Adaptive Sparse Grid Discontinuous Galerkin Method:Review and Software Implementation

This paper reviews the adaptive sparse grid discontinuous Galerkin(aSG-DG)method for computing high dimensional partial differential equations(PDEs)and its software implementation.The C++software package called AdaM-DG,implementing the aSG-DG method,is available on GitHub at https://github.com/JuntaoHuang/adaptive-multiresolution-DG.The package is capable of treating a large class of high dimensional linear and nonlinear PDEs.We review the essential components of the algorithm and the functionality of the software,including the multiwavelets used,assembling of bilinear operators,fast matrix-vector product for data with hierarchical structures.We further demonstrate the performance of the package by reporting the numerical error and the CPU cost for several benchmark tests,including linear transport equations,wave equations,and Hamilton-Jacobi(HJ)equations.

Adaptive Reclosing Scheme Based on Traveling Wave Injection For Multi-Terminal dc Grids

The hybrid dc circuit breaker(HCB)has the advantages of fast action speed and low operating loss,which is an idealmethod for fault isolation ofmulti-terminal dc grids.Formulti-terminal dc grids that transmit power through overhead lines,HCBs are required to have reclosing capability due to the high fault probability and the fact that most of the faults are temporary faults.To avoid the secondary fault strike and equipment damage that may be caused by the reclosing of the HCB when the permanent fault occurs,an adaptive reclosing scheme based on traveling wave injection is proposed in this paper.The scheme injects traveling wave signal into the fault dc line through the additionally configured auxiliary discharge branch in the HCB,and then uses the reflection characteristic of the traveling wave signal on the dc line to identify temporary and permanent faults,to be able to realize fast reclosing when the temporary fault occurs and reliably avoid reclosing after the permanent fault occurs.The test results in the simulation model of the four-terminal dc grid show that the proposed adaptive reclosing scheme can quickly and reliably identify temporary and permanent faults,greatly shorten the power outage time of temporary faults.In addition,it has the advantages of easiness to implement,high reliability,robustness to high-resistance fault and no dead zone,etc.

An Algorithm of the Adaptive Grid and Fuzzy Interacting Multiple Model

This paper studies the algorithm of the adaptive grid and fuzzy interacting multiple model （AGFIMM） for maneuvering target tracking, while focusing on the problems of the fixed structure multiple model （FSMM） algorithm＇s cost-efficiency ratio being not high and the Markov transition probability of the interacting multiple model （IMM） algorithm being difficult to determine exactly. This algorithm realizes the adaptive model set by adaptive grid adjustment, and obtains each model matching degree in the model set by fuzzy logic inference. The simulation results show that the AGFIMM algorithm can effectively improve the accuracy and cost-efficiency ratio of the multiple model algorithm, and as a result is suitable for enineering apolications.

ADAPTIVE LAYERED CARTESIAN CUT CELL METHOD FOR THE UNSTRUCTURED HEXAHEDRAL GRIDS GENERATION

Adaptive layered Cartesian cut cell method is presented to solve the difficulty of the tmstructured hexahedral anisotropic Cartesian grids generation from the complex CAD model. ＂Vertex merging algorithm based on relaxed AVL tree is investigated to construct topological structure for stereo lithography （STL） files, and a topology-based self-adaptive layered slicing algorithm with special features control strategy is brought forward. With the help of convex hull, a new points-in-polygon method is employed to improve the Cartesian cut cell method. By integrating the self-adaptive layered slicing algorithm and the improved Cartesian cut cell method, the adaptive layered Cartesian cut cell method gains the volume data of the complex CAD model in STL file and generates the unstructured hexahedral anisotropic Cartesian grids.

New Immersed Boundary Method on the Adaptive Cartesian Grid Applied to the Local Discontinuous Galerkin Method

Currently, many studies on the local discontinuous Galerkin method focus on the Cartesian grid with low computational e ciency and poor adaptability to complex shapes. A new immersed boundary method is presented, and this method employs the adaptive Cartesian grid to improve the adaptability to complex shapes and the immersed boundary to increase computational e ciency. The new immersed boundary method employs different boundary cells(the physical cell and ghost cell) to impose the boundary condition and the reconstruction algorithm of the ghost cell is the key for this method. The classical model elliptic equation is used to test the method. This method is tested and analyzed from the viewpoints of boundary cell type, error distribution and accuracy. The numerical result shows that the presented method has low error and a good rate of the convergence and works well in complex geometries. The method has good prospect for practical application research of the numerical calculation research.

