Fine Grid Size Research Articles

Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods

Abstract In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size H H and the fine grid size h h satisfy h = O ( H ( 2 k + 1 ) ⁄ ( k + 1 ) ) h={\mathcal{O}}\left({H}^{\left(2k+1)/\left(k+1)}) ( k ≥ 1 k\ge 1 ), where k k is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.

An Efficient of Overlapping Grid Method with Scattering Technique in Time Domain for Numerical Modeling

An Overlapping Grid Method (OGM) with Biquadratic Spline Interpolation in scattering technique was developed to solve the direct and inverse scattering issues. A two-dimensional (2D) numerical image model was used to analyze the accuracy of the proposed method in a direct scattering process. It was discovered that when the sub-grid, sx ∆ increased, the absolute error for the electric field amplitude will also increase. The results also discovered that as the grid size ratio increased, the absolute error of the amplitude Z E will also increase. The findings show that smaller grid spacing and a finer grid size can produce more accurate results. The Overlapping Grid Method (OGM) with Biquadratic Spline Interpolation was expanded by incorporating with Forward-Backward Time Stepping (FBTS) technique to solve inverse scattering issues. Homogenous embedded objects with a square and circular shape are used to validate the efficiency of the proposed method. The findings showed that the proposed numerical method could detect and reconstruct embedded objects in different shapes. The efficiency of the proposed method was examined by Mean Square Error (MSE) and normalizing the functional error. The findings revealed that the MSE of dielectric profiles for the proposed method were lower than the FDTD method in FBTS. The relative permittivity and conductivity profile differed by 27.06% and 20%, respectively. Hence, it was proven that the proposed method successfully solved a known drawback to the FDTD method and produced more accurate and efficient results.

High-Fidelity Energy Deposition Ignition Model Coupled With Flame Propagation Models at Engine-Like Flow Conditions

Abstract With the heightened pressure on car manufacturers to increase the efficiency and reduce the carbon emissions of their fleets, more challenging engine operation has become a viable option. Highly dilute, boosted, and stratified charge, among others, promise engine efficiency gains and emissions reductions. At such demanding engine conditions, the spark-ignition process is a key factor for the flame initiation propagation and the combustion event. From a computational standpoint, there exist multiple spark-ignition models that perform well under conventional conditions but are not truly predictive under strenuous engine operation modes, where the underlying physics needs to be expanded. In this paper, a hybrid Lagrangian–Eulerian spark-ignition (LESI) model is coupled with different turbulence models, grid sizes, and combustion models. The ignition model, previously developed, relies on coupling Eulerian energy deposition with a Lagrangian particle evolution of the spark channel, at every time-step. The spark channel is attached to the electrodes and allowed to elongate at a speed derived from the flow velocity. The LESI model is used to simulate spark ignition in a nonquiescent crossflow environment at engine-like conditions, using converge commercial computational fluid dynamics (CFD) solver. The results highlight the consistency, robustness, and versatility of the model in a range of engine-like setups, from typical with Reynolds-averaged Navier–Stokes (RANS) and a larger grid size to high fidelity with large-eddy simulation (LES) and a finer grid size. The flame kernel growth is then evaluated against Schlieren images from an optical constant volume ignition chamber with a focus on the performance of flame propagation models, such as G-equation and thickened flame model, versus the baseline well-stirred reactor model. Finally, future development details are discussed.

Enhancing acid fracture design in carbonate formation using a dynamic up-scaling procedure to convert discrete fracture network to dual continuum

For a low-permeability carbonate formation, the acid fracture process is simulated through coupling a commercial acid fracture simulator (GOHFER) to a finite volume reservoir simulator (IMEX). Unlike LGR (Local grid refinement) approach that suffers from severe convergence problems, a dynamic up-scaling procedure is employed to convert the discrete fracture network (DFN) model into a dual continuum model for our simulation. In this paper, multiple simulations are used to optimize the acid fracture schedule parameters, such as fluid volume, flow rate, perforation location, number of injection steps, and acid type, in order to maximize the effective fracture length. For four points perforation as the relevant operation case in this work, acid fracturing consumes 44 Mgal acid volume at a rate of 25 bpm at a 20 ft interval, resulting in a 3.5-fold increase in well productivity index (PI). Based on experimental relative permeability data, two models of fine and medium grid sizes are employed to be compared with the LGR method. The results have shown that these models are 22 and 54 times faster than LGR, however, the latter has a deviation of less than 5% which is quite acceptable.

