Treatment of fast moving loads in elastodynamic problems with a space-time variational formulation

The treatment of fast moving loads is faced with a variational space-time approach. A time integration algorithm and the variational treatment of fast moving loads are developed. The time integration algorithm and the treatment of moving loads are based on Hermite’s polynomials with independent field of velocities and discontinuous time derivative of the displacement field. The algorithm exhibits 6th order convergence and computational efficiency to treat problems of fast moving loads.     

Nonlinear Galerkin method for the exterior nonstationary Navier-Stokes equations

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔΩｃｏｎｔａｉｎｉｎｇｚｅｒｏｐｏｉｎｔｂｅａｓｉｍｐｌｙ＿ｃｏｎｎｅｃｔｅｄｂｏｕｎｄｅｄｏｐｅｎｓｅｔｏｆＲ2 ｗｉｔｈｓｍｏｏｔｈｂｏｕｎｄａｒｙΓａｎｄｌｅｔΩ′ｄｅｎｏｔｅｔｈｅｃｏｍｐｌｅｍｅｎｔｏｆΩ ∪Γ .ＴｈｅｅｘｔｅｒｉｏｒｎｏｎｓｔａｔｉｏｎａｒｙＮａｖｉｅｒ＿ＳｔｏｋｅｓｐｒｏｂｌｅｍｆｏｒａｆｌｕｉｄｏｃｃｕｐｙｉｎｇΩ′ｃｏｎｓｉｓｔｓｉｎｆｉｎｄｉｎｇｔｈｅｖｅｌｏｃｉｔｙ ｕ(ｘ,ｔ)ｏｆｔｈｅｆｌｕｉｄａｎｄｉｔｓｐｒｅｓｓｕｒｅ ｐ(ｘ ,…     

A numerical method for one‐dimensional compressible multiphase flows on moving meshes

This paper is devoted to the numerical approximation of a hyperbolic non‐equilibrium multiphase flow model with different velocities on moving meshes. Such goal poses several difficulties. The presence of different flow velocities in conjunction with cell velocities poses difficulties for upwinding fluxes. Another issue is related to the presence of non‐conservative terms. To solve these difficulties, the discrete equations method (J. Comput. Phys. 2003; 186 (2):361–396; J. Fluid. Mech. 2003; 495 :283–321; J. Comput. Phys. 2004; 196 :490–538; J. Comput. Phys. 2005; 205 :567–610) is employed and generalized to the context of moving cells. The complementary conservation laws, available for the mixture, are used to determine the velocities of the cells boundaries. With these extensions, an accurate and robust multiphase flow method on moving meshes is obtained and validated over several test problems with exact or experimental solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal Elastic Design for Prescribed Maximum Deflection

ABSTRACT

For the problems of the optimal elastic design with prescribed maximum deflection, a variational formulation is proposed, with reference to the one-or two-dimensional bending structures.

Necessary optimum conditions are found, and the physical features of the optimal solutions are discussed for the “absolute” minimum cost problems, and, when dealing with beams, for the solutions with piece-wise constant design function.

Some examples are solved by using numerical methods that are directly derived from the variational formulation.     

A similarity solution for the flow and heat transfer over a moving permeable flat plate in a parallel free stream

This paper presents both a numerical and analytical study in connection with the steady boundary layer flow and heat transfer induced by a moving permeable semi-infinite flat plate in a parallel free stream. Both the velocities of the flat plate and the free stream are proportional to x ^1/3. The surface temperature is assumed to be constant. The governing partial differential equations are converted into ordinary differential equations by a new similarity transformation. Numerical results for the flow and heat transfer characteristics are obtained for various values of the moving parameter, transpiration parameter and the Prandtl number. Approximate analytical solutions are also obtained when the suction or injection parameter is very large. It is found that dual solutions exist for the case when the fluid and the plate move in the opposite directions.     

New parallel method for adjacent disconnected unstructured 3D meshes

ABSTRACT

A new parallel method for simulations with non-overlapping disconnected mesh domains but adjacent boundaries is presented and studied. This technique allows simulations using 3D unstructured meshes that are independent.     

