A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems

This paper presents a node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems using a mesh of polygonal elements. The calculation of the system stiffness matrix is performed using strain smoothing technique over the smoothing cells associated with nodes, which leads to line integrations along the edges of the smoothing cells. The numerical results demonstrated that the NS-FEM possesses the following properties: (1) upper bound in the strain energy of the exact solution when a reasonably fine mesh is used; (2) well immune from the volumetric locking; (3) can use polygonal elements with an arbitrary number of sides; (4) insensitive to element distortion.     

A smoothed finite element method for shell analysis

A four-node quadrilateral shell element with smoothed membrane-bending based on Mindlin-Reissner theory is proposed. The element is a combination of a plate bending and membrane element. It is based on mixed interpolation where the bending and membrane stiffness matrices are calculated on the boundaries of the smoothing cells while the shear terms are approximated by independent interpolation functions in natural coordinates. The proposed element is robust, computationally inexpensive and free of locking. Since the integration is done on the element boundaries for the bending and membrane terms, the element is more accurate than the MITC4 element for distorted meshes. This will be demonstrated for several numerical examples.     

Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes

The present paper deals with an anisotropic mesh adaptation (AMA) of triangulation which can be employed for the numerical solution various problems of physics. AMA tries to construct an optimal triangulation of the domain of computation in the sense that an “error” of the solution of the problem considered is uniformly distributed over the whole triangulation. First, we describe the main idea of AMA. We define an optimal triangle and an optimal triangulation. Then we describe the process of optimization of the triangulation and the complete multilevel computational process. We apply AMA to a problem of CFD, namely to inviscid compressible flow. The computational results for a channel flow are presented. Received: 11 December 1997 / Accepted: 16 February 1998     

The verification of the quantities of interest based on energy norm of solutions by node-based smoothed finite element method

Verification of the quantities of interest computed with the finite element method (FEM) requires an upper bound on the strain energy, which is half of the energy norm of displacement solutions. Recently, a modified finite element method with strain smoothing, the node-based smoothed finite element method (NS-FEM), has been proposed to solve solid mechanics problems. It has been found in some cases that the energy norm formed by the smoothed strain of NS-FEM solutions bounds the energy norm of exact displacements from above. We analyze the bounding property of this method, give three kind of energy norms of solutions computed by FEM and NS-FEM, and extend them to the computation of an upper bound and a lower bound on the linear functional of displacements. By examining the bounding property of NS-FEM with different energy norms using some linear elastic problems, the advantages of NS-FEM over the traditional error estimate based methods is observed.     

A non-interactive method for the automatic generation of finite element meshes using the schwarz-christoffel transformation

This paper describes a particular conformal mapping, the Schwarz-Christoffel transformation. A practical method for evaluating the parameters defining this transformation is described and two such transformations are used to construct a mapping between any two polygons. The desirable properties of a finite element mesh are stated and a method is described which attempts to generate such a mesh in any simply-connected region, using conformal mappings. This computer method is non-interactive and attempts to produce a good mesh with the minimum of input data. Some examples are included.     

A triangular finite element with local remeshing for the large strain analysis of axisymmetric solids

In this work, a three-node triangular finite element with two degrees of freedom per node for the large strain elasto-plastic analysis of axisymmetric solids is presented. The formulation resorts to the adjacent elements to obtain a quadratic interpolation of the geometry over a patch of four elements from which an average deformation gradient is defined. Thus, the element formulation falls within the framework of assumed strain elements or more precisely of F-bar type formulations. The in-plane behavior of the element is similar to the linear strain triangle, but without the drawbacks of the quadratic triangle, e.g. contact or distortion sensitivity. The element does not suffer of volumetric locking in problems with isochoric plastic flow and the implementation is simple. It has been implemented in a finite element code with explicit time integration of the momentum equations and tools that allow the simulation of industrial processes. The widely accepted multiplicative decomposition of the deformation gradient in elastic and plastic components is adopted here. An isotropic material with non-linear isotropic hardening has been considered. Two versions of the element have been implemented based on a Total and an Updated Lagrangian Formulation, respectively. Some approximations have been considered in the latter formulation aimed to reduce the number of operations in order to increase numerical efficiency. To consider bulk forming, with large geometric changes, an automatic local remeshing strategy has been developed. Several examples are considered to assess the element performance with and without remeshing.     

