Coarse Mesh Research Articles

Assessment of variations in microbiological quality of water in the newly constructed rainwater-harvesting system storage tank

Proper harvesting of rainwater requires a good knowledge of the storage conditions and the nature of physico-chemical and microbiological contaminant loads on the harvested water. This will help to determine the nature and level of treatment required. This study assessed the variations in the microbiological quality of water stored in a newly constructed galvanized steel tank over the course of one rainy season in Afikpo, Nigeria. A 2.0 mm thick galvanized steel tank of 1.728 m3 capacity was constructed together with water collection system including coarse mesh, gutter, funnel, conduits, first-flushing device and filter. Rainwater was collected into the tank, sampled, discharged, and rinsed with chlorinated water at each sampling episode throughout the rainy season at Afikpo, from April to November 2021. The water samples were tested in the laboratory for total coliform and faecal coliform. The weekly concentrations of total coliform range between 90CFU/100 ml and 320CFU/100 ml; the mean is 177CFU/100 ml. The weekly concentrations of faecal coliform fall between 30CFU/100 ml and 261CFU/100 ml; the mean is 108CFU/100 ml. These results indicate that the concentrations of the two pathogens are far above the zero limit recommended by the World Health Organization (WHO) and Nigerian Standards for Drinking Water Quality (NSDWQ) . Hence, the rooftop-harvested rainwater is unsafe for human consumption without treatment. Boiling or chlorine treatment is needed. Public health authorities in Afikpo should sensitize the public on the dangers of consuming untreated rainwater as is the practice found among many in the area.

Evaluation of combinatorial algorithms for optimizing highly nonlinear structural problems

Optimizing highly nonlinear structural problems can be very challenging due to the large number of parameters. Classical compliance minimization does not work for such problems. Common optimization algorithms also do not find good solutions. This work evaluates both commonly used optimization algorithms and algorithms not yet used in topology optimization. The algorithms are evaluated using a simple nonlinear problem: minimizing the end displacement of a cantilever beam fixed on one side and loaded by gravity. The global optimum for a coarse mesh grid is computed by simulating nearly 60 million possible topology designs using a Brute-Force search. We use this benchmark to evaluate the computational cost and objective values of known and newly developed optimization methods. The known methods are binary-coded Genetic Algorithm, Simulated Annealing, and Free Shape Optimization. The Reduced Variable Neighborhood Search (RVNS) has not yet been applied to topology optimization. We provide two implementations of RVNS: Breadth-First Search with a limited search depth (BFSL) and with an optional restriction for the size of the simultaneously modified area (TBFSL). According to the benchmark, TBFSL is the most efficient approach. For the optimization on a finer mesh grid, TBFSL is combined with a multi-grid approach to further increase efficiency.

Integrated species distribution models fitted in INLA are sensitive to mesh parameterisation

The ever‐growing popularity of citizen science, as well as recent technological and digital developments, have allowed the collection of data on species' distributions at an extraordinary rate. In order to take advantage of these data, information of varying quantity and quality needs to be integrated. Point process models have been proposed as an elegant way to achieve this for estimates of species distributions. These models can be fitted efficiently using Bayesian methods based on integrated nested Laplace approximations (INLA) with stochastic partial differential equations (SPDEs). This approach uses an efficient way to model spatial autocorrelation using a Gaussian random field and a triangular mesh over the spatial domain. The mesh is constructed by user‐defined variables, so effectively represents a free parameter in the model. However, there is a lack of understanding about how to set these mesh parameters, and their effect on model performance. Here, we assess how mesh parameters affect predictions and model fit to estimate the distribution of the serotine bat,Eptesicus serotinus, in Great Britain. A Bayesian INLA model was fitted using five meshes of varying densities to a dataset comprising both structured observations from a national monitoring programme and opportunistic records. We demonstrate that mesh density impacted spatial predictions with a general loss of accuracy with increasing mesh coarseness. However, we also show that the finest mesh was unable to overcome spatial biases in the data. In addition, the magnitude of the covariate effects differed markedly between meshes. This confirms that mesh parameterisation is an important and delicate process with implications for model inference. We discuss how species distribution modellers might adapt their use of INLA in the light of these findings.

