Wavelet Galerkin BEM on unstructured meshes

Abstact The present paper is devoted to the fast solution of boundary integral equations on unstructured meshes by the Galerkin scheme. On the given mesh we construct a wavelet basis providing vanishing moments with respect to the traces of polynomials in the space. With this basis at hand, the system matrix in wavelet coordinates can be compressed to O(Nlog N) relevant matrix coefficients, where N denotes the number of unknowns. The compressed system matrix can be computed within suboptimal complexity by using techniques from the fast multipole method or panel clustering. Numerical results prove that we succeeded in developing a fast wavelet Galerkin scheme for solving the considered class of problems. Mathematics Subject Classification (2000) 47A20; 65F50; 65N38; 65R20; 65T60 This work is supported in part by the SFB 393 Numerical Simulation on Massive Parallel Computers funded by the Deutsche Forschungsgemeinschaft. Dedicated to George C. Hsiao on the occasion of his 70th birthday.     

A meshless Galerkin method for Stokes problems using boundary integral equations

A meshless Galerkin scheme for the simulation of two-dimensional incompressible viscous fluid flows in primitive variables is described in this paper. This method combines a boundary integral formulation for the Stokes equation with the moving least-squares (MLS) approximations for construction of trial and test functions for Galerkin approximations. Unlike the domain-type method, this scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns, thus it is especially suitable for the exterior problems. Compared to other meshless methods such as the boundary node method and the element free Galerkin method, in which the MLS is also introduced, boundary conditions do not present any difficulty in using this meshless method. The convergence and error estimates of this approach are presented. Numerical examples are also given to show the efficiency of the method.     

Convergence analysis and error estimates of the element-free Galerkin method for a class of parabolic evolutionary variational inequalities

This paper is presented for the convergence analysis of the element-free Galerkin method for a class of parabolic evolutionary variational inequalities arising from the heat-servo control problem. The error estimates illustrate that the convergence order depends not only on the number of basis functions in the moving least-squares approximation but also the relationship with the time step and the spatial step. Numerical examples verify the convergence analysis and the error estimates.     

A generalized element-free Galerkin method for Stokes problem

A generalized element-free Galerkin (GEFG) method is developed in this paper for solving Stokes problem in primitive variable form. To obtain stable numerical results for both velocity and pressure, extended terms are only introduced into the approximate space of velocity in a special way as that in the generalized finite element method. Theoretical analysis shows that the GEFG method implies a stabilized formulation similar to that in the variational multiscale element-free Galerkin (VMEFG) method. Numerical results show the efficiency of the present method and reveal that both computational errors and CPU times of the present method are less than those of the VMEFG and the finite element methods.     

On a Galerkin boundary node method for potential problems

A Galerkin boundary node method (GBNM), for boundary only analysis of partial differential equations, is discussed in this paper. The GBNM combines an equivalent variational form of a boundary integral equation with the moving least-squares (MLS) approximations for generating the trial and test functions of the variational formulation. In this approach, only a nodal data structure on the boundary of a domain is required, and boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Formulations of the GBNM using boundary singular integral equations of the second kind for potential problems are developed. The theoretical analysis and numerical results indicate that it is an efficient and accurate numerical method.     

A Galerkin finite element method for time-fractional stochastic heat equation

In this study, a Galerkin finite element method is presented for time-fractional stochastic heat equation driven by multiplicative noise, which arises from the consideration of heat transport in porous media with thermal memory with random effects. The spatial and temporal regularity properties of mild solution to the given problem under certain sufficient conditions are obtained. Numerical techniques are developed by the standard Galerkin finite element method in spatial direction, and Gorenflo–Mainardi–Moretti–Paradisi scheme is applied in temporal direction. The convergence error estimates for both semi-discrete and fully discrete schemes are established. Finally, numerical example is provided to verify the theoretical results.     

