A time-discontinuous implicit variational integrator for stress wave propagation analysis in solids

This paper proposes a time-discontinuous and space-continuous variational integration (TDSC-VI) method for accurate elastic stress wave propagation computations in one-dimensional bar. The algorithm employs a limiter, akin to classical artificial viscosity treatment, to mitigate the deleterious Gibbs jumps across the stress discontinuities, and a parametrized consistent mass to alleviate dispersion error when the stepsizes are different from the Courant stability limit, which becomes necessary for elastic unloading and internal reflections in plastic deformation problems. Stability and accuracy analyses of the proposed TDSC-VI method are carried out and compared with several well-known traditional integration algorithms. Numerical experiments are carried out with the proposed implicit and explicit methods, which show that the proposed methods perform favorably compared to the trapezoidal rule and the central difference method.     

A coupled finite element/one-dimensional wave model of stress wave propagation in a shock tube

A finite element model coupled to a numerical representation of one-dimensional wave propagation is used to model stress wave propagation in the structure of a large shock tunnel. Modelling difficulties and the method of coupling the two types of model are described. Results are presented which indicate that stress wave damper rods, attached to the tunnel, are effective as a means of controlling stress levels.     

Textbook finite element methods applied to linear wave propagation problems involving conversion and absorption

A system of two second-order ordinary differential equations describing wave propagation in a hot plasma is solved numerically by the finite element method involving standard linear and cubic elements. Evanescent short-wavelength modes do not constitute a problem because of the variational nature of the method. It is straightforward to generalize the method to systems of equations with more than two equations. The performance of the method is demonstrated on known physical situations and is measured by investigating the convergence properties. Cubic elements perform much better than linear ones. In application it is shown that global plasma oscillations might have an importance for the linear wave conversion in the ion-cyclotron range of frequency.     

A finite element formulation for transient incompressible viscous flows stabilized by local time-steps

Stabilized finite element formulations are well suited for convection dominated flows and for the solution of the incompressible Navier–Stokes equations in primitive variables. In this paper, we present a method where the structure of stabilization terms appear naturally from a least-squares minimization of the time-discretized momentum balance. Local time-steps, chosen according to the time-scales of convection–diffusion of momentum, play the role of stabilization parameters. Numerical solutions of incompressible viscous flows demonstrate the usefulness of the proposed stabilized formulation.     

On the application of a least squares residual-fitting finite element formulation to fluid flow problems

The connection between the least squares residual fitting finite element formulation and reduced integration is considered for two model problems, one of which requires an isoparametric mapping. For both problems a significant improvement in accuracy is achieved when compared with a conventional Galerkin finite element formulation. It is found that the two methods are equivalent for rectangular elements as long as an isoparametric mapping is not required. For triangular elements there is no direct link between the two methods and neither method produces any significant improvement over the use of exact numerical integration.     

A nonlinear,semi-analytical finite element analysis for nearly axisymmetric solids

The presented research is concerned with the development of the theory and accompanying computer program for a semi-analytical finite element analysis of non axisymmetrically loaded, nearly axisymmetric solids. The theoretical basis of the method together with numerical procedures for handling boundary conditions, rigid body motion and the iterative solution process are described. A finite element program for evaluating the analysis is discussed, evaluated for effectiveness and applied to several examples.The range of application of the analysis for inhomogeneous, orthotropic, nonlinear and nearly axisymmetric bodies is demonstrated by a series of examples. The substantial savings in computer time and memory as compared to a conventional finite element analysis is discussed.     

Topology optimization for transient wave propagation problems in one dimension

Structures exhibiting band gap properties, i.e., having frequency ranges for which the structure attenuates propagating waves, have applications in damping of acoustic and elastic wave propagation and in optical communication. A topology optimization method for synthesis of such structures, employing a time domain formulation, is developed. The method is extended to synthesis of pulse converting structures with possible applications in optical communication.     

Galerkin's finite element formulation of the second-order boundary-value problems

A Galerkin's finite element approach based on weighted-residual formulation is presented to find approximate solutions to obstacle, unilateral and contact second-order boundary-value problems. The approach utilizes a piece-wise linear approximations utilizing linear Langrange polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of this scheme in comparison to collocation, finite-difference and spline methods.     

An adaptive finite element scheme for transient problems in CFD

An adaptive finite element scheme for transient problems is presented. The classic h-enrichment / coarsening is employed in conjunction with a triangular finite element discretization in two dimensions. A mesh change is performed every n timesteps, depending on the Courant number employed and the number of ‘protective layers’ added ahead of the refined region. In order to simplify the refinement/ coarsening logic and to be as fast as possible, only one level of refinement/coarsening is allowed per mesh change. A high degree of vectorizability has been achieved on the CRAY XMP 12 at NRL. Several examples involving shock-shock interactions and the impact of shocks on structures demonstrate the performance of the method, indicating that considerable savings in CPU time and storage can be realized even for strongly unsteady flows.     

A finite element method for thermoviscoelastic analysis of plane problems

A stress analysis for plane problems in linear thermoviscoelasticity using a finite element formulation is presented. The method employed is based on the assumptions that (1) the material is isotropic, homogeneous and linear, (2) the stress-strain laws are expressed in the hereditary integral form, and (3) the material is thermorheologically simple, which implies that the temperature-time equivalence hypothesis is valid. The associated computer program utilizes isoparametric plane elements.The element matrices that result in the equilibrium equations involve hereditary integrals, and these are approximated by a finite difference scheme for time marching. The solutions for two problems are compared with analytical results evaluated by the integral transform method.For approximate results which require less computer time an alternative form of equilibrium equations utilizing an iterative technique is presented and an example solution is included. Finally, the effect of incompressibility is considered for an axisymmetric numerical example.     

