The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.     

Efficient simulation of multi‐body contact problems on complex geometries: A flexible decomposition approach using constrained minimization

We consider the numerical simulation of non‐linear multi‐body contact problems in elasticity on complex three‐dimensional geometries. In the case of warped contact boundaries and non‐matching finite element meshes, particular emphasis has to be put on the discretization of the transmission of forces and the non‐penetration conditions at the contact interface. We enforce the discrete contact constraints by means of a non‐conforming domain decomposition method, which allows for optimal error estimates. Here, we develop an efficient method to assemble the discrete coupling operator by computing the triangulated intersection of opposite element faces in a locally adjusted projection plane but carrying out the required quadrature on the faces directly. Our new element‐based algorithm does not use any boundary parameterizations and is also suitable for isoparametric elements. The emerging non‐linear system is solved by a monotone multigrid method of optimal complexity. Several numerical examples in 3D illustrate the effectiveness of our approach. Copyright © 2008 John Wiley & Sons, Ltd.     

Remarks on the interpretation of current non‐linear finite element analyses as differential–algebraic equations

For the numerical solution of materially non‐linear problems like in computational plasticity or viscoplasticity the finite element discretization in space is usually coupled with point‐wise defined evolution equations characterizing the material behaviour. The interpretation of such systems as differential–algebraic equations (DAE) allows modern‐day integration algorithms from Numerical Mathematics to be efficiently applied. Especially, the application of diagonally implicit Runge–Kutta methods (DIRK) together with a Multilevel‐Newton method preserves the algorithmic structure of current finite element implementations which are based on the principle of virtual displacements and on backward Euler schemes for the local time integration. Moreover, the notion of the consistent tangent operator becomes more obvious in this context. The quadratical order of convergence of the Multilevel‐Newton algorithm is usually validated by numerical studies. However, an analytical proof of this second order convergence has already been given by authors in the field of non‐linear electrical networks. We show that this proof can be applied in the current context based on the DAE interpretation mentioned above. We finally compare the proposed procedure to several well‐known stress algorithms and show that the inclusion of a step‐size control based on local error estimations merely requires a small extra time‐investment. Copyright © 2001 John Wiley & Sons, Ltd.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

An adaptive fixed mesh method for modelling and simulating of unsteady two‐phase flows with sharp physical interfaces

We present in this paper a new computational method for simulation of two‐phase flow problems with moving boundaries and sharp physical interfaces. An adaptive interface‐capturing technique (ICT) of the Eulerian type is developed for capturing the motion of the interfaces (free surfaces) in an unsteady flow state. The adaptive method is mainly based on the relative boundary conditions of the zero pressure head, at which the interface is corresponding to a free surface boundary. The definition of the free surface boundary condition is used as a marker for identifying the position of the interface (free surface) in the two‐phase flow problems. An initial‐value‐problem (IVP) partial differential equation (PDE) is derived from the dynamic conditions of the interface, and it is designed to govern the motion of the interface in time. In this adaptive technique, the Navier–Stokes equations written for two incompressible fluids together with the IVP are solved numerically over the flow domain. An adaptive mass conservation algorithm is constructed to govern the continuum of the fluid. The finite element method (FEM) is used for the spatial discretization and a fully coupled implicit time integration method is applied for the advancement in time. FE‐stabilization techniques are added to the standard formulation of the discretization, which possess good stability and accuracy properties for the numerical solution. The adaptive technique is tested in simulation of some numerical examples. With the test problems presented here, we demonstrated that the adaptive technique is a simple tool for modelling and computation of complex motion of sharp physical interfaces in convection–advection‐dominated flow problems. We also demonstrated that the IVP and the evolution of the interface function are coupled explicitly and implicitly to the system of the computed unknowns in the flow domain. Copyright © 2005 John Wiley & Sons, Ltd.     

Calculation of strict error bounds for finite element approximations of non‐linear pointwise quantities of interest

This paper deals with the verification of simulations performed using the finite element method. More specifically, it addresses the calculation of strict bounds on the discretization errors affecting pointwise outputs of interest which may be non‐linear with respect to the displacement field. The method is based on classical tools, such as the constitutive relation error and extraction techniques associated with the solution of an adjoint problem. However, it uses two specific and innovative techniques: the enrichment of the adjoint solution using a partition of unity method, which enables one to consider truly pointwise quantities of interest, and the decomposition of the non‐linear quantities of interest by means of projection properties in order to take into account higher‐order terms in establishing the bounds. Thus, no linearization is performed and the property that the local error bounds are guaranteed is preserved. The effectiveness of the approach and the quality of the bounds are illustrated with two‐dimensional applications in the context of elastic fatigue problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Shifted FSAI preconditioners for the efficient parallel solution of non‐linear groundwater flow models

The present paper investigates the performance of a shifted factorized sparse approximate inverse as a parallel preconditioner for the iterative solution to the linear systems arising in the finite element discretization of non‐linear groundwater flow models. The shift strategy is based on an inexpensive preconditioner update exploiting the structure of the coefficient matrix. The proposed algorithm is experimented with in the parallel simulation of a large‐scale real multi‐aquifer system characterized by a stochastic distribution of the hydraulic conductivity. The numerical results show that the shifted factorized sparse approximate inverse algorithm may yield an overall computational gain up to 300% with respect to the non‐shifted scheme with an excellent parallel efficiency. Copyright © 2011 John Wiley & Sons, Ltd.     

