Low-order Elements Research Articles

A finite element approach to statical and dynamical analysis of geometrically nonlinear structures

A geometrically nonlinear formulation to analyse structures in the hypotheses of large displacements and rotations and small strains is presented. In this formulation, applied to low-order elements and based on the total Lagrangian kinematics, the use of the rotation matrices is bypassed. A selective based definition of the strain tensor, used in order to avoid shear-locking problems, is effected by the linear definition of deformations because it is element reference system independent. In addition, complex manipulations required to obtain conservative descriptions and well-posed transformation matrices are avoided. Numerical tests have been carried out to validate the developed technique both in the statical and in the dynamical context.

Dynamic fracture with meshfree enriched XFEM

The enrichment of the extended finite element method (XFEM) by meshfree approximations is studied. The XFEM allows for modeling arbitrary discontinuities, but with low order elements the accuracy often needs improvement. Here, the meshfree approximation is used as an enrichment in a cluster of nodes about the crack tip to improve accuracy. Several numerical examples show that this leads to more accuracy for stress intensity factors computations, and to the capability to capture the branching point of a propagating crack from the stresses.

Development of High-Order Infinite Element Method for Stress Analysis of Elastic Bodies with Singularities

Structural analysis problems are traditionally solved using the Finite Element Method (FEM). However, when the structure of interest contains discontinuities such as cracks or re-entrant corners, a large number of elements are required to accurately reproduce the stress characteristics in the region of the discontinuities. As a result, the FEM method requires a large storage space and has a slow convergence speed. It has been shown that the Infinite Element Method (IEM) overcomes these limitations and provides a feasible means of solving various types of elasticity and singularity problems. Previous studies have generally focused on the use of IEM formulations based on low-order elements (2×2). By contrast, this study develops a high-order (3×3) IEM formulation. The solutions obtained using the proposed IEM method for various 2D elasto-static problems are compared with the results obtained using the traditional low-order IEM method and the analytical solutions presented in the literature. It is shown that the results obtained using the proposed method are more accurate than those obtained using the low-order IEM method and are in excellent agreement with the analytical solutions.

Energetic variational approach in complex fluids: Maximum dissipation principle

We discuss the general energetic variational approaches for hydrodynamic systems of complex fluids. In these energetic variational approaches, the least action principle (LAP) with action functional gives the Hamiltonian parts (conservative force) of the hydrodynamic systems, and the maximum/minimum dissipation principle (MDP), i.e., Onsager's principle, gives the dissipative parts (dissipative force) of the systems. When we combine the two systems derived from the two different principles, we obtain a whole coupled nonlinear system of equations satisfying the dissipative energy law. We will discuss the important roles of MDP in designing numerical method for computations of hydrodynamic systems in complex fluids. We will reformulate the dissipation in energy equation in terms of a rate in time by using an appropriate evolution equations, then the MDP is employed in the reformulated dissipation to obtain the dissipative force for the hydrodynamic systems. The systems are consistent with the Hamiltonian parts which are derived from LAP. This procedure allows the usage of lower order element (a continuous $C^0$ finite element) in numerical method to solve the system rather than high order elements, and at the same time preserves the dissipative energy law. We also verify this method through some numerical experiments in simulating the free interface motion in the mixture of two different fluids.

Numerical analysis of two plane wave finite element schemes based on the partition of unity method for elastic wave scattering

This work deals with a numerical model based on the partition of unity finite element method (PUFEM), for the two-dimensional time harmonic elastic wave equations. This approach consists in enriching the polynomial finite element spaces locally by superimposed shear (S) and pressure (P) plane waves. Two plane wave Finite elements schemes are particularly investigated here and compared from the point of view of levels of accuracy and conditioning. The problem considered is a horizontal S plane wave scattered by a rigid circular body, in an infinite elastic medium. A finite square domain in the vicinity of the scatterer is considered with the analytical solution of the problem imposed on its boundary through a Robin type boundary condition. An error analysis with respect to the mesh size and the plane wave enrichment is carried out and some numerical aspects, related to the conditioning and its behaviour as a function of the frequency and the number of approximating plane waves, are outlined. The aim is to gain a better practical understanding of the conditioning behaviour of the PUFEM in order to improve its reliability. The results show, for the same number of degrees of freedom per wavelength, higher order elements lead to better accuracy than low order elements. However, this is achieved in detriment of the conditioning especially when the number of degrees of freedom per wavelength is too large.

