Conventional Finite Element Method Research Articles

Elasto-plastic partial slip contact modeling of graded layers

A calculation model is developed to solve elasto-plastic partial slip contact problems of graded layer-substrate systems. With the high dependence of contact stress on the local region near the contact spot, the graded layer-substrate system of interest is modeled as a homogeneous half-plane with an embedded, finitely long inclusion. Unlike the conventional finite element method, the elasto-plastic responses are solved within a considerably small region rather than an oversized truncated domain. The computational domain allows arbitrary distributions of elasto-plastic properties and is directly discretized without tedious separation and assembling skills. The resulting disturbance fields are calculated by the numerical equivalent inclusion method. Additionally, the plastic strain is determined via the return-mapping algorithm, and the generated residual responses are directly obtained based on the explicit theoretical solutions. The conjugate gradient method is utilized to solve surface tractions, and the induced elasto-plastic fields are calculated by the technique of fast Fourier transform. Some parametric studies are subsequently conducted based on the developed model to explore the effects of property inhomogeneity and friction coefficient on the elasto-plastic response.

Numerical analysis of a space-time continuous Galerkin method for the nonlinear Sobolev equation

In this work, we shall use space-time continuous Galerkin (STCG) method to analyze the nonlinear Sobolev equation. This method is different from the conventional finite element methods, that is, it not only uses finite element to discrete spatial variables but also uses finite element to discrete temporal variable. Therefore it is easier to obtain high order accuracy in time and space than the conventional finite element methods and has the good stability. Also, it does not need time step size and space mesh parameter to satisfy the stability conditions. In this paper, we show a detailed analysis including the proof of the existence and uniqueness of the numerical solution and the a priori error estimate. Especially, the analysis method given here can be similarly applied to the more general nonlinear problems. At last, the numerical examples reveal that the iterative method is more efficient than the linearized method for the nonlinear problems with STCG method. Also, the numerical experiments show that the STCG method is more efficient than the space-time discontinuous Galerkin (STDG) method in the practical calculations.

An isoparametric quadratic boundary element for coupled stretching-bending analysis of thick laminated composite plates with transverse shear deformation

Recently, we derived the Green's function for an infinite thick laminated composite plate subjected to concentrated forces and moments. By serving this Green's function as the fundamental solutions, in this paper for the first time we develop the associated boundary element method (BEM), which is applicable to general problems of thick laminated plates with various types of deformations. For example, the in-plane stretching and out-of-plane bending deformations may be coupled together, which occurs for the unsymmetric laminated composite plates. Besides, the transverse shear deformations are also incorporated based upon the first order plate theory. To complete the entire development of the proposed BEM, all the involved issues, such as the boundary integral equations, the fundamental solutions and their derivatives, the mathematical formulation for the selected isoparametric quadratic boundary element, and the treatments of singular integrals are elaborated in this paper. The calculations of complete solutions at boundary or internal points are provided as well. The accuracy and efficiency of the proposed BEM are demonstrated through numerical examples, which show that it outperforms the conventional finite element method and is applicable for general thick laminated plates with unrestricted coupling effects.

A new variant of the fem for evaluation the strenght of structures of complex geometry with heterogeneous material structure

Structures are known to feature a complex geometry and a complex material structure which varies depending on their functional purpose, technological capabilities and design. The finite difference method, collocation method, variational method, theoretical and experimental method, finite element method, among other methods, are employed to calculate the stress-strain state (SST) of structures of complex geometry. The finite element method (FEM) has become the most common one of them all. The spline version of the finite element method (SV FEM) has proven to be an effective FEM method used to calculate the SST of structures of complex geometry. Unlike the conventional FEM method, a preliminary parametrization of the entire area of complex geometry using the parameters outlined by the canonical domain and a cubic approximation of the desired variables within each element yields consistent finite elements. The unbroken geometric continuity maintained between the elements along with the continuity of the desired quantities and their first derivatives along the line of contact of the elements makes it possible to obtain reliable results with just a small number of partitions into finite elements. One should note the application of the basics of the spline version of the finite element method in two-dimensional SV FEM-2 and three-dimensional SV FEM-3 arrangements. One should also make mention of the idea of producing a synthesized version of SV FEM. Two-dimensional and three-dimensional elements being synthesized into one element would usher in a new phase in the development of the SV FEM approach which would describe the complex structures of construction materials in addition to complex geometry. SV FEM has made it possible to evaluate the SST of structural elements of complex geometry along with the thin layers that penetrate it in all three directions. The versions of SV FEM that are available now serve to assess the SST of a broad range of various thin-walled and non-thin-walled structures of complex geometry, including multilayer structures with thin layers in all three directions, with hard and soft coatings along the edges and with local defects on the surfaces of structures. This paper presents a number of SST calculations of specific structures that have been carried out via the FEM versions described above.

