Finite Element Equations Research Articles

Parameter identification of rock mass in the time domain

A time domain rock mass parameter identification method considering the Rayleigh damping is proposed in this paper. Starting with a set of initial parameter values, the algorithm solves the finite-element (FE) equations updating the parameters in each iterative step until the result converges to a local optimum. Numerical FE analysis, displacement sensitivity analysis, and the truncated singular value decomposition method are integrated into the solution algorithm for parameter identification. The technique introduced is verified in numerical studies with in-situ experiments of rock mass structures. Our results show that the approach, compared with conventional methods, can reduce ambiguities caused by inadequate data and provide more accurate insights into the subsurface for rock mass quality evaluation.

Vibro-acoustics based sound pressure minimization in a cam-follower system using micro-geometric modifications on minimizing the noise

The systems with inherent combined rolling-sliding contacts, such as cam-followers, gearboxes, and clutches, generate undesirable noise. It affects operator well-being, environmental acoustics, and component wear during operation. The sound produced due to interaction between the cam and the follower can be reduced by micro-geometric changes, i.e., profile modification of the cam surface while maintaining the macroscopic geometry of the system. This study aims to identify an optimal profile modification to the cam, using a vibro-acoustic relationship between the contact forces, sound pressure to minimize the sound pressure. The methodology utilizes a mathematical model of a cam-follower system to estimate the contact forces for different parametric values, and an equivalent finite element model to obtain vibro-acoustic transfer functions for various profile modifications. Sound pressure is predicted for each profile based on the obtained transfer functions, and contact forces. The optimal profile is then selected based on a comparative evaluation of the predicted sound pressure levels. The outcome of this study would lead to the development of quieter systems with combined rolling-sliding contact.

Validation of a three-dimensional finite element constitutive modeling approach for a thermoplastic polyurethane calibrated with uniaxial tests

This paper presents the three-dimensional finite element constitutive modeling validation for a Thermoplastic Poly-Urethane (TPU) material. The material parameters are those given in a previous study obtained by calibrating the constitutive model with uniaxial material-level test results (1-D tensile and compression tests). Confined compression tests were performed to estimate the bulk modulus for more accurate material characterization. The parameters were validated by comparing the results of experimental tests on TPU components with those obtained by its equivalent finite element simulations. The TPU component tests comprised cyclic compression tests of (i) a TPU cylinder, (ii) a solid 100 mm diameter TPU ball, and (iii) a 100 mm diameter TPU ball with an 80 mm steel core. In all tests, the constitutive model parameters showed an excellent performance in representing the mechanical behavior of the material observed in the tests, including the nonlinear stiffness and hysteresis. Finally, a brief case study is presented to illustrate the applicability of the validated constitutive model parameters in modeling a novel TPU damper for structural control before its manufacturing and experimental testing. The validated constitutive modeling approach and available material parameters will allow the performance of reliable and early-stage finite element simulations to prototype the mechanical behavior of TPU components of different shapes and boundary conditions and thus gain relevant insight into the component level response before manufacturing and testing.

Fatigue life prediction method for bolted joints based on equivalent structural stress under tensile–compressive loading

In this study, load amplitude–life (Fa–N) curves were obtained through tensile–compressive fatigue tests of bolted joints. It was observed that the correlation coefficient squared (R2) value of the Fa–N curve with the same geometric and pre-tightening parameters was high, but the R2 value of the Fa–N curve with all the parameters was low, indicating poor correlation and inability to meet the engineering requirements. Therefore, an equivalent structural stress model for the bolted joint was first developed based on a mechanical model with a strict mathematical definition to normalize these Fa–N curves, which considered the bolted joint loads as the input conditions and integrated the geometric and pre-tightening parameters. Subsequently, a classical beam–shell equivalent finite element model of the bolted joint was constructed. The nodal loads in the bolted connection zone were coupled with the forces and moments of the beam element nodes through finite element simulation, and the equivalent structural stress (σs) of the bolted joint was then obtained based on the equivalent structural stress model. Consequently, the equivalent structural stress–life (Ss–N) curve and probabilistic stress–life (P–Ss–N) curve normalized for different Fa–N curves were obtained by fitting the data of σs and N. Lastly, the accuracy of the fatigue life prediction method based on equivalent structural stress was verified by conducting the vibration fatigue test on the bolted joint structure of the subway antenna bracket.

Nonlinear thermo-electro-mechanical responses and active control of functionally graded piezoelectric plates subjected to strong electric fields

Functionally graded piezoelectric plates (FGPPs) are often exposed to extreme thermal environments and subjected to large amplitude excitation and strong electric fields. Accurately predicting the nonlinear behaviors of FGPPs under large thermo-electro-mechanical loads remains a considerable challenge. This paper proposes a comprehensive nonlinear coupled analysis model that considers geometric nonlinearity, piezoelectric nonlinearity, and the temperature dependence of piezoelectric parameters. The total Lagrangian incremental finite element equations for FGPPs under thermo-electro-mechanical loads are derived using Hamilton's principle, by employing the first-order shear deformation theory and the von Kármán nonlinear strain-displacement relationship. The accuracy of the model is validated through comparative studies. Based on this multi-nonlinear model, we further investigate the effects of geometric nonlinearity, piezoelectric nonlinearity, and temperature dependence on the nonlinear static bending, dynamic responses, and active control of FGPPs. This study demonstrates that considering these factors enables accurate prediction of the static, dynamic, and active control behaviors of FGPPs.

A new method for the prediction of vibration responses of thin plates coupled with discontinuous connections

Due to the limitation caused by the assumption of ideal continuous line junctions between plates or the high computation cost, the existing methods cannot provide reliable prediction solutions of vibration responses of coupled plates with real connections, such as bolts and corner irons. To this end, a discontinuous energy finite element analysis (dEFEA) method with high accuracy is proposed to predict the vibration responses of discontinuously connected plates, which can revert the real connection. This method developes the coupling loss factors (CLFs) on the steady-state energy flow technique to characterize vibration transmission between plates with different connections. In addition, a discontinuous energy finite element equation of coupled plates is established to obtain the vibration responses of those. The efficacy of the dEFEA method is demonstrated by experiments and the method can improve the prediction accuracy of vibration, which is verified by the comparison between discontinuous and ideal continuous connections. To sum up, vibration transmission of connected structures varies greatly depending on the way of connection and cannot be simplified to a line connection.

Crack propagation arrest by the Joule heating in micro/nano-sized structures

The Joule heating technique is utilized to arrest the crack propagation so to protect some important structures being further damaged. In this approach there are created compression stresses at the crack tip vicinity due to a specific temperature distribution induced by the electric current. The Joule heating is amulti-physical problem which is applied here to micro/nano-sized structures. The size-effects are considered for description of both the elastic and the thermal fields. The heat conduction can’t be well described by classical Fourier’s law in nano-sized structures. A novel gradient theory is developed in such structures adopting the size effect of heat conduction. Besides, the strain gradients and double stresses are introduced into the constitutive laws. Then, the governing equations are given by partial differential equations with order of derivatives being higher than in classical theory. The principle of virtual work is the base of weak formulation and derivation of the discretized finite element equations for ageneral 2d boundary value problem with cracks. The collocation mixed FEM with large strain gradients is developed for numerical solution, where the C0 continuous approximation is applied independently to displacements and strains and also to temperature and temperature gradients. Kinematic constraints between strains and displacements are satisfied by the collocation method at selected interior points. The collocation mixed FEM is applied to solve 2D crack problems with Joule heating.

