An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence

We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. It is shown that a slightly more regularity, i.e. the mixed partial derivatives of the solution has to be bounded, is necessary for the optimal order of convergence. Numerical results are presented to support the analysis.

Operator-splitting finite element algorithms for computations of high-dimensional parabolic problems

An operator-splitting finite element method for solving high-dimensional parabolic equations is presented. The stability and the error estimates are derived for the proposed numerical scheme. Furthermore, two variants of fully-practical operator-splitting finite element algorithms based on the quadrature points and the nodal points, respectively, are presented. Both the quadrature and the nodal point based operator-splitting algorithms are validated using a three-dimensional (3D) test problem. The numerical results obtained with the full 3D computations and the operator-split 2D + 1D computations are found to be in a good agreement with the analytical solution. Further, the optimal order of convergence is obtained in both variants of the operator-splitting algorithms. (C) 2012 Elsevier Inc. All rights reserved.

Wave propagation in stiffened structures using spectrally formulated finite element

In this work, the wave propagation analysis of built-up composite structures is performed using frequency domain spectral finite elements, to study the high frequency wave responses. The paper discusses basically two methods for modeling stiffened structures. In the first method, the concept of assembly of 2D spectral plate elements is used to model a built-up structure. In the second approach, spectral finite element method (SFEM) model is developed to model skin-stiffener structures, where the skin is considered as plate element and the stiffener as beam element. The SFEM model developed using the plate-beam coupling approach is then used to model wave propagation in a multiple stiffened structure and also extended to model the stiffened structures with different cross sections such as T-section, I-section and hat section. A number of parametric studies are performed to capture the mode coupling, that is, the flexural-axial coupling present in the wave responses.

A finite element method for the Saint-Venant torsion and bending problems for prismatic beams

In this work, we present a finite element formulation for the Saint-Venant torsion and bending problems for prismatic beams. The torsion problem formulation is based on the warping function, and can handle multiply-connected regions (including thin-walled structures), compound and anisotropic bars. Similarly, the bending formulation, which is based on linearized elasticity theory, can handle multiply-connected domains including thin-walled sections. The torsional rigidity and shear centers can be found as special cases of these formulations. Numerical results are presented to show the good coarse-mesh accuracy of both the formulations for both the displacement and stress fields. The stiffness matrices and load vectors (which are similar to those for a variable body force in a conventional structural mechanics problem) in both formulations involve only domain integrals, which makes them simple to implement and computationally efficient. (C) 2014 Elsevier Ltd. All rights reserved.

An extended finite element model coupled with level set method for stable pitting corrosion

Mass balance between metal and electrolytic solution, separated by a moving interface, in stable pit growth results in a set of governing equations which are solved for concentration field and interface position (pit boundary evolution), which requires only three inputs, namely the solid metal concentration, saturation concentration of the dissolved metal ions and diffusion coefficient. A combined eXtended Finite Element Model (XFEM) and level set method is developed in this paper. The extended finite element model handles the jump discontinuity in the metal concentrations at the interface, by using discontinuous-derivative enrichment formulation for concentration discontinuity at the interface. This eliminates the requirement of using front conforming mesh and re-meshing after each time step as in conventional finite element method. A numerical technique known as level set method tracks the position of the moving interface and updates it over time. Numerical analysis for pitting corrosion of stainless steel 304 is presented. The above proposed method is validated by comparing the numerical results with experimental results, exact solutions and some other approximate solutions.

Automatic finite element formulation and assembly of hyperelastic higher order structural models

The formulation of higher order structural models and their discretization using the finite element method is difficult owing to their complexity, especially in the presence of non-linearities. In this work a new algorithm for automating the formulation and assembly of hyperelastic higher-order structural finite elements is developed. A hierarchic series of kinematic models is proposed for modeling structures with special geometries and the algorithm is formulated to automate the study of this class of higher order structural models. The algorithm developed in this work sidesteps the need for an explicit derivation of the governing equations for the individual kinematic modes. Using a novel procedure involving a nodal degree-of-freedom based automatic assembly algorithm, automatic differentiation and higher dimensional quadrature, the relevant finite element matrices are directly computed from the variational statement of elasticity and the higher order kinematic model. Another significant feature of the proposed algorithm is that natural boundary conditions are implicitly handled for arbitrary higher order kinematic models. The validity algorithm is illustrated with examples involving linear elasticity and hyperelasticity. (C) 2013 Elsevier Inc. All rights reserved.