Large eddy simulation of aircraft wake vortex with self-adaptive grid method

A self-adaptive-grid method is applied to numerical simulation of the evolu- tion of aircraft wake vortex with the large eddy simulation （LES）. The Idaho Falls （IDF） measurement of run 9 case is simulated numerically and compared with that of the field experimental data. The comparison shows that the method is reliable in the complex atmospheric environment with crosswind and ground effect. In addition, six cases with different ambient atmospheric turbulences and Brunt V^iis/il^i （BV） frequencies are com- puted with the LES. The main characteristics of vortex are appropriately simulated by the current method. The onset time of rapid decay and the descending of vortices are in agreement with the previous measurements and the numerical prediction. Also, sec-ondary structures such as baroclinic vorticity and helical structures are also simulated. Only approximately 6 million grid points are needed in computation with the present method, while the number can be as large as 34 million when using a uniform mesh with the same core resolution. The self-adaptive-grid method is proved to be practical in the numerical research of aircraft wake vortex.

Euler solution using adaptive Cartesian grid with a gridless boundary treatment

A quadtree-based adaptive Cartesian grid generator and flow solver were developed. The grid adaptation based on pressure or density gradient was performed and a gridless method based on the least-square fashion was used to treat the wall surface boundary condition, which is generally difficult to be handled for the common Cartesian grid. First, to validate the technique of grid adaptation, the benchmarks over a forward-facing step and double Mach reflection were computed. Second, the flows over the NACA 0012 airfoil and a two-element airfoil were calculated to validate the developed gridless method. The computational results indi- cate the developed method is reasonable for complex flows.

Adaptive Grid Technique Based on the Variational Principle and Its Weight Functions

Two computational cases that have analytic solutions are employed for studying the adaptive grid tech-nique based on the variational principle. The results show that for the computational case of traveling shock waves the weight function, with the 2nd-order derivation terms taken into consideration, can more effectively reduce the error than one with gradient terms. For the case of cyclonic frontogenesis, weight func-tions only related to the gradient are unable to enhance the computational accuracy while ones with the wind field and frontogenesis function taken into consideration can more reasonably arrange the grid. Com-pared with analytic solutions, the adaptive grid technique suggested in this paper can improve computational accuracy and it displays the prominent advantage of saving memory.

Real-Time Rendering of Dynamic Clouds Using Multi-Resolution Adaptive Grids

The multi-resolution adaptive grids method is proposed to solve the problems of inefficiency in the previous grid-based methods,and it can be used in clouds simulation as well as the interactive simulation between objects and clouds.Oriented bounding box(OBB)hierarchical trees of objects are established,and the resolutions of global and local grids can be selected automatically.The motion equations of fluid dynamics are simplified.Upwind difference is applied to ensure the stability of the simulation process during the discrete process of partial differential equations.To solve the speed problem of existed phase functions,the improved phase function is applied to the illumination calculation of clouds.Experimental results show that the proposed methods can promote the simulation efficiency and meet the need for the simulation of large-scale clouds scene.Real-time rendering of clouds and the interaction between clouds and objects have been realized without preprocessing stage.

ADAPTIVE HYBRID CARTESIAN GRID METHOD FOR VORTEX-DOMINATED FLOWS

An efficient compressible Euler equation solver for vortex-dominated flows is presented based on the adaptive hybrid Cartesian mesh and vortex identifying method.For most traditional grid-based Euler solvers,the excessive numerical dissipation is the great obstruction for vortex capturing or tracking problems.A vortex identifying method based on the curl of velocity is used to identify the vortex in flow field.Moreover,a dynamic adaptive mesh refinement(DAMR)process for hybrid Cartesian gird system is employed to track and preserve vortex.To validate the proposed method,a single compressible vortex convection flow is involved to test the accuracy and efficiency of DAMR process.Additionally,the vortex-dominated flow is investigated by the method.The obtained results are shown as a good agreement with the previous published data.

