Finite Element Method Research Articles

Two novel linearized energy-conserving finite element schemes for nonlinear regularized long wave equation

In this paper, two linearized second-order energy-conserving schemes for the nonlinear regularized long wave (RLW) equation are introduced, the unconditional superclose and superconvergence results are presented by using the conforming finite element method (FEM). Initially, through a skillful decomposition of the nonlinear term, two linearized second-order fully discrete schemes are developed. Compared to the previous nonlinear approaches, these schemes significantly reduce the number of iterations and improve computational efficiency; moreover, they conserve energy, and ensure the boundedness of the numerical solution in the H1-norm directly, which represents an advancement over the L∞-norm boundedness reported in prior studies. Secondly, based on the boundedness of the FE solution, the Ritz projection operator and high-precision results of the linear triangular element, the error estimates for superclose and superconvergence are derived without any restrictions on the ratio between time step size Δt and spatial mesh size h. Finally, four numerical examples are provided to confirm the accuracy of the theoretical analysis and the effectiveness of the method.

Characterization of temperature-dependent full-matrix constants of [001]c-poled Mn-doped 28PIN-43PMN-29PT single crystals using one sample

[001]c-poled Mn-doped xPb(In1/2Nb1/2)O3–(1–x–y)Pb(Mg1/3Nb2/3)O3–yPbTiO3 (PIN–PMN–PT:Mn) single crystals (SCs) in the rhombohedral phase possess not only excellent piezoelectric properties but also a high mechanical quality factor, which make them suitable for fabricating high-performance piezoelectric devices. Characterizing the temperature dependence (TD) of full-matrix constants is indispensable for their use in device design. However, there is a significant lack of relevant data. In this study, the TD of the clamped dielectric properties of [001]c-poled 28PIN-43PMN-29PT:Mn SCs was characterized using a dielectric temperature spectroscopy measurement system from 0 to 60 °C. Their elastic and piezoelectric constants were measured using resonant ultrasound spectroscopy in the same temperature range. With an increase in the temperature, all the elastic stiffness constants decreased, whereas all the clamped dielectric constants and the absolute values of all the piezoelectric stress constants increased. The characterization process required only a single sample, which ensured the consistency of the results. Furthermore, the electrical impedance spectra of the sample at 30 and 50 °C were measured and compared with those simulated using the finite-element method with the characterization results. The close agreement between them confirmed the reliability of the characterization results.

3D bioheat transfer mapping reveals nanomagnetic particles effectiveness in radiofrequency hyperthermia breast cancer treatment comparing to experimental study

Radiofrequency (RF) hyperthermia has been widely used for tumor ablation since magnetic-fluid-hyperthermia (MFH) can be utilized for increasing temperature in tumor-region as a complementary-method for hyperthermia. In this study, the effectiveness of using the magnetite-nanoparticles (Fe3O4) in RF hyperthermia for breast cancer (BC) treatment by determining 3D-temperature-distribution using bioheat-transfer-mapping was evaluated. A breast-phantom with a tumor region was placed in an RF-device with 13.56 MHz frequency in different states (with and without-nanomagnetite). Parallelly, the calculations of the RF-wave and bioheat-equation were accomplished by numerical-simulation and finite-element-method (FEM) in COMSOL-software. The temperature differences were experimentally measured at different points of the phantom with a precision of 0.1 °C, with temperature of 3.6 °C and 6.1 °C in without and with nanomagnetic conditions in tumor area, respectively, and also for normal area with temperature of 1.8 °C and 1.9 °C in non-presence and presence states of 0.05 gr magnetite for both conditions, respectively. Moreover, the difference between the simulation and the experimental results was 0.54–1.1 %. The conformity between temperature measurement in experimental and simulation studies in tumor and normal areas showed the effectiveness of the application of MNPs for RF hyperthermia in tissue equivalent breast phantom. Finally, the positive effect of 0.05 gr of MNPs on BC treatment was confirmed.

An analysis of the bilinear finite volume method for the singularly-perturbed convection-diffusion problems on Shishkin mesh

In this paper, we study the bilinear finite volume element method for solving the singularly perturbed convection-diffusion problem on the Shishkin mesh. We first prove that the finite volume element scheme is ϵ-uniformly stable. Then, based on new expression of the finite volume bilinear form and some detailed integral calculations, an ϵ-uniform error estimation is derived in the ϵ-weighted gradient norm, including the L2-norm. This error estimate is better than the known result. Moreover, we also give the L∞-error estimate near the boundary layer regions. At last, numerical experiments show the effectiveness of our method.

