Grid-based Method Research Articles

Exploring a Novel Multiple-Query Resistive Grid-Based Planning Method Applied to High-DOF Robotic Manipulators.

The applicability of the path planning strategy to robotic manipulators has been an exciting topic for researchers in the last few decades due to the large demand in the industrial sector and its enormous potential development for space, surgical, and pharmaceutical applications. The automation of high-degree-of-freedom (DOF) manipulator robots is a challenging task due to the high redundancy in the end-effector position. Additionally, in the presence of obstacles in the workspace, the task becomes even more complicated. Therefore, for decades, the most common method of integrating a manipulator in an industrial automated process has been the demonstration technique through human operator intervention. Although it is a simple strategy, some drawbacks must be considered: first, the path’s success, length, and execution time depend on operator experience; second, for a structured environment with few objects, the planning task is easy. However, for most typical industrial applications, the environments contain many obstacles, which poses challenges for planning a collision-free trajectory. In this paper, a multiple-query method capable of obtaining collision-free paths for high DOF manipulators with multiple surrounding obstacles is presented. The proposed method is inspired by the resistive grid-based planner method (RGBPM). Furthermore, several improvements are implemented to solve complex planning problems that cannot be handled by the original formulation. The most important features of the proposed planner are as follows: (1) the easy implementation of robotic manipulators with multiple degrees of freedom, (2) the ability to handle dozens of obstacles in the environment, (3) compatibility with various obstacle representations using mathematical models, (4) a new recycling of a previous simulation strategy to convert the RGBPM into a multiple-query planner, and (5) the capacity to handle large sparse matrices representing the configuration space. A numerical simulation was carried out to validate the proposed planning method’s effectiveness for manipulators with three, five, and six DOFs on environments with dozens of surrounding obstacles. The case study results show the applicability of the proposed novel strategy in quickly computing new collision-free paths using the first execution data. Each new query requires less than 0.2 s for a 3 DOF manipulator in a configuration space free-modeled by a 7291 × 7291 sparse matrix and less than 30 s for five and six DOF manipulators in a configuration space free-modeled by 313,958 × 313,958 and 204,087 × 204,087 sparse matrices, respectively. Finally, a simulation was conducted to validate the proposed multiple-query RGBPM planner’s efficacy in finding feasible paths without collision using a six-DOF manipulator (KUKA LBR iiwa 14R820) in a complex environment with dozens of surrounding obstacles.

Rapid assessment of seismic intensity based on Sina Weibo — A case study of the changning earthquake in Sichuan Province, China

Disaster information acquisition and assessment in China primarily depends on disaster-related governmental departments at all levels. As new challenges faced by disaster emergency management arise, a gap still exists in the timeliness and completeness of rapid disaster assessment. The development and popularity of social media has opened up new channels. Social media data, containing massive amounts of disaster information, has the advantages of timeliness, efficiency, and multiple spatiotemporal scales and can supplement existing methods. Based on the dominant social media platform in China, Sina Weibo, this paper proposes a method of extracting intensity information from Weibo texts, including social perceptions such as shaking, emergency reaction, mood and visible damage, which can support the rapid assessment of seismic intensity. By taking the Changning earthquake in Sichuan Province, China, as a case study, we verify the feasibility of hazard information extraction from Weibo. The results show that the intensity information derived from Weibo in the short-term can respond to a sudden earthquake promptly, then effectively identify the main earthquake affected area, locate the possible orientation close to the meizoseismal area and depict the situation in 10 min. We also propose a grid-based correction method, which synthesizes the hotness-intensity matrix and a seismic attenuation-based model. Compared with taking the seismic attenuation model directly, the proposed correction method improves the accuracy of correct recognition rate of seismic intensity assessment to 82% on average and reduces the false recognition rate significantly. The proposed framework of rapid assessment and correction reveals that the combination of social perception and hazard intensity plays a valuable and promising role as a supplement of traditional disaster assessment approach.

