Mesh Discretization Research Articles

Nonstandard finite difference method for time-fractional singularly perturbed convection–diffusion problems with a delay in time

In this work, nonstandard finite difference method is presented for the numerical solution of time-fractional singularly perturbed convection–diffusion problems with a delay in time. The time-fractional derivative is considered in the Caputo sense and discretized using Crank–Nicholson technique. Then, a nonstandard finite difference scheme is constructed on a uniform mesh discretization along the spatial direction. The parameter-uniform convergence of the proposed method is proved rigorously and shown to be ɛ-uniform convergent with order of convergence O((Δt)2) along the temporal domain and M−1 along the spatial domain. Finally, the proposed scheme is validated using model examples and the computational results are in agreement with the theoretical expectation.

Spectral Descriptors for 3D Deformable Shape Matching: A Comparative Survey.

A large number of 3D spectral descriptors have been proposed in the literature, which act as an essential component for 3D deformable shape matching and related applications. An outstanding descriptor should have desirable natures including high-level descriptive capacity, cheap storage, and robustness to a set of nuisances. It is, however, unclear which descriptors are more suitable for a particular application. This paper fills the gap by comprehensively evaluating nine state-of-the-art spectral descriptors on ten popular deformable shape datasets as well as perturbations such as mesh discretization, geometric noise, scale transformation, non-isometric setting, partiality, and topological noise. Our evaluated terms for a spectral descriptor cover four major concerns, i.e., distinctiveness, robustness, compactness, and computational efficiency. In the end, we present a summary of the overall performance and several interesting findings that can serve as guidance for the following researchers to construct a new spectral descriptor and choose an appropriate spectral feature in a particular application.

Simulating Hydrogen Diffusion in ZrH Moderator and its Impact on Coupled Neutronic-Thermal Simulation of a Heat Pipe Microreactor

Metal hydride moderators are a common material choice for moderating compact reactor designs, as their high thermal limits and high density of hydrogen make them favorable for operation. One additional attribute of metal hydrides, zirconium hydride in particular, is the relatively large mobility of hydrogen within the metal lattice, especially at high temperatures and under large thermal gradients. As hydrogen’s spatial distribution directly impacts neutron thermalization in the system, and therefore power shape, changes in hydrogen concentration of the moderator may be important to analyze. This work couples OpenMC and MOOSE in order to solve the feedback loop of the coupled power distribution-thermal distribution-hydrogen migration without relying on mesh discretization. Functional expansions are used extensively as a method of transferring spatial information, and continuously varying material tracking is used in the neutronic solver to represent the change in hydrogen concentrations. The resulting hydrogen redistribution is highlighted, as well as the resulting neutronic and thermal impact of this redistribution.

Comparison of Hexcore and Poly-Hexcore computational meshes in the aspect of air flow modeling based on the actual geometry of mining excavations

Discrete models are used in industry for many applications. In one of the most frequently used Finite Element Method (FEM) for Computational Fluid Dynamics (CFD) calculations, discrete models may be two-dimensional or three-dimensional. 2D models are used as a simplification to achieve satisfied results in the shortest computational time. 3D models, on the other hand, are used for more complex calculations. These models constitute a representation of real-world objects that have been appropriately simplified to make the calculations accurate and correct. The calculation time of a 3D model is significantly longer compared to a 2D model. For this reason, to reduce the calculation time, different types of simplifications and various types of discrete model meshes are used. In this paper, the authors made a comparison of two computational meshing technologies: Hexcore and Poly-Hexcore in the aspect of airflow modeling in mining excavations using CFD. The geometry considered in this case came from real-world excavations captured by laser scanning in the Gertruda Slant, Zloty Stok. Point cloud data was processed through feature extraction, which was subsequently utilized to create structured models of mining excavations. The results of the simulations show that taking into account such a diverse and complicated geometry and its significant lengths, reaching tens of kilometers, better results are obtained with the use of Poly-Hexcore mesh. This type of mesh allows simulations to be performed with similar accuracy in a shorter computation time. Utilizing a more modern type of mesh makes work more dynamic, which is of particular importance when conducting numerical simulations of air distribution in large and complex computational domains.

