A finite volume method for global electromagnetic induction forward modeling on collocated unstructured grids

Global electromagnetic induction provides an efficient way to probe the electrical conductivity in the Earth’s deep interior.Owing to the increasing geomagnetic data especially from high-accuracy geomagnetic satellites,inverting the Earth’s three-dimensional conductivity distribution on a global scale becomes attainable.A key requirement in the global conductivity inversion is to have a forward solver with high-accuracy and efficiency.In this study,a finite volume method for global electromagnetic induction forward modeling is developed based on unstructured grids.Arbitrary polyhedral grids are supported in our algorithms to obtain high geometric adaptability.We employ a cell-centered collocated variable arrangement which allows convenient discretization for complex geometries and straightforward implementation of multigrid technique.To validate the method,we test our code with two synthetic models and compare our finite volume results with an analytical solution and a finite element numerical solution.Good agreements are observed between our solution and other results,indicating acceptable accuracy of the proposed method.

A Finite Volume Method with Unstructured Triangular Grids for Numerical Modeling of Tidal Current

The finite volume method （FVM） has many advantages in 2-D shallow water numerical simulation. In this study, the finite volume method is used with unstructured triangular grids to simulate the tidal currents. The Roe scheme is applied in the calculation of the intercell numerical flux, and the MUSCL method is introduced to improve its accuracy. The time integral is a two-step scheme of forecast and revision. For the verification of the present method, the Stoker＇s problem is calculated and the result is compared with the mathematically analytic solutions. The comparison indicates that the method is feasible. A sea area of a port is used as an example to test the method established here. The result shows that the present computational method is satisfactory, and it could be applied to the engineering fields.

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.

An improved global-direction stencil based on the face-area-weighted centroid for the gradient reconstruction of unstructured finite volume methods

The accuracy of unstructured finite volume methods is greatly influenced by the gradient reconstruction, for which the stencil selection plays a critical role. Compared with the commonly used face-neighbor and vertex-neighbor stencils, the global-direction stencil is independent of the mesh topology, and characteristics of the flow field can be well reflected by this novel stencil. However, for a high-aspect-ratio triangular grid, the grid skewness is evident, which is one of the most important grid-quality measures known to affect the accuracy and stability of finite volume solvers. On this basis and inspired by an approach of using face-area-weighted centroid to reduce the grid skewness, we explore a method by combining the global-direction stencil and face-area-weighted centroid on high-aspect-ratio triangular grids, so as to improve the computational accuracy. Four representative numerical cases are simulated on high-aspect-ratio triangular grids to examine the validity of the improved global-direction stencil. Results illustrate that errors of this improved methods are the lowest among all methods we tested, and in high-mach-number flow, with the increase of cell aspect ratio, the improved global-direction stencil always has a better stability than commonly used face-neighbor and vertex-neighbor stencils. Therefore, the computational accuracy as well as stability is greatly improved, and superiorities of this novel method are verified.

PERTURBATIONAL FINITE VOLUME METHOD FOR THE SOLUTION OF 2-D NAVIER-STOKES EQUATIONS ON UNSTRUCTURED AND STRUCTURED COLOCATED MESHES

Based on the first-order upwind and second-order central type of finite volume (UFV and CFV) scheme, upwind and central type of perturbation finite volume (UPFV and CPFV) schemes of the Navier-Stokes equations were developed. In PFV method, the mass fluxes of across the cell faces of the control volume (CV) were expanded into power series of the grid spacing and the coefficients of the power series were determined by means of the conservation equation itself. The UPFV and CPFV scheme respectively uses the same nodes and expressions as those of the normal first-order upwind and second-order central scheme, which is apt to programming. The results of numerical experiments about the flow in a lid-driven cavity and the problem of transport of a scalar quantity in a known velocity field show that compared to the first-order UFV and second-order CFV schemes, upwind PFV scheme is higher accuracy and resolution, especially better robustness. The numerical computation to flow in a lid-driven cavity shows that the under-relaxation factor can be arbitrarily selected ranging from (0.3) to (0.8) and convergence perform excellent with Reynolds number variation from 10~2 to 10~4.

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered fi- nite volume/finite element method for solution of the two di- mensional （2D） incompressible Navier-Stokes equations on unstructured grids. An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling. The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by join- ing the centroid of cells sharing the common vertex. For the temporal integration of the momentum equations, an im- plicit second-order scheme is utilized to enhance the com- putational stability and eliminate the time step limit due to the diffusion term. The momentum equations are discretized by the vertex-centered finite volume method （FVM） and the pressure Poisson equation is solved by the Galerkin finite el- ement method （FEM）. The momentum interpolation is used to damp out the spurious pressure wiggles. The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both veloc- ity and pressure. The classic test cases, the lid-driven cavity flow, the skew cavity flow and the backward-facing step flow, show that numerical results are in good agreement with the published benchmark solutions.