An accuracy comparison of piecewise linear reconstruction techniques for the characteristic finite volume method for two‐dimensional convection‐diffusion equation

AbstractIn this paper, we apply the characteristic finite volume method (CFVM) for solving a convection‐diffusion problem on two‐dimensional triangular grids. The finite volume method is used to discretize the equation while the finite element method is applied to estimate the gradient quantities at cell faces. The numerical analysis of the convergence has been implemented for the CFVM in one‐dimension. The approximate L2 norm of the error is derived to determine the errors for the approximate solution. The accuracy of four piecewise linear reconstruction techniques, namely, Frink, Holmes‐Connell, Green‐Gauss, and least‐squares methods are investigated on structured triangular grids. Numerical evidence shows that the least‐squares method is the most accurate of all methods for smooth initial condition problems. For discontinuous initial condition problems, the Frink and the Holmes‐Connell methods give a spurious oscillating solution in the vicinity of the discontinuity upstream of the discontinuity, and the Green‐Gauss and least‐squares methods give a spurious oscillating solution in the vicinity of the discontinuity downstream of the discontinuity. Moreover, the amplitude of the oscillation could be amplified on the finer grid sizes.

Quality control of computational fluid dynamics (CFD) model of ozone reaction with human surface: Effects of mesh size and turbulence model

Indoor chemical reactions with human surfaces can meaningfully affect indoor pollutant dynamics and human exposure to reactants and products. Computational Fluid Dynamics models have been used to simulate ozone-human surface reactions in indoor environments. However, the model uncertainties due to near-human grid size and turbulence model are not well understood. The objective of this study is to assess performance of turbulence models and near-human surface grids in predicting local airflow and ozone transport near the human surface under a representative range of indoor airflow conditions. The results show that Large Eddy Simulation (LES) with a fine near-human grid size (e.g., y+ = 1) can yield reasonable results of airflow and mass transfer near the body. The SST k–ω model performs well for the simulation of near-human airspeed, whereas the standard k–ε model can predict lower airspeeds above the human head than the measurements. Both standard k–ε and SST k–ω models may underestimate the turbulent kinetic energy of the thermal plume and ozone deposition velocity at the human surface. The results also suggest that near-human grid sizes of y+ >10 can lead to unreasonable estimates of airflow and mass transfer rate near the human surface.

Evaluation of Rainfall Forecasts by Three Mesoscale Models during the Mei-Yu Season of 2008 in Taiwan. Part III: Application of an Object-Oriented Verification Method

In this study, the performances of Mei-yu (May–June) quantitative precipitation forecasts (QPFs) in Taiwan by three mesoscale models: the Cloud-Resolving Storm Simulator (CReSS), the Central Weather Bureau (CWB) Weather Research and Forecasting (WRF), and the CWB Non-hydrostatic Forecast System (NFS) are explored and compared using an newly-developed object-oriented verification method, with particular focus on the various properties or attributes of rainfall objects identified. Against a merged dataset from ~400 rain gauges in Taiwan and the Tropical Rainfall Measuring Mission (TRMM) data in the 2008 season, the object-based analysis is carried out to complement the subjective analysis in a parallel study. The Mei-yu QPF skill is seen to vary with different aspects of rainfall objects among the three models. The CReSS model has a total rainfall production closest to the observation but a large number of smaller objects, resulting in more frequent and concentrated rainfall. In contrast, both WRF and NFS tend to under-forecast the number of objects and total rainfall, but with a higher proportion of bigger objects. Location errors inferred from object centroid locations appear in all three models, as CReSS, NFS, and WRF exhibit a tendency to simulate objects slightly south, east, and northwest with respect to the observation. Most rainfall objects are aligned close to an E–W direction in CReSS, in best agreement with the observation, but many towards the NE–SW direction in both WRF and NFS. For each model, the objects are matched with the observed ones, and the results of the matched pairs are also discussed. Overall, though preliminarily, the CReSS model, with a finer grid size, emerges as best performing model for Mei-yu QPFs.