ECONOMIC GROWTH FORECAST MODEL URBAN SUPPLY CHAIN LOGISTICS DISTRIBUTION PATH DECISION USING AN IMPROVED GENETIC ALGORITHM

The proposed way of estimating the associated macroeconomic index based on a supply chain network is a novel strategy that might give useful insights for firms and organizations trying to enhance their supply chain logistics operations. To aid in this process, the suggested technique utilizes multiple regression analysis and adaptive extreme learning machine models to determine the relative importance of each indicator in the supply chain logistics decision-making process. Firms may be able to save expenses and boost economic value by employing enhanced genetic algorithms and mathematical modeling to develop a logistics distribution model for urban supply chains based on reconstructive genetic algorithms. It is possible that the paper's study of the issues plaguing the current distribution of logistics in metropolitan areas would prove valuable to organizations that are seeking to improve their own supply chain logistics procedures. This paper appears to take a thorough strategy that might assist assure the accuracy of the forecasting method by preprocessing macroeconomic indicators using imputation, classification, and other approaches to generate a time series consistency model. Some novel ways that may shed light on logistics in the supply chain include using a two-dimensional discrete mesh structure built on wireless sensors to depict the national economic development scenario and using a coding matrix to convey the extent of economic growth. A thorough and all-encompassing way that might aid organizations in making better judgments about their supply chain logistics strategies is to employ multiple regression analysis, an adaptive extreme learning machine model, and other approaches to examine the effect degree of each key indicator. Experiments demonstrating excellent and steady prediction accuracy of the algorithm model are encouraging and point to the possible usefulness of the suggested forecasting approach. The paper's contributions, including its analysis of supply chain logistics cost accounting, determination of the basic path optimization, improvement of the genetic algorithm, and design of the mathematical model of supply chain logistics distribution path decision, all look promising in terms of their potential to help businesses cut costs and boost economic value. The positive outcome of the successful design of the mathematical model demonstrates the possible efficacy of the suggested technique.

Dynamic BEM analysis of elastic anisotropic nano‐sheet with surface imperfections and cavities

AbstractThe paper considers elastic anisotropic nano‐sheet with nanotopography and containing embedded nanocavities under time‐harmonic in‐plane wave motion. The mechanical model is based on the classical elastodynamic theory for the anisotropic bulk solid and the nonclassical boundary conditions along its boundary derived by Gurtin and Murdoch. A localized constitutive equation for the half‐plane boundary is considered in the frame of surface elasticity theory. The computational tool is an efficient direct displacement boundary element method (BEM) based on the frequency‐dependent elastodynamic fundamental solution for elastic generally anisotropic material. To show the versatility of the proposed methodology, wave propagation in an elastic anisotropic heterogeneous nano‐sheet with nanorelief presented by hill‐canyon nanotopography is studied. The simulations illustrate the dependence of the wave field on the material anisotropy, on the surface elasticity properties, on the nanorelief peculiarities, on the nanocavities existence and their dynamic interaction, and on the incident wave characteristics. The obtained results reveal the potential of the developed mechanical model based on the boundary integral equations in the frame of surface elasticity theory to produce highly accurate results by using strongly reduced discretization mesh in comparison with the domain‐based methods.

NURBS-boundary-based quadtree scaled boundary finite element method study for irregular design domain

The traditional finite element analysis for irregular design domains often encounters challenges such as intricate mesh discretization and inaccurate boundary description. In this paper, we propose a quadtree scaled boundary finite element method based on NURBS curves where the boundaries can be accurately represented. Quadtree decomposition, which satisfies the 2:1 rule, is employed to rapidly subdivide the analysis domain. The scaled boundary finite element method (SBFEM) is utilized to analyze the internal elements and address the displacement incompatibility issue of hanging nodes in the quadtree. Furthermore, the boundary element is discretized into boundary curves and internal lines, whose displacement fields are respectively constructed by the NURBS shape functions and the Lagrange shape functions, and then the subsequent analysis of the boundary element is completed by SBFEM. Finally, numerical examples are tested to demonstrate the feasibility of the proposed method, which effectively enhances computational efficiency and accuracy in solving irregular design domains.

