Mesh Scheme Research Articles

An edge‐based smoothed finite element framework for partitioned simulation of vortex‐induced vibration problems

AbstractThis article proposes an edge‐based smoothed finite element method (ESFEM) for predicting vortex‐induced vibration (VIV) of multiple rigid and elastic structures. The ESFEM is applied to discretize the Navier–Stokes and elastodynamic equations with three‐node triangular (T3) element. The fluid excitation is also evaluated in the edge‐based notion. New integration points are arranged in local smoothing domains to ease the resultant approximation. Dynamic grids are moved by an efficient two‐level mesh scheme. The fluid‐structure mechanical system is formulated under the arbitrary Lagrangian–Eulerian description which enables tight coupling of interacting fields in a partitioned way. Especially, the geometric conservation law is respected for the ESFEM through a mass source term constructed in the T3 element context. The developed technique is validated against previously published data for three low‐Reynolds‐number VIV problems. Flow features and structural responses have been correctly identified therein as a result of the numerical prediction.

Accurate modeling and simulation of seepage in 3D heterogeneous fractured porous media with complex structures

The past decades have witnessed an increasing interest in numerical simulation for flow in fractured porous media. To date, most studies have focused on 2D or pseudo-3D computational models, where the impact of 3D complex structures on seepage has not been fully addressed. This work presents a method for modeling seepage in 3D heterogeneous porous media. The complex structures, typically the stochastic discrete fractures and inclusions, are able to be simulated. A mesh strategy is proposed to discretize the complex domain. In particular, a treatment on the intersected elements is developed to ensure a conforming mesh. Then, numerical discretization is provided, in which the flux interactions of fractures, inclusions and surrounding rock matrix are included. Numerical tests are performed to analyze the hydraulic characteristics of 3D fractured media. First, the developed framework is validated by comparing numerical solutions with the results of embedded discrete fracture model. Next, the effects of orientation, aperture and radius of fractures on fluid flow and equivalent permeability tensor are analyzed. The variations of pressure distribution are studied in heterogeneous and homogeneous media. Finally, the hydraulic properties of a medium with complex structures are investigated to show the difference of hydraulic feature between fractures and inclusions.

Artificial Neural Networking (ANN) Model for Drag Coefficient Optimization for Various Obstacles

For various obstacles in the path of a flowing liquid stream, an artificial neural networking (ANN) model is constructed to study the hydrodynamic force depending on the object. The multilayer perceptron (MLP), back propagation (BP), and feed-forward (FF) network models were employed to create the ANN model, which has a high prediction accuracy and a strong structure. To be more specific, circular-, octagon-, hexagon-, square-, and triangular-shaped cylinders are installed in a rectangular channel. The fluid is flowing from the left wall of the channel by following two velocity profiles explicitly linear velocity and parabolic velocity. The no-slip condition is maintained on the channel upper and bottom walls. The Neumann condition is applied to the outlet. The entire physical design is mathematically regulated using flow equations. The result is presented using the finite element approach, with the LBB-stable finite element pair and a hybrid meshing scheme. The drag coefficient values are calculated by doing line integration around installed obstructions for both linear and parabolic profiles. The values of the drag coefficient are predicted with high accuracy by developing an ANN model toward various obstacles.

Analysis of Two-Level Mesh Method for Partial Integro-Differential Equation

In this paper, we present two-level mesh scheme to solve partial integro-differential equation. The proposed method is based on a finite difference method. For the first step, we use finite difference method in time and global radial basis function (RBF) finite difference (FD) in space. For the second step, we use the finite difference method to solve the proposed problem. This two-level mesh scheme is obtained by combining the radial basis function with finite difference. We prove the stability and convergence of scheme and show that the convergence order is O τ 2 + h 2 , where τ and h are the time step size and space step size, respectively. The results of numerical examples are compared with analytical solutions to show the efficiency of proposed scheme. The numerical results are in good agreement with theoretical ones.

