Fine Uniform Grid Research Articles

GPU‐Accelerated Urban Flood Modeling Using a Nonuniform Structured Grid and a Super Grid Scale River Channel

AbstractNew remote sensing technologies, and the meter‐scale geospatial data they create, now allow for detailed urban landscape characterization, thereby advancing grid‐based hydrodynamic models. However, using a uniform fine grid over urban catchments generally result in dense grids and can lead to prohibitive computational costs. Moreover, an inability to see below the water surface and measure river bathymetry in most terrain remote sensing can severely impact local‐scale river hydraulics calculations given the significant volume of water conveyed by the channel. This paper introduces a super grid channel model which allow river channels with any width above that of the grid resolution to be simulated in 1D manner. As an extension of a previous subgrid model, this integration facilitates a seamless transition between subgrid and super grid channels, accommodating situations where channel width may surpass or fall below the grid resolution. The key contribution is the integration of the novel 1D channel representation with a nonuniform structured 2D floodplain hydrodynamic model and then coding this for application on GPU. Compared with the previous pure 2D nonuniform structured approaches, the new model presents an efficient compromise for riverine urban flooding where we are less concerned about fine‐scale details of in‐channel flow. Three tests reveal that the proposed model maintains accuracy but with significantly reduced computational cost. By leveraging GPU architectures, a ∼10× speedup compared to CPU computations is achieved, and a typical 6‐day urban flooding problem (domain size 1.42 km2) at 1 m resolution can be achieved within 10 hr on a single 8 GB GPU.

Explicit and unconditionally stable finite-difference time-domain subgridding algorithm based on hanging variables

Due to the limitation of the stability condition, the finite-difference time-domain (FDTD) method cannot efficiently deal with electromagnetic problems containing fine structures. The explicit and unconditionally stable (EUS) FDTD method can eliminate the constraint of the stability condition and improve the simulation efficiency of fine structures by filtering out the unstable modes for the system matrix. However, the EUS-FDTD method needs to solve the eigenvalues of the numerical system matrix, and the symmetry of the numerical system matrix needs to be ensured when the subgridding scheme is used to discretize targets containing fine structures. The existing EUS-FDTD subgridding method encounters some problems such as complex implementation and insufficient accuracy. In order to solve the above problems, in this work, the hanging variables subgridding (HVS) algorithm is applied to the EUS-FDTD algorithm. Starting from the symmetry of the system matrix, the stability of the hanging variables subgridding algorithm is proven, and a high-precision, stable, and easy-to-implement HVS-EUS-FDTD scheme is proposed. Numerical examples of the radiation of linear magnetic currents in free space, electromagnetic scattering of multiple dielectric objects, and a three-dimensional cavity containing a medium demonstrate the stability, high accuracy and efficiency of the proposed method. Numerical experiments show that the computational efficiency of the HVS-EUS-FDTD algorithm can be improved hundreds of times compared with that of the uniform fine grid FDTD algorithm, and the highest computational efficiency can be improved up to ratio (the size ratio of coarse grid to fine grid) times compared with that of the HVS-FDTD algorithm.

Adaptive interface-Mesh un-Refinement (AiMuR) based sharp-interface level-set-method for two-phase flow

