Crack insertion,meshing and fracture analysis of structures using tetrahedral finite elements

In this study, a method and corresponding tools are presented to insert a three-dimensional crack of a given size and location into a finite element model without any cracks using fully unstructured finite elements. For research purposes, publicly available two and three-dimensional meshing software, Triangle^© and Tetgen^©, are utilized and integrated with an in-house developed program to compatibly select and re-mesh the three-dimensional crack region of the original input model. Within the procedure, the boundary conditions and loads existing on the original model are also book kept and transferred to the new model containing the crack. Next, the new finite element model, which now contains the crack geometry, the loads and boundary conditions, is solved in a general-purpose finite element program employing enriched elements. The above procedure is demonstrated on a series of surface crack problems in finite-thickness plates including mixed-mode fracture conditions. The obtained results are compared to well-known solutions available in the literature. These comparisons showed good agreement for all cases analyzed. It is, therefore, concluded that the procedure developed is valid, efficient and yields accurate three-dimensional fracture solutions.     

Discrete maximum principle based on repair technique for diamond type scheme of diffusion problems

In this paper, two repair techniques are proposed for diamond schemes of anisotropic diffusion problems to ensure that the repaired solutions satisfy the discrete maximum principle. One of them is an extension of that in [Liska R, Shashkov M. Enforcing the discrete maximum principle for linear finite element solutions of second‐order elliptic problems. Communications in Computational Physics 2008; 3(4):852–877.] for linear finite element solutions, which is a local repair technique, and another is a new global repair technique. Both of them keep total energy conservation and are easy to be implemented in existing codes. Numerical examples show that these two repair techniques do not destroy the accuracy of solution for the diamond schemes on distorted meshes. Copyright © 2012 John Wiley & Sons, Ltd.     

Criteria of finite element algorithm for a class of parabolic equation

In finite element analysis of transient temperature field, it is quite notorious that the numerical solution may quite likely oscillate and/or exceed the reasonable scope, which violates the natural law of heat conduction. For this reason, we put forward the concept of time monotony and spatial monotony, and then derive several sufficient conditions for monotonic solutions in time dimension for 3 - D passive heat conduction equations with a group of finite difference schemes. For some special boundary conditions and regular element meshes, the lower and upper bounds for t/x ² can be obtained from those conditions so that reasonable numerical solutions are guaranteed. Spatial monotony is also discussed. Finally, the lumped mass method is analyzed. We creatively give several new criteria for the finite element solutions of a class of parabolic equation represented by heat conduction equation.     

An enthalpy‐preserving shock‐capturing term for residual distribution schemes

In this contribution, we investigate strategies to perform shock‐capturing computation of steady hypersonic flow fields by means of residual distribution schemes. The ultimate objective is the computation of flow solutions for which the correct upstream enthalpy value is recovered in the postshock region. To this end, the parallelism existing between the classical Bx scheme and the stabilized finite element techniques is exploited. The simple Lax‐Friedrichs dissipation term is leveraged to build two new residual distribution schemes. Upon testing on both inviscid and viscous steady problems, solutions obtained with one of the two schemes are shown to recover the correct upstream total enthalpy level in the postshock region. This last scheme provides also improved wall pressure and skin friction predictions; heat transfer predictions are, unfortunately, similar to those offered by the Bx scheme. A conjecture for explaining this behavior is exposed.     

悬索桥结构分析中的索-鞍座单元

为在悬索桥结构的有限元分析中真实、简洁、高效地模拟索鞍,本文建立了一类新的单元。新单元包括索段的一端固定在与其接触部分为单一半径圆弧的索鞍上,另一端分别位于索鞍两侧的两节点“左索-鞍座单元”和“右索-鞍座单元”,以及索段两端点分别位于索鞍两侧,中间一点固定于鞍座上的三节点“索-鞍座单元”,后者的鞍槽可为两不同半径圆弧的组合。根据要求的成桥状态几何参数确定结构的无应力状态时,可利用前二者进行悬索桥的单跨分析。新单元通过自动调整索与鞍座的脱离点而处于平衡状态,从而简化了计算。单元算法的推导基于有限元分析的基本原理和弹性悬链线的精确解,并利用了处于平衡状态时索与鞍座之间的内力关系。新单元可象常规单元一样直接用于成桥状态或施工过程中悬索桥结构的有限元分析。设计的算例验证了新单元的正确性,并举例说明了新单元在悬索桥结构分析中的应用。     