3D numerical simulation in acoustic-elastic coupled media with staggered-grid finite-difference method

Acoustic-elastic coupled media is often encountered in most marine explorations, and accurate simulation of acoustic-elastic coupled media is of great significance. At present, the study of acoustic-elastic coupled media still assumes that the solid of the acoustic-elastic coupled media is isotropic, but this assumption is not in accordance with the actual situation. In this paper, we derive the solid media of acoustic-elastic coupled media from isotropic media to anisotropic media, and propose an acoustic-elastic coupled medium based ontransverse isotropic media with vertical symmetric axes(VTI) to improve the accuracy of forward modeling. Based on the relationship between the Thomsen parameter and the coefficient matrix of the anisotropic elastic wave equation, we transform the Thomson parameter into a velocity model with anisotropic properties. We use a staggered grid finite difference method to simulate the propagation of a wavefield in a three-dimensional acoustic-elastic coupled media. We obtain the snapshots of the wave field when the solid of the acoustic-elastic coupled media is an isotropic medium and a VTI media. When the solid of the acoustic-elastic coupled media is considered VTI media, we can observe the qP wave and qS wave that cannot be observed in the isotropic medium from the wave field snapshot. We can also find that the seismic records obtained by the method we use are more realistic. The algorithm proposed in this paper is of great significance for high-precision ocean numerical simulation.

A method of solving the stiffness problem in Biot＇s poroelastic equations using a staggered high-order finite-difference

In modelling elastic wave propagation in a porous medium, when the ratio between the fluid viscosity and the medium permeability is comparatively large, the stiffness problem of Blot＇s poroelastic equations will be encountered. In the paper, a partition method is developed to solve the stiffness problem with a staggered high-order finite-difference. The method splits the Biot equations into two systems. One is stiff, and solved analytically, the other is nonstiff, and solved numerically by using a high-order staggered-grid finite-difference scheme. The time step is determined by the staggered finite-difference algorithm in solving the nonstiff equations, thus a coarse time step may be employed. Therefore, the computation efficiency and computational stability are improved greatly. Also a perfect by matched layer technology is used in the split method as absorbing boundary conditions. The numerical results are compared with the analytical results and those obtained from the conventional staggered-grid finite-difference method in a homogeneous model, respectively. They are in good agreement with each other. Finally, a slightly more complex model is investigated and compared with related equivalent model to illustrate the good performance of the staggered-grid finite-difference scheme in the partition method.

A Coastal Ocean Model of Semi-Implicit Finite Volume Unstructured Grid

A two-dimensional coastal ocean model based on unstructured C-grid is built, in which the momentum equation is discretized on the faces of each cell, and the continuity equation is discretized on the cell. The model is discretized by semi-implicit finite volume method, in that the free surface is semi-implicit and the bottom friction is implicit, thereby removing stability limitations associated with the surface gravity wave and friction. The remaining terms in the momentum equations are discretized explicitly by integral finite volume method and second-order Adams-Bashforth method. Tidal flow in the polar quadrant with known analytic solution is employed to test the proposed model. Finally, the performance of the present model to simulate tidal flow in a geometrically complex domain is examined by simulation of tidal currents in the Pearl River Estuary.

基于2.5D有限元的高密度电法不同装置勘探效果研究

高密度电法是一种适用性广泛的电阻率勘探方法,拥有多种电极排列装置选择。不同装置由于电极分布方式的差异,在针对不同地质目标和测区环境时往往表现出明显不同的探测能力。为探究各排列装置的特点及其适用情况,推导了2.5D电位的变分问题,采用Delaunay三角化算法实现了非结构化网格剖分,进而实现了有限元正演模拟。结合实践,对常见地质情况进行建模,分别使用温纳α、温纳β、施伦贝谢尔、偶极-偶极4种装置开展正反演对比研究。研究发现:(1)在探测未知区域单个异常体时,温纳β、施伦贝谢尔2种排列的效果更好;(2)在探测相邻较近的多个异常体时,应优先采用具有更高水平分辨率的施伦贝谢尔和偶极-偶极排列;(3)对于低阻破碎带的勘探,温纳β和偶极-偶极排列更为适用;(4)当地层具有较好分层界限时,4种排列均可体现良好的分层效果。因此,高密度电法勘探实践中应综合考虑不同排列装置的特点,根据具体情况选用合适的多种排列方式进行测量,并对采集数据进行综合对比分析,以获得更为准确的解释结果。

