Conventional Finite Element Method Research Articles

A novel hybrid boundary element for polygonal holes with rounded corners in two-dimensional anisotropic elastic solids

A novel hybrid boundary element is developed for polygonal holes in finite anisotropic elastic plates based on two different special fundamental solutions for holes. Since these special fundamental solutions satisfy traction-free condition along the hole's boundary, there is no mesh required on the boundary of polygonal holes. Various types of polygonal holes with rounded corners, such as triangles, rhombuses, ovals, pentagons, are considered by adding proper perturbation to an elliptical hole. The developed hybrid element is a mixture of two special boundary elements: one is based on the special fundamental solution derived through nonconformal mapping and the other is based on the solution derived through perturbation technique with conformal mapping. The special boundary element methods are combined through submodeling technique. First, the global model is solved using the perturbation solution. Then, using the displacements obtained from global model, an auxiliary submodel is set up and the results are evaluated with the nonconformal solution. The present method is compared and validated with conventional boundary element method and finite element method. The effect of hole curvature, material anisotropy, and loading condition on the stress distribution around the hole is presented.

Comparison of neural FEM and neural operator methods for applications in solid mechanics

AbstractMachine learning methods are progressively investigated for a large amount of applications. Recently, the solution of partial differential equations (PDE) describing problems in elastostatics came into focus. The current work investigates two neural network-based classes of methods for their solution, namely the neural finite element method (FEM) and neural operator methods. The analysis of these approaches is carried out by means of numerical experiments with linear and nonlinear material behavior where the conventional FEM serves as a benchmark. The formulation of neural FEM allows for elegant integration of finite deformation hyperelasticity at medium training effort. Here, training data are replaced by the evaluation of the equilibrium PDE at sample points. In contrast, most neural operator methods require expensive training with large data sets, but then allow for solving multiple boundary value problems with the same machine learning model. For the comparative analysis, the maximal relative error values over the whole domain and over all components of the strain tensor are evaluated as accuracy measure. The current state of research shows that none of the methods investigated reaches the accuracy and computational performance of the conventional FEM. In many standard applications, the FEM achieves an accuracy of $$10^{-6}$$ 10 - 6 , whereas the numerical tests in the present work report a relative error of order of magnitude of $$10^{-4}$$ 10 - 4 for the neural FEM and $$10^{-2}$$ 10 - 2 to $$10^{-3}$$ 10 - 3 for neural operator methods.

Dynamic Analysis of 2D FG Porous Thick Cylindrical Shells Under Thermal Shock

In this research, a thick hollow cylindrical shell made of bidirectional functionally graded open cell porous materials under internal thermal shock according to the classical theory of linear thermo-elasticity is examined for the first time. The cylinder is made of a porous cellular material and its porosity varies along both radial and axial directions continuously. The governing motion equations are obtained by using 2D-axisymmetric theory of linear thermo-elasticity rather than shell theories. This theory represents thickness stretching and gives more precise results. Graded finite element method is employed to model the problem. Applying this method rather than conventional FEM leads to more accurate results, especially for dynamic analyses. The cubic higher order element is used for dividing the solution domain. To obtain transient temperatures, Crank–Nicolson algorithm is used and then the Newmark procedure is used to derive time responses of displacement and stress components. The time history of displacements and stress components for different radial and axial power law exponents, porosity coefficient, boundary conditions, length-to-thickness ratio and two different porosity patterns are investigated in detail. The obtained results show that thermal-induced vibration is generally caused by hoop stress, and frequency and amplitude of vibrations and velocity of stress waves are considerably influenced by the porosity distribution, porosity coefficient and power law exponents in both directions.

