Mesh Ratio Research Articles

A class of quasi-variable mesh methods based on off-step discretization for the solution of non-linear fourth order ordinary differential equations with Dirichlet and Neumann boundary conditions

We propose a class of second and third order techniques based on off-step discretizations for a general non-linear ordinary differential equation of order four, subject to the Dirichlet and Neumann boundary conditions. Our approach uses only three grid points and involves the construction of a quasi-variable mesh. This type of a mesh is framed using a mesh ratio parameter $\eta>0$ whose value is chosen in accordance with the occurrence of boundary layer in the problem, and varies with the number of grid points taken. The third order technique reduces to a fourth order one when taken with $\eta=1$ . The stability and convergence analysis of the techniques are discussed over a model problem. Computational results obtained upon the application to seven linear as well as non-linear problems endorse the theoretically claimed accuracies. We also provide a comparison with the computational results using approaches of other authors, which shows that the proposed methods are better.

An exponential expanding meshes sequence and finite difference method adopted for two-dimensional elliptic equations

We demonstrate a new nonuniform mesh finite difference method to obtain accurate solutions for the elliptic partial differential equations in two dimensions with nonlinear first-order partial derivative terms. The method will be based on a geometric grid network area and included among the most stable differencing scheme in which the nine-point spatial finite differences are implemented, thus arriving at a compact formulation. In general, a third order of accuracy has been achieved and a fourth-order truncation error in the solution values will follow as a particular case. The efficiency of using geometric mesh ratio parameter has been shown with the help of illustrations. The convergence of the scheme has been established using the matrix analysis, and irreducibility is proved with the help of strongly connected characteristics of the iteration matrix. The difference scheme has been applied to test convection diffusion equation, steady state Burger’s equation, ocean model and a semi-linear elliptic equation. The computational results confirm the theoretical order and accuracy of the method.

Drag characterisation of prawn-trawl bodies

The drag of a prawn-trawl body is characterised by five design and three operational variables. The design variables comprise headline length, steepness of trawl side cut, width-to-depth mesh ratio of the trawl mouth (gape), vertical wing-end mesh count, and netting solidity; all of which effectively determine the planar twine-area of the trawl. The operational variables include towing velocity, horizontal spread, and vertical opening (headline height)—these determine the extent that the netting is exposed to relative water movement. The individual drag effects of the above variables (except for headline length, gape, and netting solidity) were systematically examined in a flume tank with prawn-trawl models built with low-stiffness full-scale netting; and the existing literature was consulted on the drag effects of gape, while drag was assumed to be proportional to twine diameter, mesh size−1 and headline length2. The developed equations in non-dimensional forms provide the basis for a drag-prediction model for a prawn trawl of any size, construction and operating conditions. Comparisons with previously published prediction equations showed considerable disagreement in some aspects, and suggest that using stiff, full-scale netting in past model experiments have produced significant model-to-full-scale prediction errors owing to the poor equivalence of twine bending-stiffness-to-netting-tension ratio.

A second-order parareal algorithm for fractional PDEs

We are concerned with using the parareal (parallel-in-time) algorithm for large scale ODEs system U ' ( t ) + A U ( t ) + d A α U ( t ) = F ( t ) arising frequently in semi-discretizing time-dependent PDEs with spatial fractional operators, where d 0 is a constant, α ? ( 0 , 1 ) and A is a spare and symmetric positive definite (SPD) matrix. The parareal algorithm is iterative and is characterized by two propagators F and G , which are respectively associated with small temporal mesh size Δt and large temporal mesh size ΔT. The two mesh sizes satisfy Δ T = J Δ t with J ? 2 being an integer, which is called mesh ratio. Let T unit f and T unit g be respectively the computational cost of the two propagators for moving forward one time step. Then, it is well understood that the speedup of the parareal algorithm, namely E , satisfies E = O ( c log ? ( 1 / ? ) ) , where c : = T unit f / T unit g and ? is the convergence factor. A larger E corresponds a more efficient parareal solver. For G = Backward-Euler and some choices of F , previous studies show that ? can be a satisfactory quantity. Particularly, for F = 2nd-order DIRK (diagonally implicit Runge-Kutta), it holds ? ? 1 3 for any choice of the mesh ratio J. In this paper, we continue to consider F = 2nd-order DIRK, but with a new choice for G , the IMEX (implicit-explicit) Euler method, where the 'implicit' and 'explicit' computation is respectively associated with A and d A α . Compared to the widely used Backward-Euler method, this choice on the one hand increases c (this point is apparent), and interestingly on the other hand it can also make the convergence factor ? smaller: ? ? 1 5 ! Numerical results are provided to support our conclusions.

