A GPU-based numerical model coupling hydrodynamical and morphological processes

Sediment transport simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model coupling hydrodynamical and morphological processes was developed to simulate water flow, sediment transport, and morphological changes. Aiming at accurately predicting the sediment transport and sediment scouring processes, the model resolved the realistic features of sediment transport and used a GPU-based parallel computing technique to the accelerate calculation. This model was created in the framework of a Godunov-type finite volume scheme to solve the shallow water equations (SWEs). The SWEs were discretized into algebraic equations by the finite volume method. The fluxes of mass and momentum were computed by the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated by the proposed a splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which could run on GPUs to substantially accelerate the computation. The aim of the work was to develop a GPU-based numerical model to simulate hydrodynamical and morphological processes. The novelty is the application of the GPU techniques in the numerical model, making it possible to simulate the sediment transport and bed evolution in a high-resolution but efficient manner. The model was applied to two cases to evaluate bed evolution and the effects of the morphological changes on the flood patterns with high resolution. This indicated that the GPU-based high-resolution hydro-geomorphological model was capable of reproducing morphological processes. The computational times for this test case on the GPU and CPU were 298.1 and 4531.2 s, respectively, indicating that the GPU could accelerate the computation 15.2 times. Compared with the traditional CPU high-grid resolution, the proposed GPU-based high-resolution numerical model improved the reconstruction speed more than 2.0–12.83 times for different grid resolutions while remaining computationally efficient.     

TWO DIMENSIONAL NUMERICAL SIMULATION OF FLOW AND GEO-MORPHOLOGICAL PROCESSES NEAR HEADLANDS BY USING UNSTRUCTURED GRID

1 INTRODUCTION In recent years, due to the increase in population and industrial developments, mankind has faced manyproblems associated with rivers, coastal waters and reservoirs. Some of these problems are flood control,water supply, power generation, and irrigation. In addition, making new hydraulic structures changesnatural conditions. Prediction of these changes is necessary for designing such constructions. For solutionof these problems usually an assessment of flow pattern, sedim…     

Smooth Particle Hydrodynamics with nonlinear Moving-Least-Squares WENO reconstruction to model anisotropic dispersion in porous media

Most numerical schemes applied to solve the advection–diffusion equation are affected by numerical diffusion. Moreover, unphysical results, such as oscillations and negative concentrations, may emerge when an anisotropic dispersion tensor is used, which induces even more severe errors in the solution of multispecies reactive transport. To cope with this long standing problem we propose a modified version of the standard Smoothed Particle Hydrodynamics (SPH) method based on a Moving-Least-Squares-Weighted-Essentially-Non-Oscillatory (MLS-WENO) reconstruction of concentrations. This scheme formulation (called MWSPH) approximates the diffusive fluxes with a Rusanov-type Riemann solver based on high order WENO scheme. We compare the standard SPH with the MWSPH for different a few test cases, considering both homogeneous and heterogeneous flow fields and different anisotropic ratios of the dispersion tensor. We show that, MWSPH is stable and accurate and that it reduces the occurrence of negative concentrations compared to standard SPH. When negative concentrations are observed, their absolute values are several orders of magnitude smaller compared to standard SPH. In addition, MWSPH limits spurious oscillations in the numerical solution more effectively than classical SPH. Convergence analysis shows that MWSPH is computationally more demanding than SPH, but with the payoff a more accurate solution, which in addition is less sensitive to particles position. The latter property simplifies the time consuming and often user dependent procedure to define the initial dislocation of the particles.     

A Godunov-Type Scheme for Atmospheric Flows on Unstructured Grids: Scalar Transport

This is the first paper in a two-part series on the implementation of Godunov-type schemes on unstructured grids for atmospheric flow simulations. Construction of a high-resolution flow solver for the scalar transport equation is described in detail. Higher-order accuracy in space is achieved via a MUSCL-type gradient reconstruction after van Leer and the monotonicity of solution is enforced by slope limiters. Accuracy in time is maintained by implementing a multi-stage explicit Runge-Kutta time-marching algorithm. The scheme is conservative and exhibits minimal numerical dispersion and diffusion. Five different benchmark test cases are simulated for the validation of the numerical scheme.     