On the isogeometric analysis of vibration and buckling of bioinspired composite plates using inverse hyperbolic shear deformation theory

Inspired by the remarkable energy absorption and damage tolerance capabilities observed in mantis shrimp crustaceans, this study delves into the intricate realm of biological composites with helicoidal structures. However, the effect of the relative orientation of fibers within the helicoidal structure matrix on the buckling and vibration characteristics still warrants comprehensive investigation for the better design of the bioinspired structures. For the first time, a Non-Uniform Rational B-Splines (NURBS)-based isogeometric analysis (IGA) framework termed NURBio is proposed to investigate the vibration and buckling characteristics of bioinspired laminated composite structures. The framework employs NURBS basis functions for exact complex geometric representation and field approximation and offers superior computational accuracy and efficiency as compared to conventional finite element method (FEM). The present analysis is underpinned by an inverse hyperbolic shear deformation theory (IHSDT) capable of accurately capturing the interlaminar stress distribution. This research investigates the performance of various bioinspired layup configurations encompassing recursive, exponential, semicircular, and Fibonacci helicoidal designs and compares them with those of unidirectional, and cross-ply laminates. A series of numerical examples on the buckling and vibration response of bioinspired composite plates are presented to demonstrate the versatility and efficacy of the proposed isogeometric framework. The influence of different layups, loading and boundary conditions, and geometric properties on the response of different bioinspired plate structures are meticulously examined. The study offers valuable insights into the design and optimization of high-performance bioinspired structures.

A review of curved crease origami: design, analysis, and applications

Origami structures with morphing behaviours and unique mechanical properties are useful in aerospace deployable structures, soft robots and mechanical metamaterials. Curved-crease origami, as one of the variants in the origami family, has a curve that connects two vertices as a crease compared to the straight crease counterpart. This feature couples the crease folding and facet bending during the folding process, providing versatile design space of mechanical metamaterials with tunable stiffness, multi-stability properties and morphing behaviours. However, current design techniques are mostly for simple geometries with intuitive construction, the modelling technique focuses on using the conventional finite element method, and the intrinsically complex geometries make specimens difficult to manufacture, which further hinders the development of curved-crease origami structures. Thus, it is valuable to review the state-of-the-art in curved-crease origami. This paper presents a review on the design methodology, analytical methods, and applications of curved-crease origami over the years, discusses their strengths, identifies future challenges and provides an outlook for the future development of the curved-crease origami concept.

Dynamic analysis of the three-phase magneto-electro-elastic (MEE) structures with the finite element method enriched by proper enrichment functions

In this study, a carefully designed enriched finite element method (EFEM) is presented to improve the solution accuracy of the conventional FEM by analyzing the dynamic behavior of the magnetic-electric-elastic (MEE) composite structures, which are frequently used in designing various smart and intelligent devices. By formulating the proper EFEM with ideal numerical performance, different enrichment functions are considered and the corresponding solution quality of different versions of the EFEM is compared and examined in great detail. When the Lagrange polynomial basis functions together with the harmonic trigonometric functions are used as enrichment functions, the obtained EFEM shows extremely powerful and ideal numerical performance, which is obviously better than the other versions of EFEM and the conventional FEM, in studying the free vibration and harmonic frequency responses of the MEE structures. Nearly exact numerical solutions for three-phase physical fields of MEE structures can be generated by the proposed EFEM even if very coarse mesh patterns are used. Intensive numerical studies are conducted to confirm and verify the excellent properties of the proposed EFEM in performing dynamic analysis of the MEE structures.

A comparative study of emerging material point method and FEM for forming simulation of textile reinforcements

For forming simulations of fabric composites, nonlinear Finite Element Method/FEM has been a long-standing tool to predict and mitigate defects such as wrinkling. However, small-time step requirements in explicit FEM codes, numerical instabilities, and large computational time are among challenges reported. This study presents an alternative fast forming simulation technique through an application of the so-called Material Point Method/MPM, which enables the use of much larger time steps along with fewer numerical instabilities. As a preliminary step towards assessment of this method, both standard 2D deformation modes and 3D hemispherical forming setups were employed, using a plain fabric weave at dry condition. The MPM results were compared to the conventional FEM simulations, as well as to the physical experiments. Notably, the MPM method showed a runtime 20 times faster than its FEM counterpart (under a comparable mesh size), yet with the same reliability in forming predictions as verified by experiments.