Organic Acid Biogeneration by <i>Aspergillus niger </i> and its Utilization for Indirect Bioleaching of Nickel Laterite Ore

This paper discusses the results of indirect bioleaching test works of nickel laterite ore by the assistance of Aspergillus niger. Series of metabolic acid biogeneration shake-flask experiments were performed to investigate the effects of substrate type, nitrogen and phosphate dosages, the presence of magnesium and manganese salts, as well as aeration of the culture media on the effectiveness of the metabolic acid biogeneration. The investigation results showed that after 16 days of incubation, the solution pH of 1.44 was obtained from the acid biogeneration using cassava starch as carbon source in the presence of 0.1% (w/w) KH2PO4, 1% (w/w) (NH4)2SO4 and 0.25 g/L of MgSO4 under aerated condition. Leaching experiments of limonite and saprolite ore samples were carried out by using the generated metabolic acids at various ore particle size distributions, solid to liquid ratio and temperature. The highest nickel extraction percentage of 88.9% was obtained from the leaching of saprolite ore at 95°C, stirring speed of 400 rpm, particle size distribution of-80+100 mesh, solid to liquid ratio of 4.65 g/mL after 240 minutes, while that of 84.9% was obtained from the leaching of limonite ore at 95°C, stirring speed of 400 rpm, particle size distribution of-80 +100 mesh and solid-liquid ratio of 5.2 g/ml after 1440 minutes. The leaching of saprolite ore by using the biogenerated acids was selective to magnesium, with co-extracted Mg of only 1.5% after 24h of the agitation leaching test.

Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces

We prove that the covering radius of an N-point subset XN of the unit sphere Sd⊂Rd+1 is bounded above by a power of the worst-case error for equal weight cubature 1N∑x∈XNf(x)≈∫Sdfdσd for functions in the Sobolev space Wps(Sd), where σd denotes normalized area measure on Sd. These bounds are close to optimal when s is close to d/p. Our study of the worst-case error along with results of Brandolini et al. motivate the definition of Quasi-Monte Carlo (QMC) design sequences for Wps(Sd), which have previously been introduced only in the Hilbert space setting p=2. We say that a sequence (XN) of N-point configurations is a QMC-design sequence for Wps(Sd) with s>d/p provided the worst-case equal weight cubature error for XN has order N−s/d as N→∞, a property that holds, in particular, for a sequence of spherical t-designs in which each design has order td points. For the case p=1, we deduce that any QMC-design sequence (XN) for W1s(Sd) with s>d has the optimal covering property; i.e., the covering radius of XN has order N−1/d as N→∞. A significant portion of our effort is devoted to the formulation of the worst-case error in terms of a Bessel kernel, and showing that this kernel satisfies a Bernstein type inequality involving the mesh ratio of XN. As a consequence we prove that any QMC-design sequence for Wps(Sd) is also a QMC-design sequence for Wp′s(Sd) for all 1≤p<p′≤∞ and, furthermore, if (XN) is a quasi-uniform QMC-design sequence for Wps(Sd), then it is also a QMC-design sequence for Wps′(Sd) for all s>s′>d/p.