Evaluation of scatter rejection and correction performance of 2D antiscatter grids in cone beam computed tomography.

We have been investigating two-dimensional (2D) antiscatter grids (2D ASGs) to reduce scatter fluence and improve image quality in cone beam computed tomography (CBCT). In this work, two different aspects of 2D ASGs, their scatter rejection and correction capability, were investigated in CBCT experiments. To correct residual scatter transmitted through the 2D ASG, it was used as a scatter measurement device with a novel method: grid-based scatter sampling. Three focused 2D ASG prototypes with grid ratios of 8, 12, and 16 were developed for linac-mounted offset detector CBCT geometry. In the first phase, 2D ASGs were used as a scatter rejection device, and the effect of grid ratio on CT number accuracy and contrast-to-noise ratio (CNR) evaluated in CBCT images. In the second phase, a grid-based scatter sampling method which exploits the signal modulation characteristics of the 2D ASG's septal shadows to measure and correct residual scatter transmitted through the grid was implemented. To evaluate CT number accuracy, the percent change in CT numbers was measured by changing the phantom from head to pelvis size and configuration. When 2D ASG was used as a scatter rejection device, CT number accuracy increased and the CT number variation due to change in phantom dimensions was reduced from 23% to 2-6%. A grid ratio of 16 yielded the lowest CT number variation. All three 2D ASGs yielded improvement in CNR, up to a factor of two in pelvis-sized phantoms. When 2D ASG prototypes were used for both scatter rejection and correction, CT number variations were reduced further, to 1.3-2.6%. In comparisons with a clinical CBCT system and a high-performance radiographic ASG, 2D ASG provided higher CT number accuracy under the same imaging conditions. When 2D ASG is used solely as a scatter rejection device, substantial improvement in CT number accuracy can be achieved by increasing the grid ratio. Two-dimensional ASGs also provided significant CNR improvement even at lower grid ratios. Two-dimensional ASGs used in conjunction with the grid-based scatter sampling method provided further improvement in CT number accuracy, irrespective of the grid ratio, while preserving 2D ASGs' capacity to improve CNR. The combined effect of scatter rejection and residual scatter correction by 2D ASG may accelerate implementation of new techniques in CBCT that require high quantitative accuracy, such as radiotherapy dose calculation and dual energy CBCT.

Evaluation and improvement of horizontal datum transformation models: a case study of Bosnia and Herzegovina

This study evaluates the applicability of different models of coordinate transformation between local and global geodetic datums. The results indicate the grid-based method as the best solution, assuming a sufficient density of tie points. Transformation based on a limited number of tie points, which do not reflect the real state of the survey basis in a particular area, have limited possibilities to reduce nonuniform and unequally distributed distortions that are usually found in the spatial data. This inevitably leads to the degradation of accuracy of transformation results, which is unacceptable. By using the optimal resolution grid model, which includes geodetic points from the area of the cadastral municipality that is being transformed, much better results are achieved compared to the model of distortion shifts used in the Federation of Bosnia and Herzegovina.

Strategies on locations of public access defibrillator: A systematic review.

Out-of-hospital cardiac arrest (OHCA) is a critical condition with poor outcomes. Although the survival rate increases in those who undergo defibrillation, the utility of on-time defibrillation among bystanders remained low. An evaluation of the deployment strategy for public access defibrillators (PADs) is necessary to increase their use and accessibility. This study was to conduct a systematic review for deployment strategies of PADs. Two authors independently searched for articles published before October 2019 from PubMed, Embase, Web of Science, and Cochrane Library. An independent librarian provided the search strategy and assisted the literature research. We included articles that were focused on the main topic, but excluded those which were missing results or that used an unclear definition. The qualitative outcomes were the utility and OHCA coverage of PADs. We performed a qualitative analysis across the studies, but a quantitative analysis was not available due to the studies' heterogeneity in design and variety of outcomes. We eventually included 15 studies. Three strategies were presented: guidelines-based, grid-based, and landmark-based. The guidelines-based deployment was common fit for OHCA events. The grid-based method increased the use of bystander defibrillation 3-fold, and 30-day survival doubled. The top 3 landmarks in the landmark-based strategy were offices (18.6%), schools (13.3%), and sports facilities (12.9%). Utility of PADs might increase if we optimize PAD location by mathematical modeling and evaluation feedback. Three deployment strategies were presented. Although the optimal method could not be fully identified, a more efficient PAD deployment could benefit the population in terms of OHCA coverage and survival among patients with OHCA.