High-strain dynamic model of large-diameter pipe piles with soil plug for vertical vibration analysis

A rigorous analytical model is developed for simulating the vibration behaviors of large-diameter open-ended pipe piles (OEPPs) and surrounding soil undergoing high-strain impact loading. To describe the soil behavior, the soil along pile shaft is divided into slip and nonslip zones and the base soil is modeled as a fictitious-soil pile (FSP) to account for the wave propagation in the soil. True soil properties are adopted and slippage at the pile-soil interface is considered, allowing realistic representation of large-diameter OEPP mechanics. The developed model is validated by comparing with conventional models and finite element method (FEM). It is further used to successfully simulate and interpret the behaviors of a steel OEPP during the offshore field test. It is found that the variation in the vertical vibrations of shaft soil along radial direction is significant for large-diameter OEPPs, and the velocity amplitudes of the internal and external soil attenuate following different patterns. The shaft soil motion may not attenuate with depth due to the soil slippage, while the wave attenuation at base soil indicates an influence depth, with a faster attenuation rate than that in the pile. The findings from the current study should aid in simulating the vibration behaviors of large-diameter OEPP-soil system under high-strain dynamic loading.

On effect of residual stress on fracture behavior of mandibular reconstruction plates

Titanium reconstruction plates are often used to bridge mandibulectomy defects in load bearing scenarios when bone grafts are not well integrated to the host bone. The residual stress within a standard reconstruction plate is generated when being bent and installed to adapt to a patient-specific anatomical contour and it may have a detrimental effect on the structural stability and reliability of the reconstruction system. This study aimed to evaluate the impact of residual stress on the mechanical strength of the reconstructed mandibular system by utilizing both conventional finite element method (FEM) and eXtended Finite Element Method (XFEM). The mechanical stresses introduced by plate pre-bending and screw tightening were first modeled computationally and the residual stress data induced by the surgical procedure was incorporated to the deformed reconstruction plate for the subsequent biomechanical evaluation. Static and cyclic loading conditions were then imposed on the mandibular plate models to further investigate two common failure types, namely overloading fracture and fatigue fracture. It is revealed that the residual stress could considerably increase the susceptibility of plate fracture. The simulation results demonstrate that the pre-stresses induced by screw tightening are more substantial than that from plate bending during the surgical procedure. The finding is of important clinical implications for surgeons who are commonly involved in selecting and preparing different forms of fixation plates for mandibular reconstruction. This study helps elucidate the key factors contributing on the failure of reconstruction plates and guide the development of more robust and durable mandibular reconstruction systems.

Boundary element method for bending-extension coupling shear deformable laminated composite plates with multiple stiffeners

In general, the laminated composite plates and stiffeners may exhibit the bending-extension coupling effect with the transverse shear deformations significantly induced. In this paper, for the first time we develop the associated boundary element method (BEM) for bending-extension coupling shear deformable laminated composite plates with multiple beams attached as stiffeners. Based upon the first order shear deformation theory, the individual boundary integral equations (BIEs) for plates and stiffeners are obtained from their corresponding matrix-form basic equations. To account for perfect bonding conditions between plate and stiffeners, we establish displacement continuity and interaction load equilibrium relations at their interfaces. To calculate the domain integrals related to unknown interaction loads, we discretize each stiffener into a series of cells using quadratic interpolation. By collecting the individual BIEs for plates and stiffeners and employing the continuity and equilibrium relations at the interfaces, we construct the entire system of linear equations to solve for all the unknown nodal quantities. Additionally, the complete solutions at the plate boundary, at internal points or along a stiffener are provided. The correctness of our present BEM formulation is verified through two representative examples by comparison with the commercial finite element software ANSYS. The numerical results demonstrate that our BEM formulation surpasses the conventional finite element method and is highly desirable for general stiffened laminated plates with bending-extension coupling and transverse shear deformations.