Error analysis of a two-grid discontinuous Galerkin method for non-linear parabolic equations

Discontinuous Galerkin (DG) approximations for non-linear parabolic problems are investigated. To linearize the discretized equations, we use a two-grid method involving a small non-linear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates in H¹-norm are obtained, O(h^r+H^r+1) where r is the order of the DG space. The analysis shows that our two-grid DG algorithm will achieve asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H^(r+1)/r). The numerical experiments verify the efficiency of our algorithm.     

Moving Mesh Discontinuous Galerkin Method for Hyperbolic Conservation Laws

In this paper, a moving mesh discontinuous Galerkin (DG) method is developed to solve the nonlinear conservation laws. In the mesh adaptation part, two issues have received much attention. One is about the construction of the monitor function which is used to guide the mesh redistribution. In this study, a heuristic posteriori error estimator is used in constructing the monitor function. The second issue is concerned with the solution interpolation which is used to interpolates the numerical solution from the old mesh to the updated mesh. This is done by using a scheme that mimics the DG method for linear conservation laws. Appropriate limiters are used on seriously distorted meshes generated by the moving mesh approach to suppress the numerical oscillations. Numerical results are provided to show the efficiency of the proposed moving mesh DG method.     

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.     

The BEM for numerical solution of partial fractional differential equations

A numerical method is presented for the solution of partial fractional differential equations (FDEs) arising in engineering applications and in general in mathematical physics. The solution procedure applies to both linear and nonlinear problems described by evolution type equations involving fractional time derivatives in bounded domains of arbitrary shape. The method is based on the concept of the analog equation, which in conjunction with the boundary element method (BEM) enables the spatial discretization and converts a partial FDE into a system of coupled ordinary multi-term FDEs. Then this system is solved using the numerical method for the solution of such equations developed recently by Katsikadelis. The method is illustrated by solving second order partial FDEs and its efficiency and accuracy is validated.     

Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

In this paper, we first split the biharmonic equation Δ² u=f with nonhomogeneous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v=Δu and then apply an hp-mixed discontinuous Galerkin method to the resulting system. The unknown approximation v _h of v can easily be eliminated to reduce the discrete problem to a Schur complement system in u _h , which is an approximation of u. A direct approximation v _h of v can be obtained from the approximation u _h of u. Using piecewise polynomials of degree p≥3, a priori error estimates of u−u _h in the broken H ¹ norm as well as in L ² norm which are optimal in h and suboptimal in p are derived. Moreover, a priori error bound for v−v _h in L ² norm which is suboptimal in h and p is also discussed. When p=2, the preset method also converges, but with suboptimal convergence rate. Finally, numerical experiments are presented to illustrate the theoretical results. Supported by DST-DAAD (PPP-05) project.     

Finite element procedures for nonlinear structures in moving coordinates. Part 1: Infinite bar under moving axial loads

This paper deals with analyzing nonlinear structures under high-speed moving loads by use of the finite element method. The stationary response of an infinite bar posed on a Winkler foundation under constant moving loads is investigated. Instead of the transient analysis, the stationary solution of this problem is obtained by solving a static system in a reference frame which moves with the load for reducing the computation cost. To overcome the difficulty due to numerical instabilities when considering very fast loads (supersonic loads), a new procedure to govern the finite element formulation in moving coordinates is proposed. Comparing numerical solutions with analytical ones shows that the proposed method is valid for all values of load speed. Last, an example of nonlinear elastic foundation is considered to outline the nonlinear effects.     

势问题的复变量无单元Galerkin方法

为提高无单元Galerkin(Element-Free Galerkin, EFG)方法的计算效率,将复变量移动最小二乘法与EFG方法结合,利用控制方程的积分弱形式并采用Lagrange乘子法引入边界条件,提出势问题的复变量无单元Galerkin(Complex Variable EFG,CVEFG)方法,并推导相关公式.与传统的EFG方法相比,该方法采用复变量移动最小二乘法可以减少试函数中的待定系数,从而减少计算量、提高计算效率. 最后,给出数值算例验证该方法的有效性.     