Broadband low-frequency sound insulation of double-panel metastructures with a perforated lattice truss-core sandwich plate

Lattice truss-core sandwich plates have drawn increasing attention owing to their excellent mechanical properties and multifunctional characteristics, but they are insufficient in low-frequency sound insulation because of their lightweight feature. To improve the sound transmission loss (STL) of such lightweight sandwich plates while without greatly increasing the structural complexity and the total mass, one faceplate of the sandwich plate is perforated so that the sandwich core can be utilized as an acoustic cavity. A homogenous plate is further introduced to construct a double-panel metastructure together with the perforated sandwich plate. For ease of analysis and design of the double-panel metastructure, a semi-analytical method is developed by combining an effective medium theory of the acoustic domain and a dynamic homogenization method of the sandwich plate. The feasibility of the developed method for efficient STL prediction under different incident angles is verified by comparing with the conventional finite element method. Numerical results show that the double-panel metastructure exhibits a remarkable sound insulation performance within a broadband low-frequency range, owing to the fact that the acoustic domain enclosed by the double-panel metastructure has an unusual equivalent bulk modulus. Then, a parametric study is conducted to reveal the sound insulation capacity and provide a guideline for tuning the STL of the double-panel metastructure. Finally, the actual sound insulation capacity of the double-panel metastructure is validated by experimental measurements of large-scale specimens.

FDU-Net: Deep Learning-Based Three-Dimensional Diffuse Optical Image Reconstruction.

Near-infrared diffuse optical tomography (DOT) is a promising functional modality for breast cancer imaging; however, the clinical translation of DOT is hampered by technical limitations. Specifically, conventional finite element method (FEM)-based optical image reconstruction approaches are time-consuming and ineffective in recovering full lesion contrast. To address this, we developed a deep learning-based reconstruction model (FDU-Net) comprised of a Fully connected subnet, followed by a convolutional encoder-Decoder subnet, and a U-Net for fast, end-to-end 3D DOT image reconstruction. The FDU-Net was trained on digital phantoms that include randomly located singular spherical inclusions of various sizes and contrasts. Reconstruction performance was evaluated in 400 simulated cases with realistic noise profiles for the FDU-Net and conventional FEM approaches. Our results show that the overall quality of images reconstructed by FDU-Net is significantly improved compared to FEM-based methods and a previously proposed deep-learning network. Importantly, once trained, FDU-Net demonstrates substantially better capability to recover true inclusion contrast and location without using any inclusion information during reconstruction. The model was also generalizable to multi-focal and irregularly shaped inclusions unseen during training. Finally, FDU-Net, trained on simulated data, could successfully reconstruct a breast tumor from a real patient measurement. Overall, our deep learning-based approach demonstrates marked superiority over the conventional DOT image reconstruction methods while also offering over four orders of magnitude acceleration in computational time. Once adapted to the clinical breast imaging workflow, FDU-Net has the potential to provide real-time accurate lesion characterization by DOT to assist the clinical diagnosis and management of breast cancer.

Towards a generalized surrogate model for truss structure analysis using graph learning

Truss analysis has been well investigated by researchers and engineers with a large number of structural analysis software that can quickly and reliably provide analysis results. However, these methods require either expensive commercial software or self-developed in-house codes based on structural expertise along with advanced programming skills. Thus, incorporating this software into other complex applications, such as an online structural analysis framework, multiple-objectives truss optimization, structural reliability, etc., is highly challenging. This study proposes a novel and efficient surrogate model for performing truss analysis based on graph theory and deep learning algorithms, which is applicable for various truss topologies with different load scenarios without requiring retraining as other Deep Learning (DL)-based counterparts. The truss connectivityinformation is expressed through adjacency matrices, while material properties, external loads, and boundary conditions are considered node features. The performance and efficiency of the proposed methods are successfully validated with numerous unseen 2D trusses, providing highly similar results compared to the conventional finite element method.

