A Grid-based integral approach for quasilinear problems

For non-homogeneous and nonlinear problems, a major difficulty in applying the Boundary Element Method is the treatment of the volume integrals that arise. A recent proposed method, the grid-based integration method (GIM), uses a 3-D uniform grid to reduce the complexity of volume discretization, i.e., the discretization of the whole domain is avoided. The same grid is also used to accelerate both surface and volume integration. The efficiency of the GIM has been demonstrated on 3-D Poisson problems. In this paper, we report our work on the extension of this technique to quasilinear problems. Numerical results of a 3-D Helmholtz problem and a quasilinear Laplace problem on a multiply-connected domain with Dirichlet boundary conditions are presented. These results are compared with analytic solutions. The performance of the GIM is measured by plotting the L₂-norm error as a function of the overall CPU time and is compared with the auxiliary domain method in the Helmholtz problem.     

二阶拟线性双曲型方程的精确边界能控性

本文以二阶拟线性双曲型方程混合初边值问题的半整体C~2解理论为基础,针对一般的二阶拟线性双曲型方程的特征根在平衡态附近的不同分布情况,分别提出了相应的一般边界条件,并采用直接构造的方法,对特征根均不为零的情况,建立了完整的局部精确边界能控性理论;对一特征根为零的情况,对一类特殊的方程建立了其精确边界能控的充分必要条件,并分别对相应的控制时间给出了估计.     

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

In this paper, high‐order systems are reformulated as first‐order systems, which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretization with 1D‐integrated radial basis function networks (1D‐IRBFN) (Numer. Meth. Partial Differential Equations 2007; 23 :1192–1210). The present method is enhanced by a new boundary interpolation technique based on 1D‐IRBFN, which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well‐known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates. Copyright © 2009 John Wiley & Sons, Ltd.     

A decoupled augmented IIM strategy for incompressible two‐phase flows with interfaces on irregular domains

A decoupled augmented immersed interface method for solving incompressible two‐phase flows involving both irregular domains and interfaces is presented. In order to impose the prescribed velocity at the boundary of the irregular domain, singular force as one set of augmented variables is introduced. The velocity components at the two‐fluid interface as another set of augmented variables are introduced to satisfy the continuity condition of the velocity across the interface so that the jump conditions for the velocity and pressure are decoupled across the interface. The augmented variables and/or the forces along the interface/boundary are related to the jumps in both pressure and velocity and the jumps in their derivatives across the interface/boundary and applied to the fluid through jump conditions. The resulting augmented equation is a couple system of these two sets of augmented variables, and the direct application of the GMRES is impractical due to larger iterations. In this work, the novel decoupling of two sets of the augmented variables is proposed, and the decoupled augmented equation is then solved by the LU or the GMRES method. The Stokes equations are discretized via the finite difference method with the incorporation of jump contributions on a staggered Cartesian grid and solved by the conjugate gradient Uzawa‐type method. The numerical results show that second‐order accuracy for the velocity is confirmed. The present method has also been applied to solve for incompressible two‐phase Navier–Stokes flow with interfaces on irregular domains. Copyright © 2011 John Wiley & Sons, Ltd.     

A velocity-decomposition formulation for the incompressible Navier–Stokes equations

The principle of velocity decomposition is used to combine field discretization and boundary-element techniques to solve for steady, viscous, external flows around bodies. The decomposition modifies the Navier–Stokes boundary-value problem and produces a Laplace problem for a viscous potential, and a new Navier–Stokes sub-problem that can be solved on the portion of the domain where the total velocity has rotation. The key development in the decomposition is the formulation for the boundary condition on the viscous potential that couples the two components of velocity. An iterative numerical scheme is described to solve the decomposed problem. Results are shown for the steady laminar flow over a sectional airfoil, a circular cylinder with separation, and the turbulent flow around a slender body-of-revolution. The results show the viscous potential is obtainable even for massively separated flows, and the field discretization must only encompass the vortical region of the total velocity.     

Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach

A surface integral formulation for light scattering on periodic structures is presented. Electric and magnetic field equations are derived on the scatterers' surfaces in the unit cell with periodic boundary conditions. The solution is calculated with the method of moments and relies on the evaluation of the periodic Green's function performed with Ewald's method. The accuracy of this approach is assessed in detail. With this versatile boundary element formulation, a very large variety of geometries can be simulated, including doubly periodic structures on substrates and in multilayered media. The surface discretization shows a high flexibility, allowing the investigation of irregular shapes including fabrication accuracy. Deep insights into the extreme near-field of the scatterers as well as in the corresponding far-field are revealed. This method will find numerous applications for the design of realistic photonic nanostructures, in which light propagation is tailored to produce novel optical effects.     

Dynamic response of shear-deformable axisymmetric orthotropic circular plate structures solved by the DQEM and EDQ based time integration schemes

The dynamic response of shear-deformable axisymmetric orthotropic circular plate structures is solved by using the DQEM to the spacial discretization and EDQ to the temporal discretization. In the DQEM discretization, DQ is used to define the discrete element model. Discrete dynamic equilibrium equations defined at interior nodes in all elements, transition conditions defined on the inter-element boundary of two adjacent elements and boundary conditions at the structural boundary form a dynamic equation system at a specified time stage. The dynamic equilibrium equation system is solved by the direct time integration schemes of time-element by time-element method and stages by stages method which are developed by using EDQ and DQ. Numerical results obtained by the developed numerical algorithms are presented. They demonstrate the developed numerical solution procedure.     

Analyzing three-dimensional potential problems with the improved element-free Galerkin method

The potential problem is one of the most important partial differential equations in engineering mathematics. A potential problem is a function that satisfies a given partial differential equation and particular boundary conditions. It is independent of time and involves only space coordinates, as in Poisson’s equation or the Laplace equation with Dirichlet, Neumann, or mixed conditions. When potential problems are very complex, both in their field variable variation and boundary conditions, they usually cannot be solved by analytical solutions. The element-free Galerkin (EFG) method is a promising method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by employing improved moving least-squares (IMLS) approximation, we derive the formulas for an improved element-free Galerkin (IEFG) method for three-dimensional potential problems. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed.     

A stable and adaptive semi-Lagrangian potential model for unsteady and nonlinear ship-wave interactions

We present an innovative numerical discretization of the equations of inviscid potential flow for the simulation of three-dimensional, unsteady, and nonlinear water waves generated by a ship hull advancing in water.The equations of motion are written in a semi-Lagrangian framework, and the resulting integro-differential equations are discretized in space via an adaptive iso-parametric collocation boundary element method, and in time via implicit backward differentiation formulas (BDF) with adaptive step size and variable order.When the velocity of the advancing ship hull is non-negligible, the semi-Lagrangian formulation (also known as Arbitrary Lagrangian Eulerian formulation or ALE) of the free-surface equations contains dominant transport terms which are stabilized with a streamwise upwind Petrov–Galerkin (SUPG) method.The SUPG stabilization allows automatic and robust adaptation of the spatial discretization with unstructured quadrilateral grids. Preliminary results are presented where we compare our numerical model with experimental results on a Wigley hull advancing in calm water with fixed sink and trim.     

模拟复杂流动的一种隐式直接力浸入边界方法

为避免复杂贴体网格的生成,该文采用一种隐式直接力浸入边界法模拟复杂边界流动问题。借助求解不可压缩N-S方程组的分步投影方法的思想,来求解基于浸入边界法的耦合系统方程。其中固体边界离散点的作用力密度通过强制满足固体边界的无滑移条件导出,进而通过δ光滑函数对固体壁面附近速度场进行二次修正。在空间离散上,对流项采用QUICK迎风格式,扩散项采用中心差分格式,采用二阶显式Adams-Bashforth法离散时间项。以雷诺数为25、40和300的圆柱绕流为基准数值算例,通过与实验结果和其他文献数值结果的对比,验证数值计算方法的可靠性。     

Isoparametric finite point method in computational mechanics

In this paper, a new meshless method, the isoparametric finite point method (IFPM) in computational mechanics is presented. The present IFPM is a truly meshless method and developed based on the concepts of meshless discretization and local isoparametric interpolation. In IFPM, the unknown functions, their derivatives, and the sub-domain and its boundaries of an arbitrary point are described by the same shape functions. Two kinds of shape functions that satisfy the Kronecker-Delta property are developed for the scattered points in the domain and on the boundaries, respectively. Conventional point collocation method is employed for the discretization of the governing equation and the boundary conditions. The essential (Dirichlet) and natural (Neumann) boundary conditions can be directly enforced at the boundary points. Several numerical examples are presented together with the results obtained by the exact solution and the finite element method. The numerical results show that the present IFPM is a simple and efficient method in computational mechanics.     

Transient pressure solution for a horizontal well in a petroleum reservoir by boundary integral methods

A novel application of the boundary integral method to horizontal well analysis in the field of petroleum engineering is presented. The transient pressure satisfies the heat equation, non‐local and non‐linear boundary conditions. The turbulent flow inside the well is modelled by considering a pressure gradient along the well. The heat potential is used and Chebyshev collocation along with a time discretization is employed. Some numerical results are presented to show the features of this new approach. Copyright © 2000 John Wiley & Sons, Ltd.     

A high-order harmonic polynomial method for solving the Laplace equation with complex boundaries and its application to free-surface flows. Part I: Two-dimensional cases

A high-order harmonic polynomial method (HPM) is developed for solving the Laplace equation with complex boundaries. The “irregular cell” is proposed for the accurate discretization of the Laplace equation, where it is difficult to construct a high-quality stencil. An advanced discretization scheme is also developed for the accurate evaluation of the normal derivative of potential functions on complex boundaries. Thanks to the irregular cell and the discretization scheme for the normal derivative of the potential functions, the present method can avoid the drawback of distorted stencils, that is, the possible numerical inaccuracy/instability. Furthermore, it can involve stationary or moving bodies on the Cartesian grid in an accurate and simple way. With the proper free-surface tracking methods, the HPM has been successfully applied to the accurate and stable modeling of highly nonlinear free-surface potential flows with and without moving bodies, that is, sloshing, water entry, and plunging breaker.     

Treatment of corners in BIE analysis of potential problems

At corners or edges in the boundary of the domain of a potential problem the local normal gradient of potential is double-valued. When Dirichlet boundary conditions are specified there are thus two unknowns at a single nodal point, and the sets of equations resulting from the usual BIE discretization are rendered indeterminate. We discuss here earlier approaches to the resolving of this problem, and describe a further approach which appears to offer some advantages. Both normal gradients can be approximated directly from local potential boundary conditions, showing the problem indeed to be formally overdetermined. This ability is discarded, in favour of yielding a robust and well-conditioned relationship between the two gradients. This, in conjunction with the BIE analysis, permits solutions of considerable accuracy to be found, including the gradients at such corner nodes. Illustrative calculations are presented for rectilinear and curvilinear domains. These show that, even with as few elements as there are corners, and thus one and a half times as many unknowns as there are nodal points, good approximations to the gradients can be obtained. The need for progressively finer discretization as a corner is approached is thus much reduced.     

Comparison of methods for wall boundary conditions as applied to the calculation of turbulent heat exchange characteristics

Methods for setting and realizing wall boundary conditions numerically in calculating turbulent flows is considered. A method for realizing weak boundary conditions on the wall with discretization of Reynolds-averaged Navier–Stokes equations by the control volume approach is discussed. The results of calculations for a number of model problems obtained within the framework of different approaches to the near-wall modeling are compared to the data of the physical experiment and the available correlation dependences. The grid dependence of the solution in using the method of near-wall functions is compared to that in using weak boundary conditions.     

Transient dynamic crack analysis in piecewise homogeneous, anisotropic and linear elastic composites by a spatial symmetric Galerkin-BEM

Transient elastodynamic analysis of two-dimensional, piecewise homogeneous, anisotropic and linear elastic solids containing interior and interface cracks is presented in this paper. To solve the initial boundary value problem, a spatial symmetric time-domain boundary element method is developed. Stationary cracks subjected to impact loading conditions are considered. Elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are implemented. The piecewise homogeneous, anisotropic and linear elastic solids are modeled by the multi-domain technique. The spatial discretization is performed by a symmetric Galerkin-method, while a collocation method is utilized for the temporal discretization. An explicit time-stepping scheme is obtained for computing the unknown boundary data. Numerical examples are presented and discussed to show the effects of the interface cracks, the material anisotropy, the material combination and the dynamic loading on the dynamic stress intensity factors.     

An analysis of monotonicity conditions in the mixed hybrid finite element method on unstructured triangulations

In this paper we consider the mixed hybrid finite element method on unstructured triangular grids and evaluate its monotonicity properties by using a non standard set of basis functions for the velocity approximation space. The mixed hybrid discretization of the steady‐state diffusion equation produces a system matrix that depends only on the inner product of the outward normals to the edges of the triangulation and not on the choice of the velocity space basis. This property is used to study the characteristics of the system matrix. It is well known that this matrix is of type M if the angles of the triangulation are not bigger than π/2. An M‐matrix has a nonnegative inverse, i.e. all the elements are nonnegative. This implies the existence of a discrete maximum principle and thus monotonicity of the discretization. We show that, when the triangulation is of Delaunay type and satisfies the property that no circumcenters of boundary elements with Dirichlet conditions lie outside the domain, the inverse of the final matrix is always positive, even in the presence of obtuse angles. Copyright © 2008 John Wiley & Sons, Ltd.     

An application of the OMLS method to domain integrals of potential problems in heterogeneous media

The present work introduces a boundary element formulation adopting an alternative procedure to remove domain cell discretization commonly used to simulate velocity correcting fields needed to account for potential problems in heterogeneous media via homogeneous fundamental solutions. To this end, an orthogonal moving least square (OMLS), borrowed from recent meshless procedures, is followed. The velocity correcting fields are interpolated using OMLS based on a number of internal points and correct the internal fluxes compensating for the non-homogeneous conductivity of the medium.Two examples are included to assess the correctness of the proposed procedure.     

A point-iterative algorithm for three-dimensional magnetic vector problems

Magnetostatic field problems are solved in three dimensions by applying a variational method that employs finite elements. Formulation through a partial differential equation allows solution for the magnetic vector potential given an inhomogeneous, orthotropic medium and a distributed current source. Three vector boundary conditions are discussed and interior sheet currents are allowed within the medium. In addition, the Lorentz condition is enforced by including a penalty term in the energy functional. A point-iterative algorithm is used to solve the set of equations resulting from finite element discretization. This method is particularily suitable for regions with regular geometry and a moderate (1,000 to 10,000) number of unknowns.     

Transient response of a semi-permeable crack between dissimilar anisotropic piezoelectric layers by time-domain BEM

This paper studied the transient response of a semi-permeable crack between two dissimilar anisotropic piezoelectric layers by a time-domain BEM with a sub-domain technique and an iterative process for the non-linear crack-face boundary conditions. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. A universal matrix-form displacement extrapolation formula and its explicit formula are used to determine the dynamic intensity factors. Several examples are presented and discussed to show the effects of the electrical crack-face boundary condition on dynamic intensity factors.