A Stable Time-Dependent Mesh Method for Generalized Credit Rating Migration Problem

AbstractThe r-adaptive difference scheme is advanced in this article for solving the generalized credit rating migration model for arbitrary volatility with multiple terminal conditions. The r-adaptive moving mesh method defines the coordinate mapping from the physical to the computational domain and then uses piece-wise polynomials to approximate the physical coordinates. The central implicit semi-discretization scheme is imposed on space, and the backward Euler time marching scheme, coupled with several moving mesh partial differential equations, is used to achieve the numerical solution. The numerical operations are performed with several examples, and the obtained results are sufficiently accurate. The convergence of the proposed scheme is second-order, which is verified with the analysis. The article also investigates the stability and convergence of the adaptive mesh discretization scheme, which are not available in the literature; the convergence of the scheme is second-order in space and first-order in time.

Effective Mechanical Properties of Periodic Cellular Solids with Generic Bravais Lattice Symmetry via Asymptotic Homogenization.

In this paper, the scope of discrete asymptotic homogenization employing voxel (cartesian) mesh discretization is expanded to estimate high fidelity effective properties of any periodic heterogeneous media with arbitrary Bravais's lattice symmetry, including those with non-orthogonal periodic bases. A framework was developed in Python with a proposed fast-nearest neighbour algorithm to accurately estimate the periodic boundary conditions of the discretized representative volume element of the lattice unit cell. Convergence studies are performed, and numerical errors caused by both voxel meshing and periodic boundary condition approximation processes are discussed in detail. It is found that the numerical error in periodicity approximation is cyclically dependent on the number of divisions performed during the meshing process and, thus, is minimized with a refined voxel mesh. Validation studies are performed by comparing the elastic properties of 2D hexagon lattices with orthogonal and non-orthogonal bases. The developed methodology was also applied to derive the effective properties of several lattice topologies, and variation of their anisotropic macroscopic properties with relative densities is presented as material selection charts.

Study on the dynamic numerical simulation of flow and combustion in hybrid rocket motors based on a discrete phase model

Numerical simulation based on the discrete phase model and dynamic mesh model can more accurately simulate the working process for hybrid rockets, including the motion and evaporation process of particles and regression process of the fuel grain. The dynamic numerical simulation of combustion and flow in hybrid rocket motors based on a discrete phase model is carried out. Aluminum and aluminum hydride are added to the internal flow field as discrete droplets. Then, a hybrid rocket is designed with 95 % hydrogen peroxide as oxidizer and 38 % aluminum hydride+20 % aluminum+42 % hydroxyl-terminated polybutadiene as the fuel and used to verify the model by firing test. A high degree of agreement between experiment and simulation findings is found. The average deviations for thrust and pressure are 3.6 % and 5.9 %, respectively. It offers the required backing for using the simulation method to investigate the flow and combustion process for the hybrid rockets. The results also demonstrate the transient characteristics of the inner flow field, including the separation and merging of vortexes in the pre-chamber. It is found that a peak of the regression rate appears at the front of the grain and moves backward. This is mainly because the vortex appears here and enhances its heat transfer. Through the discrete phase model, it is worth noting that the aluminum droplet content gradually decreases at the nozzle over time, which is consistent with the firing test. Further research on the impact of aluminum hydride content on the performance of the hybrid rockets has been carried out. The results show that the higher aluminum hydride content can lead to better performance for the hybrid rockets. The thrust, pressure, specific impulse, and characteristic velocity enhance with the increase of aluminum hydride content. An interesting discovery is that the average regression rate is maximum at 48 % aluminum hydride content. This research innovatively couples dynamic mesh and discrete phase models, which can help better understand the working process for hybrid rockets and provide important guidance for the selection of fuel formula.

