Finite Element Method Mesh Research Articles

A flux-based moving mesh method applied to solving the Poisson–Nernst–Planck equations

The moving mesh method is one of the important adaptive mesh methods which is practically useful when the mesh size and overall resolution are provided and fixed as seen in many engineering computations. We employ a moving mesh technique combined with a finite element method (FEM) to solve a widely-used charged carrier transport model, the Poisson–Nernst–Planck (PNP) equations, the solutions of which often have boundary or internal layers and sharp interfaces when the convection is dominated. Considering that flux is a significant physical quantity in designing stable and accurate numerical algorithm for transport problems such as the PNP system, we start from a relatively general functional and propose a flux-based monitor function and an adaptive step size-controlling algorithm for guiding the mesh movement in the FEM solution of the PNP equations. The numerical results illustrate that the moving mesh method is effectively addressing challenges arising from solution singularities and the convection-dominated effects. It also shows that the moving mesh finite element method we propose exhibits superior performance compared to both the traditional moving mesh finite element method and fixed mesh finite element method in some scenarios. Furthermore, it demonstrates better adherence to the physical properties of energy dissipation inherent in the PNP equation.

An SPIM-FEM adapting coupling approach for the analysis of quasi-brittle media

This paper presents an adaptive coupling approach between meshless Smoothed Point Interpolation Methods (SPIMs) and the Finite Element Method (FEM) for the physically nonlinear analysis of quasi-brittle media. The nonlinear behaviour is represented by scalar damage and smeared-crack models. In the proposed adaptive coupling approach, the domain is initially discretised with a relatively coarse FEM mesh. During the solution process, FEM patches are replaced by meshless discretisations according to certain selection criteria. Refinement is also considered.

A new crack-tip element for the logarithmic stress-singularity of Mode-III cracks in spring interfaces

A new crack-tip finite element able to improve the accuracy of Finite Element Method (FEM) solutions for cracks growing along the Winkler-type spring interfaces between linear elastic adherents is proposed. The spring model for interface fracture, sometimes called Linear-Elastic (perfectly) Brittle Interface Model (LEBIM), can be used, e.g., to analyse fracture of adhesive joints with a thin adhesive layer. Recently an analytical expression for the asymptotic elastic solution with logarithmic stress-singularity at the interface crack tip considering spring-like interface behaviour under fracture Mode III was deduced by some of the authors. Based on this asymptotic solution, a special 5-node triangular crack-tip finite element is developed. The generated special singular shape functions reproduce the radial behaviour of the first main term and shadow terms of the asymptotic solution. This special element implemented in a FEM code written in Matlab has successfully passed various patch tests with spring boundary conditions. The new element allows to model cracks in spring interfaces without the need of using excessively refined FEM meshes, which is one of the current disadvantages in the use of LEBIM when stiff spring interfaces are considered. Numerical tests carried out by h-refinement of uniform meshes show that the new singular element consistently provides significantly more accurate results than the standard finite elements, especially for stiff interfaces, which could be relevant for practical applications minimizing computational costs. The new element can also be used to solve other problems with logarithmic stress-singularities.

Rapid patient-specific FEM meshes from 3D smart-phone based scans

Objective. The objective of this study was to describe and evaluate a smart-phone based method to rapidly generate subject-specific finite element method (FEM) meshes. More accurate FEM meshes should lead to more accurate thoracic electrical impedance tomography (EIT) images. Approach. The method was evaluated on an iPhone® that utilized an app called Heges, to obtain 3D scans (colored, surface triangulations), a custom belt, and custom open-source software developed to produce the subject-specific meshes. The approach was quantitatively validated via mannequin and volunteer tests using an infrared tracker as the gold standard, and qualitatively assessed in a series of tidal-breathing EIT images recorded from 9 subjects. Main results. The subject-specific meshes can be generated in as little as 6.3 min, which requires on average 3.4 min of user interaction. The mannequin tests yielded high levels of precision and accuracy at 3.2 ± 0.4 mm and 4.0 ± 0.3 mm root mean square error (RMSE), respectively. Errors on volunteers were only slightly larger (5.2 ± 2.1 mm RMSE precision and 7.7 ± 2.9 mm RMSE accuracy), illustrating the practical RMSE of the method. Significance. Easy-to-generate, subject-specific meshes could be utilized in the thoracic EIT community, potentially reducing geometric-based artifacts and improving the clinical utility of EIT.