Simulation and experimental study on lubrication of high-speed reducer of electric vehicle

Purpose The lubrication of the high-speed reducer of an electric vehicle is investigated. The specific contents include visualization of the flow field inside reducer, lubrication evaluation of bearings and efficiency experiment. Design/methodology/approach The flow field inside reducer at five working conditions: straight, uphill, downhill, left lean and right lean is simulated by smoothed particle hydrodynamics (SPH). According to the instantaneous number of particles through bearings, the lubrication states of bearings are evaluated. The test platform is set up to measure the efficiency of the reducer. Findings The flow field inside the reducer is obtained, the lubrication of bearings needs to be improved, the efficiency of the electric vehicle reducer meets the requirement. Originality/value The SPH method is used to simulate lubrication instead of using the traditional grid-based finite volume method. A novel method to evaluate the lubrication of bearings is proposed. The method and conclusions can guide electric vehicle reducer design.

A unified scatter rejection and correction method for cone beam computed tomography.

Scattered radiation is a major cause of image quality degradation in flat panel detector-based cone beam CT (CBCT). While recently introduced 2D antiscatter grids reject the majority of scatter fluence, the small percentage of scatter fluence still transmitted to the detector remains a major challenge for implementation of quantitative imaging techniques such as dual energy imaging in CBCT. Additionally, this residual scatter is also a major source of grid-induced artifacts, which impedes implementation of 2D grids in CBCT. We therefore present a new method to achieve both robust scatter rejection and residual scatter correction using a 2D antiscatter grid; in doing so, we expand the role of 2D grids from mere scatter rejection devices to scatter measurement devices. In our method, the radiopaque septa of the 2D grid emulate a micro array of beam-stops placed on the detector which introduce spatially periodic septal shadows. By selecting sufficiently thin grid septa, the primary intensity can be reduced while preserving the uniformity of scatter intensity. This enables us to correlate the modulated pixel signal intensity in septal shadows with local scatter intensity. Our method then exploits this correlation to measure and remove residual scatter intensity from projections. No assumptions are made about the object being imaged. We refer to this as Grid-based Scatter Sampling (GSS). In this work, we evaluate the principle of signal modulation with grid septa, the accuracy of scatter estimates, and the effect of the GSS method on image quality using simulations and measurements. We also implement the GSS method experimentally using a 2D grid prototype. Our results demonstrate that the GSS method increased CT number accuracy and reduced image artifacts associated with scatter. With 2D grid and residual scatter correction, HU nonuniformity was reduced from 65HU to 30HU in pelvis sized phantoms, and HU variations due to change in phantom size were reduced from 59HU to 20HU, when compared to use of only a 2D grid. With residual scatter correction via GSS method, grid-induced ring artifacts were suppressed, leading to a 41% reduction in noise. The shape of the modulation transfer function (MTF) was preserved before and after suppression of ring artifacts. Our grid-based scatter sampling method enables utilization of a 2D grid as a scatter measurement and correction device. This method significantly improves quantitative accuracy in CBCT, further reducing the image quality gap between CBCT and multi-detector CT. By correcting residual scatter with the proposed method, grid-induced line artifacts in projections and associated ring artifacts in CBCT images were also suppressed with no compromise of spatial resolution.