Simulation of the dynamic cracking of brittle materials using a nonlocal damage model with an effective strain rate effect

In this study, a nonlocal multiscale damage model that considers the effect of a strain rate is presented. Moreover, this model is combined with the scaled boundary finite element method (SBFEM) and quadtree mesh for dynamic crack propagation in quasi-brittle materials. The scaling centre of each SBFEM subdomain was selected as the material point, and the bond connecting two material points was referred to as the material bond. Microscopic damage was quantified based on the stretch rate of the material bond, and macroscale topological damage was calculated by taking the weighted average of the microscopic damage over the bonds within the influence domain. The bell distribution function is utilised as the weight function in the calculation of the nonlocal weight function. To account for the impact of the strain rate during dynamic loading, an effective rate R is introduced to enhance multiscale damage by modifying the damage threshold and bond softening coefficient throughout the analysis process. By analysing the energetic degradation function of the phase field, which establishes a connection between the macroscale topological damage and energy-based damage, a nonlocal multiscale damage model was incorporated into the framework of the SBFEM. Quadtree mesh discretization was employed to achieve rapid and high-quality multilevel mesh generation by effectively utilising the hanging nodes. Three typical dynamic cracking examples were simulated using the proposed model, and the results were compared with the relevant experimental results and references. The results show that the method is suitable for accurate dynamic cracking simulation in quasi-brittle materials. Moreover, the results indicate that the proposed method can spontaneously simulate dynamic crack bifurcation without introducing additional crack bifurcation criteria. For a compact tensile test of concrete under different loading rates, the method proposed in this study can accurately simulate the corresponding failure modes. The results also show that without considering the effect of the strain rate, the damage develops rapidly in an unconstrained state, which is inconsistent with the actual case.

Strain based damage indicators for finite element analysis of reinforced concrete structures

Numerical analysis conducted using a finite element model provides a quantitative and detailed evaluation of damage in concrete. As a result, the performance verification based on the local damage obtained numerically is considered an effective method for achieving a rational design of RC structures. Stresses and strains computed from numerical analysis are suited to indicators for representing local damage in concrete instead of sectional forces calculated from design equations; however, the performance evaluation using stresses and strains lack versatility owing to their dependency on constitutive equations and mesh discretization in numerical models. This study presents two types of strain-based damage indicators for performance verification in the design of RC structures using nonlinear finite element analysis. The second strain invariant of the deviatoric strain tensor is used to assess tensile damage, such as the opening of flexural cracks and the development of shear cracking in concrete, whereas the normalized accumulated strain energy is used to evaluate compressive damage, i.e., concrete crushing. These scalar indicators have high general usefulness, with calculation as non-local values by a weighted average, to reduce their mesh dependency. Furthermore, numerical analyses were performed to evaluate the damage condition and failure modes of RC beams, and the proposed damage indicators helped accurately understand flexure and shear failure. Lastly, we propose threshold values of the damage indicators for the performance verification of safety, represented by the load-carrying capacity in the design of RC structures.

Grounded source transient electromagnetic 3D forward modeling with the spectral-element method and its application in hydraulic fracturing monitoring

A long wire with large current source transient electromagnetic (TEM) monitoring, with a large detection depth, low cost, safety, and environmental protection, has unique advantages in the testing and identification of unconventional reservoir fluid and the evaluation of stimulated reservoir volume. So, the TEM 3D forward modeling method has become a research hotspot. Although the finite-element method (FEM) is a type of numerical algorithm that has been widely applied in three-dimensional (3D) electromagnetic field forward modeling, the efficiency and accuracy of FEM require further improvement in order to meet the demand of fast 3D inversion. By increasing the order of the basis function and adjusting the principle of mesh discretization, the precision of the mixed-order spectral-element (SEM) result will be increased. The backward Euler scheme is an unconditionally stable technique which can ignore the impact of the scale of the time step. To achieve a better description of the nonlinear electromagnetic (EM) response of the grounded source TEM method and to optimize the efficiency and accuracy/precision of the 3D TEM forward modeling method significantly, we proposed the use of 3D TEM forward modeling based on the mixed-order SEM and the backward Euler scheme, which can obtain more accurate EM results with fewer degrees of freedom. To check its accuracy and efficiency, the 1D and 3D layered models are applied to compare the SEM results with the semi-analytical and FEM solutions. In addition, we analyzed the accuracy and efficiency of the SEM method for different types of order basis functions. Finally, we calculated the long-wire source TEM response for a practical 3D earth model of a shale gas reservoir for fracturing monitoring and tested the feasibility of the TEM method in a hydraulic fracturing monitoring area to further demonstrate the flexibility of the SEM method.