Solution of the nonlinear transport equation using modified Picard iteration

The transport and fate of reactive chemicals in groundwater is governed by equations which are often difficult to solve due to the nonlinear relationship between the solute concentrations for the liquid and solid phases. The nonlinearity may cause mass balance errors during the numerical simulation in addition to numerical errors for linear transport system. We have generalized the modified Picard iteration algorithm of Celia et al.⁵ for unsaturated flow to solve the nonlinear transport equation. Written in a ‘mixed-form’ formulation, the total solute concentration is expanded in a Taylor series with respect to the solution concentration to linearize the transport equation, which is then solved with a conventional finite element method. Numerical results of this mixed-form algorithm are compared with those obtained with the concentration-based scheme using conventional Picard iteration. In general, the new solver resulted in negligible mass balance errors (< ∥10⁻⁸∥%) and required less computational time than the conventional iteration scheme for the test examples, including transport involving highly nonlinear adsorption under steady-state as well as transient flow conditions. In contrast, mass balance errors resulting from the conventional Picard iteration method were higher than 10% for some highly nonlinear problems. Application of the modified Picard iteration scheme to solve the nonlinear transport equation may greatly reduce the mass balance errors and increase computational efficiency.     

Three‐dimensional numerical modeling of the Bulle effect: the nonlinear distribution of near‐bed sediment at fluvial diversions

The Bulle effect is a phenomenon in which a disproportionately higher amount of near‐bed sediment load at a fluvial diversion moves into the diverted channel, even for cases in which the proportion of water (with respect to the main flow) entering the diversion channel is relatively small. This phenomenon has wide‐ranging implications for both engineered and natural systems: from efficient design of channels to redirect water and sediment for reclaiming sinking deltas, designing navigational channels that do not need frequent dredging, to morphological evolution of river bifurcations. The first ever, and one of the most extensive set of experiments conducted to explore this phenomenon, were conducted by Bulle in 1926 . In the current study the experiments conducted by Bulle have been simulated using an open‐source, free‐surface finite‐element‐based hydrodynamic solver. The main objectives were to explore to what extent the complex phenomenon of the Bulle effect at the scale of a laboratory experiment can be simulated accurately using Reynolds‐averaged Navier–Stokes (RANS)‐based hydrodynamic solver, and to understand the details of the hydrodynamics that Bulle could not analyze through his experiments. The hydrodynamics captured by the simulations were found to match the observations made by Bulle through his experiments, and the distributions of sediment at the diversion predicted by the numerical simulations were found to match the general trend observed in the laboratory experiments. The results from the numerical simulations were also compared with existing one‐dimensional models for sediment distribution at bifurcations, and the three‐dimensional numerical model was found to perform appreciably better. This is expected due to the complex flow features at the diversion, which can only be captured satisfactorily using a three‐dimensional hydrodynamic model. Copyright © 2017 John Wiley & Sons, Ltd.     

A Godunov-Type Scheme for Atmospheric Flows on Unstructured Grids: Euler and Navier-Stokes Equations

In recent years there has been a growing interest in using Godunov-type methods for atmospheric flow problems. Godunov's unique approach to numerical modeling of fluid flow is characterized by introducing physical reasoning in the development of the numerical scheme (van Leer, 1999). The construction of the scheme itself is based upon the physical phenomenon described by the equation sets. These finite volume discretizations are conservative and have the ability to resolve regions of steep gradients accurately, thus avoiding dispersion errors in the solution. Positivity of scalars (an important factor when considering the transport of microphysical quantities) is also guaranteed by applying the total variation diminishing condition appropriately. This paper describes the implementation of a Godunov-type finite volume scheme based on unstructured adaptive grids for simulating flows on the meso-, micro- and urban-scales. The Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver used to calculate the Godunov fluxes is described in detail. The higher-order spatial accuracy is achieved via gradient reconstruction techniques after van Leer and the total variation diminishing condition is enforced with the aid of slope-limiters. A multi-stage explicit Runge-Kutta time marching scheme is used for maintaining higher-order accuracy in time. The scheme is conservative and exhibits minimal numerical dispersion and diffusion. The subgrid scale diffusion in the model is parameterized via the Smagorinsky-Lilly turbulence closure. The scheme uses a non-staggered mesh arrangement of variables (all quantities are cell-centered) and requires no explicit filtering for stability. A comparison with exact solutions shows that the scheme can resolve the different types of wave structures admitted by the atmospheric flow equation set. A qualitative evaluation for an idealized test case of convection in a neutral atmosphere is also presented. The scheme was able to simulate the onset of Kelvin-Helmholtz type instability and shows promise in simulating atmospheric flows characterized by sharp gradients without using explicit filtering for numerical stability.     

