Grid Mesh Size Research Articles

On the phase field based model for the crystalline transition and nucleation within the Lagrange multiplier framework

Understanding the complexity of the nucleation and transition between the crystalline and quasicrystalline is significant because the structural incommensurability is anisotropic and of significance in revealing material properties. This paper reports the two- and three-dimensional nucleation and transition investigation from quasicrystals to crystals using a phase field method. The investigation starts with the Lifshitz–Petrich (LP) model, which is derived from the Landau theory for the exploration of critical nuclei. By the variational derivation, we construct two phase field models with tenth- and eighth-order, which provide the possibility of exploring the transition under stable phase states. In order to dissipate the original Lifshitz–Petrich energy, we apply a Lagrange multiplier method to modify the two models and solve them by the Fourier spectral method. Whereas the nonlinearity leads to expensive computational burden and extra stiffness, the designed algorithm can effectively avoid the numerical oscillations caused by rigidity and keep an O(Nlog⁡N) computational complexity, where N is the mesh grid size. To further demonstrate the robustness and advantages of the proposed method for handling phase-field modeling of crystalline structures, we compare its performance with other methods for constructing unconditionally stable methods. Our method can be directly implemented on a GPU for acceleration and achieves multiple times faster performance compared to CPU-only alternatives. Various numerical tests have been used to validate that our method works well for revealing the transition between different stable states during the nucleation process.

Cyclic behavior of existing RC bridge piers strengthened by exterior ECC and BFRP/ECC composite jacketing at pier base

Engineered cementitious composites (ECC) are a class of high-performance fiber-reinforced cementitious composites. This study proposes using ECC and BFRP grid-reinforced ECC (BFRP/ECC) as external jackets at the base of existing RC bridge piers to enhance seismic performance. Seven 1/5-scale RC bridge piers were tested under reversed cyclic loading, including one unmodified pier as reference and six reinforced piers with varying jacket thickness, height, interface, and BFRP grid mesh size. The test results demonstrate that both ECC and BFRP/ECC composite jacketing effectively enhance ductility. The hysteretic response of the reinforced piers with different interfacial properties, however, varied significantly. The piers with interfacial shear keys exhibited enhanced flexural strength, ductility, and energy dissipation capacity but sustained severe damage as jacketing thickness and height increased. Conversely, the piers with smooth interfaces showed insignificant improvements in flexural strength but significant enhancement in ductility with marginal damage. This behavior can be attributed to the fact that the exterior jacketing in specimens with interfacial shear keys was under bi-axial tension-compression stress state due to flexural compression and lateral expansion of the plastic hinge, while the jacket in specimens with smooth interfaces acted solely as lateral confinement.

Mesh size effect on finite source inversion with 3-D finite-element modelling

SUMMARY Three-dimensional finite-element models, which can handle the stress perturbations caused by subsurface mechanical heterogeneities and fault interactions, have been combined with the finite source inversion to estimate the coseismic slip distribution over the fault plane. However, the mesh grid for discretizing the governing equations in the finite-element model significantly affects the numerical accuracy. In this study, we performed kinematic finite source inversion with idealized (regular observation point array; M1A–M1D) and regional (GEONET, GPS Earthquake Observation Network System stations in Japan; M2A–M2H) models with different mesh sizes to quantitatively analyse the effect of the mesh grid size around the fault plane on the inverted fault slip distribution. Synthetic observation data vectors obtained from the finest models (M1A and M2A) are compared with those from the coarser models (M1B–M1D and M2B–M2H), which were adopted to construct Green's function matrix. We found that the coarser mesh models derived a smaller surface displacement, leading to a decrease in the norm of Green's function matrix, which in turn influences the fault slip magnitude from the finite source inversion. Finally, we performed the source inversion for the fault slip distribution of the 2011 Mw 9.0 Tohoku–Oki earthquake using the coseismic surface displacements recorded at the GEONET and seafloor stations and finite-element modelling. By reducing the mesh size on the fault, we confirmed that the estimated magnitude of fault slip converged to approximately 80 m, which is consistent with the range of fault slip amounts from previous studies based on the Okada model. At least 0.88 million total domain elements and a 6.7 km2 mesh size on the fault plane with an area of 240 × 720 km2 are required for the convergence of the fault slip. Furthermore, we found that the location of the maximum fault slip is less sensitive to the mesh size, implying that source inversion based on a coarse mesh model (i.e. less than 0.5 million elements and > ∼60 km2 mesh size) can quickly provide the rough fault slip distribution.