New Modeling and Simulation Techniques Optimize Completion Design and Well Spacing

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 194367, “Optimize Completion Design and Well Spacing With the Latest Complex Fracture Modeling and Reservoir-Simulation Technologies—A Permian Basin Case Study With Seven Wells,” by Hongjie Xiong, SPE, and Songxia Liu, University Lands, and Feng Feng, SPE, Texas A&M University, et al., prepared for the 2019 SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, 5–7 February. The paper has not been peer reviewed. Proper lateral and vertical well spacing is critical for efficient development of unconventional reservoirs. Much research has focused on lateral well spacing but little on vertical spacing, which is challenging for stacked-bench plays such as the Permian Basin. Following a previous, successful single-well study in paper SPE 189855, the authors have performed a seven-well case study in which the latest complex fracture modeling and reservoir-simulation technologies have been applied. This synopsis will concentrate on the methodology behind the study; the reader is encouraged to view the complete paper for specific comparisons of completion designs. Introduction Complex-fracture-modeling tools are used frequently to study well spacing. Most research has focused on lateral well spacing rather than on vertical spacing, though the industry has seen many fracturing hits and hydraulic communications between wellbores placed vertically in stacked plays. Field pilot tests have been used extensively to test and optimize lateral and vertical well spacings and to optimize well-completion designs. These pilot tests, however, take considerable time to implement and are very expensive. In this multiple-well study, the authors used an established work flow to study the fracture interaction between wellbores and lateral and vertical well spacings. A calibrated model was then used to optimize well-completion designs for the Wolfcamp formation. Work Flow The work flow to build and calibrate the complex-fracture-network and reservoir-performance-simulation models, and then use the models to conduct sensitivity analysis on the fracture networks, resulted from different completion designs and corresponding well performance. Because some critical data sets are spread throughout a wide areA&Mdash;including core data, comprehensive well-logging data, and geomechanical properties—the authors first built a regional geological model, then sliced the sector models from the regional model. This enabled use of available data that were spread sparsely across the area. On the basis of the well locations, a sector model was built by cutting a section from the regional geological model. In the process, the properties were repopulated with a finer grid size. Previous studies discussed using the seismic data to build a discrete fracture network (DFN). For the authors of this paper, however, the DFN represents not only the natural fracture networks but also includes geological bedding or layering information. Those beddings or layerings may be mechanical-weakness connections, which influence hydraulic-fracturing propagations. The detailed well-completion information was then plugged in. Seven-Well Case-History-Modeling Setup Seven wells drilled and completed in four different zones within the Wolfcamp formation were studied. A sector model was cut from the full-field (regional) model on the basis of the well locations, considering the possible fracture-propagation-modeling needs. The sector model covers a length of 12,600 ft, a width of 4,200 ft, and a thickness of 3,000 ft. In the process, a much finer grid size was used (refined from 300×300 ft in the regional model to 33×33 ft in the sector model) with consideration of perforation cluster spacings. Then, all formation properties were repopulated to the smaller grid cells with the same property models used in the regional model. The wellbores and perforations then were input into the sector model.

Use of Precise Area Fraction Model for Fine Grid DEM Simulation of ICFB with Large Particles

The heterogeneous structures in a gas–solid fluidized bed can be resolved in discrete element simulation so long as the grid is fine enough. In order to conveniently calculate mean porosity in fine grid simulations, a precise area fraction model is given for two-dimensional simulations. The proposed area fraction model is validated by the discrete element simulation test on a small-scale internal circulation fluidized system of large particles, using a fine grid size of two particle diameters. Simulations show that the discrete element method can perform well in modelling time-varying waveforms for the physical quantities in an internal circulating fluidized bed, employing the precise gas area fraction model. This thought of precise calculation can be generalized to construct a volume fraction porosity model for three-dimensional simulation by use of the similar symmetry of a rectangular grid. Moreover, to construct these area and volume fraction models is to enrich and perfect the underlying model of fine grid simulation.

Revisiting tin melting for phase change model verification

Model verification is necessary before numerical models can be applied to produce meaningful results. For solid-liquid phase change modelling involving convection, pure gallium and tin melting have been widely used as reference for verification. It was later found that contrasting observations have been reported on the flow structure of both metals in the liquid region during the phase change process. Some researchers have reported monocellular while others reported multicellular structures in past works. In this work, tin melting problem was revisited by extending the results to flow structure visualization with Line Integral Convolution (LIC) plots to confirm the flow structure for tin melting thus pure metals in general. Enthalpy-porosity formulation coupled with Finite-Volume Method (FVM) was used to solve the set of governing equations which represented the problem at Prandtl Number = 0.02, Stefan Number = 0.01 and Rayleigh Number = 2.5 x 105. The location of solid-liquid interface and LIC plots at different times were presented. At initial state, the solid-liquid interface was closely similar for all grid sizes but as time progresses, finer grids provided improved solutions as expected. Reasonable fine grid size must be selected for solid-liquid phase change models to ensure complete physics of the problems are captured and eventually yield acceptable numerical results. The LIC plots confirmed that the flow structure is multicellular. Future phase change models referring to pure metal melting problem for verification should obtain similar flow structure to be considered acceptable.

