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.

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.

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.

Multiple Targets Localization Algorithm Based on Covariance Matrix Sparse Representation and Bayesian Learning

The multi-source passive localization problem is a problem of great interest in signal pro-cessing with many applications.In this paper,a sparse representation model based on covariance matrix is constructed for the long-range localization scenario,and a sparse Bayesian learning algo-rithm based on Laplace prior of signal covariance is developed for the base mismatch problem caused by target deviation from the initial point grid.An adaptive grid sparse Bayesian learning targets localization(AGSBL)algorithm is proposed.The AGSBL algorithm implements a covari-ance-based sparse signal reconstruction and grid adaptive localization dictionary learning.Simula-tion results show that the AGSBL algorithm outperforms the traditional compressed-aware localiza-tion algorithm for different signal-to-noise ratios and different number of targets in long-range scenes.

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.

3D variable-grid full-waveform inversion on GPU

Full-waveform inversion(FWI)is a powerful tool to reconstruct subsurface geophysical parameters with high resolution.As3 D surveys become widely implemented,corresponding 3 D processing techniques are required to solve complex geological cases,while a large amount of computation is the most challenging problem.We propose an adaptive variable-grid 3 D FWI on graphics processing unit devices to improve computational efficiency without losing accuracy.The irregular-grid discretization strategy is based on a dispersion relation,and the grid size adapts to depth,velocity,and frequency automatically.According to the transformed grid coordinates,we derive a modified acoustic wave equation and apply it to full wavefield simulation.The 3 D variable-grid modeling is conducted on several 3 D models to validate its feasibility,accuracy and efficiency.Then we apply the proposed modeling method to full-waveform inversion for source and residual wavefield propagation.It is demonstrated that the adaptive variable-grid FWI is capable of decreasing computing time and memory requirements.From the inversion results of the 3 D SEG/EAGE overthrust model,our method retains inversion accuracy when recovering both thrust and channels.

A Wavelet-Free Approach for Multiresolution-Based Grid Adaptation for Conservation Laws

In recent years the concept of multiresolution-based adaptive discontinuous Galerkin(DG)schemes for hyperbolic conservation laws has been developed.The key idea is to perform a multiresolution analysis of the DG solution using multiwavelets defined on a hierarchy of nested grids.Typically this concept is applied to dyadic grid hierarchies where the explicit construction of the multiwavelets has to be performed only for one reference element.For non-uniform grid hierarchies multiwavelets have to be constructed for each element and,thus,becomes extremely expensive.To overcome this problem a multiresolution analysis is developed that avoids the explicit construction of multiwavelets.

Shipborne radar maneuvering target tracking based on the variable structure adaptive grid interacting multiple model

The trajectory of a shipbome radar target has a certain complexity, randomness, and diversity. Tracking a strong maneuvering target timely, accurately, and effectively is a key technology for a shipbome radar tracking system. Combining a variable structure interacting multiple model with an adaptive grid algorithm, we present a variable structure adaptive grid inter- acting multiple model maneuvering target tracking method. Tracking experiments are performed using the proposed method for five maneuvering targets, including a uniform motion - uniform acceleration motion target, a uniform acceleration motion - uni- form motion target, a serpentine locomotion target, and two variable acceleration motion targets. Experimental results show that the target position, velocity, and acceleration tracking errors for the five typical target trajectories are small. The method has high tracking precision, good stability, and flexible adaptability.

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

In order to suppress the failure of preserving positivity of density or pres-sure,a positivity-preserving limiter technique coupled with h-adaptive Runge-Kutta discontinuous Galerkin(RKDG)method is developed in this paper.Such a method is implemented to simulate flows with the large Mach number,strong shock/obstacle interactions and shock diffractions.The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented.This ap-proach directly uses the cell solution polynomial of DG finite element space as the interpolation formula.The method is validated by the well documented test ex-amples involving unsteady compressible flows through complex bodies over a large Mach numbers.The numerical results demonstrate the robustness and the versatility of the proposed approach.

Transition of Liesegang Precipitation Systems: Simulations with an Adaptive Grid PDE Method

The dynamics of the Liesegang type pattern formation is investigated in a centrally symmetric two-dimensional setup.According to the observations in real experiments,the qualitative change of the dynamics is exhibited for slightly different initial conditions.Two kinds of chemical mechanisms are studied;in both cases the pattern formation is described using a phase separation model including the CahnHilliard equations.For the numerical simulations we make use of an adaptive grid PDE method,which successfully deals with the computationally critical cases such as steep gradients in the concentration distribution and investigation of long time behavior.The numerical simulations show a good agreement with the real experiments.

Marker-Based,3-D Adaptive Cartesian Grid Method for Multiphase Flow Around Irregular Geometries

Computational simulations of multiphase flow are challenging because many practical applications require adequate resolution of not only interfacial physics associated with moving boundaries with possible topological changes,but also around three-dimensional,irregular solid geometries.In this paper,we highlight recent efforts made in simulating multiphase fluid dynamics around complex geometries,based on an Eulerian-Lagrangian framework.The approach uses two independent but related grid layouts to track the interfacial and solid boundary conditions,and is capable of capturing interfacial as well as multiphase dynamics.In particular,the stationary Cartesian grid with time dependent,local adaptive refinement is utilized to handle the computation of the transport equations,while the interface shape and movement are treated by marker-based triangulated surface meshes which freely move and interact with the Cartesian grid.The markers are also used to identify the location of solid boundaries and enforce the no-slip condition there.Issues related to the contact line treatment,topological changes of multiphase fronts during merger or breakup of objects,and necessary data structures and solution techniques are also highlighted.Selected test cases including spacecraft fuel tank flow management and liquid plug flow dynamics are presented.

Study on grid adaptability testing methodology for wind turbines

The use of wind power has grown rapidly in recent years.Wind power is a clean source of energy,but can have negative impacts on the distribution grid.The influence of large-scale wind power integration on the safe and stable operation of a power system cannot be ignored.It is necessary and urgent to achieve grid adaptability for wind turbines in China.Using a 35 kV/6 MVA grid simulator,the performance of a grid is investigated by simulation.Typical grid disturbances such as voltage deviation,frequency fluctuation,voltage unbalance,and distortion can be simulated.A grid adaptability testing methodology was developed and applied to a doubly fed wind turbine with a focus on analyzing real test data to ascertain its three-phase voltage unbalance adaptability,which was successfully demonstrated.The methodology can also be used to guide other grid adaptability tests.

Evaluation of Turbulence Models in Predicting Hypersonic and Subsonic Base Flows Using Grid Adaptation Techniques

The flows behind the base of a generic rocket, at both hypersonic and subsonic flow conditions, are numerically studied. The main concerns are addressed to the evaluation of turbulence models and the using of grid adaptation techniques. The investigation focuses on two configurations, related to hypersonic and subsonic experiments. The applicability tests of different turbu- lence models are conducted on the level of two-equation models calculating the steady state solution of the Reynolds-averaged Navier-Stokes（RANS） equations. All used models, the original Wilcox k-co, the Menter shear-stress transport （SST） and the ex- plicit algebraic Reynolds stress model（EARSM） formulation, predict an asymmetric base flow in both cases caused by the support of the models. A comparison with preliminary experimental results indicates a preference for the SST and EARSM results over the results from the older k-co model. Sensitivity studies show no significant influence of the grid topology or the location of the laminar to turbulent transition on the base flow field, but a strong influence of even small angles of attack is reported from the related experiments.

A fast and automatic full-potential finite volume solver on Cartesian grids for unconventional configurations

To meet the requirements of fast and automatic computation of subsonic and transonic aerodynamics in aircraft conceptual design,a novel finite volume solver for full potential flows on adaptive Cartesian grids is developed in this paper.Cartesian grids with geometric adaptation are firstly generated automatically with boundary cells processed by cell-cutting and cell-merging algorithms.The nonlinear full potential equation is discretized by a finite volume scheme on these Cartesian grids and iteratively solved in an implicit fashion with a generalized minimum residual（GMRES） algorithm.During computation,solution-based mesh adaptation is also applied so as to capture flow features more accurately.An improved ghost-cell method is proposed to implement the non-penetration wall boundary condition where the velocity-potential of a ghost cell is modified by an analytic method instead.According to the characteristics of the Cartesian grids,the Kutta condition is applied by specially computing the gradients on Kutta-faces without directly assigning the potential jump to cells adjacent wake faces,which can significantly improve the solution converging speed.The feasibility and accuracy of the proposed method are validated by several typical cases of sub/transonic flows around an ONERA M6 wing,a DLR-F4 wing-body,and an unconventional figuration of a blended wing body（BWB）.The validation cases demonstrate a fast convergence with fully automatic grid treatment and computation,and the results suggest its capacity in application for aircraft conceptual design.

A Geometric Space-Time Multigrid Algorithm for the Heat Equation

We study the time-dependent heat equation on its space-time domain that is discretised by a k-spacetree.k-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint.The paper presents a full approximation storage geometric multigrid implementation for this setting that combines the smoothing properties of multigrid for the equation’s elliptic operator with a multiscale solution propagation in time.While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded,the holistic approach promises to exhibit a better parallel scalability than classical time stepping,adaptive dynamic refinement in space and time fall naturally into place,as well as the treatment of periodic boundary conditions of steady cycle systems,on-time computational steering is eased as the algorithm delivers guesses for the solution’s long-term behaviour immediately,and,finally,backward problems arising from the adjoint equation benefit from the the solution being available for any point in space and time.

An Adaptive,Finite Difference Solver for the Nonlinear Poisson-Boltzmann Equation with Applications to Biomolecular Computations

We present a solver for the Poisson-Boltzmann equation and demonstrate its applicability for biomolecular electrostatics computation.The solver uses a level set framework to represent sharp,complex interfaces in a simple and robust manner.It also uses non-graded,adaptive octree grids which,in comparison to uniform grids,drastically decrease memory usage and runtime without sacrificing accuracy.The basic solver was introduced in earlier works[16,27],and here is extended to address biomolecular systems.First,a novel approach of calculating the solvent excluded and the solvent accessible surfaces is explained;this allows to accurately represent the location of the molecule’s surface.Next,a hybrid finite difference/finite volume approach is presented for discretizing the nonlinear Poisson-Boltzmann equation and enforcing the jump boundary conditions at the interface.Since the interface is implicitly represented by a level set function,imposing the jump boundary conditions is straightforward and efficient.

A three-dimensional solid-liquid two-phase turbulence model with the effect of vegetation in non-orthogonal curvilinear coordinates

A three-dimensional k-ε-Ap solid-liquid two-phase two-fluid model with the effect of vegetation is solved numerically with a finite-volume method on an adaptive grid to study water-sediment movements and bed evolution in vegetated channels. The additional drag force and additional turbulence generation due to vegetation are added to the relevant control equations for simulating the interaction between vegetation and flow. The flow structure and the bed-topography changes in a 60° partly vegetated channel bend are calculated by the model. The numerical results agree well with the measured ones. Calculated and measured results show that the primary flow velocity reduces much in the vegetation zone and increases in the non-vegetation zone, the secondary flow velocity weakens in the vegetation zone and strengthens in the non-vegetation zone, the sediment movement and bed-topography change also weaken in the vegetation zone and strengthen in the non-vegetation zone, a well-planed vegetation arrangement can improve bank stabilization program, and the k-ε-Ap model can deal with bed-load transport with a more reasonable method than the one-fluid model.