Asymptotic numerical methods for unilateral contact

New algorithms based upon the asymptotic numerical method (ANM) are proposed to solve unilateral contact problems. ANM leads to a representation of a solution path in terms of series or Padé approximants. To get a smooth solution path, a hyperbolic relation between contact forces and clearance is introduced. Three key points are discussed: the influence of the regularization of the contact law, the discretization of the contact force by Lagrange multipliers and prediction–correction algorithms. Simple benchmarks are considered to evaluate the relevance of the proposed algorithms. Copyright © 2006 John Wiley & Sons, Ltd.     

ANM for stationary Navier–Stokes equations and with Petrov–Galerkin formulation

This paper deals with the use of the asymptotic numerical method (ANM) for solving non‐linear problems, with particular emphasis on the stationary Navier–Stokes equation and the Petrov–Galerkin formulation. ANM is a combination of a perturbation technique and a finite element method allowing to transform a non‐linear problem into a succession of linear ones that admit the same tangent matrix. This method has been applied with success in non‐linear elasticity and fluid mechanics. In this paper, we apply the same kind of technique for solving Navier–Stokes equation with the so‐called Petrov–Galerkin weighting. The main difficulty comes from the fact that the non‐linearity is no more quadratic and it is not evident, in this case, to be able to compute a large number of terms of the perturbation series. Several examples of fluid mechanic are presented to demonstrate the performance of such a method. Copyright © 2001 John Wiley & Sons, Ltd.     

A higher order compact finite difference algorithm for solving the incompressible Navier–Stokes equations

On the basis of the projection method, a higher order compact finite difference algorithm, which possesses a good spatial behavior, is developed for solving the 2D unsteady incompressible Navier–Stokes equations in primitive variable. The present method is established on a staggered grid system and is at least third‐order accurate in space. A third‐order accurate upwind compact difference approximation is used to discretize the non‐linear convective terms, a fourth‐order symmetrical compact difference approximation is used to discretize the viscous terms, and a fourth‐order compact difference approximation on a cell‐centered mesh is used to discretize the first derivatives in the continuity equation. The pressure Poisson equation is approximated using a fourth‐order compact difference scheme constructed currently on the nine‐point 2D stencil. New fourth‐order compact difference schemes for explicit computing of the pressure gradient are also developed on the nine‐point 2D stencil. For the assessment of the effectiveness and accuracy of the method, particularly its spatial behavior, a problem with analytical solution and another one with a steep gradient are numerically solved. Finally, steady and unsteady solutions for the lid‐driven cavity flow are also used to assess the efficiency of this algorithm. Copyright © 2011 John Wiley & Sons, Ltd.     

A general conjugate‐gradient‐based predictor–corrector solver for explicit finite‐element contact

A Lagrange‐multiplier based approach is presented for the general solution of multi‐body contact within an explicit finite element framework. The technique employs an explicit predictor step to permit the detection of interpenetration and then utilizes a corrector step, whose solution is obtained with a pre‐conditioned matrix‐free conjugate gradient projection method, to determine the Lagrange multipliers necessary to eliminate the predicted penetration. The predictor–corrector algorithm is developed for deformable bodies based upon the central difference method, and for rigid bodies from momentum and energy conserving approaches. Both frictionless and Coulomb‐based frictional contact idealizations are addressed. The technique imposes no time‐step constraints and quickly mitigates velocity discontinuities across closed interfaces. Special attention is directed toward contact between rigid bodies. Algorithmic moment arms conserve the translational and angular momentums of the system in the absence of external loads. Elastic collisions are captured with a two‐phase predictor–corrector approach and a geometrically approximate velocity jump criterion. The first step solves the inelastic contact problem and identifies inactive constraints between rigid bodies, while the second step generates the necessary velocity jump condition on the active constraints. The velocity criterion is shown to algorithmically preserve the system kinetic energy for two unconstrained rigid bodies. Copyright © 1999 John Wiley & Sons, Ltd. This paper was produced under the auspices of the U.S. Government and it is therefore not subject to copyright in the U.S.     

