Mesh Skeleton Research Articles

A numerical investigation on coupling of conforming and hybridizable interior penalty discontinuous Galerkin methods for fractured groundwater flow problems

The present paper focuses on the numerical modeling of groundwater flows in fractured porous media using the codimensional model description. Therefore, fractures are defined explicitly as a (d−1)-dimensional geometric object immersed in a d-dimensional region and can act arbitrarily as a drain or a barrier. We numerically investigate a novel numerical strategy combining distinctive classes of conforming and nonconforming high-order Galerkin methods, both eligible for static condensation. This procedure is here particularly relevant, leading to a smaller and sparser final system with coupled degrees of freedom solely on the mesh skeleton. Precisely, we combine an inspired hybridizable interior penalty discontinuous Galerkin (HIP) formulation inside the bulk region and a standard continuous Galerkin (CG) approximation on the fracture network. The distinctive discretization of corresponding PDEs and the coupling strategy are rigorously exposed, and the local and global matrix assemblies are detailed. Extensive numerical experiments are then achieved to prove the model’s performances for 2D/3D analytic and realistic benchmarks. Qualitative comparisons are also considered with other discretization methods and commercial software, such as comsol.

Penalty‐free discontinuous Galerkin method

AbstractIn this article, we present a new high‐order discontinuous Galerkin (DG) method, in which neither a penalty parameter nor a stabilization parameter is needed. We refer to this method as penalty‐free DG. In this method, the trial and test functions belong to the broken Sobolev space, in which the functions are in general discontinuous on the mesh skeleton and do not meet the Dirichlet boundary conditions. However, a subset can be distinguished in this space, where the functions are continuous and satisfy the Dirichlet boundary conditions, and this subset is called admissible. The trial solution is chosen to lie in an augmented admissible subset, in which a small violation of the continuity condition is permitted. This subset is constructed by applying special augmented constraints to the linear combination of finite element basis functions. In this approach, all the advantages of the DG method are retained without the necessity of using stability parameters or numerical fluxes. Several benchmark problems in two dimensions (Poisson equation, linear elasticity, hyperelasticity, and biharmonic equation) on polygonal (triangles, quadrilateral, and weakly convex polygons) meshes as well as a three‐dimensional Poisson problem on hexahedral meshes are considered. Numerical results are presented that affirm the sound accuracy and optimal convergence of the method in the norm and the energy seminorm.

Experimental study on mechanical properties and damage mechanism of self-stress grouted crushed coal of different grain sizes

The reinforcement mechanism of self-stress grouted crushed coal of different grain sizes is not clear. In this study, crushed coal of 5∼10, 10–20 and 20∼30 mm grain was selected. It obtained that the grain size was proportional to the width of sedimentary fissure among coal blocks and inversely proportional to its distribution range. As coal grain size increases, the uniaxial compression strength (UCS) of grouted specimens decrease and the AE parameters increase, respectively. The slurry forms a mesh skeleton structure in the 5∼10 mm specimen, significantly increasing its bearing capacity. However, in the 20∼30 mm specimen, the porous coal block is the main carrier, reducing the bearing capacity. Swelling needle-like Aft crystals produced closely occludes the coal, increasing the binding ability and overall compactness; almost none of the coal-slurry intersections are damaged during uniaxial compression. The main damage areas of 10∼20 mm and 20∼30 mm grouted specimens are inside coal by SEM, and the cracks increase with the rise of coal grain size. The trends of damage variable and cumulative energy counts curves are approximately the same, further confirming the reliability of the analysis of the AE experimental results. This paper can lay the foundation for theoretical research and field practice of self-stress grouting reinforcement.

Spatially-assembled binary carbon anode synergizing directional electron transfer and enriched microbe accommodation for wastewater treatment and energy conversion: From simulation to experiments

