HighlightsAn adaptive bidirectional ES-FEM-SPH coupling algorithm with element-particle transformation is proposed for thermo-fluid-solid simulation in laser powder bed fusion.The algorithm includes a nodal mass normalization scheme, a ghost particle coupling algorithm, and a bidirectional transformation algorithm between finite elements and particles.Bidirectional conversion between mesh-free Lagrangian SPH and Lagrangian FEM is realized for the first time with mass conservation.The method enables fully coupled simulation of the temperature field, melt flow dynamics, and thermal stress evolution throughout the LPBF process.Laser powder bed fusion (LPBF) poses significant simulation challenges due to its highly nonlinear thermo-fluid-solid coupling. To address this, we propose an adaptive framework coupling the edge-based smoothed finite element method (ES-FEM) and smoothed particle hydrodynamics (SPH) via a bidirectional element-particle transformation algorithm. This integration leverages ES-FEM for modeling solid thermo-mechanical responses and SPH for resolving melt pool dynamics, enabling fully coupled simulation of temperature, fluid flow, and stress within a unified model. The framework comprises three key components: a nodal mass normalization scheme ensuring conservation during transformations, a ghost particle algorithm for solid-fluid heat transfer and interaction, and a bidirectional finite-element-to-particle conversion mechanism. This work represents the first implementation of bidirectional coupling between mesh-free Lagrangian SPH and Lagrangian FEM. The validation against benchmark cases confirms the framework’s accuracy in capturing transient thermal, hydrodynamic, and mechanical behavior. It successfully reproduces key LPBF phenomena, including melt pool morphology, Marangoni flows, and residual stress evolution, demonstrating its suitability for high-fidelity LPBF process simulation. It should be noted that the current ES-FEM-SPH framework has not taken into account the recoil pressure, evaporation, and the interaction between the powder and the molten pool. The powder is regarded as a rigid body. Future work will focus on incorporating these neglected physical factors to further improve the predictive capability of the proposed framework.

Numerical algorithm of concave polygon elements in solid mechanics: Theoretical analysis and performance evaluation of S-FEM and VEM

Abstract A selective cell-based smoothed finite element method (CSFEM) is incorporated into a partitioned semi-implicit coupling scheme for fluid-structure interaction (FSI) based on four-node quadrilateral element. The selective CSFEM invokes full and reduced integrations at different steps of the characteristic-based split (CBS) scheme that solves the Navier-Stokes equations in a fractional-step manner. Because the semi-implicit coupling framework is also underpinned by the CBS scheme, an hourglass control is adopted to stabilize both the partitioned algorithm and selective CSFEM over under-integrated smoothing cells. The elastodynamics equations are computed via the conventional CSFEM. The developed technique is validated against available data for different FSI examples. In addition to satisfactory results, it can save more computer time than the original semi-implicit coupling algorithm using the standard CSFEM.

This study presents a new framework that combines Bayesian inference, Artificial Neural Networks (ANNs), and numerical and experimental modal analysis to determine material properties in structural plates and detect flaws. The Smoothed Finite Element Method (SFEM) is used as the forward solver along with experimental modal analysis to accurately characterize the plate’s material properties. The material properties derived through Bayesian inference are used to create a natural frequency dataset for predicting flaws within the plate. This dataset is used to train ANNs, which are the primary tool for solving the inverse problem. Two different ANN models are developed to predict a complex star-shaped flaw and multiple circular flaws with an accuracy exceeding 96%. This comprehensive framework not only improves flaw detection capabilities but also demonstrates the effectiveness of SFEM in Bayesian inference for material parameter estimation and flaw data generation. The proposed methodology shows significant promise for advancing structural health monitoring and maintenance practices.

Purpose This research aims to apply the smoothed finite element method (S-FEM) to perform the static strength analysis of a metro train bogie frame and to investigate its computational accuracy when compared to the traditional FEM. Design/methodology/approach The S-FEM, known for enhancing numerical simulation accuracy using linear tetrahedral elements, is applied to analyze the complexity of the bogie frame. A three-dimensional structure model of a metro bogie frame is constructed, and various loading conditions are simulated to assess its strength. In this study, we adopt the edge-based smoothed finite element method (ES-FEM) and the face-based smoothed finite element method (FS-FEM) and validate them using relevant standards. Stress and deformation distributions of the bogie frame are analyzed to ensure compliance with strength requirements. Findings Comparative analyses with the conventional FEM demonstrate that the S-FEM yields superior accuracy and convergence results in predicting the static strength of the bogie frame. Originality/value This research provides an in-depth analysis of the strength of a complex structure like the bogie frame using S-FEM specifically the ES-FEM and FS-FEM. The S-FEM serves as an effective and accurate approach for static strength analysis of mechanical structures and their practical applications in engineering design and analysis.