IoT-Driven Experimental Framework for Advancing Electrical Impedance Tomography

This research paper focuses on the current emphasis on the latest industrial revolution, particularly the innovative integration of artificial intelligence and the Internet of Things (IoT). The study explores the seamless integration of Electrical Impedance Tomography (EIT) with IoT, presenting a groundbreaking framework where impedance-based sensing plays a vital role in enhancing the dynamic and adaptable qualities of IoT ecosystems. This contribution facilitates intelligent decision-making and real-time monitoring. The research investigates the application of non-invasive Electrical Impedance Tomography for the rapid identification of minor changes in the electrical impedance of the body or a simulated object. Electrodes positioned at the ends of the phantom’s cylinder measure impedance changes through the application of a high-frequency, low-current signal. Image reconstruction employs both forward and inverse solutions, utilizing a triangular finite element method (FEM) mesh to determine conductivity distribution based on recommended phantom models. The integration of IoT enables data capture, enhancing accessibility through remote monitoring. The novel IoT system proves advantageous for various engineering research applications, providing easily monitored parameters in both commercial and clinical contexts.

Deep learning‐accelerated multiscale approach for granular material modeling

AbstractThe hierarchical finite element method (FEM)–discrete element method (DEM) multiscale approach is a powerful tool for solving geotechnical boundary value problems. However, despite parallel computing can be resorted to, the high computational cost remains an insurmountable barrier to its practical application in engineering‐scale problems. As an alternative, a deep learning model (DLM) is employed to replace the representative volume element (RVE) in the DEM. The complex constitutive responses of sand under various loading paths can be reproduced by leveraging the powerful learning capacity of the DLM. During the numerical computations, the DLM is integrated into the Gauss integration points of the FEM mesh, where it receives strains as input and returns the prediction of stresses to advance the calculation. The applicability and accuracy of the approach were examined with three BVPs, in which the sources of error for both global and local performances were systematically analyzed. The feasibility of the approach in accounting for the inherent anisotropy of sand was also investigated. The results demonstrated that the incorporation of the DLM can achieve a significant computational acceleration of up to two orders of magnitude while maintaining a high degree of accuracy.

Numerical Analysis of the Unsteady Mixed Convection of a Nanofluid in a Concentric Tube Heat Exchanger

In the present work, a numerical investigation of the unsteady mixed convection and entropy generation of a nanofluid in an annular cylindrical space is presented using the Buongiorno’s two-phase flow model. It deals with a concentric tube heat exchanger where the inner cylinder rotates with a constant frequency and is maintained at hot temperature, while the outer cylinder is cold. The aim of the present investigation is to highlight the effects of some parameters on the hydrodynamic, thermal and mass behavior of the considered nanofluid as well as on the system irreversibility, namely: the inertia (1 ⩽ Re ⩽ 20), the buoyancy (0 ⩽ Ri ⩽ 5), the mass diffusion (0.1 ⩽ Le ⩽ 10) and the vertical positions of the inner cylinder (-0.4 ⩽ H ⩽ 0.4). Moreover, at specific parameters, an optimal position in terms of heat transfer has been determined. The flow of the nanofluid is two-dimensional and governed by the equations of continuity, momentum, energy as well as volume fraction conservation. After performing a finite element method mesh test and validation with the literature, the Nusselt number and the entropy generation are discussed. The results show that the heat transfer rate and entropy generation increase with increasing values of Richardson and Reynolds number, especially when positioning the inner cylinder in the lower part. On the other hand, the nanoparticles migration under the thermophoretic diffusion decrease with the increase of the Lewis number, which consequent decrease of the heat transfer rate.

