Modelling and numerical analysis of heat treatments on aluminium parts

A theoretical model suitable for the modelling and analysis of heat treatment problems is described. The presented formulations are implemented incrementally with a non‐linear rate‐dependent constitutive model, adequate to the simulation of a wide range temperature processes. The large deformation model is based on the additive decomposition of the strain rate tensor in elastic, thermal and viscoplastic parts. The flow rule is a function of the equivalent stress and the deviatoric stress tensor of the temperature field and of a set of internal state variables. The thermal problem is solved by a prediction/correction algorithm. The prediction, which uses a semi‐implicit integration, provides a first estimation of the temperature field. The correction stage, performed with a Newton–Raphson implicit scheme, improves the solution until a satisfactory result is found. The thermomechanical coupled problem is solved with a staggered approach. The finite element formulations for coupled analysis of deformation and heat transfer given in this article are validated and applied to the process of quenching of aluminium parts into boiling water. Copyright © 2006 John Wiley & Sons, Ltd.     

Substructured formulations of nonlinear structure problems – influence of the interface condition

We investigate the use of non‐overlapping domain decomposition (DD) methods for nonlinear structure problems. The classic techniques would combine a global Newton solver with a linear DD solver for the tangent systems. We propose a framework where we can swap Newton and DD so that we solve independent nonlinear problems for each substructure and linear condensed interface problems. The objective is to decrease the number of communications between subdomains and to improve parallelism. Depending on the interface condition, we derive several formulations that are not equivalent, contrarily to the linear case. Primal, dual and mixed variants are described and assessed on a simple plasticity problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Fast Computation of Stationary Inviscid Flow Around an Airfoil

The parallel performance of an implicit solver for the Euler equations on a structured grid is discussed. The flow studied is a two-dimensional transonic flow around an airfoil. The spatial discretization involves the MUSCL scheme, a higher-order Total Variation Diminishing scheme. The solver described in this paper is an implicit solver that is based on quasi Newton iteration and approximate factorization to solve the linear system of equations resulting from the Euler Backward scheme. It is shown that the implicit time-stepping method can be used as a smoother to obtain an efficient and stable multigrid process. Also, the solver has good properties for parallelization comparable with explicit time-stepping schemes. To preserve data locality domain decomposition is applied to obtain a parallelizable code. Although the domain decomposition slightly affects the efficiency of the approximate factorization method with respect to the number of time steps required to attain the stationary solution, the results show that this hardly affects the performance for practical purposes. The accuracy with which the linear system of equations is solved is found to be an important parameter. Because the method is equally applicable for the Navier-Stokes equations and in three-dimensions, the presented combination of efficient parallel execution and implicit time-integration provides an interesting perspective for time-dependent problems in computational fluid dynamics.     

A simple explicit–implicit finite element tearing and interconnecting transient analysis algorithm

A simple explicit–implicit finite element tearing and interconnecting (FETI) algorithm (AFETI‐EI algorithm) is presented for partitioned transient analysis of linear structural systems. The present algorithm employs two decompositions. First, the total system is partitioned via spatial or domain decomposition to obtain the governing equations of motions for each partitioned domain. Second, for each partitioned subsystem, the governing equations are modally decomposed into the rigid‐body and deformational equations. The resulting rigid‐body equations are integrated by an explicit integrator, for its stability is not affected by step‐size restriction on account of zero‐frequency contents (ω = 0). The modally decomposed partitioned deformation equations of motion are integrated by an unconditionally stable implicit integration algorithm. It is shown that the present AFETI‐EI algorithm exhibits unconditional stability and that the resulting interface problem possesses the same solution matrix profile as the basic FETI static problems. The present simple dynamic algorithm, as expected, falls short of the performance of the FETI‐DP but offers a similar performance of implicit two‐level FETI‐D algorithm with a much cheaper coarse solver; hence, its simplicity may offer relatively easy means for conducting parallel analysis of both static and dynamic problems by employing the same basic scalable FETI solver, especially for research‐mode numerical experiments. Copyright © 2011 John Wiley & Sons, Ltd.     