Two-grid finite element method with an H2N2 interpolation for two-dimensional nonlinear fractional multi-term mixed sub-diffusion and diffusion wave equation

<abstract><p>In this paper, we studied the two-grid method (TGM) for two-dimensional nonlinear time fractional multi-term mixed sub-diffusion and diffusion wave equation. A fully discrete scheme with the quadratic Hermite and Newton interpolation (H2N2) method was considered in the temporal direction and the expanded finite element method is used to approximate the spatial direction. In order to reduce computational time, a dual grid method based on Newton iteration was constructed with order $ \alpha\in(0, 1) $ and $ \beta\in(1, 2) $. The global convergence order of the two-grid scheme reaches $ O(\tau^{3-\beta}+h^{r+1}+H^{2r+2}) $, where $ \tau $, $ H $ and $ h $ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. The error estimation and stability of the fully discrete scheme were derived. Theoretical analysis shows that the two grid algorithms maintain asymptotic optimal accuracy while saving computational costs. In addition, numerical experiments further confirmed the theoretical results.</p></abstract>

A simple experiment to study the array theorem using Fraunhofer diffraction of a two-dimensional grating

A transmission electron microscopy (TEM) grid comprises an array of identical square apertures. In this study, the Fraunhofer diffraction pattern of a TEM grid was obtained by a 532 nm wavelength diode laser. It was compared with the Fraunhofer diffraction pattern computed by using the array theorem and employed to estimate the mesh size of the TEM grid.

On the Importance of Model Selection for CFD Analysis of High Temperature Gas-Solid Reactive Flow; Case Study: Post Combustion Chamber of HIsarna Off-Gas System

In this paper a CFD analysis of HIsarna off-gas system for post combustion of CO-H2-carbon particle mixture is presented to evaluate the effect of different sub-models and parameters on the accuracy of predictions and simulation time. The effects of different mesh type, mesh grid size, radiation models, turbulent models, kinetic mechanism, turbulence chemistry interaction models, including and excluding gas-solid reactions, number of reactive solid particles are investigated in detail. Based on the accuracy of the predictions and agreement with counterpart measured values, the best combination is selected and conclusions are derived. It was found that radiation and turbulence chemistry interaction model have a major effect on the temperature and composition profile prediction along the studied off-gas system, compared to the variations in other models. The effect of these two models becomes even more evident when the temperature and fuel content of the flue gas are high.

Unsteady dissipation scaling in static- and active-grid turbulence

A new time-dependent analysis of the global and local fluctuating velocity signals in grid turbulence is conducted to assess the scaling laws for non-equilibrium turbulence. Experimental datasets of static- and active-grid turbulence with different Rossby numbers $R_o({=}U/\varOmega M$ : $U$ is the mean velocity, $\varOmega$ is the mean rotation rate and $M$ is the grid mesh size) are considered. Although the global (long-time-averaged) non-dimensional dissipation rate $C_\varepsilon$ is independent of the Reynolds number $Re_\lambda$ based on the global Taylor microscale, the local (short-time-averaged) non-dimensional dissipation rate $\left \langle C_\varepsilon (t_i) \right \rangle$ ( $t_i$ is the local time) both in the static- and active-grid turbulence clearly show the non-equilibrium scaling $\left \langle C_\varepsilon (t_i)\right \rangle / \sqrt {Re_0} \propto \left \langle Re_\lambda (t_i) \right \rangle ^{-1}$ ( $\left \langle Re_\lambda (t_i) \right \rangle$ and $Re_0$ are the Reynolds numbers based on the local Taylor microscale $\lambda (t_i)$ and the global integral length scale, respectively), which has only been confirmed for global statistics in the near field of grid turbulence. The local value of $\left \langle L(t_i) / \lambda (t_i) \right \rangle$ ( $L(t_i)$ is the local integral length scale) shifts from the equilibrium to non-equilibrium scaling as $\left \langle Re_\lambda (t_i) \right \rangle$ increases, further confirming that the non-equilibrium scalings are recovered for local statistics both in the static- and active-grid turbulence. The local values of $\left \langle C_\varepsilon (t_i) \right \rangle$ and $\left \langle L(t_i) / \lambda (t_i) \right \rangle$ follow the theoretical predictions for global statistics (Bos & Rubinstein, Phys. Rev. Fluids, vol. 2, 2017, 022601).

A direct integral pseudospectral method for solving a class of infinite-horizon optimal control problems using Gegenbauer polynomials and certain parametric maps

<abstract><p>We present a novel direct integral pseudospectral (PS) method (a direct IPS method) for solving a class of continuous-time infinite-horizon optimal control problems (IHOCs). The method transforms the IHOCs into finite-horizon optimal control problems in their integral forms by means of certain parametric mappings, which are then approximated by finite-dimensional nonlinear programming problems (NLPs) through rational collocations based on Gegenbauer polynomials and Gegenbauer-Gauss-Radau (GGR) points. The paper also analyzes the interplay between the parametric maps, barycentric rational collocations based on Gegenbauer polynomials and GGR points and the convergence properties of the collocated solutions for IHOCs. Some novel formulas for the construction of the rational interpolation weights and the GGR-based integration and differentiation matrices in barycentric-trigonometric forms are derived. A rigorous study on the error and convergence of the proposed method is presented. A stability analysis based on the Lebesgue constant for GGR-based rational interpolation is investigated. Two easy-to-implement pseudocodes of computational algorithms for computing the barycentric-trigonometric rational weights are described. Three illustrative test examples are presented to support the theoretical results. We show that the proposed collocation method leveraged with a fast and accurate NLP solver converges exponentially to near-optimal approximations for a coarse collocation mesh grid size. The paper also shows that typical direct spectral/PS and IPS methods based on classical Jacobi polynomials and certain parametric maps usually diverge as the number of collocation points grow large if the computations are carried out using floating-point arithmetic and the discretizations use a single mesh grid, regardless of whether they are of Gauss/Gauss-Radau type or equally spaced.</p></abstract>

Local-Scale Weather Forecasts over a Complex Terrain in an Early Warning Framework: Performance Analysis for the Val d’Agri (Southern Italy) Case Study

Forecasting applications based on hourly meteorological predictions for weather variables are nowadays used in energy market operations, planning of gas and power supply, and renewable energy, among others. Available meteorological and climatological data, as well as critical thresholds of rainfall, may also have a key role in the hazard classification, related to slope instabilities of pipelines and critical infrastructures along routes. The present study concerns the performance of a weather forecast model in the framework of an early warning system (EWS) application, which supports the integrity management of oil and gas pipelines. This EWS has been applied on to a specific area: the Val d’Agri basin in the Basilicata region of Southern Italy, which is extensively affected by several landslides and floods. The hourly precipitation forecasts are provided by a dedicated meteorological model, the KALM-HD, using two different horizontal resolutions, 1.25 and 5 km, to analyze possible influences of the mesh grid size as well. On this area, several weather stations were specifically deployed to obtain observed data in a region where hydrogeological hazards are relevant for asset management. A comparison among observations and the KALM-HD scaled forecasts on six of these weather stations is presented to assess the model performance. Besides, precipitation, temperature, and wind speed are evaluated as well. The forecasting analysis is performed considering two years of data both on an overall and seasonal basis. Results show that the KALM-HD performs well with the 1.25 km grid, particularly on temperature and wind speed variables. Since weather stations can be gathered in two main sets depending on their positions, differences arise in the forecast quality of these two groups, related to orography and thermal effects, whose detection is difficult in the typical narrow valleys characterizing the area of study. This issue prevalently influences temperatures and local winds, which, these latter, are generally underestimated, while precipitation is mainly driven by synoptic circulation and its interaction with mesoscale meteorological features.

