Smoothed finite element method for topology optimization involving incompressible materials

It is well known that the finite element method (FEM) suffers severely from the volumetric locking problem for incompressible materials in topology optimization owing to its numerical ‘overly stiff’ property. In this article, two typical smoothed FEMs with a certain softened effect, namely the node-based smoothed finite element method (NS-FEM) and the cell-based smoothed finite element method, are formulated to model the compressible and incompressible materials for topology optimization. Numerical examples have demonstrated that the NS-FEM with an ‘overly soft’ property is fairly effective in tackling the volumetric locking problem in topology optimization when both compressible and incompressible materials are involved.     

F‐bar aided edge‐based smoothed finite element method using tetrahedral elements for finite deformation analysisof nearly incompressible solids

A new smoothed finite element method (S‐FEM) with tetrahedral elements for finite strain analysis of nearly incompressible solids is proposed. The proposed method is basically a combination of the F‐bar method and edge‐based S‐FEM with tetrahedral elements (ES‐FEM‐T4) and is named ‘F‐barES‐FEM‐T4’. F‐barES‐FEM‐T4 inherits the accuracy and shear locking‐free property of ES‐FEM‐T4. At the same time, it also inherits the volumetric locking‐free property of the F‐bar method. The isovolumetric part of the deformation gradient ( F ^iso) is derived from the F of ES‐FEM‐T4, whereas the volumetric part ( F ^vol) is derived from the cyclic smoothing of J(=det( F )) between elements and nodes. Some demonstration analyses confirm that F‐barES‐FEM‐T4 with a sufficient number of cyclic smoothings suppresses the pressure oscillation in nearly incompressible materials successfully with no increase in DOF. Moreover, they reveal that our method is capable of relaxing the corner locking issue arising at the corner in the cylinder barreling analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

A mixed finite volume formulation for determining the small strain deformation of incompressible materials

A finite volume formulation for determining small strain deformations in incompressible materials is presented in detail. The formulation includes displacement and hydrostatic pressure variables. The displacement field varies linearly along and across each cell face. The hydrostatic pressure field associated with each face is uniform. The cells that discretize the structure are geometrically unrestricted, each cell can have an arbitrary number of faces. The formulation is tested on a number of linear elastic plane strain benchmark problems. This testing reveals that when meshes of multifaceted cells are employed to represent the structure then locking behaviour is exhibited, but when triangular cells are used then accurate predictions of the displacement and stress fields are produced. Copyright © 1999 John Wiley & Sons, Ltd.     

On using linear elements in incompressible plane strain problems: a simple edge based approach for triangles

In this paper a simple iterative method is presented for finite element solution of incompressible plane strain problems using linear elements. Instead of using a mixed formulation approach, we use an equivalent displacement/velocity approach in an iterative manner. Control volumes are taken for regions which are to exhibit incompressible behaviour. For triangular elements the control volume is chosen as the area built on the parts of each pair of elements at the sides of an edge. In this case, elements are let to exchange volume. It is shown that the proposed edge based approach removes the deficiency of the linear triangular elements i.e. locking effect. Similar edge based approach is applied to the linear quadrilateral elements. However, if the control volume is chosen as the element volume the formulation gives similar results as the discontinuous mixed formulation using one pressure point without exhibiting instability behaviour. The formulation is based on decomposition of the displacement/velocity field into deviatoric and volumetric parts. The volumetric part is iteratively eliminated without confronting locking or instability phenomenon. The iterative procedure is very cheap and simple to be implemented in any FEM code. Several examples are given to demonstrate the performance of the procedure. Copyright © 2004 John Wiley & Sons, Ltd.     

Fully tensorial nodal and modal shape functions for triangles and tetrahedra

This paper presents nodal and modal shape functions for triangle and tetrahedron finite elements. The functions are constructed based on the fully tensorial expansions of one‐dimensional polynomials expressed in barycentric co‐ordinates. The nodal functions obtained from the application of the tensorial procedure are the standard h‐Lagrange shape functions presented in the literature. The modal shape functions use Jacobi polynomials and have a natural global C⁰ inter‐element continuity. An efficient Gauss–Jacobi numerical integration procedure is also presented to decrease the number of points for the consistent integration of the element matrices. An example illustrates the approximation properties of the modal functions. Copyright © 2005 John Wiley & Sons, Ltd.     