非均匀网格有限差分法大地电磁测深法正演模拟

为解决有限差分法大地电磁正演模拟中模拟精度与计算效率之间的矛盾,采用非均匀网格有限差分进行了大地电磁响应计算。基于二维大地电磁方程,推导出非均匀网格剖分下的TM极化模式的大地电磁方程的差分格式。通过均匀半空间模型和低阻异常体模型的正演模拟,验证了非均匀网格差分法的正确性,非均匀网格差分法可以准确反映出复杂模型的大地电磁响应特征,识别异常体分布范围。

THE FINITE VOLUME PROJECTION METHOD WITH HYBRID UNSTRUCTURED TRIANGULAR COLLOCATED GRIDS FOR INCOMPRESSIBLE FLOWS

In this article a finite volume method is proposed to solve viscous incompressible Navier-Stokes equations in two-dimensional regions with corners and curved boundaries. A hybrid collocated-grid variable arrangement is adopted, in which the velocity and pressure are stored at the centroid and the circumcenters of the triangular control cell, respectively. The cell flux is defined at the mid-point of the cell face. Second-order implicit time integration schemes are used for convection and diffusion terms. The second-order upwind scheme is used for convection fluxes. The present method is validated by results of several viscous flows.

2001年昆仑山口西MS8.1地震自发破裂过程数值模拟研究

2001年11月14日,青藏高原的东昆仑断裂带发生昆仑山口西MS8.1地震,现场考察和破裂过程反演结果表明这是一次破裂过程非常复杂的走滑地震事件.研究这次地震的自发破裂过程,对于认识大陆型特大地震的发生机理具有重要意义.本文利用曲线网格有限差分法对2001年昆仑山口西MS8.1地震的动力学破裂过程进行研究.首先建立能反映这次地震主要特征的三维非平面断层模型,并以地质考察、以及根据GPS和InSAR观测数据、远场地震记录得到的反演结果为约束,通过数值模拟重现了这次地震破裂过程的主要特征.在此基础上,进一步讨论了背景应力场、断层几何、摩擦系数对断层面滑移量分布、震源时间函数、破裂传播速度的影响.研究结果表明,昆仑山口西MS8.1地震发展成为具有超长破裂尺度的大型地震的重要原因可能是:青藏高原最大水平主压应力方向沿东向的顺时针旋转特征和断层面较低的动摩擦系数.并且沿昆仑山口西地震断层面的动摩擦系数可能是非均匀的,由此产生了沿断层走向复杂变化的应力降,从而导致超剪切破裂的发生,并控制了断层面滑移量分布.

复杂地质条件下浅海区电性源瞬变电磁法三维响应特征

与频域电磁法相比,瞬变电磁法可有效区分空气波和海底电磁响应,在浅海油气藏探测中具有良好的应用前景。近海海域地质条件一般较为复杂,普遍具有强切割地形及复杂构造,使电磁场响应特征变得异常复杂,给数据解释工作带来极大困难。本文基于时域有限元法,采用非结构化网格对复杂地质模型进行剖分,通过构建时域有限元方程并结合偶极子离散的长导线源近似技术以及后退欧拉离散技术,实现了复杂地质条件下浅海区电性源瞬变电磁三维正演模拟。在验证算法精度后,通过复杂地质模型的三维正演,分析了不同海水深度对空气波与海底油气藏目标体的影响,在此基础上,进一步分析了不同围岩电阻率以及海底地形对油气藏目标体分辨率的影响,结果表明:在浅海条件下,脉冲响应受空气波影响较大,阶跃响应受空气波影响较小,随着深度增大空气波影响变小,对油气藏的分辨率也降低;围岩电阻率及海底复杂地形对电性源瞬变电磁影响严重。

四叉树网格线源频率域电磁法二维正演研究

可控源电磁法正演面临的问题,主要表现在源的奇异性和边界条件的处理困难.本文首先利用面积分将源的模拟精度转化为网格的离散精度,然后利用四叉树网格局部任意加密的特性,有效解决了奇异源模拟精度不高的问题.由于大地中的电磁场迅速衰减,在距源很远的侧边界,地下的电磁场主要是由空气中的电磁场感应产生的,而非从源经由大地传播而来,因此在地中使用一维平面波侧边界条件比一阶吸收侧边界条件更为合理,精度更高.通过算例分析,我们还发现空气侧边界的场强叠加了大地介质感应的二次场而显著改变,因此空气侧边界使用基于一次场的一阶吸收边界条件,是导致线源正演边界处计算精度不高的主因.本文中,我们使用四叉树网格将边界发散式向外扩展,以较少的网格覆盖更大的边界区域,来确保核心区的计算精度.研究还发现,线源全区视电阻率能较好地反映地下电性结构的变化,且趋势形态与大地电磁卡尼亚视电阻率相似,使用成熟的大地电磁反演技术对线源全区视电阻率进行反演,可以得到地下电性结构较好的近似模型.特别是倾子计算的全区视电阻率也能反映地下结构的变化,这为无电场观测的地下结构探测提供了可能.但二者之间仍存在差别,线源全区视电阻率还具有一定的几何测深能力,需开发基于线源全区视电阻率正演的二维/三维反演技术.

