Numerical modelling of the propagation of fast landslides using the finite element method

This paper proposes a two‐dimensional (2D) model for the analysis of the propagation of fast landslides involving a fluidized material such as debris and mud flows, flowslides and avalanching flows. The model is based on the Navier–Stokes depth‐integrated equations. To incorporate the effect of steep slopes and centrifugal forces due to the high velocities characterizing the flowslides and the bed curvature, a curvilinear system of reference is used. The corresponding equations of motion are complemented by depth‐averaged constitutive equations and bed friction laws. The resulting set of differential equations are solved using the two‐step Taylor–Galerkin algorithm. This algorithm has been used by the authors to solve hydraulic and dam‐break problems using the finite element method. Owing to the importance of the source term compared to the advection component, the proposed algorithm follows a splitting scheme using a fourth‐order Runge–Kutta method for integrating the friction and slope components. The performance of the overall approach has been checked in a number of examples. The analysis of the results provides insights into the key elements of the model and shows the adequacy of the method to solve real problems where merging and splitting of the flow occur. Copyright © 2004 John Wiley & Sons, Ltd.     

New two‐dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes

In this paper, we introduce an extension of Van Leer's slope limiter for two‐dimensional discontinuous Galerkin (DG) methods on arbitrary unstructured quadrangular or triangular grids. The aim is to construct a non‐oscillatory shock capturing DG method for the approximation of hyperbolic conservative laws without adding excessive numerical dispersion. Unlike some splitting techniques that are limited to linear approximations on rectangular grids, in this work, the solution is approximated by means of piecewise quadratic functions. The main idea of this new reconstructing and limiting technique follows a well‐known approach where local maximum principle regions are defined by enforcing some constraints on the reconstruction of the solution. Numerical comparisons with some existing slope limiters on structured as well as on unstructured meshes show a superior accuracy of our proposed slope limiters. Copyright © 2004 John Wiley & Sons, Ltd.     

Symmetric BEM formulations for 2-D steady-state and transit potential problems

This paper presents a new concept for symmetric boundary element method (SBEM) applicable to 2-D steady-state and transit potential problems. Two kinds of SBEM formulations are derived. Symmetry is obtained simply through matrix manipulation, and no hypersingularity appears. Therefore, SBEM is much easier than the traditional symmetric Galerkin BEM. Compared with the traditional asymmetric BEM, the present SBEM can reduce the computational cost for time domain problems only. However, when applied to BEM/FEM coupling procedure, SBEM can reduce the computational cost for both steady-state and time domain problems. Three numerical examples are included to illustrate the effectiveness and accuracy of the present formulations.     

Application of Galerkin Method to Dynamical Behavior of Viscoelastic Timoshenko Beam with Finite Deformation

The motion equations governing the dynamical behavior of a viscoelasticTimoshenko beam with finite deformation are derived and simplified byGalerkin method. The viscoelastic material is assumed to obey thethree-dimensional fractional derivative constitutive relation. Thedynamical behaviors of the simplified systems with order 1 and order 2are numerically computed and compared by using the computational methodpresented by the authors. The dynamical behaviors of the systems areuniform qualitatively, but there is a little deviation quantitatively.And the truncated system with order 1 is safer than the one of order 2.It is also shown that the lower order system is reasonable. Theinfluences of the load parameter and the fractional derivative parameter(material parameter) on the deflection of the beam are consideredrespectively. The numerical methods in nonlinear dynamics, such as phasediagram, and Poincaré section, are applied to reveal dynamical behaviorsof the nonlinear viscoelastic Timoshenko beam. There are plenty ofdynamical behaviors, such as periodicity, bifurcation, quasi-periodicityand chaos in the dynamical system.     