Road Boundary-Enhanced Automatic Background Filtering for Roadside Lidar Sensors

The roadside-deployed lidar sensor provides a solution to obtain the real-time, high-resolution micro traffic data (HRMTD) of all road users in the mixed traffic situation (connected vehicles and unconnected vehicles both exist on the roads). Background filtering is a necessary and important step for the HRMTD collection to serve the connected vehicle. Without excluding the background points, the accuracy of Euclidean-based object clustering and tracking algorithms can be reduced. The widely used method-3D-density-statistic filtering (3D-DSF) for roadside lidar background filtering can effectively exclude the background points for free-flow conditions. However, the performance of 3D-DSF can be greatly influenced by congested traffic conditions. This article presents a revised 3D-DSF algorithm to automatically extract background points by involving the road-boundary information. This new method is named <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">road-boundary-enhanced, 3D-density statistic filtering</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3D-DSFRB</i> ). This algorithm involves the boundary of the historical trajectories of road users as the region of interest (ROI) to enhance the accuracy of background filtering. A revised grid-based method was developed for road-boundary ID. The 3D-DSF was only applied for the area outside of the ROI. Within the ROI, only ground surface was excluded. Case studies were conducted to evaluate the effectiveness of the 3D-DSFRB algorithm. The results showed that the 3D-DSFRB can filter background points for both free-flow conditions and congested traffic conditions. The time cost of the 3D-DSFRB was also reduced compared to the 3D-DSF. Compared to the state of the art, the 3D-DSFRB improved the accuracy of background filtering.

Investigation of Cooperative Modes for Collective Molecules Using Grid-Based Principal Component Analysis.

A grid-based principal component analysis method (GBPCA) has been developed and implemented to investigate the modes of collective molecules present in the cube in the grid system from their trajectories using molecular dynamics (MD) simulation. This method is applied to the simulations of water, methane, and hydrated proteins. In the cases of single molecules, GBPCA demonstrates that while individual molecules interact with other molecules and move randomly, the collective molecules nevertheless produce cooperative principal component (PC) modes. Notably, these PC modes of the collective molecules formed vorticial or symmetrical structures, with some resembling torus-like structures. Such structures were observed not only in simulations with models that reproduced real molecules, such as water or methane, but also in simulations with artificial molecules that were modified to interact weakly or only repulsively. On the contrary, molecules without any interactions did not exhibit the cooperative PC modes that lead to the formation of specific structures. These results imply that when molecules subject to intermolecular forces are located in a space in which they can interact with one another, the PC modes of the collective molecules will form vorticial and torus-like structures. In the cases of hydrated proteins, the results reveal that the collective molecules in the protein and water molecules showed cooperative PC modes, forming vorticial or torus-like structures, as in the cases of the water or methane molecules. The PC modes of the water molecules close to the protein were influenced by the protein motion and indicated cooperative modes with the protein. Moreover, proteins with the same folds had similar PC mode structures. Besides, the PC modes of the proteins composed of multiple domains appeared independently in the protein domain. Hence, it can be understood that the cooperative motions of the collective molecules of the protein and surrounding water molecules contribute strongly to the protein's structure and function. These results are expected to help our understanding of the dynamics of molecules in the liquid state.

Design of a New Lightweight Accurate Grid-Based Localization Method for WASN

Existing solutions designed for acoustic target localization based on the received signal energy in WSN suffer either from low localization accuracy due to suboptimal convergence or high computations requirements due to the application of complex algorithms. The main contribution of this article is the design of a new lightweight energy-based acoustic source localization solution intended for implementation in wireless sensors. The novelty of this solution resides in devising lightweight computation of the cost function in the location estimator algorithm and the application of the Tabu metaheuristic method to enhance the localization accuracy. In terms of localization accuracy, the proposed scheme outperforms many existing solutions with reduced localization error by up to 88% while using only three energy measurements, which saves around 85% of node energy. Furthermore, the performance analysis results showed a stable behavior of the proposed algorithm across a wide range of parameters variations of the energy-decay model, which demonstrates the robustness of the scheme.

