A locally analytic technique applied to grid generation by elliptic equations

One technique for obtaining grids for irregular geometries is to solve sets of elliptic partial differential equations. The solution of the partial differential equations yields a grid which discretizes the physical solution domain and also a transformation for the irregular physical domain to a regular computational domain. Expressing the governing equation of interest in the computational domain requires the derivatives of the physical to computational domain transformation, i.e., the metrics. These metrics are typically determined by numerical differentiation, which is a potential source of error. The locally analytic method uses the analytic solution of the locally linearized equation to develop numerical stencils. Thus, the locally analytic method allows numerical differentiation with no loss of accuracy. In this paper, the locally analytic method is applied to the solution of the Poisson and Brackbill–Saltzman equations. Comparison with an exact solution shows the locally analytic method to be more accurate than the finite difference method, both in solving the partial differential equation and evaluating the metrics. However, it is more computationally expensive.     

Plane elasticity problems by barycentric rational interpolation collocation method and a regular domain method

Barycentric rational interpolation collocation method (BRICM) for solving plane elasticity problems with high accuracy is presented. The plane elasticity problems on a circular or rectangular domain can be solved directly by BRICM. Embedded the irregular domain into a regular (circular or rectangular) domain, the governing equations of plane elasticity on regular domain are discretized by the differentiation matrices based on barycentric rational interpolation to form a system of algebraic equations. Discrete boundary conditions are obtained using barycentric rational interpolation. The irregular boundary conditions are imposed by the additional method to form an over-constraint linear system of algebraic equations. Numerical experiments are presented to illustrate the efficiency and high computing precision of proposed method.     

Shape optimization using time evolution equations

In this study, we deal with a numerical solution based on time evolution equations to solve the optimization problem for finding the shape that minimizes the objective function under given constraints. The design variables of the shape optimization problem are defined on a given original domain on which the boundary value problems of partial differential equations are defined. The variations of the domain are obtained by the time integration of the solution to derive the time evolution equations defined in the original domain. The shape gradient with respect to the domain variations are given as the Neumann boundary condition defined on the original domain boundary. When the constraints are satisfied, the decreasing property of the objective function is guaranteed by the proposed method. Furthermore, the proposed method is used to minimize the heat resistance under a total volume constraint and to solve the minimization problem of mean compliance under a total volume constraint.     

The use of Stokes' fundamental solution for the boundary only element formulation of the three-dimensional Navier–Stokes equations for moderate Reynolds numbers

This paper presents a boundary element formulation for the permanent Navier–Stokes equations in which the well-known closed-form fundamental solution for the steady Stokes equations is employed. In this way, from the integral representation formulae for the Stokes' equations, an integral equation is found in which the original non-linear convective terms of the Navier–Stokes equations appear as a domain integral. Additionally, the method of dual reciprocity is used to transform the domain integral to boundary integrals (this method is closely related to the method of particular integrals also used in the literature to transform domain integrals to boundary integrals). Numerical results are presented for the three-dimensional internal flow in a cylindrical container with a rotating cover, in which the accuracy of the method is shown.     

Boundary element analysis in thermoelasticity with and without internal cells

In this paper, a set of internal stress integral equations is derived for solving thermoelastic problems. A jump term and a strongly singular domain integral associated with the temperature of the material are produced in these equations. The strongly singular domain integral is then regularized using a semi‐analytical technique. To avoid the requirement of discretizing the domain into internal cells, domain integrals included in both displacement and internal stress integral equations are transformed into equivalent boundary integrals using the radial integration method (RIM). Two numerical examples for 2D and 3D, respectively, are presented to verify the derived formulations. Copyright © 2003 John Wiley & Sons, Ltd.     