Combined interior‐point method and semismooth Newton method for frictionless contact problems

AbstractIn the present paper, a solution scheme is proposed for frictionless contact problems of linear elastic bodies, which are discretized using the finite element method with lower order elements. An approach combining the interior‐point method and the semismooth Newton method is proposed. In this method, an initial active set for the semismooth Newton method is obtained from the approximate optimal solution by the interior‐point method. The simplest node‐to‐node contact model is considered in the present paper, that is, pairs of matching nodes exist on the contact surfaces. However, the discussions can be easily extended to a node‐to‐segment or segment‐to‐segment contact model. In order to evaluate the proposed method, a number of illustrative examples of the frictionless contact problem are shown. The proposed combined method is compared with the interior‐point method and the semismooth Newton method. Two numerical examples that are difficult to solve using the semismooth Newton method are solved effectively using the proposed combined method. It is shown that the proposed method converges within far fewer iterations than the semismooth Newton methods or the interior‐point method. Copyright © 2009 John Wiley & Sons, Ltd.

Structural dynamical analysis by a lengths-based description of the small strains in the finite displacements regime

In this work we present a geometrically nonlinear formulation to analyze the motion of structures in the case of large displacements and rotations and small strains. In this formulation, applied to low-order elements and based on the total Lagrangian kinematic description, the use of the rotation matrices is bypassed. In this way a selective based definition of the strain tensor, used in order to avoid shear-locking problems, is effected by the linear definition of deformations because it is independent of element reference system. In addition, complex manipulations required to obtain conservative descriptions and well-posed transformation matrices are avoided. Numerical tests have been carried out to evaluate the validity of the developed method.

Non-linear finite element analysis of composite beams with interlayer slips

This article presents a new non-linear finite element formulation for the analysis of two-layer composite plane beams with interlayer slips. The element is based on the corotational method. The main interest of this approach is that different linear elements can be automatically transformed to non-linear ones. To avoid curvature locking that may occur for low order element(s), a local linear formulation based on the exact stiffness matrix is used. Five numerical applications are presented in order to assess the performance of the formulation.

Efficient preconditioning for the discontinuous Galerkin finite element method by low-order elements

We derive and analyze a block diagonal preconditioner for the linear problems arising from a discontinuous Galerkin finite element discretization. The method can be applied to second-order self-adjoint elliptic boundary value problems and exploits the natural decomposition of the discrete function space into a global low-order subsystem and components of higher polynomial degree. Similar to results for the p-version of the conforming FEM, we prove for the interior penalty discontinuous Galerkin discretization that the condition number of the preconditioned system is uniformly bounded with respect to the mesh size of the triangulation. Numerical experiments demonstrate the performance of the method.

A discontinuous Galerkin formulation of a model of gradient plasticity at finite strains

The work presented here constitutes an extension to the finite-strain regime of a discontinuous Galerkin based, strain gradient plasticity formulation presented in Djoko et al. [J.K. Djoko, F. Ebobisse, A.T. McBride, B.D. Reddy, A discontinuous Galerkin formulation for classical and gradient plasticity – Part 1: formulation and analysis, Comput. Methods Appl. Mech. Engrg. 196 (2007) 3881–3897; J.K. Djoko, F. Ebobisse, A.T. McBride, B.D. Reddy, A discontinuous Galerkin formulation for classical and gradient plasticity – Part 2: algorithms and numerical analysis, Comput. Methods Appl. Mech. Engrg. 197 (2007) 1–21]. The focus here is on algorithmic and computational aspects of the formulation at finite strains. The adoption of a logarithmic hyperelastic–plastic formulation preserves the essential features of the infinitesimal formulation. This key ingredient allows the predictor–corrector solution algorithms developed for the infinitesimal gradient formulation to be extended readily to the finite-strain regime. The use of low-order elements is essential to contain the computational expense of the formulation but these elements are prone to locking. The method of enhanced assumed strains for geometrically nonlinear problems is utilised to circumvent this limitation. The form of the consistent tangent modulus is derived for the case of gradient plasticity. Two numerical examples are presented to illustrate aspects of the approximation scheme and the algorithm, as well as features of the model of gradient plasticity.