A local refinement purely meshless scheme for time fractional nonlinear Schrödinger equation in irregular geometry region* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11501495, 51779215, and 11672259), the Postdoctoral Science Foundation of China (Grant Nos. 2015M581869 and 2015T80589), and the Jiangsu Government Scholarship for Overseas Studies, China (Grant No. JS-2017-227).

A local refinement hybrid scheme (LRCSPH-FDM) is proposed to solve the two-dimensional (2D) time fractional nonlinear Schrödinger equation (TF-NLSE) in regularly or irregularly shaped domains, and extends the scheme to predict the quantum mechanical properties governed by the time fractional Gross–Pitaevskii equation (TF-GPE) with the rotating Bose–Einstein condensate. It is the first application of the purely meshless method to the TF-NLSE to the author’s knowledge. The proposed LRCSPH-FDM (which is based on a local refinement corrected SPH method combined with FDM) is derived by using the finite difference scheme (FDM) to discretize the Caputo TF term, followed by using a corrected smoothed particle hydrodynamics (CSPH) scheme continuously without using the kernel derivative to approximate the spatial derivatives. Meanwhile, the local refinement technique is adopted to reduce the numerical error. In numerical simulations, the complex irregular geometry is considered to show the flexibility of the purely meshless particle method and its advantages over the grid-based method. The numerical convergence rate and merits of the proposed LRCSPH-FDM are illustrated by solving several 1D/2D (where 1D stands for one-dimensional) analytical TF-NLSEs in a rectangular region (with regular or irregular particle distribution) or in a region with irregular geometry. The proposed method is then used to predict the complex nonlinear dynamic characters of 2D TF-NLSE/TF-GPE in a complex irregular domain, and the results from the posed method are compared with those from the FDM. All the numerical results show that the present method has a good accuracy and flexible application capacity for the TF-NLSE/GPE in regions of a complex shape.

Enhancement of accuracy in double sided incremental forming by compensating tool path for machine tool errors

Incremental Sheet Forming (ISF) is a dieless forming process and has capability to form complex 3D geometries. Double Sided Incremental Forming (DSIF) is the most flexible variant of ISF as it uses two tools, one on either side of the sheet (one tool forms the sheet and other acts as support). However, forming the components with acceptable accuracy is one of the challenges as it depends on many factors such as sheet and tool(s) deflections as well as machine tool errors. If these deflections/deviations are not compensated in tool path, accuracy of the component gets affected and support tool may lose contact or squeeze the sheet while forming components. In the present work, grid-based method is used to measure the machine tool error at discrete points of the work envelope. To enhance the accuracy and support tool contact condition, machine tool errors are compensated in tool path along with tool and sheet deflection compensations developed earlier. To demonstrate the effectiveness of developed methodology on the accuracy, components having different geometric characteristics as well as opening sizes are formed with and without compensating for machine tool errors in tool path. Results indicate that deviation in symmetry within the same contour reduced from 850 μm to less than 40 μm. In addition, support tool has maintained contact throughout support surface with maximum force of 50 N when machine tool errors are compensated.

Proposal of Novel Icing Simulation Using a Hybrid Grid- and Particle-Based Method

ABSTRACTIcing on aircraft can drastically reduce aerodynamic performance and lead to serious accidents. Therefore, prediction of the accreted ice shape and area and its effects on aerodynamic performance is crucial during the design phase of an aircraft. However, numerical simulations based on conventional grid-based methods such as the finite volume method cannot accurately reproduce the complex ice shapes, which involve horn growth, feather growth, air voids, and severe surface roughness. In the present study, instead of the grid-based method, a hybrid grid- and particle-based method was newly proposed and applied to the icing problem on a NACA0012 airfoil. The explicit moving particle semi-implicit method was employed as the particle-based method due to its short computing time. The numerical simulations effectively reproduced feather-shaped ice, air voids, and surface roughness. Finally, by computing the flow around the iced airfoil, it was confirmed that flow separation around the leading edge occurred due to the ice layer, which resulted in a thicker boundary layer and wake and an increase in the drag coefficient of approximately 70% after a residence time of only 60 seconds.