Cross-scale optimization of advanced materials for micro and nano structures based on strain gradient theory

Recently developed cross-scale optimization methods are mainly the macro equivalent calculation of performances based on the traditional homogenization method. However, the traditional homogenization is limited to the classic continuum of Cauchy–Boltzmann. Therefore, it is inadequate to interpret the size dependence of the optimal result. Hence, a new cross-scale optimization is proposed based on Wei–Hutchinson strain gradient theory by employing the non-local homogenization model, which could describe and explain the size dependence during optimization process when considering micro structures. The topological optimization procedure simultaneously has the ability of coupled computing by using subdomain parameterized coarse meshes. The numerical computations involved in the entire model can be solved in one iteration, which helps to eliminate mesh dependencies and greatly reduce the computation time. It is shown that the final stiffness of the optimized periodic structure can be significantly increased by considering the strain gradient theory compared with the classic homogenization scheme in the process of cross-scale optimization.

Multiscale steady discrete unified gas kinetic scheme with macroscopic coarse mesh acceleration using preconditioned Krylov subspace method for multigroup neutron Boltzmann transport equation.

A multiscale steady discrete unified gas kinetic scheme with macroscopic coarse mesh acceleration [accelerated steady discrete unified gas kinetic scheme (SDUGKS)] is proposed to improve the convergence of the original SDUGKS for an optically thick system in solving the multigroup neutron Boltzmann transport equation (NBTE) to analyze the distribution of fission energy in the reactor core. In the accelerated SDUGKS, by solving the coarse mesh macroscopic governing equations (MGEs) derived from the moment equations of the NBTE, the numerical solutions of the NBTE on fine meshes at the mesoscopic level can be rapidly obtained from the prolongation of the coarse mesh solutions of the MGE. Furthermore, the use of the coarse mesh can greatly reduce the computational variables and improve the computational efficiency of the MGE. The biconjugate gradient stabilized Krylov subspace method with the modified incomplete LU preconditioner and the lower-upper symmetric-Gauss-Seidel sweeping method are implemented to solve the discrete systems of the macroscopic coarse mesh acceleration model and mesoscopic SDUGKS to further improve the numerical efficiency. Numerical solutions validate good numerical accuracy and high acceleration efficiency of the proposed accelerated SDUGKS for the complicated multiscale neutron transport problems.

Direct Guaranteed Lower Eigenvalue Bounds with Optimal a Priori Convergence Rates for the Bi-Laplacian

An extra-stabilised Morley finite element method (FEM) directly computes guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplace Dirichlet eigenvalues. The smallness assumption $\min\{\lambda_h,\lambda\}h_{\max}^{4}$ $\le 15.0864$ on the maximal mesh-size $h_{\max}$ makes the computed $k$-th discrete eigenvalue $\lambda_h\le \lambda$ a lower eigenvalue bound for the $k$-th Dirichlet eigenvalue $\lambda$. This holds for multiple and clusters of eigenvalues and serves for the localisation of the bi-Laplacian Dirichlet eigenvalues in particular for coarse meshes. The analysis requires interpolation error estimates for the Morley FEM with explicit constants in any space dimension $n\ge 2$, which are of independent interest. The convergence analysis in $3$D follows the Babuska-Osborn theory and relies on a companion operator for the Morley finite element method. This is based on the Worsey-Farin 3D version of the Hsieh-Clough-Tocher macro element with a careful selection of center points in a further decomposition of each tetrahedron into 12 sub-tetrahedra. Numerical experiments in 2D support the optimal convergence rates of the extra-stabilised Morley FEM and suggest an adaptive algorithm with optimal empirical convergence rates.