Modelling dam-break flows over mobile beds using a 2D coupled approach

Dam-break flows usually propagate along rivers and floodplains, where the processes of fluid flow, sediment transport and bed evolution are closely linked. However, the majority of existing two-dimensional (2D) models used to simulate dam-break flows are only applicable to fixed beds. Details are given in this paper of the development of a 2D morphodynamic model for predicting dam-break flows over mobile beds. In this model, the common 2D shallow water equations are modified, so that the effects of sediment concentrations and bed evolution on the flood wave propagation can be considered. These equations are used together with the non-equilibrium transport equations for graded sediments and the equation of bed evolution. The governing equations are solved using a matrix method, thus the hydrodynamic, sediment transport and morphological processes can be jointly solved. The model employs an unstructured finite volume algorithm, with an approximate Riemann solver, based on the Roe-MUSCL scheme. A predictor–corrector scheme is used in time stepping, leading to a second-order accurate solution in both time and space. In addition, the model considers the adjustment process of bed material composition during the morphological evolution process. The model was first verified against results from existing numerical models and laboratory experiments. It was then used to simulate dam-break flows over a fixed bed and a mobile bed to examine the differences in the predicted flood wave speed and depth. The effects of bed material size distributions on the flood flow and bed evolution were also investigated. The results indicate that there is a great difference between the dam-break flow predictions made over a fixed bed and a mobile bed. At the initial stage of a dam-break flow, the rate of bed evolution could be comparable to that of water depth change. Therefore, it is often necessary to employ the turbid water governing equations using a coupled approach for simulating dam-break flows.     

Two-dimensional numerical modeling of dam-break flows over natural terrain using a central explicit scheme

A two-dimensional (2D) numerical model has been developed to solve shallow water equations for simulation of dam-break flows. The spatial derivatives are discretized using a well-balanced explicit central upwind conservative scheme. The scheme is Riemann solver free and guarantees the positivity of the flow depth over complex topography if the Courant number is kept less than 0.25. The time integration is performed by Euler’s scheme. The model is verified against analytical results for water surface elevation and discharge for three benchmark test cases. A good agreement between analytical solutions and computed results is observed. The property of well-balancing in still water over an uneven bottom is also confirmed. The model is then validated by simulating a laboratory experiment in which a dam break flow propagates over a triangular obstacle. The model performance was found to be satisfactory. A dam break laboratory experimental test case on a frictionless horizontal bottom is also simulated for 2D validation of the model, and good agreement between simulation and the experimental data is observed. The suitability of the proposed model for real life applications is demonstrated by simulating the Malpasset dam-break event, which occurred in 1959 in France. The computed arrival time of the flood wave front and the maximum flow depths at various observation points matched well with the measurements on a 1/400 scale physical model. The overall performance indicates that this model can be applied for simulation of dam-break waves in real life cases.     

Development and validation of a morphological model for multiple sediment classes

The complexity of sediment dynamics in aquatic systems can be better understood by applying numerical models. The development of a comprehensive morphological model is presented in this paper.The model aims to predict the sediment transport and bed evolution in natural systems composed of different sediment types. The morphological model was implemented in MOHID, a modelling system that solves the three-dimensional hydrodynamics and advection-diffusion transport of suspended sediments. Multiple sediment classes were taken into account(non-cohesive and cohesive) considering the effects of sediment mixtures and bed consolidation on resistance to erosion. To represent bottom stratigraphy, the bottom column can be divided into several layers. The key points of the simulated processes are discussed in this paper. Model results are assessed in six test cases through comparison with analytic solutions or experimental data. The outcomes demonstrate the model's capacity to simulate the transport dynamics of non-cohesive and cohesive sediments. The speed up of morphological changes by an acceleration factor permitted modelling bed evolution for long time periods.Moreover, a test case for the Tagus Estuary demonstrated the model's capacity for generating realistic sediment distribution based on the local hydrodynamic conditions. Limitations in the availability of bed composition data can be overcome by considering a warm-up run to provide realistic initial conditions for further predictions of morphological developments.     

Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed

In this paper, we study the numerical approximation of the two-dimensional morphodynamic model governed by the shallow water equations and bed-load transport following a coupled solution strategy. The resulting system of governing equations contains non-conservative products and it is solved simultaneously within each time step. The numerical solution is obtained using a new high-order accurate centered scheme of the finite volume type on unstructured meshes, which is an extension of the one-dimensional PRICE-C scheme recently proposed in Canestrelli et al. (2009) [5]. The resulting first-order accurate centered method is then extended to high order of accuracy in space via a high order WENO reconstruction technique and in time via a local continuous space–time Galerkin predictor method. The scheme is applied to the shallow water equations and the well-balanced properties of the method are investigated. Finally, we apply the new scheme to different test cases with both fixed and movable bed. An attractive future of the proposed method is that it is particularly suitable for engineering applications since it allows practitioners to adopt the most suitable sediment transport formula which better fits the field data.     