Density Difference Grid Design in a Point-Mass Filter

The paper deals with the Bayesian state estimation of nonlinear stochastic dynamic systems. The stress is laid on the point-mass filter, solving the Bayesian recursive relations for the state estimate conditional density computation using the deterministic grid-based numerical integration method. In particular, the grid design is discussed and the novel density difference grid is proposed. The proposed grid design covers such regions of the state-space where the conditional density is significantly spatially varying, by the dense grid. In other regions, a sparse grid is used to keep the computational complexity low. The proposed grid design is thoroughly discussed, analyzed, and illustrated in a numerical study.

A survey for variable young stars with small telescopes – III. Warm spots on the active star V1598 Cyg

ABSTRACT Magnetic spots on low-mass stars can be traced and characterized using multiband photometric light curves. Here, we analyse an extensive data set for one active star, V1598 Cyg, a known variable K dwarf which is either pre-main sequence and/or in a close binary system. Our light curve contains 2854 photometric data points, mostly in V, Rc, Ic, but also in U, B, and Hα, with a total baseline of about 4 yr, obtained with small telescopes as part of the HOYS project. We find that V1598 Cyg is a very fast rotator with a period of 0.8246 d and varying amplitudes in all filters, best explained as a signature of strong magnetic activity and spots. We fit the photometric amplitudes in V, Rc, Ic and use them to estimate spot properties, using a grid-based method that is also propagating uncertainties. We verify the method on a partial data set with high cadence and all five broad-band filters. The method yields spot temperatures and fractional spot coverage with typical uncertainties of 100 K and 3–4 per cent, respectively. V1598 Cyg consistently exhibits spots that are a few hundred degrees warmer than the photosphere, most likely indicating that the light curve is dominated by chromospheric plage. The spot activity varies over our observing baseline, with a typical time-scale of 0.5–1 yr, which we interpret as the typical spot lifetime. Combining our light curve with archival data, we find a six year cycle in the average brightness, that is probably a sign of a magnetic activity cycle.

Validity of the static-exchange approximation for inner-shell photoionization of polyatomic molecules

Author(s): Marante, CA; Greenman, L; Trevisan, CS; Rescigno, TN; McCurdy, CW; Lucchese, RR | Abstract: The simple single-channel static-exchange approximation completely ignores correlation between the continuum and molecular ion electrons. In molecular systems with symmetry equivalent atoms, the single-channel approximation can seriously fail in core ionization when using delocalized orbitals to represent the core hole states. We present cross sections and molecular frame photoelectron angular distributions with both localized and delocalized core orbitals in CF4 F (1s) ionization. We show that only a full coupled-channel calculation can recover an accurate description of the physics of inner-shell photoionization when using delocalized orbitals, whereas nearly the same result can be obtained from independent single-channel static-exchange calculations when localized core orbitals are used. A grid-based variational method described here makes such single-channel calculations possible on larger systems without local-exchange approximations. Illustrative calculations on the core ionization of SF6 are presented to illustrate the power of the grid-based method.

The formation of H···X hydrogen bond, C···X carbon-halide or Si···X tetrel bonds on the silylene-halogen dimers (X = F or Cl): intermolecular strength, molecular orbital interactions and prediction of covalency

The optimized geometries of the CH4Si···F2, C2H6Si···F2, CH4Si···Cl2 and C2H6Si···Cl2 complexes were obtained at the MP2/aug-cc-pVDZ level of theory. By means of the results of structure, vibration modes, molecular topology, bonding energy, thermodynamic stability and molecular orbital analysis, the interaction strength was examined by taking into account the presence of the σ-hole effect manifested on the halogen molecules. The topography of the quantum theory of atoms in molecules, the charge transfer determined by means of the natural bond orbitals as well as by the electrostatic potentials using a grid-based method and the graphical visualizer of non-covalent interaction were used to unveil if these systems are dominated by hydrogen bond, carbon-halide bond or tetrel bond. At last, the intermolecular covalency was analyzed through the symmetry-adapted perturbation theory at light of the EInd(R) /EElst(1) ratio and interaction distance.