A finite element method for plane strain deformations of incompressible solids

A finite element method for incompressible deformation is formulated from the virtual work equation based on deviatoric quantities. Particular emphasis is given to nonlinear material behavior. The incompressibility constraint is imposed on the admissible displacement field by direct elimination of nodal displacements. The resulting stiffness matrix is symmetric and positive definite. Once a convergent solution for the displacement is obtained, the hydrostatic stress is determined from the principle of virtual work. A variety of illustrative examples are presented. The efficiency, economy and limitations of the method are discussed.     

A finite element method for the solution of rotary pumps

We present in this paper a numerical strategy for the simulation of rotary positive displacement pumps, taking as an example a gear pump. While the two gears of the pump are rotating, the intersection between them changes in time. Therefore, the computational domain should be recomputed in some way at each time step. The strategy used here consists in dividing a cycle into a certain number of time steps and obtaining different computational meshes for each of these time steps. The coupling between two consecutive time steps is achieved by interpolating the flow unknowns in a proper way. This geometrical decomposition enables one to have a plain control over the mesh, particularly in the zones of interest, which are the gap between the gears and the casing, and the engagement and disengagement zones of the gears.     

A transient finite element solution method for the Navier-Stokes equations

This paper is concerned with the discrete finite element formulation and numerical solution of transient incompressible viscous flow in terms of the primitive variables. A restricted variational principle is introduced as equivalent to the momentum equations and the Poisson equation for pressure. The latter is introduced to replace the continuity equation, and thus the incompressibility condition is realized only asymptotically; i.e. through the iterative process. An incomplete cubic interpolation function is used for both the velocities and pressure within a triangular finite element. The discrete equations are integrated in time with backward finite differences. We illustrate the similarity between the (ψ,ζ) finite difference method and the ($\text{u}\text{,p}$) finite element method by calculations on the driven square cavity problem.     

Large deflection analyses of skew plates using hybrid/mixed finite element method

A new hybrid/mixed shell element is developed using oblique coordinate systems to analyze the large deflection behavior of skew plate with various skew angles, length to width ratios, thicknesses and supported edges under uniformly distributed and concentrated loads. The results obtained from the new element are compared with available theoretical and numerical solutions. An excellent agreement is achieved even for coarse meshes. The accuracy and efficiency of the proposed element are demonstrated.     

A finite element method for Stokes equation using discrete singularity expansion

The numerical method presented in this paper for Stokes equation with corner singularity includes mainly two steps. Firstly, we solve a simple eigenvalue problem, which is one dimension less than the original problem, to obtain the discrete expansion of the singularity near the corner. Secondly, we combine the approximation of the singularity and standard finite element basis functions to construct special finite element spaces, and solve the original problem in the special spaces on a conventional mesh. The numerical examples show the effectiveness of this method.     

A new strategy for stress analysis using the finite element method

In this paper the authors examine the effectiveness of the Powell-Toint strategy for evaluating the Hessian of the potential energy surface of a finite element model that can be used for linear stress analysis and transient response predictions of structures. Cases for which the Powell-Toint strategy may be cost-effective with the conventional method of stress analysis are identified.     

On the numerical solution of the stokes equations using the finite element method

A numerical solution of the stationary Stokes equations is considered based on the work of Crouzeix and Raviart [1]. The finite element method is used to discretize the partial differential equations, and a direct discretization of the velocity field and pressure is given which is applicable in both two and three dimensions. It is shown that not every arbitrary element can be used, and a condition is given to check whether or not an element is admissible. The system of linear equations is solved using the method of Powell and Hestenes for constrained optimization (see [2]).     

Superconvergence analysis of nonconforming finite element method for time-fractional nonlinear parabolic equations on anisotropic meshes

In this paper, we prove a novel result of the consistency error estimate with order $O\left(h^2\right)$ for $E{Q}_1^{rot}$ element (see Lemma 2) on anisotropic meshes. Then, a linearized fully discrete Galerkin finite element method (FEM) is studied for the time-fractional nonlinear parabolic problems, and the superclose and superconvergent estimates of order $O(\tau +h^2)$ in broken $H^1$-norm on anisotropic meshes are derived by using the proved character of $E{Q}_1^{rot}$ element, which improve the results in the existing literature. Numerical results are provided to confirm the theoretical analysis.     