The time finite element as a robust general scheme for solving nonlinear dynamic equations including chaotic systems

Schemes that can be proven to be unconditionally stable in the linear context can yield unstable solutions when used to solve nonlinear dynamical problems. Hence, the formulation of numerical strategies for nonlinear dynamical problems can be particularly challenging. In this work, we show that time finite element methods because of their inherent energy momentum conserving property (in the case of linear and nonlinear elastodynamics), provide a robust time-stepping method for nonlinear dynamic equations (including chaotic systems). We also show that most of the existing schemes that are known to be robust for parabolic or hyperbolic problems can be derived within the time finite element framework; thus, the time finite element provides a unification of time-stepping schemes used in diverse disciplines. We demonstrate the robust performance of the time finite element method on several challenging examples from the literature where the solution behavior is known to be chaotic. (C) 2015 Elsevier Inc. All rights reserved.

The time finite element as a robust general scheme for solving nonlinear dynamic equations including chaotic systems

Schemes that can be proven to be unconditionally stable in the linear context can yield unstable solutions when used to solve nonlinear dynamical problems. Hence, the formulation of numerical strategies for nonlinear dynamical problems can be particularly challenging. In this work, we show that time finite element methods because of their inherent energy momentum conserving property (in the case of linear and nonlinear elastodynamics), provide a robust time-stepping method for nonlinear dynamic equations (including chaotic systems). We also show that most of the existing schemes that are known to be robust for parabolic or hyperbolic problems can be derived within the time finite element framework; thus, the time finite element provides a unification of time-stepping schemes used in diverse disciplines. We demonstrate the robust performance of the time finite element method on several challenging examples from the literature where the solution behavior is known to be chaotic. (C) 2015 Elsevier Inc. All rights reserved.

A hybrid finite element formulation for flexible multibody dynamics

This work deals with the transient analysis of flexible multibody systems within a hybrid finite element framework. Hybrid finite elements are based on a two-field variational formulation in which the displacements and stresses are interpolated separately yielding very good coarse mesh accuracy. Most of the literature on flexible multibody systems uses beam-theory-based formulations. In contrast, the use of hybrid finite elements uses continuum-based elements, thus avoiding the problems associated with rotational degrees of freedom. In particular, any given three-dimensional constitutive relations can be directly used within the framework of this formulation. Since the coarse mesh accuracy as compared to a conventional displacement-based formulation is very high, the scheme is cost effective as well. A general formulation is developed for the constrained motion of a given point on a line manifold, using a total Lagrangian method. The multipoint constraint equations are implemented using Lagrange multipliers. Various kinds of joints such as cylindrical, prismatic, and screw joints are implemented within this general framework. Hinge joints such as spherical, universal, and revolute joints are obtained simply by using shared nodes between the bodies. In addition to joints, the formulation and implementation details for a DC motor actuator and for prescribed relative rotation are also presented. Several example problems illustrate the efficacy of the developed formulation.

Universal crack closure integral for SIF estimation

Numerical analysis of cracked structures often involves numerical estimation of stress intensity factors (SIFs) at a crack tip/front. A newly developed formulation called universal crack closure integral (UCCI) for the evaluation of potential energy release rates (PERRs) and the corresponding SIFs is presented in this paper. Unlike the existing element dedicated forms of crack closure integrals (MCCI, VCCI) with application limited to finite element analysis, this new numerical SIF/PERR estimation technique is independent of the basic stress analysis procedure, making it universally applicable. The second merit of this procedure is that it avoids the generally error-producing zones close to the crack tip/front singularity. The UCCI procedure, based on Irwin's original CCI, is formulated and explored using a simple 2D problem of a straight crack in an infinite sheet. It is then applied to some three-dimensional crack geometries with the stresses and displacements obtained from a boundary element program.