Structural alternation of tandem dye-sensitized solar cells based on mesh-type of counter electrode

Tandem dye-sensitized solar cells (DSCs) are very effective to improve light absorption characteristics and overall performance. The structure of conventional tandem DSC is an assembly of two independent DSCs. Therefore, additional TiO2 layer, Pt film, and transparent conductive oxide (TCO) electrodes weaken incident light to the bottom cell and complicate the fabrication as compared with standard DSCs. Here, this work proposed the structural alternation of tandem DSC as a solution. Mesh type of counter electrode was inserted between top and bottom cells instead of TCO electrodes. Two photo electrodes shared electrolyte and counter electrode in this structure. High aperture ratio of mesh increased light penetration into bottom cell and led to the performance improvement. Structural alternation also simplified the fabrication. It could be fabricated like standard DSCs. After dye arrangement and TiO2 layer of bottom cell were controlled, the photovoltaic performance of proposed tandem DSC was enhanced and it was higher than conventional tandem DSC. Finally, the long-term stability of proposed tandem DSC was secured by the control of sealing walls.

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Dual‐phase‐lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher‐order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher‐order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher‐order accurate and unconditionally stable compact finite difference scheme for solving one‐dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two‐dimensional case and develop a fourth‐order accurate compact finite difference method in space coupled with the Crank–Nicolson method in time, where the Robin's boundary condition is approximated using a third‐order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth‐order in space and second‐order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1742–1768, 2015

A novel subgridding algorithm for the FDTD method combined with an adaptively adjusted time-stepping scheme

In a finite-difference time-domain (FDTD) calculation, relatively fine spatial gridding is required to resolve regions containing fine structural features. To reduce computation time, subgridding algorithms can be used to efficiently handle configurations that feature multiscale dimensions: Such algorithms can maintain the desired accuracy while avoiding the computation of the entire FDTD space at a finer mesh resolution. A novel FDTD subgridding algorithm with adaptively adjusted time-steps is proposed in this article. The separated temporal and spatial interfaces for the proposed FDTD subgridding algorithm with a typical 1:3 mesh ratio are described in detail. Higher mesh ratios were achieved using the adaptively adjusted time-stepping procedure. Accuracy of the proposed method was verified by comparing the results with those of the conventional FDTD method involving a fine mesh resolution. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 57:613–616, 2015

Convergence Analysis for Three Parareal Solvers

We analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem $\mathbf{u}'+A\mathbf{u}=g$. The coarse propagator $\mathcal{G}$ is fixed to the backward-Euler method and three time integrators are chosen for the $\mathcal{F}$-propagator: the trapezoidal rule, the third-order diagonal implicit Runge--Kutta (RK) (DIRK) method, and the fourth-order Gauss RK method. It is well known that the Parareal-Euler algorithm using the backward-Euler method for $\mathcal{F}$ and $\mathcal{G}$ converges rapidly, but less is known when one uses for $\mathcal{F}$ the trapezoidal rule, or the fourth-order Gauss RK method, especially when the mesh ratio $J(=\Delta T/\Delta t)$ is small. We show that for a specified $\lambda_{\max}$(the maximal eigenvalue of $A$ or its upper bound), there exists some critical $J_{\rm cri}$ such that the parareal solvers derived from these three choices of $\mathcal{F}$ converge as fast as Parareal-Euler, provided $J\geq J_{\rm cri}$. Precisely, for $\mathcal{F}$ the trapezoidal rule and the fourth-order Gauss RK method, $J_{\rm cri}$ depends on $\Delta T, \Delta t$, and $\lambda_{\max}$ and we present concise formulas to calculate $J_{cri}$, while for $\mathcal{F}$ the third-order DIRK method, $J_{cri}=4$, independently of these parameters. Numerical examples with applications in fractional PDEs and uncertainty quantification are presented to support the theoretical predictions.

PENGARUH UKURAN PARTIKEL DAN KOMPOSISI TERHADAP SIFAT KEKUATAN BENTUR KOMPOSIT EPOKSI BERPENGISI SERAT DAUN NANAS

This research was aimed to investigate the effect of pineapple leafs particle size and pineapple leaf fiber composition of the impact strength of epoxy composites filled with pineapple leaf fibers. The composites were made by hand lay up method by mixing epoxy and pineapple leaf fiber with particle size variation of 30 mesh, 40 mesh, 70 mesh, 100 mesh, and volume fraction ratio between filler and matrix 5/95, 10/90, 15/85 (v/v). Mechanical properties wich tested was impact strength and supported with SEM analysis. The results obtained show that the addition pineapple leaf fiber as filler in epoxy composites generally increase the impact strength of the composites, with best performance shown by 100 mesh particle size variation with ratio 90/10 (v/v) which from SEM analysis show that this variation having better filler distribution.

