A BEM based domain decomposition method for nonlinear analysis of elastic space membranes

In this paper the Domain Decomposition Method (DDM) is developed for nonlinear analysis of both flat and space elastic membranes of complicated geometry which may have holes. The domain of the projection of the membrane on the xy plane is decomposed into non-overlapping subdomains and the membrane problem is solved sequentially in each subdomain starting from zero displacements on the virtual boundaries. The procedure is repeated until the traction continuity conditions are also satisfied on the virtual boundaries. The membrane problem in each subdomain is solved using the Analog Equation Method (AEM). According to this method the three coupled strongly nonlinear partial differential equations, governing the response of the membrane, are replaced by three uncoupled linear membrane equations (Poisson's equations) subjected to fictitious sources under the same boundary conditions. The fictitious sources are established using a meshless BEM procedure. Example problems are presented, for both flat and space membranes, which illustrate the method and demonstrate its efficiency and accuracy.     

Ponding on floating membranes

Membranes subjected to ponding loads and floating on a liquid are analyzed. The initially flat membrane, which may be prestressed by edge in-plane tractions or displacements, is subjected to the weight of a liquid (e.g. rain water) filling the space created by the deflection of the membrane. Large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious sources. The problem is strongly nonlinear. In addition to the geometrical nonlinearity, the ponding problem is itself nonlinear, because the ponding load and the liquid reaction are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed which converge to the equilibrium state of the membrane. Example problems are presented, which illustrate the method and demonstrate its efficiency. The method has all the advantages of the pure BEM.     

Dynamic analysis of nonlinear membranes by the analog equation method: a boundary-only solution

 A boundary-only solution is presented for dynamic analysis of elastic membranes under large deflections. The solution procedure is based on the analog equation method (AEM). According to this method, the three coupled nonlinear second order hyperbolic partial differential equations in terms of displacements, which govern the response of the membrane, are replaced with three Poisson's quasi-static equations under fictitious time dependent sources. The fictitious sources are established using a BEM-based procedure and the displacements as well as the stress resultants at any point are evaluated from their integrals representations. Numerical examples are presented which illustrate the method. Received 16 December 2000 / Accepted 25 April 2002     

Nonlinear analysis of elastic space cable-supported membranes

In this paper a solution method is presented for the coupled problem of elastic flat or space membranes supported by elastic flexible cables. Both membrane and cable undergo large deflections. Starting from the minimal surface the membrane is prestressed by imposed boundary displacements under the self-weight. Then an iterative procedure is employed, which consists in solving the membrane and the cable large deflection problems separately in each iteration step and checking the continuity of displacements and forces between membrane and cable. The procedure is repeated until convergence is achieved. Both membrane and cable problems are solved using the analog equation method (AEM). The displacements as well as the stress resultants are evaluated at any point of the membrane and the cable from the integral representations of the solution of the analog equations, which are used as mathematical formulae. Example problems are presented, for both flat and space membranes, which illustrate the method and demonstrate its efficiency and accuracy.     

Post-buckling analysis of viscoelastic plates with fractional derivative models

The post-buckling response of thin plates made of linear viscoelastic materials is investigated. The employed viscoelastic material is described with fractional order time derivatives. The governing equations, which are derived by considering the equilibrium of the plate element, are three coupled nonlinear fractional partial evolution type differential equations in terms of three displacements. The nonlinearity is due to nonlinear kinematic relations based on the von Kármán assumption. The solution is achieved using the analog equation method (AEM), which transforms the original equations into three uncoupled linear equations, namely a linear plate (biharmonic) equation for the transverse deflection and two linear membrane (Poisson’s) equations for the inplane deformation under fictitious loads. The resulting initial value problem for the fictitious sources is a system of nonlinear fractional ordinary differential equations, which is solved using the numerical method developed recently by Katsikadelis for multi-term nonlinear fractional differential equations. The numerical examples not only demonstrate the efficiency and validate the accuracy of the solution procedure, but also give a better insight into this complicated but very interesting engineering plate problem     

Geometrically nonlinear analysis of elastic membranes with embedded rigid inclusions