Transient elastodynamic analysis of nonhomogeneous anisotropic plane bodies

A boundary-only BEM procedure is employed to solve the transient dynamic analysis of nonhomogeneous anisotropic plane elastic bodies. The response of such bodies is governed by two coupled linear, second-order hyperbolic PDEs with spatially dependent coefficients. The lack of a reliable 2D time-domain elastodynamic fundamental solution is overcome using the principle of the Analog Equation, a method by which the equations of motion of the problem are substituted by two coupled quasi-static Poisson-type equations having as nonhomogeneous terms the components of a fictitious time-dependent load distribution in the specified domain. The standard BEM is employed for the solution of the substitute equations. To avoid the appearance of the domain integral in the integral representation of the solution, the fictitious load distribution is approximated by multiquadrics with unknown time-dependent expansion coefficients, which are calculated at discrete timepoints by collocating the equations of motion at a predefined set of domain interpolation nodes. The obtained numerical results by the proposed method demonstrate its stability and accuracy over other numerical methods.     

Radial integration BEM for transient heat conduction problems

In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.     

Solution of non‐linear boundary integral equations in complex geometries with auxiliary integral subtraction

The boundary integral equation that results from the application of the reciprocity theorem to non‐linear or non‐homogeneous differential equations generally contains a domain integral. While methods exist for the meshless evaluation of these integrals, mesh‐based domain integration is generally more accurate and can be performed more quickly with the application of fast multipole methods. However, polygonalization of complex multiply‐connected geometries can become a costly task, especially in three‐dimensional analyses. In this paper, a method that allows a mesh‐based integration in complex domains, while retaining a simple mesh structure, is described. Although the technique is intended for the numerical solution of more complex differential equations, such as the Navier–Stokes equations, for simplicity the method is applied to the solution of a Poisson equation, in domains of varying complexity. It is shown that the error introduced by the auxiliary domain subtraction method is comparable to the discretization error, while the complexity of the mesh is significantly reduced. The behaviour of the error in the boundary solution observed with the application of the new method is analogous to the behaviour observed with conventional cell‐based domain integration. Copyright © 2002 John Wiley & Sons, Ltd.     

A meshless solution for two dimensional density-driven groundwater flow

Density-driven groundwater flow is a complicated nonlinear problem in groundwater hydraulics. The local boundary integral method is a promising meshless scheme that is used for solving several difficult problems in different areas. This method applies the boundary integral equations to the local domain around every node. The nodes can be randomly distributed in the domain and on the global boundary. Therefore, this method is characterised as meshless. The unknown potentials and concentrations in all of the nodes are approximated by interpolation to obtain a system of linear equations. Solving this system of equations leads to the numerical solution for the main problem. In this paper, a combination of the radial basis function interpolation and the local boundary element method is used to solve groundwater flow problem combined with the transport of pollution, which also influences the density of groundwater.     

A meshless local boundary integral equation method for dynamic anti-plane shear crack problem in functionally graded materials

This paper presents a meshless local boundary integral equation method (LBIEM) for dynamic analysis of an anti-plane crack in functionally graded materials (FGMs). Local boundary integral equations (LBIEs) are formulated in the Laplace-transform domain. The static fundamental solution for homogeneous elastic solids is used to derive the local boundary–domain integral equations, which are applied to small sub-domains covering the analyzed domain. For the sub-domains a circular shape is chosen, and their centers, the nodal points, correspond to the collocation points. The local boundary–domain integral equations are solved numerically in the Laplace-transform domain by a meshless method based on the moving least–squares (MLS) scheme. Time-domain solutions are obtained by using the Stehfest's inversion algorithm. Numerical examples are given to show the accuracy of the proposed meshless LBIEM.     

Application of a direct Trefftz method with domain decomposition to 2D potential problems

This article presents an application of a direct Trefftz method with domain decomposition method to the two-dimensional potential problem. In the direct Trefftz methods, regular T-complete functions satisfying the governing equations are taken as the weighting functions and then, the boundary integral equations are derived from the weighted residual expressions of the governing equations. Since the T-complete functions are regular, the final equations are also regular and therefore, much simpler than the ordinary boundary element methods employing the singular fundamental solutions. Their computational accuracy, however, is dependent on the condition number of the coefficient matrices of the algebraic system of equations. So, for improving the accuracy, we introduce the domain decomposition method to the direct Trefftz methods. The present method is applied to the two-dimensional potential problem in order to confirm the validity.     