Newtonian nudging for a Richards equation-based distributed hydrological model

The objective of data assimilation is to provide physically consistent estimates of spatially distributed environmental variables. In this study a relatively simple data assimilation method has been implemented in a relatively complex hydrological model. The data assimilation technique is Newtonian relaxation or nudging, in which model variables are driven towards observations by a forcing term added to the model equations. The forcing term is proportional to the difference between simulation and observation (relaxation component) and contains four-dimensional weighting functions that can incorporate prior knowledge about the spatial and temporal variability and characteristic scales of the state variable(s) being assimilated. The numerical model couples a three-dimensional finite element Richards equation solver for variably saturated porous media and a finite difference diffusion wave approximation based on digital elevation data for surface water dynamics. We describe the implementation of the data assimilation algorithm for the coupled model and report on the numerical and hydrological performance of the resulting assimilation scheme. Nudging is shown to be successful in improving the hydrological simulation results, and it introduces little computational cost, in terms of CPU and other numerical aspects of the model’s behavior, in some cases even improving numerical performance compared to model runs without nudging. We also examine the sensitivity of the model to nudging term parameters including the spatio-temporal influence coefficients in the weighting functions. Overall the nudging algorithm is quite flexible, for instance in dealing with concurrent observation datasets, gridded or scattered data, and different state variables, and the implementation presented here can be readily extended to any of these features not already incorporated. Moreover the nudging code and tests can serve as a basis for implementation of more sophisticated data assimilation techniques in a Richards equation-based hydrological model.     

A finite volume upwind scheme for the solution of the linear advection–diffusion equation with sharp gradients in multiple dimensions

A finite volume upwind numerical scheme for the solution of the linear advection equation in multiple dimensions on Cartesian grids is presented. The small-stencil, Modified Discontinuous Profile Method (MDPM) uses a sub-cell piecewise constant reconstruction and additional information at the cell interfaces, rather than a spatial extension of the stencil as in usual methods. This paper presents the MDPM profile reconstruction method in one dimension and its generalization and algorithm to two- and three-dimensional problems. The method is extended to the advection–diffusion equation in multiple dimensions. The MDPM is tested against the MUSCL scheme on two- and three-dimensional test cases. It is shown to give high-quality results for sharp gradients problems, although some scattering appears. For smooth gradients, extreme values are best preserved with the MDPM than with the MUSCL scheme, while the MDPM does not maintain the smoothness of the original shape as well as the MUSCL scheme. However the MDPM is proved to be more efficient on coarse grids in terms of error and CPU time, while on fine grids the MUSCL scheme provides a better accuracy at a lower CPU.     

Numerical methods for reactive transport on rectangular and streamline-oriented grids

Coupling advection-dominated transport to reactive processes leads to additional requirements and limitations for numerical simulation beyond those for non-reactive transport. Particularly, both monotonicity avoiding the occurence of negative concentrations, and high-order accuracy suppressing artificial diffusion, are necessary to study accurately the reactive interactions of compounds transported in groundwater. These requirements are met by non-linear Eulerian methods. Two cell-centered Finite Volume schemes are presented for the simulation of advection-dominated reactive transport. The first scheme is based on rectangular grids, whereas the second scheme requires streamline-oriented grids the generation of which is explained in an accompanying paper. Although excellent results for conservative transport are obtained by the scheme for rectangular grids, some artificial transverse mixing occurs in the case of multi-component transport. This may lead to erroneous reaction rates if the compounds interact. The transport scheme for streamline-oriented grids, on the other hand, avoids artificial transverse mixing. A quantitative comparison is given by two test cases. A conservative tracer simulation for a five-spot configuration in a heterogeneous aquifer shows a high coincidence of the breakthrough curves obtained for the two methods, whereas a test case of two reacting compounds shows significant differences. In this test case, a rate of convergence with respect to the overall reaction rates lower than first-order is calculated for the rectangular grid.     