Comparing outcomes of sheet grafting with 1:1 mesh grafting in patients with thermal burns: A randomized trial

BackgroundIn many units, the standard mesh ratio is 1.5:1, but in our unit we have a 1:1 mesher, which does not expand the skin but provides regular fenestrations. There is some evidence that the unexpanded 1.5:1 meshed graft compares favourably with sheet grafts from a cosmetic perspective whilst reducing the risk of graft failure secondary to a subgraft haematoma, but none comparing the 1:1 meshed graft with the sheet graft.We conducted a randomized trial to compare surgical outcomes in unfenestrated sheet grafts with 1:1 meshed grafts. MethodsAll patients aged ≥16 years undergoing skin grafts with either a sheet or a 1:1 mesh for burn reconstruction were included. Patients on steroids, those with conditions that impair healing, and burns >20% were excluded. Patients were randomized into the sheet grafting or mesh graft using a computer-generated allocation system. The mean percentage of graft loss was assessed by a Visitrak overlay system. At 3–4 months, 7–8 months and at 1 year, photos were taken for scar assessment using the Vancouver Scar Score (VSS). ResultsOut of 72 patients, 48 patients (24 sheet vs. 24 mesh) completed the trial at 12 months. The mean age was 58 years (range 21–90). There was no total loss of graft in either group. The mean percentage of graft loss due to haematoma formation was higher in the sheet graft group (10%) compared to the 1:1 mesh group (6%) (P<0.062). The VSS score was 5 in both groups at 12 months. There was no significant difference in scar quality between the treatment groups. ConclusionThese results show that the 1:1 mesh graft is superior to the sheet graft with regard to graft loss, although this result is not statistically significant. There are comparable findings in terms of cosmetic perspective at 12 months post-operatively in both arms of the trial.

Modeling stress wave propagation in rocks by distinct lattice spring model

In this paper, the ability of the distinct lattice spring model (DLSM) for modeling stress wave propagation in rocks was fully investigated. The influence of particle size on simulation of different types of stress waves (e.g. one-dimensional (1D) P-wave, 1D S-wave and two-dimensional (2D) cylindrical wave) was studied through comparing results predicted by the DLSM with different mesh ratios (lr) and those obtained from the corresponding analytical solutions. Suggested values of lr were obtained for modeling these stress waves accurately. Moreover, the weak material layer method and virtual joint plane method were used to model P-wave and S-wave propagating through a single discontinuity. The results were compared with the classical analytical solutions, indicating that the virtual joint plane method can give better results and is recommended. Finally, some remarks of the DLSM on modeling of stress wave propagation in rocks were provided.

A fast Galerkin method for parabolic space–time boundary integral equations

An efficient scheme for solving boundary integral equations of the heat equation based on the Galerkin method is introduced. The parabolic fast multipole method (pFMM) is applied to accelerate the evaluation of the thermal layer potentials. In order to remain attractive for a wide range of applications, a key issue is to ensure efficiency for a big variety of temporal to spatial mesh ratios. Within the parabolic Galerkin FMM (pGFMM) it turns out that the temporal nearfield can become very costly. To that end, a modified fast Gauss transform (FGT) is developed. The complexity and convergence behavior of the method are analyzed and numerically investigated on a range of model problems. The results demonstrate that the complexity is nearly optimal in the number of discretization parameters while the convergence rate of the Galerkin method is preserved.