The enriched finite element method-virtual crack closure technique for cracked structures

In the present work, the enriched finite element method (EFEM) and virtual crack closure technology (VCCT) are applied to model the fracture mechanics problems of structures. EFEM-VCCT is used for improving the accuracy and efficiency of the standard FEM in solving the fracture parameters of cracked structures. The models with different structures, crack lengths and meshes are calculated and compared with the finite element method-VCCT (FEM-VCCT). By introducing the interpolation cover function, the computational efficiency and accuracy of EFEM-VCCT are improved. This improvement is confirmed by a series of numerical examples. The problems of crack initiation and crack growth under different materials and loading conditions are analyzed. The high accuracy of EFEM-VCCT under a relatively coarse mesh is demonstrated through a series of numerical examples. In the numerical analysis of fracture mechanics, EFEM-VCCT can be considered a very good alternative to FEM-VCCT due to its high efficiency and accuracy.

Investigation of Very Large Eddy Simulation Method for Applications of Supersonic Turbulent Combustion

The very large eddy simulation (VLES) method was investigated for supersonic reacting flows in the present work. The advantages and characteristics of the VLES model and the widely used improved delayed detached eddy simulation (IDDES) method were revealed through a supersonic ramped-cavity cold flow. Compared to the IDDES model, the VLES model transformed from RANS mode to LES mode faster, resulting in a smaller gray region caused by the mode transition. However, the original volume-averaging truncation length scale could lead to poor predictions of the velocity profiles and wall pressure distribution. By introducing a hybrid truncation length scale combining the maximum grid length and the shear layer adaptive (SLA) length with different coefficients, the accuracy of the VLES method was significantly improved, and the issue of the low shear layer position was solved. Moreover, owing to the resolution control function, the VLES method could adaptively model more turbulent kinetic energy and maintain a good accuracy in a coarser mesh. Finally, the modified VLES method was applied in conjunction with a hybrid combustion model constructed by the partially stirred reactor (PaSR) model and the Ingenito supersonic combustion model (ISCM) in simulations of the supersonic flame in the DLR scramjet combustor. After introducing the correction of the molecular collision frequency by the ISCM, the results obtained by the hybrid combustion model were more consistent with the experimental results, especially for the time-averaging temperature profile in the ignition zone.

A hybrid implicit-explicit discontinuous Galerkin spectral element time domain (DG-SETD) method for computational elastodynamics

SUMMARY A hybrid implicit-explicit (IMEX) discontinuous Galerkin spectral element time domain (DG-SETD) algorithm is proposed to simulate 3D elastic wave propagation in inhomogeneous media. In this method, the original problem can be divided into a number of well designed subdomains, and the mesh generation of different subdomains is completely independent, thus allowing flexible spatial discretization of complex geometry. The neighboring subdomains are connected by a Riemann transmission condition (RTC), and spectral elements with different interpolation orders can be used in different subdomains to maximize the computational efficiency of multiscale problems to facilitate parallel computing for different subdomains. In particular, the explicit or implicit time iteration scheme can be appropriately selected for a subdomain according to the size of its mesh elements to increase the global time step increment, thus giving higher computational efficiency: For subdomains with coarse meshes, the explicit time integration scheme is used and the time step increment is limited by the Courant−Friedrichs−Lewy (CFL) stability condition; for subdomain with fine structures and thus fine meshes, an implicit time integration scheme is used so that a large time step increment can be used without affecting the stability. In addition, a jump condition of displacement and velocity is introduced to accurately simulating fractures and faults, including lossless and viscous fractures with plane, curve or cross structures. This avoids the volume modeling of the extremely thin fracture structures, and effectively reduces the number of degrees of freedoms (DoFs) in the discretized system without the loss of accuracy. The accuracy, robustness and efficiency of the DG-SETD algorithm are demonstrated by quantitative comparisons of the waveforms with the commercial finite element software COMSOL.