A hybrid-stress element based on Hamilton principle

A novel hybrid-stress finite element method is proposed for constructing simple 4-node quadrilateral plane elements, and the new element is denoted as HH4-3fl here. Firstly, the theoretical basis of the traditional hybrid-stress elements, i.e., the Hellinger-Reissner variational principle, is replaced by the Hamilton variational principle, in which the number of the stress variables is reduced from 3 to 2. Secondly, three stress parameters and corresponding trial functions are introduced into the system equations. Thirdly, the displacement fields of the conventional bilinear isoparametric element are employed in the new models. Finally, from the stationary condition, the stress parameters can be expressed in terms of the displacement parameters, and thus the new element stiffness matrices can be obtained. Since the required number of stress variables in the Hamilton variational principle is less than that in the Hellinger-Reissner variational principle, and no additional incompatible displacement modes are considered, the new hybrid-stress element is simpler than the traditional ones. Furthermore, in order to improve the accuracy of the stress solutions, two enhanced post-processing schemes are also proposed for element HH4-3β. Numerical examples show that the proposed model exhibits great improvements in both displacement and stress solutions, implying that the proposed technique is an effective way for developing simple finite element models with high performance.     

Crack separation energy-rates for the DBCS model under biaxial modes of loading

A Modified version of the Dugdale-Bilby-Cottrell-Swinden (DBCS) model simulating the effect of plasticity at the tip of a crack in an infinite region was used by kfouri and rice (1978) to calculate the crack separation energy-rate G^Δ corresponding to a finite crack growth step Δa during plane strain mode I crack extension. The loading consisted of a remote uniaxial tension σp applied normally to the plane of the crack. Using Rice's path-independent integral J to characterize the applied load in the crack tip region, and assuming the length R of the crack tip plastic zone to be small compared with the length of the crack, an analytical expression was derived relating the ratios (J/G^Δ) and (2a/R) for small values of (2a/R). The analytical solution was incomplete in itself in that the value assumed in the plastic region of the DBCS model for the normal stress Y acting on the extending crack surfaces was the product of the yield stress in uniaxial tension σ_Y and an unknown parameter C, the value of which depends on the effect of the local hydrostatic stresses in the case of plane strain conditions. The analytical solution was compared with a numerical solution obtained from a plane strain elastic-plastic finite element analysis on a centre-cracked plate (CCP) of material obeying the von Mises yield criterion. The value used for the yield stress was 310 MN/m² and moderate isotropic linear hardening was applied with a tangent modulus of 4830 MN/m². A uniaxial tension σp was applied on the two appropriate sides of the plate. The comparisons showed that the analytical and finite element solutions were mutually consistent and they enabled the value of C to be established at 1.91. In the present paper similar comparisons are made between the analytical solution and the finite element solutions for the CCP of the same material under different biaxial modes of loading. By assuming further that the form of the analytical solution does not depend on the details of the geometry and of the loading at remote boundaries, a comparison has also been made with the results of a finite element analysis on a compact tension specimen (CTS) made of the same material as the CCP. The different values of C obtained in each case are consistent with investigations by other authors on the effect of load biaxiality on crack tip plasticity.     

Numerical solutions to regularized long wave equation based on mixed covolume method

The mixed covolume method for the regularized long wave equation is developed and studied. By introducing a transfer operator γh , which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.     

