Application of a vortex method to free surface flows

Vortex methods have found wide applications in various practical problems. The use of vortex methods in free surface flow problems, however, is still very limited. This paper demonstrates a vortex method for practical computation of non-linear free surface flows produced by moving bodies. The method is a potential flow formulation which uses the exact non-linear free surface boundary condition at the exact location of the instantaneous free surface. The position of the free surface, on which vortices are distributed, is updated using a Lagrangian scheme following the fluid particles on the free surface. The vortex densities are updated by the non-linear dynamic boundary condition, derived from the Euler equations, with an iterative Lagrangian numerical scheme. The formulation is tested numerically for a submerged circular cylinder in unsteady translation. The iteration is shown to converge for all cases. The results of the unsteady simulations agree well with classical linearized solutions. The stability of the method is also discussed.     

Dynamics of cable truss via polynomial shape functions

A model for the in-plane dynamic behaviour of a biconcave cable structure, subject to large static deformations and potentially slack harnesses is proposed, based on polynomial shape functions, in line with the classical Ritz method. The model provides a semi-analytical approach to the calculation of natural frequencies and modal shapes of the structure. The proposed formulation leads to an eigenvalue problem, based on a reduced number of degrees of freedom compared with equivalent FEM solutions, providing the basis for fast and accurate sensitivity analysis. The behaviour of the deformed structure is analysed in detail to understand the non-linear effects of non-symmetric mass and load distribution and slack harnesses on natural frequencies and corresponding modal shapes. Results confirm the relevance of the non-linear effects, due to the statically loaded configuration, on the linear vibrations of the structure, in particular evidencing the influence of the slackening of harnesses on modal shapes. Results are compared to analytical models, where available (single sagged cable), and to FEM solutions (for cable trusses with non-uniform mass and load distribution and potentially slack harnesses), providing good agreement.

     

Non-linear dynamic analysis of tensegrity structures using a co-rotational method

A new formulation is presented for the non-linear dynamic analysis of space truss structures. The formulation is based on the dynamics of 3D co-rotational rods. In the co-rotation method, the rigid body modes are assumed to be separated from the total deformations at the local element level. In this paper a new co-rotational formulation is proposed based on the direct derivation of the inertia force vector and the tangent dynamic matrix. A closed-form equation is derived for the calculation of the inertia force, the tangent dynamic matrix, the mass matrix and the gyroscopic matrix. The new formulation is used to perform dynamic analysis of example tensegrity structures. The developed formulation is applicable to tensegrity structures with non-linear effects due to internal mechanisms or geometric non-linearities, and is applied to two numerical examples. The efficiency of the proposed approach is compared to the conventional Lagrangian method, and savings in computation of about 55%, 54% and 37% were achieved.     

A robust shell element in meshfree SPH method

With the incorporation of total Lagrangian smoothed particle hydrodynamics(SPH) method equation and moving least square(MLS) function,the traditional SPH method is improved regarding the stability and consistency.Based on Mindlin-Ressiner plate theory,the SPH method simulating dynamic behavior via one layer of particles is applied to plate’s mid-plane,i.e.,a SPH shell model is constructed.Finally,through comparative analyses on the dynamic response of square,stiffened shells and cylindrical shells under various strong impact loads with common finite element software,the feasibility,validity and numerical accuracy of the SPH shell method are verified.Consequently,further researches on SPH shell may well pave the way towards solving problems involving dynamic plastic damage,tearing or even crushing.     

Finite deformation plate theory and large rigid-body motions

A refined non-linear first-order theory of multilayered anisotropic plates undergoing finite deformations is elaborated. The effects of the transverse shear and transverse normal strains, and laminated anisotropic material response are included. On the basis of this theory, a simple and efficient finite element model in conjunction with the total Lagrangian formulation and Newton-Raphson method is developed. The precise representation of large rigid-body motions in the displacement patterns of the proposed plate elements is also considered. This consideration requires the development of the strain-displacement equations of the finite deformation plate theory with regard to their consistency with the arbitrarily large rigid-body motions. The fundamental unknowns consist of six displacements and 11 strains of the face planes of the plate, and 11 stress resultants. The element characteristic arrays are obtained by using the Hu-Washizu mixed variational principle. To demonstrate the accuracy and efficiency of this formulation and compare its performance with other non-linear finite element models reported in the literature, extensive numerical studies are presented.     

