Finite elements in space and time for parallel computing of viscoelastic deformation

The efficient parallel computation of time dependent problems, e.g. parabolic problems of viscoelastic material deformation, underlies the “bottleneck” of the serial approach in time. The usual method of lines, also called semidiscretization, leads to an iterative calculation in time, i.e. a sequential solution of the spatial problems for all time steps. Due to that, only one spatial problem can be solved in parallel at a certain time step. For an efficient parallelization, it is necessary to compute the whole problem in a distributed way. Furthermore, both h- and p-adaptive approximation should be possible in time and space. For these purposes, in addition to the spatial FE-discretization, a continuous finite element discretization in time is used. Thus, one obtains a total algebraic equation system in space and time, whose solution has to be parallelized efficiently, and h- and p-adaptivity in time and space within the frame of the overall Galerkin-process has to be realized. The present paper treats symmetric and non-symmetric formulations of two different viscoelastic three-parameter models. The new numerical approach concerns first for the Malvern Model (generalized Maxwell Model). The numerical examples for the new non-symmetric formulation and the traditional semidiscretization show the advantage (with respect to convergence to the problem solution) of the new finite element approach with simultaneous discretizations in time and space. But the algebraic systems are bad-conditioned such that parallel iterative solvers with various preconditions are not efficient. The symmetric formulation for the Malvern Model can be obtained for the one-dimensional case only. A numerical example showed the good iterative solvability of the symmetric formulation. Therefore, in order to obtain a symmetric formulation in the 3D-case the generalized Kelvin–Voigt Model was chosen as an alternative one. It should be mentioned that the numerical examples show both the effectiveness of parallel computation and the efficiency of h- and p-adaptation (p-adaptation yields the higher rate of convergence than h-adaptation). Received 19 April 1998     

Finite elements in space and time for dynamic contact problems

Dynamic obstacle and Signorini problems are discretized by continuous and discontinuous finite elements in space and time. The resulting discrete problems are attributed to a standard implicit time‐stepping scheme through relaxation of impact phenomena and suitable numerical integration. The method can cope with dynamic contact problems, which is shown by an analysis of a model problem. Moreover, numerical examples demonstrate that it is actually able to approximate the solution of dynamic contact problems, which are not fully covered by the theory. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite elements analysis of viscoelastic fracture

A numerical procedure based on the finite elements method and Schapery's formulation is proposed to determine the critical condition of cracks in viscoelastic structures. Some initial results trying to couple fracture mechanics with continuum damage mechanics also are presented.     

Finite elements in space and time for the analysis of generalised visco‐elastic materials

Numerical analysis of linear visco‐elastic materials requires robust and stable methods to integrate partial differential equations in both space and time. In this paper, symmetric space–time finite element operators are derived for the first time for elementary linear elastic spring and linear viscous dashpot. These can thereafter be assembled in parallel and in series to simulate an arbitrarily complex linear visco‐elastic behaviour. The flexibility of the proposed method allows the formulation of the behaviour, which closely reflects physical processes. An efficient algorithm is proposed to use the generated elementary matrices in a way that is comparable with finite difference schemes, in terms of both processor and memory costs. This unconditionally stable and convergent procedure is equally valid for space domains in which geometry or material properties evolve with time. Copyright © 2013 John Wiley & Sons, Ltd.     

A generalized model for the uniaxial isothermal deformation of a viscoelastic body

Summary Using the notion of a fractional derivative we formulate a new model for a uniaxial deformation of a visco-elastic body. The basic assumption is that all derivatives ₍₎ with respect to time of the stress depend (with specified weighting factor) on all derivatives ₍₎ with respect to time of the strain (multiplied with another weighting factor), for 01. In this respect our model is a generalization of the Zener model, i.e., it is a Zener fractional model with infinitely many terms. The relation between stress and strain is given in explicit form. For two specific choices of parameters the behavior of the model under suddenly applied stress (creep) and suddenly applied strain (stress relaxation) are examined.     

Finite elements with embedded displacement discontinuity: a generalized variable formulation

This paper is intended to bring a contribution towards a satisfactory simulation of those fracture phenomena which result in the appearance and development of discrete cracks. To this purpose, a general mixed finite element formulation is proposed, based on the concept of generalized variables in Prager's sense. The displacement field inside an element is modelled by the sum of two contributions: a regular (continuous) part which is governed by standard shape functions, and a possibly discontinuous one which is introduced soon after a suitable criterion is satisfied. The formulation is first specialized to a one‐dimensional case, then a triangular element for two‐dimensional problems is described in detail. Analytical and numerical examples are presented in order to clarify the formulation and to point out the essential role of inter‐element conformity. Copyright © 2000 John Wiley & Sons, Ltd.     