A novel hybrid flux finite element model to steady heat conduction in porous materials

A new hybrid finite-element model was developed for heat flux and temperature analyses in porous material. The model in this paper was based on an assumed heat flux hybrid finite-element method, which is a two-field functional. An independent heat flux field that meets a priori inner thermal equilibrium was assumed inside the element, and a temperature field was assumed on the cell boundary. Temperature was used as the Lagrange multiplier to introduce the continuous condition of heat flux at the hole edge into the functional, and a modified complementary energy functional was obtained. The element flux field that was built up of polynomial terms and reciprocal terms from the stream function and that can consider the shape of pores in location coordinates was raised for the first time. Several numerical examples are discussed and compared with conventional finite-element methods to validate the accuracy and efficiency of the proposed approach.

Smooth particle hydrodynamics: a meshless approach for structural mechanics

This article provides an overview of the smooth particle hydrodynamics (SPH) approach and its mathematical modeling. SPH is a numerical technique based on a mesh-free Lagrangian scheme for evaluating the continuum mechanics problems. This method is suitable in the case of continuum objects undergoing large deformation, as conventional finite element methods are unreliable due to mesh failure and convergence issues. It is a widely used approach in the field of astrophysics, fluid mechanics, structural mechanics, soil mechanics, automobiles, and so on. A numerical example is also considered in this research paper to demonstrate the applicability of the method. The simulation process was achieved using LS-Dyna explicit solver software, and plots related to cutting and thrust forces, von Mises stress, plastic strain, temperature distribution, and so on, were obtained. Also, the effect of Time-Scaling Factor (TSSFAC) on SPH simulations was observed in this research.

The M3 pressure-differential wave energy converter in Persian shores; numerical modeling and design

Following a double objective, this paper develops a numerical model for the M3 fixed-bottom pressure-differential wave energy converter (WEC) and then employs the model in the design of the WEC with available wave data for three Persian shores. To reach the first objective, using the potential theory, a simple numerical method, based on the conventional finite element method (FEM), has been developed for calculating the added mass and damping coefficients of the WEC. This paper shows that a complex engineering problem can be solved easily by making adjustments and making simple assumptions. These assumptions, though some of which may not be applicable to other types of WECs, are shown to be very effective in the modeling of the WEC considered in this paper. By comparing the results of the proposed method with those previously reported in the literature, the obtained results are validated. With a simple and practical numerical tool in hand, on the sequel and as the second objective, the energy absorption of the WEC is investigated based on the wave data in the Persian Gulf and the Gulf of Oman. For this purpose, several cases are considered for the dimensions of the WEC. Other effective parameters on the energy absorption of the WEC, i.e. the stiffness and damping coefficient of the power take-off (PTO) system, are optimized for each dimensional case to achieve the maximum absorbed power. A comparison of the absorbed power in different dimensional cases suggests appropriate WEC dimensions and the corresponding stiffness and damping coefficient of the PTO system for different application locations.

Surrogate-Based Physics-Informed Neural Networks for Elliptic Partial Differential Equations

The purpose of this study is to investigate the role that a deep learning approach could play in computational mechanics. In this paper, a convolutional neural network technique based on modified loss function is proposed as a surrogate of the finite element method (FEM). Several surrogate-based physics-informed neural networks (PINNs) are developed to solve representative boundary value problems based on elliptic partial differential equations (PDEs). According to the authors’ knowledge, the proposed method has been applied for the first time to solve boundary value problems with elliptic partial differential equations as the governing equations. The results of the proposed surrogate-based approach are in good agreement with those of the conventional FEM. It is found that modification of the loss function could improve the prediction accuracy of the neural network. It is demonstrated that to some extent, the deep learning approach could replace the conventional numerical method as a significant surrogate model.

The Method of Optical Paths for the Numerical Solution of Integrated Photonics Problems

A number of topical problems of integrated photonics are reduced to oblique incidence of radiation on a plane-parallel scatterer. For such problems, a method for integrating Maxwell’s equations along the direction of beam propagation is proposed. As a result, the original two-dimensional problem is reduced to a one-dimensional problem, and it is solved using recently proposed one-dimensional bicompact schemes. This significantly reduces the computational cost compared with the conventional two-dimensional finite difference and finite element methods. The proposed method is verified by solving test problems for which exact solutions are known.