用GDQ空-时半解析法分析圆弧拱强迫振动

从圆弧拱的强迫振动控制方程出发,在空间域采用广义微分求积法,将广义微分求积法中的节点参数在时域内取为响应的时间级数,通过时域配点求解该六阶偏微分方程得到全域内的响应位移场,分析了圆弧拱的平面内强迫振动问题。运用Matlab语言编制圆弧拱强迫振动时程分析的计算程序,并进行了相应的算例分析,数值计算结果表明该方法对于求圆弧拱的强迫振动问题具有较高的精度和效率。     

3D mass-redistributed finite element method in structural–acoustic interaction problems

A 2D mass-redistributed finite element method (MR-FEM) for pure acoustic problems was recently proposed to reduce the dispersion error. In this paper, the 3D MR-FEM is further developed to solve more complicated structural–acoustic interaction problems. The smoothed Galerkin weak form is adopted to formulate the discretized equations for the structure, and MR-FEM is applied in acoustic domain. The global equations of structural–acoustic interaction problems are then established by coupling the MR-FEM for the acoustic domain and the edge-based smoothed finite element method for the structure. The perfect balance between the mass matrix and stiffness matrix is able to improve the accuracy of the acoustic domain significantly. The gradient smoothing technique used in the structural domain can provide a proper softening effect to the “overly-stiff” FEM model. A number of numerical examples have demonstrated the effectiveness of the mass-redistributed method with smoothed strain.     

A boundary element method without internal cells for solving viscous flow problems

In this paper, a new boundary element method without internal cells is presented for solving viscous flow problems, based on the radial integration method (RIM) which can transform any domain integrals into boundary integrals. Due to the presence of body forces, pressure term and the non-linearity of the convective terms in Navier–Stokes equations, some domain integrals appear in the derived velocity and pressure boundary-domain integral equations. The body forces induced domain integrals are directly transformed into equivalent boundary integrals using RIM. For other domain integrals including unknown quantities (velocity product and pressure), the transformation to the boundary is accomplished by approximating the unknown quantities with the compactly supported fourth-order spline radial basis functions combined with polynomials in global coordinates. Two numerical examples are given to demonstrate the validity and effectiveness of the proposed method.     

Investigation on a Two-dimensional Generalized Thermal Shock Problem with Temperature-dependent Properties

The dynamic response of a two-dimensional generalized thermoelastic problem with temperature-dependent properties is investigated in the context of generalized thermoelasticity proposed by Lord and Shulman. The governing equations are formulated, and due to the nonlinearity and complexity of the governing equations resulted from the temperature-dependent properties, a numerical method, i.e., finite element method is adopted to solve such problem. By means of virtual displacement principle, the nonlinear finite element equations are derived. To demonstrate the solution process, a thermoelastic half-space subjected to a thermal shock on its bounding surface is considered in detail. The nonlinear finite element equations for this problem are solved directly in time domain. The variations of the considered variables are obtained and illustrated graphically. The results show that the effect of the temperature-dependent properties on the considered variables is to reduce their magnitudes, and taking the temperature-dependence of material properties into consideration in the investigation of generalized thermoelastic problem has practical meaning in predicting the thermoelastic behaviors accurately. It can also be deduced that directly solving the nonlinear finite element equations in time domain is a powerful method to deal with the thermoelastic problems with temperature-dependent properties.     