Geometrically non-linear analysis of arbitrary elastic supported plates and shells using an element-based Lagrangian shell element

In the present paper, the ELF (element-based Lagrangian formulation) 9-node ANS (assumed natural strain) shell element was combined with the spring element for geometrically non-linear analysis of plates and shells sustained by arbitrary elastic edge supports that are subjected to variation in loading.This particular spring element serves as tool for modeling an arbitrary elastic edge support with 6 DOF (degrees of freedom). The elastic edge support was modeled by combining different spring models. The ANS method was used to overcome shear and membrane locking problems inherent in some thin plate and shell problems. In the formulation of the ELF characteristic arrays, the expression of element strains was adopted in the framework of the element natural coordinates. The non-linear analysis results of idealized edge supports were validated against the reference solutions available in the literature. As a result of the numerical test, the combination of the ELF 9-node shell element and spring element shows an exceptional performance for non-linear analysis of plates and shells under elastic edge supports.     

多相流动的光滑粒子流体动力学方法研究综述

光滑粒子流体动力学(smoothed particle hydrodynamics, SPH)具有粒子方法的无网格和全拉格朗日特征, 适用于具有界面大变形、不连续性和多物理场的多相流的高精度模拟. SPH方法模拟多相流已有大量报道, 具体的实现方式也大不相同. 本文首先阐述了采用SPH方法模拟流体的基本控制方程, 以及求解过程中需要考虑的流体压力求解、表面张力、固体边界等问题. 整理和总结了基于SPH方法进行多相流模拟的主要实现方式: (1)双流体模型的拉格朗日求解器: 两相离散为两组独立SPH粒子, 并用显式相间作用耦合两相; (2)多相SPH方法: SPH方法对多相流模拟的自然延伸, 相间作用由SPH参数隐式描述; (3) SPH与其他离散方法的耦合: 差异较大的两相各自采用不同离散方法, 发挥不同拉格朗日方法的优点; (4) SPH和基于网格方法的耦合: 网格方法处理简单的单相流动主体, 获得精度和效率间的平衡. 另外, 还在模拟参数物理化等方面论述了与SPH方法模拟多相流相关的一些改进和修正方法, 并在最后讨论和建议了提高多相流SPH模拟效率和精度的措施.      

Incompressible SPH simulation of water entry of a free‐falling object

In this paper an incompressible smoothed particle hydrodynamics (Incom‐SPH) model is used to simulate the interactions between the free surface flow and a moving object. Incom‐SPH method is a two‐step semi‐implicit hydrodynamic formulation of the SPH algorithm and is capable of accurately treating the free surface deformations and impact forces during the solid–fluid interactions. For a free‐falling object, its motion is tracked by an additional Lagrangian algorithm based on Newton's law to couple with the Incom‐SPH program. The developed model is employed to investigate the water entry of a free‐falling wedge. The accuracy of the computations is validated by the good agreement in predicting the relevant hydrokinematic and hydrodynamic parameters. Finally, a numerical test is performed to study the influence of spatial resolution on the water entry features. The Incom‐SPH modeling coupled with the solid–fluid interaction algorithm could provide a promising computational tool to predict the slamming problems in coastal and offshore engineering. Copyright © 2008 John Wiley & Sons, Ltd.     

Incompressible SPH simulation of wave breaking and overtopping with turbulence modelling

In this paper a truly incompressible version of the smoothed particle hydrodynamics (SPH) method is presented to investigate the surface wave overtopping. SPH is a pure Lagrangian approach which can handle large deformations of the free surface with high accuracy. The governing equations are solved based on the SPH particle interaction models and the incompressible algorithm of pressure projection is implemented by enforcing the constant particle density. The two‐equation k–ε model is an effective way of dealing with the turbulence and vortices during wave breaking and overtopping and it is coupled with the incompressible SPH numerical scheme. The SPH model is employed to reproduce the experiment and computations of wave overtopping of a sloping sea wall. The computations are validated against the experimental and numerical data found in the literatures and good agreement is observed. Besides, the convergence behaviour of the numerical scheme and the effects of particle spacing refinement and turbulence modelling on the simulation results are also investigated in further detail. The sensitivity of the computed wave breaking and overtopping on these issues is discussed and clarified. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite elements based on a first-order shear deformation moderate rotation shell theory with applications to the analysis of composite structures