Three time integration methods for incompressible flows with discontinuous Galerkin Boltzmann method

This paper presents three time integration methods for incompressible flows with finite element method in solving the lattice-BGK Boltzmann equation. The space discretization is performed using nodal discontinuous Galerkin method, which employs unstructured meshes with triangular elements and high order approximation degrees. The time discretization is performed using three different kinds of time integration methods, namely, direct, decoupling and splitting. From the storage cost, temporal accuracy, numerical stability and time consumption, we systematically compare three time integration methods. Then benchmark fluid flow simulations are performed to highlight efficient time integration methods. Numerical results are in good agreement with others or exact solutions.     

Development of a meshless Galerkin boundary node method for viscous fluid flows

In this paper, a meshless Galerkin boundary node method is developed for boundary-only analysis of the interior and exterior incompressible viscous fluid flows, governed by the Stokes equations, in biharmonic stream function formulation. This method combines scattered points and boundary integral equations. Some of the novel features of this meshless scheme are boundary conditions can be enforced directly and easily despite the meshless shape functions lack the delta function property, and system matrices are symmetric and positive definite. The error analysis and convergence study of both velocity and pressure are presented in Sobolev spaces. The performance of this approach is illustrated and assessed through some numerical examples.     

A new numerical algorithm for 2D moving boundary problems using a boundary element method

Phase change problems are of practical importance and can be found in a wide range of engineering applications. In the present paper, two proposed numerical algorithms are developed; the first one is general for phase change problems, while the second one is for ablation problems. The boundary elements method is used as a mathematical tool in conjunction with the proposed algorithms. Two test examples were solved and the results agree with the physics of the problems.     

A special BEM for elastostatic analysis of building floor slabs on columns

This work presents a boundary element formulation for the analysis of building floor slabs, without beams, in which columns are coupled with the plate. An alternative formulation of boundary element method is presented, which considers three nodal displacements values (w, ∂w/∂n and ∂w/∂s) for the nodes at the boundary of the plate. In this formulation three boundary equations are written for all nodes at the boundary and in the domain of the plate. As the nodes of the column-plate connections are also represented by three nodal values, all these structural elements can be easily coupled. It is supposed that the cross-sections of the columns remain flat after the deflection and consequently the assumption of linear variation of the stress in the plate-column contact surface is also valid.     

A spacetime discontinuous Galerkin method for hyperbolic heat conduction

Non-Fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation. For applications involving very short length or time scales, hyperbolic conduction models better represent the physical thermal transport processes. This paper reviews the Maxwell-Cattaneo-Vernotte modification of the Fourier conduction law and describes its implementation within a spacetime discontinuous Galerkin (SDG) finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time. A causal, advancing-front meshing procedure enables a patch-wise solution procedure with linear complexity in the number of spacetime elements. An h-adaptive scheme and a special SDG shock-capturing operator accurately resolve sharp solution features in both space and time. Numerical results for one spatial dimension demonstrate the convergence properties of the SDG method as well as the effectiveness of the shock-capturing method. Simulations in two spatial dimensions demonstrate the proposed method’s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place.     

An algorithm for velocity calculation in close proximity to body surfaces in panel methods

To model incompressible flow over a body of arbitrary geometry when using vortex methods, it is necessary to construct an irrotational field to impose the impermeability condition at the surface of the object. In order to achieve this impermeability, this paper uses a boundary integral equation based on the single-layer representation for the velocity potential. Specifically, we formulate this exterior Neumann problem in terms of a source/sink boundary integral equation. The solution to this integral equation is then coupled with an interpolation procedure which smoothes the transition between near-wall and interior regimes. We describe the numerical scheme embedding this strategy and discuss its accuracy and efficiency. For validation purposes, we consider the potential and vortical flow over a circular cylinder, for which an analytical solution and the commonly used method of images are available.