On the numerical solution of non-linear reservoir flow models with gravity

Based on a two-phase fluid model for immiscible displacement in a porous medium, we develop and analyse numerical solution techniques for certain non-linear phenomena. Two different solution strategies for the treatment of gravity effects, which are non-trivial to model by existing solution techniques and may be of great influence in many practical flow situations, are presented. The solution procedures are based on an operator-splitting technique, combining the modified method of characteristics with finite element techniques and adaptive grid refinement.     

Formulation and solution of the non-linear,damped eigenvalue problem for skeletal systems

This paper presents and discusses an Arnoldi-based eigensolution technique for evaluating the complex natural frequencies and mode shapes from frequency dependent quadratic eigenproblems associated with vibration analysis of damped structures. The new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices. This modelling provides the ability to represent a frequency dependent damping matrix in vibration analysis of skeletal systems. The eigensolution methodology presented here is based upon the ability to evaluate a specific set of parametrized curves for the non-linear eigenvalue problem at given values of the parameter. Numerical examples illustrate that this method, used in conjunction with a secant interpolation, accurately evaluates the complex natural frequencies and modes of the quadratic non-linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models.     

On the solution of non-linear drying problems in capillary porous media through integral transformation of Luikov equations

The generalized integral transform technique is employed to provide hybrid numerical–analytical solutions to the non-linear Luikov equations, that govern drying within capillary porous media. The coupled equations for moisture and temperature distributions are integral transformed to eliminate the spatial co-ordinates, yielding an ordinary differential system for the time variation of the transformed potentials, which is readily solved through scientific subroutine libraries with automatic error control schemes. The analytical inversion formula is then recalled to provide explicit expressions for the original potentials at any desired position. The convergence behaviour of the proposed eigenfunction expansions is illustrated, and a parametric study is performed, for the drying of a slab subjected to non-linear radiative boundary conditions.     

A qualitative analysis of the behaviour of the Galerkin equations relevant to non-linear eddy current problems

The Galerkin equations relevant to the eddy currents induced in an iron body are considered. These equations are obtained by formulating the field problem in terms of the magnetic vector potential and by applying the Galerkin method. They are shown to have a unique steady-state solution if a certain condition on the magnetic constituitive relationship is satisfied. In particular a T-periodic source gives rise to a unique T-periodic solution to which all other solutions converge asymptotically independently from the initial conditions. Under the same condition the exponential decay of the ‘transients’ is shown, and an explicit lower and upper bound for its rate is given. These structural properties allow us to exclude a priori that qualitatively different asymptotic behaviours, including even chaotic solutions, may occur. Numerical simulation, when based on qualitative information of this type, enables us to obtain the quantitative properties in an efficient manner. In order to demonstrate the practical use of these results some numerical experiments are presented.     

On the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models

Free vibration analysis is a major part of any dynamic analysis. Natural frequencies and related mode shapes may be obtained from free vibration analysis as the solutions of generalized eigenproblems. Although the eigensolutions of large‐scale structures require large computational efforts, these solutions may be achieved simply for symmetric structures. We present an efficient method for the decomposition of generalized eigenproblems related to finite element models with cyclic symmetry (having nodes at the axis of symmetry) into eigensubproblems with significantly smaller dimensions. This decomposition is obtained by block diagonalization of a matrix with a special pattern known as a block circulant, using the concept of the Kronecker product and similarity transformations. The proposed method is applied to three finite element models discretized by triangular and four‐node quadrilateral plate and shell elements, and its efficiency, accuracy and simplicity are evaluated. Copyright © 2009 John Wiley & Sons, Ltd.     

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.     

New exact solutions of the Tzitzéica-type equations in non-linear optics using the expa function method

One specific class of non-linear evolution equations, known as the Tzitzéica-type equations, has received great attention from a group of researchers involved in non-linear science. In this article, new exact solutions of the Tzitzéica-type equations arising in non-linear optics, including the Tzitzéica, Dodd–Bullough–Mikhailov and Tzitzéica–Dodd–Bullough equations, are obtained using the exp_a function method. The integration technique actually suggests a useful and reliable method to extract new exact solutions of a wide range of non-linear evolution equations.     