Calculation of transient temperature fields with finite elements in space and time dimensions

A program is demonstrated which apart from linear finite elements in time also includes elements with shape functions of the second and third degree. The algorithm for discretization in the time dimension is described and, using the example of a parabolic time element, the coefficients required to form the global system are given. By various test examples the efficiency of the process is examined by comparison with the customary difference method. Generally, with finite elements in time, the solution has better stability. Comparing the time required for calculation with the accuracy of the solution it would appear that in examining problems where boundary conditions are constant in time, higher order time elements are no improvement over the linear time element. However, for the purpose of reproducing periodic processes, higher order time elements offer an advantage in that one is not limited to linear variations of the boundary conditions within the element. Thus, for example, the temperature curve for parabolic variation of the surface temperature can be reproduced with close approximation by two time elements per period and a shape function of the third degree.     

General boundary elements solution for ageing viscoelastic structures

A general boundary element formulation for the stress analysis of bodies with ageing viscoelastic constitutive relations such as concrete or biological materials is presented. It is shown that in the case of a constant Poisson modulus, the domain integral vanishes. Otherwise, the domain integrals are taken to the boundary by means of the dual reciprocity method. Computational details and examples are given. Copyright © 2001 John Wiley & Sons, Ltd.     

Constitutive equations for orthotropic nonlinear viscoelastic behaviour using a generalized Maxwell model Application to wood material

This paper presents a nonlinear viscoelastic orthotropic constitutive equation applied to wood material. The proposed model takes into account mechanical and mechanosorptive creep via a 3D stress ratio and moisture change rate for a cylindrical orthotropic material. Orthotropic frame is based on the grain direction (L), radial (R) and hoop (T) directions, which are natural wood directions. Particular attention is taken to ensure the model to fulfill the necessary dissipation conditions. It is based on a rheological generalized Maxwell model with two elements in parallel in addition with a single linear spring taking into account the long term response. The proposed model is implemented in the finite element code ABAQUS/Standard^® via a user subroutine UMAT and simple example is shown to demonstrate the capability of the proposed model. Future works would deal with damage and fracture prediction for wooden structures submitted to climate variations and mechanical loading.     

Finite element analysis of viscoelastic composite structures based on a micromechanical material model

The stress and creep analysis of structures made of micro-heterogeneous composite materials is treated as a two-scale problem, defined as a mechanical investigation on different length scales. Reinforced composites show by definition a heterogeneous texture on the microlevel, determined by the constitutive behaviour of the matrix material and the embedded fibres as well as the characteristics of the bonding properties in the interphase. All these heterogeneities are neglected by the finite element analysis of structural elements on the macroscale, since a ficticious and homogeneous continuum with averaged properties is assumed. Therefore, the constitutive equations of the substitute material should well reflect the mechanical behaviour of the existing micro-heterogeneous composite in an average sense.The paper at hand starts with the brief outline of a micromechanical model, named generalized method of cells (GMC), which provides the macrostress responses due to macrostrain processes as well as the homogenised constitutive tensor of the substitute material. The macroscopic stresses and strains are obtained as volume averages of the corresponding microfields within a representative volume element. The effective material tensor constitutes the mapping between the macro-strains and the macro-stresses. The cells method is used for the homogenisation of the unidirectionally reinforced single layers of laminates made of viscoelastic resins and flexibly embedded elastic fibres. The algorithm for the homogenisation of the constitutive properties runs simultaneously to the finite element analysis at each point of numerical integration and provides the macro-stresses and the homogenised constitutive properties. The validity of the proposed two-scale simulation is investigated by solving boundary value problems and comparing the numerical results for the structures to the experimental data of creep and relaxation tests or analytical solutions.     

Finite elements for field problems in cylindrical co-ordinates

This paper deals with the extension of the finite element method as applied to the solution of the Laplace or wave equation to cylindrical co-ordinate systems. The base matrices required for solving problems governed by these equations are derived for circular polar, elliptic cylinder and parabolic cylinder co-ordinates. The matrices allow problems whose boundaries are described as co-ordinate surfaces in cylinder co-ordinates to be attacked directly by the finite element method. The subject is discussed from the point of view of one interested in electromagnetic wave propagation in uniform waveguide structures.     