A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL FOR BI-DIRECTIONAL PEDESTRIAN FLOWS

In this paper, a reactive dynamic user equilibrium model is extended to simulate two groups of pedestrians traveling on crossing paths in a continuous walking facility. Each group makes path choices to minimize the travel cost to its destination in a reactive manner based on instantaneous information. The model consists of a conservation law equation coupled with an Eikonal-type equation for each group. The velocity-density relationship of pedestrian movement is obtained via an experimental method. The model is solved using a finite volume method for the conservation law equation and a fast-marching method for the Eikonal-type equation on unstructured grids. The numerical results verify the rationality of the model and the validity of the numerical method. Based on this continuum model, a number of results, e.g., the formation of strips or moving clusters composed of pedestrians walking to the same destination, are also observed.

The parallel 3D magnetotelluric forward modeling algorithm

3D 磁电机的工作量碲的向前当模特儿的算法那么大传统的连续算法花费一极其大计算时间。然而，向前为算法建模的 3D 能在频率域处理数据，它对平行计算很合适。与 MPI 并且基于 3D 磁电机的流动的分析的优点碲的连续前面的算法，我们建议平行计算的想法并且使用它。三个理论模型被测试，实行效率处于不同状况被比较。结果显示向前为计算建模的平行 3D 是正确的，效率极大地被改进。这个方法对大尺寸合适地球物理的计算。

A Nominally Second-Order Cell-Centered Finite Volume Scheme for Simulating Three-Dimensional Anisotropic Diffusion Equations on Unstructured Grids

We present a finite volume based cell-centered method for solving diffusion equations on three-dimensional unstructured grids with general tensor conduction.Our main motivation concerns the numerical simulation of the coupling between fluid flows and heat transfers.The corresponding numerical scheme is characterized by cell-centered unknowns and a local stencil.Namely,the scheme results in a global sparse diffusion matrix,which couples only the cell-centered unknowns.The space discretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces.It is characterized by the introduction of sub-face normal fluxes and sub-face temperatures,which are auxiliary unknowns.A sub-cellbased variational formulation of the constitutive Fourier law allows to construct an explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered temperature and the adjacent sub-face temperatures.The elimination of the sub-face temperatures with respect to the cell-centered temperatures is achieved locally at each node by solving a small and sparse linear system.This system is obtained by enforcing the continuity condition of the normal heat flux across each sub-cell interface impinging at the node under consideration.The parallel implementation of the numerical algorithm and its efficiency are described and analyzed.The accuracy and the robustness of the proposed finite volume method are assessed by means of various numerical test cases.

“数值传热学”中贴体网格与非结构化网格上的有限容积法对比教学

提出了一套贴体网格和非结构化网格有限容积法的对比教学方法。该方法基于典型案例引入以及对照式、综合式辨析策略,给出了对不规则区域有限容积法涉及贴体网格和非结构化网格内涵的差异,两者对有限容积法实施过程差异,控制方程的异同,控制方程变换的实质以及物理问题适应性5个重要内容进行综合辨析教学的要点及实施过程。该对比教学方法可为“数值传热学”或“计算流体力学”的教学或自学提供参考。

基于非结构外推多重网格法的带地形二维大地电磁各向异性有限元正演

基于非结构化网格的大地电磁各向异性有限元正演模拟若采用经典迭代法,收敛速度慢,不适用于复杂地电模型的正演计算。多重网格法作为经典迭代法的改进算法,是求解椭圆方程离散线性系统的一种有效算法。然而,经典多重网格法依赖于嵌套的正交网格,无法直接应用于非结构化网格问题的求解。笔者提出一种基于半结构化网格,即通过对一个初始的非结构网格进行逐层二分加密,并利用外推瀑布式多重网格算法(EXCMG)快速求解各向异性介质中二维大地电磁有限元正演大规模复线性系统。EXCMG算法运用三角形网格上的外推和高次插值技术,构造新的多网格延拓算子,通过两层粗网格上的数值解构造下层密网上有限元解的高阶逼近,作为多网格磨光算子BiCGStab的迭代初值,加速迭代收敛。对国际标准测试模型(COMMEMI-2D1和COMMEMI-2D4)运用该方法进行测试,得到的视电阻率和相位相对误差均在1%以内,求解时间较BiCGStab、聚合型代数多网格法(AGMG)大幅缩短。EXCMG算法具有较好的拓展性和适应性,可处理复杂地电模型、任意起伏地形和各向异性问题。