A high‐order compact scheme for solving the 2D steady incompressible Navier‐Stokes equations in general curvilinear coordinates

SummaryIn this paper, a high‐order compact finite difference algorithm is established for the stream function‐velocity formulation of the two‐dimensional steady incompressible Navier‐Stokes equations in general curvilinear coordinates. Different from the previous work, not only the stream function and its first‐order partial derivatives but also the second‐order mixed partial derivative is treated as unknown variable in this work. Numerical examples, including a test problem with an analytical solution, three types of lid‐driven cavity flow problems with unusual shapes and steady flow past a circular cylinder as well as an elliptic cylinder with angle of attack, are solved numerically by the newly proposed scheme. For two types of the lid‐driven trapezoidal cavity flow, we provide the detailed data using the fine grid sizes, which can be considered the benchmark solutions. The results obtained prove that the present numerical method has the ability to solve the incompressible flow for complex geometry in engineering applications, especially by using a nonorthogonal coordinate transformation, with high accuracy.

Precipitation Characteristics in the Community Atmosphere Model and Their Dependence on Model Physics and Resolution

AbstractPrecipitation amount (A), frequency (F), intensity (I), and duration (D) are important properties of precipitation, but their estimates are sensitive to data resolution. This study investigates this resolution dependence, and the influences of different model physics, by analyzing simulations by the Community Atmospheric Model (CAM) version 4 (CAM4) and version 5 (CAM5) with varying grid sizes from ~0.25 to 2.0°. Results show that both CAM4 and CAM5 greatly overestimate F and D but underestimate I at all resolutions, despite realistic A. These biases partly result from too much parameterized (convective) precipitation with high F and D but low I. Different cloud microphysics schemes contribute to the precipitation differences between CAM4 and CAM5. The A, F, I, and D of convective and nonconvective precipitation react differently to grid‐size decreases, leading to the large decreases in F and D but increases in the I for total precipitation as model resolution increases. This resolution dependence results from the increased probability of precipitation over a larger area (area aggregation effect, which is smaller than in observations) and the varying performance of model physics under changing resolution (model adjustment effect), which roughly enhances the aggregation‐induced dependence. Finer grid sizes not only increase resolved precipitation, which has higher intensity and thus improves overall precipitation intensity in CAM, but also reduce the area aggregation effect. Thus, the long‐standing drizzling problem in climate models may be mitigated by increasing model resolution and modifying model physics to suppress parameterized convective precipitation and enhance resolved nonconvective precipitation.

How does spatial resolution affect model performance? A case for ensemble approaches for marine benthic mesophotic communities

AbstractAimTo investigate how changing grid size can alter model predictions of the distribution of mesophotic taxa and how it affects different modelling methods.LocationNingaloo Marine Park, Western Australia.TaxonBenthic mesophotic taxa: corals, macroalgae and sponges.MethodsWe determined the distributions of the major benthic taxonomic groups: corals, macroalgae and sponges, using a number of modelling techniques and an ensemble using the ‘sdm’ R package. A range of grid sizes were used (10, 50, 100 and 250 m) to identify how model predictions were altered. Models were evaluated using the area under the curve of a receiver operator characteristic plot (AUC) and the true skill statistic (TSS) using a spatially independent dataset.ResultsGrid size had a large effect on model performance across the taxonomic groups. Model outputs were compared to null surfaces and 88.8% of models performed significantly better than null. Distribution of corals was best predicted using the finest grid size (10 m) regardless of modelling method, although a model ensemble produced the best results (AUC = 0.80, TSS = 0.52). Macroalgae and sponges were better predicted at coaster grids sizes (250 m). Again, ensembles performed well for both macroalgae (AUC = 0.83, TSS = 0.63) and sponges (AUC = 0.88, TSS = 0.66). Model ensembles maintained high accuracy across grid sizes and were consistently the best, or second‐best, performing method.Main conclusionsThis study has shown how grid size should be considered when producing distribution models. Identifying the most relevant grid size and being aware of the influence it may have will provide more accurate predictions of the distributions of taxa. Ensemble methods maintained good performance across scenarios and thus provide a useful tool for conservation and management especially where single modelling methods showed high levels of variability.