Speeding up the solution of thermal EHL problems

This paper considers the fast solution of large, dense linear systems arising from thermal EHL problems.We present an efficient preconditioner for these, based on a novel combination of wavelet compression with the Schur complement. Numerical results illustrate the effectiveness of this approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Application of the mixed approximation to the solution of two-dimensional problems of the theory of small elastoplastic strains using the finite element method

A triangular finite element that provides the stability and convergence of the mixed approximation is used for the solution of two-dimensional boundary-value problems of the theory of small elastoplastic strains. A system of resolving matrix equations of a mixed type is presented for the solution of which a three-layer iteration algorithm with a preconditioning matrix is used. Numerical results for the solution of model problems obtained by the classical and combined finite element methods are compared. __________ Translated from Problemy Prochnosti, No. 2, pp. 124–136, March–April, 2006.     

On Some Approximate Methods for the Tensile Instabilities of Thin Annular Plates

A thin annular plate is subjected to a uniform tensile field at its inner edge which leads to compressive circumferential stresses. When the intensity of the applied field is strong enough, elastic buckling occurs circumferentially, leading to a wrinkling pattern. Using a linear non-homogeneous pre-bifurcation state, the linearised eigenvalue problem describing this instability is cast as a fourth-order linear differential equation with variable coefficients. This problem is investigated numerically and it is shown that the simple application of the Galerkin technique reported in the literature leads to gross errors in the corresponding approximations. Several novel mathematical features of the eigenvalue problem are included as well.     

An efficient numerical method for the solution of sliding contact problems

In this paper, an efficient numerical method to solve sliding contact problems is proposed. Explicit formulae for the Gauss–Jacobi numerical integration scheme appropriate for the singular integral equations of the second kind with Cauchy kernels are derived. The resulting quadrature formulae for the integrals are valid at nodal points determined from the zeroes of a Jacobi polynomial. Gaussian quadratures obtained in this manner involve fixed nodal points and are exact for polynomials of degree 2n ? 1, where n is the number of nodes. From this Gauss–Jacobi quadrature, the existing Gauss–Chebyshev quadrature formulas can be easily derived. Another apparent advantage of this method is its ability to capture correctly the singular or regular behaviour of the tractions at the edge of the region of contact. Also, this analysis shows that once if the total normal load and the friction coefficient are given, the external moment M and contact eccentricity e (for incomplete contact) in fully sliding contact are uniquely determined. Finally, numerical solutions are computed for two typical contact cases, including sliding Hertzian contact and a sliding contact between a flat punch with rounded corners pressed against the flat surface of a semi‐infinite elastic solid. These results provide a demonstration of the validity of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

Identification of diffusion parameters in a non-linear convection–diffusion equation using adaptive homotopy perturbation method

This paper is concerned with the numerical identification of diffusion parameters in a non-linear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. In order to overcome the defect of the local convergence of traditional methods, an adaptive homotopy perturbation method is applied to solve this parameter identification inverse problem. The adaptive homotopy perturbation method provides a simple way to adapt computational refinement to the choice of the homotopy parameter. Numerical simulations illustrate that the proposed algorithm is globally convergent and computationallyefficient.     

Scaling,sensitivity and stability in the numerical solution of quadratic eigenvalue problems

The most common way of solving the quadratic eigenvalue problem (QEP) (λ² M + λD + K)x = 0 is to convert it into a linear problem (λX + Y)z = 0 of twice the dimension and solve the linear problem by the QZ algorithm or a Krylov method. In doing so, it is important to understand the influence of the linearization process on the accuracy and stability of the computed solution. We discuss these issues for three particular linearizations: the standard companion linearization and two linearizations that preserve symmetry in the problem. For illustration we employ a model QEP describing the motion of a beam simply supported at both ends and damped at the midpoint. We show that the above linearizations lead to poor numerical results for the beam problem, but that a two‐parameter scaling proposed by Fan, Lin and Van Dooren cures the instabilities. We also show that half of the eigenvalues of the beam QEP are pure imaginary and are eigenvalues of the undamped problem. Our analysis makes use of recently developed theory explaining the sensitivity and stability of linearizations, the main conclusions of which are summarized. As well as arguing that scaling should routinely be used, we give guidance on how to choose a linearization and illustrate the practical value of condition numbers and backward errors. Copyright © 2007 John Wiley & Sons, Ltd.     