Multi-grid domain decomposition approach for solution of Navier–Stokes equations in primitive variable form

A new multi-grid (two-grid) pseudospectral element method has been carried out for solution of incompressible flow in terms of primitive variable formulation. The main objective of the proposed method is to apply the multi-grid technique solving the incompressible flow problems associated with three commonly encountered multi-grid environments. In domain decomposition terminology, it includes (I) partially overlapped subdomains, each of which has same types of grids; (II) partially overlapped subdomains, each of which has different types of grids; (III) local adaptive subdomains fully overlapped with the original computational domain (composite grids). The approach for flow problems, complex geometry or not, is to first divide the computational domain into a number of subdomains with the inter-overlapping area (partially or fully overlapped). In categories I and II, the fine-grid or coarse-grid subdomains can be defined by their representation, while in category III the fine-grid or coarse-grid subdomains are defined as usual. Next, implement the Schwarz Alternating Procedure (SAP) to exchange the data among subdomains, where the coarse-grid correction is used to remove the high frequency error that occurs when the data interpolation from the fine-grid subdomain to the coarse-grid subdomain is conducted. The strategy behind the coarse-grid correction is to adopt the operator of the divergence of velocity field, which intrinsically links the pressure equation, into this process. The solution of each subdomain can be efficiently solved by the direct (or iterative) eigenfunction expansion technique or preconditioned method with the least storage requirement, i.e. O(N²) in 2-D. Numerical results of (i) driven cavity flow (Re = 100,400) with Cartesian grids (category I) in each subdomain, (ii) driven cavity flow (Re = 3200) with local adaptive grids (category III) in each subdomain, and (iii) flow over a cylinder (Re = 250) with ‘O’ grids in one subdomain and Cartesian grids in another (category II) will be presented in the paper to account for the versatility of the proposed multi-grid method.     

Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations

Procedures based on group representation theory, allowing the exploitation of geometrical symmetry in symmetric Galerkin BEM formulations, are investigated. In particular, this investigation is based on the weaker assumption of partial geometrical symmetry, where the boundary has two disconnected components, one of which is symmetric; e.g. this can be very useful for defect identification problems. The main development is expounded in the context of 3D Neumann elastostatic problems, considered as model problems; and then extended to SGBIE formulations for Dirichlet and/or scalar problems. Both Abelian and non‐Abelian finite symmetry groups are considered. The effectiveness of the present approach is demonstrated through numerical examples, where both partial and complete symmetry are considered, in connection with both Abelian and non‐Abelian symmetry groups. Copyright © 2003 John Wiley & Sons, Ltd.     

Thermal Analysis of CNT-Based Nano-Composites by Element Free Galerkin Method

This paper deals with the thermal analysis of carbon nanotube (CNT) based composites by meshless element free Galerkin method. Cylindrical representative volume element (cylindrical RVE) has been chosen to evaluate the thermal properties of nano-composites using multi-domain and simplified approaches. The values of temperature have been calculated at different points and plotted against RVE length and RVE radius. A sensitivity analysis of RVE as well as CNT dimensions has been carried out in detail. The present computations show that the equivalent thermal conductivity is a function of CNT length, CNT radius, RVE length and RVE radius. Based on present numerical simulations, an approximate formula is proposed to calculate the equivalent thermal conductivity of nano-composites. The results obtained by simplified approach have been found in good agreement with those obtained by multi-domain approach.     

间断Galerkin方法的最优误差限

本文考虑了数性双曲方程的离散Ｇａｌｅｒｋｉｎ方法。如果精确到光滑的并且同是几乎均匀的，证明了对于分层常数元的最优的和超收敛的误差限。     

A conservative discontinuous Galerkin scheme for the 2D incompressible Navier–Stokes equations

In this paper we consider a conservative discretization of the two-dimensional incompressible Navier–Stokes equations. We propose an extension of Arakawa’s classical finite difference scheme for fluid flow in the vorticity–stream function formulation to a high order discontinuous Galerkin approximation. In addition, we show numerical simulations that demonstrate the accuracy of the scheme and verify the conservation properties, which are essential for long time integration. Furthermore, we discuss the massively parallel implementation on graphic processing units.     

水平随机地震激励下弹性圆拱的半解析法

结合虚拟激励法与Galerkin法,研究弹性圆拱在水平随机地震作用下随机响应的半解析解.在建立圆拱平面内动力平衡微分方程的基础上,通过选取适当的试函数,应用Galerkin法将动力平衡微分方程转化为线性常微分方程组.通过设定虚拟荷载,采用确定性方法求解响应量的功率谱密度函数的近似解,得到圆拱随机振动问题的闭合解.该方法无须计算拱的振型,对非正交阻尼同样适用.通过算例分析和与有限元计算结果的比较,验证了该方法的计算精度.当采用的试函数与圆拱振型接近时,采用较少的试函数就能获得较高的精度,该方法是一种简便、高效的方法.