In this paper the deformation of membranes containing rigid inclusions is analyzed. These rigid inclusions can significantly change the entire stress distribution in the membrane and therefore create major difficulties for the design. The initially flat membrane, which may be prestretched by boundary in-plane tractions or displacements, is subjected to externally applied loads and to the weight of the rigid inclusions. The composite system is examined in cases where its deformation reaches a state for which the undeformed and deformed shapes are substantially different. In such cases large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious domain source densities. The problem is strongly nonlinear [Katsikadelis JT, Nerantzaki MS. The boundary element method for nonlinear problems. Eng Anal Boundary Elements 1999;23:365–73]. In addition to the geometrical nonlinearity, the problem is itself nonlinear, since the membrane's reactions on the boundary of the rigid inclusions are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed for calculation of deformed membrane's configuration, which converge to the final equilibrium state of the membrane with the given external applied loads. Several example problems are presented, which illustrate the method and demonstrate its accuracy and efficiency. The method employed for the solution is boundary only with all the advantages of the pure BEM.     

The analog equation method for large deflection analysis of membranes. A boundary-only solution

 In this paper the analog equation method (AEM) is applied to nonlinear analysis of elastic membranes with arbitrary shape. In this case the transverse deflections influence the inplane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear. The present formulation, being in terms of the three displacements components, permits the application of geometrical inplane boundary conditions. The membrane is prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of the membranes. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary. Received 21 November 2000     

The ponding problem on elastic membranes: an analog equation solution

 In this paper, the nonlinear response of elastic membranes with arbitrary shape under partial and full ponding loads has been analyzed. Large deflections are considered, which result from nonlinear kinematic relations. The problem is formulated in terms of the displacements components and the three coupled nonlinear governing equations are solved using the analog equation method (AEM). The membrane may be prestressed either by prescribed boundary displacements or tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants at any point of the membrane are evaluated from their integral representations. In addition to the geometrical nonlinearity, the ponding problem is itself nonlinear, because the ponding load depends on the deflection surface that it produces. Iterative schemes are developed which converge to the equilibrium state of the membrane under the ponding loads. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary. Received 28 July 2001     

Exact static element stiffness matrices of non‐symmetric thin‐walled curved beams

An evaluation procedure of exact static stiffness matrices for curved beams with non‐symmetric thin‐walled cross section are rigorously presented for the static analysis. Higher‐order differential equations for a uniform curved beam element are first transformed into a set of the first‐order simultaneous ordinary differential equations by introducing 14 displacement parameters where displacement modes corresponding to zero eigenvalues are suitably taken into account. This numerical technique is then accomplished via a generalized linear eigenvalue problem with non‐symmetric matrices. Next, the displacement functions of displacement parameters are exactly calculated by determining general solutions of simultaneous non‐homogeneous differential equations. Finally an exact stiffness matrix is evaluated using force–deformation relationships. In order to demonstrate the validity and effectiveness of this method, displacements and normal stresses of cantilever thin‐walled curved beams subjected to tip loads are evaluated and compared with those by thin‐walled curved beam elements as well as shell elements. Copyright © 2004 John Wiley & Sons, Ltd.     

State space solution of non-axisymmetric Biot consolidation problem for multilayered porous media

In this paper, the non-axisymmetric Biot consolidation problem for multilayered porous media is studied. Taking stresses, pore pressure and displacements at layer interfaces as basic unknown functions, two sets of partial differential equations, which are independent each other, are formulated. Using Fourier expansion, Laplace transforms and Hankel transforms with respect to the circumferential, time and radial coordinates, respectively, the partial differential equations presented are reduced to the ordinary differential equations. Transfer matrices describing the transfer relation between the state vectors for a finite layer are derived explicitly in the transform space. Using the transfer matrices presented, three cases are studied for the lower surface: (1) permeable rough rigid base, (2) impermeable rough rigid base, and (3) poroelastic half space. The explicit solution in the transform space is presented. Considering the continuity condition at layer interfaces, the solutions of the non-axisymmetric Biot consolidation problems for multilayered semi-infinite porous media are presented in the integral form. The time histories of displacements, stresses and pore pressure are obtained by solving a linear equation system for discrete values of Laplace-Hankel transform inversions.     

Analysis of cylindrical shell panels. A meshless solution

The Meshless Analog Equation Method, a purely meshless method, is applied to the static analysis of cylindrical shell panels. The method is based on the concept of the analog equation of Katsikadelis, which converts the three governing partial differential equations in terms of displacements into three substitute equations, two of second order and one fourth order, under fictitious sources. The fictitious sources are represented by series of radial basis functions of multiquadric type. Thus the substitute equations can be directly integrated. This integration allows the representation of the sought solution by new radial basis functions, which approximate accurately not only the displacements but also their derivatives involved in the governing equations. This permits a strong formulation of the problem. Thus, inserting the approximate solution in the differential equations and in the associated boundary conditions and collocating at a predefined set of mesh-free nodal points, a system of linear equations is obtained, which gives the expansion coefficients of radial basis functions series that represent the solution. The minimization of the total potential of the shell results in the optimal choice of the shape parameter of the radial basis functions. The method is illustrated by analyzing several shell panels. The studied examples demonstrate the efficiency and the accuracy of the presented method.     

