Accelerating the discontinuous Galerkin method for seismic wave propagation simulations using multiple GPUs with CUDA and MPI

We have successfully ported an arbitrary high-order discontinuous Galerkin method for solving the three-dimensional isotropic elastic wave equation on unstructured tetrahedral meshes to multiple Graphic Processing Units (GPUs) using the Compute Unified Device Architecture (CUDA) of NVIDIA and Message Passing Interface (MPI) and obtained a speedup factor of about 28.3 for the single-precision version of our codes and a speedup factor of about 14.9 for the double-precision version. The GPU used in the comparisons is NVIDIA Tesla C2070 Fermi, and the CPU used is Intel Xeon W5660. To effectively overlap inter-process communication with computation, we separate the elements on each subdomain into inner and outer elements and complete the computation on outer elements and fill the MPI buffer first. While the MPI messages travel across the network, the GPU performs computation on inner elements, and all other calculations that do not use information of outer elements from neighboring subdomains. A significant portion of the speedup also comes from a customized matrix–matrix multiplication kernel, which is used extensively throughout our program. Preliminary performance analysis on our parallel GPU codes shows favorable strong and weak scalabilities.     

砂土液化变形的有限元-无网格耦合方法

通过构造Biot固结理论u-p方程的无网格伽辽金-有限元耦合方法,对砂土液化变形问题进行了数值模拟。对于饱和砂土,采用Oka等提出的弹塑性本构模型,同时采用更新的Lagrange计算格式推导了控制方程。耦合方法能够发挥有限元和无网格各自的优点,既避免了由于单元变形扭曲而引起的计算中断,也可节约计算时间,算例验证了该方法在地震液化问题中的有效性。     

Higher-order compositional modeling with Fickian diffusion in unstructured and anisotropic media

We present advances in compositional modeling of two-phase multi-component flow through highly complex porous media. Higher-order methods are used to approximate both mass transport and the velocity and pressure fields. We employ the Mixed Hybrid Finite Element (MHFE) method to simultaneously solve, to the same order, the pressure equation and Darcy's law for the velocity. The species balance equation is approximated by the discontinuous Galerkin (DG) approach, combined with a slope limiter. In this work we present an improved DG scheme where phase splitting is analyzed at all element vertices in the two-phase regions, rather than only as element averages. This approximation is higher-order than the commonly employed finite volume method and earlier DG approximations. The method reduces numerical dispersion, allowing for an accurate capture of shock fronts and lower dependence on mesh quality and orientation. Further new features are the extension to unstructured grids and support for arbitrary permeability tensors (allowing for both scalar heterogeneity, and shear anisotropy). The most important advancement in this work is the self-consistent modeling of two-phase multi-component Fickian diffusion. We present several numerical examples to illustrate the powerful features of our combined MHFE–dg method with respect to lower-order calculations, ranging from simple two component fluids to more challenging real problems regarding CO₂ injection into a vertical domain saturated with a multi-component petroleum fluid.     

地震波动方程的局部间断有限元方法数值模拟

近年来,随着地震波数值模拟对计算精度和效率的要求越来越高,间断有限元方法开始受到越来越多的关注.本文中,针对具有吸收边界条件的二维地震声波波动方程,作者提出了一种基于局部间断有限元方法的数值模拟算法.该算法在空间上使用局部间断有限元方法进行离散,在时间上采用了显式蛙跳格式.在这种时空离散的组合方式下,每个时间步上,此算法在空间剖分的每个单元上的求解计算是相互独立的,因而具有极高的并行性.通过数值算例,我们将该算法与连续有限元方法进行了比较.结果表明,本算法不仅具有对起伏构造的良好适应性,而且在计算效率和计算精度等方面,都具有优越性.     

Dynamic p-enrichment schemes for multicomponent reactive flows

We present a family of p-enrichment schemes. These schemes may be separated into two basic classes: the first, called fixed tolerance schemes, rely on setting global scalar tolerances on the local regularity of the solution, and the second, called dioristic schemes, rely on time-evolving bounds on the local variation in the solution. Each class of p-enrichment scheme is further divided into two basic types. The first type (the Type I schemes) enrich along lines of maximal variation, striving to enhance stable solutions in “areas of highest interest.” The second type (the Type II schemes) enrich along lines of maximal regularity in order to maximize the stability of the enrichment process. Each of these schemes are tested on three model systems. The first is an academic exact system where basic analysis is easily performed. Then we discuss a pair of application model problems arising in coastal hydrology. The first being a contaminant transport model, which addresses a declinature problem for a contaminant plume with respect to a bay inlet setting. And the second, a multicomponent chemically reactive flow model of estuary eutrophication arising in the Gulf of Mexico.     

