Free surface and surface tension effects on submerged bodies

Summary The Oseen problem for the steady motion of an object beneath a free surface with surface tension under the action of gravity is formulated. The Green's tensor for the problem is used to convert the boundary value problem to a coupled pair of integral equations for the stresses which the fluid exerts on the object. For the special case of a flat plate, these integral equations are analyzed asymptotically for small velocities and deep immersion. This yields a Fredholm equation of the second kind with Cauchy kernel, which hasa well-known solution. The results indicate the effect of surface tension on the stress singularities at the edges of the plate, and modification of the lift and drag due to the free surface.     

饱和地基上含刚核弹性圆板的摇摆振动分析

基于饱和土弹性波动方程,研究了饱和地基上含刚核弹性圆板在摇摆谐和力矩作用下的振动特性。首先应用Hankel积分变换求解该饱和土波动方程,然后按混合边值条件建立起一组描述含刚核弹性圆板振动的对偶积分方程,并将其化为第二类Fredholm积分方程进行数值求解。此外,给出了含刚核弹性圆板在饱和地基上振动的阻抗函数随无量纲频率的变化曲线,并考察了土的渗透系数、弹性板含刚域的大小以及板的柔度等参数对阻抗函数的影响,得出了一些有意义的结论。     

Simply supported plates by the boundary integral equation method

Plates governed by Kirchhoff's equation have been analysed by the boundary integral equation method using the fundamental solution of the biharmonic equation. In the case of supported plates, the boundary conditions permit the uncoupling of the field equation into two harmonic equations that originate, due to the nature of the fundamental solution, easier integration kernels and a simpler system of equations. The calculation of bending and twisting moments and transverse shear force can be formed, combining derivatives of the integral equation which defines the expression of the deflection on any point of the plate. The uncoupling of the biharmonic equation into two Poisson's equations involves the discretization of the domain of the studied problems. Nevertheless, the unknown quantity of the problem does not appear in the domain integrations for which a refined discretization is unnecessary. In the paper, however, a numerical alternative is considered to express the domain integral by means of boundary integrals. In this way, we need only discretize the boundary of the plate, making it necessary to solve a supplementary system of equations in order to calculate the coefficients of the approximation carried out.     

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.     

Coupled horizontal and rocking vibrations of a rigid circular plate on a transversely isotropic bi-material interface

This paper investigates the coupled rocking and horizontal vibratory response of a rigid circular plate embedded in viscoelastic, transversely isotropic, three-dimensional unbounded media. The boundary-value problem corresponding to the case of distributed horizontal and rocking ring loads at a bi-material interface is solved to obtain the required influence functions for the solution of the present problem. The case of an embedded rigid plate is formulated in terms of a discretized integral equation, which couples the rigid body displacements of the plate with the tractions acting over its contact surface through a set of displacement influence functions. The system of resulting discretized integral equations is solved numerically. The solution results in the tractions over each disc element. This paper carefully takes into account the coupling of the rocking and horizontal responses of the plates that is typical of non-homogeneous interfaces, i.e., their horizontal displacements due to rocking moments and their rotations due to horizontal loads. The dynamic direct and cross compliances of the embedded plate are shown for different governing parameters such as frequency of excitation and bi-material configuration. The present results are useful to the study of dynamic response of deeply buried foundations and anchors in non-homogeneous soils.     

An adaptive fixed mesh method for modelling and simulating of unsteady two‐phase flows with sharp physical interfaces

We present in this paper a new computational method for simulation of two‐phase flow problems with moving boundaries and sharp physical interfaces. An adaptive interface‐capturing technique (ICT) of the Eulerian type is developed for capturing the motion of the interfaces (free surfaces) in an unsteady flow state. The adaptive method is mainly based on the relative boundary conditions of the zero pressure head, at which the interface is corresponding to a free surface boundary. The definition of the free surface boundary condition is used as a marker for identifying the position of the interface (free surface) in the two‐phase flow problems. An initial‐value‐problem (IVP) partial differential equation (PDE) is derived from the dynamic conditions of the interface, and it is designed to govern the motion of the interface in time. In this adaptive technique, the Navier–Stokes equations written for two incompressible fluids together with the IVP are solved numerically over the flow domain. An adaptive mass conservation algorithm is constructed to govern the continuum of the fluid. The finite element method (FEM) is used for the spatial discretization and a fully coupled implicit time integration method is applied for the advancement in time. FE‐stabilization techniques are added to the standard formulation of the discretization, which possess good stability and accuracy properties for the numerical solution. The adaptive technique is tested in simulation of some numerical examples. With the test problems presented here, we demonstrated that the adaptive technique is a simple tool for modelling and computation of complex motion of sharp physical interfaces in convection–advection‐dominated flow problems. We also demonstrated that the IVP and the evolution of the interface function are coupled explicitly and implicitly to the system of the computed unknowns in the flow domain. Copyright © 2005 John Wiley & Sons, Ltd.     