Bioelectrochemical systems (BESs) hold prospects in wastewater energy and resource recovery. Anode optimization is important for simultaneous enhancement of wastewater energy conversion and effluent quality in BESs. In this study, a multi-physics model coupling fluid flow, organic degradation and electrochemical process was constructed to guide the design and optimization of BES anodes. Based on the multi-physics simulation, spatially-assembled binary carbon anodes composed of three-dimensional carbon mesh skeleton and granular activated carbon were proposed and established. The granular activated carbon conducive to microbe accommodation played a vital role in improving effluent water quality, while the carbon mesh skeleton favoring electron collection and transfer could enhance the bioelectricity output. With an average chemical oxygen demand (COD) removal rate of 0.442 kg m−3 d−1, a maximum power density of 20.6 W m−3 was achieved in the optimized composite anode BES, which was 25% and 154% higher than carbon mesh skeleton BES and granular activated carbon BES. Electroactive bacteria were enriched in composite anodes and performed important functions related to microbial metabolism and energy production. The spatially-assembled binary carbon anode with low carbon mesh packing density was more cost-effective with a daily energy output per anode cost of 221 J d−1 RMB−1. This study not only provides a cost-efficient alternative anode to simultaneously improve organic degradation and power generation performance, but also demonstrates the potential of multi-physics simulation in offering theoretical support and prediction for BES configuration design as well as optimization.

A posteriori error estimates for primal hybrid finite element methods applied to Poisson problem

This paper presents a new fully computable a posteriori error estimates for the primal hybrid finite element methods based on equilibrated flux and potential reconstructions. The reconstructed potential is obtained from a local L2 orthogonal projection of the numerical solution on a continuous function space over the mesh skeleton. The equilibrated flux is the solution of a local mixed problem with a Neumann boundary condition given by the Lagrange multipliers of the primal hybrid finite element solution. For that, a divergence-consistent finite element pair is used. The upper and lower bounds of the error estimator are proved, and numerical results illustrate the efficiency of the error indicators.

A face-centred finite volume approach for coupled transport phenomena and fluid flow

We present a particular derivation of the face-centred finite volume (FCFV) method and study its performance in non-linear, coupled transport problems commonly encountered in geoscientific and geotechnical applications. The FCFV method is derived from the hybridisable discontinuous Galerkin formulation, using a constant degree of approximation for the discretization of the unknowns defined on the mesh faces (edges in two dimensions). The piecewise constant degrees of freedom are determined in a global problem over the mesh skeleton. Then, the solution and its gradient are recovered at the cells centroid in a set of element-by-element independent postprocesses, both exhibiting linear convergence. The formulation of the transient advection-diffusion-reaction equation is presented in detail and the numerical analysis under challenging advective/diffusive regimes is studied. Finally, we use several numerical examples to illustrate the advantages and limitations of the FCFV method to solve problems of geoscientific and geotechnical relevance governed by the non-linear coupling between advection-diffusion-reactive transport and Stokes flow. Our results show that the FCFV method is an attractive and highly competitive alternative to other commonly used methods.

Research on Road Characteristics and the Microscopic Mechanism of Lime–Phosphogypsum-Stabilized Red Clay

In this paper, mixtures with different proportions of lime, phosphogypsum, and red clay were prepared, and the road properties and micromechanisms of lime–phosphogypsum-stabilized red clay were investigated by unconfined compressive strength test, water stability test, swelling test, shrinkage test, XRD quantitative analysis, and scanning electron microscope analysis. The results showed that the unconfined compressive strength of the mix increased and then decreased with the increase of phosphogypsum content. With the increase of age, the growth was faster in the first 14 days and basically stabilized in the last 14 days. The mixture has poor water stability, large absolute swelling rate, and low linear shrinkage. The reason for the increase of strength is that the reaction of lime, phosphogypsum, and red clay produces ettringite, and the cementing substance gels form a three-dimensional mesh skeleton structure; the excess of ettringite will cause the skeleton to expand and break, and the strength decreases.

A posteriori error approximation in discontinuous Galerkin method on polygonal meshes in elliptic problems

The paper presents a posteriori error approximation concept based on residuals in the two-dimensional discontinuous Galerkin (DG) method. The approach is relatively simple and effective in application, and it takes advantage of some unique properties of the DG method. The error function is constructed in an enriched approximation space, utilizing the hierarchical nature of the basis functions. Among many versions of the DG method, the most popular one is based on the interior penalty approach. However, in this paper a DG method with finite difference (DGFD) is utilized, where the continuity of the approximate solution is enforced by finite difference conditions applied on the mesh skeleton. In the DG methods arbitrarily shaped finite elements can be used, so in this paper the meshes with polygonal finite elements are considered, including quadrilateral and triangular elements. Some benchmark examples are presented, in which Poisson’s and linear elasticity problems are considered. The examples use various mesh densities and approximation orders to evaluate the errors. The error estimation maps, generated for the discussed tests, indicate a good correlation with the exact errors. In the last example, the error approximation concept is applied for an adaptive hp mesh refinement.