Generalized finite element analysis of three-dimensional heat transfer problems exhibiting sharp thermal gradients

In this paper, heat transfer problems exhibiting sharp thermal gradients are analyzed using the classical and generalized finite element methods. The effect of solution roughness on the ability of the methods to obtain accurate approximations is investigated. Convergence studies show that low order (linear and quadratic) elements require strongly refined meshes for acceptable accuracy. We propose a generalized FEM with global–local enrichments for the class of problems investigated in the paper. In this procedure, a global solution space defined on a coarse mesh is enriched through the partition of unity framework of the generalized FEM with solutions of local boundary value problems. The local problems are defined using the same procedure as in the global–local FEM, where boundary conditions are provided by a coarse scale global solution. Coarse, uniform, global meshes are acceptable even at regions with thermal spikes that are orders of magnitude smaller than the element size. Convergence on these discretizations was achieved even when no or limited convergence was observed in the local problems. Two approaches are proposed to improve the boundary conditions prescribed on local problems and their convergence. The use of the corresponding improved local solutions as enrichments for the global problem extends the range of target error level for the enriched global problem. The two-way information transfer provided by the proposed generalized FEM is appealing to several classes of problems, especially those involving multiple spatial scales. The proposed methodology brings the benefits of generalized FEM to problems where limited or no information about the solution is known a priori.

A Practical Variant of the Semismooth Newton Method for Frictionless Contact Problems

In this paper, a simple solution scheme for frictionless contact problems of linear elastic bodies is proposed. The proposed method is a variant of the semismooth Newton method. The frictionless contact problem, which is discretized using the finite element method with lower-order elements and a node-to-node contact model, is considered, although the proposed method can be extended to a node-to-segment or segment-to-segment contact model. The present method can be implemented by slightly modifying a computer program for the semismooth Newton method. In the iterative loop of the proposed method, a symmetric linear structural problem with multi-point constraints is solved. Therefore, the proposed method can be applied to large-scale contact problems. Good convergence of the proposed method compared to the semismooth Newton method is demonstrated in some examples.

The p‐version of the FEM for computational contact mechanics

AbstractContact analyses are being performed in various engineering applications. Here, like in most other fields, FE codes are based on low order elements using linear or quadratic shape functions. The intention of this paper is to show that finite elements with shape functions of high polynomial degree (p‐FEM) are a very attractive alternative to low order elements, even for computational contact mechanics. One of the advantages is the possibility to enhance the element formulation with the blending function method in order to accurately discretize the given geometry, which leads in combination with high convergence rates to very efficient computations. In order to solve the problem of frictionless contact, a penalty formulation is applied in this work. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Non-locking Tetrahedral Finite Element for Surgical Simulation.

To obtain a very fast solution for finite element models used in surgical simulations low order elements such as the linear tetrahedron or the linear under-integrated hexahedron must be used. Automatic hexahedral mesh generation for complex geometries remains a challenging problem, and therefore tetrahedral or mixed meshes are often necessary. Unfortunately the standard formulation of the linear tetrahedral element exhibits volumetric locking in case of almost incompressible materials. In this paper we extend the average nodal pressure tetrahedral element proposed by Bonet and Burton for a better handling of multiple material interfaces. The new formulation can handle multiple materials in a uniform way, with better accuracy, while requiring only a small additional computation effort. We discuss some implementation issues and show how easy an existing TLED (Total Lagrangian Explicit Dynamics) algorithm can be modified in order to support the new element formulation. The performance evaluation of the new element shows the clear improvement in reaction forces and displacements predictions compared to the average nodal pressure element in case of models consisting of multiple materials.

A tale of two elements: [formula omitted] and RT0

The P 1 NC - P 1 and RT 0 finite element schemes are among the most promising low order elements for use in unstructured mesh marine and lake models. They are both free of spurious elevation modes, have good dispersive properties and have a relatively low computational cost. In this paper, we derive both finite element schemes in the same unified framework and discuss their respective qualities in terms of conservation, consistency, propagation factor and convergence rate. We also highlight the impact that the local variables placement can have on the model solution. The main conclusion that we can draw is that the choice between elements is highly application dependent. We suggest that the P 1 NC - P 1 element is better suited to purely hydrodynamical applications while the RT 0 element might perform better for hydrological applications that require scalar transport calculations.