Numerical Solution of 2nd Order Boundary Value Problems with Dirichlet, Neumann and Robin Boundary Conditions using FDM

In many fields of science and engineering, to determine the harmonic motion, damped and forced variation, current from electric circuit, 2nd order ODE is required to solve. Solving the ODE with complicated boundary condition that occur in engineering problems is a great challenges analytically. Therefore, numerical technique finite difference method (FDM) is very popular and important for solving the boundary value problems. In this article three different conditions as Dirichlet, Neumann and Robin (mixed) boundary conditions are applied in initial-boundary problem. FDM is used to solve ODE boundary value problems. Error calculation, stability, convergence are also explained. To test the accuracy numerical solutions are verified with analytical solution and error is calculated at each point for different mesh grid size as mesh grid size is decreased result will give the accuracy.

Interscale transfer of turbulent energy in grid-generated turbulence with low Reynolds numbers

Direct numerical simulations are performed for grid-generated turbulence with low Reynolds numbers to study the interscale transfer of turbulent energy. The Reynolds numbers based on the uniform velocity and grid mesh size, M, are set to ReM=5000, 9000, and 15000. The results show that, for ReM=9000 and ReM=15000, the turbulent dissipation constant increases in the upstream region but it becomes nearly constant in the region of x/M>15. In contrast, for ReM=5000, it continues to increase and do not level off even in the downstream region. Scale-by-scale analysis using the Karman–Howarth–Monin–Hill equation indicates that, the non-linear transfer term balances with the dissipation term for all the cases. Instead, it is found that, in the downstream region of the ReM=5000 case, where the turbulent Reynolds number becomes Reλ<25, the contribution of the advection term varies depending on the streamwise positions. Further analysis for the one-point statistics indicates that, in such conditions, the coherences among the advection term, dissipation term, and diffusion term become small at small scales, while they are generally large in the other cases including the upstream region for ReM=5000. These indicate that energy cascade mechanism in the downstream region for ReM=5000 is different from the cases of 30<Reλ<60, even though both of them are generally considered as small Reλ, and it is caused by the increase of the dissipation scale range appearing in the transition period to the final decay.

High-frequency estimation of the Lévy-driven Graph Ornstein-Uhlenbeck process

We consider the Graph Ornstein-Uhlenbeck (GrOU) process observed on a non-uniform discrete timegrid and introduce discretised maximum likelihood estimators with parameters specific to the whole graph or specific to each component of the graph. Under a high-frequency sampling scheme, we study the asymptotic behaviour of those estimators as the mesh size of the observation grid goes to zero. We prove two stable central limit theorems to the same distribution as in the continuously-observed case under both finite and infinite jump activity for the Lévy driving noise. In addition to providing the consistency of the estimators, the stable convergence allows us to consider probabilistic sparse inference procedures on the edges themselves when a graph structure is not explicitly available. It also preserves its asymptotic properties. In particular, we also show the asymptotic normality and consistency of an Adaptive Lasso scheme. We apply the new estimators to wind capacity factor measurements, i.e. the ratio between the wind power produced locally compared to its rated peak power, across fifty locations in Northern Spain and Portugal. We compare those estimators to the standard least squares estimator through a simulation study extending known univariate results across graph configurations, noise types and amplitudes.

