Meshfree co‐rotational formulation for two‐dimensional continua

In this paper, a meshfree co‐rotational formulation for two‐dimensional continua is proposed. In a co‐rotational formulation, the motion of a body is separated into rigid motion and strain‐producing deformation. Traditionally, this has been done in the setting of finite elements for beams and shell‐type elements. In the present work every node in a meshfree discretized domain has its own co‐rotating coordinate system. Three key ingredients are established in order to apply the co‐rotational formulation: (i) the relationship between global and local variables, (ii) the angle of rotation of a typical co‐rotating coordinate system, and (iii) a variationally consistent tangent stiffness matrix. An algorithm for the co‐rotational formulation based on load control is provided. Maximum‐entropy basis functions are used to discretize the domain and stabilized nodal integration is implemented to construct the global system of equations. Numerical examples are presented to demonstrate the validity of the meshfree co‐rotational formulation. Copyright © 2009 John Wiley & Sons, Ltd.     

Level‐Sets fields,placement and velocity based formulations of contact‐impact problems

By introducing unknown Sign‐like fields of Level‐Sets type, the Signorini‐Moreau dynamic contact conditions are set merely as boundary equations. From this setting, a continuous hybrid weak–strong formulation for dynamic contact between deformable solids is derived and a new Lagrangian formulation (we call stabilized) generalizing both the classical and augmented ones is obtained. Friction phenomena are treated similarly. In the global problem, the irregular Sign‐like fields stand for the intrinsic contact unknown ones. This problem is discretized by means of time, space and collocation schemes. Some numerical experimentations are carried out, showing the potential of our developments. Copyright © 2006 John Wiley & Sons, Ltd.     

An L∞–Lp mesh‐adaptive method for computing unsteady bi‐fluid flows

This paper discusses the contribution of mesh adaptation to high‐order convergence of unsteady multi‐fluid flow simulations on complex geometries. The mesh adaptation relies on a metric‐based method controlling the L ^p‐norm of the interpolation error and on a mesh generation algorithm based on an anisotropic Delaunay kernel. The mesh‐adaptive time advancing is achieved, thanks to a transient fixed‐point algorithm to predict the solution evolution coupled with a metric intersection in the time procedure. In the time direction, we enforce the equidistribution of the error, i.e. the error minimization in L ^∞ norm. This adaptive approach is applied to an incompressible Navier–Stokes model combined with a level set formulation discretized on triangular and tetrahedral meshes. Applications to interface flows under gravity are performed to evaluate the performance of this method for this class of discontinuous flows. Copyright © 2010 John Wiley & Sons, Ltd.     

An efficient solution method for incompressible N‐S equations using non‐orthogonal collocated grid

An efficient strategy for the solution of N‐S Equations using collocated, non‐orthogonal grids is presented. The governing equations have been discretized in the physical plane itself without co‐ordinate transformation, thereby retaining the lucidity of the basic finite volume method. The non‐orthogonal terms and QUICK type corrections for the convective terms in the momentum equations are treated explicitly, while the other terms are taken in implicit form. In the pressure correction equation, the non‐orthogonal terms have been dropped altogether. The discretized equations have been solved by the preconditioned conjugate gradient square method. The specific combination of above steps has resulted in better convergence properties as compared to those of existing algorithms, even for highly skewed grids. The scheme has been validated against benchmark solutions such as lid‐driven flow in square and skewed cavities and experi mental results of flow over a single cylinder. Its applicability has also been illustrated for flow through a bank of staggered cylinders, with anti‐symmetric inlet and outlet boundary conditions. Copyright © 1999 John Wiley & Sons, Ltd.     