Theoretical modeling and experimental investigation of in-phase resonant MEMS mirrors with cascaded structures

This paper proposes an efficient nonlinear one-dimensional (1D) compact mass-damper-spring model to predict the dynamic response of electrostatic resonant micro-electro-mechanical system (MEMS) mirrors with cascaded structures. The time-dependent damping moment due to viscous shear and pressure drag is computed using semi-empirical analytical equations for comb-drive structures and device frames. Nonlinear electrostatic force induced by the comb drives is efficiently acquired based on the hybrid method. The optimized device is fabricated using MEMS fabrication processes based on a 4-inch silicon-on-insulator wafer. The proposed compact model with the measured key parameters from the fabricated device shows excellent capability to accurately predict nonlinear dynamic responses of the fabricated device, including parametric excitation and hysteretic frequency response, with an average error of less than 5%. In particular, our 1D model is three orders of magnitude faster than the conventional finite element method model (0.8 s versus 1 h), enabling efficient system-level optimization of the critical design parameters. Based on the parametric study, electrode gap distance and torsion spring width are found to be two critical design parameters and dimensional analysis is conducted for design optimization with scan angle enhancing from 16.8° to 24° compared with the first design.

Artificial intelligence-based surrogate model for computation of the electric field of high voltage transmission line ceramic insulator with corona ring

Purpose The ionization of the air surrounding the phase conductor in high-voltage transmission lines results in a phenomenon known as the Corona effect. To avoid this, Corona rings are used to dampen the electric field imposed on the insulator. The purpose of this study is to present a fast and intelligent surrogate model for determination of the electric field imposed on the surface of a 120 kV composite insulator, in presence of the Corona ring. Design/methodology/approach Usually, the structural design parameters of the Corona ring are selected through an optimization procedure combined with some numerical simulations such as finite element method (FEM). These methods are slow and computationally expensive and thus, extremely reducing the speed of optimization problems. In this paper, a novel surrogate model was proposed that could calculate the maximum electric field imposed on a ceramic insulator in a 120 kV line. The surrogate model was created based on the different scenarios of height, radius and inner radius of the Corona ring, as the inputs of the model, while the maximum electric field on the body of the insulator was considered as the output. Findings The proposed model was based on artificial intelligence techniques that have high accuracy and low computational time. Three methods were used here to develop the AI-based surrogate model, namely, Cascade forward neural network (CFNN), support vector regression and K-nearest neighbors regression. The results indicated that the CFNN has the highest accuracy among these methods with 99.81% R-squared and only 0.045468 root mean squared error while the testing time is less than 10 ms. Originality/value To the best of the authors’ knowledge, for the first time, a surrogate method is proposed for the prediction of the maximum electric field imposed on the high voltage insulators in the presence Corona ring which is faster than any conventional finite element method.

Implicit direct time integration of the heat conduction problem in the Method of Matched Sections

The paper is devoted to further elaboration of the Method of Matched Sections as a new branch of finite element method in application to the transient 2D temperature problem. The main distinction of MMS from conventional FEM consist in that the conjugation is provided between the adjacent sections rather than in the nodes of the elements. Important feature is that method is based on approximate strong form solution of the governing differential equations called here as the Connection equations. It is assumed that for each small rectangular element the 2D problem can be considered as the combination of two 1D problems – one is x-dependent, and another is y-dependent. Each problem is characterized by two functions – the temperature, , and heat flux . In practical realization for rectangular finite elements the method is reduced to determination of eight unknowns for each element – two unknowns on each side, which are related by the Connection equations, and requirement of the temperature continuity at the center of element. Another salient feature of the paper is an implementation of the original implicit time integration scheme, where the time step became the parameter of shape function within the element, i.e. it determines the behavior of the Connection equations. This method was early proposed by first author for number of 1D problem, and here in first time it is applied for 2D problems. The number of tests for rectangular plate exhibits the remarkable properties of this “embedded” time integration scheme with respect to stability, accuracy, and absence of any restrictions as to increasing of the time step.

A Coupled Finite-Boundary Element Method for Efficient Dynamic Structure-Soil-Structure Interaction Modeling

This paper introduces an innovative approach to numerically model Structure–Soil-Structure Interaction (SSSI) by integrating the Boundary Element Method (BEM) and the Finite Element Method (FEM) in a coupled manner. To assess the accuracy of the proposed method, a comparative study is undertaken, comparing its outcomes with those generated by the conventional FEM technique. Alongside accuracy, the computational efficiency aspect is crucial for the analysis of large-scale SSSI problems. Hence, the computational performance of the coupled BEM–FEM method undergoes a thorough examination and is compared with that of the standalone FEM method. The results from these comparisons illustrate the superior capabilities of the proposed method in comparison to the FEM method. The novel approach provides more reliable results compared to traditional FEM methods, serving as a valuable tool for engineers and researchers involved in structural analysis and design.

Bending-extension coupling analysis of shear deformable laminated composite curved beams with non-uniform thickness

In this paper, the boundary-based finite element method (BFEM) is developed for the bending-extension coupling analysis of shear deformable laminated composite curved beams with non-uniform thickness. The curved beams are discretized into a series of uniform straight segments. Based upon the first order shear deformation theory and the reciprocal theorem of Betti and Rayleigh, the boundary integral equations (BIEs) associated with each beam segment are derived and expressed in explicit form. The fundamental solutions required in the BIEs are obtained explicitly from the associated Green’s function. Since no singular integrals, numerical integrations and function interpolations are involved in the BIEs, our proposed BFEM provides an exact solution for a straight beam with uniform thickness, and only one single element with two end nodes is required for our analysis. Even only an approximate solution is provided for a curved beam with non-uniform thickness, BFEM converges faster than the conventional finite element method, and the results are in well agreement with those obtained by the commercial finite element software. Through the illustrated numerical examples, its generality is shown by: (1) stacking sequence of laminates: symmetric or unsymmetric; (2) beam geometries: straight, stepped, or curvilinear with uniform or non-uniform thickness; (3) loading types: axial force, transverse force and/or bending moment; distributed or concentrated; and (4) supporting conditions: clamped, simply supported, or free; with or without internal supports.

A particle finite element method approach to model shear cutting of high-strength steel sheets

AbstractShear cutting introduces residual strains, notches and cracks, which negatively affects edge-formability. This is especially relevant for forming of high-strength sheets, where edge-cracking is a serious industrial problem. Numerical modelling of the shear cutting process can aid the understanding of the sheared edge damage and help preventing edge-cracking. However, modelling of the shear cutting process requires robust and accurate numerical tools that handle plasticity, large deformation and ductile failure. The use of conventional finite element methods (FEM) may give rise to distorted elements or loss of accuracy during re-meshing schemes, while mesh-free methods have tendencies of tensile instability or excessive computational cost. In this article, the authors propose the particle finite element method (PFEM) for modelling the shear cutting process of high-strength steel sheets, acquiring high accuracy results and overcoming the stated challenges associated with FEM. The article describe the implementation of a mixed axisymmetric formulation, with the novelty of adding a ductile damage- and failure model to account for material fracture in the shear-cutting process. The PFEM shear-cutting model was validated against experiments using varying process parameters to ensure the predictive capacity of the model. Likewise, a thorough sensitivity analysis of the numerical implementation was conducted. The results show that the PFEM model is able to predict the process forces and cut edge shapes over a wide range of cutting clearances, while efficiently handling the numerical challenges involved with large material deformation. It is thus concluded that the PFEM implementation is an accurate predictive tool for sheared edge damage assessment.

Structural Reliability and Optimization Using Differential Geometric Approaches

This study investigates how differential geometry ideas can be used to effectively carry out structural optimization and reliability analysis. Strong mathematical representations and methods for examining intricate surfaces and forms are provided by differential geometry. The basic ideas of differential geometry, such as tensors, manifolds, and curvature, are initially introduced in the work. Then, to account for ambiguities in the geometry, the probability theory in tangent spaces is developed. As a result, structural reliability can be determined using propagating uncertainty. To enable reliability-based design optimization, differential geometry representations are linked with optimization techniques. The proposed differential geometry-based approach is applied in a number of case studies to trusses, airplane wings, car bodies, and ship hulls. The outcomes show a significant increase in productivity and scalability compared to conventional finite element methods. The article offers new tools for dealing with uncertainty.

Free Vibration Analysis of Laminated Sandwich Plates Using Wavelet Finite Element Method

The accuracy of a wavelet depends on the choice of the mother wavelet adopted. The present work aims to predict the free vibration behavior of laminated sandwich plates using wavelet finite element (WFE). Different kinds of mother wavelets, namely, B-spline wavelet on the interval (BSWI), Gaussian, Haar, Daubechies 6 (db6), Biorthogonal 3.7 (bior3.7), Coiflet5 (coif5), Symlets (sym8), Morlet, Mexican hat (Mh), and Meyer mother wavelets, are employed in WFE for predicting the frequencies. Both symmetric and unsymmetric laminates are studied using the proposed approaches. A wide range of problems, including the influence of the geometric and material properties and end conditions on the free vibration behavior of the laminated sandwich plates, are solved. The effectiveness of the WFE over the conventional finite element method in terms of computational efficiency is discussed. In conclusion, BSWI-based WFE method (WFEM) is found to be the most accurate and computationally efficient in predicting the free vibration behavior of laminated sandwich plates. The accuracy of the WFEM depends widely on the type of mother wavelet adopted.

Modelling of grain size effects in progressive microforming using CPFEM

Progressive microforming is widely recognized as one of the most efficient and desirable methods of mass production in micromanufacturing arena. To predict the deformation behaviour and size effects of materials induced in progressive microforming, the finite element method (FEM) employed for modelling the microforming process needs to account for microstructure details and deformation/failure mechanisms of the materials. This led to the development of the novel crystal plasticity finite element method (CPFEM) and cohesive zone model (CZM). Previous research and application of CPFEM have been mainly limited to simple deformations such as uniaxial tension and compression, whereas the new method can provide physical insights into how the grain size affects the interplay between crystallographic slip and mechanical twinning in complex microforming, and further material deformation during sheet blanking. A case study was conducted to manufacture a hexagonal socket part using a three-step progressive microforming system, with the comparison between experiments and CPFEM simulations focusing on microstructure evolution, deformation load, and product quality. The CPFEM was found to be more reliable than the conventional FEM in predicting complex deformation, particularly in microstructure and texture evolution, dimensional accuracy and irregular geometries. Results showed that the total height of the part increases with the decreasing grain size, while the head diameter rises with grain size. Simulations successfully anticipated the distributions of dead metal zones and shear bands and identified hole and rollover geometries and corresponding grain size effects. In conclusion, this research facilitates the understanding of grain size effects on the deformation behaviour in progressive microforming and presents a novel approach and strategy for modelling, prediction, and product quality assurance in complex micro deformation and forming processes.

Physics-informed neural networks based on source term decoupled and its application in discharge plasma simulation

In recent years, the artificial intelligence computing paradigm represented by physics-informed neural networks (PINNs) has received great attention in the field of plasma numerical simulation. However, the plasma chemical system considered in related research is relatively simplified, and the research on solving the more complex multi-particle low-temperature fluid model based on PINNs is still blank. In more complex chemical systems, the coupling relationship between particle densities and between particle densities and mean electron energy become more intricate. Therefore, the applicability of PINNs in dealing with sophisticated reaction systems needs further exploring and improving. In this work, we propose a general PINN framework (source term decoupled PINNs, Std-PINNs) for solving multi-particle low-temperature plasma fluid model. By introducing equivalent positive ions and replacing each particle transport equation with the current continuity equation as a physical constraint, Std-PINN splits the entire solution process into the training processes of two neural networks, realizing the decoupling of the source term of the heavy particle transport equation from the electron density and mean electron energy, which greatly reduces the complexity of neural network training. In this work, the application of Std-PINNs to solving multi-particle low-temperature plasma fluid models is demonstrated through two classic discharge cases with different complexity of reaction systems (low-pressure argon glow discharge and atmospheric-pressure helium glow discharge) and the performance of Std-PINN is compared with that of conventional PINN and finite element method (FEM). The results show that the training results output from the traditional PINN are completely incorrect due to the strong coupling correlation of each physical variable through the source terms of each particle transport equation, while the <i>L</i><sub>2</sub> relative error between Std-PINN and FEM results can reach up to ~10<sup>–2</sup> , thus verifying the feasibility of Std-PINN in simulating multi-particle plasma fluid model. Std-PINN expands the application of deep learning method to modeling complex physical systems and provides new ideas for conducting low-temperature plasma simulations. In addition, this study provides novel insights into the field of artificial intelligence scientific computing: the mathematical form that describes the state of a physical system is not unique. By introducing equivalent physical variables, equations suitable for neural network solutions can be derived and combined with observable data to simplify problems.