Analysis Of Curved Laminated Beams Of Bimodulus Composite-Materials

Composite materials exhibiting different moduli in tension and in compression, commonly called as bimodular composites are being used in many engineering fields. A finite element analysis is carried out for small deflection static behavior of laminated curved beams of bi modulus materials for both solid and hollow circular cross-sections using an iterative procedure. The finite element has 16 d.o.f. and uses the displacement field in terms of first order Hermite in terpolation polynomials. The neutral surface, i.e. the locus of points having zero axial strain is found to vary drastically depending on the loading, lay up schemes and radius of curvature. As il lustrations, plots of the cross-sections of the ruled neutral-surface are presented for some of the investigated cases. Using this element a few problems of curved laminated beams of bimodulus materials are solved for both solid and hollow circular cross-sections.

An FETI-preconditioned conjuerate gradient method for large-scale stochastic finite element problems

In the spectral stochastic finite element method for analyzing an uncertain system. the uncertainty is represented by a set of random variables, and a quantity of Interest such as the system response is considered as a function of these random variables Consequently, the underlying Galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled. The solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and/or the level of uncertainty is increased This paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual-primal finite element tearing and interconnecting method equipped with a Krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right-hand sides. Preliminary performance results on a Linux Cluster suggest that the proposed Solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification Of realistic systems tractable.

Shortcomings of discontinuous-pressure finite element methods on a class of transient problems

Past studies that have compared LBB stable discontinuous- and continuous-pressure finite element formulations on a variety of problems have concluded that both methods yield Solutions of comparable accuracy, and that the choice of interpolation is dictated by which of the two is more efficient. In this work, we show that using discontinuous-pressure interpolations can yield inaccurate solutions at large times on a class of transient problems, while the continuous-pressure formulation yields solutions that are in good agreement with the analytical Solution.

Modelling of solute transport in a mild heterogeneous porous medium using stochastic finite element method: Effects of random source conditions

Randomness in the source condition other than the heterogeneity in the system parameters can also be a major source of uncertainty in the concentration field. Hence, a more general form of the problem formulation is necessary to consider randomness in both source condition and system parameters. When the source varies with time, the unsteady problem, can be solved using the unit response function. In the case of random system parameters, the response function becomes a random function and depends on the randomness in the system parameters. In the present study, the source is modelled as a random discrete process with either a fixed interval or a random interval (the Poisson process). In this study, an attempt is made to assess the relative effects of various types of source uncertainties on the probabilistic behaviour of the concentration in a porous medium while the system parameters are also modelled as random fields. Analytical expressions of mean and covariance of concentration due to random discrete source are derived in terms of mean and covariance of unit response function. The probabilistic behaviour of the random response function is obtained by using a perturbation-based stochastic finite element method (SFEM), which performs well for mild heterogeneity. The proposed method is applied for analysing both the 1-D as well as the 3-D solute transport problems. The results obtained with SFEM are compared with the Monte Carlo simulation for 1-D problems.

Parametric finite element analyses of geocellsupported embankments

This paper presents the results from parametric finite element analyses of geocell-supported embankments constructed on weak foundation soils. A composite model is used to numerically simulate the improvement in the strength and stiffness of the soil as a result of geocell confinement. The shear strength of the geocell-encased soil is obtained as a function of the additional confining pressure due to the geocell encasement considering it as a thin cylinder subjected to internal pressure. The stiffness of the geocell-encased soil is obtained from the stiffness of the unreinforced soil and the tensile modulus of the geocell material using an empirical equation. The validity of the model is verified by simulating the laboratory experiments on model geocell-supported embankments. Parametric finite element analyses of the geocell-supported embankments are carried out by varying the dimensions of the geocell layer, the tensile strength of the material used for fabricating the geocell layer, the properties of the infill soil, and the depth of the foundation layer. Some important guidelines for selecting the geocell reinforcement to support embankments on weak foundation soils are established through these numerical studies.