Retarding avalanches in motion with net structures

Steel wire rope nets have become a common protection measure against snow avalanches in Europe, as they can prevent a release in potential starting zones. A novel approach in this context, is to retard the movement of an avalanche after it has been initiated. A full scale structure, the so-called Snowcatcher, was installed and instrumented with several load measuring pins, which record the dynamic loads caused by an avalanche. The motivation of the measurements is to observe the influence of net structures on snow avalanches. In the lab, scaled granular experiments were performed in two set-ups, investigating the influence of i) the net barrier angle and ii) the mesh size of the net. For both set ups various experiments with different chute inclinations were performed. The results from measuring the front velocities and flow depths showed that higher chute angles are accompanied with both, higher flow velocities and Froude numbers. Experiments with different net barrier angles showed that the effectivity increases with higher chute inclinations. Furthermore the results indicate that different barrier angles slightly influence the effectivity, e.g. for small chute inclinations, nets perpendicular to the flow direction lead to lower effectivities than inclined nets. Experiments with different mesh sizes indicate a velocity dependency of the effectivity corresponding to a certain ratio of mesh to grain size. Smaller mesh sizes in the range of the maximal particle grain size lead to an obstruction of the net, acting as a solid barrier and therefore reaching best effectivity, notwithstanding overflows. For large mesh sizes the effectivity of the net barrier increases with a higher velocity of the flow.

Mesh ratios for best-packing and limits of minimal energy configurations

For $N$-point best-packing configurations $\omega_N$ on a compact metric space $(A,\rho)$, we obtain estimates for the mesh-separation ratio $\gamma(\omega_N,A)$, which is the quotient of the covering radius of $\omega_N$ relative to $A$ and the minimum pairwise distance between points in $\omega_N$. For best-packing configurations $\omega_N$ that arise as limits of minimal Riesz $s$-energy configurations as $s\to \infty$, we prove that $\gamma(\omega_N,A)\le 1$ and this bound can be attained even for the sphere. In the particular case when N=5 on $S^2$ with $\rho$ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid $\omega_5^*$, that is the limit (as $s\to \infty$) of 5-point $s$-energy minimizing configurations. Moreover, $\gamma(\omega_5^*,S^2)=1$.

Accurate numerical method for solving dual-phase-lagging equation with temperature jump boundary condition in nano heat conduction

Dual-phase-lagging (DPL) equation with temperature jump boundary condition shows promising for analyzing nano heat conduction. For solving it, development of higher-order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nano-scale, using a higher-order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, in this article we present a higher-order accurate and unconditionally stable compact finite difference scheme based on the ratio of relaxation times (0⩽B⩽1 and B>1). The method is illustrated by three numerical examples including a 2D case.

A fifth order accurate geometric mesh finite difference method for general nonlinear two point boundary value problems

Second order boundary value problems are treated using fifth order accurate geometric mesh finite difference approximations. The method uses evaluation at two off steps, three neighbouring knots and it reduces to simple five terms recurrence relations. The formal convergence analysis and error estimate reveals that the truncation errors of the new discretizations are fifth order accurate. Some physical problems are solved to demonstrate efficiency and reliability of the proposed numerical method. The root mean square errors for different values of mesh ratio parameter have been given in tables to support the utility of the method.

Analysis of Non-Standard Gear Contact Fatigue Life

A model of gear contact has been established based on Hertz theory in this thesis, Analysis of non-standard gear contact fatigue life test process . For ha* take 1,1.15,1.25 and c* take 0.25,0.5 respectively ,the gear generates contact fatigue failure, then calculate the meshing ratio under different tooth high coefficients, and gear pairs state with fatigue testing time by the finite element method and orthogonal test. These data are compared with the results of specific experiments verified. They description that tooth height coefficients and headspace coefficients change have a great impact on gear contact fatigue life.

Time-engineeringed biphasic drug release by electrospun nanofiber meshes

A drug-loaded nanofiber mesh which could achieve time-engineeringed biphasic release was fabricated through sequential electrospinning. The drug to polymer ratio of each single mesh was allocated and designed before the tri-layered meshes were created. The resultant meshes had the following construction: (i) the first drug-loaded mesh (top side), (ii) the second drug-loaded mesh (second side), and (iii) the third drug-loaded mesh (bottom side). The drug release speed and duration were controlled by designing morphological features of the electrospun meshes such as the fiber diameter and mesh thickness. An in vitro release experiment revealed that the tri-layered construction with distinct morphological features of each component mesh can provide biphasic drug release. The time-engineeringed dual release system using the multilayered electrospun nanofiber meshes was proved to be a useful formulation when achieving controlled drug release at different times.