Decoupling joint and elastic accelerations in deformable mechanical systems using non‐linear recursive formulations

The aim of this paper is to develop non‐linear recursive formulations for decoupling joint and elastic accelerations, while maintaining the non‐linear inertia coupling between rigid body motion and elastic deformation in deformable mechanical systems. The inertia projection schemes used in most existing recursive formulations for the dynamic analysis of deformable mechanisms lead to dense coefficient matrices in the equations of motion. Consequently, there are strong dynamic couplings between the joint and elastic coordinates. When the number of elastic degrees of freedom increases, the size of the coefficient matrix in the equations of motion becomes large. Consequently, the use of these recursive formulations for solving the joint and elastic accelerations becomes less efficient. In this paper, the non‐linear recursive formulations have been used to decouple the elastic and joint accelerations in deformable mechanical systems. The relationships between the absolute, elastic and joint variables and generalized Newton–Euler equations are used to develop systems of loosely coupled equations that have sparse matrix structure. By using the inertia matrix structure of deformable mechanical systems and the fact that joint reaction forces associated with elastic coordinates do represent independent variables, a reduced system of equations whose dimension is dependent of the number of elastic degrees of freedom is obtained. This system can be solved for the joint accelerations as well as for the joint reaction forces. The use of the approaches developed in this investigation is illustrated using deformable open‐loop serial robot and closed‐loop four‐bar mechanical systems. Copyright © 2007 John Wiley & Sons, Ltd.     

Comparison of matrix-free acceleration techniques in compressible Navier–Stokes calculations

Six different preconditioning methods to accelerate the convergence rate of Krylov-subspace iterative methods are described, implemented and compared in the context of matrix-free techniques. The acceleration techniques comprehend Krylov-subspace iterative methods; invariant subspace-based methods and matrix approximations: Jacobi, LU-SGS, Deflated GMRES; Augmented GMRES; polynomial preconditioner and FGMRES/Krylov. The relative behaviour of the methods is explained in terms of the spectral properties of the resulting iterative matrix. The employed code uses a Newton–Krylov approach to iteratively solve the Euler or Navier–Stokes equations, for a supersonic ramp or a viscous compressible double-throat flow. The linear system approximate solver is the GMRES method, in either the restarted or FGMRES variants. The results show the better performance of the methods that approximate the iterative matrix, such as Jacobi, LU-SGS and FGMRES/Krylov. Copyright © 2004 John Wiley & Sons, Ltd.     

Crystal plasticity with Jacobian-Free Newton–Krylov

The objective of this work is to study potential benefits of solving crystal plasticity finite element method (CPFEM) implicit simulations using the Jacobian-Free Newton–Krylov (JFNK) technique. Implicit implementations of CPFEM are usually solved using Newton’s method. However, the inherent non-linearity in the flow rule model that characterizes the crystal slip system deformation on occasions would require considerable effort to form the exact analytical Jacobian needed by Newton’s method. In this paper we present an alternative using JFNK. As it does not require an exact Jacobian, JFNK can potentially decrease development time. JFNK approximates the effect of the Jacobian through finite differences of the residual vector, allowing modified formulations to be studied with relative ease. We show that the JFNK solution is identical to that obtained using Newton’s method and produces quadratic convergence. We also find that preconditioning the JFNK solution with the elastic tensor provides the best computational efficiency.     