Modeling pipe-soil interaction under vertical downward relative offset using B-spline material point method

To analyze the pipeline response under permanent ground deformation, the evolution of resistance acting on the pipe during the vertical downward offset is an essential ingredient. However, the efficient simulation of pipe penetration into soil is challenging for the conventional finite element (FE) method due to the large deformation of the surrounding soils. In this study, the B-spline material point method (MPM) is employed to investigate the pipe-soil interaction during the downward movement of rigid pipes buried in medium and dense sand. To describe the density- and stress-dependent behaviors of sand, the J2-deformation type model with state-dependent dilatancy is adopted. The effectiveness of the model is demonstrated by element tests and biaxial compression tests. Afterwards, the pipe penetration process is simulated, and the numerical outcomes are compared with the physical model tests. The effects of pipe size and burial depth are investigated with an emphasis on the mobilization of the soil resistance and the failure mechanisms. The simulation results indicate that the bearing capacity formulas given in the guidelines can provide essentially reasonable estimates for the ultimate force acting on buried pipes, and the recommended value of yield displacement may be underestimated to a certain extent.

Transfer Learning-Based Coupling of Smoothed Finite Element Method and Physics-Informed Neural Network for Solving Elastoplastic Inverse Problems

In practical engineering applications, there is a high demand for inverting parameters for various materials, and obtaining monitoring data can be costly. Traditional inverse methods often involve tedious computational processes, require significant computational effort, and exhibit slow convergence speeds. The recently proposed Physics-Informed Neural Network (PINN) has shown great potential in solving inverse problems. Therefore, in this paper, we propose a transfer learning-based coupling of the Smoothed Finite Element Method (S-FEM) and PINN methods for the inversion of parameters in elastic-plasticity problems. The aim is to improve the accuracy and efficiency of parameter inversion for different elastic-plastic materials with limited data. High-quality small datasets were synthesized using S-FEM and subsequently combined with PINN for pre-training purposes. The parameters of the pre-trained model were saved and used as the initial state for the PINN model in the inversion of new material parameters. The inversion performance of the coupling of S-FEM and PINN is compared with the coupling of the conventional Finite Element Method (FEM) and PINN on a small data set. Additionally, we compared the efficiency and accuracy of both the transfer learning-based and non-transfer learning-based methods of the coupling of S-FEM and PINN in the inversion of different material parameters. The results show that: (1) our method performs well on small datasets, with an inversion error of essentially less than 2%; (2) our approach outperforms the coupling of conventional FEM and PINN in terms of both computational accuracy and computational efficiency; and (3) our approach is at least twice as efficient as the coupling of S-FEM and PINN without transfer learning, while still maintaining accuracy. Our method is well-suited for the inversion of different material parameters using only small datasets. The use of transfer learning greatly improves computational efficiency, making our method an efficient and accurate solution for reducing computational cost and complexity in practical engineering applications.

A new family of high‐order triangular weight functions for mass lumping in vibration analysis

AbstractIn the conventional finite element method, only a few types of elements can directly use the weight functions to generate a proper, diagonally lumped mass matrix (LMM), which satisfies mass conservation and positive definiteness. This study proposes a new set of weight functions that can be efficiently constructed and used for generating a proper LMM based on classical 6‐, 9‐, and 10‐node triangular elements. Furthermore, we prove that these new elements are all complete and consistent, which ensures the convergence of the numerical solution. Finally, some modal and forced/free vibration analyses are conducted to demonstrate the computational accuracy and validation of the proposed LMM.