A heterogeneous parallel model of unstructured mesh finite element method based on CPU+GPU

Most of the existing numerical simulation programs using the unstructured mesh finite element are based on the traditional multicore processor architecture. With the increase of the number of computing meshes, the computing time is increasing, which leads to the common multicore CPU cluster can’t meet the high computing demand of complex applications. In order to adapt to the trend of the heterogeneous development of high-performance computers, a heterogeneous parallel model of unstructured mesh finite element method is proposed in this paper. It can transplant the unstructured mesh finite element program framework to heterogeneous platform better and faster. The model realizes the efficient utilization of the multicore CPU by hierarchical parallelization, and realizes the efficient utilization of GPU by heterogeneous parallel rewriting for time-consuming computing hotspot. Finally, the model is applied to the parallel transplantation of CPU + GPU heterogeneous platform for the thermal radiation effect program. The results show that the model can reduce the programming difficulty and has good portability and extensibility.

Numerical calculation and fast method for the magnetohydrodynamic flow and heat transfer of fractional Jeffrey fluid on a two-dimensional irregular convex domain

In this paper, we study the magnetohydrodynamic (MHD) flow and heat transfer of the fractional Jeffrey fluid in a straight channel, of which the cross section is a two-dimensional irregular convex domain. We consider a spatial fractional operator to modify the classical Fourier's law of thermal conduction, and obtain the time-space fractional coupled model. The fractional coupled model is solved numerically by the L2−1σ method in the temporal direction and the unstructured mesh finite element method in the spatial direction. Besides, we prove the stability and convergence of the numerical scheme. In order to reduce the computational time, a fast method is proposed. Finally, a numerical example is given to verify the theoretical analysis and the efficiency of the fast method. Furthermore, an example is considered to discuss the effects of the fractional parameters on the MHD flow and heat transfer of the fractional Jeffrey fluid.

Energy stable finite element strategy for simulating spreading, sliding and rolling flow dynamics of viscoelastic droplets

Moving mesh finite element method (FEM) for simulating dynamics of viscoelastic droplets on inclined surfaces is proposed. Viscoelasticity is incorporated into the governing system employing the Oldroyd–B constitutive model. The supporting (inclined) surface is allowed to have non–homogeneous properties incorporated into the mathematical model through the generalized Navier boundary conditions (GNBC). The droplet motion (sliding and/or rolling) is handled by employing arbitrary Lagrangian Eulerian (ALE) framework. Energy balance is derived from the governing system and it is a starting point in the derivation of the numerical scheme. The overall numerical strategy is designed in such a way that a counterpart of the (continuous) energy balance holds on the discrete level. This ensures that no spurious energy is introduced into the discrete system which, in turn, guarantees the stability of the scheme in the energy norm. The framework proposed is very general and encapsulates both two and three dimensional scenarios. Numerical studies in this work focus primarily on the three dimensional scenarios since such are significantly more challenging and seem to be much scarcer in the literature. The newly proposed numerical strategy is validated on several examples to confirm the theoretical predictions. The role of viscoelasticity in the overall droplet dynamics is briefly investigated and the behaviors of Newtonian and non–Newtonian droplets are compared.

3D Multi-Ion Corrosion Model in Hierarchically Structured Cementitious Materials Obtained from Nano-XCT Data.

Corrosion of steel reinforcements in concrete constructions is a worldwide problem. To assess the degradation of rebars in reinforced concrete, an accurate description of electric current, potential and concentrations of various species present in the concrete matrix is necessary. Although the concrete matrix is a heterogeneous porous material with intricate microstructure, mass transport has been treated in a homogeneous material so far, modifying bulk transport coefficients by additional factors (porosity, constrictivity, tortuosity), which led to so-called effective coefficients (e.g., diffusivity). This study presents an approach where the real 3D microstructure of concrete is obtained from high-resolution X-ray computed tomography (XCT), processed to generate a mesh for finite element method (FEM) computations, and finally combined with a multi-species system of transport and electric potential equations. This methodology allows for a more realistic description of ion movements and reactions in the bulk concrete and on the rebar surface and, consequently, a better evaluation of anodic and cathodic currents, ultimately responsible for the loss of reinforcement mass and its location. The results of this study are compared with a state-of-the-art model and numerical calculations for 2D and 3D geometries.