Computing elastic moduli on 3-D X-ray computed tomography image stacks

A numerical task of current interest is to compute the effective elastic properties of a random composite material by operating on a 3D digital image of its microstructure obtained via X-ray computed tomography (CT). The 3-D image is usually sub-sampled since an X-ray CT image is typically of order 1000³ voxels or larger, which is considered to be a very large finite element problem. Two main questions for the validity of any such study are then: can the sub-sample size be made sufficiently large to capture enough of the important details of the random microstructure so that the computed moduli can be thought of as accurate, and what boundary conditions should be chosen for these sub-samples? This paper contributes to the answer of both questions by studying a simulated X-ray CT cylindrical microstructure with three phases, cut from a random model system with known elastic properties. A new hybrid numerical method is introduced, which makes use of finite element solutions coupled with exact solutions for elastic moduli of square arrays of parallel cylindrical fibers. The new method allows, in principle, all of the microstructural data to be used when the X-ray CT image is in the form of a cylinder, which is often the case. Appendix A describes a similar algorithm for spherical sub-samples, which may be of use when examining the mechanical properties of particles. Cubic sub-samples are also taken from this simulated X-ray CT structure to investigate the effect of two different kinds of boundary conditions: forced periodic and fixed displacements. It is found that using forced periodic displacements on the non-geometrically periodic cubic sub-samples always gave more accurate results than using fixed displacements, although with about the same precision. The larger the cubic sub-sample, the more accurate and precise was the elastic computation, and using the complete cylindrical sample with the new method gave still more accurate and precise results. Fortran 90 programs for the analytical solutions are made available on-line, along with the parallel finite element codes used.     

A Dorodnitsyn finite element formulation for laminar boundary layer flow

The Dorodnitsyn boundary later formulation is given a finite element interpretation and found to generate very accurate and economical solutions when combined with an implicit, non-iterative marching scheme in the downstream direction. The algorithm is of order (Δ²u, Δx) whether linear or quadratic elements are used across the boundary layer. Solutions are compared with a Dorodnitsyn spectral formulation and a conventional finite difference formulation for three Falkner-Skan pressure gradient cases and the flow over a circular cylinder. With quadratic elements the Dorodnitsyn finite element formulation is approximately five times more efficient than the conventional finite difference formulation.     

A locally conservative,discontinuous least‐squares finite element method for the Stokes equations

Conventional least‐squares finite element methods (LSFEMs) for incompressible flows conserve mass only approximately. For some problems, mass loss levels are large and result in unphysical solutions. In this paper we formulate a new, locally conservative LSFEM for the Stokes equations wherein a discrete velocity field is computed that is point‐wise divergence free on each element. The central idea is to allow discontinuous velocity approximations and then to define the velocity field on each element using a local stream‐function. The effect of the new LSFEM approach on improved local and global mass conservation is compared with a conventional LSFEM for the Stokes equations employing standard C⁰ Lagrangian elements. Copyright © 2011 John Wiley & Sons, Ltd.     

HIGH-ORDER-ACCURATE SCHEMES FOR INCOMPRESSIBLE VISCOUS FLOW

We present new finite difference schemes for the incompressible Navier–Stokes equations. The schemes are based on two spatial differencing methods; one is fourth-order-accurate and the other is sixth-order accurate. The temporal differencing is based on backward differencing formulae. The schemes use non-staggered grids and satisfy regularity estimates, guaranteeing smoothness of the solutions. The schemes are computationally efficient. Computational results demonstrating the accuracy are presented. © 1997 by John Wiley & Sons, Ltd.     

A new finite element method for strain gradient theories and applications to fracture analyses

A new compatible finite element method for strain gradient theories is presented. In the new finite element method, pure displacement derivatives are taken as the fundamental variables. The new numerical method is successfully used to analyze the simple strain gradient problems – the fundamental fracture problems. Through comparing the numerical solutions with the existed exact solutions, the effectiveness of the new finite element method is tested and confirmed. Additionally, an application of the Zienkiewicz–Taylor C1 finite element method to the strain gradient problem is discussed. By using the new finite element method, plane-strain mode I and mode II crack tip fields are calculated based on a constitutive law which is a simple generalization of the conventional J₂ deformation plasticity theory to include strain gradient effects. Three new constitutive parameters enter to characterize the scale over which strain gradient effects become important. During the analysis the general compressible version of Fleck–Hutchinson strain gradient plasticity is adopted. Crack tip solutions, the traction distributions along the plane ahead of the crack tip are calculated. The solutions display the considerable elevation of traction within the zone near the crack tip.     

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high‐order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space‐time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element‐local space‐time Galerkin finite element predictor on moving curved meshes to obtain a high‐order accurate one‐step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tⁿ is connected to the new one at t^n + 1 by straight edges, hence providing unstructured space‐time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature‐free integration, in which the space‐time boundaries of the control volumes are split into simplex sub‐elements that yield constant space‐time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub‐elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high‐order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.     