Nonlinear dynamics analysis of hydraulic turbochargers in reverse osmosis desalination plants

The hydraulic turbocharger plays a vital role in harnessing the energy stored in brine within reverse osmosis desalination plants. To optimize the efficiency and durability of this equipment, it is crucial to develop accurate dynamic models of the turbocharger rotor. An improved understanding of rotor dynamics enables the integration of innovative technologies such as Hydraulic Energy Management Integration, effectively enhancing efficiencies in systems characterized by small capacities and high rotational speeds. This study presents a dynamic modeling methodology for the hydraulic turbocharger. The analysis involves approximating the turbocharger rotor with an equivalent finite element shaft line model. Verification of the model’s natural frequency is conducted using three-dimensional finite element analysis, employing the ANSYS modal analysis module. Computational fluid dynamics is employed to evaluate the fluid forces, while the Reynolds equation is utilized to assess the journal bearing forces. The resulting model is employed to investigate the nonlinear dynamics of the rotor, examining the impact of various system parameters, including rotational speed, unbalance forces, and shaft geometrical parameters. The results highlight the significance of balancing the turbine and pump disks for optimal performance. Furthermore, the research demonstrates that increasing the shaft length reduces the rotor’s threshold speed, while increasing the shaft diameter initially raises the threshold speed until it reaches a critical value. Beyond this critical value, further increases in shaft diameter lead to a decrease in the threshold speed.

Design and crashworthiness evaluation of corrugated honeycomb with multi-directional energy absorption capacity

In this study, three novel multi-directional energy-absorbing honeycombs were designed to meet the requirements in the crash of uncertain directions, which are named as bow-shaped honeycomb (BSHC), staggered honeycomb (SGHC) and corrugated honeycomb (CGHC). These innovative designs can significantly narrow the huge gap of the energy absorption capacity between the in-plane and out-of-plane directions of traditional honeycombs. Compression tests were conducted in three orthogonal directions. The BSHC is found to have the smallest densification strain but the highest plateau stress in each direction. The SGHC can only balance the energy absorption between out-of-plane and in-plane-x directions. The CGHC demonstrates a better densification strain and the highest multi-directional energy absorption coefficient. The detailed and equivalent finite element models of CGHC were further established and validated, and both exhibited high accuracy. Finally, a honeycomb anti-climber, with only about half length of the traditional guided honeycomb anti-climber, was designed and equipped with metro vehicles. Simulations were conducted under eccentric collision scenario. The results demonstrated that the CGHC anti-climber was capable of orderly deformation in the axial direction (out-of-plane direction) while effectively resisting the vertical (in-plane-y direction) force during collision. The energy absorption capacity of CGHC anti-climber was significantly enhanced as compared to the HEHC anti-climber under eccentric collision scenario.

FE/PDE: a novel approach applied to PC plate structure with multi-scale and multi-physics field coupling

For the more straightforward and more efficient solution of phononic crystal (PC) plate frequency band structure, transmission curve, and vibration mode, in this paper, related theories based on spatial Fourier series expansions, combined with Bloch’s theorem, a novel approach to solve the structural governing equations of PC plate is proposed by using the partial differential equations (PDE) module in the finite element software COMSOL. It is named the FE/PDE (Finite element and partial differential equations) method. The method’s accuracy is verified by comparing the results with those obtained from the traditional method. Systematic elucidation of the application of the method to probe the properties of multi-scale, multi-physics field coupled PC plate. In order to demonstrate the flexibility and scientific validity of the method, a novel nano-piezoelectric PC plate structure is proposed and solved. The method is simple, computationally efficient, and applicable, and provides a new method for investigating the properties of PC plates.