A reduced‐order representation of the Poincaré–Steklov operator: an application to coupled multi‐physics problems

This work investigates a model reduction method applied to coupled multi‐physics systems. The case in which a system of interest interacts with an external system is considered. An approximation of the Poincaré–Steklov operator is computed by simulating, in an offline phase, the external problem when the inputs are the Laplace–Beltrami eigenfunctions defined at the interface. In the online phase, only the reduced representation of the operator is needed to account for the influence of the external problem on the main system. An online basis enrichment is proposed in order to guarantee a precise reduced‐order computation. Several test cases are proposed on different fluid–structure couplings. Copyright © 2016 John Wiley & Sons, Ltd.     

A coupled BEM and arbitrary Lagrangian–Eulerian FEM model for the solution of two‐dimensional laminar flows in external flow fields

This paper describes a new computational model developed to solve two‐dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three‐step finite element method. The arbitrary Lagrangian–Eulerian method is incorporated into the model, to solve the moving boundary problems. The present model is applied to simulate various external flow problems like flow across circular cylinder, acceleration and deceleration of the circular cylinder moving in a still fluid and vibration of the circular cylinder induced by the vortex shedding. The simulation results are found to be very reasonable and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

An assessment of using the predictor–corrector technique to solve reactive transport equations

We examined, through comparison among the full‐coupling (FC), operator‐splitting (OS), and predictor–corrector (PC) techniques, the effectiveness of using the PC technique to solve depth‐averaged reactive transport equations in the shallow water domain. Our investigation has led to three major conclusions. Firstly, both the OS and PC techniques can efficiently solve reactive transport equations because the advection–diffusion transport equations are solved outside the non‐linear iteration loop and the reaction equations are solved node by node. However, these two techniques may risk sacrificing computational accuracy. Secondly, the OS or PC technique incorporated with the Lagrangian–Eulerian (LE) approach can handle boundary sources more precisely than alternatively with the conventional Eulerian (CE) approach. Thirdly, with the LE approach incorporated, the numerical results from the three techniques agreed highly with one another except when diffusion became significant. In this case, the PC technique's result still matched well with the FC technique's result, but differences between the OS and FC techniques' results arose as diffusion increased. Based on this study, we recommend to apply as a first step the PC technique to solving reactive transport equations with respect to both computational efficiency and accuracy. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical simulation of flapping wings using a panel method and a high‐order Navier–Stokes solver

The design of efficient flapping wings for human engineered micro aerial vehicles (MAVs) has long been an elusive goal, in part because of the large size of the design space. One strategy for overcoming this difficulty is to use a multifidelity simulation strategy that appropriately balances computation time and accuracy. We compare two models with different geometric and physical fidelity. The low‐fidelity model is an inviscid doublet lattice method with infinitely thin lifting surfaces. The high‐fidelity model is a high‐order accurate discontinuous Galerkin Navier–Stokes solver, which uses an accurate representation of the flapping wing geometry. To compare the performance of the two methods, we consider a model flapping wing with an elliptical planform and an analytically prescribed spanwise wing twist, at size scales relevant to MAVs. Our results show that in many cases, including those with mild separation, low‐fidelity simulations can accurately predict integrated forces, provide insight into the flow structure, indicate regions of likely separation, and shed light on design–relevant quantities. But for problems with significant levels of separation, higher‐fidelity methods are required to capture the details of the flow field. Inevitably high‐fidelity simulations are needed to establish the limits of validity of the lower fidelity simulations.Copyright © 2012 John Wiley & Sons, Ltd.     

A bridge between projection methods and SIMPLE type methods for incompressible Navier–Stokes equations

A bridge is built between projection methods and SIMPLE type methods (Semi‐Implicit Method for Pressure‐Linked Equation). A general second‐order accurate projection method is developed for the simulation of incompressible unsteady flows by employing a non‐linear update of pressure term as Θⁿ?pⁿ⁺¹+(I?Θⁿ)?pⁿ, where Θⁿ is a coefficient matrix, which may depend on the grid size, time step and even velocity. It includes three‐ and four‐step projection methods. The standard SIMPLE method is written in a concise formula for steady and unsteady flows. It is proven that SIMPLE type methods have second‐order temporal accuracy for unsteady flows. The classical second‐order projection method and SIMPLE type methods are united within the framework of the general second‐order projection formula. Two iteration algorithms of SIMPLE type methods for unsteady flows are described and discussed. In addition, detailed formulae are provided for general projection methods by using the Runge–Kutta technique to update the convective term and Crank–Nicholson scheme for the diffusion term. Copyright © 2007 John Wiley & Sons, Ltd.     