The first-order shear deformation moderate rotation shell theory of Schmidt and Reddy [R. Schmidt and J. N. Reddy, J. Appl. Mech. 55, 611–617 (1988)] is used as a basis for the development of finite element models for the analysis of the static, geometrically non-linear response of anisotropic and laminated structures. The incremental, total Lagrangian formulation of the theory is developed, and numerical solutions are obtained by using the isoparametric Lagrangian 9-node and Serendipity 8-node shell finite elements. Various integration schemes (full, selective reduced, and uniformly reduced integration) are applied in order to detect and to overcome the effects of shear and membrane locking on the predicted structural response. A number of sample problems of isotropic, orthotropic, and multi-layered structures are presented to show the accuracy of the present theory. The von Kármán-type first-order shear deformation shell theory and continuum 2D theory are used for comparative analyses.     

一种可考虑界面不连续的改进SPH方法

SPH方法模拟工程问题时通常遇到不连续的物理量,因此有必要引入不连续的SPH方法.本文基于Taylor展开公式推导了2D和3D的不连续SPH公式.针对越过材料界面不连续物理量的计算,给出了大变形计算中确定不连续位置的方法,基于Taylor展开公式,从理论上给出了确定不连续公式中x_(k)点的方法,并用数值方法验证了此方法的有效性.比较和讨论了初始SPH方法,CSPM方法与不连续SPH方法处理不连续量的效果.结果显示不连续SPH方法在计算不连续量时有较大的优势.     

A one-dimensional meshfree particle formulation for simulating shock waves

In this paper, a one-dimensional meshfree particle formulation is proposed for simulating shock waves, which are associated with discontinuous phenomena. This new formulation is based on Taylor series expansion in the piecewise continuous regions on both sides of a discontinuity. The new formulation inherits the meshfree Lagrangian and particle nature of SPH, and is a natural extension and improvement on the traditional SPH method and the recently proposed corrective smoothed particle method (CSPM). The formulation is consistent even in the discontinuous regions. The resultant kernel and particle approximations consist of a primary part similar to that in CSPM, and a corrective part derived from the discontinuity. A numerical study is carried out to examine the performance of the formulation. The results show that the new formulation not only remedies the boundary deficiency problem but also simulates the discontinuity well. The formulation is applied to simulate the shock tube problem and a 1-D TNT slab detonation. It is found that the proposed formulation captures the shock wave at comparatively lower particle resolution. These preliminary numerical tests suggest that the new meshfree particle formulation is attractive in simulating hydrodynamic problems with discontinuities such as shocks waves.Received: 8 October 2002, Accepted: 10 May 2003, Published online: 15 August 2003     

HIA: A Hybrid Integral Approach to model incompressible isotropic hyperelastic materials – Part 2: Finite element analysis

The present paper is devoted to the modeling of finite deformations of hyperelastic bodies by using the Hybrid Integral Approach (HIA) model introduced by the Authors. This model combines the affine behavior of chains included in the molecular 8-chains density, the interleaving constraint part and an original phenomenological integral density which allows to balance the mismatch between model and experimental data. A total Lagrangian formulation is adopted to describe the geometrical and material non-linearities and a finite element implementation is performed in the FER software. Four test examples, including homogeneous and non-homogeneous deformations, as well as complex computation involving dynamic contact and impact are proposed to show the applicability of the developed approach and the efficiency of the model.     

SPH computation of plunging waves using a 2‐D sub‐particle scale (SPS) turbulence model

The paper presents a 2‐D large eddy simulation (LES) modelling approach to investigate the properties of the plunging waves. The numerical model is based on the smoothed particle hydrodynamics (SPH) method. SPH is a mesh‐free Lagrangian particle approach which is capable of tracking the free surfaces of large deformation in an easy and accurate way. The Smagorinsky model is used as the turbulence model due to its simplicity and effectiveness. The proposed 2‐D SPH–LES model is applied to a cnoidal wave breaking and plunging over a mild slope. The computations are in good agreement with the documented data. Especially the computed turbulence quantities under the breaking waves agree better with the experiments as compared with the numerical results obtained by using the k–ε model. The sensitivity analyses of the SPH–LES computations indicate that both the turbulence model and the spatial resolution play an important role in the model predictions and the contributions from the sub‐particle scale (SPS) turbulence decrease with the particle size refinement. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite deformations of Ogden's materials under impact loading

The present paper is devoted to the modeling of finite deformations of a hyperelastic body described by Ogden's model under contact/impact conditions. Frictional contact problems are solved by means of the bi-potential method. The first order algorithm is applied to integrate the equation of motion. The total Lagrangian formulation is adopted to describe the geometrically non-linear behavior. For the finite element implementation, the explicit expression of the tangent operator is derived including the case of repeated eigenvalues. A numerical example is given to illustrate efficiency and accuracy of the method.     