Accurate coarse mesh simulation of delamination in composites using a novel hp-adaptive cohesive element

Cohesive elements are routinely used to simulate delamination between plies in composites. However, the allowable maximum size of these elements is limited, owing to the presence of a very small process zone ahead of the cohesive crack-tip, which needs to be discretised with multiple elements for obtaining an accurate failure response. This restricts their applicability beyond analysing delamination in coupon scale problems, due to the need for a very fine mesh and associated large problem size. In this paper, a novel hp cohesive element is developed to address this problem. This element is capable of simultaneous mesh refinement and elevation of interpolation order, locally and adaptively around the propagating cohesive crack tip. Additional degrees of freedom associated with the hp refinement are handled by extra nodes already supplied in the initial mesh file, that are selectively activated by the program as required. Several example problems are presented showing the ability of this method to improve the prediction of delamination onset and growth in extremely coarse meshes with a reduced computational time.

A novel Cell-Centered nodal integral method for the Convection-Diffusion equation

The nodal integral methods (NIMs) are coarse mesh methods for solving partial differential equations (PDEs) efficiently and accurately. Here, a new variant of the NIM is proposed for the solution of the two-dimensional convection–diffusion equation. The earlier versions of the NIM for fluid flow problems used surface-averaged variables. In contrast, the current scheme is developed in terms of cell-centered values using different treatments of cell interface conditions. The advantage of this scheme is that coupling with other physics is straightforward. Another distinctive feature of the current approach is that the temporal derivative is explicitly addressed, resulting in a system of differential–algebraic equations (DAEs) of index-1. The system of DAEs is then solved using Backward Difference Formula, which provides higher accuracy in time compared to traditional NIM. A few problems with the known analytical solution are solved to demonstrate the efficacy of the current simplified approach.

A mixed element scheme for the Helmholtz transmission eigenvalue problem for anisotropic media

In this paper, we study the Helmholtz transmission eigenvalue problem for inhomogeneous anisotropic media with the index of refraction in two and three dimensions. Starting with the nonlinear fourth-order formulation established by Cakoni et al 2009 J. Integral Equ. Appl. 21 203–27, we present an equivalent mixed formulation for this problem with auxiliary variables, followed by finite element discretization. Using the proposed scheme, we rigorously show that the optimal convergence rate for the transmission eigenvalues on both convex and nonconvex domains can be expected. With this scheme, we obtain a sparse generalized eigenvalue problem whose size is too demanding, even with a coarse mesh that its smallest few real eigenvalues fail to be solved by the shift and invert method. We partially overcome this critical issue by deflating nearly all of the eigenvalues with huge multiplicity, resulting in a marked reduction in the matrix size without deteriorating the sparsity. Extensive numerical examples are reported to demonstrate the effectiveness and efficiency of the proposed scheme.

A layer-wise plasticity-based approach for the analysis of reinforced concrete shell structures using a mixed finite element

The analysis of reinforced concrete shell structures accounting for material nonlinearity is addressed. The structural response is numerically evaluated using a mixed shell finite element and a plasticity-based material behaviour. The finite element is a quadrilateral with four nodes and is based on self-equilibrated assumed stresses. The kinematic fields are interpolated only along the element boundary by polynomials up to the third order. The generalised shell stresses are evaluated through layer-wise integration of the Cauchy pointwise stresses. This allows appropriated three-dimensional elastoplastic constitutive equation to be employed and to include multiple reinforcing layers. The integration of the constitutive laws is performed using a dual decomposition method which preserves the assumed stress interpolation. Results, including civil engineering applications, show that the proposed approach is reliable and accurate in evaluating the static nonlinear equilibrium path. Additionally, the mixed finite element shows a quadratic rate of convergence in the collapse load and low error even for coarse meshes.

A fast adaptive PD-FEM coupling model for predicting cohesive crack growth

In this study, nonlocal theory of peridynamic (PD) is coupled with finite element method (FEM) to simulate cohesive crack growth in quasi-brittle materials. The adaptive dynamic relaxation method is adopted to implement quasi-static loading conditions. The advantage of peridynamic method in the modeling of crack propagation is taken into consideration to be coupled with that of finite elements which is computationally less expensive and covers various boundary value problems. In this novel method, the whole domain of the problem is initially dominated by FEM solver with coarse mesh grids. Coupling process is then introduced to those regions which have critical conditions according to damage criterion. The capability of the proposed coupling method in the modeling of cohesive crack growth in quasi-brittle materials is demonstrated. In comparison to other conventional PD-FEM coupling methods, the computational cost of the proposed method is considerably improved and total running time of the solution is significantly attenuated. The validity of the method is confirmed through comparing the results with experimental ones.