The enhanced assumed strain method for the isogeometric analysis of nearly incompressible deformation of solids

Isogeometric analysis has recently become very popular for the numerical modeling of structures and fluids. Among other potential features, advantages of using a non‐uniform rational B‐splines (NURBS)‐based isogeometric analysis over the traditional finite element method include the possibility of using higher‐order polynomials for the basis functions of the approximation space, which may be easily built on a recursive (hierarchical) fashion as well as higher convergence ratio. Nevertheless, NURBS‐based isogeometric analysis suffers from the same problems depicted by other methods when it comes to reproduce isochoric deformations, that is, it shows volumetric locking, especially for low‐order basis functions. Similar remedies as those that have been proposed for the finite element method may be appropriate for integration in the NURBS‐based isogeometric analysis and some have already been tried with success. In this work, the analysis of the underlying space of incompressible deformations of a NURBS‐based isogeometric approximation is performed with the main objective of understanding the likelihood of volumetric locking. As a remedy, the enhanced assumed strain methodology is blended with the NURBS‐based isogeometric analysis to alleviate the volumetric locking associated with incompressible deformations. The solution includes a stabilization term derived directly from a penalized form of the classical Veubeke–Hu–Washizu three‐field variational principle. Copyright © 2012 John Wiley & Sons, Ltd.     

A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

In this paper, a two‐dimensional displacement‐based meshfree‐enriched FEM (ME‐FEM) is presented for the linear analysis of compressible and near‐incompressible planar elasticity. The ME‐FEM element is established by injecting a first‐order convex meshfree approximation into a low‐order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker‐delta property on the element boundaries. The gradient matrix of ME‐FEM element satisfies the integration constraint for nodal integration and the resultant ME‐FEM formulation is shown to pass the constant stress test for the compressible media. The ME‐FEM interpolation is an element‐wise meshfree interpolation and is proven to be discrete divergence‐free in the incompressible limit. To prevent possible pressure oscillation in the near‐incompressible problems, an area‐weighted strain smoothing scheme incorporated with the divergence‐free ME‐FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu–Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near‐incompressible problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Triangles and tetrahedra in explicit dynamic codes for solids

Explicit dynamic codes which are used currently for the study of plastic deformations in impact, or with some modification for metal forming, suffer two serious limitations. First, only quadrilateral or hexahedral linear elements can be used thus limiting the possibilities of adaptive refinement and adaptive meshing. Second, even with the use of such elements, special devices such as reduced integration must be introduced to avoid locking and reduce costs. These necessitate complex hour glass control, mending-type procedures. The main difficulties are those due to the need of treatment of (almost) incompressible deformation modes. Recently, similar difficulties have been overcome in the context of fluid dynamics soil dynamics and we show here how the processes introduced there can be adopted effectively to the present problem, thus allowing an almost unrestricted choice of element interpolations. © 1998 John Wiley & Sons, Ltd.     

Finite cover method for linear and non‐linear analyses of heterogeneous solids

We introduce the finite cover method (FCM) as a generalization of the finite element method (FEM) and extend it to analyse the linear and non‐linear mechanical behaviour of heterogeneous solids and structures. The name ‘FCM’ is actually an alias for the manifold method (MM) and the basic idea of the method has already been established for linear analyses of structures with homogeneous materials. After reviewing the concept of physical and mathematical covers for approximating functions in the FCM, we present the formulation for the static equilibrium state of a structure with arbitrary physical boundaries including material interfaces. The problem essentially involves the discontinuities in strains, and possibly has the discontinuities in displacement caused by interfacial debonding or rupture of material interfaces. We simulate such non‐linear mechanical behaviour after presenting simple numerical examples that demonstrate the equivalence between the approximation capabilities of the FCM and those of the FEM. Copyright © 2003 John Wiley & Sons, Ltd.     