New variational principles in plasticity and their application to composite materials

I , a variational method for bounding the effective properties of nonlinear composites with isotropic phases, proposed recently by ponte castañeda (J. Mech. Phys. Solids 39, 45, 1991), is given full variational principle status. Two dual versions of the new variational principle are presented and their equivalence to each other, and to the classical variational principles, is demonstrated. The variational principles are used to determine bounds and estimates for the effective energy functions of nonlinear composites with prescribed volume fractions in the context of the deformation theory of plasticity. The classical bounds of Voigt and Reuss for completely anisotropic composites are recovered from the new variational principles and are given alternative, simpler forms. Also, use of a novel identity allows the determination of simpler forms for nonlinear Hashin-Shtrikman bounds, and estimates, for isotropic, particle-reinforced composites, as well as for transversely isotropic, fiber-reinforced composites. Additionally, third-order bounds of the Beran type are determined for the first time for nonlinear composites. The question of the optimality of these bounds is discussed briefly.     

Moving velocities of precursor soliton generation in two-layer flow

Based on the AfKdV equation of Xu et al.^[1], a theory on the velocities of the precursor soliton generation in two-layer flow over topography is presented in the present paper. Moving velocities of precursor solitons, of the first zero-crossing of the tailing wavetrain and of the flow behind the topography are found theoretically. It is shown that for a given topography, when its moving velocities are at the resonant points, we have the following rules: the ratio of the moving velocity of the precursor solitons to that of the first zero-crossing of the tailing wavetrain equals −4/3. At the same time, the ratio of the width of generating region of the precursor solitons to that of the depressed water region equals also −4/3. The theoretical results are examined by means of numerical calculation. The comparison between the theoretical and numerical results are found in good agreement. For different stratified parameters of two-layer flow, the velocities of the precursor soliton generation are also predicted in terms of the present theoretical results. The project supported by the National Natural Science Foundation of China under the Grant No. 49776284     

Dynamics analysis of a parallel hip joint simulator with four degree of freedoms (3R1T)

For a hip joint simulator with a 3SPS+1PS spatial parallel manipulator as the core module, a formulation based on the Kane equation was derived for the dynamic characteristics of the simulator from the kinematics analysis of the model. The relationships of the velocities and accelerations between the moving platform and active branched-chains were deduced. The velocity and angular velocity components of the moving platform were served as the generalized velocities. And the dynamic model was established by obtaining the generalized active forces and inertial forces. Then the driving forces and powers of the active branched-chains and the constraint reaction forces of the intermediate branched-chain were simulated in the numerical method. The results showed that the active driving forces of the branched-chains reached their respective maximum when the moving platform rotated into 0.13^° around X-axis, 2^° around Y-axis, and 18^° around Z-axis. And the intermediate branched-chain needed to balance the driving and inertia forces, as well as support the moving platform and load the force of hydraulic cylinder. Therefore, the maximum constraint reaction force of the intermediate branched-chain is along the Z-axis. The research works provided a theoretical basis for the design of the active branched-chains driving system and the structural parameters of the intermediate branched-chain, as well as for the control system.     

A structure-preserving algorithm for time-scale non-shifted Hamiltonian systems

The variational calculus of time-scale non-shifted systems includes both the traditional continuous and traditional significant discrete variational calculus. Not only can the combination of $\Delta$ and $?$ derivatives be beneficial to obtaining higher convergence order in numerical analysis, but also it prompts the time-scale numerical computational scheme to have good properties, for instance, structure-preserving. In this letter, a structure-preserving algorithm for time-scale non-shifted Hamiltonian systems is proposed. By using the time-scale discrete variational method and calculus theory, and taking a discrete time scale in the variational principle of non-shifted Hamiltonian systems, the corresponding discrete Hamiltonian principle can be obtained. Furthermore, the time-scale discrete Hamilton difference equations, Noether theorem, and the symplectic scheme of discrete Hamiltonian systems are obtained. Finally, taking the Kepler problem and damped oscillator for time-scale non-shifted Hamiltonian systems as examples, they show that the time-scale discrete variational method is a structure-preserving algorithm. The new algorithm not only provides a numerical method for solving time-scale non-shifted dynamic equations but can be calculated with variable step sizes to improve the computational speed.     