Efficient XFEM approach for the analysis of thin-walled beams

This paper presents a new approach to the analysis of thin-walled beams. The proposed model is developed by utilizing the eXtended Finite Element Method (XFEM). Global enrichment functions are added to a standard 3D Finite Element (FE) coarse mesh to enhance its performance and reduce the number of degrees of freedom. These analytical functions provide an approximate solution in the longitudinal direction of the beam. The resulting model bridges the gap between the Finite Prism Method (FPM), which provides an efficient approximation to beam bending with analytical functions but has also well-known limitations, and a detailed 3D FE model, which can accurately resolve thin-walled structures but is computationally demanding.Our method can be easily extended for beams with non-prismatic cross-sections (e.g., tapered beams) or with perforations, which makes it more general than other semi-analytical methods, such as the FPM. However, these extensions are not in the scope of this paper. Additionally, one can improve the solution by sectional and/or longitudinal mesh refinements.The performance of the model is investigated on two benchmark problems. The results show that the proposed method captures different global and sectional deformation modes of thin-walled beams when subjected to torsion, including shear-lag and distortion cross-section. Furthermore, convergence studies demonstrate the superior performance of the proposed method as compared to the FPM, and a 3D FE model with a coarse mesh.

An Efficient FDTD Method Based on Subgridding Technique and One-Step Leapfrog ADI-FDTD

In this letter, a 3-D subgridding finite-difference time-domain (FDTD) approach is proposed. The calculation domain is divided into regions with dense meshes and regions with coarse meshes. By applying the proposed subgridding technique to dense grid regions, memory and computation resources can be significantly reduced. Furthermore, the Courant–Friedrich–Levy (CFL) limitation of dense mesh regions is broken by the one-step leapfrog alternately direction-implicit (ADI)-FDTD. Coarse and dense regions can adopt a uniform time step, which is limited only by the standard FDTD CFL condition for coarse meshes. In comparison with the standard FDTD approach and the subgridding FDTD approach, numerical results demonstrate the efficiency and accuracy of the proposed method.

A Kernel-Free Boundary Integral Method for 2-D Magnetostatics Analysis

Performing magnetostatic analysis accurately and efficiently is very important for multi-objective optimization of electromagnetic device designs. In this research, a kernel-free boundary integral method (KFBIM) has been introduced for solving magnetic computations in a toroidal core geometry in 2-D. This study is very relevant in the design and optimization of toroidal inductors or transformers used in electrical systems, where lighter weight, higher inductance, higher efficiency, and lower leakage flux are required. The governing partial differential equations (PDEs) have been formulated as a system of the Fredholm integral equations of the second kind. Unlike traditional boundary integral methods or boundary element methods, KFBIM does not require an analytical form of Green’s function for evaluating integrals via numerical quadrature. Instead, KFBIM computes integrals by solving an equivalent interface problem on a Cartesian mesh. Compared with traditional finite difference methods for solving the governing PDEs directly, KFBIM produces a well-conditioned linear system. Therefore, the KFBIM requires only a fixed number of iterations when an iterative method [e.g., generalized minimal residual method (GMRES)] is applied for solving the linear system, and the numerical solution is not sensitive to computer round-off errors. The obtained results are then compared with a commercial finite element solver (ANSYS), which shows excellent agreement. It should be noted that, compared with FEM, the KFBIM does not require a body-fit mesh and can achieve high accuracy with a coarse mesh. In particular, the calculations of the magnetic potential and the tangential field intensity on the boundaries are more stable and exhibit almost no oscillations.