A model for anisotropic viscoelastic damage in composites

A model for continuous damage combined with viscoelasticity is proposed. The starting point is the formulation connecting the elastic properties to the tensor of damage variables. A hardening law associated with the damage process is identified from available experimental information and the rate-type constitutive equations are derived. This elastic damage formulation is used to formulate an internal variable approximation to viscoelastic damage in the form of a non-linear Kelvin chain. Elastic and viscoelastic equations are implemented into a finite element procedure. The code is verified by comparison with closed-form solutions in simplified configurations, and validated by fitting results of experimental creep tests.     

Alternative time marching process for BEM and FEM viscoelastic analysis

Based on the weighted residual technique, both Finite Element and Boundary Element alternative procedures for viscoelastic analysis are proposed. After imposing the space approximations, applying the kinematical relations for material and strain velocities at the approximation level, the time integration is carried out using appropriate operators. The Kelvin‐Voigt viscoelastic model is implemented in order to validate the idea. The Newmark β time integral scheme is applied to the Finite Element procedure while the Houbolt scheme is applied to the Boundary Elements, allowing the consideration of dynamic analysis in future works. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite elements for the vibration of cones and cylinders

Elemental mass matrices have been produced for the vibration of conical and cylindrical shells, based on a semi-analytical approach. Frequencies and modes of vibration have been compared with existing solutions and also with experimental results obtained from other sources. Good agreement has been found between theory and experiment for thin-walled circular cylinders and cones, a cone-cylinder combination, and a cooling tower model. A theoretical investigation was also made on the vibration of a circular cylinder when subjected to uniform pressure.     

Finite elements for elasticity with microstructure and gradient elasticity

We present a general finite element discretization of Mindlin's elasticity with microstructure. A total of 12 isoparametric elements are developed and presented, six for plane strain conditions and six for the general case of three‐dimensional deformation. All elements interpolate both the displacement and microdeformation fields. The minimum order of integration is determined for each element, and they are all shown to pass the single‐element test and the patch test. Numerical results for the benchmark problem of one‐dimensional deformation show good convergence to the closed‐form solution. The behaviour of all elements is also examined at the limiting case of vanishing relative deformation, where elasticity with microstructure degenerates to gradient elasticity. An appropriate parameter selection that enforces this degeneration in an approximate manner is presented, and numerical results are shown to provide good approximation to the respective displacements and strains of a gradient elastic solid. Copyright © 2007 John Wiley & Sons, Ltd.     

Gasless combustion-driven heating elements for materials experiments in space

The novel concept of using gasless combustible mixtures as heating elements for materials processing in space and in ground-based microgravity facilities is presented. The unique properties of metal-sulfur combustible compositions (i.e., high flame temperatures, low ignition temperatures, liquid combustion products, nonporous charges, and gasless reactions) make them ideally suited for such heating applications. Heating elements based on metal-sulfur combustion have an energy density more than order of magnitude greater than electrical batteries, can be easily integrated with processing samples, and can operate under high pressures and in different gaseous environments. Demonstration prototypes of the gasless combustion-driven furnaces have already demonstrated peak temperatures close to 2300 K and heating rates above 200 °C/s.     

Quadrature rules for triangular and tetrahedral elements with generalized functions

Quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains. These quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries. Specifically, the integrands considered here consist of a quadratic polynomial multiplied by a Heaviside or Dirac delta function operating on a linear polynomial. This form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function. Exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values. Compared with methods involving explicit integration over subdomains, the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical Jacobian information describing the dependence of the integrals on the nodal level set values. The accuracy of the method is demonstrated for a simple conduction problem with the Neumann and Robin‐type boundary conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite elements for piezoelectric vibrations with open electricboundaries

The problem of a piezoelectric body vibrating in the free space of infinite extent has been posed as a problem with open electric boundaries. A three-dimensional finite element analysis of piezoelectric vibrations has been complemented by the modeling of the external electrostatic field. The infinite exterior region that adheres to Laplace's equation is represented as a single “superelement” obtained by the ballooning of the outer boundary. The spectral transformation Lanczos method is used to find approximations to some solutions of the generalized eigenvalue problem arising from the finite element discretization of interior and exterior regions. The proposed approach was tested on simple vibrators with different mesh discretization and element order. The method produces numerically stable results, and was used to estimate the influence of the exterior leakage field on electromechanical properties of free piezoelectric vibrations. Numerical tests show that for some materials, vibrator geometries and modes, favorable to the coupling with the exterior field, the resonant frequency and electrical parameter shifts due to open electric boundaries may be as great as several percent