Measuring soil coverage using image feature descriptors and the decision tree learning algorithm

The quantification and identification of ground cover plays a key role in erosion modelling, weed measurement, plant disease identification and other environmental applications. Currently, a variety of methods are used to mechanically classify digital images for ground cover. Only a few of these methods can distinguish green vegetation, straw/dormant vegetation, and exposed soil using only the Red-Green-Blue (RGB) spectrum. This research presents an approach to classifying ground cover using standard JPEG images and readily available Matlab (2018b) functions. The approach uses block segmentation, as opposed to pixel-wise or object-based segmentation, and compares multiple machine learning approaches with varying pixel block size and classification acceptance thresholds. The most successful classification approach found through this study was the decision tree algorithm with a 70-pixel block size and 60% classification acceptance threshold. Images were reduced to three feature descriptors: colour, texture, and oriented gradients to represent the respective RGB spectrum for an image. Both the training set and test set images used in this research came from field and greenhouse studies done between 2016 and 2019. The produced classifications were compared to manual coverage classifications using Samplepoint, a grid-based method, with R-squared values of 0.86 for green vegetation, 0.87 for straw/dormant vegetation, and 0.96 for exposed soil, respectively. This method showed strong performance for images containing exposed soil and either green vegetation or straw/dormant vegetation. The method was less effective for images with large quantities of both green vegetation and straw/dormant vegetation likely due to their similar shape.

A Coarea Formulation for Grid-Based Evaluation of Volume Integrals

Abstract We present a new method for computing volume integrals based on data sampled on a regular Cartesian grid. We treat the case where the domain is defined implicitly by an inequality, and the input data include sampled values of the defining function and the integrand. The method employs Federer’s coarea formula (Federer, 1969, Geometric Measure Theory, Grundlehren der mathematischen Wissenschaften, Springer) to convert the volume integral to a one-dimensional quadrature over level set values where the integrand is an integral over a level set surface. Application of any standard quadrature method produces an approximation of the integral over the continuous range as a weighted sum of integrals over level sets corresponding to a discrete set of values. The integral over each level set is evaluated using the grid-based approach presented by Yurtoglu et al. (2018, “Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains,” J. Comput. Inf. Sci. Eng., 18, p. 3). The new coarea method fills a need for computing volume integrals whose integrand cannot be written in terms of a vector potential. We present examples with known results, specifically integration of polynomials over the unit sphere. We also present Saye’s (2015, “High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles,” SIAM J. Sci. Comput., 37) example of integrating a logarithmic integrand over the intersection of a bounding box with an open domain implicitly defined by a trigonometric polynomial. For the final examples, the input data is a grid of mixture ratios from a direct numerical simulation of fluid mixing, and we demonstrate that the grid-based coarea method applies to computing volume integrals when no analytical form of the implicit defining function is given. The method is highly parallelizable, and the results presented are obtained using a parallel implementation capable of producing results at interactive rates.

Towards Reducing Computational Costs of Finite Difference Time Domain Algorithm in Plasmonic Optical Properties Modelling of Metal Nanoparticles

Finite difference time domain (FDTD) method is a grid-based, robust and straightforward method to model and study the optical properties of metal nanoparticles (MNPs). However, high computational costs of FDTD including simulation time and memory demand mitigate the interest in this algorithm. In this paper, FDTD algorithm is reviewed and reasons of high computational cost requirement in FDTD are investigated. Computational costs are directly characterised by the resolution and size of FDTD grid (known as Yee grid). High FDTD grid resolution is essentially required in MNPs plasmonic modelling to achieve promising accuracy. This results in increasing the dimension of FDTD operational parameters (stored in FDTD matrices) which increases memory and simulation time required and consumed by the algorithm.