一种基于Hamel形式的无条件稳定动力学积分算法

Time integration algorithm is a key issue in solving dynamical system. An unconditionally stable Hamel generalized α method is proposed to solve the instability issue arising in the time integration of dynamic equations and to eliminate the pseudo high order harmonics incurred by the spatial discretization of finite element simultaneously. Therefore, the development of numerical integration algorithm to solve the above-mentioned problems has important theoretical and application value. The algorithm proposed in this paper is developed based on the moving frame method and Hamel’s field variational integrators along with the strategy to construct an unconditionally stable Hamel generalized α method. It is shown that a new numerical formalism with higher accuracy can be derived under the same framework of the unconditional stable algorithm established through a special variational formalism and variational integrators. The above-mentioned formalism can be extended from general linear space to Lie group by utilizing the moving frame method and the Lie group formalism of the Hamel generalized α method has been obtained. Both the convergence and stability of the algorithm are discussed, and some numerical examples are presented to verify the conclusion. It is demonstrated by the theoretical analysis that the Hamel generalized α method proposed in the paper is unconditionally stable, second-order accurate and can quickly filter out pseudo high-frequency harmonics. Both conventional and proposed methods have been applied to numerical examples respectively. Comparisons between results of numerical examples show that the aforementioned advantages of the proposed method in terms of accuracy, dissipation and stability are tested and verified. At the same time, it can be developed that new numerical integration algorithms with even higher order accuracy. The scheme can also be proposed, which is suitable for both general linear space and Lie group space. A new way for constructing variational integrators is also obtained in this paper.      

Dynamic parallel Galerkin domain decomposition procedures with grid modification for parabolic equation

Dynamic parallel Galerkin domain decomposition procedures with grid modification for semi‐linear parabolic equation are given. These procedures allow one to apply different domain decompositions, different grids, and interpolation polynomials on the sub‐domains at different time levels when necessary, in order to capture time‐changing localized phenomena, such as, propagating fronts or moving layers. They rely on an implicit Galerkin method in the sub‐domains and simple explicit flux calculation on the inter‐domain boundaries by integral mean method to predict the inner‐boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the sub‐domains and across inter‐boundaries. The explicit nature of the flux prediction induces a time step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. Stability and convergence analysis in L²‐norm are derived for these procedures. The experimental results are presented to confirm the theoretical results. Copyright © 2010 John Wiley & Sons, Ltd.     

Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals

The paper presents new continuous and discrete variational formulations for the homogenization analysis of inelastic solid materials undergoing finite strains. The point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro-structure as a generalized standard medium in terms of an energy storage and a dissipation function. Consistent with this type of finite inelasticity we develop a new incremental variational formulation of the local constitutive response, where a quasi-hyperelastic micro-stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. We specify the local variational formulation for a distinct setting of multi-surface inelasticity and develop a numerical solution technique based on a time discretization of the internal variables. The existence of the quasi-hyperelastic stress potential allows the extension of homogenization approaches of finite elasticity to the incremental setting of finite inelasticity. Focussing on macro-deformation-driven micro-structures, we develop a new incremental variational formulation of the global homogenization problem for generalized standard materials at finite strains, where a quasi-hyperelastic macro-stress potential is obtained from a global minimization problem with respect to the fine-scale displacement fluctuation field. It is shown that this global minimization problem determines the state of the micro-structure for finite increments of time. We consider three different settings of the global variational problem for prescribed displacements, non-trivial periodic displacements and prescribed stresses on the boundary of the micro-structure and develop numerical solution methods based on a spatial discretization of the fine-scale displacement fluctuation field. Representative applications of the proposed minimization principles are demonstrated for a constitutive model of crystal plasticity and the homogenization problem of texture analysis in polycrystalline aggregates.     