An error‐estimate‐free and remapping‐free variational mesh refinement and coarsening method for dissipative solids at finite strains

A variational h‐adaptive finite element formulation is proposed. The distinguishing feature of this method is that mesh refinement and coarsening are governed by the same minimization principle characterizing the underlying physical problem. Hence, no error estimates are invoked at any stage of the adaption procedure. As a consequence, linearity of the problem and a corresponding Hilbert‐space functional framework are not required and the proposed formulation can be applied to highly non‐linear phenomena. The basic strategy is to refine (respectively, unrefine) the spatial discretization locally if such refinement (respectively, unrefinement) results in a sufficiently large reduction (respectively, sufficiently small increase) in the energy. This strategy leads to an adaption algorithm having O(N) complexity. Local refinement is effected by edge‐bisection and local unrefinement by the deletion of terminal vertices. Dissipation is accounted for within a time‐discretized variational framework resulting in an incremental potential energy. In addition, the entire hierarchy of successive refinements is stored and the internal state of parent elements is updated so that no mesh‐transfer operator is required upon unrefinement. The versatility and robustness of the resulting variational adaptive finite element formulation is illustrated by means of selected numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.     

Multi‐time‐step explicit–implicit method for non‐linear structural dynamics

We present a method with domain decomposition to solve time‐dependent non‐linear problems. This method enables arbitrary numeric schemes of the Newmark family to be coupled with different time steps in each subdomain: this coupling is achieved by prescribing continuity of velocities at the interface. We are more specifically interested in the coupling of implicit/explicit numeric schemes taking into account material and geometric non‐linearities. The interfaces are modelled using a dual Schur formulation where the Lagrange multipliers represent the interfacial forces. Unlike the continuous formulation, the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements, velocities and accelerations at the interfaces. We show that, within the framework of the Newmark family of numeric schemes, continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged. To prove this stability, we use an energy method, i.e. a global method over the whole time interval, in order to verify the algorithms properties. Then, we propose to extend this to non‐linear situations in the following cases: implicit linear/explicit non‐linear, explicit non‐linear/explicit non‐linear and implicit non‐linear/explicit non‐linear. Finally, we present some examples showing the feasibility of the method. Copyright © 2001 John Wiley & Sons, Ltd.     

The least‐squares meshfree method for elasto‐plasticity and its application to metal forming analysis

A new meshfree method for the analysis of elasto‐plastic deformation is presented. The method is based on the proposed first‐order least‐squares formulation for elasto‐plasticity and the moving least‐squares approximation. The least‐squares formulation for classical elasto‐plasticity and its extension to an incrementally objective formulation for finite deformation are proposed. In the formulation, equilibrium equation and flow rule are enforced in least‐squares sense, i.e. their squared residuals are minimized, and hardening law and loading/unloading condition are enforced pointwise at each integration point. The closest point projection method for the integration of rate‐form constitutive equation is inherently involved in the formulation, and thus the radial‐return mapping algorithm is not performed explicitly. The proposed formulation is a mixed‐type method since the residuals are represented in a form of first‐order differential system using displacement and stress components as nodal unknowns. Also the penalty schemes for the enforcement of boundary and frictional contact conditions are devised and the reshaping of nodal supports is introduced to avoid the difficulties due to the severe local deformation near contact interface. The proposed method does not employ structure of extrinsic cells for any purpose. Through some numerical examples of metal forming processes, the validity and effectiveness of the method are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

Higher‐order accurate time‐step‐integration algorithms by post‐integration techniques

In this paper, a framework to construct higher‐order‐accurate time‐step‐integration algorithms based on the post‐integration techniques is presented. The prescribed initial conditions are naturally incorporated in the formulations and can be strongly or weakly enforced. The algorithmic parameters are chosen such that unconditionally A‐stable higher‐order‐accurate time‐step‐integration algorithms with controllable numerical dissipation can be constructed for linear problems. Besides, it is shown that the order of accuracy for non‐linear problems is maintained through the relationship between the present formulation and the Runge–Kutta method. The second‐order differential equations are also considered. Numerical examples are given to illustrate the validity of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

A three‐dimensional hybrid meshfree‐Cartesian scheme for fluid–body interaction

A numerical method based on a hybrid meshfree‐Cartesian grid is developed for solving three‐dimensional fluid–solid interaction (FSI) problems involving solid bodies undergoing large motion. The body is discretized and enveloped by a cloud of meshfree nodes. The motion of the body is tracked by convecting the meshfree nodes against a background of Cartesian grid points. Spatial discretization of second‐order accuracy is accomplished by the combination of a generalized finite difference (GFD) method and conventional finite difference (FD) method, which are applied to the meshfree and Cartesian nodes, respectively. Error minimization in GFD is carried out by singular value decomposition. The discretized equations are integrated in time via a second‐order fractional step projection method. A time‐implicit iterative procedure is employed to compute the new/evolving position of the immersed bodies together with the dynamically coupled solution of the flow field. The present method is applied on problems of free falling spheres and tori in quiescent flow and freely rotating spheres in simple shear flow. Good agreement with published results shows the ability of the present hybrid meshfree‐Cartesian grid scheme to achieve good accuracy. An application of the method to the self‐induced propulsion of a deforming fish‐like swimmer further demonstrates the capability and potential of the present approach for solving complex FSI problems in 3D. Copyright © 2011 John Wiley & Sons, Ltd.     

Modeling of flow‐induced sound in porous materials

The impact of flow on a structure plays a crucial part when considering structural behavior, for example, in aviation. As structural vibrations (also denominated as structural sound) propagate within a structure, sound radiation is a likely consequence. To reduce the emission of noise, the use of poroelastic material is investigated. The approach consists in applying a poroelastic layer on the surface submitted to flow, as such utilizing the damping properties of poroelastic material. To predict flow‐induced sound, a computational model has been developed to account for (1) flow‐induced sound immission into a structure; (2) sound propagation; and (3) possible resulting sound radiation. Consistent formulation of the interactions between the components—that is, flow, poroelastic material, elastic structure, and acoustic fluid—allows to apply different simulation techniques for each component and thus to exploit each method's advantages. The key aspect of this work is the formulation of the interface conditions to couple flow with poroelastic material. The proposed and implemented coupling conditions are studied. The given example shows a possible application and demonstrates the effectiveness of poroelastic material to reduce flow‐induced sound emission. Copyright © 2014 John Wiley & Sons, Ltd.     

Transient non‐linear heat conduction–radiation problems—a boundary element formulation

A novel boundary‐only formulation for transient temperature fields in bodies of non‐linear material properties and arbitrary non‐linear boundary conditions has been developed. The option for self‐irradiating boundaries has been included in the formulation. Heat conduction equation has been partially linearized by Kirchhoff's transformation. The result has been discretized by the dual reciprocity boundary element method. The integral equation of heat radiation has been discretized by the standard boundary element method. The coupling of the resulting two sets of equations has been accomplished by static condensation of the radiative heat fluxes arising in both sets. The final set of ordinary differential equations has been solved using the Runge–Kutta solver with automatic time step adjustment. The algorithm proved to be robust and stable. Numerical examples are included. Copyright © 1999 John Wiley & Sons, Ltd.     

Isogeometric shape optimization for quasi‐static processes

A framework to solve shape optimization problems for quasi‐static processes is developed and implemented numerically within the context of isogeometric analysis (IGA). Recent contributions in shape optimization within IGA have been limited to static or steady‐state loading conditions. In the present contribution, the formulation of shape optimization is extended to include time‐dependent loads and responses. A general objective functional is used to accommodate both structural shape optimization and passive control for mechanical problems. An adjoint sensitivity analysis is performed at the continuous level and subsequently discretized within the context of IGA. The methodology and its numerical implementation are tested using benchmark static problems of optimal shapes of orifices in plates under remote bi‐axial tension and pure shear. Under quasi‐static loading conditions, the method is validated using a passive control approach with an a priori known solution. Several applications of time‐dependent mechanical problems are shown to illustrate the capabilities of this approach. In particular, a problem is considered where an external load is allowed to move along the surface of a structure. The shape of the structure is modified in order to control the time‐dependent displacement of the point where the load is applied according to a pre‐specified target. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite element analysis of time‐dependent semi‐infinite wave‐guides with high‐order boundary treatment

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ??. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C⁰ semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ??. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

An interior‐point algorithm for elastoplasticity

The problem of small‐deformation, rate‐independent elastoplasticity is treated using convex programming theory and algorithms. A finite‐step variational formulation is first derived after which the relevant potential is discretized in space and subsequently viewed as the Lagrangian associated with a convex mathematical program. Next, an algorithm, based on the classical primal–dual interior point method, is developed. Several key modifications to the conventional implementation of this algorithm are made to fully exploit the nature of the common elastoplastic boundary value problem. The resulting method is compared to state‐of‐the‐art elastoplastic procedures for which both similarities and differences are found. Finally, a number of examples are solved, demonstrating the capabilities of the algorithm when applied to standard perfect plasticity, hardening multisurface plasticity, and problems involving softening. Copyright © 2006 John Wiley & Sons, Ltd.     

A mixed‐enhanced finite‐element for the analysis of laminated composite plates

This paper presents a new 4‐node finite‐element for the analysis of laminated composite plates. The element is based on a first‐order shear deformation theory and is obtained through a mixed‐enhanced approach. In fact, the adopted variational formulation includes as variables the transverse shear as well as enhanced incompatible modes introduced to improve the in‐plane deformation. The problem is then discretized using bubble functions for the rotational degrees of freedom and functions linking the transverse displacement to the rotations. The proposed element is locking free, it does not have zero energy modes and provides accurate in‐plane/out‐of‐plane deformations. Furthermore, a procedure for the computation of the through‐the‐thickness shear stresses is discussed, together with an iterative algorithm for the evaluation of the shear correction factors. Several applications are investigated to assess the features and the performances of the proposed element. Results are compared with analytical solutions and with other finite‐element solutions. Copyright © 1999 John Wiley & Sons, Ltd.     

Time‐domain hybrid formulations for wave simulations in three‐dimensional PML‐truncated heterogeneous media

We are concerned with the numerical simulation of wave motion in arbitrarily heterogeneous, elastic, perfectly‐matched‐layer‐(PML)‐truncated media. We extend in three dimensions a recently developed two‐dimensional formulation, by treating the PML via an unsplit‐field, but mixed‐field, displacement‐stress formulation, which is then coupled to a standard displacement‐only formulation for the interior domain, thus leading to a computationally cost‐efficient hybrid scheme. The hybrid treatment leads to, at most, third‐order in time semi‐discrete forms. The formulation is flexible enough to accommodate the standard PML, as well as the multi‐axial PML. We discuss several time‐marching schemes, which can be used à la carte, depending on the application: (a) an extended Newmark scheme for third‐order in time, either unsymmetric or fully symmetric semi‐discrete forms; (b) a standard implicit Newmark for the second‐order, unsymmetric semi‐discrete forms; and (c) an explicit Runge–Kutta scheme for a first‐order in time unsymmetric system. The latter is well‐suited for large‐scale problems on parallel architectures, while the second‐order treatment is particularly attractive for ready incorporation in existing codes written originally for finite domains. We compare the schemes and report numerical results demonstrating stability and efficacy of the proposed formulations. Copyright © 2014 John Wiley & Sons, Ltd.     

