Smoothed Finite Element Method Research Articles

Motor magnetic field analysis using the edge-based smooth finite element method (ES-FEM)

This paper proposes a smooth finite element method (S-FEM) for efficient and accurate analysis of motor magnetic fields. The edge-based smooth finite element (ES-FEM) formulations are derived for two-dimensional triangular element meshes suitable for multi-material motor structures, and then a magnetic flux density calculation method appropriate for S-FEM to calculate nonlinear electromagnetic fields is introduced. In the calculation of motor magnetic field, ES-FEM has a proper softening effect on the stiffness matrix of the finite elements, which makes the calculation more accurate and more efficient than FEM. The magnetic flux intensity calculation method introduced in this paper is not only appropriate for ES-FEM to calculate the nonlinear electromagnetic field of motor, but also can improve the calculation accuracy of magnetic flux density. In the calculation of nonlinear magnetic field of motor, ES-FEM can also improve the calculation accuracy of magnetic field analysis compared with FEM. Several numerical examples demonstrate the improving effect of the ES-FEM discussed above on the motor's magnetic field calculation.

An accurate coupled method for analysis of transcranial magneto-acoustic-electrical stimulation

In this paper, a numerical method for simulating transcranial magneto-acoustic-electrical stimulation (TMAES) is presented. The smoothed finite element method (SFEM) is applied to calculate the magnetic field and the Rayleigh integral (RI) is used for solving the ultrasonic field. Three-node triangular and four-node tetrahedral elements are set as the background cells for constructing the smoothing domains (SDs). A gradient smoothing technique (GST) is applied over each SD to obtain the magnetic flux density. The discretized system equations are obtained by using the generalized smoothed Galerkin weakform. To solve the sound pressure, the ultrasonic transducer is discretized into a series of micro-elements. The acoustic pressure in the problem domain is then equal to the superposition of that generated by the individual micro-element. Finally, the current density can be calculated by the magneto-acoustic coupling effect. Numerical examples demonstrate that SFEM can improve accuracy, convergence rate, and efficiency to 10, 1.8, and 4 times those of FEM, respectively. In the calculation of coupled fields, SFEM-RI also improves the accuracy by 2.5 times. Additionally, the presented method possesses the property of insensitivity to mesh distortion.

A cell-based smoothed finite element model for the analysis of turbulent flow using realizable k-ε model and mixed meshes

Smoothed finite element method (S-FEM) has been widely employed in computational mechanics, particularly in computational solid mechanics (CSM) and, to a lesser extent, in computational fluid dynamics (CFD). The present work focuses on the development of S-FEM for solving three-dimensional incompressible turbulent flow problems using the realizable k-ε model. The Streamline Upwind Petrov-Galerkin approach combined with Stabilized Pressure Gradient Projection (SUPG/SPGP) was used to restrain/control the spatial oscillation and instability in conjunction with mixed meshes employed to discretize complex geometries. The presented turbulent cell-based S-FEM (CS-FEM) was validated under several common flow types, including mesh turbulence, turbulent channel flow, turbulent flow past a circular cylinder, flow separation around an obstacle, flow in the dimpled channel, and flow in the upper human airway. In particular, the presented work exhibited the latent capability of the CS-FEM in solving turbulent flows near boundary regions and strong convection regions. Meanwhile, the results of the CS-FEM were compared to those of the lowest equal-order finite element method (i.e. P1–P1 or Q1–Q1 elements) in simulating turbulent flows, and it was found that the CS-FEM had better agreement with other published works.

Modal Analysis of a High-Speed Train Gearbox Housing Using Smoothed Finite Element Method

The gearbox is a crucial component of the power transmission system in high-speed trains, and realistically calculating its natural frequency is vital in avoiding system resonance. In recent decades, numerical techniques have become essential in designing engineering structures, and the finite element method (FEM) has gained widespread recognition and acceptance for its robustness and effectiveness. However, the FEM technique using linear 4-node tetrahedral elements (T4) performs poorly, leading to low accuracy and unreliable results. Consequently, complex hexahedral or second-order elements are often required for better performance. In this paper, we employ the smoothed finite element method (S-FEM), which combines the gradient smoothing technique with the standard FEM to improve calculation accuracy while still using linear tetrahedral elements. We apply S-FEM to accurately obtain the natural frequencies and mode shapes of a high-speed train gearbox using automatically generated linear tetrahedral elements. A test of tolerance to mesh distortion is also carried out on the gearbox. The results demonstrate that S-FEM can significantly enhance the computational accuracy of low-order 4-node tetrahedral elements without altering the mesh topology. It is expected that S-FEM can be an alternative to FEM in the modal analysis of complex structures in the railroad industry.