A numerical method for calculating the long-term settlement of three-dimensional subgrade under numerous traffic loads

Settlement problems caused by various traffic loads significantly impact infrastructure sustainability. Accurately predicting the long-term settlement of subgrade is crucial for improving infrastructure resilience. This study determined the element strain of soil using an explicit strain model, based on the bounding surface model and associated flow rules. Using the method of generalized shear strain equivalent, a cyclic cumulative factor is proposed for determining the magnitude of cumulative strain increment. The direction of plastic strain increment was determined using the bounding surface model combined with associated flow rules. A cumulative strain calculation method integrating the bounding surface and explicit strain models under general stress states was established. This computational method, combined with equivalent finite elements, can be used for calculating the long-term settlement of three-dimensional subgrade under numerous cyclic loads at general stress states through a user-defined subroutine provided by the commercial finite element modeling (FEM) software. The reliability of the method was verified through unit tests.

An optimized CIP-FEM to reduce the pollution errors for the Helmholtz equation on a general unstructured mesh

The continuous interior penalty finite element method (CIP-FEM) has shown promise in reducing pollution errors when numerically simulating the Helmholtz equation with high wave numbers. However, its reliance on structured meshes largely limits its applicability. To address this limitation, this paper presents a novel approach to CIP-FEM on general unstructured meshes. The interior penalty parameters of the CIP-FEM are determined in an offline phase through minimizing the residual obtained by substituting the plane waves into the CIP finite element equation. To enhance accuracy, a quasi-Newton algorithm is employed to correct the penalty parameters and minimize errors of the CIP finite element approximations to the plane waves. The calculated interior penalty parameters can be saved, enabling the online solutions of CIP-FEM to be obtained by reusing these parameters for arbitrary sources during practical computations. Numerical experiments are presented to demonstrate the significant reduction of pollution errors for both linear and higher-order CIP-FEMs on unstructured meshes. Furthermore, the proposed algorithm successfully simulates high-frequency acoustic fields in a three-dimensional vehicle cabin with greatly reduced errors within certain fixed computational time, demonstrating its practical effectiveness in real-world scenarios.

Finite Element Analysis for Linear Viscoelastic Materials Considering Time-Dependent Poisson’s Ratio: Variable Stiffness Method

For linear viscoelastic materials, this paper proposes a finite element analysis method based on an integral constitutive relationship that can simultaneously consider the relaxation behavior of the elastic modulus and the creep Poisson’s ratio. Firstly, the generalized Maxwell model is employed to depict the relaxation characteristics of the elastic modulus, while the generalized Kelvin model is used to represent the creep Poisson’s ratio. Subsequently, the element relaxation stiffness matrix is established, thereby forming a convolutional finite element equation. Furthermore, the recursive calculation of the convolutional integral is derived, and the calculation steps of the finite element for viscoelasticity considering the time-dependent nature of both the elastic modulus and Poisson’s ratio are established. Finally, the accuracy of the proposed algorithm is verified through two numerical examples with linear viscoelastic material. The results indicate that the proposed variable stiffness method for the finite element analysis of linear viscoelastic materials can simultaneously consider the changes in the elastic modulus and Poisson’s ratio over time, thereby establishing a new path and idea for the more accurate simulation of viscoelastic materials’ mechanical properties. Compared with the initial strain method for linear viscoelastic materials, the variable stiffness method proposed in this paper effectively avoids the assumption of constant stress during the micro time interval, thus significantly enhancing the finite element calculation accuracy of linear viscoelastic materials. The proposed method establishes a simulation algorithm that matches existing commercial software with viscoelastic material experiments by considering the elastic modulus and Poisson’s ratio as material parameters.

A thermodynamically consistent phase field model for brittle fracture in graded coatings under thermo-mechanical loading