Enriched physics-informed neural networks for 2D in-plane crack analysis: Theory and MATLAB code

In this paper, a method based on the physics-informed neural networks (PINNs) is presented to analyze 2D in-plane crack problems in the linear elastic fracture mechanics. Instead of using a discretizing mesh, the PINNs-based method is meshless and can be trained on batches of discretely sampled collocation points. In order to capture the asymptotic behavior of the near-tip displacement and stress fields, the standard PINNs formulation is enriched here by including the crack-tip asymptotic functions such that the local behavior in the crack-tip region can be modeled accurately without a nodal refinement. The trainable parameters of the enriched PINNs are learned to satisfy the governing partial differential equations of the cracked elastic solid and the corresponding boundary conditions. It is found that the incorporation of the crack-tip enrichment functions in the PINNs is substantially simpler and more feasible than in the conventional finite element method (FEM) or boundary element method (BEM). The present PINNs-based algorithm is tested and verified by several representative benchmark examples for different loading and crack-mode types. Numerical results show that the present PINNs-based method can provide highly accurate stress intensity factors (SIFs) with only few degrees of freedom. For interested readers, a self-contained MATLAB code and data-sets supplementing this paper are also provided in the Supplementary material.

Dynamic Response Analysis of High-Speed Maglev-Guideway System

PurposeMaglev train travel is an efficient, modern and unconventional mode of transportation of passengers that has many advantages over the conventional railway transportation. In practice, maglev trains are primarily used for city transportation and connection with the airports. They often travel on elevated guideway bridges. This paper is concerned with developing a computationally efficient and accurate numerical method for the dynamic response of a maglev train traversing an “infinitely” long multi-span guideway bridge.MethodsThis study is based on numerical analysis in the time domain. Each guideway span is modelled as a simply supported beam with rotational springs connecting to the adjacent spans. The maglev vehicle is modelled by employing the multi-body system. The vehicle and the guideway are coupled via the electromagnetic force. In the numerical analysis, a computational scheme in conjunction with the MEM is proposed for the global time-domain simulations.ResultsThe accuracy of the proposed computational model is validated by comparison with available data for a maglev test line in the literature. Thereafter, parametric studies are conducted to examine the effects of train speed, stiffness of the suspension system and the coupling connection between adjacent guideway beams, and guideway irregularity on the dynamic response of the train. Results show that a stiffer guideway coupling connection helps to reduce the vertical acceleration of the car body and the vertical displacement of the guideway for the parameters considered in the study.ConclusionsThe computational model presented in this study in conjunction with the moving element method has the advantage of computationally efficient analysis and accurate prediction of the dynamic responses of a maglev train traversing an “infinitely” long multi-span guideway bridge over the conventional finite-element method, especially when the train is travelling at high speeds. The method can be extended to further consider more complex and practical cases.

Extended Finite Element Method for Calculation of Eddy Currents at High Frequencies

The extended finite element method is used for calculating eddy currents at high frequencies, which reduces the number of unknowns in conductive materials. In this study, 2D and 3D models are analyzed numerically, and the solutions are compared with the solution obtained using the conventional finite element method. The proposed method is shown to give the accurate eddy current distribution and Joule losses for a wide range of frequencies even when the skin depth is smaller than the size of elements in a mesh.

Direct FE2 analysis of additively manufactured parts with voids

This study challenges the Direct FE2 method to model voids in additively manufactured parts. Two sets of polylactic acid (PLA) tensile coupons—without any voids and with 23% voids—were manufactured using fused deposition modelling (FDM) and assessed under monotonic loading. An in-house Python script was developed to implement the modelling steps of Direct FE2 in ABAQUS commercial package. A homogenised medium was used in conventional finite element (FE) method whereas discrete modelling of voids was adopted in Direct FE2. Despite using the same isotropic elastoplastic material model with progressive damage, the response of Direct FE2 was noticeably softer. Although stress concentration around the voids were illustrated in Direct FE2, high error values demand further refinements in selecting the representative volume. The applications of this approach can be extended to modelling defects in other complex parts such as auxetic/sandwich structures and composites.

Thermoelectric Joule heating in crack problems in nano-sized structures

Crack problems in nano-sized elastic structures under the Joule heating are analyzed by the finite element method (FEM). The variational principle is applied for derivation of coupled electrothermomechanical governing equations. The temperature field is described by classical Fourier′s law and it is not influenced by mechanical field. Transient heat conduction is considered since it is the slowest process among all other coupled fields here. Quasi-static conditions are considered for mechanical and electrical fields. Due to higher derivatives in governing equations the conventional FEM cannot be applied. Therefore, the mixed FEM with the C0 continuous interpolation for displacements and strains is developed. The influence of the material characteristic length on variation of crack opening displacements in finite nano-sized structures under an electric current load is investigated.