Prediction of bearing capacities and fracture processes in open-hole plates using a hybrid model of peridynamics and FEM

Holes or cutouts are unavoidable during the construction and service of ocean engineering structures. However, open holes in a plate can cause large stress concentrations that can have a great impact on the structural strength and make the structure more prone to brittle failure. Therefore, a new hybrid model that combines peridynamics (PD) and the finite element method (FEM) was proposed to predict the bearing capacity and fracture process in plates with open holes. The surface effect of PD was removed by directly coupling PD grids with FEM meshes without an overlapping zone, and the improved bond-based PD model was utilized for the regions where failure was expected. Additionally, a new fracture criterion was given to evaluate the damage, and the accuracy of the proposed hybrid model was verified through three examples. Ultimately, by calculating and comparing the bearing capacity and fracture process of brittle plates with different open hole arrangements, it was concluded that the number and contribution of open holes had a substantial effect on the bearing capacity and crack propagation path and that reasonable circular hole arrangements could increase the load capacity by more than 50%. Several reliable open hole arrangements were suggested, and they may provide a reference for the future design of structures with good reliability.

An improved DEM-based mesoscale modeling of bimrocks with high-volume fraction

Block-in-matrix rocks (bimrocks), a typical heterogeneous and discrete geomaterial, are widely found in nature and may cause problems in the design and construction of projects. The generation of mesoscale bimrocks with a high volumetric block proportion (VBP) is still a challenge. This paper proposes a novel mesoscale method that can generate bimrocks models with high VBPs and can generate high-quality mesh using the finite element method (FEM). The Rigid Block discrete element method was then applied to simulate uniaxial compression tests based on the FEM mesh. The agreement between the numerical and experimental results demonstrates that the method is reasonable and effective. In addition, the effect of VBP, strength of the block-matrix interface, and friction coefficient of rock blocks on the uniaxial compressive strength of bimrocks was also investigated.

Simulation toolkit for digital material characterization of large image-based microstructures

In this paper, an efficient image-based simulation toolkit for material characterization is presented, which is scalable to work from personal computers to workstations. The effective thermal conductivity, elasticity, and permeability are evaluated employing a computational homogenization framework based on the Finite Element Method (FEM). Two complementary open-source packages are presented: one developed in Python, which can convert digital images into voxel meshes (pyTomoviewer); the other developed in Julia, that can run numerical simulations to compute effective material properties (chpack). Also, a CUDA C version of chpack is provided (chfem_gpu). They were designed to deal with large multi-phase models, so strategies were devised to minimize their memory footprint, while avoiding a high toll on execution time. The voxel-based approach significantly simplifies the FEM meshes and allows efficient matrix-free implementations. In that sense, to handle large linear systems of equations, the element-by-element (EBE) technique is adopted, in conjunction with a low-memory implementation of the Preconditioned Conjugate Gradient (PCG) method. The code was thoroughly tested on an artificial geometry made of a square array of cylinders, for which analytical solutions exist, as well as on a real micro-tomographic reconstruction of FiberFormTM, a carbon preform commonly used in thermal protection systems.

Brainstorm-DUNEuro: An integrated and user-friendly Finite Element Method for modeling electromagnetic brain activity