The fast methods of high-frequency wave scattering in complex media and structures

In this review work, we give a review of recent works of high-frequency electromagnetic (EM) wave scattering in complex media and structures. The EM wave scattering from electrically large scatterers remains an important and challenging problem. We introduce the quadratic patches to discretize the electrically large scatterers and present the fast physical optics (FPO) method to solve the PO radiation integral. The computational workload and accuracies of the physical optics (PO) scattered fields from the quadratic patches are error-controllable and frequency-independent, respectively. To remedy the limitations of the accuracies of the PO scattered fields in the lit region of the electrically large scatterers, we consider the Fock current from the convex scatterers to cover the contributions of creeping wave fields in the shadow region. For the coated material from the perfectly conducting structure, we propose the multi-level technique together with the PO method to calculate the scattered field in an efficient way. In the process of mesh discretization for electrically large scatterers, we adopt the quadratic patches and further extend them to adaptive patches. For the multi-scale scatterers, we use the hybrid method of moment and FPO (MoM-FPO) method to calculate the scattered field from the multi-scale scatterers.

Performance analysis and material distribution optimization for sound barriers using a semianalytical meshless method

AbstractWith the increase in car ownership, traffic noise pollution has increased considerably and is one of the most severe types of noise pollution that affects living standards. Noise reduction by sound barriers is a common protective measure used in this country and abroad. The acoustic performance of a sound barrier is highly dependent on its shape and material. In this paper, a semianalytical meshless Burton–Miller‐type singular boundary method is proposed to analyze the acoustic performance of various shapes of sound barriers, and the distribution of sound‐absorbing materials on the surface of sound barriers is optimized by combining a solid isotropic material with a penalization method. The acoustic effect of the sound‐absorbing material is simplified as the acoustical impedance boundary condition. The objective of optimization is to minimize the sound pressure in a given reference plane. The volume of the sound‐absorbing material is used as a constraint. The density of the nodes covered with the sound‐absorbing material is used as the design variable. The method of moving asymptotes was used to update the design variables. This model completely avoids the mesh discretization process in the finite element method and requires only boundary nodes. In addition, the approach also does not require the singular integral calculation in the boundary element method. The method is illustrated and validated using numerical examples to demonstrate its accuracy and efficiency.

Parameter uniform higher order numerical treatment for singularly perturbed Robin type parabolic reaction diffusion multiple scale problems with large delay in time

In this paper, we address a class of boundary layer originated singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions having large time delay; for the higher order numerical analysis. The boundary layer originated nonuniform mesh, is generated by adaptive moving mesh approach based on equidistribution principle. The mesh is obtained by solving a nonlinear system; involving the discrete problem and the discrete nonlinear mesh generating functions. We provide the theoretical mesh structure. The difference scheme for Robin type boundary conditions having normalized flux; uses a combination of space and time discretizations so that the order of convergence in space is higher than the usual linear accurate upwind discretization in space. The present theoretical convergence analysis requires less regularity of the solution and proves to be exactly second order parameter uniform accurate in space. Numerical experiments highly validate the theoretical findings and also show that the present approach is better than the well-known upwind discretizations.

Numerical Simulations of Carbon-Fibre Impregnation with a Polymer as an Anisotropic Permeability Medium.

The impregnation process of carbon fibres with polymers is challenging to model due to the system's complexity, particularly concerning the following aspects: the complex rheology of the polymeric matrices and the presence of solid, continuous fibres, both with anisotropic properties, and the interaction between solid and fluid, which can change the displacement of fibres into a cyclic dependence. In this work, an interesting approach was considered by setting the fibres as a porous medium whose properties were calculated with microscale/macroscale cycle modelling. In the microscale modelling stage, two main assumptions can be made: (i) a homogeneous distribution with a representative cell or (ii) a stochastic distribution of fibres. The solution to the abovementioned flow and fibre distribution problem can severely differ with only a slight change in a single parameter for a given set of processing parameters. Therefore, the influence of some of them during the fibre impregnation process was evaluated, allowing a shortcut for the polymer through a gap between fibres and the bottom wall of the extrusion die. The range of investigated values regarding the gap enables one to cover good impregnation conditions up to the occurrence of the shortcut and consequent poor impregnation quality. These studies were performed with numerical simulations with circa 126,000 degrees of freedom, considering the discretisation mesh elements and the unknowns (pressure and two velocity components).

