A hierarchical finite element for geometrically non‐linear vibration of doubly curved,moderately thick isotropic shallow shells

A p‐version, hierarchical finite element for doubly curved, moderately thick, isotropic shallow shells is derived and geometrically non‐linear free vibrations of panels with rectangular planform are investigated. The geometrical non‐linearity is due to large displacements, and the effects of the rotatory inertia and transverse shear are considered. The time domain equations of motion are obtained by applying the principle of virtual work and the d'Alembert's principle. These equations are mapped to the frequency domain by the harmonic balance method, and are finally solved by a predictor–corrector method. The convergence properties of the element proposed and the influence of several parameters on the dynamic response are studied. These parameters are the shell's thickness, the width‐to‐length ratio, the curvature‐to‐width ratio and the ratio between curvature radii. The first and higher order modes are analysed. Some results are compared with results published or calculated using a commercial finite element package. It is demonstrated that with the proposed element low‐dimensional, accurate models are obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

A method of finite element tearing and interconnecting and its parallel solution algorithm

A novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented. The spatial domain is partitioned into a set of totally disconnected subdomains, each assigned to an individual processor. Lagrange multipliers are introduced to enforce compatibility at the interface nodes. In the static case, each floating subdomain induces a local singularity that is resolved in two phases. First, the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution. Next, the contributions of these modes are related to the Lagrange multipliers through an orthogonality condition. A parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and Lagrange multipliers, which completes the solution of the problem. When implemented on local memory multiprocessors, this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring. It is also suitable for parallel/vector computers with shared memory. Moreover, unlike parallel direct solvers, it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations.     

A Computational Study of Stabilized, Low-order C 0 Finite Element Approximations of Darcy Equations

We consider finite element methods for the Darcy equations that are designed to work with standard, low order C ⁰ finite element spaces. Such spaces remain a popular choice in the engineering practice because they offer the convenience of simple and uniform data structures and reasonable accuracy. A consistently stabilized method [20] and a least-squares formulation [18] are compared with two new stabilized methods. The first one is an extension of a recently proposed polynomial pressure projection stabilization of the Stokes equations [5,13]. The second one is a weighted average of a mixed and a Galerkin principles for the Darcy problem, and can be viewed as a consistent version of the classical penalty stabilization for the Stokes equations [8]. Our main conclusion is that polynomial pressure projection stabilization is a viable stabilization choice for low order C ⁰ approximations of the Darcy problem.     

Mixed finite element method for generalized convection-diffusion equations based on an implicit characteristic-based algorithm

Summary A mixed finite element method for generalized convection-diffusion equations is proposed. The primitive variable with its spatial gradient and the diffusion flux are interpolated as independent variables. The variational (weak) form of the governing equations is given on the basis of the extended Hu-Washizu three-field variational principle. The mixed element is formulated with stabilized one point quadrature scheme and particularly implicit characteristic-based algorithm for eliminating spurious numerical oscillations. The numerical results illustrate good performances in accuracy and efficiency of the proposed mixed element in comparison with standard finite element.     

Slope stability analysis based on elasto‐plastic finite element method

The paper deals with two essential and related closely processes involved in the finite element slope stability analysis in two‐dimensional problems, i.e. computation of the factors of safety (FOS) and location of the critical slide surfaces. A so‐called ?–v inequality, sin ??1 – 2v is proved for any elasto‐plastic material satisfying Mohr–Coulomb's yield criterion. In order to obtain an FOS of high precision with less calculation and a proper distribution of plastic zones in the critical equilibrium state, it is stated that the Poisson's ratio v should be adjusted according to the principle that the ?–v inequality always holds as reducing the strength parameters c and ?. While locating the critical slide surface represented by the critical slide line (CSL) under the plane strain condition, an initial value problem of a system of ordinary differential equations defining the CSL is formulated. A robust numerical solution for the initial value problem based on the predictor–corrector method is given in conjunction with the necessary and sufficient condition ensuring the convergence of solution. A simple example, the kinematic solution of which is available, and a challenging example from a hydraulic project in construction are analysed to demonstrate the effectiveness of the proposed procedures. Copyright © 2005 John Wiley & Sons, Ltd.     

A parallel domain decomposition BEM for diffusion problems with surface reactions

A parallel domain decomposition boundary element method (BEM) is developed for the solution of three-dimensional multispecies diffusion problems. The chemical species are uncoupled in the interior of the domain but couple at the boundary through a nonlinear surface reaction equation. The method of lines is used whereby time is discretized using the finite difference method and space is discretized using the boundary element method. The original problem is transformed into a sequence of nonhomogeneous modified Helmholtz equations. A Schwarz Neumann–Neumann iteration scheme is used to satisfy interfacial boundary conditions between subdomains. A segregated solver based on a quasi-predictor–corrector time integrator is used to satisfy the nonlinear boundary conditions on the reactive surfaces. The accuracy and parallel efficiency of the method is demonstrated through a benchmark problem.     