This paper presents a thermodynamically consistent phase field formulation designed for investigating quasi-static brittle fracture under thermo-mechanical loading conditions in functionally graded materials. The governing equations for linear momentum and crack evolution are derived by minimizing the free energy functional, which includes elastic strain energy and surface energy, with respect to displacement field and the phase field variable, respectively. A thermodynamically consistent formulation is employed to establish the governing equation for heat conduction. The proposed formulation is converted into a set of finite element equations using the Ritz method and implemented in COMSOL Multiphysics, a commercial finite element software which allows for writing governing equations in their strong form. A segregated solver is utilized to solve iteratively the coupled problem. The proposed model is verified against existing mechanical and thermo-mechanical phase field models, demonstrating good agreement with the literature. To explore the model’s capabilities, a graded layer composed of Zirconia and a Titanium alloy with a surface crack undergoing thermo-mechanical loading is considered as a case study for this investigation. Additionally, the model incorporates a thermal conductivity degradation proposed. A parametric study is performed to investigate the effect of varying parameters like volume fraction, thermal loading rate, and the thermal conductivity degradation on the load–displacement curve of the graded layer. These findings contribute to the understanding of thermo-mechanical behavior of surface graded layers and serve as a foundation for addressing more complex thermo-mechanical problems.

Finite element boundary element numerical simulation study on acoustic vibration coupling problems

Excessive structural resonance response in the acoustic vibration coupling environment can cause adverse consequences such as structural damage, local instability, and component failure. Accurately predicting the vibration response of instruments under external sound fields helps to avoid structural performance degradation or damage caused by vibration, and ensures the reliability and safety of the system, which is of great significance for the effective load of instrument equipment. This paper constructs a finite element boundary element equation for the acoustic vibration coupling problem, and adopts a series of optimization measures for it. In response to the low computational efficiency of finite element boundary element equations and their inability to meet the computational requirements of complex models, this paper introduces the acoustic vibration reciprocity theorem and the fast directional boundary element method to compensate for the aforementioned shortcomings. In addition, the study used the weighted residual method to construct a data exchange algorithm to ensure energy conservation during the transfer process. The research focuses on typical structures of spacecraft and conducts simulation analysis and application research on acoustic vibration coupling problems. The simulation showcases that the introduction of the reciprocity theorem markedly reduces the calculation time and memory occupied by the equation, with a required calculation time of 1.23 h and a data file size of approximately 299.56 MB generated by the calculation; The overall error of the data exchange process is below 10−4. Scanning frequency analysis is conducted on the surface and local openings of the model structure, and it is found that the displacement of the selected excitation point is in good agreement with the analytical solution. This study effectively reduces simulation time and memory requirements through the proposed strategy, ensures energy conservation data exchange algorithms, improves the reliability and safety of structural design, and has important practical value for evaluating the structural performance of key equipment such as spacecraft.

Anti-galloping analysis of iced quad bundle conductor based on compound damping cables

Based on the strain displacement relationship of spatial curved beam theory, a nonlinear dynamic equation for a hybrid model of iced bundle conductor galloping analysis with three translational degrees of freedom and one rotational degree of freedom was established. The element independence is numerically studied, and the impact of modal truncation on the galloping response is analyzed to verify the accuracy of the model. In addition, the compound damping cable is applied to transmission lines for the first time to prevent galloping, and a corresponding nonlinear vibration control finite element equation for the iced quad bundle conductor with a compound damping cable structure under wind load is developed. The study also examines the influence of various parameters on galloping amplitude. The results show that the hybrid model accurately predicts the galloping response of transmission lines. The compound damping cable can effectively suppress the galloping of the quad bundle conductor, and the vibration reduction effect reaches more than 85%. Increasing the installation height of the compound damping cable improves its damping effect. At the same installation height, when the horizontal installation position is close to the mid-span of the conductor, the damping effect first increases and then decreases, suggesting an optimal installation position exists. Moreover, properly increasing the stiffness of the primary cable and reducing the stiffness of the return spring can improve the vibration reduction effect of the compound damping cable. Additionally, if the damping coefficient and the mass of the primary cable are properly increased, the vibration reduction effect of the compound damping cable will be better.