Analytical solution for upscaling hydraulic conductivity in anisotropic heterogeneous formations

Modern geological modeling techniques represent anisotropic heterogeneous formations by high-resolution grids, which can be computationally prohibitive. This motivates the upscaling process that scales-up properties defined at a fine-scale system to equivalent properties defined at a coarse-scale system. In general, analytical methods are very efficient but limited to assumptions and approximations, whereas numerical methods are more robust albeit more time-consuming.In this work, we developed an analytical method to approximate numerical solutions in a finite difference scheme with periodic boundary conditions for two-dimensional problem. Using perturbation expansion techniques and Fourier analysis, the method generates explicit formulas of tensorial equivalent conductivity considering heterogeneity and anisotropy of two-dimensional space, as well as geometry of gridblocks. It is applicable for various cases with different covariance/variagram models and a wide range of log-conductivity variances, correlation lengths, rotation angles, anisotropy ratios of fine grid conductivity, anisotropy ratios of fine grid size, and the number of fine gridblocks in a coarse gridblock. The analytical method matched well with the numerical method for the estimation of the conductivity tensor, hydraulic head, and discharge velocity. The coefficients in the analytical method need to be computed only once for any given statistics, which makes the proposed method much more efficient than the numerical method.

Two-grid economical algorithms for parabolic integro-differential equations with nonlinear memory

In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization. It is shown that these algorithms are as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size H and the fine grid size h satisfy H=O(hr−1r). Especially for PIDEs with nonlinear memory defined by a lower order nonlinear operator, our two-grid algorithm can save significant storage and computing time. Numerical experiments are given to confirm the theoretical results.

Transformation Optics-Based Finite Difference Time Domain Algorithm for Scattering From Object With Thin Dielectric Coating

A proper dielectric coating can reduce the electromagnetic scattering of the conducting object significantly so that it cannot be detected by the radar. However, when the object contains a thin coating, fine grid size is needed to discretize the thin coating in the conventional finite-difference time-domain (FDTD) algorithm, which increases the amount of memory and computational time significantly. To overcome this dilemma, we present a transformation optics-based FDTD (TO-FDTD) algorithm to accelerate the solution of electromagnetic scattering from objects with thin dielectric coatings. Two kinds of novel TO-FDTD models are proposed in this paper for a coated cylinder and a coated arbitrary polygonal cylinder, respectively. Through coordinate transformation, the size of the object remains unchanged while its thin coating is enlarged to a thicker one, meaning that it can be simulated by the FDTD algorithm with uniform coarse grids instead of fine grids. The transformed material parameters become inhomogeneous and anisotropic in the transformed region, which can be obtained by solving a Jacobian transformation matrix. We then develop a stable FDTD algorithm for solving anisotropic Maxwell’s equations. Bistatic scatterings of coated cylinders and a coated polygonal cylinder are solved by the TO-FDTD algorithm proposed in this paper, respectively. The result of the TO-FDTD algorithm matches well with the exact value and the result of the commercial software Comsol. The computational efficiency and accuracy of the proposed TO-FDTD algorithm are validated by numerical experiments. Numerical results show that the TO-FDTD algorithm has higher computational accuracy than the conventional FDTD algorithm that fails to simulate the absorbing property of the coating, when the same coarse grid size is used in the simulation. Under the same level of accuracy, the proposed TO-FDTD method can improve the computational efficiency by 62-63 times than the conventional FDTD method with fine grids in the simulations in the paper.