Three-Dimensional Dual-Mesh Inversions for Sparse Surface-to-Borehole TEM Data

The surface-to-borehole transient electromagnetic (SBTEM) method can provide images at higher resolution for deep earth because its receivers are close to targets. However, as usually the boreholes distribute sparsely, the limited EM data can result in an “equivalent trap” in SBTEM inversions, i.e., the data are well-fitted, but the model is not properly recovered. To overcome this non-unique problem, we propose a dual-mesh three-dimensional (3D) SBTEM inversion scheme. We first use a coarse mesh to obtain a rough resistivity distribution near the borehole, and then we map the coarse mesh attribute into a fine one and capture details from the fine mesh inversion. We test our method on both synthetic data and survey data acquired in Daye, Hubei Province, China. Numerical experiments show that our dual-mesh inversion strategy can better recover the location and resistivity of targets.

The consistent boundary element method for potential and elasticity: Part III — Topologically challenging numerical assessments for 2D problems

The collocation boundary element method is consistently outlined in a companion (Part I) paper for the general three-dimensional, static case of elasticity on the basis of a weighted-residuals statement that leads to the Somigliana’s identity. Arbitrary rigid-body displacements, as for elasticity, are naturally taken into account, and traction force parameters are always in balance independently of problem scale and mesh discretization. For generally curved boundaries, the correct definition of traction force interpolation functions enables the enunciation of a general convergence theorem, the introduction of patch and cut-out tests and, not least, a considerable simplification of the numerical implementations. Simple code schemes are proposed in a second companion (Part II) paper for 2D problems of potential and elasticity, which rely exclusively on Gauss–Legendre quadrature and lead to arbitrarily high – actually only machine-precision dependent – computational accuracy of all results of interest independently of a problem’s geometry and topology. We present here numerical results and convergence assessments – including numerical illustration of convergence Theorem 1 of the companion paper I – for 2D potential and elasticity problems with very challenging topology issues and even for subnanometer source–field distances — maybe with approximations due to a coarse mesh discretization but never introducing unduly singularities. We dare say the present results cannot be replicated by any code implementation other than the ones of our own, which resort to no ad hoc means beyond the problem’s correct mathematics. In fact, we present the objective and controllable tools of separately assessing a mesh discretization for a given practical problem in terms of precision, accuracy and liability to round-off-errors. We also evaluate the distance threshold for a source point to be considered sufficiently close to demand the proposed exact mathematical treatment, as, for large distances, only Gauss–Legendre quadrature is required, be it in terms of quadrature adaptivity of even a fast multipole scheme.

A collection device for various-sized microparticles that uses four serial acoustic separations: Working toward microplastic emission prevention

Small plastic debris particles less than 5 mm in size called microplastics (MPs) are an emerging global ecological issue. This study developed a high enrichment microfluidic device to collect various-sized microparticles that uses four serial acoustic separations. To adopt the device for use with up to 200-µm MPs, the microchannel was widened by lowering the excitation frequency. The microfluidic device was designed for a 3.2-fold enrichment at each junction and a total 105-fold enrichment at the four junctions. The microfluidic network of the device was designed on the basis of a hydraulic-electric analogy and it worked even though only a single pump was used without multiple precise flow controllers. The collection performance of the device was evaluated based on the total collection rate that in turn was based on microscopic observations at the four junctions and the actual collection rate and the actual enrichment ratio obtained by measuring the effluents from the outlets. First, the device was evaluated separately using microparticles of 5, 10, 15, 25, 50, and 200 µm in diameter. The total collection rates were over 90% except for the 5-µm microparticles which seemed to be too small to manipulate acoustically at the lowered frequency. Finally, the device was evaluated with two example mixtures representing small MPs ranging from 200 µm down to 25 µm and very small MPs ranging from 25 µm down to 10 µm. Both the total collection rates and the actual collection rates ranged from 70% to 90%. However, the actual enrichment ratios ranged from half the designed value of 105 to the design value itself as some microparticles were slowed down and a few seemed to become trapped and attached to the microchannel walls by acoustic radiation force. Therefore, the microfluidic device was judged to be applicable to MP removal applications after prefiltration through a coarse mesh, while the MP analysis applications were judged to require improvement to inhibit the microparticle attachment used in 2D focusing. In conclusion, the serial acoustic separations could be a promising approach to highly enrich and remove various-sized MPs from environmental samples.