梯形板的非线性动力分析

基于弹性板的几何非线性动力平衡方程,首先建立了一个坐标变换将梯形板域变换到正方形域,并将控制方程及其相应的边界条件变换到该正方形域内,然后通过引入中间变量将控制方程降阶,并利用问题的数学物理关系将边界条件进一步简化,在正方形域内对板的控制方程应用伽辽金法使问题变为时间域的非线性动力方程,最后应用参数摄动法得到了梯形板的几何非线性自由振动和动力响应,所得计算结果可供工程设计人员参考。     

桥梁抖振时域和频域分析的一致性研究

为考查时域分析方法用于桥梁抖振分析的可行性和可靠性,基于相同的分析参数,分别采用时域和频域分析方法对一斜拉桥的抖振响应进行分析,并从结构振动型态、抖振响应均方根及抖振响应功率谱密度函数三个方面对时域和频域分析结果的一致性进行了较详细的比较。时域抖振响应分析中,采用谱解法模拟了斜拉桥的脉动风场,抖振力采用准定常表达形式,自激力采用Y.K. Lin 表达形式。由于自激力的存在,结构的运动方程为非线性,提出了一种迭代方法来考虑自激力引起的非线性。采用Newmark-β方法进行积分计算。频域抖振分析采用多模态耦合分析方法。时域及频域抖振响应分析结果的一致性较好,这表明了大跨度斜拉桥时域抖振响应分析的可行性和可靠性。     

Iterative solution of a hybrid method for Maxwell's equations in the frequency domain

A hybrid method for solution of Maxwell's equations of electromagnetics in the frequency domain is developed as a combination between the method of moments and the approximation in physical optics. The equations are discretized by a Galerkin method and solved by an iterative block Gauss–Seidel method. The convergence of the iterations is studied theoretically and in numerical experiments. The accuracy of the hybrid method is compared to the method of moments for a cylinder with an incident field for different wavenumbers. Copyright © 2003 John Wiley & Sons, Ltd.     

Fast Computation of Stationary Inviscid Flow Around an Airfoil

The parallel performance of an implicit solver for the Euler equations on a structured grid is discussed. The flow studied is a two-dimensional transonic flow around an airfoil. The spatial discretization involves the MUSCL scheme, a higher-order Total Variation Diminishing scheme. The solver described in this paper is an implicit solver that is based on quasi Newton iteration and approximate factorization to solve the linear system of equations resulting from the Euler Backward scheme. It is shown that the implicit time-stepping method can be used as a smoother to obtain an efficient and stable multigrid process. Also, the solver has good properties for parallelization comparable with explicit time-stepping schemes. To preserve data locality domain decomposition is applied to obtain a parallelizable code. Although the domain decomposition slightly affects the efficiency of the approximate factorization method with respect to the number of time steps required to attain the stationary solution, the results show that this hardly affects the performance for practical purposes. The accuracy with which the linear system of equations is solved is found to be an important parameter. Because the method is equally applicable for the Navier-Stokes equations and in three-dimensions, the presented combination of efficient parallel execution and implicit time-integration provides an interesting perspective for time-dependent problems in computational fluid dynamics.     

A sub‒domain smoothed Galerkin method for solid mechanics problems

A sub?domain smoothed Galerkin method is proposed to integrate the advantages of mesh?free Galerkin method and FEM. Arbitrarily shaped sub?domains are predefined in problems domain with mesh?free nodes. In each sub?domain, based on mesh?free Galerkin weak formulation, the local discrete equation can be obtained by using the moving Kriging interpolation, which is similar to the discretization of the high?order finite elements. Strain smoothing technique is subsequently applied to the nodal integration of sub?domain by dividing the sub?domain into several smoothing cells. Moreover, condensation of DOF can also be introduced into the local discrete equations to improve the computational efficiency. The global governing equations of present method are obtained on the basis of the scheme of FEM by assembling all local discrete equations of the sub?domains. The mesh?free properties of Galerkin method are retained in each sub?domain. Several 2D elastic problems have been solved on the basis of this newly proposed method to validate its computational performance. These numerical examples proved that the newly proposed sub?domain smoothed Galerkin method is a robust technique to solve solid mechanics problems based on its characteristics of high computational efficiency, good accuracy, and convergence. Copyright © 2014 John Wiley & Sons, Ltd.