Adaptive interface-Mesh un-Refinement (AiMuR) based Sharp-Interface Level-Set-Method (SI-LSM) is proposed for both uniform and non-uniform Cartesian-Grid. The AiMuR involves interface location based dynamic un-refinement (with merging of the four control volumes) of the Cartesian grid away from the interface. The un-refinement is proposed for the interface solver only. A detailed numerical methodology is presented for the AiMuR and ghost-fluid method based SI-LSM. Advantage of the novel as compared to the traditional SI-LSM is demonstrated with a detailed qualitative as well as quantitative performance study, involving the SI-LSMs on both coarse grid and fine grid, for three sufficiently different two-phase flow problems: dam break, breakup of a liquid jet and drop coalescence. A superior performance of AiMuR based SI-LSM is demonstrated - the AiMuR on a coarser non-uniform grid ( $$NU_{c}^{AiMuR}$$ ) is almost as accurate as the traditional SI-LSM on a uniform fine grid ( $$U_{f}$$ ) and takes a computational time almost same as that by the traditional SI-LSM on a uniform coarse grid ( $$U_{c}$$ ). The AMuR is different from the existing Adaptive Mesh Refinement (AMR) as the former involves only mesh un-refinement while the later involves both refinement and un-refinement of the mesh. Moreover, the proposed computational development is significant since the present adaptive un-refinement strategy is much simpler to implement as compared to that for the commonly used adaptive refinement strategies. The proposed numerical development can be extended to various other multi-physics, multi-disciplinary and multi-scale problems involving interfaces.

Regularizing the 3-D teleseismic wavefield for receiver function imaging using a radial basis function

SUMMARY Receiver function is important for imaging crustal and upper-mantle discontinuities. However, sparsely scattered stations could introduce imaging artefacts or misinterpretations for complex structures. We regularize a 3-D teleseismic wavefield to reduce these artefacts using radial basis function interpolation. First, we evaluated the feasibility of wavefield regularization with several typical models using synthetic data. The results demonstrate the high reliability of our method for recovering local 2-D and 3-D structures, even when seismic stations intentionally miss 95 per cent of a uniform fine grid. Then, we applied this method to sparsely deployed stations in Northeast China. The waveforms reconstructed from surrounding stations show good consistency with the observed waveforms; furthermore, the imaging results using the regularized data are highly comparable with the reference results obtained by using a dense 2-D seismic array of 60 stations (with a spacing of 10–17 km), even though our input data were mainly contributed by only 9 stations (with an average spacing of ∼80 km). Our results show better continuity of the 3-D topography of the subsurface compared with that obtained by the traditional method. Our regularization method could significantly improve the spatial resolution of receiver function imaging for both sparse and dense distributions of seismic stations, especially for imaging relatively complex structures with lateral variations.

Dispersion and Stability Analysis for TLM Unstructured Block Meshing

The insatiable demand to optimize and engineer new functionalities in electromagnetic devices has risen their geometrical complexity to unprecedented levels. This usually results in computationally multiscale problems with some tiny components of great importance to the overall behavior of these devices. Block meshing is a powerful technique for treating such problems. The usage of variable mesh size is adaptable for adequately representing the geometrical details without exhausting the computational resources with a uniform fine grid in the entire computational domain. In addition, block meshing allows the use of cubic cells only for which time step in maximum and velocity error is minimal in each subregion. In this article, we present a mathematical formulation for stability and dispersion analysis when using block meshing in the transmission-line matrix (TLM) method. These relations permit us to compute the maximum mesh size and time step that guarantee a tolerable level of numerical dispersion, hence minimizing the computational expenditures. Moreover, we study the case of adopting the local time step and demonstrate the origins of the instability that may appear in this case. Finally, some numerical experiments are presented to show the advantages of the proposed approach when using TLM block meshing. A similar procedure can be used for FDTD or FIT with block meshing.

Lid driven flow field statistics: A non-conforming finite element Simulation

In this paper, we implement the non-conforming finite elements for the simulation of two dimensional incompressible Newtonian fluid flows. The classical driven cavity benchmark problem is simulated on a very fine uniform grid. In order to discretize the governing partial differential equations (PDEs) we employ the rotated bilinearQ˜1element for velocity and piecewise constant Q0element for the pressure on the general quadrilateral meshes. In L2-norm this pair is 2ndorder accurate for velocity and provides 1st order accuracy for pressure approximations. For the solution of non-linear systems arising from the discretization, Newton’s method is used as an outer nonlinear iteration and for the inner linear sub problems, the direct solver UMFPACK is utilized. The results have been compared with the reference data present in the literature and a good agreement is found.