Quasi-brittle and Brittle Fracture Simulation Using Phase-field Method based on Cell-based Smoothed Finite Element Method

Quasi-brittle and Brittle Fracture Simulation Using Phase-field Method based on Cell-based Smoothed Finite Element Method

Smoothed finite element methods in simulation of active contraction of myocardial tissue samples

In modelling and simulation of cardiac mechanics, tetrahedral meshes are often used due to the easy availability of efficient meshing algorithms. This is beneficial in particular when complex geometries such as cardiac structures are considered. The gold standard in simulating the cardiac cycle is to solve the mechanical balance equations with the finite element method (FEM). However, using linear shape functions in the FEM in combination with nearly-incompressible material models is known to produce overly stiff approximations, whereas higher order elements are computationally more expensive.To overcome these problems, smoothed finite element methods (S-FEMs) have been proposed by Liu and co-workers. So far, S-FEMs in 3D have been utilised only in simulations of passive mechanics. In the present work, different S-FEMs are for the first time used for simulation of an active cardiac contraction on three-dimensional myocardial tissue samples. Further, node-based S-FEM (NS-FEM), face-based S-FEM (FS-FEM) and selective FS/NS-FEM are for the first time implemented as user subroutine in the commercial software Abaqus.Our results confirm that all S-FEMs perform softer than linear FEM and volumetric locking is reduced. The FS/NS-FEM produces solutions with the relative error in maximum displacement and rotation being less than 5% with respect to the reference solution obtained by the quadratic FEM for all considered mesh sizes, although linear shape functions are used. We therefore conclude that in particular FS/NS-FEM is an efficient and accurate numerical method in the simulation of an active cardiac muscle contraction.

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.

An improved M-SPEM for modeling complex hydroelastic fluid-structure interaction problems

Hydroelastic fluid-structure interaction (FSI) problems usually involve nonlinear free surface flows, large structure deformations and changing interfaces, which present great challenges for numerical modeling. To deal with violent FSIs, the authors proposed a multi-resolution smoothed particle element method (M-SPEM), which adaptively simulates local fluid regions with an improved smoothed particle hydrodynamics (SPH) method and models other solid/fluid regions with a smoothed finite element method (S-FEM). Both computational accuracy in local fluid domain and entire efficiency can be improved by M-SPEM compared with the conventional partitioned FSI solvers based on SPH-FEM coupling schemes. In this work, an improved M-SPEM is developed and extended to simulate the complex hydroelastic FSI problems with structural deformations considered in the numerical model. To solve the accuracy reduction problem in the multi-resolution simulations, we developed a velocity and position correction technique for the generated fluid and virtual particles. The computational results demonstrated that the new correction technique can effectively improve the computational accuracy of M-SPEM while the corrected schemes produce results very close to the reference ones. The positions of the generated particles should also be appropriately optimized during the element-particle conversion process. With both accuracy and efficiency improved, the improved M-SPEM is proven to be suitable to deal with some challenging hydroelastic FSI problems.

