考虑粘结面厚度的局部粘结单元法

传统无厚度粘结单元法CFEM (Cohesive finite element method)在模拟脆性材料断裂方面具有很强的优势,但也存在很大问题.一是单元尺寸增大,收敛性变差;二是单元尺寸变小,模型刚度发生折减.为了克服这两个问题,发展了考虑厚度的局部粘结单元法,即在裂纹可能扩展区插入具有一定厚度的粘结面单元.粘结面单元采用拓展虚内键本构(Augmented virtual internal bond)描述.由于考虑了厚度,粘结面交叉处会形成多边形空缺.为了弥补这一空缺,将其看作多边形键元胞,采用离散虚内键模型(Discretized virtual internal bond)对其建模,保证了模型的几何完整性.模拟结果表明,本文方法有效,克服了传统CFEM方法的刚度折减问题,提高了计算稳定性和收敛性.     

On correct account of finite rotations in finite plasticity theory

The non-uniqueness of the trantition from nonobjective constitutive relations to objective ones with the use of the principle of material frame-indifference (PMFI) is shown. To eliminate it, the concept of finite strain without rotations (FSWR) for a given material type and each strain component (elastic, plastic) is introduced. In FSWR the rotation is excluded with respect to the natural preferred configuration for a given material. Considered are a simple solid, a liquid, a monocrystal, a polycrystal and a composite. The proecedure is proposed for consistent generalization of known infinitesimal relations for finite strains and rotations. The structure of constitutive relations is derived for anisotropic elasto-plastic mono- and polycrystalline materials.     

理性有限元

提出了与常现有限元迥然不同的理性有限元列式。其方法论的差别在于理性有限元充分考虑了力学的微分方程，用方程的解来逼近单元内部场．即以力学的需求为主导，再用相应的数学方法推导。并用理住平面四边形元RQ4为典型予以表述。数值结果表明理性有限元的优越性质。     

A finite strain, finite band method for modeling ductile fracture

We present a finite deformation generalization of the finite thickness embedded discontinuity formulation presented in our previous paper [A.E. Huespe, A. Needleman, J. Oliver, P.J. Sánchez, A finite thickness band method for ductile fracture analysis, Int. J. Plasticity 25 (2009) 2349-2365]. In this framework the transition from a weak discontinuity to a strong discontinuity can occur using a single constitutive relation which is of importance in a range of applications, in particular ductile fracture, where localization typically precedes the creation of new free surface. An embedded weak discontinuity is introduced when the loss of ellipticity condition is met. The resulting localized deformation band is given a specified thickness which introduces a length scale thus providing a regularization of the post-localization response. The methodology is illustrated through several example problems emphasizing finite deformation effects including the development of a cup-cone failure in round bar tension.     

Footbridge between finite volumes and finite elements with applications to CFD

The aim of this paper is to introduce a new algorithm for the discretization of second‐order elliptic operators in the context of finite volume schemes on unstructured meshes. We are strongly motivated by partial differential equations (PDEs) arising in computational fluid dynamics (CFD), like the compressible Navier–Stokes equations. Our technique consists of matching up a finite volume discretization based on a given mesh with a finite element representation on the same mesh. An inverse operator is also built, which has the desirable property that in the absence of diffusion, one recovers exactly the finite volume solution. Numerical results are also provided. Copyright © 2001 John Wiley & Sons, Ltd.     

A combined finite volumes - finite elements method for a low-Mach model

In this paper, we develop a finite volumes - finite elements method based on a time splitting to simulate some low-Mach flows. The mass conservation equation is solved by a vertex-based finite volume scheme using a τ-limiter. The momentum equation associated with the compressibility constraint is solved by a finite element projection scheme. The originality of the approach is twofold. First, the state equation linking the temperature, the density, and the thermodynamic pressure is imposed implicitly. Second, the proposed combined scheme preserves the constant states, in the same way as a similar one previously developed for the variable density Navier-Stokes system. Some numerical tests are performed to exhibit the efficiency of the scheme. On the one hand, academic tests illustrate the ability of the scheme in term of convergence rates in time and space. On the other hand, our results are compared to some of the literature by simulating a transient injection flow as well as a natural convection flow in a cavity.     