EAS concept for higher‐order finite shell elements to eliminate volumetric locking

AbstractThe deficiency of volumetric locking phenomena in finite elements using higher‐order shell element formulations based on Lagrangean polynomials and a linear finite shell kinematics cannot be avoided by the existent enhanced assumed strain (EAS) concept established for low‐order elements. In this paper a consistent modification of the EAS concept is proposed to extend its applicability to higher‐order shell elements. This modification, affecting the transversal normal strain for polynomial orders p>1, eliminates pathological modes caused by volumetric locking. The efficiency of the proposed extended EAS method is demonstrated by means of eigenvalue analyses and two representative numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.

Personality and drugs of choice

Although drug preference has been studied from the perspectives of individual differences and comorbid psychopathology, research incorporating both of these levels of analysis has been limited. In the current study, inpatients in a substance abuse treatment facility who reported preferences for alcohol, marijuana, methamphetamines, cocaine, heroin, or crack cocaine were compared in terms of their scores on measures of higher-order personality variables and psychopathology constructs representing lower-order elements of these factors. Results suggested that a broad externalizing dimension differentiated heroin users from alcohol, marijuana, and cocaine users. With the single exception that crack users remained more paranoid than alcohol users, psychopathology variables did not provide more specific discriminatory ability after broad personality factors were controlled. Implications for substance use research are discussed, with a focus on the utility of integrating individual differences and psychopathology constructs.

A Least-Squares Finite Element Method for the Magnetostatic Problem in A Multiply Connected Lipschitz Domain

A new least-squares finite element method is developed for the curl-div magnetostatic problem in Lipschitz and multiply connected domains filled with anisotropic nonhomogeneous materials. In order to deal with possibly low regularity of the solution, local $L^2$ projectors are introduced to standard least-squares formulation and applied to both curl and div operators. Coercivity is established by adding suitable mesh-dependent bilinear terms. As a result, any continuous finite elements (lower-order elements are enriched with suitable bubbles) can be employed. A desirable error bound is obtained: $||{\bf u}-{\bf u}_h||_0\le C\,||{\bf u}-\tilde{{\bf u}}||_0$, where ${\bf u}_h$ and $\tilde{{\bf u}}$ are the finite element approximation and the finite element interpolant of the exact solution ${\bf u}$, respectively. Numerical tests confirm the theoretical results.

Three Dimensional Eddy Current Analysis by Multi-Order Vector Finite Elements

In this paper, we propose new method based on multi-order vector finite elements in three dimensional eddy current analysis. Edges, faces, and volume of the multi-order element have orders of shape functions, respectively. In order to realize high efficiency of numerical calculation, high-order elements are assigned in the important area of the eddy current model, and low-order elements in the surrounding area, and multi-order elements in the middle area to bond two areas, respectively. The results obtained by multi-order elements are compared with those by full high-order elements and by full low-order elements.

Modeling of Belt-Drives Using a Large Deformation Finite Element Formulation

In this paper, the applicability of the absolute nodal coordinate formulation for the modeling of belt-drive systems is studied. A successful and effective analyzing method for belt-drive systems requires the exact modeling of the rigid body inertia during an arbitrary rigid body motion, accounting of shear deformation, description of highly nonlinear deformations, and a simple as well as realistic description of the contact. The absolute nodal coordinate formulation meets the challenge and is a promising approach for the modeling of belt-drive systems. In this study, a recently proposed two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation has been modified to obtain a belt-like element. In the original element, a continuum mechanics approach is applied to the exact displacement field of the shear deformable beam. The belt-like element allows the user to control the axial and bending stiffness through the use of two parameters. In this study, the interaction between the belt and the pulleys is modeled using an elastic approach in which the contact is accounted for by the inclusion of a set of external forces that depend on the penetration between the belt and pulley. When using the absolute nodal coordinate formulation, the contact forces can be distributed over the length of the element due to the use of high-order polynomials. This is different from other approaches that are used in the modeling of belt-drives. Static and dynamic analysis are used in this study to show the performance of the distributed contact force model and the proposed belt-like element, which is able to model highly nonlinear deformations. Applying these two contributions to the modeling of belt-drive systems, instead of contact forces applied at nodes and low-order elements, leads to a considerable reduction in the degrees of freedom.