Saturated‐unsaturated seepage simulation and hydrological dam models for quantifying the heterogeneity to hydrological state variables: Impacts of auxiliary structures and leakage anomalies

AbstractSeepage analysis is the central issue in solving the unconfined seepage problems from complex earth‐rock dams. The precision and efficiency of seepage simulation are the major technical barriers to exploring the heterogeneities of hydrological state variables in dams. This paper addresses the heterogeneity of the saturated‐unsaturated zones of dams and develops the smoothed finite element method (S‐FEM) capable of solving the seepage problem in domains with arbitrary geometry and continuously varied permeability. The gradient smoothing technique, in which the area integrals are transformed into the line integrals around edges of smoothing cells, is used to obtain the element matrices, which is used to simplify the solution of seepage problem. To be specific, we employ Van Genuchten method to modify the hydraulic conductivity of unsaturated zones medium and Signorini condition to convert the outflow boundary into the water head boundaries. The precision of the proposed method together with the importance of considering the unsaturated zones are first deliberated by the verification tests. Then, to quantify the impacts on the hydrological state variables, we focus on the heterogeneity of the entire zones through exploratory tests. The results show that the presence of leakage anomalies and auxiliary dam structures has a significant effect on the redistribution of hydrological state variables, which convinces that the findings may lay the foundation for the development of multi‐parameter seepage inversion and the derivation of coupled hydro‐geophysical inversion using the most appropriate starting model.

Enhancing Computational Accuracy in Surrogate Modeling for Elastic–Plastic Problems by Coupling S-FEM and Physics-Informed Deep Learning

Physics-informed neural networks (PINNs) provide a new approach to solving partial differential equations (PDEs), while the properties of coupled physical laws present potential in surrogate modeling. However, the accuracy of PINNs in solving forward problems needs to be enhanced, and solving inverse problems relies on data samples. The smoothed finite element method (S-FEM) can obtain high-fidelity numerical solutions, which are easy to solve for the forward problems of PDEs, but difficult to solve for the inverse problems. To the best of the authors’ knowledge, there has been no prior research on coupling S-FEM and PINN. In this paper, a novel approach that couples S-FEM and PINN is proposed. The proposed approach utilizes S-FEM to synthesize high-fidelity datasets required for PINN inversion, while also improving the accuracy of data-independent PINN in solving forward problems. The proposed approach is applied to solve linear elastic and elastoplastic forward and inverse problems. The computational results demonstrate that the coupling of the S-FEM and PINN exhibits high precision and convergence when solving inverse problems, achieving a maximum relative error of 0.2% in linear elasticity and 5.69% in elastoplastic inversion by using approximately 10,000 data points. The coupling approach also enhances the accuracy of solving forward problems, reducing relative errors by approximately 2–10 times. The proposed coupling of the S-FEM and PINN offers a novel surrogate modeling approach that incorporates knowledge and data-driven techniques, enabling it to solve both forward and inverse problems associated with PDEs with high levels of accuracy and convergence.

An SFEM Abaqus UEL for Nonlinear Analysis of Solids

In this paper, three different smoothed finite element method (SFEM), viz., node-based smoothed finite element method (NS-FEM), face-based smoothed finite element method (FS-FEM) and [Formula: see text]-finite element method ([Formula: see text]-FEM) are adopted for 3D solids undergoing large deformation. The common feature of all these techniques is the introduction of smoothed strain which is written as a weighted average of the compatible strain field over smoothing domains. The choice of smoothing domain is what differentiates them. The spatial discretization can be based on the simplest and automatically genera-table four-node tetrahedral elements and aforementioned techniques have shown to yield accurate results even on a coarser discretization. To take the advantages of the SFEM, it is beneficial to the FEM community to have it implemented in the widely used Abaqus[Formula: see text] software. Such an implementation is challenging because the neighboring SFEM elements are interconnected in the smoothed strain matrices in the elemental level. In this work, the above-mentioned SFEM models are implemented in the commercial software Abaqus using the softwares’ user element (UEL) feature. The challenges during the definition and the assembly of the smoothing domains are effectively addressed in this work. The developed UEL and the associated files can be downloaded from https://github.com/nsundar/3DSFEM. The implementation is validated against benchmark examples and the robustness is demonstrated with complicated real-life problems, viz., tire patch contact with road and simulation of human thumb.