A control volume finite element method for three-dimensional three-phase flows

A novel control volume finite element method with adaptive anisotropic unstructured meshes is presented for three-dimensional three-phase flows with interfacial tension. The numerical framework consists of a mixed control volume and finite element formulation with a new P₁DG-P₂ elements (linear discontinuous velocity between elements and quadratic continuous pressure between elements). A “volume of fluid” type method is used for the interface capturing, which is based on compressive control volume advection and second-order finite element methods. A force-balanced continuum surface force model is employed for the interfacial tension on unstructured meshes. The interfacial tension coefficient decomposition method is also used to deal with interfacial tension pairings between different phases. Numerical examples of benchmark tests and the dynamics of three-dimensional three-phase rising bubble, and droplet impact are presented. The results are compared with the analytical solutions and previously published experimental data, demonstrating the capability of the present method.     

弱不连续问题高阶有限元离散系统的GAMG法

弱不连续问题(如含夹杂问题)是固体力学计算中的一类重要问题。高阶有限元方法由于其具有更好的逼近效果,是确保数值解在界面保持较高精度的计算方法之一。但与线性元相比,高阶单元需要更多的计算机存储单元,具有更高的计算复杂性。本文利用两水平算法的思想,将高阶有限元离散系统化归于线性元离散系统的求解,为弱不连续问题高阶有限元离散系统设计了一种新的基于几何与分析信息的代数多重网格(GAMG)法,并应用于圆形求解域含单夹杂问题的高阶有限元离散系统的求解。数值试验结果表明,相比于常用GAMG法,新方法的迭代次数基本不依赖于问题规模、单元阶次以及杨氏模量的间断性,CPU计算时间得到明显改善,具有更好的计算效率和鲁棒性,可大大提高弱不连续问题有限元分析的整体效率。     

Trigonometric function used to formulate a multi-nodal finite tubular element

It is presented an alternative formulation to solve the problem of the deformation analysis for tubular element under pinching loads. The solution is based on a new displacement field defined from a total set of trigonometric functions. The solution is developed in a multi-nodal finite tubular ring element with a total of eight degrees of freedom per section considered. The purpose of this paper is to provide an easy alternative formulation when compared with a complex finite shell element or beam element analysis for the same application. Several case studies presented have been compared and discussed with numerical analyses results reported by other authors and the results obtained with a shell element from a Cosmos/M^® programme.     

The application of compatible stress iterative method in dynamic finite element analysis of high velocity impact

There is a common difficulty in elastic-plastic impact codes such as EPIC ^[2,3].NONSAP ^[4],etc. Most of these codes use the simple linear functions usually taken from static problem to represent the displacement components. In such finite element formulation, the stress components are constant in each element and they are discontinuous in any two neighboring elements. Therefore, the bases of using the virtual work principle in such elements are unreliable. In this paper, we introduce a new method, namely, the compatible stress iterative method, to eliminate the above-said difficulty. The calculated examples show that the calculation using the new method in dynamic finite element analysis of high velocity impact is valid and stable, and the element stiffness can be somewhat reduced.This is a part of the author's Ph. D dissertation under the supervision of Professor Chien Wei-zang.     

缆索吊装系统有限元分析中的滑轮单元

为精确模拟结构中的滑轮，建立了滑轮支承一段索的单元，它考虑了滑轮的半径. 该``滑轮单元'的中间节点在滑轮的中心，另两个节点为索段位于滑轮两侧的端点. 新单元通过自动调整索在滑轮两侧的长度，使单元达到平衡状态. 单元算法的推导基于有限元分析的基本原理，并利用了处于平衡状态时单元内力之间的关系. 介绍新单元的推导过程，算例验证了它的正确性，并显示了其在含有滑轮索结构有限元分析中的高效性和方便性.     

Nonlinear dynamics of three-dimensional belt drives using the finite-element method

In this paper, new nonlinear dynamic formulations for belt drives based on the three-dimensional absolute nodal coordinate formulation are developed. Two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. The first element is a thin-plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects. The other three-dimensional element used in this investigation is a cable element obtained from a more general three-dimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling systematic examination of the effect of bending on the nonlinear dynamics of belt drives. The finite-element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The use of the formulations developed in this investigation is demonstrated using two-roller belt-drive system. The results obtained using the two finite-element formulations are compared and the convergence of the two finite-element solutions is examined.