Diffusion law and diffusion model for backfill grouting in loess shield tunnel at different soil moisture

Loess is a special type of soil whose properties are significantly affected by water. However, the grout diffusion law for backfill grouting in loess shield tunnels remains unknown. Based on a visual model experimental device, three experiments were conducted with 10%, 20%, and 30% loess moisture. A finite discrete element method was used to verify the grout diffusion mode, and parameters such as the tunnel buried depth, grout viscosity, and elastic modulus were considered to analyse the grout diffusion law. Experiments and numerical simulations show that the screening diffusion of grout occurs at low loess moisture, whereas splitting diffusion occurs at high loess moisture. The farthest splitting diffusion distance decreases as the tunnel buried depth, grout viscosity, and elastic modulus increase. In addition, based on capillary theory and geotechnical strength criteria, screening diffusion and splitting diffusion models were established. This study investigated the grout diffusion law and grout diffusion model, providing a reference for the design and construction of loess shield tunnels.

Soft fingers with variable stiffness for space gripping tasks: An assessment

The miniaturization of satellites and the problem of uncontrolled space debris demand highly flexible methods for space grippers. Soft grippers have surged as viable alternatives to overcome the limitations of traditional rigid grippers, such as lack of adaptability and dexterity. Although existent models have shown promising features, most have a high part count or make use of actuation systems that are inadequate for space, like pneumatic pumps or temperature-dependent materials. This work introduces novel designs of soft fingers that address the inherent challenges of grasping objects in space while overcoming these limitations. The proposed fingers use compliant structures based on several geometries of metamaterials. A detailed compliance and compressibility analysis is conducted using finite element methods to highlight the performance and behavior of each proposed design. Prototypes were fabricated, and their ability to exhibit different modes of actuation (variability in stiffness) by tendon compression was confirmed. The most apt design was selected and showcased in a three-fingered gripper to demonstrate the ability to grasp a given set of geometries with different sizes.

Analysis of load-displacement curves of an adhesive-reinforced composite patch repaired plate using the combination of XFEM and CZM techniques

Since the development of the eXtended Finite Element Method (XFEM), Cohesive Zone Modelling (CZM), and the Virtual Crack Closure Technique (VCCT) for damage analysis in repaired structures, it has now become easy to validate experimental tensile tests carried out on damaged and repaired plates with a composite patch. In most cases, these tensile tests can be expensive and time-consuming due to surface preparation, cleaning, polishing, adhesive application, and the crosslinking time of the adhesive, which must be carefully considered. The analysis of damage in structures assembled by adhesive bonding is crucial to predicting their behavior under various loading conditions. Moreover, the analysis of experimental curves is in most cases translated a global behavior without taking into account the details that can influence the results. With the introduction of techniques (XFEM, CZM and VCCT), it is now possible to model different forms of damage, accounting for several mechanical and geometric parameters of the plate, patch, and adhesive. This work aims to analyze the behavior under tensile loading of a 2024-T3 aluminum plate presenting an initial crack of different lengths emanating from a central circular notch. This plate is repaired by a carbon/epoxy type patch through an adekit A140 adhesive. The combined use of XFEM to track damage in the plate and CZM for adhesive debonding allowed the acquisition of tensile load-elongation curves and the assessment of damage inflicted by loading on the plate. The results demonstrate that the combined use of XFEM and CZM techniques enables the identification of the adhesive debonding impact on the load transfer to the patch and consequently on the strength of the repaired structure. Furthermore, these highlight that the decrease in the effectiveness of the composite patch with increasing crack size is directly linked to the progressive deterioration of both the plate and the adhesive strength.

Two-grid finite element methods for space-fractional nonlinear Schrödinger equations

Two-grid finite element methods for space-fractional nonlinear Schrödinger equations

Thermal analysis of nonlinear mixed convection effects within wavy enclosure

The analysis in this study focusses on the transfer of heat within a wavy cavity in the presence of nonlinear mixed convection and MHD. We conduct the investigation under conditions where the upper wall of the cavity remains cold, the lower wall receives heat, and the left and right wavy walls remain adiabatic. Using the PDE solver in COMSOL, the non-dimensional system is simplified, employing the finite element method. The study aims to explore the impact of parameters such as nanoparticle concentration, Hartmann number, Reynolds number, Richardson number, and nonlinear mixed convection parameters on fluid flow and heat transfer. An artificial neural network has been employed to examine the behaviour of the Nusselt number within a cavity. The isotherms reveal that the temperature is higher near the lower wall for all parameters due to the heated source. Analysis of the streamline plots indicates that heat increases as it moves towards the upper wall, influenced by the Hartman number and volume fraction parameter. The use of ANN in the study forecasts the accuracy, smoothness, and convergence of the data through plots of mean squared error (MSE), error histogram, regression analysis, and transition state plot. The authors believe that using artificial neural networks to explore the heat transfer rate under the assumptions of MHD and nonlinear mixed convection is a novel approach that has not been previously investigated. Additionally, researchers can utilise ANN concepts to predict intricate cavity formations through multiphysics. This approach will require less human effort and computing time to analyse the heat response of any dataset.