Closed-form Solutions in Optimal Design of Structures with Nonlinear Behavior

ABSTRACT

Optimal design problems for flexural systems with a nonlinear constitutive law are considered, in the presence of constraints on displacements. A general nonlinear holonomic moment-curvature relationship is assumed and a direct variational method is applied in order to obtain optimality criteria. Accordingly, a general method of solution is proposed and some examples are solved.     

A finite element formulation satisfying the discrete geometric conservation law based on averaged Jacobians

In this article, a new methodology for developing discrete geometric conservation law (DGCL) compliant formulations is presented. It is carried out in the context of the finite element method for general advective–diffusive systems on moving domains using an ALE scheme. There is an extensive literature about the impact of DGCL compliance on the stability and precision of time integration methods. In those articles, it has been proved that satisfying the DGCL is a necessary and sufficient condition for any ALE scheme to maintain on moving grids the nonlinear stability properties of its fixed‐grid counterpart. However, only a few works proposed a methodology for obtaining a compliant scheme. In this work, a DGCL compliant scheme based on an averaged ALE Jacobians formulation is obtained. This new formulation is applied to the θ family of time integration methods. In addition, an extension to the three‐point backward difference formula is given. With the aim to validate the averaged ALE Jacobians formulation, a set of numerical tests are performed. These tests include 2D and 3D diffusion problems with different mesh movements and the 2D compressible Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd.     

Elastic perfectly plastic structures with temperature dependent elastic coefficients

We study the evolution of elastic perfectly plastic structures where the elastic coefficients depend on temperature, as they are subjected to classical loading and given variation of the temperature field. We prove variational theorems for the instantaneous fields of velocities and stress rates, and establish the generalized differential equation for the evolution of the stress field. To cite this article: B. Halphen, C. R. Mecanique 333 (2005).     

Hamel 框架下几何精确梁的离散动量守恒律

Hamel场变分积分子是一种研究场论的数值方法, 可以通过使用活动标架规避几何非线性带来的计算复杂度, 同时数值上具有良好的长时间数值表现和保能动量性质. 本文在一维场论框架下, 以几何精确梁为例, 从理论上探究Hamel场变分积分子的保动量性质. 具体内容包括: 利用活动标架法对几何精确梁建立动力学模型, 通过变分原理得到其动力学方程, 利用其动力学方程及Noether定理得到系统动量守恒律; 将几何精确梁模型离散化, 通过变分原理得到其Hamel场变分积分子, 利用Hamel场变分积分子和离散Noether定理得到离散动量守恒律, 并给出离散动量的一阶近似表达式; Hamel场变分积分子可在计算中利用系统对称性消除系统运动带来的非线性问题, 但此框架中离散对流速度、离散对流 应变及位形均不共点, 而这种错位导致离散动量中出现级数项, 本文对几何精确梁的离散动量与连续形式的关系及其应 用进行了讨论, 并通过算例验证了结论. 上述证明方法也同样适用一般经典场论场景下的Hamel场变分积分子. Hamel场变分积分子的动量守恒为进一步研究其保结构性质提供了参考依据.      

MULTI-VALUED QUASI VARIATIONAL INCLUSIONS IN BANACH SPACES

The purpose is to introduce and study a class of more general multivalued quasi variational inclusions in Banach spaces. By using the resolvent operator technique some existence theorem of solutions and iterative approximation for solving this kind of multivalued quasi variational inclusions are established. The results generalize, improve and unify a number of Noor‘s and others‘ recent results.     

Interaction of viscous drops in a yield stress material

The interaction of Newtonian drops of various volumes moving in yield stress material is investigated experimentally. Tetrachloroethylene drops move in a rectangular reservoir filled with neutralized 0.07% w/w Carbopol gel under the action of gravity. For initially vertically aligned drop pairs, we present time evolution of separation distance, velocities during the interaction and conditions for coalescence of the drops, which depend on the volumes of the drops and the initial separation between them. For the asymmetric interaction of the pairs, we present interaction patterns, which have been used for estimation of the size of the yielded region and its shape around the leading drop.