A gradient smoothing method (GSM) with directional correction for solid mechanics problems

A novel gradient smoothing method (GSM) is proposed in this paper, in which a gradient smoothing together with a directional derivative technique is adopted to develop the first- and second-order derivative approximations for a node of interest by systematically computing weights for a set of field nodes surrounding. A simple collocation procedure is then applied to the governing strong-from of system equations at each node scattered in the problem domain using the approximated derivatives. In contrast with the conventional finite difference and generalized finite difference methods with topological restrictions, the GSM can be easily applied to arbitrarily irregular meshes for complex geometry. Several numerical examples are presented to demonstrate the computational accuracy and stability of the GSM for solid mechanics problems with regular and irregular nodes. The GSM is examined in detail by comparison with other established numerical approaches such as the finite element method, producing convincing results.     

A unified dual‐primal finite element tearing and interconnecting approach for incompressible Stokes equations

A unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI‐DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI‐DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two‐dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI‐DP algorithms represented under the same framework.Copyright © 2012 John Wiley & Sons, Ltd.     

Stochastic inverse heat conduction using a spectral approach

An adjoint‐based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data, process conditions and measurement noise. The ill‐posed stochastic inverse problem is restated as a conditionally well‐posed L₂ optimization problem. The gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field. The L₂ optimization problem is solved using a conjugate‐gradient approach. Accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems. Copyright © 2004 John Wiley & Sons, Ltd.     

A boundary integral approach for plane analysis of electrically semi-permeable planar cracks in a piezoelectric solid

A plane electro-elastostatic problem involving arbitrarily located planar stress free cracks which are electrically semi-permeable is considered. Through the use of the numerical Green's function for impermeable cracks, the problem is formulated in terms of boundary integral equations which are solved numerically by a boundary element procedure together with a predictor–corrector method. The crack tip stress and electric displacement intensity factors can be easily computed once the boundary integral equations are properly solved.     

ODDLS: A new unstructured mesh finite element method for the analysis of free surface flow problems

This paper introduces a new stabilized finite element method based on the finite calculus (Comput. Methods Appl. Mech. Eng. 1998; 151 :233–267) and arbitrary Lagrangian–Eulerian techniques (Comput. Methods Appl. Mech. Eng. 1998; 155 :235–249) for the solution to free surface problems. The main innovation of this method is the application of an overlapping domain decomposition concept in the statement of the problem. The aim is to increase the accuracy in the capture of the free surface as well as in the resolution of the governing equations in the interface between the two fluids. Free surface capturing is based on the solution to a level set equation. The Navier–Stokes equations are solved using an iterative monolithic predictor–corrector algorithm (Encyclopedia of Computational Mechanics. Wiley: New York, 2004), where the correction step is based on imposing the divergence‐free condition in the velocity field by means of the solution to a scalar equation for the pressure. Examples of application of the ODDLS formulation (for overlapping domain decomposition level set) to the analysis of different free surface flow problems are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Matrix differential equation and higher-order numerical methods for problems of non-linear creep,viscoelasticity and elasto-plasticity

The constitutive equation is assumed in a very general form which includes as special cases non-linear creep, incremental elasto-plasticity as well as viscoelasticity represented by a chain of n standard solid models. Subdividing the structure into N finite elements, the problem of structural analysis is formulated with a system of 6N(n + 1) ordinary non-linear first-order differential equations in terms of the components of stresses and strains in the elements. This formulation enables one to apply Runge–Kutta methods or the predictor–corrector methods.     

The variational formulation and solution of problems of finite-strain elastoplasticity based on the use of a dissipation function

The solution of initial-boundary value problems involving finite elastoplastic deformations is discussed. The formulation considered differs from conventional formulations in that the evolution law is expressed in terms of the dissipation function. A generalized midpoint rule is used to obtain an incremental problem, a variational form of which is derived. The finite element method is used for spatial discretization, and an algorithm to solve the resulting discrete problem is developed. This algorithm has the predictor–corrector structure common to most solution procedures for problems in plasticity. Methods for imposing the plastic incompressibility constraint are investigated. Solutions to two axisymmetric examples obtained using the proposed algorithm are presented and compared with those obtained by other authors.     

New predictor/corrector algorithms with improved energy balance for a recent formulation of dynamic vehicle/structure interaction

A new class of predictor/corrector algorithms is proposed to solve the complex system of differential equations that arises from a Galerkin spatial discretization of the equations of motion in a recent formulation of dynamic vehicle/structure interaction. The applicability of the concept of a building-block vehicle/structure interaction model developed in our previous work—where the vehicle nominal motion is not prescribed a priori, but is part of the unknown motion of the system—is demonstrated through the construction of a simple vehicle model. In the new algorithms, the presence of the accelerations of the vehicle component is eliminated in the predictor structural equations, making these equations different from the corrector structural equations. The special treatment of the predicted axial motion that provides an artificial damping to eliminate unstable oscillations in the numerical results as proposed in the old algorithms is avoided. Accurate results from numerical simulations using the new algorithms are obtained, and there are no unstable oscillations that were observed in some other predictor/corrector schemes. The system energy balance is also better preserved compared with the old algorithms.     