Forced vibration assessment of moderately thick carbon nanotube-reinforced composite plates

Carbon nanotubes, distributed in composite sheets with various arrangements, can lead to the development of materials with functionally graded properties. In the present study, the method of spectral components was used to investigate the free vibration frequencies and dynamic responses of single-layer sheets reinforced with carbon nanotubes. The spectral component method offers significant advantages for high-frequency dynamic problems, as it requires fewer fine elements to solve boundary conditions. Based on the first-order shear theory, the governing equations of sheet vibration are derived. The discrete Fourier transform is applied to convert the differential equations from the time domain to the frequency domain. Using dynamic shape functions obtained from the exact solutions, the stiffness matrix of the spectral element is constructed. Dynamic frequency responses are then derived, and time-domain responses are obtained using the inverse Fourier transform. The study extracted the exact natural frequencies of functionally graded sheets and verified them with results from the literature, showing high accuracy with a minimal number of elements. The spectral finite element method of fundamental natural frequencies for different ratios of modulus of elasticity and width-to-thickness ratios obtained were investigated and compared with the results of research. The forced vibration was addressed, demonstrating that the method efficiently captures dynamic responses under different carbon nanotube distributions, with spectral displacements verified through numerical comparison. These findings demonstrate the capability and efficiency of the spectral finite element method for analyzing carbon nanotube-reinforced composite plates.

Investigation of the History-dependent Nonlinear Micro-viscoplasticity Behavior of Dual-phase Steel using Crystal Plasticity Finite Element Method

Investigation of the History-dependent Nonlinear Micro-viscoplasticity Behavior of Dual-phase Steel using Crystal Plasticity Finite Element Method

A novel temporal finite element method to solve static viscoelastic problems

A novel temporal finite element method is presented to solve static viscoelastic problems, which is more flexible and appropriate to describe temporal variation of displacement than conventional finite difference or numerical integral methods. By using virtual work principle, a spatial finite element based governing equation is derived in terms of displacement and its derivatives. Then two temporal finite element models are developed using Gurtin variational principle and weighted residual technique. A kind of hybrid shape functions with polynomial and trigonometric basis is stressed to give more flexible descriptions of time varying variables. A criterion of stability analysis is derived, which can numerically be conducted when the constitution of shape functions and step size are prescribed. A recursive algorithm toward end is developed by which the temporal FE analysis can be conducted via a matrix power product with the initial condition, instead of step marching. The proposed approach is available for the viscoelastic model described by linear differential equations with order h ≤ 2, and can conveniently be combined with well-developed numerical algorithms to tackle with boundary value problems, such as FEM, SBFEM etc. Various numerical examples, including those with static/harmonic loads, creep, stress singularity and heterogeneous structures, etc. are provided to illustrate the efficiency of the proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with analytical or ABAQUS-based solutions.

Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

A globally divergence-free weak Galerkin finite element method with IMEX-SAV scheme for the Kelvin-Voigt viscoelastic fluid flow model with high Reynolds number

A globally divergence-free weak Galerkin finite element method with IMEX-SAV scheme for the Kelvin-Voigt viscoelastic fluid flow model with high Reynolds number

Numerical investigation of patch geometry effect on the fatigue life of aluminum panels containing cracks repaired with CFRP composite patch using XFEM and CZM approach

This study investigated numerically the effect of using composite patches with various geometric shapes to improve the fatigue life of aluminum panels of AA7075-T6 and AA2024-T3 containing cracks and compared with the available experimental results. The Cohesive Zone Model (CZM) and Extended Finite Element Method (XFEM) were applied to static and fatigue analyses. Four types of rectangular, trapezoidal, left-oriented and right-oriented triangular composite patches were used in the numerical analysis. With the combination of XFEM and CZM methods, and using the Paris equation, the fatigue crack propagation and fatigue life of aluminum panels containing cracks can be reasonably predicted. A composite patch with a suitable shape and geometry, fatigue life of the repaired panel can be improved considerably. With a 204% improvement in fatigue life in comparison to the unrepaired panel, the repaired sample made of aluminum alloy AA2024-T3 alloy achieved the most significant improvement in fatigue life. In AA7075-T6 alloy samples repaired with composite patches, the samples repaired with rectangular and trapezoidal patches experienced the highest fatigue life increases, respectively with 342 and 290% compared to the unrepaired samples. The comparison of numerical simulation results with the available experiment results shows a good agreement.