Framework for non-coherent interface models at finite displacement jumps and finite strains

This paper deals with a novel constitutive framework suitable for non-coherent interfaces, such as cracks, undergoing large deformations in a geometrically exact setting. For this type of interface, the displacement field shows a jump across the interface. Within the engineering community, so-called cohesive zone models are frequently applied in order to describe non-coherent interfaces. However, for existing models to comply with the restrictions imposed by (a) thermodynamical consistency (e.g., the second law of thermodynamics), (b) balance equations (in particular, balance of angular momentum) and (c) material frame indifference, these models are essentially fiber models, i.e. models where the traction vector is collinear with the displacement jump. This constraints the ability to model shear and, in addition, anisotropic effects are excluded. A novel, extended constitutive framework which is consistent with the above mentioned fundamental physical principles is elaborated in this paper. In addition to the classical tractions associated with a cohesive zone model, the main idea is to consider additional tractions related to membrane-like forces and out-of-plane shear forces acting within the interface. For zero displacement jump, i.e. coherent interfaces, this framework degenerates to existing formulations presented in the literature. For hyperelasticity, the Helmholtz energy of the proposed novel framework depends on the displacement jump as well as on the tangent vectors of the interface with respect to the current configuration – or equivalently – the Helmholtz energy depends on the displacement jump and the surface deformation gradient. It turns out that by defining the Helmholtz energy in terms of the invariants of these variables, all above-mentioned fundamental physical principles are automatically fulfilled. Extensions of the novel framework necessary for material degradation (damage) and plasticity are also covered.     

Homogenization in finite thermoelasticity

A homogenization framework is developed for the finite thermoelasticity analysis of heterogeneous media. The approach is based on the appropriate identifications of the macroscopic density, internal energy, entropy and thermal dissipation. Thermodynamical consistency that ensures standard thermoelasticity relationships among various macroscopic quantities is enforced through the explicit enforcement of the macroscopic temperature for all evaluations of temperature dependent microscale functionals. This enforcement induces a theoretical split of the accompanying micromechanical boundary value problem into two phases where a mechanical phase imposes the macroscopic deformation and temperature on a test sample while a subsequent purely thermal phase on the resulting deformed configuration imposes the macroscopic temperature gradient. In addition to consistently recovering standard scale transition criteria within this framework, a supplementary dissipation criterion is proposed based on alternative identifications for the macroscopic temperature gradient and heat flux. In order to complete the macroscale implementation of the overall homogenization methodology, methods of determining the constitutive tangents associated with the primary macroscopic variables are discussed. Aspects of the developed framework are demonstrated by numerical investigations on model microstructures.     

Coupling of finite volume and finite element subdomains using different time integrators

This paper proposes a coupling strategy that can be used for transient fluid–structure interaction. The objective of this paper is to propose a time integrator coupling strategy, which ensures good properties to couple typical solid and typical fluid time integrators in linear cases. It is evaluated on a 1‐D toy problem only dedicated to the study of the quality of time integrators coupling. The structure is discretized by the linear finite element method and solved in time by the Newmark scheme, whereas the finite difference and finite volume methods are used for the fluid subdomain. By projecting the fluid equations on the eigenvector, we obtain a compatibility relation that corresponds to the characteristic line that transfers the fluid information from the inside to the fluid–structure interface. With this appropriate compatibility relation and the solid equations, the interface status is predicted for the next time step, ensuring zero interface energy. Hence, the order of accuracy and the stability are preserved to the minimum level of the two parts, the structure and the fluid. Furthermore, the coupling strategy allows incompatible time steps. Some numerical results are obtained for the 1‐D linear problem, and a good agreement with the analytical solution has been found. Copyright © 2013 John Wiley & Sons, Ltd.     