Cut-cell Cartesian meshes for incompressible laminar and turbulent flows based on n-sided Cell-based Smoothed FEM

This study investigates the static bending behavior of Functionally Graded (FG) sandwich spherical shells using a triangular shell element that integrates the Edge-based Smoothed Finite Element Method (ES-FEM) with the Mixed Interpolation of Tensorial Components technique (MITCi), referred to as the ES-MITC3i approach. In this approach, the stiffness matrix is formulated based on a strain smoothing technique, where the smoothing domains are established by grouping neighboring MITC3i triangular elements that share common edges. The strain smoothing process enhances both the accuracy and convergence rate compared to the conventional MITC3i formulation. The material characteristics of the FG shells are assumed to follow a power-law distribution of constituent volume fractions through the shell thickness. Numerical simulations confirm that the ES-MITC3i approach effectively avoids shear locking and delivers highly accurate results when compared with benchmark solutions reported in existing literature.

A Fully Edge-Based Smoothed Finite Element Method for Free Vibration Analysis of Functionally Graded Plates

This paper delves into the application of the alpha finite element method ([Formula: see text]FEM) in the simulation of electromagnetic fields within electric machines, with a particular focus on permanent magnet synchronous motors. Electromagnetic simulation analysis based on the finite element method (FEM) has become a critical tool for the design and optimization of electric machines. Traditional FEM, which employ linear triangular elements, often encounter challenges regarding computational accuracy and efficiency in complex electromagnetic scenarios. [Formula: see text]FEM, a variant within the smooth finite element method (S-FEM) family, enhances the capabilities of conventional FEM by scaling physical coordinates or strain gradients in the Jacobian matrix with an adjustable factor “[Formula: see text].” This modification yields more precise numerical solutions, rendering [Formula: see text]FEM particularly effective in dynamic problems such as vibration, acoustics, and plate analysis. Widely acknowledged for its stability and convergence properties, [Formula: see text]FEM is utilized in this study to analyze computational performance in simulating the electromagnetic fields of motor. Numerical examples presented confirm that [Formula: see text]FEM achieves higher computational accuracy with the same mesh model compared to conventional FEM.

As a critical component of a roadheader, the performance of the pick directly affects operational efficiency in mining and tunneling. Thermo-elastic coupling is found to influence the wear characteristics of conical picks. In this paper, an effective thermo-elastic coupling model was proposed for numerical thermomechanical analysis in conical picks. Based on a simple linear tetrahedral mesh, three S-FEM models including ES-FEM, NS-FEM and FS-FEM were implemented using different smoothing strategies with gradient smoothing operations on the temperature gradient and displacement gradient to obtain accurate solutions. Detailed verification studies were conducted using the standard finite element method (FEM) results, with dense mesh as reference solutions. The results show that ES-FEM significantly reduces the computational errors of temperature, displacement and stress under the same meshing conditions, which errors are only 17%, 16% and 34% of FEM, respectively. In addition, the numerical displacement results were used to predict the wear of picks made of three materials under given conditions. It is found that the minimum error in the ES-FEM prediction is only one–seventh of that found in FEM. These findings underscore ES-FEM as a robust numerical simulation tool for predicting conical pick wear under actual conditions. Finally, comparisons of thermo-elastic coupling, static, and dynamic analyses were also conducted. The results indicate that temperature significantly influences pick wear than mechanical forces.