Spatially Driven Chemical Species Tomography With Size-Adaptive Hybrid Meshing Scheme

This paper develops a size-adaptive hybrid meshing scheme for Chemical Species Tomography (CST) that is driven by the customized spatial resolution of the sensing region. Traditionally, the entire sensing region in CST is uniformly discretized with the empirically determined density of the meshes. Such a discretization results in a) waste of computational efforts on the less spatially resolved location; and b) much severer rank deficiency. To solve the above-mentioned issues, we introduce, for the first time, a size-adaptive hybrid meshing scheme for CST. Driven by the spatial resolution, dense meshes are deployed in the region of interest (RoI) to detail the target flow field while sparse ones are deployed out of the RoI to fully consider the physically existing laser absorption. The proposed scheme is numerically validated using a CST sensor with 128 laser beams. The visual and quantitative metric comparisons show that the proposed hybrid-size meshing scheme outperforms the traditionally uniform-size meshing scheme, giving 35% lower image error and 38% less significant dislocation at a typical 35 dB signal-to-noise ratio in the RoI. The proposed hybrid-size meshing scheme significantly facilitates the reconstruction of the industrial combustion processes where the combustion zone is bypassed by cooling flows. In these scenarios, the proposed scheme can adapt a finer resolution to detail the combustion zone, while maintaining the integrity of the physical model by less resolved reconstruction of the bypass flows.

Two-level method part-scale thermal analysis of laser powder bed fusion additive manufacturing

Numerical simulations of a complete laser powder bed fusion (LPBF) additive manufacturing (AM) process are extremely challenging, or even impossible, to achieve without a radical model reduction of the complex physical phenomena occurring during the process. However, even when we adopt a reduced model with simplified physics, the complex geometries of parts usually produced by the LPBF AM processes make this kind of analysis computationally expensive. In fact, small geometrical features—which might be generated when the part is designed following the principle of the so-called design for AM, for instance, by means of topology optimization procedures—often require complex conformal meshes. Immersed boundary methods offer an alternative to deal with this kind of complexity, without requiring complicated meshing strategies. The two-level method lies within this family of numerical methods and presents a flexible tool to deal with multi-scale problems. In this contribution, we apply a modified version of the recently introduced two-level method to part-scale thermal analysis of LPBF manufactured components. We first validate the proposed part-scale model with respect to experimental measurements from the literature. Then, we apply the presented numerical framework to simulate a complete LPBF process of a topologically optimized structure, showing the capability of the method to easily deal with complex geometrical features.

Mesh scheme for a phase transition problem with time-fractional derivative

Abstract The time-fractional phase transition problem, formulated in enthalpy form, is studied. This nonlinear problem with an unknown moving boundary includes, as an example, a mathematical model of one-phase Stefan problem with the latent heat accumulation memory. The posed problem is approximated by the backward Euler mesh scheme. The unique solvability of the mesh scheme is proved and a priori estimates for the solution are obtained. The properties of the mesh problem are studied, in particular, an estimate of movement rate for the mesh phase transition boundary is established. The proved estimate make it possible to localize the phase transition boundary and split the mesh scheme into the sum of a nonlinear problem of small algebraic dimension and a larger linear problem. This information can be used for further construction of efficient algorithms for implementing the mesh scheme. Several algorithms for implementing mesh scheme are briefly discussed.

Damage modelling of carbon fibre composite crush tubes: Numerical simulation and experimental validation of drop weight impact

Angle-ply carbon fibre reinforced polymer (CFRP) crush tubes were tested in drop weight impact experiments. High-fidelity computational modelling of the dynamic impact response was performed using explicit finite element analysis in LS-DYNA. Ply-by-ply and fibre-aligned meshing was used for the composite lamina wherein the intralaminar damage was treated with a 3D rate- and pressure-dependent continuum damage mechanics (CDM) model, implemented as a user material. Delamination was modelled using cohesive tiebreak contacts to deal with the mis-matched nodes caused by the use of a structured, fibre-aligned mesh. The material aligned meshing scheme was shown to be required to capture the intralaminar splits observed in the ±45∘ plies. The numerical predictions showed excellent correlation with experimental measurements in terms of both the damage mechanisms and macroscopic material behaviour of the drop weight. In the simulation, interlaminar friction between delaminated plies seemed to be a main contributor of energy dissipation. Parameter sensitivity analysis showed that the interaction between the delamination fracture energy and friction can substantially influence the results and stable crushing load, in particular. For the scenario studied here, a regular structured mesh (unaligned) was shown to be insufficient for simulating realistic crack paths despite producing reasonable predictions of the force–displacement and absorbed energy.

A free‐cutting mesh strategy for optimal shape synthesis in magnetics

The authors propose an innovative technique for dealing with optimal shape design problems that exploits the flexibility of the virtual element method in generating meshes composed of general polygonal and polyhedral elements. Virtual element method and finite element method can coexist on the same discretized domain; therefore, the possibility of dealing with hanging nodes and gluing sub-domain meshes is ensured. Accordingly, the shape synthesis of a magnetic pole is considered as the case study. It is shown that the proposed technique is effective in handling the shape variations dictated by an algorithm of evolutionary optimisation.