Exact finite element method

In this paper,a new method,exact element method for constructing finite element,ispresented.It can be applied to solve nonpositive definite or positive definite partialdifferential equation with arbitrary variable coefficient under arbitrary boundarycondition.Its convergence is proved and its united formula for solving partial differentialequation is given.By the present method,a noncompatible element can be obtained and thecompatibility conditions between elements can be treated very easily.Comparing the exactelement method with the general finite element method with the same degrees of freedom,the high convergence rate of the high order derivatives of solution can be obtained.Threenumerical examples are given at the end of this paper,which indicate all results canconverge to exact solution and have higher numerical precision.     

Introduction of turbulent model in a mixed finite volume/finite element method

The aim of this work is to present a new numerical method to compute turbulent flows in complex configurations. With this in view, a k-? model with wall functions has been introduced in a mixed finite volume/finite element method. The numerical method has been developed to deal with compressible flows but is also able to compute nearly incompressible flows. The physical model and the numerical method are first described, then validation results for an incompressible flow over a backward-facing step and for a supersonic flow over a compression ramp are presented. Comparisons are performed with experimental data and with other numerical results. These simulations show the ability of the present method to predict turbulent flows, and this method will be applied to simulate complex industrial flows (flow inside the combustion chamber of gas turbine engines). The main goal of this paper is not to test turbulence models, but to show that this numerical method is a solid base to introduce more sophisticated turbulence model.     

The random variational principle in finite deformation of elasticity and finite element method

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Standing waves of finite amplitude on surface of ideal liquid of finite depth

The nonlinear problem of plane gravitational standing waves of finite amplitude on the surface of an ideal incompressible liquid of infinite depth has been solved analytically [1], It was also solved in [2] using a new method, in which the dimensionless velocity potential, the profile of the free surface, and the frequency were expressed as power series in the parameter , equal to the ratio between the the amplitude and the wavelength. The results of these two papers agree. Below, the method of [2] is used to study plane standing surface waves of finite amplitude on the surface of a liquid of finite depth. The frequency, the profile of the free surface, and the velocity potential are expressed as power series in the small parameter . The solution is obtained in the third approximation. An expression for the amplitude dependence of the frequency is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 38–43, March–April, 1978.     

Coupling finite difference method with finite particle method for modeling viscous incompressible flows

A hybrid approach to couple finite difference method (FDM) with finite particle method (FPM) (ie, FDM-FPM) is developed to simulate viscous incompressible flows. FDM is a grid-based method that is convenient for implementing multiple or adaptive resolutions and is computationally efficient. FPM is an improved smoothed particle hydrodynamics (SPH), which is widely used in modeling fluid flows with free surfaces and complex boundaries. The proposed FDM-FPM leverages their advantages and is appealing in modeling viscous incompressible flows to balance accuracy and efficiency. In order to exchange the interface information between FDM and FPM for achieving consistency, stability, and convergence, a transition region is created in the particle region to maintain the stability of the interface between two methods. The mass flux algorithm is defined to control the particle creation and deletion. The mass is updated by N-S equations instead of the interpolation. In order to allow information exchange, an overlapping zone is defined near the interface. The information of overlapping zone is obtained by an FPM-type interpolation. Taylor-Green vortices and lid-driven shear cavity flows are simulated to test the accuracy and the conservation of the FDM-FPM hybrid approach. The standing waves and flows around NACA airfoils are further simulated to test the ability to deal with free surfaces and complex boundaries. The results show that FDM-FPM retains not only the high efficiency of FDM with multiple resolutions but also the ability of FPM in modeling free surfaces and complex boundaries.     

Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

This paper deals with the non-linear viscoelastodynamics of three-dimensional rotating structure undergoing finite displacement. In addition, the non-linear dynamics is studied with respect to geometrical and mechanical perturbations. On part of the boundary of the structure, a rigid body displacement field is applied which moves the structure in a rotation motion. A time-dependent Dirichlet condition is applied to another part of the boundary. For instance, this corresponds to the cycle step of a helicopter rotor blade. A surface force field is applied to the third part of the boundary and depends on the time history of the structural displacement field. For example, this might corresponds to general unsteady aerodynamics forces applied to the structure. The objective of this paper is to model the non-linear dynamic behavior of such a rotating viscoelastic structure undergoing finite displacements, and to allow small geometrical and mechanical (mass, constitutive equations) perturbations analysis to be performed. The model is constructed by the introduction of a reference configuration which is deduced from the non-linear steady boundary value problem. A constitutive equation deduced from the Coleman and Noll theory concerning the viscoelasticity in finite displacement is used. Thereafter, the weak formulation of the boundary value problem is constructed and discretized using the finite element method. In order to simplify the mathematical study of the equations, multilinear forms are introduced in the algebraic calculation and their mathematical properties are presented.     

基于有限断裂法和比例边界有限元法的裂纹扩展模拟

基于有限断裂法和比例边界有限元法提出了一种裂缝开裂过程模拟的数值模型。采用基于有限断裂法的混合断裂准则作为起裂及扩展的判断标准,当最大环向应力和能量释放率同时达到其临界值时,裂缝扩展。结合多边形比例边界有限元法,可以半解析地求解裂尖区域附近的应力场和位移场,在裂尖附近无需富集即可获得高精度的解。计算能量释放率时,只需将裂尖多边形内的裂尖位置局部调整,无需改变整体网格的分布,网格重剖分的工作量降至最少。裂缝扩展步长通过混合断裂准则确定,避免了人为假设的随意性,并可以实现裂缝变步长扩展的模拟,更符合实际情况。通过对四点剪切梁的复合型裂缝扩展过程的模拟,对本文模型进行了验证,并应用于重力坝模型的裂缝扩展模拟,计算结果表明,本文提出的模型简单易行且精度较高。     

Nonlinear quasi-conforming finite element method

The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasiconforming finite element. First, the incremental principle of stationary potential energy is discussed. Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.     

A mixed finite element/finite volume approach for solving biodegradation transport in groundwater

A numerical model for the simulation of flow and transport of organic compounds undergoing bacterial oxygen- and nitrate-based respiration is presented. General assumptions regarding microbial population, bacteria metabolism and effects of oxygen, nitrogen and nutrient concentration on organic substrate rate of consumption are briefly described. The numerical solution techniques for solving both the flow and the transport are presented. The saturated flow equation is discretized using a high-order mixed finite element scheme, which provides a highly accurate estimation of the velocity field. The transport equation for a sorbing porous medium is approximated using a finite volume scheme enclosing an upwind TVD shock-capturing technique for capturing concentration-unsteady steep fronts. The performance and capabilities of the present approach in a bio-remediation context are assessed by considering a set of test problems. The reliability of the numerical results concerning solution accuracy and the computational efficiency in terms of cost and memory requirements are also estimated. © 1998 John Wiley & Sons, Ltd.     

Buckling of structures with finite prebuckling deformations—a perturbation,finite element analysis

In some technically important structures, finite prebuckling displacements have a profound effect on the bifurcation load. To ignore these displacements, as is done in most instability analyses, is to invite major errors, usually on the unsafe side. A method is presented which approximates this effect without the necessity of solving nonlinear equations. The general theory is developed for any elastic body under conservative loads. The governing equations are subsequently discretized by a finite element approach and it is shown that for planar framed structures, the second order approximation to the buckling load can be found in terms of the standard linear and geometric stiffness matrices of structural analysis; the solution procedure does not require iterations. For illustrative purposes, a computer program was developed for planar structures and the results are compared to the exact solution for the buckling of shallow circular arches.     

平面非协调动态有限元法

本文通过引入高阶非协调位移模式，从Ｈｅｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ泛函出发，给出了非协调动态有限元的一般列式，得到了相应的收敛准则。算例表明本文方法简单、有效