Análise da influência da malha numérica na modelagem hidrodinâmica do lago Paranoá utilizando o MIKE 3

Purpose: This work aims to evaluate the effects of the different meshes constructed in MIKE 3 software on the simulation and calibration results of the model. Theoretical framework: 3D hydrodynamic models, such as MIKE 3, provide the closest representation of reality by simulating the gradients in the three spatial dimensions and solutioning the Navier-Stokes equations. In these models, meshes are used to represent complex geometries. An efficient computational mesh is required to allow convergence and stability of the solution of the equations and, furthermore, of the modelling result. Method/design/approach: Simulation of four meshes with distinct discretization, calibration, comparison, and assessment of the model performance for these four conceptual models considering: mesh’s number of elements, simulation time, mean absolute error (MAE), coefficient of determination (R2), and relative difference. Results and conclusions: For the meshes adopted for comparison, refinement only in the “throat” (region near the dam) did not show significant influences on the results that would justify its use, considering the high computational cost. Therefore, in this case, a sparse mesh and without refinement can be used in detriment of a mesh with refinement only in the “throat”. Research implication: Understand how different meshes discretization can significantly alter simulation time and highlight that optimized simulation requires an equilibrium between simulation time and mesh discretization to maintain model’s performance. Originality/value: Understanding and quantifying the influence of the discretization of the model's mesh on the simulation time and the performance of the model allows the optimization of the modeling, considering the cost-effectiveness of different discretizations leading to smaller simulation time with similar performance.

Infrared BRDF measurement based on projection reconstruction with attenuated aperture filter effects

The Bidirectional Reflectance Distribution Function (BRDF) is a basic function that describes the spatial reflectivity of an opaque surface. Existing BRDF measurements are related to the size of the detector aperture because the values are not the true BRDF but an average over the aperture, and high spatial resolution and high SNR cannot be achieved simultaneously. In addition, the detection positions of the aperture cannot perfectly cover the entire hemisphere, and the measurement is not complete. In this paper, BRDF measurements were combined with projection reconstruction, and the mapping between the detected signals and the BRDF over a discrete hemispherical mesh was constructed. The spatial resolution of the reconstruction was the same as that of the discrete mesh, rather than the aperture size, and the aperture filter effect was attenuated. Simulations and a frosted gold plate measurement experiment were performed to verify the effectiveness of the proposed reconstruction method.

A parallel geometric multigrid method for adaptive topology optimization

This work presents an efficient parallel geometric multigrid (GMG) implementation for preconditioning Krylov subspace methods solving differential equations using non-conforming meshes for discretization. The approach does not constrain such meshes to the typical multiscale grids used by Cartesian hierarchical grid methods, such as octree-based approaches. It calculates the restriction and interpolation operators for grid transferring between the non-conforming hierarchical meshes of the cycle scheme. Using non-Cartesian grids in topology optimization, we reduce the mesh size discretizing only the design domain and keeping the geometry of boundaries in the final design. We validate the GMG method operating on non-conforming meshes using an adaptive density-based topology optimization method, which coarsens the finite elements dynamically following a weak material estimation criterion. The GMG method requires the generation of the hierarchical non-conforming meshes dynamically from the one used by the adaptive topology optimization to analyze to the one coarsening all the mesh elements until the coarsest level of the mesh hierarchy. We evaluate the performance of the adaptive topology optimization using the GMG preconditioner operating on non-conforming meshes using topology optimization on a fine-conforming mesh as the reference. We also test the strong and weak scaling of the parallel GMG preconditioner with two three-dimensional topology optimization problems using adaptivity, showing the computational advantages of the proposed method.