Analysis of early-age behaviour of textile-reinforced cementitious matrix composites (TRC) using different measurements techniques

This study aims at investigating the early-age behaviour of textile-reinforced cementitious matrix composites. Six different configurations with two types of reinforcements (glass and carbon meshes) and three different reinforcement layers were tested. Experimental measurements were conducted using various measurement techniques: optical fibre sensors, thermocouples, thermal cameras, and Fourier transform infrared spectroscopy analysis. The different phases of expansion and shrinkage were monitored during the first 12 h and after 15 days from casting. The effects of air contact, textile, and mesh size of reinforcement grids were evaluated and discussed. Then, the effect of shrinkage on the mechanical tensile behaviour of these composites was identified and analysed. Moreover, a chemical analysis was performed to determine the influences of the constituents, the cement hydration process and related heat release on the early-age behaviour of these cementitious matrix composites.

Suppression of High-Frequency Electron Beam Modulation in a Plasma-Filled Diode

The article presents the results of recent experimental studies of spontaneous electron beam modulation with a frequency of several tens of megahertz in a plasma-filled diode. The diode comprises a plasma cathode with a grid-stabilized emission boundary and a plasma anode with an open (conditionally electrodeless) movable boundary. The cathode plasma is formed by a cathodic arc and the anode plasma by toroidal plasma generators based on an anode-layer closed drift thruster. According to our previous studies, the beam current starts to modulate as the arc current goes above 50–55 A at an accelerating voltage of 12–22 kV. Here, we analyze how such electron beam modulation and its frequency are influenced by the grid geometry at a current of about 100–110 A and what methods can be used for its removal. The idea of removal consists in decreasing the effect of the accelerating voltage on the cathode plasma, for example, via a negative current feedback. It is shown that the most efficient modulation suppression is attained when the negative feedback is frequency-dependent and that the modulation frequency decreases to its complete disappearance as the grid mesh size is decreased.

Analysis of influence of mesh partition on Mike21 calculation in flood impact assessment

In order to study the impact of different grid mesh sizes on the calculation results of the model in the analysis of flood impact assessment, this paper takes the newly constructed bridge between the Niushou and Xiangyang sections of the Hanjiang River as an example. It simulates the influences of bridge construction on river flow under the six schemes of setting the bridge pier as a solid boundary, no internal mesh generation, and not participating in numerical calculation, and dividing the bridge pier local mesh size into 2m, 4m, 6m, 8m, and 10m. The results show that different grid sizes in the project have an effect on the river water level and the range of water level changes. The smaller the local grid size of the project relative to the building size, the greater the maximum water level change value, and the closer it is to the calculated value of the bridge pier as a solid boundary. By adding piers module of the model to generalize the project, the water level change in the model results is significantly smaller. Under the same water level change conditions, the larger the mesh size, the greater the scope of the project. For projects located in important river sections with significant water blocking effects, it is recommended that buildings be used as solid boundaries without participating in generating meshes or modifying local terrain to simulate the effects of water flow. The results are more in line with actual conditions.

A three-dimensional slope stability analysis method based on finite element method stress analysis

A new three-dimensional slope stability analysis method based on stress calculation by the finite element method is proposed. This method conforms to the uniform assumption of the sliding direction of the rigid limit equilibrium method and can easily be used calculate the three-dimensional sliding direction. By using this method, both the local and global safety factors of the slip surface can be calculated according to the integral mean value theorem. Additionally, an improved ellipsoidal slip surface construction method is proposed for roughly searching for the minimum safety factor and the critical slip surface. A two-stage particle swarm optimization is used to search for the critical slip surface more accurately. Two numerical examples of fixed slip surfaces are provided to confirm the effectiveness of the proposed method and its applicability to nonspherical slip surfaces. The influence of the slip surface mesh grid size, boundary conditions, Poisson's ratio, and dilatancy angle on the calculation results is revealed. Another two numerical examples of searching for the critical slip surface are then provided, and the results of the proposed method are compared with the results of the strength reduction method to confirm the validity of the proposed method.