A Node-Based Smoothed Finite Element Method with Linear Gradient Fields for Elastic Obstacle Scattering Problems

A Node-Based Smoothed Finite Element Method with Linear Gradient Fields for Elastic Obstacle Scattering Problems

Three-dimensional growth simulation of swellable soft materials based on CS-FEM

In this paper, a three-dimensional numerical framework for modeling growth of swellable soft materials at large deformation is established based on the cell-based smooth finite element method, and the multiplicative decomposition scheme of deformation gradient is given. The second P-K stress and Green's strain tensor are selected as work conjugate pairs, and the corresponding mathematical expressions of stiffness matrix and geometric stiffness matrix are derived. The numerical method is implemented based on Matlab platform, and the isotropic and anisotropic growth behaviors of swellable soft materials are simulated respectively. The results show that anisotropic growth will inhibit the deformation of expandable soft materials compared with isotropic growth. The simulation results are compared with the calculation results in the existing literature. The comparison results show that the characteristics and the morphological mode are in good agreement, which proves the effectiveness of the numerical framework in simulating the growth behavior of expandable soft materials at large deformation, and is able to reveal the mechanical mechanism of the plant growth phenomenon in nature.

Arbitrary polygon mesh for elastic and elastoplastic analysis of solids using smoothed finite element method

This work extends the smoothed finite elements method (S-FEM) to the use of arbitrarily complicated meshes, including nonconvex polygonal meshes for the first time. A novel formulation for arbitrarily polygonal elements based on the S-FEM is presented for analyses of elastic and elastoplastic problems. This enable the use of polygonal elements that are of conventional quadrilateral, convex polygonal, concave polygon elements and n-sides concave polygonal elements. This is achieved by making use of the exceptional unique features of S-FEM in terms of stability, accommodation to mesh distortion, and free from differentiation in strain field computation. The ear clipping technique is used to automatically generate smoothing domains without any dummy nodes. The stability in using such a complicated mesh is ensured by the gradient smoothing technique in S-FEM, and no additional stability control measure is required. In addition, domain integration is converted to boundary integral along sub-triangular domain, which requires only the shape function values at the field nodes. This avoiding coordinate mapping, and hence it is a Jacobian free formulation. The von Mises elastoplastic model is implemented in our S-FEM by decomposing the tangent stiffness matrix into three parts: elastic stiffness matrix, strain–displacement matrix and the derivative of the stress–strain matrix. The first two matrices are calculated only once at the initial stage, and only the derivative of the stress–strain matrix is updated in each iteration for capture the nonlinearity. The numerical results show that the proposed method can handle the general non-convex elements effectively with good accuracy and robustness. It offers an effective tool in dealing with polycrystalline metallic materials with crystal grains of arbitrary shapes.

Preconditioning for finite element methods with strain smoothing

Strain smoothing methods such as the smoothed finite element methods (S-FEMs) and the strain-smoothed element~(SSE) method have successfully improved the performance of finite elements, and there have been numerous applications of them in finite element analysis. For the sake of efficient applications to large-scale problems, it is important to develop a mathematically and numerically well-elaborated iterative solver for the strain smoothing methods. In this paper, inspired by the spectral properties of the strain smoothing methods, we propose efficient ways of preconditioning for the methods. First, we analyze the spectrums of the stiffness matrices of the edge-based S-FEM and the SSE method. Then, we propose an improved two-level additive Schwarz preconditioner for the strain smoothing methods by modifying local solvers appropriately. For the sake of convenience of implementation, an alternative form of the preconditioner is also proposed by defining the coarse-scale operation in terms of the standard FEM. We verify our theoretical results through numerical experiments.

Numerical simulation of repetitive transcranial magnetic stimulation by the smoothed finite element method

This paper presents a series of smoothed finite element methods (SFEM) for repetitive transcranial magnetic stimulation (rTMS) based on the quasi-static electromagnetic equations. The problem domain is first discretized into a set of tetrahedral elements and linear shape functions are used to interpolate the field variables. Then the smoothing domains (SDs) are constructed based on the edges, nodes, and faces of the background mesh. The smoothed gradient of electric potential and smoothed magnetic flux density over each SD are obtained by using the gradient smoothing technique (GST). The generalized smoothed Galerkin weakform is utilized to derive the discretized system equations. Numerical examples, including both spherical and realistic shaped heads, illustrate that the SFEM has the following important characteristics: (1) easier pre-processing; (2) better accuracy; (3) faster convergence; (4) higher computational efficiency.