An adaptive strategy for the bivariate solution of finite element problems using multivariate nested Padé approximants

Most engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present paper, an adaptive approach is proposed as a way to estimate the solution of such multivariate finite element problems. It is based upon the integration of so‐called nested Padé approximants within the finite element procedure. This procedure includes an effective control of the approximation error, which enables adaptive refinements of the converged intervals upon reconstruction of the solution. The main advantages lie in a potential reduction of the computational effort and the fact that the level of a priori knowledge required about the solution in order to have an accurate, efficient, and well‐sampled estimate of the solution is low. The approach is introduced for bivariate problems, for which it is validated on elasto‐poro‐acoustic problems of both academic and more industrial scale. It is argued that the methodology in general holds for more than two variables, and a discussion is opened about the truncation refinements required in order to generalize the results accordingly. Copyright © 2014 John Wiley & Sons, Ltd.     

Low‐Reynolds‐number flow around an oscillating circular cylinder using a cell viscousboundary element method

Flow fields from transversely oscillating circular cylinders in water at rest are studied by numerical solutions of the two‐dimensional unsteady incompressible Navier–Stokes equations adopting a primitive‐variable formulation. These findings are successfully compared with experimental observations. The cell viscous boundary element scheme developed is first validated to examine convergence of solution and the influence of discretization within the numerical scheme of study before the comparisons are undertaken. A hybrid approach utilising boundary element and finite element methods is adopted in the cell viscous boundary element method. That is, cell equations are generated using the principles of a boundary element method with global equations derived following the procedures of finite element methods. The influence of key parameters, i.e. Reynolds number Re, Keulegan–Carpenter number KC and Stokes' number β, on overall flow characteristics and vortex shedding mechanisms are investigated through comparisons with experimental findings and theoretical predictions. The latter extends the study into assessment of the values of the drag coefficient, added mass or inertia coefficient with key parameters and the variation of lift and in‐line force results with time derived from the Morison's equation. The cell viscous boundary element method as described herein is shown to produce solutions which agree very favourably with experimental observations, measurements and other theoretical findings. Copyright © 2001 John Wiley & Sons, Ltd.     

Fully implicit Lagrange–Newton–Krylov–Schwarz algorithms for boundary control of unsteady incompressible flows

We develop a parallel fully implicit domain decomposition algorithm for solving optimization problems constrained by time‐dependent nonlinear partial differential equations. In particular, we study the boundary control of unsteady incompressible Navier–Stokes equations. After an implicit discretization in time, a fully coupled sparse nonlinear optimization problem needs to be solved at each time step. The class of full space Lagrange–Newton–Krylov–Schwarz algorithms is used to solve the sequence of optimization problems. Among optimization algorithms, the fully implicit full space approach is considered to be the easiest to formulate and the hardest to solve. We show that Lagrange–Newton–Krylov–Schwarz, with a one‐level restricted additive Schwarz preconditioner, is an efficient class of methods for solving these hard problems. To demonstrate the scalability and robustness of the algorithm, we consider several problems with a wide range of Reynolds numbers and time step sizes, and we present numerical results for large‐scale calculations involving several million unknowns obtained on machines with more than 1000 processors. Copyright © 2012 John Wiley & Sons, Ltd.     

A methodology for the generation of low‐cost higher‐order methods for linear dynamics

This work presents a methodology which generates efficient higher‐order methods for linear dynamics by improving the accuracy properties of Nørsett methods towards those of Padé methods. The methodology is based on a simple and low‐cost iterative procedure which is used to implement a set of higher‐order methods with controllable dissipation. A sequence of improved solutions is obtained which correspond to algorithms offering an effective compromise between the efficiency of Nørsett methods and the accuracy of Padé methods. Moreover, a direct control over high‐frequency dissipation is possible by means of an algorithmic parameter. Numerical tests are reported which confirm that this set of algorithms is really attractive for linear dynamic analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