Stokes flow around cylinders in a bounded two‐dimensional domain using multipole‐accelerated boundary element methods

The multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from O(N²) (iterative solvers) or O(N³) (gaussian elimination) to O(N log N) or O(N), where N is the number of nodes in the discretized system. Such a reduction in the growth of the calculation time and data storage is crucial in applications where N is large, such as when modelling the macroscopic behaviour of suspensions of particles. In such cases, a minimum of 1000 particles is needed to obtain statistically meaningful results, leading to systems with N of the order of 10 000 for the smallest problems. When only boundary velocities are known, the indirect boundary element formulation for Stokes flow results in Fredholm equations of the second kind, which generally produce a well‐posed set of equations when discretized, a necessary requirement for iterative solution methods. The direct boundary element formulation, on the other hand, results in Fredholm equations of the first kind, which, upon discretization, produce ill‐conditioned systems of equations. The model system here is a two‐dimensional wide‐gap couette viscometer, where particles are suspended in the fluid between the cylinders. This is a typical system that is efficiently modelled using boundary element method simulations. The multipolar technique is applied to both direct and indirect formulations. It is found that the indirect approach is sufficiently well‐conditioned to allow the use of fast multipole methods. The direct approach results in severe ill‐conditioning, to a point where application of the multipole method leads to non‐convergence of the solution iteration. Copyright © 1999 John Wiley & Sons, Ltd.     

Non‐linear exact geometry 12‐node solid‐shell element with three translational degrees of freedom per node

This paper presents the finite rotation exact geometry (EG) 12‐node solid‐shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9‐parameter shell model by employing a new concept of sampling surfaces (S‐surfaces) inside the shell body. We introduce three S‐surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid‐shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid‐body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non‐linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

A Cartesian ghost‐cell multigrid Poisson solver for incompressible flows

In this paper, a simple Cartesian ghost‐cell multigrid Poisson solver is proposed for simulating incompressible fluid flows. The flow field is discretized efficiently on a rectangular mesh, in which solid bodies are immersed. A small number of ghost mesh cells and their symmetric image cells are distributed in the vicinity of the solid boundary. With the aid of the ghost and image cells, the Dirichlet and Neumann boundary conditions can be implemented effectively. Chorin's fractional‐step projection method is adopted for the coupling of velocity and pressure for the solution of the Navier–Stokes equations. Point‐wise Gauss–Seidel iteration is used to solve the pressure Poisson equation. To speed up the convergence of the solution to the corresponding linear system, sub‐level coarse meshes embedded with ghost and image cells are also introduced and operated in a sequential V‐cycle. Several test cases including the classical ideal incompressible flow around a cylinder, a lid‐driven cavity flow and viscous flow past a fixed/rotating cylinder are presented to demonstrate the accuracy and efficiency of the current approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Large‐eddy simulation of separated leading‐edge flow in general co‐ordinates

An incompressible separated transitional boundary‐layer flow on a flat plate with a semi‐circular leading edge has been simulated and a very good agreement with the experimental data has been obtained, demonstrating how this technique may be applied even when finite difference formulae are used in the periodic direction. The entire transition process has been elucidated and vortical structures have been identified at different stages during the transition process. Efficient numerical methods for the large‐eddy simulation (LES) of turbulent flows in complex geometry are developed. The methods used are described in detail: body‐fitted co‐ordinates with the contravariant velocity components of the general Navier–Stokes equations discretized on a staggered mesh with a dynamic subgrid‐scale model in general co‐ordinates. The main source of computational expense in simulations for incompressible flows is due to the solution of a Poisson equation for pressure. This is especially true for flows in complex geometry. Fourier techniques can be employed to speed up the pressure solution significantly for a flow which is periodic in one dimension. With simple conditions fulfilled, it is possible to Fourier transform a discrete elliptic equation such as the Poisson equation for the pressure field, decomposing the problem into a set of two‐dimensional problems of similar type (Poisson‐like). Even when a complex geometry and body‐fitted curvilinear co‐ordinates are used in the other two dimensions, as in the present case, the resulting Fourier‐transformed 2D problems are much more efficiently solved than the 3D problem by iterative means. Copyright © 2000 John Wiley & Sons, Ltd.