approaches, namely, continuum finite element method (FEM) and linked discrete-finite element method (linked DEM-FEM). In the continuum FEM approach, two intendedly chosen simple material models, linear elastic and porous elastic models, are attempted. The comparative study reveals that the majority of resultant characteristics show satisfactory agreement amongst the numerical predictions and the experimental measurements. The simple elastic continuum FEM models can hence be a useful alternative in modelling such problems with mild structural flexibility under a monotonic loading scenario. However, precise prediction of some characteristics, such as lateral pressure ratio, may demand more elaborated material model or parameter selection. The enhancements needed for each numerical approach in order to achieve an improved result are further discussed.

Effective Measures to Improve Current Collection Quality for Double Pantographs and Catenary Based on Wave Propagation Analysis

The pantograph installed on the train roof is responsible for collecting the electrical energy via the sliding contact with the catenary constructed along the railroad. To enhance the carrying capacity of the high-speed train, multiple pantographs are normally mounted on the EMU (Electrical Multiple Unit train) to interact simultaneously with the catenary. Especially in China, the double pantographs-catenary interaction is gradually becoming the mainstream in the newly built high-speed network. The biggest challenge of double-pantograph operation is the deterioration of the current collection quality of the trailing pantograph. The mechanical wave excited by the leading pantograph propagates along the contact wire and disturbs the trailing pantograph. This paper attempts to propose effective measures to improve the current collection quality of the trailing pantograph. To improve the understanding of the wave propagation in the contact wire excited by two pantographs, the double pantographs-catenary model is established using a FEM (Finite Element Method) approach. Through the analysis of the contact wire uplift response excited by a single moving force, the optimal interval of double pantographs is discussed. The results indicate that the bad interval appears at the velocity peak of the contact wire uplift, whereas, the good interval appears at the valley value of the contact wire uplift velocity. Based on this idea, the formula of optimal interval of double pantographs is proposed, and its validity is verified using the parameters of the European and China high-speed networks. Then the damper is introduced to the steady arm to reduce the wave intensity. The simulation results show that the slight steady arm damping has a positive effect on the performance of the double pantographs-catenary interaction.

GPU Finite Element Method Computation Strategy Without Mesh Coloring

Abstract A GPU-adapted Finite Element Method (A-FEM) implementation without using a mesh coloring strategy to avoid race conditions is presented. All the elements are simultaneously processed, and the local data is computed and consolidated into the device memory in a new adapted Coordinate Format data structure (a-COO). The comparison between the proposed solution and two GPU Element-by-Element Finite Element Method (EbE-FEM) approaches shows promising results since it is no longer necessary the mesh coloring step. This new approach is well suited for problems that need re-meshing and consequently would need re-coloring, e.g., adaptive mesh refinement.

Simulation assisted process chain design for the manufacturing of bulk hybrid shafts with tailored properties

To manufacture semi-finished hybrid workpieces with tailored properties, a finite element simulation assisted process chain design was investigated. This includes the process steps of cross wedge rolling, hot geometry inspection, induction hardening, and fatigue testing. The process chain allows the utilisation of material combinations such as high-strength steels with low-cost and easy to process steels. Here, plasma transferred arc welding is applied to supply the process chain with hybrid specimen featuring different steel grades. An overview of the numerical approaches to consider the various physical phenomena in each of the process steps is presented. The properties of the component behaviour were investigated via the finite element method (FEM) and theoretical approaches. At first, the manufacturing of a hybrid workpiece featuring a near net shape geometry with improved mechanical properties due to recrystallising the weld was computed, using the example of a cross wedge rolling process. The rolling process was designed by means of FEM to determine suitable process parameters and to reduce experimental testing. An optical multi-scale geometry inspection of the hot workpiece is meant to be carried out after each manufacturing step to detect potential undesired forming or cooling-induced deformations. Due to the heat transfer from the hot component to the ambient medium, an optical measurement is affected by the developing inhomogeneous refractive index field in air. To gain a basic understanding of the refractive index field and induced light deflection effects, computations were conducted using heat transfer and ray tracing simulations. According to the proposed process route, a subsequent local heat treatment of the hybrid component is required to adapt the mechanical properties by a spray cooling assisted induction hardening. The heat treatment step was computed via a 2D FEM calculation. After finishing by machining, the hybrid material shafts are examined in fatigue tests under load conditions. To predict the component’s lifetime under rolling contact fatigue, a damage accumulation model was combined with an FE simulation. The resulting residual stress state after quenching and the geometry after the finishing process were used as input data for the fatigue life calculations.