The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems

This paper deals with an application of the boundary element method to the analysis of nonlinear sloshing problems, namely nonlinear oscillations of a liquid in a container subjected to forced oscillations. First, the problem is formulated mathematically as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions, assuming the fluid to be inviscid and incompressible and the flow to be irrotational. Next, the governing equation (Laplace equation) and boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by employing the Galerkin method. Two dynamic boundary condition is reduced to a weighted residual equation by employing the Galerkin method. Two equations thus obtained are discretized by the use of the finite element method spacewise and the finite difference method timewise. Collocation method is employed for the discretization of the integral equation. Due to the nonlinearity of the problem, the incremental method is used for the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the conventional finite element method and also with existing analytical solutions of the nonlinear theory. Good agreements are obtained, and this indicates the availability of the boundary element method as a numerical technique for nonlinear free surface fluid problems.     

A computational approach for predicting the hydroelasticity of flexible structures based on the pressure Poisson equation

This work is concerned with the modeling of the interaction of fluid flow with flexible solid structures. The flow under consideration is governed by the Navier–Stokes equations for incompressible viscous fluids and modeled with low‐order velocity–pressure finite elements. The motion of the fluid domain is accounted for by the arbitrary Lagrangian–Eulerian formulation. The structure is represented by means of an appropriate standard finite element formulation. The spring smooth analogy is used to mesh control. The time integrating algorithm is based on the predictor–multi‐corrector algorithm. An important aspect of the present work is the introduction of a new monolithic approach based on the fluid pressure Poisson equation (PPE) to solve the hydroelasticity problem of an incompressible viscous fluid with an elastic body that is vibrating due to flow excitation. The PPE is derived to be consistent with the coupled system equation for the fluid–structure interaction (FSI). Based on this approach, an efficient monolithic method is adopted to simulate hydroelasticity between the flexible structure and the flow. The fluid pressure is implicitly derived to satisfy the incompressibility constraint, and the other unknown variables are explicitly derived. The coefficient matrix of the PPE for the FSI becomes symmetric and positive definite. To demonstrate the performance of the proposed approach, two working examples, a beam immersed in incompressible fluid and a guide vane of a Francis turbine passage, were used. The results show the validity of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Boundary integral equation method for two dissimilar elastic inclusions in an infinite plate

This paper provides a numerical solution for an infinite plate containing two dissimilar elastic inclusions, which is based on complex variable boundary integral equation (CVBIE). The original problem is decomposed into two problems. One is an interior boundary value problem (BVP) for two elastic inclusions, while other is an exterior BVP for the matrix with notches. After performing discretization for the coupled boundary integral equations (BIEs), a system of algebraic equations is formulated. The inverse matrix technique is suggested to solve the relevant algebraic equations, which can avoid using the assembling of some matrices. Several numerical examples are carried out to prove the efficiency of suggested method and the hoop stress along the interface boundary is evaluated.     

Influence of the electric current waveform on the dynamic response of the electrified composites

The influence of the electric current waveform (DC, AC and pulsed currents) on the dynamic electromagnetic, thermal, and impact response of the composite plate is studied. The analysis includes solving Maxwell’s equations in the electrified composite plate to determine an electric-current induced magnetic field and heat transfer equation to estimate the electric-current-induced heating. In addition, the dynamic mechanical response of the electrified composite plate subjected to impact and various electromagnetic loads (DC, AC, pulsed electric currents and a constant magnetic field) is analyzed by solving a coupled system of equations of motion and Maxwell’s equations in the composite plate. The results show that the dynamic response of the plate is highly dependent on the characteristics of the electromagnetic field, and the pulsed electromagnetic fields are most effective in reducing vibrations caused by the application of dynamic mechanical loads.     

Analysis of the effects of rough surfaces in compressible thin film flow by homogenization

A classical problem in lubrication theory is to predict the pressure distribution in a thin fluid film between two surfaces which are in relative motion. If one of the surfaces is rough, then the distance between the surfaces is rapidly oscillating. This leads to that the governing Reynolds partial differential equation involves rapidly oscillating coefficients. The branch in mathematics which considers such types of equations is known as homogenization. In this paper we study the effects of surface roughness for a special type of compressible fluid. In particular, we derive homogenization results connected to the friction force and the load carrying capacity.     

Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law, whereas slip is described by a Coulomb-type friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complementary function, and a semismooth Newton method is used to solve the system of equations. For the discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite volume discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2D and 3D, showing that the convergence rates of the finite volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.     

Non-orthogonal stagnation-point flow of a micropolar fluid

This paper considers the problem of steady two-dimensional flow of a micropolar fluid impinging obliquely on a flat plate. The flow under consideration is a generalization of the classical modified Hiemenz flow for a micropolar fluid which occurs in the boundary layer near an orthogonal stagnation point. A coordinate decomposition transforms the full governing equations into a primary equation describing the modified Hiemenz flow for a micropolar fluid and an equation for the tangential flow coupled to the primary solution. The solution to the boundary-value problem is governed by two non-dimensional parameters: the material parameter K and the ratio of the microrotation to skin friction parameter n. The obtained ordinary differential equations are solved numerically for some values of the governing parameters. The primary consequence of the free stream obliqueness is the shift of the stagnation point toward the incoming flow.     

Einführung in moderne Galerkin-Randeiementmethoden mit einer Anwendung aus dem Maschinenbau

The Galerkin-type boundary element method (BEM) is an discretization procedure for integral equations, represents itself however compared with classical integral equation methods as an universal tool for the solution of practical engineering problems and can be coupled very easily with finite element substructures. The BEM, whose main advantage lies in the fact that only a surface mesh must be generated, is superior to FEM in special applications, i.e. in elastostatics (notch problems) and fracture mechanics. In this paper the individual steps to solving an elliptical boundary value problem of 3-D linear elasticity theory by way of an equivalent system of boundary integral equations will be explained. For the mathematical investigation of elliptical differential equations and integral equations, the theory of Sobolev spaces has proved to be especially suitable. Basic terms to Sobolev spaces will be introduced so that the reader does not have to refer to textbooks for new terms. The transformation of elliptical boundary value problems to systems of singular and hypersingular integral equations will be explained with help of a Calderón projector, which is defined by using fundamental solutions. The discretization of the obtained integral equations with the Galerkin-type BEM will be presented. Finally the approximation of non-linear problems by using the Galerkin-type BEM will be shown. A numerical test for a strength problem will be discussed shortly.     

流体介质中圆板受冲击载荷时的弹塑性非线性动力响应

提出了一种有限元结合积分变化的新方法，用以分析无粘性流体介质中，受冲击载荷的圆板的弹塑性响应。结合轴对称有限元方法，根据节点速度，应用Lagrange插值得到圆板任意时刻的速度函数，并根据速度函数结合积分变换求取流体阻力。通过在每一个积分步上，交互求解结构方程和流体阻力，最终得到圆板的变形和应力时间历程。计算结果同LS-DYNA进行了比较，表面该方法切实可行，且适应面广。在最后以一个半径为的圆板为例进行了分析，并讨论了流体对板受冲击时的动力学行为的影响。     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

Direct boundary integral equations for elastodynamics in 3-D half-spaces

Vector boundary integral equations (BIE's) based on Somigliana's integral formula are presented with Stokes' (full-space) and Lamb's (half-space) fundamental tensors (Green's functions), for radiation and scattering of time-harmonic elastic waves by bodies embedded in or laying on the surface of a three-dimensional homogeneous, isotropic, linear-elastic half-space. Numerical work is based on BIEs free from principal-value integrals. Whereas the Stokes'-tensor BIE requires discretization of the infinite half-space surface, all discretization is confined to (finite) surfaces of the body when Lamb's tensors are used. The nonuniqueness of the integral equation solution at fictitious eigenfrequencies is addressed. Numerical results are presented for a rigid circular footing, a rigid hemispherical foundation and a fully embedded spherical cavity.     

浸没的圆柱壳在冲击波作用下动响应的Galerkin解法

研究了无粘、无旋和不可压缩流体中两端简支圆柱壳在给定冲击波作用下的动响应。圆柱壳的运动方程中考虑了流体动压力和冲击波压力的共同作用，通过将冲击波压力分布函数表示为Ｆｏｕｒｉｅｒ级数有限项形式，并利用Ｇａｌｅｒｋｉｎ方法对耦合方程进行数值求解，得到了圆柱壳在冲击波作用下的位移响应特性