Numerical analysis of history-dependent variational-hemivariational inequalities

In this paper, numerical analysis is carried out for a class of history-dependent variational-hemivariational inequalities by arising in contact problems. Three different numerical treatments for temporal discretization are proposed to approximate the continuous model. Fixed-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size. A special temporal discretization is introduced for the history-dependent operator, leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time step-size. As for spatial approximation, the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity assumptions. Numerical examples are presented to illustrate the theoretical results.

Nonparametric estimation of the kernel function of symmetric stable moving average random functions

We estimate the kernel function of a symmetric alpha stable ( $$S\alpha S$$ ) moving average random function which is observed on a regular grid of points. The proposed estimator relies on the empirical normalized (smoothed) periodogram. It is shown to be weakly consistent for positive definite kernel functions, when the grid mesh size tends to zero and at the same time the observation horizon tends to infinity (high-frequency observations). A simulation study shows that the estimator performs well at finite sample sizes, when the integrator measure of the moving average random function is $$S\alpha S$$ and for some other infinitely divisible integrators.

Effects of Various Earth Grid Configurations on Ground Potential Rise Caused by Lightning Strike

Ground Potential Rise (GPR) caused by lightning strike is a potential hazard for electrical equipment inside an oil and gas refinery plant. In order to mitigate the risk, horizontal grounding grid is applied. The best mitigation is to install a grounding grid with mesh size as small as possible. This condition requires a high cost. In order to obtain the optimal mesh size, a series of simulation of a grounding grid with mesh size variations on GPR caused by lightning strike has been carried out. CDEGS software was used to observe the GPR with various mesh size from 6.5 x 6.5 m to 20 x 20 m. Simulation results show that the maximum transient GPR rises as the grounding grid mesh size is increased, while the GPR distribution throughout the grounding grid area does not change much for different mesh sizes. In the other hand, decreasing the grid size would mean that more conductors are required, hence the cost would increase accordingly. The result shows that grid sizes from 6.5 x 6.5 m up to 20 x 20 m have no significant difference in term of GPR. In term of cost, 10 x 10 m does not show significant difference with 20 x 20 m, on the other hand, there is a significant difference for grid sizes 1 x 1 m to 10 x 10 m. From the results, grid sizes between 10 x 10 m up to 20 x 20 m are still applicable as stated in Petronas Technical. To comply with proper GPR value, additional protection devices are needed to protect the electrical equipment from potential damage.Keywords –GPR, grounding grid, mesh size

Numerical simulation of the hydrodynamic behavior and the synchronistic oxidation and reduction in an internal circulation micro-electrolysis reactor

The filter packing of traditional iron–carbon micro-electrolysis reactors is prone to hardening and passivation during operation. In this study, internal loop airlift reactors were used to contain iron–carbon fillings in order to solve these problems and realize a synchronistic oxidation and reduction environment in an internal circulation micro-electrolysis (ICE) reactor. The hydrodynamics of the novel gas–liquid–solid three-phase reactor were investigated through a computational approach. A 2D and 3D transient Eulerian–Eulerian multiphase model was carried out with the standard k − ε turbulence model. Important design parameters of the ICE reactor were identified: the solid velocity, radial profiles of the axial gas holdup and water distribution. The effects of the mesh grid size sensitivity, superficial gas velocity, riser diameter (Dr), height–diameter ratio (H/D), draft tube axial height (Hd) and number of water distribution pipes were considered. The key observations were as follows. The minimum fluidization velocity was 1 m·s−1. With the increase in the number of distribution pipes, the distribution of water was more uniform, but the solid flow resistance increased. When Dr = 30 mm, H/D = 4:1, Hd = 90 mm, and the number of distribution pipes was 4, the hydrodynamic performance was observed to be better. The dissolved oxygen distribution was simulated based on the optimum structural parameters. The experimental results have the same trend as the simulation results, and synchronistic oxidation and reduction was verified in the ICE reactor. Therefore, the optimum structural design and operating parameters and the three-phase system theory were obtained through the CFD simulation, and these findings will provide a basis for the design and enlargement of the ICE reactor.