Anisotropic metrics for finite element meshes using a posteriori error estimates: Poisson and Stokes equations

In this paper, metrics derived from a posteriori error estimates for the Poisson problem and for the Stokes system solved by some finite element methods are presented. Numerical examples of mesh adaptation in two dimensions of the space are given and show that these metrics detect the singular behavior of the solution, in particular its anisotropy.     

On numerical stabilization in the solution of Saint-Venant equations using the finite element method

Solving the Saint-Venant equations by using numerical schemes like finite difference and finite element methods leads to some unwanted oscillations in the water surface elevation. The reason for these oscillations lies in the method used for the approximation of the nonlinear terms. One of the ways of smoothing these oscillations is by adding artificial viscosity into the scheme. In this paper, by using a suitable discretization, we first solve the one-dimensional Saint-Venant equations by a finite element method and eliminate the unwanted oscillations without using an artificial viscosity. Second, our main discussion is concentrated on numerical stabilization of the solution in detail. In fact, we first convert the systems resulting from the discretization to systems relating to just water surface elevation. Then, by using M-matrix properties, the stability of the solution is shown. Finally, two numerical examples of critical and subcritical flows are given to support our results.     

A numerical solution of the Navier-Stokes equations using the finite element technique

The finite element discretisation technique is used to effect a solution of the Navier- Stokes equations. Two methods of formulation are presented, and a comparison of the effeciency of the methods, associated with the solution of particular problems, is made. The first uses velocity and pressure as field variables and the second stream function and vorticity. It appears that, for contained flow problems the first formulation has some advantages over previous approaches using the finite elemental method[1,2].     

A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua

A general method called FE² has been introduced which consists in describing the behavior of heterogeneous structures using a multiscale finite element model. Instead of trying to build differential systems to establish a stress-strain relation at the macroscale, a finite element computation of the representative volume element is carried out simultaneously. Doing so does not require any constitutive equations to be written at the macroscopic scale: all non-linearities come directly from the microscale.In this paper, we describe how this method can be used in the context of generalized continua. For such continua, constitutive equations are very difficult to write, and a new set of material is difficult to fit to experimental data. The use of FE² models bypasses this problem because no analytical equation is needed at the macroscale.An academic application is presented to show that generalized continua are necessary when the size of the heterogeneities increases, and that FE² models behave well compared to a reference solution.     

A general topology-based framework for adaptive insertion of cohesive elements in finite element meshes

Large-scale simulation of separation phenomena in solids such as fracture, branching, and fragmentation requires a scalable data structure representation of the evolving model. Modeling of such phenomena can be successfully accomplished by means of cohesive models of fracture, which are versatile and effective tools for computational analysis. A common approach to insert cohesive elements in finite element meshes consists of adding discrete special interfaces (cohesive elements) between bulk elements. The insertion of cohesive elements along bulk element interfaces for fragmentation simulation imposes changes in the topology of the mesh. This paper presents a unified topology-based framework for supporting adaptive fragmentation simulations, being able to handle two- and three-dimensional models, with finite elements of any order. We represent the finite element model using a compact and “complete” topological data structure, which is capable of retrieving all adjacency relationships needed for the simulation. Moreover, we introduce a new topology-based algorithm that systematically classifies fractured facets (i.e., facets along which fracture has occurred). The algorithm follows a set of procedures that consistently perform all the topological changes needed to update the model. The proposed topology-based framework is general and ensures that the model representation remains always valid during fragmentation, even when very complex crack patterns are involved. The framework correctness and efficiency are illustrated by arbitrary insertion of cohesive elements in various finite element meshes of self-similar geometries, including both two- and three-dimensional models. These computational tests clearly show linear scaling in time, which is a key feature of the present data-structure representation. The effectiveness of the proposed approach is also demonstrated by dynamic fracture analysis through finite element simulations of actual engineering problems.

用有限元法计算特征值问题的一种新的动态凝聚方法

有限元法是常用的建模方法,由于所建模型具有较大的自由度,通常需要进行降阶处理.一般来讲,模型前几阶特征值和特征向量可以较精确地得到,利用所得到的特征值和主振型分量(在特征向量中与所给定的主自由度对应的振型分量),本文提出了一种新的动态凝聚方法,该方法是通过迭代方式,利用所得到的特征值和主振型分量对Guyan降阶法所得到的降阶模型进行修正.与同类方法相比,本文方法具有较高的计算精度和很小的计算量,且迭代收敛的稳定性很好.最后本文给出了一个计算实例.