Optimal Geometric Multigrid Preconditioners for HDG-P0 Schemes for the reaction-diffusion equation and the Generalized Stokes equations

We present the lowest-order hybridizable discontinuous Galerkin schemes with numerical integration (quadrature), denoted as HDG-P0 for the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. Here by lowest order, we mean that the (hybrid) finite element space for the global HDG facet degrees of freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A discontinuous piecewise linear space is used for the approximation of the local primal unknowns. We give the optimal a priori error analysis of the proposed HDG-P0 schemes, which hasn’t appeared in the literature yet for HDG discretizations as far as numerical integration is concerned. Moreover, we propose optimal geometric multigrid preconditioners for the statically condensed HDG-P0 linear systems on conforming simplicial meshes. In both cases, we first establish the equivalence of the statically condensed HDG system with a (slightly modified) nonconforming Crouzeix–Raviart (CR) discretization, where the global (piecewise-constant) HDG finite element space on the mesh skeleton has a natural one-to-one correspondence to the nonconforming CR (piecewise-linear) finite element space that live on the whole mesh. This equivalence then allows us to use the well-established nonconforming geometry multigrid theory to precondition the condensed HDG system. Numerical results in two- and three-dimensions are presented to verify our theoretical findings.

C 0-hybrid high-order methods for biharmonic problems

Abstract We devise and analyze $C^0$-conforming hybrid high-order (HHO) methods to approximate biharmonic problems with either clamped or simply supported boundary conditions. $C^0$-conforming HHO methods hinge on cell unknowns that are $C^0$-conforming polynomials of order $(k+2)$ approximating the solution in the mesh cells and on face unknowns, which are polynomials of order $k\ge 0$ approximating the normal derivative of the solution on the mesh skeleton. Such methods deliver $O(h^{k+1})$$H^2$-error estimates for smooth solutions. An important novelty in the error analysis is to lower the minimal regularity requirement on the exact solution. The technique to achieve this has a broader applicability than just $C^0$-conforming HHO methods, and to illustrate this point, we outline the error analysis for the well-known $C^0$-conforming interior penalty discontinuous Galerkin methods as well. The present technique does not require a $C^1$-smoother to evaluate the right-hand side in case of rough loads; loads in $W^{-1,q}$, $q>\frac {2d}{d+2}$, are covered, but not in $H^{-2}$. Finally, numerical results including comparisons to various existing methods showcase the efficiency of the proposed $C^0$-conforming HHO methods.

Ultrafast microwave-induced synthesis of lithiophilic oxides modified 3D porous mesh skeleton for high-stability Li-metal anode

The wide application of Li metal anodes is hindered by the inhomogeneous growth of Li dendrites. Herein, we propose an ultrafast microwave-induced synthesis method for a lithiophilic 3D porous mesh skeleton converted by commercial brass mesh. Due to the lithiophilic oxides (FeO and CuO) granular layer and 3D porous structure, this novel skeleton can be used as a current collector to achieve a high-stability of Li metal anode. A dendrite-free deposition morphology Li metal anode is obtained with noticeable cycling stability up to 1000 h (2500 cycles) at 5 mA cm−2 current density with 1 mA h cm−2 capacity. This skeleton is used to assemble the full-cell with the commercial LiCoO2 (LCO) cathode. A stable full-cell is obtained with 87.2 % capacity retention after 500 cycles at 0.5C. The microwave-induced synthesis method in this work can prepare 3D lithiophilic skeleton in a brief period, which provides guidance for the design of Li metal anode.

HHO Methods for the Incompressible Navier-Stokes and the Incompressible Euler Equations

We propose two Hybrid High-Order (HHO) methods for the incompressible Navier-Stokes equations and investigate their robustness with respect to the Reynolds number. While both methods rely on a HHO formulation of the viscous term, the pressure-velocity coupling is fundamentally different, up to the point that the two approaches can be considered antithetical. The first method is kinetic energy preserving, meaning that the skew-symmetric discretization of the convective term is guaranteed not to alter the kinetic energy balance. The approximated velocity fields exactly satisfy the divergence free constraint and continuity of the normal component of the velocity is weakly enforced on the mesh skeleton, leading to H-div conformity. The second scheme relies on Godunov fluxes for pressure-velocity coupling: a Harten, Lax and van Leer approximated Riemann Solver designed for cell centered formulations is adapted to hybrid face centered formulations. The resulting numerical scheme is robust up to the inviscid limit, meaning that it can be applied for seeking approximate solutions of the incompressible Euler equations. The schemes are numerically validated performing steady and unsteady two dimensional test cases and evaluating the convergence rates on h-refined mesh sequences. In addition to standard benchmark flow problems, specifically conceived test cases are conducted for studying the error behaviour when approaching the inviscid limit.

Visualization pipeline of autonomous driving scenes based on FCCR-3D reconstruction

Visual sensor-based human–computer interaction can provide environmental perception data of traffic road scenes for mechatronic intelligent vehicles. The existing three-dimensional (3D) reconstruction visualization schemes are mainly based on structured scene reconstruction with multiview geometry and are offline operations. It is difficult for large-scale autonomous driving scenes with wide baselines to ensure consistent reconstruction results in the wide field-of-view area and the center of the light-of-sight area. It is also difficult to achieve uniform reconstruction results for regions with drastic depth changes and inconsistent texture fineness. Therefore, we propose a 3D reconstruction-based visualization pipeline for autonomous driving scenes of wide baseline. First, the stereo geometric model is initialized. The parameters of distortion correction and polar calibration are calibrated. Second, the proposed feature chain code resampling (FCCR) is used to uniformly extract the features for regions with inconsistent texture fineness and depth variation. The filtered 3D data are obtained based on the stereo geometric model. Finally, the texture mapping of the mesh skeleton of 3D data is carried out by the bilinear combination of surrounding multilevel gradual textures. The visualization results of scene rendering and scene roaming are realized. The pipeline has been effectively verified on the simulated experimental platform, real vehicle test, and public dataset. It can meet the real-time 3D reconstruction visualization of wide-baseline scenes. The reconstruction texture is uniform, and the detail response is rich. We can present the intuitive, 3D, real-time, graphical physical quantities of scene data for drivers, intelligent vehicles, AR-HUD, and control systems. The method we present has research value and application significance for the practical engineering of electronic images.

An improved Multiscale FEM for the free vibrations of heterogeneous solids

We present an improvement of the Multiscale Finite Element Method (MsFEM). MsFEM incorporates fine-scale features of a domain through special oscillatory coarse-scale trial functions and its essential step is an online computation of these special functions defined using local auxiliary problems. Our enhancement makes use of the dynamic condensation that transmits the construction of the oscillatory basis functions exclusively to the coarse mesh skeleton (element interfaces) where we defined special 1D eigenproblems. Consequently, both the approximation error and computational cost are reduced. Moreover, arbitrary coarse mesh element geometries, like polytopal ones, may be used without any additional modifications. We illustrate the accuracy and convergence of the improved method with higher order approximation through selected plane strain examples.

A hybrid-mixed finite element method for single-phase Darcy flow in fractured porous media

We present a hybrid-mixed finite element method for a novel hybrid-dimensional model of single-phase Darcy flow in a fractured porous media. In this model, the fracture is treated as a (d−1)-dimensional interface within the d-dimensional fractured porous domain, for d=2,3. Two classes of fracture are distinguished based on the permeability magnitude ratio between the fracture and its surrounding medium: when the permeability in the fracture is (significantly) larger than in its surrounding medium, it is considered as a conductive fracture; when the permeability in the fracture is (significantly) smaller than in its surrounding medium, it is considered as a blocking fracture. The conductive fractures are treated using the classical hybrid-dimensional approach of the interface model where pressure is assumed to be continuous across the fracture interfaces, while the blocking fractures are treated using the recent Dirac-δ function approach where normal component of Darcy velocity is assumed to be continuous across the interface. Due to the use of Dirac-δ function approach for the blocking fractures, our numerical scheme allows for nonconforming meshes with respect to the blocking fractures. This is the major novelty of our model and numerical discretization. Moreover, our numerical scheme produces locally conservative velocity approximations and leads to a symmetric positive definite linear system involving pressure degrees of freedom on the mesh skeleton only. As an application, we extend the idea to a simple transport model. The performance of the proposed method is demonstrated by various benchmark test cases in both two- and three-dimensions. Numerical results indicate that the proposed scheme is highly competitive with existing methods in the literature

Development of MoS2/cellulose aerogels nanocomposite with superior application capability for selective lead(II) capture