Modelling local failure around hard-rock tunnels based on a linked multiscale mesh strategy

Research on local failure around hard-rock tunnels is the basis for revealing the failure mechanism of surrounding rock, predicting the ground excavation response and formulating effective support strategies. However, to characterize the failure morphology at a high resolution, a global uniform fine mesh numerical model may lead to a high computational time and space burdens. This study aims to develop a linked multiscale mesh modelling that establishes a high-resolution square mesh only near the free face around a tunnel instead of a global uniform fine-scale mesh. Then, the linked multiscale mesh strategy is implemented with a meso-damage smear numerical method to model local failure around hard-rock tunnels. The simulation results of laboratory- and engineering-scale examples indicate that the proposed strategy can save considerable computational cost; moreover, the downscaling refinement range and pattern have almost no effect on the surrounding rock failure results. It is shown that the combination of the linked multiscale mesh strategy and damage smear mechanical method presented in this study can be effectively employed for tunnel lightweight modelling, and this approach is very well suited for modelling hard-rock tunnel stability problems.

Combined multi-scale mesh and full-matrix inversion for enhancing time-domain breast diffuse optical tomography.

Time-domain diffuse optical tomography can efficiently reconstruct both absorption and reduced scattering coefficients but is heavily limited by the ill-posedness in its inverse problem and low spatial resolution. To deal with these adversities, the truncated singular value decomposition (TSVD)-based whole-weighting-matrix inversion scheme can be a particularly suitable implementation. Unfortunately, TSVD is subject to a storage challenge for three-dimensional imaging of a bulk region, such as breast. In this paper, a multi-scale mesh strategy based on computed tomography (CT) anatomical geometry is adopted to solve the storage challenge, where a fine mesh is used in forward calculation to ensure accuracy, and a coarse mesh in the inversion process to enable TSVD-based inversion of the whole-weighting matrix. We validate the proposed strategy using simulated data for a single lesion model from clinical positron emission tomography images of a breast cancer patient, and further, for a complex model that is constructed by setting dual lesions at different separations in the CT breast geometry.

Three-dimensional gravity inversion based on optimization processing from edge detection

Gravity inversion requires much computation, and inversion results are often non-unique. The first problem is often due to the large number of grid cells. Edge detection method, i.e., tilt angle method of analytical signal amplitude (TAS), helps to identify the boundaries of underground geological anomalies at different depths, which can be used to optimize the grid and reduce the number of grid cells. The requirement of smooth inversion is that the boundaries of the meshing area should be continuous rather than jagged. In this paper, the optimized meshing strategy is improved, and the optimized meshing region obtained by the TAS is changed to a regular region to facilitate the smooth inversion. For the second problem, certain constraints can be used to improve the accuracy of inversion. The results of analytic signal amplitude (ASA) are used to delineate the central distribution of geological bodies. We propose a new method using the results of ASA to perform local constraints to reduce the non-uniqueness of inversion. The guided fuzzy c-means (FCM) clustering algorithm combined with priori petrophysical information is also used to reduce the non-uniqueness of gravity inversion. The OpenAcc technology is carried out to speed up the computation for parallelizing the serial program on GPU. In general, the TAS is used to reduce the number of grid cells. The local weighting and priori petrophysical constraint are used in conjunction with the FCM algorithm during the inversion, which improves the accuracy of inversion. The inversion is accelerated by the OpenAcc technology on GPU. The proposed method is validated using synthetic data, and the results show that the efficiency and accuracy of gravity inversion are greatly improved by using the proposed method.

Modeling Large-Size Round Steel Plates With Surface PEEC Method for Lightning Transient Analysis

The round steel plate is the basic component of the external floating roof tank structure. Modeling large-size round steel plates is essential for analyzing the lightning response of external floating roof tanks. This article builds time-domain equivalent circuit models for round steel plates excited by the lightning current. First, a meshing scheme is presented to generate sector and cylindrical surface cells, which make use of the distribution characteristics of the currents and voltages on the round steel plates. Then, the surface partial element equivalent circuit (PEEC) approach is applied to model the plate. A method of calculating the equivalent surface impedance is proposed, which converts the frequency dependence of the skin effect into impedance based on the time domain. Finally, the time-domain surface PEEC approach is validated by the indirect time-domain solution based on the Fourier transform and the commercial software using the finite-difference time-domain technique, respectively. The comparisons show that the calculation results of different methods are consistent with each other. However, the proposed method in this article can greatly enhance the computational efficiency while meeting the accuracy requirements. It has great significance to the study of lightning protection for external floating roof tank.