Efficient homogenization of honeycomb sandwich panels using orthotropic core simplification and Finite Element-based method: A comparative study

Composite materials, particularly honeycomb composites, are widely utilized in various industries, including aerospace, due to their high energy absorption against the impact and exceptional strength-to-weight ratio. This study aims to leverage the plastic and elastic properties of these materials to develop a simplified numerical model that incorporates orthotropic properties for core modeling. By doing so, the need for detailed honeycomb structure modeling is eliminated, resulting in reduced computational costs and time. A comprehensive three-dimensional finite element model, accounting for structural intricacies, is presented based on experimental data from a reputable source (isotropic model) and its equivalent finite element model (orthotropic model). The model is validated by the experimental results, demonstrating good agreement. The study also investigates parameters such as energy absorption, the internal energy of the core and faces, maximum displacement, and maximum contact force under low-velocity impact scenarios with spherical and cylindrical projectiles. These findings highlight the effectiveness of the orthotropic model, particularly in showcasing greater energy absorption in the core of the sandwich panel when subjected to a cylindrical impactor.

Target strength analysis using reflection coefficient of a submarine with low-frequency anechoic coatings

As the technological development of modern weapons systems progresses, attack and defense against the weapons systems of the past can become useless. In the case of underwater weapon systems, improvements in sonar and torpedo technology necessitate enhancements in submarine stealth performance. The stealth performance of a submarine is analyzed through the acoustic target strength (TS), which is indicated by the degree of sound wave reflection in the sonar equation. Various methods have been researched to analyze the TS of submarines, but methods considering the Anechoic coating attached to submarines are rare. Coating-related research has mainly focused on the absorption characteristics of coatings attached to flat surfaces. Therefore, there is an increasing need for research to analyze the sound absorption performance of anechoic coatings considering the curved surface of a submarine. In this study, we propose a method to verify the stealth performance of a submarine by analyzing the TS according to the azimuth angle by applying periodic inclusion of anechoic coating (PIAC) to the submarine. The PIAC based on the meta-structure is designed to form a bandgap in the low-frequency region by considering the resonance of the cylindrical Inclusion and the air cavity, applying the Bloch Theorem and the Brillouin zone, as well as the periodic arrangement characteristics. The PIAC was attached to the submarine model and modeled, and the absorption and reflection coefficients according to the azimuth angle were calculated using the finite element equation. The TS analysis included the process of determining the effective area by combining Geometrical Optics (GO) based on ray tracing and Physical Optics (PO) using the Kirchhoff approximation. The stealth performance was verified by comparing the TS analysis results according to the azimuth angle of the submarine equipped with the existing Alberich coating and the submarine equipped with PIAC at the same frequency.

Local multigrid method for solving adaptive finite element equations of three-dimensional elasticity problems

Local multigrid method for solving adaptive finite element equations of three-dimensional elasticity problems

Approximate global mode method for flutter analysis of folding wings

The aircraft with folding wings benefits from the ability to transform its folding wings to adapt to various flight missions but suffers from the change of the wings’ aeroelastic characteristics as a result. To clarify the change, it is necessary to study the dynamical modeling of the folding wing and also its aeroelastic response. This paper proposes a novel approximate global mode method (AGMM) for dealing with complex boundary conditions and models a folding wing consisting of separate rectangular plates for nonlinear flutter analysis. A high-precision low-dimensional dynamical model of the folding wing is developed, which is first validated by comparing the modal results obtained from the AGMM model with those obtained from an equivalent finite element model and is also experimentally validated by a comparison of forced vibration responses. On this basis, an AGMM model of the folding wing in supersonic aerodynamic loads based on the piston theory is further developed. The critical flutter velocity at different folding angles and the aeroelastic response are comprehensively studied by simulations based on the model. The result shows that the first 7th-order modes are sufficient for obtaining a converged critical flutter velocity, which varies with geometric parameters and potentially increases by 714 %. In addition, the low-order torsional modes of the wing contribute more to the vibration than the other modes during flutter. This work provides a new way of developing advanced dynamical modeling methods of folding and combinational structures for their dynamics design, optimization, and system control.