Efficient surface water flow simulation on static Cartesian grid with local refinement according to key topographic features

Aiming at improving the computational efficiency without accuracy losses for surface water flow simulation, this paper presents a structured but non-uniform grid system incorporated into a Godunov-type finite volume scheme. The proposed grid system can detect the key topographic features in the computational domain where high-resolution mesh is in need for reliably solving the shallow water equations. The mesh refinement is automatically carried out in these areas while the mesh in the rest of the domain remains coarse. The criterion determining the refinement is suggested by a dimensionless number with a fixed value of 0.2 after sensitivity analysis. Three laboratory and field-scale test cases are employed to demonstrate the performance of the model for flow simulations on the new non-uniform grids. In all of the tests, the grid system is shown to successfully generate high-resolution mesh only in those areas with abruptly changing topographic features that dominate the flooding processes. To produce numerical solutions of similar accuracy, the non-uniform grid based model is able to accelerate by about two times comparing with the fine uniform grid based counterpart.

Local Time-Step TLM Unstructured Block Meshing for Electromagnetic and Bio-Thermal Applications

Many practical problems are of multiscale nature. This issue appears when relatively small regions inside the computational domain have a large impact on the overall behavior of the structure. This requires using small mesh size in these regions of interest for adequately representing their geometrical details. Using uniform fine grid everywhere is computationally very exhaustive. The use of irregular structured grids may lead to some memory cost reduction as compared to uniform fine mesh. However, it imposes a smaller time step and yields higher dispersion as cell aspect ratio deviates from unity. The alternate approach is to use block meshing which allows fine discretization in regions with fine details and larger mesh size in quasi homogenous regions. The main disadvantage of this technique is that a global time step corresponding to the Courant–Friedrichs–Lewy (CFL) limit in the smallest cells should be used to avoid long-term instability. In this paper, we present a time-domain transmission-line matrix (TLM) algorithm for interfacing block regions that allows the use of local time steps. Hence, cubic cells can be optimally used in all regions with substantial coarseness error reduction. Applied to both electromagnetic and heat transfer TLM models, simulation experiments are presented to show the behavior and performances of the algorithm.

A parallel and adaptive hybridized discontinuous Galerkin method for anisotropic nonhomogeneous diffusion

The diffusion equation in anisotropic and nonhomogeneous media arises in the study of flow through porous media with sharp material interfaces. We discuss the solution of this problem by a hybrid discontinuous Galerkin (HDG) method. The method can be applied in three steps. First, we use a condensation technique to derive the scalar variable and the flux inside each element in terms of the numerical trace on the faces of that element. Then we form a global system of equations to solve for these numerical traces. We then solve the equation inside each element for the internal unknowns using the obtained numerical traces in the global solve step. Similar to other DG variants, HDG is a locally conservative method, and a noticeable share of calculations are performed independently within each element. In a mesh with pth order elements (p≥0), this method gives p+1 order of accuracy for smooth solutions for both the scalar variable and the flux. Moreover, by using a simple post-processing technique, one can reach an accuracy of order p+2 for the scalar variable. To be able to handle problems with sharp material discontinuities, we use an adaptive refinement strategy, with octree grid structure. Hence, we avoid a larger global system which would arise from a fine uniform grid. In this process, we refine those elements with highest gradient of flux, at two sides of their faces. We have also utilized shared and distributed memory parallelism to enhance the performance of the method. The method is implemented using different modules of deal.II (Bangerth and Kanschat, 1999), PETSc (Balay et al., 2015), p4est (Burstedde et al., 2011) and Hypre (Falgout and Yang, 2002). To demonstrate the accuracy, efficiency, scalability, and flexibility of the method, several two and three dimensional numerical experiments are studied.

AN ADAPTIVE FINITE DIFFERENCE METHOD USING FAR-FIELD BOUNDARY CONDITIONS FOR THE BLACK-SCHOLES EQUATION

We present an accurate and efficient numerical method for solving the Black-Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid gen- eration: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demon- strate the accuracy and efficiency of the proposed method. Theresults show that the computational time is reduced substantially with the ac- curacy being maintained.

Multiscale/fractional step schemes for the numerical simulation of the rotating shallow water flows with complex periodic topography

In this paper, we study several multiscale/fractional step schemes for the numerical solution of the rotating shallow water equations with complex topography. We consider the case of periodic boundary conditions (f-plane model). Spatial discretization is obtained using a Fourier spectral Galerkin method. For the schemes presented in this paper we consider two approaches. The first approach (multiscale schemes) is based on topography scale separation and the numerical time integration is function of the scales. The second approach is based on a splitting of the operators, and the time integration method is function of the operator considered (fractional step schemes). The numerical results obtained are compared with the explicit reference scheme (Leap-Frog scheme). With these multiscale/fractional step schemes the objective is to propose new schemes giving numerical results similar to those obtained using only one uniform fine grid N×N and a time step Δt, but with a CPU time near the CPU time needed when using only one coarse grid N1×N1, N1<N and/or a time step Δt′>Δt.

Numerical Investigation of Magnetic Field Effects on Mixed Convection Flow in a Nanofluid-filled Lid-driven Cavity

In this work, the stencil adaptive method is applied to investigate the effects of a magnetic field on mixed convection of Al2O3-water nanofluid in a square lid-driven cavity. The incompressible Navier-Stokes equations are solved by an adaptive mesh method which has superior numerical advantages compared to the traditional method on the uniform fine grid. The main objective of this study is to investigate the influence of several pertinent parameters such as the Reynolds number, the Hartmann number and the solid particle volume fraction on the heat transfer performance of the nanofluid. Based on the obtained numerical results, the heat transfer rate increases with an increase of the Reynolds number but, it decreases with an increase of the Hartmann number. Also, the results indicate that heat transfer of the nanofluid could be either enhanced or mitigated with respect to that of the base fluid depending on the Reynolds number.

A new grid structure for domain extension

We present an efficient grid structure that extends a uniform grid to create a significantly larger far-field grid by dynamically extending the cells surrounding a fine uniform grid while still maintaining fine resolution about the regions of interest. The far-field grid preserves almost every computational advantage of uniform grids including cache coherency, regular subdivisions for parallelization, simple data layout, the existence of efficient numerical discretizations and algorithms for solving partial differential equations, etc. This allows fluid simulations to cover large domains that are often infeasible to enclose with sufficient resolution using a uniform grid, while still effectively capturing fine scale details in regions of interest using dynamic adaptivity.

Application of coarse-grid computational fluid dynamics on indoor environment modeling: Optimizing the trade-off between grid resolution and simulation accuracy

Computational fluid dynamics has been playing an important role in building design and indoor environment study for decades. However, the computing cost of grid-independent computational fluid dynamics prevents its application from real-time simulation in many areas, such as building emergency evacuation and hourly-based energy simulation. Although coarse-grid computational fluid dynamics has the potential of being as fast as or faster than real time and can provide valuable information for decision making, the credibility of coarse-grid computational fluid dynamics results is questioned due to the unknown error scale it brings. This study investigates the grid-induced error, evaluates the potential computing cost saving by using a coarse grid, and provides a guideline for optimizing the trade-off between grid resolution and computing cost. The numerical error caused by coarse grid can be minimized by appropriately adapting the distribution of grid size. Following the guideline of coarse-grid specifications, coarse-grid computational fluid dynamics can provide informative prediction that is comparable to a grid-independent result on building environment modeling. The computing cost of computational fluid dynamics with an optimized coarse grid is usually orders of magnitude less than that with uniform fine grid.

An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods

An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources. The interface is represented by the zero level set of a Lipschitz function φ(x,y). Our adaptive mesh refinement is done within a small tube of |φ(x,y)|≤ δ with finer Cartesian meshes. The discrete linear system of equations is solved by a multigrid solver. The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically, therefore, reduce the size of the linear system of the equations. Numerical examples presented show the efficiency of the grid refinement strategy.

Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations

In this paper, we propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework [4,5] with the high order finite difference weighted essentially non-oscillatory (WENO) method in space and the total variation diminishing (TVD) Runge–Kutta (RK) method in time (WENO-RK) [18,10] by a high order coupling. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. The high order coupling of AMR and WENO-RK is accomplished by high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from coarse to fine grid solutions, and at ghost points. The resulting AMR-WENO method is accurate, robust and efficient, due to the mesh adaptivity and very high order spatial and temporal accuracy. We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes. In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.

A Fine Grid Solution for a Lid-Driven Cavity Flow Using Multigrid Method

This article presents the numerical solution of two dimensional fluid flow problem using multigrid method (MGM) full approximation scheme (FAS). Laminar flow in a square cavity with a moving wall is calculated for different Reynolds number on a fine uniform grid of 513×513. The finite volume method through the concepts of colocated grid and SIMPLEC algorithm has been applied. The method is fully conservative and uses higher order schemes (QUICK scheme) for convective fluxes and second order central differencing for diffusion fluxes. The numerical solutions of 2D incompressible flow in a driven cavity are presented. The multigrid implementation is documented in detail. The steady driven cavity solutions are computed for Re ≤ 10000 with maximum absolute residual less than 10−8. Detailed results are presented and the percentage deviation of midplane velocity from the established results is also calculated.

Comparison of Local Grid Refinement Methods for MODFLOW

Many ground water modeling efforts use a finite-difference method to solve the ground water flow equation, and many of these models require a relatively fine-grid discretization to accurately represent the selected process in limited areas of interest. Use of a fine grid over the entire domain can be computationally prohibitive; using a variably spaced grid can lead to cells with a large aspect ratio and refinement in areas where detail is not needed. One solution is to use local-grid refinement (LGR) whereby the grid is only refined in the area of interest. This work reviews some LGR methods and identifies advantages and drawbacks in test cases using MODFLOW-2000. The first test case is two dimensional and heterogeneous; the second is three dimensional and includes interaction with a meandering river. Results include simulations using a uniform fine grid, a variably spaced grid, a traditional method of LGR without feedback, and a new shared node method with feedback. Discrepancies from the solution obtained with the uniform fine grid are investigated. For the models tested, the traditional one-way coupled approaches produced discrepancies in head up to 6.8% and discrepancies in cell-to-cell fluxes up to 7.1%, while the new method has head and cell-to-cell flux discrepancies of 0.089% and 0.14%, respectively. Additional results highlight the accuracy, flexibility, and CPU time trade-off of these methods and demonstrate how the new method can be successfully implemented to model surface water-ground water interactions.

Solving parabolic problems using local defect correction in combination with the finite volume method

AbstractWe present a method for solving partial differential equations characterized by highly localized properties in which the local defect correction (LDC) algorithm for time‐dependent problems is combined with a finite volume discretization. At each time step, LDC computes a numerical solution on a composite grid, a union of a global uniform coarse grid and a local uniform fine grid. The main feature of the method is that the discrete conservation property, typical of the finite volume approach is preserved on the composite grid. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Galerkin finite element methods for wave problems

We compare here the accuracy, stability and wave propagation properties of a few Galerkin methods. The basic Galerkin methods with piecewise linear basis functions (called G1FEM here) and quadratic basis functions (called G2FEM) have been compared with the streamwise-upwind Petrov Galerkin (SUPG) method for their ability to solve wave problems. It is shown here that when the piecewise linear basis functions are replaced by quadratic polynomials, the stencils become much larger (involving five overlapping elements), with only a very small increase in spectral accuracy. It is also shown that all the three Galerkin methods have restricted ranges of wave numbers and circular frequencies over which the numerical dispersion relation matches with the physical dispersion relation — a central requirement for wave problems. The model one-dimensional convection equation is solved with a very fine uniform grid to show the above properties. With the help of discontinuous initial condition, we also investigate the Gibbs’ phenomenon for these methods.