Multiscale method of modelling surface texture with mass-conserving cavitation model

To solve the multiscale problem of modelling surface texture with mass-conserving cavitation, a new method combining the finite cell method with the Fischer–Burmeister–Newton–Schur algorithm (FCM-FBNS) was developed. Numerical experiments confirmed that the proposed method has strong robustness and a high computational efficiency. When the number of degrees of freedom (DOFs) was 360,000, the calculation difference between FCM-FBNS and the finite-element method (FEM) was < 1 %, and the computation time was approximately 1 % of that of the FEM. FCM-FBNS is suitable for parallel processing, and the computation time increases nearly linearly with an increase in the number of DOFs in most cases. Furthermore, FCM-FBNS has a flexible meshing strategy for discretising complex geometries.

Developments in remeshing‐free fatigue crack growth simulation including a new adaptive virtual crack closure technique

AbstractThe present study aims to evaluate and improve the remeshing‐free fatigue crack growth (FCG) simulation and life estimation through two developing FEM‐VCCT and XFEMPN‐VCCT algorithms in Abaqus. First, new energy‐based forms of the Paris, Walker, and Elber FCG models are implemented by a user subroutine, a novel systematic meshing strategy is proposed, and some challenging features of these algorithms are investigated. Despite the success of this methodology in enhancing efficiency, decreasing the run‐time, and preventing potential errors, it is observed that the XFEMPN‐VCCT algorithm has some fundamental errors such as a serious overestimation in FCG life prediction. Thus, a novel “adaptive VCCT” is introduced and implemented by a Python script in Abaqus in the form of a new XFEMPN‐AVCCT algorithm. It is finally concluded that the adaptive VCCT can significantly enhance the accuracy of FCG simulation and life estimation.

A semi-empirical mesh strategy for CFD simulation of indoor airflow

Proper meshing is critical for accurate computational fluid dynamics (CFD) simulation. In the three stages of CFD simulation, pre-processing accounts for about 40–60% of the total time. In this study, the performance of the grid for a large number of numerical simulations was analyzed, and base mesh sizes were recommended for different indoor geometric models. The simulation results for temperature, velocity, mesh resolution and computation time were compared, and mesh types were recommended for different models. Considering factors such as wall function, wall y+, face quality of the mesh, grid volume change rate and prism layer setting, a semi-empirical mesh strategy was presented. This strategy was verified in three cases with different geometric complexity. The verification results showed that the mesh strategy can yield reasonable simulation results. The root-mean-square error of the velocity simulation results of the optimized grid can be reduced by nearly 20%. This strategy can save nearly 50% of the mesh adjustment time. Through analysis and fitting of the grid results, the evaluation criteria for this mesh strategy were obtained. The criteria can be used to evaluate the cases using this mesh strategy.

A segmented heat source for efficiently calculating the residual stresses in laser powder bed fusion process

To improve the efficiency and accuracy of the numerical simulation for the laser powder bed fusion (LPBF) process, an optimized segmented heat source (OSHS) model is proposed in this work. The results show that the geometrical size and temperature distribution contour of the calculated cross-sectional molten pool are almost the same as those calculated by the moving heat source (MHS). Under the same mesh scheme, the calculation efficiency of the OSHS model improves about 9 times to that of the traditional MHS model. Moreover, the OSHS model can achieve stable calculations with the medium-size elements grid, which can reduce the required calculation time to one-third of the fine mesh scheme and achieve almost the same thermal-mechanical calculation result. This model was successfully applied on the simulation of the LPBF single layer multi-tracks thermal-mechanical numerical simulation. To verify the calculated results, the sample of the same size was prepared using 316L stainless steel by LPBF technology. The surface residual stress components of the as-built sample were measured by X-Ray Diffraction (XRD). The calculated residual stress curves were in good agreement with the measured results. The simulation and experimental results indicated that the primary surface residual stress component was the tensile stress along the laser scanning direction, which was about 280 MPa. The shear stress was concentrated at the corners of the as-built sample. Therefore, the OSHS model proposed in this work greatly improved the calculation efficiency and obtained very accurate results for the thermal-mechanical simulation of the LPBF process.