Conservation properties of a time FE method—part II: Time‐stepping schemes for non‐linear elastodynamics

In the present paper one‐step implicit integration algorithms for non‐linear elastodynamics are developed. The discretization process rests on Galerkin methods in space and time. In particular, the continuous Galerkin method applied to the Hamiltonian formulation of semidiscrete non‐linear elastodynamics lies at the heart of the time‐stepping schemes. Algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals. We newly introduce the ‘assumed strain method in time’ which enables the design of energy–momentum conserving schemes and which can be interpreted as temporal counterpart of the well‐established assumed strain method for finite elements in space. The numerical examples deal with quasi‐rigid motion as well as large‐strain motion. Copyright © 2001 John Wiley & Sons, Ltd.     

Energy conserving and dissipative time finite element schemes for N‐body stiff problems

Petrov–Galerkin finite element method is adopted to develop a family of temporal integrators, which preserves the feature of energy conservation or numerical dissipation for non‐linear N‐body dynamical systems. This leads to an enhancement of numerical stability and the integrators may therefore offer some advantage for the numerical solution of stiff systems in long‐term simulations. Dynamically tuneable numerical integration is exploited to improve the accuracy of the time‐stepping schemes. Representative simulations for simple non‐linear systems show the performance of the schemes in controlling over or damping out unresolved high frequencies. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamic inflation of non‐linear elastic and viscoelastic rubber‐like membranes

The present paper deals with the dynamic inflation of rubber‐like membranes.The material is assumed to obey the hyperelastic Mooney's model or the non‐linear viscoelastic Christensen's model. The governing equations of free inflation are solved by a total Lagrangian finite element method for the spatial discretization and an explicit finite‐difference algorithm for the time‐integration scheme. The numerical implementation of constitutive equations is highlighted and the special case of integral viscoelastic models is examined in detail. The external force consists in a gas flow rate, which is more realistic than a pressure time history. Then, an original method is used to calculate the pressure evolution inside the bubble depending on the deformation state. Our numerical procedure is illustrated through different examples and compared with both analytical and experimental results. These comparisons yield good agreement and show the ability of our approach to simulate both stable and unstable large strain inflations of rubber‐like membranes. Copyright © 2001 John Wiley & Sons, Ltd.     

c‐Type method of unified CAMG and FEA. Part 1: Beam and arch mega‐elements—3D linear and 2D non‐linear

Computer‐aided mesh generation (CAMG) dictated solely by the minimal key set of requirements of geometry, material, loading and support condition can produce ‘mega‐sized’, arbitrary‐shaped distorted elements. However, this may result in substantial cost saving and reduced bookkeeping for the subsequent finite element analysis (FEA) and reduced engineering manpower requirement for final quality assurance. A method, denoted as c‐type, has been proposed by constructively defining a finite element space whereby the above hurdles may be overcome with a minimal number of hyper‐sized elements. Bezier (and de Boor) control vectors are used as the generalized displacements and the Bernstein polynomials (and B‐splines) as the elemental basis functions. A concomitant idea of coerced parametry and inter‐element continuity on demand unifies modelling and finite element method. The c‐type method may introduce additional control, namely, an inter‐element continuity condition to the existing h‐type and p‐type methods. Adaptation of the c‐type method to existing commercial and general‐purpose computer programs based on a conventional displacement‐based finite element method is straightforward. The c‐type method with associated subdivision technique can be easily made into a hierarchic adaptive computer method with a suitable a posteriori error analysis. In this context, a summary of a geometrically exact non‐linear formulation for the two‐dimensional curved beams/arches is presented. Several beam problems ranging from truly three‐dimensional tortuous linear curved beams to geometrically extremely non‐linear two‐dimensional arches are solved to establish numerical efficiency of the method. Incremental Lagrangian curvilinear formulation may be extended to overcome rotational singularity in 3D geometric non‐linearity and to treat general material non‐linearity. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite element implementation of non‐linear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control

Implicit stress integration algorithms have been demonstrated to provide a robust formulation for finite element analyses in computational mechanics, but are difficult and impractical to apply to increasingly complex non‐linear constitutive laws. This paper discusses the performance of fully explicit local and global algorithms with automatic error control used to integrate general non‐linear constitutive laws into a non‐linear finite element computer code. The local explicit stress integration procedure falls under the category of return mapping algorithm with standard operator split and does not require the determination of initial yield or the use of any form of stress adjustment to prevent drift from the yield surface. The global equations are solved using an explicit load stepping with automatic error control algorithm in which the convergence criterion is used to compute automatically the coarse load increment size. The proposed numerical procedure is illustrated here through the implementation of a set of elastoplastic constitutive relations including isotropic and kinematic hardening as well as small strain hysteretic non‐linearity. A series of numerical simulations confirm the robustness, accuracy and efficiency of the algorithms at the local and global level. Published in 2001 by John Wiley & Sons, Ltd.     