Robust control volume finite element methods for numerical wave tanks using extreme adaptive anisotropic meshes

SummaryMultiphase inertia‐dominated flow simulations, and free surface flow models in particular, continue to this day to present many challenges in terms of accuracy and computational cost to industry and research communities. Numerical wave tanks and their use for studying wave‐structure interactions are a good example. Finite element method (FEM) with anisotropic meshes combined with dynamic mesh algorithms has already shown the potential to significantly reduce the number of elements and simulation time with no accuracy loss. However, mesh anisotropy can lead to mesh quality‐related instabilities. This article presents a very robust FEM approach based on a control volume discretization of the pressure field for inertia dominated flows, which can overcome the typically encountered mesh quality limitations associated with extremely anisotropic elements. Highly compressive methods for the water‐air interface are used here. The combination of these methods is validated with multiphase free surface flow benchmark cases, showing very good agreement with experiments even for extremely anisotropic meshes, reducing by up to two orders of magnitude the required number of elements to obtain accurate solutions.

A coupled Integral Transform Method - Finite Element Method approach to model the Soil Structure Interaction of finite (3D) and length invariant (2.5D) systems

The prediction of vibrations in an unbounded medium and their interaction with structures on the ground surface or within the soil requires adequate approaches. Therefore the numerical methods addressing the dynamic Soil Structure Interaction (SSI) have to account for both, the infinite extent of the soil by satisfying the radiation condition for the wave propagation towards infinity and the detailed modeling of complex geometries and material properties of the considered structures. In this contribution an efficient domain decomposition method to model the three dimensional SSI is presented using a coupled Integral Transform Method (ITM) - Finite Element Method (FEM) approach. Based on the Lamé differential equation, fundamental solutions for systems with one boundary surface, as a halfspace or a fullspace with either spherical or cylindrical cavity, can be derived using the ITM. These are superposed to determine solutions of systems with two boundary surfaces: the halfspace with spherical or cylindrical cavity. The FEM is used to model the structure possibly within a limited volume of surrounding soil, which then is coupled to the ITM substructure on the cylindrical or spherical interaction surface. The practical applicability and high accuracy of the method is demonstrated by comparison with the analytical solution of fundamental systems and the investigation of typical SSI problems such as a mitigation measure in the transmission path and a rigid surface footing on a homogeneous soil.

Mechanical behavior of carbon nanotube reinforced polymethylmethacrylate foam: A multi-scale finite element method approach

In the present study, considering the random distribution of carbon nanotubes in polymethylmethacrylate (PMMA) polymer, a multi-scale finite element method has been developed to investigate the mechanical behavior of these materials. In order to make realistic assumptions, the interface zone between the nanotubes and the matrix is simulated using the interfacial zone model (IZM), with its parameters obtained by calibrating the finite element model results with the results of experimental tests. Tensile tests and scanning electron microscopy (SEM) were used to study the mechanical properties and morphology of the PMMA foam samples reinforced with 0 wt%, 1 wt% and 2 wt% carbon nanotubes. Experimental results show that by adding 1 wt% and 2 wt% carbon nanotubes to PMMA foam, the tensile strength of the foam increases by 8% and 22%, respectively, compared to pure foam. Based on the results of the finite element models, the optimum percentage of the carbon nanotubes is 3.3 wt%, which gives the optimum modulus and tensile strength of 87 MPa and 4.02 GPa, respectively. The finite element model developed in the present study shows good agreement with the experimental results. Therefore, the impact of effective parameters on the mechanical properties of the foam can be easily studied based on the finite element method (FEM) results.

Modeling of Magnetic Field and Transients in a Novel Permanent Magnet Valve Actuator.

This paper concerns mathematical modelling of dynamic performances to a new permanent magnet electromagnetic valve actuator (PMEVA). Both static and transient characteristics were simulated by using the finite element method (FEM) and field-circuit approach. The magnetic force values versus the excitation current and the position of the valve actuator movable part have been determined. Our concept of the mover positioning relative to the radial magnets is quite novel. PMEVA parameters are satisfied for implementation in combustion engines. Transients in the device have also been analyzed for no-load and for the nominal burden of the actuator. The indications of the position sensors and the excitation current waves were simulated and measured for the step voltage supply. The calculation results were verified experimentally, and a good conformity has been observed. The advantages of our actuator are simple construction, short time of the switching, the current supplying being needed only at the runner extreme positions, and simple controlling. Additionally, PMEVA design can be extended to support the simultaneous operation of four valves.