On the solution of eigenvalue problems of laminated non-homogeneous orthotropic circular shells with clamped edges subjected to hydrostatic pressure

This article presents a method to study the free vibration and stability of laminated homogeneous and non-homogeneous orthotropic cylindrical, truncated and complete conical shells of general staking with clamped edges under a hydrostatic pressure. Based on the Love first approximation theory, the basic relations, the modified Donnell-type stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells, the material properties of which vary piecewise continuously in the thickness direction. To solve this problem an unknown parameter λ was included in the approximation functions. Applying Galerkin methods, the buckling pressures and fundamental natural frequencies of laminated homogeneous and non-homogeneous orthotropic conical shells are obtained. The parameter λ which is included in the obtained formulas is obtained from the minimum conditions of critical stresses and frequencies. The different generalized values are obtained for the parameter λ for buckling pressures and frequencies of cylindrical shells, truncated and complete conical shells. The appropriate formulas for single-layer and laminated cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the influences of the degree of non-homogeneity, the number and ordering of layers and the variations of conical shell characteristics on the critical hydrostatic pressure and natural frequencies are investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

Numerical solution of inverse nodal problem with an eigenvalue in the boundary condition

In this study, we consider Sturm–Liouville problem with a boundary condition depending on spectral parameter. We apply the nodal points as input data and calculate the approximate solution of the inverse nodal problem using Chebyshev polynomials of the first kind. Finally, we illustrate the numerical results by providing several examples.     

Proper generalized decomposition solution of the parameterized Helmholtz problem: application to inverse geophysical problems

The identification of the geological structure from seismic data is formulated as an inverse problem. The properties and the shape of the rock formations in the subsoil are described by material and geometric parameters, which are taken as input data for a predictive model. Here, the model is based on the Helmholtz equation, describing the acoustic response of the system for a given wave length. Thus, the inverse problem consists in identifying the values of these parameters such that the output of the model agrees the best with observations. This optimization algorithm requires multiple queries to the model with different values of the parameters. Reduced order models are especially well suited to significantly reduce the computational overhead of the multiple evaluations of the model. In particular, the proper generalized decomposition produces a solution explicitly stating the parametric dependence, where the parameters play the same role as the physical coordinates. A proper generalized decomposition solver is devised to inexpensively explore the parametric space along the iterative process. This exploration of the parametric space is in fact seen as a post‐process of the generalized solution. The approach adopted demonstrates its viability when tested in two illustrative examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Testing complete and compact radial basis functions for solution of eigenvalue problems using the boundary element method with direct integration

This paper analyses the performance of the main radial basis functions in the formulation of the Boundary Element Method (DIBEM). This is an alternative for solving problems modeled by non-adjoint differential operators, since it transforms domain integrals in boundary integrals using radial basis functions. The solution of eigenvalue problem was chosen to performance evaluation. Natural frequencies are calculated numerically using several radial functions and their accuracy is evaluated by comparison with the available analytical solutions and with the Finite Element Method as well. The standard radial basis functions have presented similar performance to compact radial functions, being even slightly superior.     

An implicit meshless scheme for the solution of transient non-linear Poisson-type equations

A meshfree point collocation method is used for the numerical simulation of both transient and steady state non-linear Poisson-type partial differential equations. Particular emphasis is placed on the application of the linearization method with special attention to the lagging of coefficients method and the Newton linearization method. The localized form of the Moving Least Squares (MLS) approximation is employed for the construction of the shape functions, in conjunction with the general framework of the point collocation method. Computations are performed for regular nodal distributions, stressing the positivity conditions that make the resulting system stable and convergent. The accuracy and the stability of the proposed scheme are demonstrated through representative and well-established benchmark problems.     

On the use of space-time finite elements in the solution of elasto-dynamic fracture problems

The use of a discontinuous Galerkin (DG) formulation for the solution of dynamic fracture problems in linear elastic media with and without cohesive zones is explored. The results are compared with closed-form as well as numerical solutions available from the literature. The effectiveness of the space-time finite element method in the study of dynamic fracture problems is demonstrated, especially in those cases in which dynamic fracture occurs along with time discontinuous loading.     

Green''s function method solution of the Reissner''s plate problem

Green's functions can play a significant role in the development of numerical procedures based on the BEM. For that reason the construction of Green's functions and matrices is of great practical importance. In this study, the extension of the Green's function formalism is introduced to Reissner's plate theory, which accounts for the effect of transverse normal stress and transverse shear deformation. The development of Green's matrix for the governing boundary value problem is described based on the separation of variables. The explicit expressions for the entries of Green's matrix are derived. The algorithm is developed for computation of the stress-strain for a rectangular plate. A validation example is presented, illustrating an equal level of accuracy attained for both displacement and stress.