Forecasting-Based Adaptive Optimized Dispatch in Smart Grid Online

The power grid is a fusion of technologies in energy systems, and how to adjust and control the output power of each generator to balance the load of the grid is a crucial issue. As a platform, the smart grid is for the convenience of the implementation of adaptive control generators using advanced technologies. In this paper, we are introducing a new approach, the Central Lower Configuration Table, which optimizes dispatch of the generating capacity in a smart grid power system. The dispatch strategy of each generator in the grid is presented in the configuration table, and the scenario consists of two-level agents. A central agent optimizes dispatch calculation to get the configuration table, and a lower agent controls generators according to the tasks of the central level and the work states during generation. The central level is major optimization and adjustment. We used machine learning to predict the power load and address the best optimize cost function to deal with a different control strategy. We designed the items of the cost function, such as operations, maintenances and the effects on the environment. Then, according to the total cost, we got a new second-rank-sort table. As a result, we can resolve generator’s task based on the table, which can also be updated on-line based on the environmental situation. The signs of the driving generator’s controller include active power and system’s f. The lower control level agent carries out the generator control to track f along with the best optimized cost function. Our approach makes optimized dispatch algorithm more convenient to realize, and the numerical simulation indicates the strategy of machine learning forecast of optimized power dispatch is effective.

A self-adaptive Model based on Multi-Objective Programming for Grid Resource Management

The growing computational power requirements of grand challenge applications have positioned computational grid as promising next generation computing platform. However, resource management and application with varied requirements in grid environment continue to be a complex undertaking. In order to address complex resource management issues, we provide a self-adaptive model, which is based on multi-objective programming. The model make use of virtues of market mechanism efficiently, meanwhile, the shortcomings of market mechanism, such as too frequent fluctuations of price, are avoided by means of the method of changing prices after trading. Through using atom allocation of resource group, the cooperating allocation is improved, and some problems, such as deadlock of resource and inefficiently occupying resource, are solved. What's more important, efficiently using various resources in grid system is guaranteed through importing multi-objective programming mechanism in our resource management solution. A frame of resource allocation is given at first, then, the mathematical model of the method is constructed. An algorithm is proposed to get the approximate solution in this paper.

ADAPTIVE FLOW SOLUTION BASED ON MATRIX ERROR

An adaptive method for the solution of compressible flows is described. The idea results from the desire for an efficient grid system,and an accurate and robust solution method are used to resolve flow features of the interest. The adaptation flow solution is proposed,including the detection of flow features based on the matrix error; the mesh adaptation using the mesh movement,the mesh refinement,the mesh coarsening,and their combination. The feature detection based on the matrix error can maintain the high resolution property for shock waves of the one-dimensional approximate Riemann solver and the higher order reconstruction scheme. The high grid efficiency is obtained with the anisotropically directional grid corresponding to feature directions,and the error of the flow-field is averaged. The procedure and its application to flow solutions of shock waves are described. Results validate that the method is reliable.

An Adaptive Local Grid Nesting-based Genetic Algorithm for Multi-earth Observation Satellites' Area Target Observation

The Scheduling of the Multi-EOSs Area Target Observation(SMEATO)is an EOS resource schedul-ing problem highly coupled with computational geometry.The advances in EOS technology and the ex-pansion of wide-area remote sensing applications have increased the practical significance of SMEATO.In this paper,an adaptive local grid nesting-based genetic algorithm(ALGN-GA)is proposed for developing SMEATO solutions.First,a local grid nesting(LGN)strategy is designed to discretize the target area into parts,so as to avoid the explosive growth of calculations.A genetic algorithm(GA)framework is then used to share reserve information for the population during iterative evolution,which can generate high-quality solutions with low computational costs.On this basis,an adaptive technique is introduced to determine whether a local region requires nesting and whether the grid scale is sufficient.The effectiveness of the proposed model is assessed experimentally with nine randomly generated tests at different scales.The results show that the ALGN-GA offers advantages over several conventional algorithms in 88.9%of instances,especially in large-scale instances.These fully demonstrate the high efficiency and stability of the ALGN-GA.