Stable and time-dissipative finite element methods for the incompressible Navier–Stokes equations in advection dominated flows

This paper examines a new Galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of 1-D scalar advection–diffusion and that leads to non-oscillatory solutions as the streamline upwinding algorithms for 2-D scalar advection–diffusion and incompressible Navier–Stokes. This method retains the satisfaction of the Babuska–Brezzi condition and, thus, leads to optimal performance in the incompressible limit. This method, when, combined with the recently proposed linear unconditionally stable algorithms of Simo and Armero (1993), yields a method for solution of the incompressible Navier–Stokes equations ideal for either diffusive or advection-dominated flows. Examples from scalar advection–diffusion and the solution of the incompressible Navier–Stokes equations are presented.     

A high‐level programming‐language implementation of topology optimization applied to steady‐state Navier–Stokes flow

We present a versatile high‐level programming‐language implementation of non‐linear topology optimization. Our implementation is based on the commercial software package FEMLAB, and it allows a wide range of optimization objectives to be dealt with easily. We exemplify our method by studies of steady‐state Navier–Stokes flow problems, thus extending the work by Borrvall and Petersson on topology optimization of fluids in Stokes flow (Int. J. Num. Meth. Fluids 2003; 41 :77–107). We analyse the physical aspects of the solutions and how they are affected by different parameters of the optimization algorithm. A complete example of our implementation is included as FEMLAB code in an appendix. Copyright © 2005 John Wiley & Sons, Ltd.     

On a high‐order Newton linearization method for solving the incompressible Navier–Stokes equations

The present study aims to accelerate the non‐linear convergence to incompressible Navier–Stokes solution by developing a high‐order Newton linearization method in non‐staggered grids. For the sake of accuracy, the linearized convection–diffusion–reaction finite‐difference equation is solved line‐by‐line using the nodally exact one‐dimensional scheme. The matrix size is reduced and, at the same time, the CPU time is considerably saved owing to the reduction of stencil points. This Newton linearization method is computationally efficient and is demonstrated to outperform the classical Newton method through computational exercises. Copyright © 2005 John Wiley & Sons, Ltd.     

Padé approximants and the modal connection: Towards increased robustness for fast parametric sweeps

To increase the robustness of a Padé‐based approximation of parametric solutions to finite element problems, an a priori estimate of the poles is proposed. The resulting original approach is shown to allow for a straightforward, efficient, subsequent Padé‐based expansion of the solution vector components, overcoming some of the current convergence and robustness limitations. In particular, this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of Padé approximants. The choice of these approximants, as shown in the present work, is theoretically supported by the Montessus de Ballore theorem, concerning the convergence of a series of approximants with fixed denominator degrees. Key features and originality of the proposed approach are (1) a component‐wise expansion which allows to specifically target subsets of the solution field and (2) the a priori, simultaneous choice of the Padé approximants and their associated interval of convergence for an effective and more robust approximation. An academic acoustic case study, a structural‐acoustic application, and a larger acoustic problem are presented to demonstrate the potential of the approach proposed.     

Parallelization in time for thermo‐viscoplastic problems in extrusion of aluminium

The ParaReal algorithm (C.R. Acad. Sci. Paris 2001; 332 :1–6) is a parallel approach for solving numerically systems of ordinary differential equations by exploiting parallelism across the steps of the numerical integrator. The method performs well for dissipative problems and problems of fluid–structure interaction (Int. J. Numer. Methods Engng 2003; 58 :1397–1434). We consider here a convergence analysis for the method and we report the performance achieved from the parallelization of a Stokes/Navier–Stokes code via the ParaReal algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Bifurcation points and bifurcated branches by an asymptotic numerical method and Padé approximants

The aim of this work is to develop a reliable and fast algorithm to compute bifurcation points and bifurcated branches. It is based upon the asymptotic numerical method (ANM) and Padé approximants. The bifurcation point is detected by analysing the poles of Padé approximants or by evaluating, along the computed solution branch, a bifurcation indicator well adapted to ANM. Several examples are presented to assess the effectiveness of the proposed method, that emanate from buckling problems of thin elastic shells. Especially problems involving large rotations are discussed. Copyright © 2004 John Wiley & Sons, Ltd.