The boundary element method applied to a moving free boundary problem

In this paper a boundary problem is considered for which the boundary is to be determined as part of the solution. A time‐dependent problem involving linear diffusion in two spatial dimensions which results in a moving free boundary is posed. The fundamental solution is introduced and Green’s Theorem is used to yield a non‐linear system of integral equations for the unknown solution and the location of the boundary. The boundary element method is used to obtain a numerical solution to this system of integral equations which in turn is used to obtain the solution of the original problem. Graphical results for a two‐dimensional problem are presented. Published in 1999 by John Wiley & Sons, Ltd.     

A 3D finite element approach for the coupled numerical simulation of electrochemical systems and fluid flow

A comprehensive finite element method for three‐dimensional simulations of stationary and transient electrochemical systems including all multi‐ion transport mechanisms (convection, diffusion and migration) is presented. In addition, non‐linear phenomenological electrode kinetics boundary conditions are accounted for. The governing equations form a set of coupled non‐linear partial differential equations subject to an algebraic constraint due to the electroneutrality condition. The advantage of a convective formulation of the ion‐transport equations with respect to a natural application of homogeneous flux boundary conditions is emphasized. For one of the numerical examples, an analytical solution for the coupled problem is provided, and it is demonstrated that the proposed computational approach is robust and provides accurate results. Copyright © 2011 John Wiley & Sons, Ltd.     

A semi‐analytical locally transversal linearization method for non‐linear dynamical systems

An numeric‐analytical, implicit and local linearization methodology, called the locally transversal linearization (LTL), is developed in the present paper for analyses and simulations of non‐linear oscillators. The LTL principle is based on deriving the locally linearized equations in such a way that the tangent space of the linearized equations transversally intersects that of the given non‐linear dynamical system at that particular point in the state space where the solution vector is sought. For purposes of numerical implementation, two different numerical schemes, namely LTL‐1 and LTL‐2 schemes, based on the LTL methodology are presented. Both LTL‐1 and LTL‐2 procedures finally reduce the given set of non‐linear ordinary differential equations (ODEs) to a set of transcendental algebraic equations valid over a short interval of time or over a short segment of the evolving trajectories as projected on the phase space. While in the LTL‐1 scheme the desired solution vector at a forward time point enters the linearized differential equations as an unknown parameter, in the LTL‐2 scheme a set of unknown residues enters the linearized system as parameters. A limited set of examples involving a few well‐known single‐degree‐of‐freedom (SDOF) non‐linear oscillators indicate that the LTL methodology is capable of accurately predicting many complicated non‐linear response patterns, including limit cycles, quasi‐periodic orbits and even strange attractors. Copyright © 2001 John Wiley & Sons, Ltd.     

Nonlinear dynamic analysis of heterogeneous orthotropic membranes by the analog equation method

In this paper the analog equation method, a BEM-based method, is employed to analyze the dynamic response of flat heterogeneous orthotropic membranes of arbitrary shape, undergoing large deflections. The problem is formulated in terms of the three displacement components. Due to the heterogeneity of the membrane, the elastic constants are position dependent and consequently the coefficients of the partial differential equations governing the dynamic equilibrium of the membrane are variable. Using the concept of the analog equation, the three-coupled nonlinear second order hyperbolic partial differential equations are replaced with three uncoupled Poisson's quasi-static equations with fictitious time dependent sources. The fictitious sources are represented by radial basis functions series and are established using a BEM-based procedure. Both free and forced vibrations are considered. Membranes of various shapes are analyzed to illustrate the merits of the method as well as its applicability, efficiency and accuracy. The proposed method is boundary-only in the sense that the discretization and the integration are restricted on the boundary. Therefore, it maintains all the advantages of the pure BEM.     