A finite element formulation for beams with thin walled cross-sections

The paper presents a finite element model for calculation of stresses and deformations of beams with thin walled cross-sections. The beam model takes into account deformations due to shear. Warping is accounted for by a modified sector coordinate formulation. As interpolation functions between the seven degrees of freedom at each node are used the analytical solutions for the special case of a double symmetric cross-section. Therefore, depending on the external loading, each prismatic beam can in most cases be treated as a single element. The assembly of the beam elements to the global model is performed by use of transition matrices which assures compatibility between the elements in the sence of least squares.     

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.     

Finite element stress formulation for dynamic elastic-plastic analysis

The solution to wave propagation problems in solids with elastic-plastic material properties is obtained by using the finite element method directly in terms of the stresses. A variational principle due to Gurtin is modified by including a plastic strain tensor in the constitutive relationship. The resulting finite element equations, which represent the strain-displacement equations written in terms of the stresses, are simultaneous integral equations in time. With a transformation of variables, a set of simultaneous differential equations is obtained of the form$\text{H}\text{s}\text{?}\text{+ Qs+ Ve}{}_{\mathrm{p}}\text{= q(t)}$, where $\text{H}$ is a symmetric positive-semidefinite matrix, and $\text{Q}$ is a symmetric positive-definite matrix. The stresses and the plastic strains are represented by $\text{s}\text{?}$ and $\text{e}{}_{\mathrm{p}}$, respectively.Finite element equations are developed for an axisymmetric ring element with an arbitrary quadrilateral cross section in which the stresses and the plastic strains vary linearly along the sides of the elements. The equations are numerically integrated with respect to time by Newmark's generalized acceleration method.An iterative procedure is presented, which uses the finite element strain-displacement equations and the plasticity relationships, to determine the state of stress at the end of the time step. Several examples are used to demonstrate the solution technique for elastic and elastic-plastic problems.     

A quadrilateral finite difference plate element for nonlinear transient analysis of panels

A quadrilateral plate element for the analysis of nonlinear transient response of panels has been developed based on the variational finite difference method for an irregular mesh. Due to the superior computational characteristics of the variational finite difference method with a lesser degree of continuity constraint on the interpolation functions and the use of lower-order polynomials allowing faster numerical integration methods to be implemented, this plate element is quite competitive or perhaps even superior when compared with the conforming finite elements. Three illustrative problems have been solved using this plate element to demonstrate its capability and accuracy in analyzing the large deformation response of panels subject to dynamic loadings.Very favorable correlation was observed between analysis and experiment on large deformations of elasticplastic rectangular plates subject to intensive impulsive loadings. Similar correlation was also observed for circular plates modeled with this quadrilateral plate element under impulsive loadings. Finally, the large dynamic deformations of composite rectangular panels of graphite-epoxy, boron-epoxy, glass-epoxy, and isotropic material were analyzed and found in good agreement with other analytical results.     

Unconditionally stable concurrent procedures for transient finite element analysis

A new class of algorithms for transient finite element analysis which is amenable to an efficient implementation in parallel computers is proposed. The suitability of the method for parallel computation stems from the fact that, given an arbitrary partition of the finite element mesh, each subdomain in the partition can be processed over a time step independently and simultaneously with the rest. Both element-by-element and coarse partitions of the mesh are discussed. For the former, the proposed algorithms are shown to have the structure of an explicit scheme. In particular, no global equation solving effort is involved in the update procedure. However, in contrast to explicit schemes the proposed algorithms are shown to be unconditionally stable over a certain range of the algorithmic parameters. In structural dynamics problems, good accuracy is obtained with a constant time step integration. For heat conduction problems accuracy limitations suggest the use of a step-changing technique. When this is done, numerical tests indicate the good behavior of the method. The case in which the mesh is partitioned into a small number of subdomains, typically as many as processors in the computer, is also explored in detail. Good accuracy is obtained over a wide range of time steps. Finally, extensions to second- and higher-order accuracy methods are discussed.     

Finite element formulations for transient dynamic problems in solids using explicit time integration

This paper presents the displacement, mixed and stress formulations of the finite element method when applied to transient dynamic problems of solids. The formulations are chosen so that explicit time integration may be used. Large deformations are considered for these formulations, and infinitesimal strain assumptions are employed with the stress formulation. Displacement formulations are well-known, but the mixed formulations presented provide a viable alternative. The stress formulation has not proven successful for the large deformation problem, but when infinitesimal strains are assumed, the formulation is attractive. A problem of an internally pressurized ring is solved in order to evaluate the different proposed formulations.     

A simple algorithm for three-dimensional finite element analysis of contact problems

This paper addresses the numerical solution of three-dimensional frictionless contact problems by a finite element method. The two-body contact problem is considered in the context of fully non-linear kinematics. The impenetrability constraint is satisfied via a classical penalty formulation. The contacting surfaces are discretized by means of projections of the interacting element faces onto suitably chosen flat surfaces. Attention is focused on the efficiency of the overall algorithm. Numerical simulations are conducted for a series of test problems to assess the performance of the proposed methodology.     

A finite element formulation for multilayered and thick plates

A bilinear isoparametric finite element concept is used for the numerical analysis of multilayered plates. The underlying theory used allows for transverse shear and normal strains in each layer, thus extending the analysis to very thick plates and laminates. To illustrate the versatility of the multilayered element, three examples are presented and the results are compared with available exact solutions.