Progressive collapse analysis through strength degradation and fracture in the Mixed Lagrangian Formulation

The dynamic analysis of progressive collapse faces a great number of obstacles that often lead to the collapse of the analysis prior to the actual analysis of collapse. Hence, the Mixed Lagrangian Formulation that has been shown to be very robust was adopted as a framework to accommodate such analysis. By modifying the loading function and the numerical scheme, the capabilities of this framework were extended to account for strength degradation and fracture, while some insight to its behavior is introduced as well. The examples presented show a very robust and stable behavior of the numerical scheme in terms of the time step size required, even in cases where a sudden fracture takes place. Copyright © 2009 John Wiley & Sons, Ltd.     

基于并行化直接解法的频率域可控源电磁三维正演

电磁法的三维数值模拟是一个对数值算法和计算机硬件要求都非常高的问题.对常用的微分类方法如有限单元法和有限差分法而言, 求解最后所得的大型线性方程组是至关重要的一步, 直接影响到正演算法的实用性.如何高效、稳定且准确地解线性方程长期以来一直是被探讨的问题.本文实现了基于线性系统直接求解技术的频率域可控源电磁(CSEM)三维正演.使用交错网格有限体积法(FV)来离散化关于二次电场的 Helmholtz 方程; 使用直接解法取代传统的迭代解法来求解离散线性系统, 即对系统矩阵进行完全LU分解, 具体通过调用大规模并行矩阵直接求解器(MUMPS)来实现.基于理论模型做了一系列数值实验, 首先证明了直接解法的高精度和稳定性, 并考察了其内存需求、计算时间和并行可伸缩性等主要计算性能, 最后检验了所开发的算法快速模拟多场源 CSEM 问题的能力以及对常规海洋和陆地CSEM模拟的有效性.     

Numerical Computation of Nonlinear Inelastic Waves in Soil

— Godunov's method, a numerical method for solving conservation laws, is applied to nonlinear and inelastic wave propagation in soil. The solution is restricted to the one-dimensional case. An approximate Riemann solver for Godunov's method is presented. The capability of the numerical method is shown by a comparison with the analytical solution of a linear inelastic wave propagation. Finally the behaviour of the nonlinear inelastic soil is described by a hypoplastic constitutive law.     

Mesh type tradeoffs in 2D hydrodynamic modeling of flooding with a Godunov-based flow solver

The effect of mesh type on the accuracy and computational demands of a two-dimensional Godunov-type flood inundation model is critically examined. Cartesian grids, constrained and unconstrained triangular grids, constrained quadrilateral grids, and mixed meshes are considered, with and without local time stepping (LTS), to determine the approach that maximizes computational efficiency defined as accuracy relative to computational effort. A mixed-mesh numerical scheme is introduced so all grids are processed by the same solver. Analysis focuses on a wide range of dam-break type test cases, where Godunov-type flood models have proven very successful. Results show that different mesh types excel under different circumstances. Cartesian grids are 2–3 times more efficient with relatively simple terrain features such as rectilinear channels that call for a uniform grid resolution, while unstructured grids are about twice as efficient in complex domains with irregular terrain features that call for localized refinements. The superior efficiency of locally refined, unstructured grids in complex terrain is attributable to LTS; the locally refined unstructured grid becomes less efficient using global time stepping. These results point to mesh-type tradeoffs that should be considered in flood modeling applications. A mixed mesh model formulation with LTS is recommended as a general purpose solver because the mesh type can be adapted to maximize computational efficiency.     

Simulation of sediment motions using a discrete particle model in the inner surf and swash-zones

A volume-of-fluid Navier–Stokes solver (RIPPLE) was used to simulate inner surf and swash zone flow with a 3 s wave period and wave height of 0.14 m on a planar, 1:10 sloping beach (Iribarren number of 1.0). In addition to other hydrodynamic information, RIPPLE was used to provide high-resolution predictions of the pressure gradient and fluid velocity in the horizontal and vertical dimensions that served as forcing to a discrete particle model (DPM). Sediment transport processes in the inner surf and swash zones were simulated for a thin veneer of sediment particles over a 5 m test section in the DPM. Coupling between RIPPLE and the DPM was one-way such that particle–particle and fluid–particle interactions in the DPM did not provide feedback to alter the flow predicted by RIPPLE. The numerical simulation showed strong sediment suspension localized under vortices that reach the bed. Interestingly, the bulk of the sediment located in the small-scale vortex originated from locations nearly 0.2 m landward. These findings suggest that (1) sediment motion for a single swash event can be significant, (2) that sediment measured in suspension likely originates from locations other than the bed directly below the suspension plume suggesting the importance of sediment advection and (3) that sparse cross-shore measurements in the field will only sporadically capture localized suspension events.