Defect dynamics modeling of mesoscale plasticity

The collective motion of defects and their interaction are the basic building blocks for plastic deformation and corresponding mechanical behaviors of crystalline metals. Especially, dislocations among various defects are the “carrier” of plastic deformation in many crystalline materials, particularly ductile materials. To get a fundamental understanding of plastic deformation mechanisms, it calls for an integrated computational platform to simultaneously capture detailed defects characteristics across several length scales together with corresponding macroscopic mechanical response. In this paper, we present a three-dimensional mesoscale defect dynamics model to directly couple the three dimensional discrete dislocation dynamics model with continuum finite element method, aiming at capturing both size dependent plasticity at micron-, and submicron scale and constitutive behaviors at larger scales where such size-dependence disappear. Using non-singular dislocation theories, our model could accurately consider both short- and long-range elastic interactions between multiple dislocation segments with even higher computational efficiency than traditional dislocation dynamics simulations, together with the careful consideration of crystal/material rotation in the coupled framework. In addition, our model could directly model dislocation nucleation from stress concentrators such as a void, crack and indentor tip, which could allow us to investigate various defects’ motion and their mutual interactions, predicting macroscopic mechanical response of complex structures. The developed concurrently coupled model could also consider multiphysical phenomena by solving coupled governing equations in finite element framework, which could shed light on complex defect behaviors under various physical environments.

OXFORD-UMAT: An efficient and versatile crystal plasticity framework

The crystal plasticity-based finite element method is widely used, as it allows complex microstructures to be simulated and allows direct comparison with experiments. This paper presents the OXFORD-UMAT for Abaqus®, a novel crystal plasticity code that is publicly available online for researchers interested in using crystal plasticity. The model is able to simulate a wide range of materials and incorporates two different solvers based on the solution of slip increments and Cauchy stress with variants of state update procedures including explicit, semi-implicit, and fully-implicit for computational efficiency that can be set by the user based on the application. Constitutive laws are available for a range of materials with single or multiple phases for slip, creep, strain hardening, and back stress. The model includes geometrically necessary dislocations that can be computed using finite element interpolation functions by four alternative methods, including the total form with and without a correction for the dislocation flux, a widely used rate form, and a slip-gradient formulation. In addition, the initial strengthening and subsequent softening seen in irradiated materials can also be simulated with the model. The analysis is available in 2D (plane stress and plane strain) and 3D, including linear and quadratic elements. Here we include full derivations of the key equations used in the code and then demonstrate the capability of the code by modeling single-crystal and large-scale polycrystal cases. Comparison of OXFORD-UMAT with other available crystal plasticity codes for Abaqus® reveals the efficiency of the proposed approach, with the backup solver offering greater versatility for handling convergence issues commonly found in practical applications.

Modeling and Analysis of Taipei 101 Building Based on ANSYS

Abstract. Taipei 101 building is used to be the highest building in the world [1], which is also famous for its unique design and competent performance in responding to disasters nowadays. To apply the finite element method in studying the dynamic behaviour of Taipei 101 building, ANSYS is employed as a powerful tool to build the building's structural model. This simulation enables us to analyse the natural frequency of Taipei 101 building, which is crucial in understanding and predicting its vibrational characteristics under various conditions. By conducting this analysis, insights into the building's response to wind, tsunamis, earthquakes, or other external forces can be gained, thereby enhancing its structural safety.

Prediction of the sound transmission loss of shape-varied sonic crystals: A transfer matrix approach.

This paper proposes a transfer matrix method (TMM) for modeling sonic crystals to predict the transmission loss of noise exiting an air extraction system. Because the crystals may be of different shapes (e.g., square, circular, or standardized airfoil profile to minimize airflow resistance) and must account for thermo-viscous losses, a discrete version of the TMM is used. Similar to the finite element method, a discretization of the geometry is first performed. Each element is modeled with a transfer matrix (TM) that includes the local thermo-viscous losses which attenuate the sound wave. For each element in parallel, the parallel TMM is employed. For the subsequently created elements in series, the classic TMM is used. This generates a global TM from which the sound transmission loss of the crystal network is deduced. The predictions obtained by the proposed method are compared to measurements in an acoustic tube for three different shapes of sonic crystals. The results show that a geometric tortuosity correction is necessary for the predicted bandgap center frequency to match the measurement. A correction is proposed, but this requires a possible refinement for more complicated profiles.

Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh-Stokes equations

Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh-Stokes equations