A Topology Optimization Method Based on the Edge-Based Smoothed Finite Element Method

In this paper, we combined the edge-based smoothed finite element method (ES-FEM) with topology optimization. The edge-based gradient smoothing operation was introduced to overcome the accuracy-loss of the classical finite element method raised by the coarse mesh and “overly stiff” phenomenon. By employing the ES-FEM, design variables can be related to the smoothed edge, thus more design variables can be adaptively obtained without additional remeshing. Two classical topology optimization problems were considered, namely compliance minimization and stress-constrained topology optimization. We presented several numerical examples, among which the compliance minimization examples illustrated the potential of the proposed method, and the advantages of applying such a numerical method in topology optimization were demonstrated through the stress-constrained topology optimization.

N-Side Cell-Based Smoothed Finite Element Method for Incompressible Flow with Heat Transfer Problems

In this paper, an n-sided cell-based smoothed finite element method (nCS-FEM) is presented to solve the convective heat transfer in incompressible viscous fluid flow. The Characteristic-based-split (CBS) scheme is used to stabilize the spatial and pressure oscillations of CS-FEM that occur during the process of solving incompressible Navier-Stokes equations. Meanwhile, the characteristic-Galerkin method in CBS is adopted as the convective stabilization in the heat transfer equation. In order to verify the proposed method, typical numerical examples are used to test the computational accuracy of CS-FEM. Meanwhile, the investigation of the robustness of highly distorted element is conducted in a natural convection test case. The calculation results of numerical examples show that CS-FEM is qualified to solve the steady and unsteady convective heat transfer problems for incompressible laminar flow with high reliability. The flow across tube banks with complex geometry in the heat sinks is also simulated to prove the capabilities of the proposed method. Well agreed comparison with the results of CFD software, Fluent, proves that the present method can effectively simulate realistic and complex flow with heat transfer problems.

A Simple Automatic Hexahedron Mesh Generation and Polyhedral Smoothed Finite Element Method for Mechanics Problems

A new fully automatic pure hexahedron mesh, called lotus mesh, generation technique of an arbitrary 2D/3D geometric domain and polyhedral smoothed finite element method (PSFEM) for efficient numerical simulation are presented. The lotus mesh algorithm is mainly based on the signed distance function (SDF) and the density function to generate nodes and control node density, respectively. We improved the generality of the SDF-based mesh method, realizing the meshing of 2D arbitrary polygons and 3D complex structures, and propose geometry-based adaptive meshing methods. In addition, to improve the stability and accuracy of lotus mesh, PSFEM based on smoothed finite method (SFEM) is proposed. Unlike the existing SFEM, this paper adopts an indirect way to calculate the stiffness matrix, which first calculates the transition matrix on the smoothed domain, and then maps it back to the real stiffness matrix through the transformation matrix, to address stiffness integral operation of the 3D complex polyhedron. Several demonstration examples are presented to verify the lotus mesh and its accuracy. • A fully automatic pure hexahedron mesh, lotus mesh, is proposed, whose algorithm structure is simple and easy to implement. • Mesh generation method based on signed distance function is improved to realize the meshing of 2D arbitrary polygons and 3D complex structures with geometry-based adaptivity. • Polyhedral smoothed finite element method is proposed, which has higher accuracy using lotus mesh than conventional finite element method.

A dynamic analysis of a conical pick during rock-cutting process based on the smooth finite element method

In this paper, the smooth finite element methods (S-FEMs) for accurate and reliable dynamic analysis are proposed to improve the wear resistance of road header picks. Experimentally measured forces and smoothing operations are used to establish a numerical model of conical picks subjected to dynamic forces that changes with time during the rock-cutting process. The S-FEMs are extended for dynamics problems of rock picks, so that tetrahedral elements can be used for model generation. The proposed dynamic analysis procedure has been found with several features, including upper bounds for dynamic problems of the Node-based S-FEM, “ultra-accuracy” in both displacement and energy norms in dynamics solutions of the Edge-based S-FEM. These features are beneficial to practical engineering problems whose exact solutions are difficult to obtain, and hence offer an efficient numerical tool for investigating the effects of actual parameters on the performance of picks subjected to various dynamic forces.