A compatible boundary element approach with geologic modeling techniques to model transient fluid flow in heterogeneous systems

Multi-scale and arbitrarily shaped heterogeneous systems based on geological interpretation of structures, facies, and architectures in a piece-wise manner of variation are solved by a boundary element method (BEM) with systematic partition and coupling strategies. This technique extends the BEM to be compatible with geological models to obtain highly accurate pressure and production transients of multiple wells without finer grid sizes or smaller time steps. Comparisons to analytical solutions, existed BEM approaches, and semianalytical methods have validated this work in aspects of subdividing strategies and modeling accuracy. In this approach, two options are available to divide the reservoir into locally homogeneous subsystems: first is to partition along existed interface boundaries between different geo-bodies or structures and thus result in multiple irregular subsystems; the second is to subdivide the domain into identical rectangular subsystems and a few irregular subsystems as needed, and assign heterogeneity like finite difference or element simulators do. Each subsystem with uniformly average properties is then formulated in integral equations and systematically coupled with adjacent subsystems along interface boundaries to obtain solutions. Examples of calculating pressure and production transients of a fluvial system demonstrate its compatibility with both object and pixel based geologic modeling techniques. Pressure derivatives are very sensitive to geological facies with different rock properties. This work shows promises to develop BEM-based simulators.

Exploring the Locally Low Dimensional Structure in Solving Random Elliptic PDEs

We propose a stochastic multiscale finite element method (StoMsFEM) to solve random elliptic partial differential equations with a high stochastic dimension. The key idea is to simultaneously upscale the stochastic solutions in the physical space for all random samples and explore the low stochastic dimensions of the stochastic solution within each local patch. We propose two effective methods for achieving this simultaneous local upscaling. The first method is a high-order interpolation method in the stochastic space that explores the high regularity of the local upscaled quantities with respect to the random variables. The second method is a reduced-order method that explores the low rank property of the multiscale basis functions within each coarse grid patch. Our complexity analysis shows that, compared with the standard FEM on a fine grid, the StoMsFEM can achieve computational savings on the order of $(H/h)^{d}/(\log(H/h))^k$, where $H/h$ is the ratio between the coarse and the fine grid sizes, $d$ ...

SU-F-T-331: The Effect of Optimization Grid Size On Helical Tomotherapy for Nasopharyngeal Carcinama

Purpose: To investigate the effect of plan optimization grid size on helical tomotherapy for nasopharyngeal carcinoma (NPC) Methods: Ten patients of NPC were enrolled. For each patient, three plans with different dose optimization grid size (fine, normal, coarse) were created by Helical Tomotherapy Planning Station (version: 5.0, GPU Acceleration and VoLo).The time consumed by 500 optimization iterations, the monitor units and the dose volume histogram factors were compared. Paired-Wilcoxon test was used in statistical analysis. Results: The iterations time, monitor units and plan quality have significant difference from various optimization grid size: the mean iterations time of fine optimization grid size(138.89min) is 3.02 times, 6.29times to normal (45.99min) and coarse (22.11min) respectively; The monitor units of fine(6811.90MU) is 1.25% (6896.90MU, p=0.285) and 6.28% (7239.90MU, p=0.005)lower than normal and coarse; the mean homogeneity index (HI) of fine is 9.32% and 68.25% better than normal (p=0.075) and coarse (p=0.005); the conformity index (CI) of fine is 5.86% and 16.90% better than normal(p=0.005) and coarse (p=0.005). Compared to normal and coarse, fine optimization grid size reduced D2%,D98% of PTV by 0.47Gy, 3.43Gy and 0.11Gy, 0.23Gy respectively. Larger grid size caused a mean dose increase to the structure of parotids. There are no significant difference between normal and fine for the maximum dose of spinal cord, brainstem and optic nerves, but both fine and normal have obvious advantages than coarse. Three optimization grid size have no significant difference on the maximum dose of lens and eyes. Conclusion: In NPC helical tomotherapy planning, larger optimization grid size required higher MUs to achieve the same target coverage. Fine optimization grid size is recommended while normal will improve calculation efficiency. Considering the disadvantage of coarse optimization grid size, it is not recommended to be used.

A Newton type linearization based two grid method for coupling fluid flow with porous media flow

In this paper, we propose a two-grid finite element method for solving the mixed Navier–Stokes/Darcy model with the Beavers–Joseph–Saffman interface condition. After solving a coupled nonlinear problem on a coarse grid, we sequentially solve decoupled and linearized subproblems on a fine grid and then correct the solution on the same grid. Compared with the existing work on the two-grid methods for the coupled model, our two-grid method allows a much higher order scaling between the coarse grid size H and the fine grid size h. Specifically, if a k-th order discretization is applied, by using h=H2k+1k for k=1,2 and h=Hk+1k−1 for k≥3, the final step two-grid solution errors in the energy norm are still optimal. Numerical experiments are also given to confirm the theoretical analysis.