Human brain activity generates scalp potentials (electroencephalography – EEG), intracranial potentials (iEEG), and external magnetic fields (magnetoencephalography – MEG). These electrophysiology (e-phys) signals can often be measured simultaneously for research and clinical applications. The forward problem involves modeling these signals at their sensors for a given equivalent current dipole configuration within the brain. While earlier researchers modeled the head as a simple set of isotropic spheres, today's magnetic resonance imaging (MRI) data allow for a detailed anatomic description of brain structures and anisotropic characterization of tissue conductivities. We present a complete pipeline, integrated into the Brainstorm software, that allows users to automatically generate an individual and accurate head model based on the subject's MRI and calculate the electromagnetic forward solution using the finite element method (FEM). The head model generation is performed by integrating the latest tools for MRI segmentation and FEM mesh generation. The final head model comprises the five main compartments: white-matter, gray-matter, CSF, skull, and scalp. The anisotropic brain conductivity model is based on the effective medium approach (EMA), which estimates anisotropic conductivity tensors from diffusion-weighted imaging (DWI) data. The FEM electromagnetic forward solution is obtained through the DUNEuro library, integrated into Brainstorm, and accessible with either a user-friendly graphical interface or scripting. With tutorials and example data sets available in an open-source format on the Brainstorm website, this integrated pipeline provides access to advanced FEM tools for electromagnetic modeling to a broader neuroscience community.

Path Generation Strategy and Wire Arc Additive Manufacturing of Large Aviation Die with Complex Gradient Structure.

To realize automatic wire arc additive manufacturing (WAAM) of a large aviation die with a complex gradient structure, a new contour-parallel path generation strategy was proposed and practically applied. First, the planar curve was defined as a vertical slice of a higher-dimensional surface and a partial differential equation describing boundary evolution was derived to calculate the surface. The improved Finite Element Method (FEM) and Finite Difference Method (FDM) were used to solve this partial differential equation. Second, a cross section of a large aviation die was used to test the path-generation algorithms. The results show that FEM has a faster solving speed than FDM under the same solving accuracy because the solving domain of FEM mesh was greatly reduced and the boundary mesh could be refined. Third, the die was divided into three layers: base layer, transition layer (Fe-based material) and strengthening layer (Co-based material) according to the difference of the temperature and stress field, and corresponding WAAM process parameters has been discussed. The optimum welding parameters are obtained as follows: voltage is 28 V, wire feeding speed is 8000 mm/min and welding speed is 450 mm/min. Finally, the path generation strategy was practically applied to the remanufacture of the large aircraft landing gear die with a three-layer structure. The application test on aircraft landing gear dies justified the effectiveness of the algorithms and strategy proposed in this paper, which significantly improved the efficiency of the WAAM process and the service life of large aviation dies with complex gradient structures. The microstructure of the fusion zone shows that the base metal and welding material can be fully integrated into the welding process.

Phase field simulation of the instability and splitting processes of elliptical inclusions in interconnects due to anisotropic interface diffusion under electric and stress fields

Electromigration and stress migration induced failure of thin-film metal interconnects is one of the most challenging material reliability issues for microelectronic circuits toward ultra-large-scale integrated circuits. Based on the theory of anisotropic interface diffusion, a modified Cahn–Hilliard phase field model is established to elucidate the instability and splitting processes of elliptical inclusions under the multi-physics field. The reliability of the model is verified by comparing numerical and theoretical solutions for the evolution of circular inclusions under electric and stress fields, respectively. The numerical results elaborate on the role of the conductivity ratio, the elastic modulus ratio, the aspect ratio, the electric field, the stress field, the linewidth, and the anisotropic interface diffusion mobility on morphological evolution using an adaptive mesh finite element method. The numerical results show that the larger the electric and stress fields, the greater the aspect ratio larger than 1 or smaller than 1, and the more easily the elliptical inclusions split into several small inclusions or get destabilized. The smaller the linewidth, the easier it is for the inclusions to migrate toward the edge of the line, severely reducing the conductivity of the line. Under anisotropic interface diffusion, lower misorientations favor a steady-state, whereas higher values render the inclusion unstable, splitting, or bifurcating into more small ones. Moreover, the splitting time of the elliptical inclusion decreases with an increase in the electric field, the stress field, and the misorientations, then increases, and subsequently decreases with an increase in the aspect ratio.