This paper presents a statistical framework for identifying circular flaws in structures using natural frequency data and Bayesian inference, explicitly addressing uncertainties arising from modeling errors and measurement noise. In this approach, the circular flaw is characterized by parameters such as the center coordinates and radius. The natural frequencies of the structure, measured under known boundary conditions, serve as the input data for the identification process. The smoothed finite element method (SFEM) forward model predicts the natural frequency shifts due to the presence of flaws and is integrated into the analysis. By combining observed frequency data with prior knowledge, Bayes’ theorem is employed to refine the probability distributions of the flaw parameters. The Markov chain Monte Carlo (MCMC) algorithm is utilized to sample from the posterior distributions of the parameters, ensuring robust uncertainty quantification. A numerical case study validates the proposed method, highlighting its accuracy and effectiveness in detecting and characterizing circular flaws.

A hybrid method of coupling phase field model and linear elastic model to simulate fracture using cell-based smooth finite element method and finite element method

Abstract A modified inner-element edge-based smoothed finite element method (IES-FEM) is developed and integrated with ABAQUS using a user-defined element (UEL) in this study. Initially, the smoothing domain discretization of IES-FEM is described and compared with ES-FEM. A practical modification of IES-FEM is then introduced that used the technique employed by ES-FEM for the nodal strain calculation. The differences in the strain computation among ES-FEM, IES-FEM, and FEM are then discussed. The modified IES-FEM exhibited superior performance in displacement and a slight advantage in stress compared to FEM using the same mesh according to the results obtained from both the regular and irregular elements. The robustness of the IES-FEM to severely deformed meshes was also verified.

An Adaptive Cell-Based Smoothed Finite Element Method with Arbitrary Polygonal Elements for Coupled Thermo-Mechanical Analysis

An Adaptive Cell-Based Smoothed Finite Element Method with Arbitrary Polygonal Elements for Coupled Thermo-Mechanical Analysis

An Adaptive Cell-Based Smoothed Finite Element Method with Arbitrary Polygonal Elements for Coupled Thermo-Mechanical Analysis

ABSTRACTSmoothed Finite Element Method (S‐FEM) has been widely used in many engineering simulations. However, there are still a lot of theoretical problems to be solved, especially with regard to composite materials and convergence proof. Based on a novel least‐squares approximation and conservation property, we extend S‐FEM to heterogeneous materials. Firstly, the orthogonality, Softening Effect, and energy function are checked. Secondly, the interpolation error in the maximum norm is estimated in any dimension. Consequently, we get a theoretical convergence rate which has been sought since the year 2010. When restricted to one‐dimensional problems, we construct a special test function to prove the superconvergence: (1) S‐FEM flux is exact in the meaning of element‐wise integral; (2) numerical flux is exact at some point in each element; (3) physical flux can be quadratically approximated at the center of each element. At last, we present two numerical experiments: (1) conventional S‐FEM fails in high‐contrast composite materials while our new scheme performs well; (2) our flux converges quadratically.

A theoretical proof of superiority of Smoothed Finite Element Method over the conventional FEM

The object-oriented programming is widely used in different fields of finite element analysis (FEA). The commonly used object-oriented codes were almost written by C++. Due to the inability to automatically manage memory and the lack of gigantic standard library and a third-party library for C++, building matrix classes are needed and a strong programming ability for program developers is also necessary. The conventional FEM method still has many disadvantages such as poor accuracy, hard stiffness matrix, mesh dependence and lack of upper bound solution. The smoothed FEM (S-FEM) method proposed by Liu et al. can efficiently overcome these disadvantages and is therefore widely used in FEA. In this work, the elastic-plastic CS-FEM method is extended and implemented using Python programming based on the framework proposed [H. Li and Y. Xiao, Implementation and numerical simulation on object-oriented elastic-plastic finite element method based on Python, J. Syst. Simul. 36(5) (2024) 1107–1117] The resulting object-oriented elastic-plastic CS-FEM framework is then applied to FEA of elastic-plastic mechanics problems. The correctness of the resulting framework and the high computational accuracy and good mesh adaptability of CS-FEM models have been verified by comparing the corresponding results with an analytical solution or ABAQUS results. The object-oriented elastic-plastic CS-FEM framework provides a more efficient and adaptable approach to FEA, and can also overcome many limitations of the conventional FEM method and C++ programming. The use of Python and the CS-FEM method offers improved computational accuracy, mesh adaptability and development efficiency, and this will make it a valuable tool for solving practical problems in mechanics and engineering.

Assessment of RANS turbulence models based on the cell-based smoothed finite element model for prediction of turbulent flow