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.     

Comparison of high‐order curved finite elements

Several finite element techniques used in domains with curved boundaries are discussed and compared, with particular emphasis in two issues: the exact boundary representation of the domain and the consistency of the approximation. The influence of the number of integration points on the accuracy of the computation is also studied. Two‐dimensional numerical examples, solved with continuous and discontinuous Galerkin formulations, are used to test and compare all these methodologies. In every example shown, the recently proposed NURBS‐enhanced finite element method (NEFEM) provides the maximum accuracy for a given spatial discretization, at least one order of magnitude more accurate than classical isoparametric finite element method (FEM). Moreover, NEFEM outperforms Cartesian FEM and p‐FEM, stressing the importance of the geometrical model as well as the relevance of a consistent approximation in finite element simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical analysis of dissolution processes in cementitious materials using discontinuous and continuous Galerkin time integration schemes

The present paper is concerned with the numerical integration of non‐linear reaction–diffusion problems by means of discontinuous and continuous Galerkin methods. The first‐order semidiscrete initial value problem of calcium leaching of cementitious materials, based on a phenomenological dissolution model, an electrolyte diffusion model and the spatial p‐finite element discretization, is used as a highly non‐linear model problem. A p‐finite element method is used for the spatial discretization. In the context of discontinuous Galerkin methods the semidiscrete mass balance and the continuity of the primary variables are weakly formulated within time steps and between time steps, respectively. Continuous Galerkin methods are obtained by the strong enforcement of the continuity condition as special cases. The introduction of a natural time co‐ordinate allows for the application of standard higher order temporal shape functions of the p‐Lagrange type and the well‐known Gauss–Legendre quadrature of associated time integrals. It is shown, that arbitrary order accurate integration schemes can be developed within the framework of the proposed temporal p‐Galerkin methods. Selected benchmark analyses of calcium dissolution demonstrate the robustness of these methods with respect to pronounced changes of the reaction term and non‐smooth changes of Dirichlet boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

A comparison of transformation methods for evaluating two‐dimensional weakly singular integrals

Accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique. In this paper, a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of Laplace's equation in three dimensions and Poisson's equation in two dimensions. Both are two‐dimensional weakly singular integrals and are evaluated using (in a product fashion) methods which have recently been used for evaluating one‐dimensional weakly singular integrals arising in the boundary element method. The methods used are based on various polynomial transformations of conventional Gaussian quadrature points where the transformation polynomial has zero Jacobian at the singular point. Methods which split the region of integration into sub‐regions are considered as well as non‐splitting methods. In particular, the newly introduced and highly accurate generalized composite subtraction of singularity and non‐linear transformation approach (GSSNT) is applied to various two‐dimensional weakly singular integrals. A study of the different methods reveals complex relationships between transformation orders, position of the singular point, integration kernel and basis function. It is concluded that the GSSNT method gives the best overall results for the two‐dimensional weakly singular integrals studied. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical approximation of optimal control for distributed diffusion Hopfield neural networks

This work presents a numerical approximation of optimal control problems for non‐linear distributed Hopfield Neural Network equations with diffusion term. For one spatial dimensional case, a semi‐discrete numerical algorithm was constructed to find optimal control variable using finite element discretization, updated conjecture gradient iteration method. Furthermore, experiments demonstration will be implemented to show the effectiveness and stability through 3D graphics simulations. Copyright © 2006 John Wiley & Sons, Ltd.     

The probability density evolution method for dynamic response analysis of non‐linear stochastic structures

The probability density evolution method (PDEM) for dynamic responses analysis of non‐linear stochastic structures is proposed. In the method, the dynamic response of non‐linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one‐dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark‐Beta time‐integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single‐degree‐of‐freedom non‐linear conservative system and dynamic responses of an 8‐storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non‐linear stochastic structures are usually irregular and far from the well‐known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

An efficient finite element time‐domain formulation for the elastic second‐order wave equation: A non‐split complex frequency shifted convolutional PML

The perfectly matched layer (PML) technique has demonstrated very high efficiency as absorbing boundary condition for the elastic wave equation recast as a first‐order system in velocity and stress in attenuating non‐grazing bulk and surface waves. This paper develops a novel convolutional PML formulation based on the second‐order wave equation with displacements as the only unknowns to annihilate spurious reflections from near‐grazing waves. The derived variational form allows for the use of e.g. finite element and the spectral element methods as spatial discretization schemes. A recursive convolution update scheme of second‐order accuracy is employed such that highly stable, effective time integration with the Newmark‐beta (implicit and explicit with mass lumping) method is achieved. The implementation requires minor modifications of existing displacement‐based finite element software, and the stability and efficiency of the proposed formulation is verified by relevant two‐dimensional benchmarks that accommodate bulk and surface waves. Copyright © 2011 John Wiley & Sons, Ltd.