Assumed‐deformation gradient finite elements with nodal integration for nearly incompressible large deformation analysis

An assumed‐strain finite element technique for non‐linear finite deformation is presented. The weighted‐residual method enforces weakly the balance equation with the natural boundary condition and also the kinematic equation that links the elementwise and the assumed‐deformation gradient. Assumed gradient operators are derived via nodal integration from the kinematic‐weighted residual. A variety of finite element shapes fits the derived framework: four‐node tetrahedra, eight‐, 27‐, and 64‐node hexahedra are presented here. Since the assumed‐deformation gradients are expressed entirely in terms of the nodal displacements, the degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials and no volumetric/deviatoric split is required. The consistent tangent operator is inexpensive and symmetric. Furthermore, the material update and the tangent moduli computation are carried out exactly as for classical displacement‐based models; the only deviation is the consistent use of the assumed‐deformation gradient in place of the displacement‐derived deformation gradient. Examples illustrate the performance with respect to the ability of the present technique to resist volumetric locking. A constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler–Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.     

Finite element simulation of dynamic crack propagation process using an arbitrary Lagrangian Eulerian formulation

In this paper, the arbitrary Lagrangian Eulerian formulation is employed for finite element modelling of dynamic crack propagation problem. The application phase simulation of computational dynamic fracture is applied to model by which the crack propagation history and variation of crack velocity are predicted using the material dynamic fracture toughness. The dynamic solution of problem is accomplished using the implicit time integration method. The convective terms due to mesh‐material motion are taken into account via the convection equation. A robust and efficient mesh motion technique, that its equations need not to be solved at every time step, is employed in Eulerian phase. The mesh connectivity is preserved during the analysis. So, the successive remeshing of model is eliminated. When the dynamic fracture criterion is satisfied for crack growth, the presented algorithm allows the crack to advance by splitting the material particle at the crack tip. The dynamic energy release rate is calculated at each time step to determine dynamic stress intensity factor. The predicted results are compared with those obtained through the experimental study and remeshing technique cited in the literature. The proposed computational algorithm leads to an accurate and efficient simulation of dynamic crack propagation process.     

Velocity‐based formulations for standard and quasi‐incompressible hypoelastic‐plastic solids

We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are derived and tested for non‐linear solid mechanics problems. The so‐called V‐element is based on a standard velocity approach, while a mixed velocity–pressure formulation is used for the VP and the VPS elements. The two‐field problem is solved via a two‐step Gauss–Seidel partitioned iterative scheme. First, the momentum equations are solved in terms of velocity increments, as for the V‐element. Then, the constitutive relation for the pressure is solved using the updated velocities obtained at the previous step. For the VPS‐element, the formulation is stabilized using the finite calculus method in order to solve problems involving quasi‐incompressible materials. All the solid elements are validated by solving two‐dimensional and three‐dimensional benchmark problems in statics as in dynamics. Copyright © 2016 John Wiley & Sons, Ltd.     

有限元方法在双头电火花加工机床设计中的应用

鉴于双头电火花加工机床部件结构复杂,很难建立力学模型采分析,随着计算机技术的不断发展,有限元方法已经广泛应用于工程计算,而Y轴横梁是三轴双头电火花加工机床的关键部件,因此应用有限元方法对其进行静力学分析,得到Y轴横梁在不同位置时导轨滑块处所受的载重以及主轴头端面的变形量,为直线导轨的选择和安装提供了理论依据.     

三维动态八结点有限单元的建立及理论推导

本文为扩大动态有限单元法的应用范围,在 J.S.Przemieniecki 和 K.K.Gupta 等人的研究基础上建立了一种新的三维动态八结点单元,并对其进行了理论推导。通过算例验证了本文理论结果的正确性。     

有限元与包络解调相结合的碰摩故障诊断

利用有限元与包络解调方法研究碰摩故障下静子振动信号的变化规律。分析静子振动机理,碰摩时静子受到冲击力作用,会引发静子高频固有振动,提出可采用包络解调法进行碰摩故障的诊断。针对静子结构特点,利用有限元方法进行静子碰摩振动动力学分析,然后进行静子振动信号的包络分析,从而完成分析诊断过程。最后分别进行单点碰摩与局部碰摩故障模拟实验,对实测静子振动信号进行包络解调分析,仿真与实验结果表明静子的高频固有振动信号可以揭示碰摩故障的发生。     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

Patch‐averaged assumed strain finite elements for stress analysis

A finite element model for linear‐elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain‐displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four‐node and linear hexahedral eight‐node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf‐sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd.