An iterative method to solve the heat transfer problem under the non-linear boundary conditions

The aim of the paper is to determine the approximation of the tangential matrix for solving the non-linear heat transfer problem. Numerical model of the strongly non-linear heat transfer problem based on the theory of the finite element method is presented. The tangential matrix of the Newton method is formulated. A method to solve the heat transfer with the non-linear boundary conditions, based on the secant slope of a reference function, is developed. The contraction mapping principle is introduced to verify the convergence of this method. The application of the method is shown by two examples. Numerical results of these examples are comparable to the ones solved with the Newton method and the commercial software COMSOL for the heat transfer problem under the radiative boundary conditions.     

一种模拟大规模高频声场的双层奇异边界法

大规模高频声场的数值模拟是一项非常有计算挑战性的课题. 为了解决传统边界型离散方法由于全局支撑的满阵限制, 不易应用于大规模高频声场模拟的计算瓶颈, 本文提出了一种用于模拟大规模高频声场的双层奇异边界法. 在该方法中, 通过引入双层结构, 细网格上的全局支撑的满阵被转化为局部支撑的大规模稀疏矩阵, 传统奇异边界法模拟大规模问题时所面临的高计算量以及过度存储需求遂得以解决. 其次, 双层奇异边界法仅通过粗网格评估远场作用, 且独立于特定的插值核函数. 相较于快速多级方法, 该方法具有更强的适应性和灵活性, 且多层结构使该方法具有一定的预调节作用, 非常适合求解具有大规模、高秩、高条件数特点的高频波矩阵. 在其后的散射球模型算例中, 双层奇异边界法配置10万个节点, 成功模拟了无量纲波数高达160的声散射问题. 在对于人头模型的声散射特性分析中, 双层奇异边界法比COMSOL软件计算速度快了约78.13\mathrm{\%}. 当配置8万个节点时, 双层奇异边界法成功模拟了频率高达25 kHz 的工况, 该频率已远远超出了人耳的听力极限.      

Analysis of linear elastic wave propagation in piping systems by a combination of the boundary integral equations method and the finite element method

The combined methodology of boundary integral equations and finite elements is formulated and applied to study the wave propagation phenomena in compound piping systems consisting of straight and curved pipe segments with compact elastic supports. This methodology replicates the concept of hierarchical boundary integral equations method proposed by L. I. Slepyan to model the time-harmonic wave propagation in wave guides, which have components of different dimensions. However, the formulation presented in this article is tuned to match the finite element format, and therefore, it employs the dynamical stiffness matrix to describe wave guide properties of all components of the assembled structure. This matrix may readily be derived from the boundary integral equations, and such a derivation is superior over the conventional derivation from the transfer matrix. The proposed methodology is verified in several examples and applied for analysis of periodicity effects in compound piping systems of several alternative layouts.     

Transient heat conduction analysis using the NURBS-enhanced scaled boundary finite element method and modified precise integration method

The Non-uniform rational B-spline(NURBS)enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this paper.The scaled boundary finite element method is a semi-analytical technique,which weakens the governing differential equations along the circumferential direction and solves those analytically in the radial direction.In this method,only the boundary is discretized in the finite element sense leading to a reduction of the spatial dimension by one with no fundamental solution required.Nevertheless,in case of the complex geometry,a huge number of elements are generally required to properly approximate the exact shape of the domain and distorted meshes are often unavoidable in the conventional finite element approach,which leads to huge computational efforts and loss of accuracy.NURBS are the most popular mathematical tool in CAD industry due to its flexibility to fit any free-form shape.In the proposed methodology,the arbitrary curved boundary of problem domain is exactly represented with NURBS basis functions,while the straight part of the boundary is discretized by the conventional Lagrange shape functions.Both the concepts of isogeometric analysis and scaled boundary finite element method are combined to form the governing equations of transient heat conduction analysis and the solution is obtained using the modified precise integration method.The stiffness matrix is obtained from a standard quadratic eigenvalue problem and the mass matrix is determined from the low-frequency expansion.Finally the governing equations become a system of first-order ordinary differential equations and the time domain response is solved numerically by the modified precise integration method.The accuracy and stability of the proposed method to deal with the transient heat conduction problems are demonstrated by numerical examples.     

Dynamic simulation of the hydrodynamic interaction among immersed particles in Stokes flow

A numerical method for the dynamic simulation of the hydrodynamic interaction among particles in Stokes flow is developed. The method couples the quasi-static Stokes equations for the fluid with the equilibrium equations for the particles. The boundary element method is used to represent the velocity at a general field point in terms of surface velocities and stresses. However, neither the stresses nor the velocities are assumed to be known on the surface of the particles. Kinematic equations relating the linear and angular velocities at the centroids of the particles to the surface velocities are combined with the discretized boundary element equations and the equilibrium equations to generate a system of linear equations. The associated coefficient matrix is correspondent to the grand resistance matrix which relates the velocities of the particles to a given geometry.     

精化8—20节点块体元的综合一致质量阵及几种质量阵构成方法的讨论

针对精化８－２０节点块体元构造了几种不同类型的质量阵，通过对几种支撑条件下的梁进行的振动特性分析，对其进行了比较研究，提出了构造精化８－２０节点块体元质量阵的新途径—综合一致质量阵法。结果表明，这种综合一致质量阵方法是几种质量阵形成方法中对提高动力分析精度最为有效的方法。     

结构边界约束修改后的重分析方法

在结构设计优化中经常将结构边界约束作为设计优化对象,结构边界约束的修改通常导致系统的求解规模发生改变,使得快速准确分析修改后结构的响应成为一个挑战。本文发展了逐次矩阵逆(SMI)方法,提出了一种适合各种结构边界约束(包括初始结构中的约束)修改的快速重分析算法。该方法利用边界约束修改后对应刚度矩阵的对称性,有效缩减了计算量。数值算例表明,本文方法能够快速给出精确的重分析结果。     

Moving immersed boundary method

A novel implicit immersed boundary method of high accuracy and efficiency is presented for the simulation of incompressible viscous flow over complex stationary or moving solid boundaries. A boundary force is often introduced in many immersed boundary methods to mimic the presence of solid boundary, such that the overall simulation can be performed on a simple Cartesian grid. The current method inherits this idea and considers the boundary force as a Lagrange multiplier to enforce the no‐slip constraint at the solid boundary, instead of applying constitutional relations for rigid bodies. Hence excessive constraint on the time step is circumvented, and the time step only depends on the discretization of fluid Navier‐Stokes equations, like the CFL condition in present work. To determine the boundary force, an additional moving force equation is derived. The dimension of this derived system is proportional to the number of Lagrangian points describing the solid boundaries, which makes the method very suitable for moving boundary problems since the time for matrix update and system solving is not significant. The force coefficient matrix is made symmetric and positive definite so that the conjugate gradient method can solve the system quickly. The proposed immersed boundary method is incorporated into the fluid solver with a second‐order accurate projection method as a plug‐in. The overall scheme is handled under an efficient fractional step framework, namely, prediction, forcing, and projection. Various simulations are performed to validate current method, and the results compare well with previous experimental and numerical studies.     

圆柱壳自由振动特性分析方法研究

对传统解析法进行了改进,直接将复特征根代入边界条件构造了特征方阵;并基于圆柱壳Flügge 模型推导了位移形式的传递函数矩阵.在此基础上对比了改进的解析法、传递函数法、有限元法和实验方法获得的两端自由边界条件下圆柱壳的固有频率和模态.研究结果表明:改进的解析方法其特征方阵表达式简单且易于实现;同时通过与有限元和实验方法的对比,验证了改进解析法和位移形式传递函数矩阵的正确性;传递函数法相对于传统解析法对舍入误差敏感,可以通过增加字长的方式进行改善.     

在无单元法里直接准确施加位移边界条件和材料不连续条件

在无单元伽辽金法(EFG)里，由于其滑动最小二乘近似位移函数不满足Kronecker条件，使得它不能准确地施加本质边界条件和材料不连续条件，从而极大地限制了EFG法的发展和进一步应用。本文在位移边界和不同材料交界面的离散结点上采用实际的结点位移值，提出了一种准确施加位移边界和材料不连续条件的方法，该方法实施简单、稳定、求解精度高，而且其推导得出的整体刚度矩阵具有正定、对称和带状分布的特点，可以和有限单元法一样，直接利用各种成熟、高效的线性方程组解法求解系统平衡方程。数值算例结果表明了文中理论和方法的正确性和可靠性。     

基于整体位移模式的平面孔洞结构数值模拟方法

针对平面孔洞问题提出了一种新的数值模拟方法。本文通过水平集方法引入孔洞边界、力边界和位移边界水平集函数,利用边界水平集函数来构造边界试探项,将试探空间表示为二元幂级数与边界试探项的线性组合;同时提出一种基于水平集方法的位移边界条件施加方法,利用位移边界水平集函数来构造满足位移边界条件的近似位移场,并给出了相应的刚度矩阵和载荷矩阵表达式。与FEM、XFEM、无网格法等方法相比,该方法无需将求解域离散,具有较低的计算成本、特性良好的刚度矩阵和较为广泛的适用性。数值算例验证了该方法的有效性。     

Precise integration method for solving singular perturbation problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.     

A practical method for the evaluation of eigenfunctions from compound matrix variables in finite elastic bifurcation problems

We show how the compound matrix method can be extended to give eigenfunctions as well as generalised eigenvalues to bifurcation problems in non-linear elasticity. When the incremental problem is formulated in terms of displacements only there are significant difficulties that arise from the non-trivial boundary conditions. In order to avoid these problems we adopt a Stroh formulation of the incremental problem. This then produces trivial boundary conditions for the compound matrix eigenvalue problem and more importantly known initial conditions for the compound matrix eigenfunction problem. This results in a straightforward and robust calculation for the eigenfunctions.     

Coupling of assumed stress finite element and boundary element methods with stress-traction equilibrium

In this study, the stress based finite element method is coupled with the boundary element method in two different ways. In the first one, the ordinary distribution matrix is used for coupling. In the second one, the stress traction equilibrium is used at the interface line of both regions as a new coupling process. This new coupling procedure is presented without a distribution matrix. Several case studies are solved for the validation of the developed coupling procedure. The results of case studies are compared with the distribution matrix coupling, displacement based finite element method, assumed stress finite element method, boundary element method, ANSYS and analytical results whenever possible. It is shown that the coupling of the stress traction equilibrium with assumed stress finite elements gives as accurate results as those by the distribution matrix coupling.     

Amended influence matrix method for removal of rigid motion in the interior BVP for plane elasticity

A conventional complex variable boundary integral equation(CVBIE) in plane elasticity is provided. After using the Somigliana identity between a particular fundamental stress field and a physical stress field, an additional integral equality is obtained. By adding both sides of this integral equality to both sides of the conventional CVBIE, the amended boundary integral equation(BIE) is obtained. The method based on the discretization of the amended BIE is called the amended influence matrix method.With this method, for the Neumann boundary value problem(BVP) of an interior region,a unique solution for the displacement can be obtained. Several numerical examples are provided to prove the efficiency of the suggested method.     

Analytical solutions for single- and multi-span functionally graded plates in cylindrical bending

A recently developed plate theory using the concept of shape function of the transverse coordinate parameter is extended to determine the stress distribution in an orthotropic functionally graded plate subjected to cylindrical bending. The transfer matrix method is presented to derive the shape function. The equations governing the plate deformation are then solved analytically using the transfer matrix method for arbitrary boundary conditions. For a simply supported functionally graded plate, a comparison of the present solution with the exact elasticity solution, the first- and third-order shear deformation plate theories is presented and discussed. It is demonstrated that the present method yields more accurate stresses than the first- and third-order shear deformation theories. The effect of boundary conditions and inhomogeneity of material on the displacements and stresses in functionally graded plates are investigated. A multi-span functionally graded plate with arbitrary boundary conditions is further considered to demonstrate the efficiency of the present method.     

An analytical approach for calculating natural frequencies of finite one-dimensional acoustic metamaterials

In this paper, an analytical matrix method is presented to drive closed-form characteristic equations for natural frequencies of finite monoatomic and diatomic metamaterials with various boundary conditions. Here, we extend the matrix method introduced by Louck for monoatomic lattice chains. The proposed method is used to calculate the vibration frequencies of the monoatomic metamaterials with fixed–fixed, fixed-free and free-free boundary conditions. In addition, the natural frequencies of fixed–fixed diatomic metamaterials are calculated. The existence of band gaps in the frequencies of the metamaterials is numerically shown.

     

敷设多孔吸声材料声腔特征值分析的径向积分边界元法

由于Helmholtz方程的基本解是频率的函数,因此传统边界元法在处理声场特征值问题时具有天生的缺陷。本文采用Laplace方程基本解生成积分方程,通过径向积分法将在此过程中产生的域积分项转化为边界积分。此方法克服了传统边界元法系数矩阵对频率的依赖,同时克服了特解积分法对特解的依赖,并通过对表面声导纳的多项式逼近,将敷设多孔吸声材料声腔特征值问题转化为矩阵多项式,从而避免了复杂的非线性求解。通过数值算例验证了算法的有效性。     

断裂力学的半解析相似边界元法

本文首先对弹性力学的相似边界元法进行了研究，推导了相应的计算公式。与传统的边界元法相比，相似边界元法由于只需在少数单元上进行数值积分，大大减少了计算量。在此基础上，对断裂力学问题，利用裂纹尖端位移场的解析表达式将裂纹尖端节点未知量转化为几个待定常数，提出了半解析相似边界元法，可大大减少最终形成的线性代数方程组的系数矩阵的阶数，进一步减小计算量。最后给出了算例，说明了本文方法的有效性。     

二维系统传递矩阵法

以薄板横向振动的振动特性问题为例提出了二维系统传递矩阵法. 从质点和无质量梁的传递 矩阵出发,建立了按列排列的薄板子结构的传递矩阵. 用二维系统传递矩阵法获得了整个板 的总体传递方程,从而可得到板在任意一种边界条件下的特征方程. 数值求解了两种情况下 薄板横向振动的固有振动特性. 计算结果表明,用该方法可用于研究类似薄板的二维系统的 动力学问题,且无需建立系统的总体动力学方程.