Frontal impact on a coach, door sub-system numerical modelling

A coach is a heavy load vehicle used for long distances. It is a passenger transport defined as a M3 Class III vehicle. When developing new coach structures, the study of each sub-systems becomes fundamental in order to pre-validate the structure for improved driver and currier safety. In particular, a coach needs a careful study mainly due to the higher travelling speed. Currently there are few available standards to be followed. R-29 is the obligated regulation for trucks but it can also can be considered for a coach. Thus, when considering a frontal impact, the R-29 regulation must be followed. Since this study will be focused on the passenger door sub-system, it will be assumed that only 3% of the total impact energy from an external mass weighting 1500 kg will be transmitted to the door. It is expected that the coach front pillar will absorb the largest quantity of energy.The Finite Element Method (FEM) can be used to predict crash test results and improve the driver and currier safety. The Pam Crash FEM software, provided by ESI Group was used. This software is extensively used for crash simulation in the automotive industry. A dynamic study using the explicit method was performed. Belytschko-Tsay shell elements with reduced integration were used to construct the FEM mesh. The door is considered to be manufactured with three different materials. The tubular frame uses an S355 J2H steel alloy and the smaller components an EN 10130 DC01 steel alloy. The large frontal panel is manufactured in the EN AW 5754 H111 aluminium alloy.

BFEMP: Interpenetration-free MPM–FEM coupling with barrier contact

This paper introduces BFEMP, a new approach for monolithically coupling the Material Point Method (MPM) with the Finite Element Method (FEM) through barrier energy-based particle–mesh frictional contact using a variational time-stepping formulation. The fully implicit time integration of the coupled system is recast into a barrier-augmented unconstrained nonlinear optimization problem. A modified line-search Newton’s method is adopted to strictly prevent material points from penetrating the FEM domain, ensuring convergence and feasibility regardless of the time step size or the mesh resolutions. The proposed coupling scheme also reduces to a new approach for imposing separable frictional kinematic boundaries for MPM when all nodal displacements in the FEM domain are prescribed with Dirichlet boundary conditions. Compared to standard implicit time integration, the extra algorithmic components associated with the contact treatment only depend on simple point-segment (or point-triangle in 3D) geometric queries which robustly handle arbitrary FEM mesh boundaries represented with codimension-1 simplices. Experiments and analyses are performed to demonstrate the robustness and accuracy of the proposed method.

Fast multi-resolution 3D inversion of potential fields with application to high-resolution gravity and magnetic anomaly data from the Eastern Goldfields in Western Australia

We present a fast inversion algorithm tailored for high-resolution potential field (gravity or magnetic) anomaly maps. The algorithm design objectives are to optimize performance with respect to the size of problem that can be solved and computation speed. It is based on a finite element method (FEM) discretization of both the inversion and forward problems using unstructured tetrahedral meshes with coarsening at the far field. This allows a significant reduction in computational costs from a structured mesh with the same ground level resolution, as well as supporting geometrical features such as topography. Minimization of the cost function uses the Lagrange minimization approach and the resulting system of partial differential equations is solved using the integral preconditioned conjugate gradient method (I-PCG). I-PCG works naturally with an unstructured mesh FEM and can effectively precondition with the Hessian of the regularization term. In contrast to established approaches, minimization of the cost function occurs prior to discretization and using I-PCG avoids the need to invert a large dense sensitivity matrix, a major obstacle to solving large 3D problems using parallel computers. Fast solvers for the forward problem, the adjoint problem and the preconditioner make use of the smoothed aggregation algebraic multi-grid method for the preconditioner in another PCG iteration (AMG-PCG). Parallel implementation uses a domain decomposition approach to support scalability with spatial extent. The proposed method is applied to a synthetic example and tohigh-resolution data set of rastered gravity and magnetic anomaly maps from the Eastern Goldfields in Western Australia with the finest resolution of over 1 million data points and about 60 million cells for the 3D inversion. For this field application we demonstrate that the proposed algorithm achieves computation time scaling with grid size, an indication of optimal algorithm performance and delivers strong scalability with number of cores. The gravity inversion result is compared to the inversion result using the Broyden Fletcher Goldfarb Shanno (BFGS) method on a structured grid.