Adaptive smoothed particle hydrodynamics for high strain hydrodynamics with material strength

This paper presents the implementation of an adaptive smoothed particle hydrodynamics (ASPH) method for high strain Lagrangian hydrodynamics with material strength. In ASPH, the isotropic kernel in the standard SPH is replaced with an anisotropic kernel whose axes evolve automatically to follow the mean particle spacing as it varies in time, space, and direction around each particle. Except for the features inherited from the standard SPH, ASPH can capture dimension-dependent features such as anisotropic deformations with a more generalized elliptical or ellipsoidal influence domain. Two numerical examples, the impact of a plate against a rigid surface and the penetration of a cylinder through a plate, are investigated using both SPH and ASPH. The comparative studies show that ASPH has better accuracy than the standard SPH when being used for high strain hydrodynamic problems with inherent anisotropic deformations. PACS 46.15.-x, 83.10.Rs, 83.50.-v     

General non-linear finite element analysis of thick plates and shells

A non-linear finite element analysis is presented, for the elasto-plastic behavior of thick shells and plates including the effect of large rotations. The shell constitutive equations developed previously by the authors [Voyiadjis, G.Z., Woelke, P., 2004. A refined theory for thick spherical shells. Int. J. Solids Struct. 41, 3747–3769] are adopted here as a base for the formulation. A simple C⁰ quadrilateral, doubly curved shell element developed in the authors’ previous paper [Woelke, P., Voyiadjis, G.Z., submitted for publication. Shell element based on the refined theory for thick spherical shells] is extended here to account for geometric and material non-linearities. The small strain geometric non-linearities are taken into account by means of the updated Lagrangian method. In the treatment of material non-linearities the authors adopt: (i) a non-layered approach and a plastic node method [Ueda, Y., Yao, T., 1982. The plastic node method of plastic analysis. Comput. Methods Appl. Mech. Eng. 34, 1089–1104], (ii) an Iliushin’s yield function expressed in terms of stress resultants and stress couples [Iliushin, A.A., 1956. Plastichnost’. Gostekhizdat, Moscow], modified to investigate the development of plastic deformations across the thickness, as well as the influence of the transverse shear forces on plastic behaviour of plates and shells, (iii) isotropic and kinematic hardening rules with the latter derived on the basis of the Armstrong and Frederick evolution equation of backstress [Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial Bauschinger effect. (CEGB Report RD/B/N/731). Berkeley Laboratories. R&D Department, California.], and reproducing the Bauschinger effect. By means of a quasi-conforming technique, shear and membrane locking are prevented and the tangent stiffness matrix is given explicitly, i.e., no numerical integration is employed. This makes the current formulation not only mathematically consistent and accurate for a variety of applications, but also computationally extremely efficient and attractive.     

Simulations of single bubbles rising through viscous liquids using Smoothed Particle Hydrodynamics

The shapes and dynamics of single bubbles rising through viscous fluids are studied using the SPH (Smoothed Particle Hydrodynamics) approach. This fully Lagrangian, particle-based method is applied to compute the complete two-phase flow, both inside the bubbles as well as around them. For that purpose, a multi-phase formulation of the SPH method that can address large density differences is retained, while surface tension effects are explicitly accounted for through a CSF (Continuum Surface Force) model. Numerical simulations have been performed for different regimes (corresponding to different relative importance of surface tension, viscosity and buoyancy effects) and the predicted topological changes as well as the terminal velocity and drag coefficients of bubbles are validated. The numerical outcomes are assessed not only with respect to reference experimental data but also with respect to other numerical methods, namely the Front-Tracking and the Lattice Boltzmann Methods. It is believed that this study corresponds to a new application of SPH approaches for two-phase flow simulations and results reveal the interest of this method to capture fine details of gas–liquid systems with deformable and rapidly changing interfaces.