Three‐dimensional coupled heat,moisture, and air transfer in a deformable unsaturated soil

A three‐dimensional numerical model is presented for three‐phase flow (moisture, air, and heat) in a deformable partly saturated soil with deformation calculated via a non‐linear elastic theory. The present work is an extension of a two‐dimensional analysis presented by Thomas and He. The objective of this work is the solution of problems of greater geometric complexity. The mathematical formulation of this coupled problem consists of four governing equations, developed from the principles of mass and energy conservations as well as the stress equilibrium equation. Darcy's flow law is used to describe the motion of liquid and air in the porous medium, and a Philip and de Vries type vapour flow approach is employed in the formulation. A Galerkin finite element method coupled with a finite difference recurrence relationship is used to obtain simultaneous solutions to the governing equations where pore liquid, pore air pressures, temperature and displacements are the primary variables. The method allows the non‐linear nature of the soil parameters to be modelled. Three‐dimensional 20‐noded isoparametric elements are used to simulate different types of cases for the verification of the work. Results are presented of the application of the new model to four problems, two of which are isothermal and two heating simulations. The three‐dimensional nature of the results achieved is highlighted. Copyright © 1999 John Wiley & Sons, Ltd.     

The analog equation method for large deflection analysis of heterogeneous orthotropic membranes: a boundary-only solution

In this paper, the analog equation method (AEM) is applied to nonlinear analysis of heterogeneous orthotropic membranes with arbitrary shape. In this case, the transverse deflections influence the in-plane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear with variable coefficients. The present formulation, being in terms of the three displacement components, permits the application of geometrical in-plane boundary conditions. The membrane may be prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation, the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method, and demonstrate its efficiency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of heterogeneous anisotropic membranes. The method has all the advantages of the pure BEM, since the discretization and integration are limited only to the boundary.     

Non‐linear analysis of thin‐walled members of open cross‐section

The paper presents a means of determining the non‐linear stiffness matrices from expressions for the first and second variation of the Total Potential of a thin‐walled open section finite element that lead to non‐linear stiffness equations. These non‐linear equations can be solved for moderate to large displacements. The variations of the Total Potential have been developed elsewhere by the authors, and their contribution to the various non‐linear matrices is stated herein. It is shown that the method of solution of the non‐linear stiffness matrices is problem dependent. The finite element procedure is used to study non‐linear torsion that illustrates torsional hardening, and the Newton–Raphson method is deployed for this study. However, it is shown that this solution strategy is unsuitable for the second example, namely that of the post‐buckling response of a cantilever, and a direct iteration method is described. The good agreement for both of these problems with the work of independent researchers validates the non‐linear finite element method of analysis. Copyright © 1999 John Wiley & Sons, Ltd.     

On the modelling of incompressibility in linear and non‐linear elasticity with the master–slave approach

The master–slave approach is adapted to model the kinematic constraints encountered in incompressibility. The method presented here allows us to obtain discrete displacement and pressure fields for arbitrary finite element formulations that have discontinuous pressure interpolations. The resulting displacements satisfy exactly the incompressibility constraints in a weak sense, and are obtained by solving a system of equations with the minimum (independent) degrees of freedom. In linear analysis, the method reproduces the well‐known stability results for inf–sup compliant elements, and permits to compute the pressure modes (physical or spurious) when they exist. By rewriting the equilibrium equations of a hyperelastic material, the method is extended to non‐linear elasticity, while retaining the exact fulfilment of the incompressibility constraints in a weak sense. Problems with analytical solution in two and three dimensions are tested and compared with other solution methods. Copyright © 2007 John Wiley & Sons, Ltd.     

Effects of nonlinear strain components on the buckling response of stiffened shear-deformable composite plates

A detailed investigation of the weight of each non linear term of the Green–Lagrange strain displacement equation is presented, with reference to the buckling of orthotropic, both flat and prismatic, Mindlin plates. Usually in the literature, in buckling analysis only the second order terms related to the out-of-plane displacement are considered. Such heuristic simplification, known as von Kármán hypothesis, starts by the consideration that the buckling mode of a flat plate is described by dominant out-of-plane displacement and disregards the non-linear terms of the Green–Lagrange strain tensor depending on the in plane displacement components, whose role is confined to first order, say pre-critical, deformation. The present paper shows that disregarding the non linear terms related to the in-plane strain–displacement is equivalent to neglect shear induced rotation. In the work, the governing equations are derived using the principle of strain energy minimum and the differential equations solution is gained by using the general Levy-type method. The obtained results show that the von Kármán model overestimates the critical load when, in buckling mode, magnitudes of shear rotation, in-plane and out-of-plane displacements are comparable.