Discontinuous Galerkin methods for modeling Hurricane storm surge

Storm surge due to hurricanes and tropical storms can result in significant loss of life, property damage, and long-term damage to coastal ecosystems and landscapes. Computer modeling of storm surge can be used for two primary purposes: forecasting of surge as storms approach land for emergency planning and evacuation of coastal populations, and hindcasting of storms for determining risk, development of mitigation strategies, coastal restoration and sustainability.Storm surge is modeled using the shallow water equations, coupled with wind forcing and in some events, models of wave energy. In this paper, we will describe a depth-averaged (2D) model of circulation in spherical coordinates. Tides, riverine forcing, atmospheric pressure, bottom friction, the Coriolis effect and wind stress are all important for characterizing the inundation due to surge. The problem is inherently multi-scale, both in space and time. To model these problems accurately requires significant investments in acquiring high-fidelity input (bathymetry, bottom friction characteristics, land cover data, river flow rates, levees, raised roads and railways, etc.), accurate discretization of the computational domain using unstructured finite element meshes, and numerical methods capable of capturing highly advective flows, wetting and drying, and multi-scale features of the solution.The discontinuous Galerkin (DG) method appears to allow for many of the features necessary to accurately capture storm surge physics. The DG method was developed for modeling shocks and advection-dominated flows on unstructured finite element meshes. It easily allows for adaptivity in both mesh (h) and polynomial order (p) for capturing multi-scale spatial events. Mass conservative wetting and drying algorithms can be formulated within the DG method.In this paper, we will describe the application of the DG method to hurricane storm surge. We discuss the general formulation, and new features which have been added to the model to better capture surge in complex coastal environments. These features include modifications to the method to handle spherical coordinates and maintain still flows, improvements in the stability post-processing (i.e. slope-limiting), and the modeling of internal barriers for capturing overtopping of levees and other structures. We will focus on applications of the model to recent events in the Gulf of Mexico, including Hurricane Ike.     

Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: A comparative study

This work deals with a comparison of different numerical schemes for the simulation of contaminant transport in heterogeneous porous media. The numerical methods under consideration are Galerkin finite element (GFE), finite volume (FV), and mixed hybrid finite element (MHFE). Concerning the GFE we use linear and quadratic finite elements with and without upwind stabilization. Besides the classical MHFE a new and an upwind scheme are tested. We consider higher order finite volume schemes as well as two time discretization methods: backward Euler (BE) and the second order backward differentiation formula BDF (2). It is well known that numerical (or artificial) diffusion may cause large errors. Moreover, when the Péclet number is large, a numerical code without some stabilising techniques produces oscillating solutions. Upwind schemes increase the stability but show more numerical diffusion. In this paper we quantify the numerical diffusion for the different discretization schemes and its dependency on the Péclet number. We consider an academic example and a realistic simulation of solute transport in heterogeneous aquifer. In the latter case, the stochastic estimates used as reference were obtained with global random walk (GRW) simulations, free of numerical diffusion. The results presented can be used by researchers to test their numerical schemes and stabilization techniques for simulation of contaminant transport in groundwater.     

Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures

Contrast in capillary pressure of heterogeneous permeable media can have a significant effect on the flow path in two-phase immiscible flow. Very little work has appeared on the subject of capillary heterogeneity despite the fact that in certain cases it may be as important as permeability heterogeneity. The discontinuity in saturation as a result of capillary continuity, and in some cases capillary discontinuity may arise from contrast in capillary pressure functions in heterogeneous permeable media leading to complications in numerical modeling. There are also other challenges for accurate numerical modeling due to distorted unstructured grids because of the grid orientation and numerical dispersion effects. Limited attempts have been made in the literature to assess the accuracy of fluid flow modeling in heterogeneous permeable media with capillarity heterogeneity. The basic mixed finite element (MFE) framework is a superior method for accurate flux calculation in heterogeneous media in comparison to the conventional finite difference and finite volume approaches. However, a deficiency in the MFE from the direct use of fractional flow formulation has been recognized lately in application to flow in permeable media with capillary heterogeneity. In this work, we propose a new consistent formulation in 3D in which the total velocity is expressed in terms of the wetting-phase potential gradient and the capillary potential gradient. In our formulation, the coefficient of the wetting potential gradient is in terms of the total mobility which is smoother than the wetting mobility. We combine the MFE and discontinuous Galerkin (DG) methods to solve the pressure equation and the saturation equation, respectively. Our numerical model is verified with 1D analytical solutions in homogeneous and heterogeneous media. We also present 2D examples to demonstrate the significance of capillary heterogeneity in flow, and a 3D example to demonstrate the negligible effect of distorted meshes on the numerical solution in our proposed algorithm.     

A positive definite continuous FEM model for advection

This work introduces a continuous and sign-preserving finite element model (FEM) for advection problems. The model is formulated by integrating flux corrected transport technique (FCT) with the characteristic based (CB) FEM. Low order solution for the FCT has the form of element contribution and derives from an upwind monotone scheme with the minimum necessary diffusion for positivity. Selected numerical experiments illustrate the efficacy of the method by comparing results with some successful models, by demonstrating an approximate conservation of the accuracy of original CB algorithm once the complete procedure is operating, and by testing the model for severe time dependent advective flows.     

Winkler地基上材料非线性矩形薄板主参数共振-主共振研究

研究Winkler地基上材料非线性矩形薄板的参数共振-主共振问题。建立Winkler地基上材料非线性矩形薄板的动力学方程。利用Galerkin方法将其转化为非线性振动方程。应用非线性振动的多尺度法求得系统满足主参数共振-主共振条件的近似解,并进行数值计算。分析激励、调谐值、阻尼系数、非线性参数、几何参数对共振响应曲线的影响。