Residential Energy Scheduling for Variable Weather Solar Energy Based on Adaptive Dynamic Programming

The residential energy scheduling of solar energy is an important research area of smart grid. On the demand side, factors such as household loads, storage batteries, the outside public utility grid and renewable energy resources, are combined together as a nonlinear, time-varying, indefinite and complex system, which is difficult to manage or optimize. Many nations have already applied the residential real-time pricing to balance the burden on their grid. In order to enhance electricity efficiency of the residential micro grid, this paper presents an action dependent heuristic dynamic programming(ADHDP) method to solve the residential energy scheduling problem. The highlights of this paper are listed below. First,the weather-type classification is adopted to establish three types of programming models based on the features of the solar energy. In addition, the priorities of different energy resources are set to reduce the loss of electrical energy transmissions.Second, three ADHDP-based neural networks, which can update themselves during applications, are designed to manage the flows of electricity. Third, simulation results show that the proposed scheduling method has effectively reduced the total electricity cost and improved load balancing process. The comparison with the particle swarm optimization algorithm further proves that the present method has a promising effect on energy management to save cost.

CFD-based hull optimization in calm water using adaptive grid deformation method

This paper presents an adaptive grid deformation technique for optimizing ship hull forms using computational fluid dynamics(CFD).The proposed method enables accurate and smooth updates of the hull surface and 3-D CFD grids in response to design variables.This technique incorporates a two-level point-transformation approach to move the grid points by a few design points.Initially,generic B-splines are utilized to transform grid points according to the displacements of the control points within a defined control box.This ensures surface modification accuracy and smoothness,similar to those provided by non-uniform rational B-splines.Subsequently,radial basis functions are used to interpolate the movements of the control points with a limited set of design points.The developed method effectively maintains the mesh quality and simulation efficiency.By applying this method to surface and grid adaptation,a regression model is proposed in the form of a second-order polynomial to represent the relationship between the geometric parameters and design variables.This polynomial is then used to introduce geometric constraints.Furthermore,a radial basis function surrogate model for the calm-water resistance is constructed to approximate the objective function.An enhanced optimization framework is proposed for CFD–based hull optimization and applied to KVLCC2 to validate its feasibility and efficiency.

GENERATION AND APPLICATION OF UNSTRUCTURED ADAPTIVE MESHES WITH MOVING BOUNDARIES

This paper presents a method to generate unstructured adaptive meshes with moving boundaries and its application to CFD. Delaunay triangulation criterion in conjunction with the automatic point creation is used to generate 2 D and 3 D unstructured grids. A local grid regeneration method is proposed to cope with moving boundaries. Numerical examples include the interactions of shock waves with movable bodies and the movement of a projectile within a ram accelerator, illustrating an efficient and robust mesh generation method developed.

Optimization of Air Route Network Nodes to Avoid ″Three Areas″ Based on An Adaptive Ant Colony Algorithm

Air route network(ARN)planning is an efficient way to alleviate civil aviation flight delays caused by increasing development and pressure for safe operation.Here,the ARN shortest path was taken as the objective function,and an air route network node(ARNN)optimization model was developed to circumvent the restrictions imposed by″three areas″,also known as prohibited areas,restricted areas,and dangerous areas(PRDs),by creating agrid environment.And finally the objective function was solved by means of an adaptive ant colony algorithm(AACA).The A593,A470,B221,and G204 air routes in the busy ZSHA flight information region,where the airspace includes areas with different levels of PRDs,were taken as an example.Based on current flight patterns,a layout optimization of the ARNN was computed using this model and algorithm and successfully avoided PRDs.The optimized result reduced the total length of routes by 2.14% and the total cost by 9.875%.

An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

This paper develops a high-order adaptive scheme for solving nonlinear Schrödinger equa-tions.The solutions to such equations often exhibit solitary wave and local structures,which make adaptivity essential in improving the simulation efficiency.Our scheme uses the ultra-weak discontinuous Galerkin(DG)formulation and belongs to the framework of adaptive multiresolution schemes.Various numerical experiments are presented to demon-strate the excellent capability of capturing the soliton waves and the blow-up phenomenon.

A SELF－ADAPTIVE FINITE ELEMENT METHOD FOR SOLVING 2－D EULER EQUATIONS ON THE UNSTRUCTURED MESHES

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