The development of highly efficient and practical adsorbents for selective lead(II) capture remains an urgent challenge to be overcome. Herein, a new nanocomposite (W-MoS2@CA) was prepared by confining MoS2 nanosheets with widened interlayer spacing in the cross-linked cellulose aerogels. W-MoS2@CA combines the excellent handling, flow characteristics, and porous structure of aerogels with the specific affinity and accessible interior spaces of MoS2 toward lead cation. The adsorbent shows outstanding adsorption capacity and fast kinetics for lead removal. Benefiting from the high affinity between MoS2 and lead, W-MoS2@CA exhibits excellent adsorption selectivity even in the solution containing multiple coexisting heavy metal ions and high concentrations of competing mineral cations. The centimeter/millimeter-sized of the 3D cellulose mesh skeleton significantly improves the hydraulics property of nanocomposite, enabling it to be conveniently used in the flow system. The considerable treatment volume in fixed-bed indicated that W-MoS2@CA is of great potential for lead removal in practice. In addition, W-MoS2@CA can be readily regenerated with negligible adsorption capacity loss after 5 adsorption − desorption cycles. These results confirm that W-MoS2@CA has a great prospect in the water remediation of lead-removing, and this work may provide a new avenue for the application of MoS2 as an effective adsorbent in practical water/wastewater treatment.

Performances of Hybridized-, Embedded-, and Weighted-Interior Penalty Discontinuous Galerkin Methods for Heterogeneous and Anisotropic Diffusion Problems

The present paper discusses families of Interior Penalty Discontinuous Galerkin (IP) methods for solving heterogeneous and anisotropic diffusion problems. Specifically, we focus on distinctive schemes, namely the Hybridized-, Embedded-, and Weighted-IP schemes, leading to final matrixes of different sizes and sparsities. Both the Hybridized- and Embedded-IP schemes are eligible for static condensation, and their degrees of freedom are distributed on the mesh skeleton. In contrast, the unknowns are located inside the mesh elements for the Weighted-IP variant. For a given mesh, it is well-known that the number of degrees of freedom related to the standard Discontinuous Galerkin methods increases more rapidly than those of the skeletal approaches (Hybridized- and Embedded-IP). We then quantify the impact of the static condensation procedure on the computational performances of the different IP classes in terms of robustness, accuracy, and CPU time. To this aim, numerical experiments are investigated by considering strong heterogeneities and anisotropies. We analyze the fixed error tolerance versus the run time and mesh size to guide our performance criterion. We also outlined some relationships between these Interior Penalty schemes. Eventually, we confirm the superiority of the Hybridized- and Embedded-IP schemes, regardless of the mesh, the polynomial degree, and the physical properties (homogeneous, heterogeneous, and/or anisotropic).

Multiscale modeling and simulation methods for electromagnetic and multiphysics problems

Multiscale modeling and simulation methods for electromagnetic and multiphysics problems

Seismic and axial compression performance of insulated concrete sandwich wall

A new type of composite insulated concrete sandwich (ICS) wall is proposed. The wall consists of a three-dimensional spatial steel wire mesh skeleton, insulating core material and inner and outer concrete layers, all of which form a synergistic composite wall panel to work as the main stress-bearing member in the construction of multistorey high-rise buildings. To analyse the mechanical properties of the wall body, its failure mode was investigated. The wall's hysteresis curve, strength, stiffness and deformation were all analysed through a low reversed cyclic load test on two specimens, followed by an evaluation of its seismic performance. The deformability, failure mode, strain and ultimate bearing capacity of the specimens under the action of axial compression were also studied through an axial compression test on three specimens. The hysteresis test indicated that, in comparison with an ordinary shear wall, the ICS wall, connected through effective construction measures, had superior overall working performance under the same reinforcement ratio and its ductility and energy-dissipating capacity were improved. The results of the axial compression test also confirmed that the ICS wall had excellent axial compressive resistance, bending and deformation resistance, and high stability.

Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian

For the discretization of the integral fractional Laplacian $(-\Delta)^s$, $0 < s < 1$, based on piecewise linear functions, we present and analyze a reliable weighted residual a posteriori error estimator. In order to compensate for a lack of $L^2$-regularity of the residual in the regime $3/4 < s < 1$, this weighted residual error estimator includes as an additional weight a power of the distance from the mesh skeleton. We prove optimal convergence rates for an $h$-adaptive algorithm driven by this error estimator. Key to the analysis of the adaptive algorithm are local inverse estimates for the fractional Laplacian.