A Legendre spectral element method for the Stefan problem

In this paper we present a Legendre spectral element method for solution of multi-dimensional unsteady change-of-phase Stefan problems. The spectral element method is a high-order (p-type) finite element technique, in which the computational domain is broken up into general (curved) quadrilateral macroelements, and the solution, data and geometry are expanded within each element in terms of tensor-product Lagrangian interpolants. The discrete equations are generated by a Galerkin formulation followed by Gauss–Lobatto Legendre quadrature, for which it is shown that exponential convergence to smooth solutions is obtained as the polynomial order of fixed elements is increased. The spectral element equations are inverted by conjugate gradient iteration, in which the matrix-vector products are calculated efficiently using tensor-product sum-factorization. To solve the Stefan problem numerically, the heat equations in the liquid and solid phases are transformed to fixed domains applying an interface-local time-dependent immobilization transformation technique. The modified heat equations are discretized using finite differences in time, resulting at each time step in a Helmholtz equation in space that is solved using Legendre spectral element elliptic discretizations. The new interface position is then computed using a variationally consistent flux treatment along the phase boundary, and the solution is projected upon the corresponding updated mesh. The rapid convergence rate and stability of the method are discussed, and numerical results are presented for a one-dimensional Stefan problem using both a semi-implicit and a fully implicit time-stepping scheme. Finally, a two-dimensional Stefan problem with a complex phase boundary is solved using the semi-implicit scheme.     

A block iterative finite element algorithm for numerical solution of the steady–state,compressible Navier–Stokes equations

An iterative method for numerically solving the time independent Navier–Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss–Seidel principle in block form to the systems of the non-linear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C⁰-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.     

Dual‐grid‐based tree/cotree decomposition of higher‐order interpolatory H(∇∧, Ω) basis

This work extends the zeroth‐order tree/cotree (TC) decomposition method into higher order (HO) interpolatory elements and develops the constraints operator required for the elimination of spurious solutions for general HO spectral basis. Earlier methods explicitly enforce the divergence condition that requires a mixed finite element (FE) formulation with both H¹ and H(?∧) expansions and involves repeated solutions of the Poisson equation. A recent approach, which avoids the mixed formulation and the Poisson problem, uses TC decomposition of edge DoF over the primal graph and construction of integration and gradient matrices. The approach is easily applied to HO hierarchical elements but becomes quite complex for HO spectral elements. In the presence of internal DoF, it is difficult to utilize the primal graph for an explicit decomposition of the spectral DoF. In contrast, this work utilizes the dual grid, resulting in an explicit decomposition of DoF and construction of constraint equations from a fixed element matrix. Thus, mixed formulation and the Poisson problems are avoided while eliminating the need for evaluation of integration and gradient matrices. The proposed constraints matrix is element‐geometry independent and possesses an explicit sparsity formulation reducing the need for dynamic memory allocation. Numerical examples are included for verification. Copyright © 2010 John Wiley & Sons, Ltd.     

Treatment of internal constraints by mixed finite element methods: Unification of concepts

A general method to generate assumed stress and strain fields within the context of mixed finite element methods is presented. The assumed fields are constructed in such a way that internal constraints are satisfied a priori. Consequently, the locking behaviour commonly observed in finite element solutions of problems with internal constraints is avoided. To this end, the assumed stress and strain fields are constructed to satisfy a priori the homogeneous part of the equilibrium equations, thus avoiding Fraeijs de Veubeke's limitation principle. Results obtained using the proposed methodology on a nearly incompressible plane strain problem and thin plate application using a shear deformable theory are indicated.     

A multiplicative finite strain finite element framework for the modelling of semicrystalline polymers and polycarbonates

This paper presents a phenomenological model for the simulation and analysis of stress‐induced orientational hardening in semicrystalline polymers and polycarbonates at finite strains. The notion of intermediate (local) stress‐free configuration is used to develop a set of constitutive equations, and its relation to the multiple natural (stress‐free) configurations in the class of materials being considered here is discussed. A hyperelastic stored energy function, written with respect to the intermediate stress‐free configuration is presented to model the finite elastic response. It is then combined with the J₂‐flow theory to model the finite inelastic response. The isochoric constraint during inelastic deformation is treated via an exact multiplicative decomposition of the deformation gradient into volume‐preserving and spherical parts. The numerical solution algorithm is based on the use of operator splitting technique that results in a product formula algorithm with elastic‐predictor/inelastic‐corrector components. Numerical results are presented to show the behaviour of the model. Copyright © 2000 John Wiley & Sons, Ltd.