A mesh refinement scheme for fourth order bi-harmonic equation of mixed boundary-value elastic problems

Fourth order bi-harmonic equation is extensively used for stress-strain analysis of mixed boundary-value elastic problems. However, currently existing uniform mesh scheme based on finite difference method (FDM) needs vast amount of computational resources and efforts for an acceptable solution. Therefore, in this study, a mesh refinement (MR) scheme based on FDM is developed to solve fourth order bi-harmonic equation effectively. The developed MR scheme allows high resolution computation in sub-domains of interest and relatively low resolution in other regions which overcomes the memory exhausting problems associating with the traditional uniform mesh based FDM. In this paper, sub-domain that needs high resolution (mesh refinement) are identified based on gradient of stress and displacement vectors. A very high gradient in any region signifies the need of fine mesh because coarse grained meshes are not adequate to capture the sharply changing stresses or displacements. Once the sub-domains of interest are identified, the mesh refinement is done by splitting course meshes into smaller meshes. Several new stencils are created to satisfy the fourth order by harmonic equation and associated boundary conditions over the various sizes of meshes. The developed MR scheme has been applied to solve several classical mixed boundary-value elastic problems to show its applicability. In addition, the validity, effectiveness, and superiority of the MR scheme have been established by comparing of obtained solutions with uniform mesh results, finite element method (FEM) results, and the well-known analytical results. Our results show that the developed MR scheme can provide a more reliable and accurate result than the conventional uniform mesh scheme with a reduced number of equations, thus, saves a huge amount of computational memory.

Full compression response of FG-based scaffolds with varying porosity via an effective numerical scheme

The triply periodic minimal surfaces (TPMS) can be employed to realize the variable porosity gradient in different directions and has a great application prospect in the orthopedic implant field. In the present scrutiny, two kinds of gyroid-based TPMS gradient scaffolds (skeletal-gyroid and sheet-gyroid) in different orientations are established and manufactured by utilizing selective laser melting (SLM) technology with Ti-6Al-4V powders. The meshing of the TPMS based on the voxelization loses its surface smoothness and brings the geometric defects of steps. Therefore, an optimized smoothing-tetrahedral mesh strategy (STMS) to finite element analysis (FEA) is proposed to explore the graded scaffolds’ overall deformation response and energy absorption efficiency. The obtained results reveal that the mechanical properties of the samples are slightly altered after heat treatment. The sheet-gyroid (Sh-G) structure has a higher elastic modulus and compressive strength, more stable energy absorption efficiency, and higher total energy absorption compared with the skeletal-gyroid (Sk-G). The fracture morphology of the Sk-G exhibits two distinct regions: a typical dimple of ductile fracture and a brittle fracture characterized by the flow pattern, while the Sh-G samples only demonstrate brittle fractures. The STMS and Johnson-Cook plastic damage model correctly simulated the gradient porous scaffolds’ plastic deformation and failure behavior after yielding. Furthermore, the stress-strain curves of the simulation agree well with the experimental results such that the minimum elastic modulus and yield strength discrepancies in order are 0.5 and 2.8%. The STMS improves the simulation accuracy of the TPMS structure and provides a solid foundation for designing and applying artificial bone scaffolders in the near future.

Spatial resolution of different discretizations over long-time for the Dirac equation with small potentials

We compare the long-time error bounds and spatial resolution of finite difference methods with different spatial discretizations for the Dirac equation with small electromagnetic potentials characterized by ɛ∈(0,1] a dimensionless parameter. We begin with the simple and widely used finite difference time domain (FDTD) methods, and establish rigorous error bounds of them, which are valid up to the time at O(1/ɛ). In the error estimates, we pay particular attention to how the errors depend explicitly on the mesh size h and time step τ as well as the small parameter ɛ. Based on the results, in order to obtain “correct” numerical solutions up to the time at O(1/ɛ), the ɛ-scalability (or meshing strategy requirement) of the FDTD methods should be taken as h=O(ɛ1/2) and τ=O(ɛ1/2). To improve the spatial resolution capacity, we apply the Fourier spectral method to discretize the Dirac equation in space. Error bounds of the resulting finite difference Fourier pseudospectral (FDFP) methods show that they exhibit uniform spatial errors in the long-time regime, which are optimal in space as suggested by the Shannon’s sampling theorem. Extensive numerical results are reported to confirm the error bounds and demonstrate that they are sharp.