Efficient implicit simulation of incremental sheet forming

In single point incremental forming (SPIF), the sheet is incrementally deformed by a small spherical tool following a lengthy tool path. The simulation by the finite element method of SPIF requires extremely long computing times that limit the application to simple academic cases. The main challenge is to perform thousands of load increments modelling the lengthy tool path with elements that are small enough to model the small contact area. Because of the localised deformation in the process, a strong nonlinearity is observed in the vicinity of the tool. The rest of the sheet experiences an elastic deformation that introduces only a weak nonlinearity because of the change of shape. The standard use of the implicit time integration scheme is inefficient because it applies an iterative update (Newton–Raphson) strategy for the entire system of equations. The iterative update is recommended for the strong nonlinearity that is active in a small domain but is not required for the large part with only weak nonlinearities. It is proposed in this paper to split the finite element mesh into two domains. The first domain models the plastically deforming zone that experiences the strong nonlinearity. It applies a full nonlinear update for the internal force vector and the stiffness matrix every iteration. The second domain models the large elastically deforming zone of the sheet. It applies a pseudolinear update strategy based on a linearization at the beginning of each increment. Within the increment, it reuses the stiffness matrix and linearly updates the internal force vector. The partly linearized update strategy is cheaper than the full nonlinear update strategy, resulting in a reduction of the overall computing. Furthermore, in this paper, adaptive refinement is combined with the two domain method. It results in accelerating the standard SPIF implicit simulation of 3200 shell elements by a factor of 3.6. Copyright © 2011 John Wiley & Sons, Ltd.     

Accelerating iterative solution methods using reduced‐order models as solution predictors

We propose the use of reduced‐order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large‐scale numerical simulation. The acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps. The algorithm basically consists of two projection steps: (1) projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions, and (2) solving the governing equations in this reduced space and projecting the solution back on the original, high dimensional one. We applied the algorithm to numerical models for simulation of two‐phase flow through heterogeneous porous media. In particular we considered implicit‐pressure explicit‐saturation (IMPES) schemes and investigated the scope to accelerate the iterative solution of the pressure equation, which is by far the most time‐consuming part of any IMPES scheme. We achieved a substantial reduction in the number of iterations and an associated acceleration of the solution. Our largest test problem involved 93 500 variables, in which case we obtained a maximum reduction in computing time of 67%. The method is particularly attractive for problems with time‐varying parameters or source terms. Copyright © 2006 John Wiley & Sons, Ltd.     

An Eulerian finite‐volume scheme for large elastoplastic deformations in solids

Conservative formulations of the governing laws of elastoplastic solid media have distinct advantages when solved using high‐order shock capturing methods for simulating processes involving large deformations and shock waves. In this paper one such model is considered where inelastic deformations are accounted for via conservation laws for elastic strain with relaxation source terms. Plastic deformations are governed by the relaxation time of tangential stresses. Compared with alternative Eulerian conservative models, the governing system consists of fewer equations overall. A numerical scheme for the inhomogeneous system is proposed based upon the temporal splitting. In this way the reduced system of non‐linear elasticity is solved explicitly, with convective fluxes evaluated using high‐order approximations of Riemann problems locally throughout the computational mesh. Numerical stiffness of the relaxation terms at high strain rates is avoided by utilizing certain properties of the governing model and performing an implicit update. The methods are demonstrated using test cases involving large deformations and high strain rates in one‐, two‐, and three‐dimensions. Copyright © 2009 John Wiley & Sons, Ltd.     

Efficient iteration schemes for anisotropic hyperelasto‐plasticity

Numerical issues arising when integrating hyperelasto‐plastic constitutive equations with elastic anisotropy, stemming from anisotropic damage, and plastic anisotropy as represented by kinematic hardening are discussed. In particular, solution algorithms for the corresponding non‐linear system of equations due to implicit integration algorithms are addressed. It is shown that algorithms like staggered iteration and quasi‐Newton techniques are superior to a pure Newton technique when the cpu time is compared. However, the drawback of the staggered and the quasi‐Newton technique is that rather small time steps must be taken to ensure convergence, which can be of importance when applying complex constitutive models in a finite element programme. Copyright © 2005 John Wiley & Sons, Ltd.     

Computational modeling of cardiac electrophysiology: A novel finite element approach

The key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element‐based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global–local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell‐specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh–Nagumo model for oscillatory pacemaker cells and the Aliev–Panfilov model for non‐oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non‐linear system of equations is solved with an incremental iterative Newton–Raphson solution procedure. Since this framework only introduces one single scalar‐valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two‐dimensional patient‐specific cardiac excitation problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Trust‐region based return mapping algorithm for implicit integration of elastic‐plastic constitutive models

A new method for the solution of the non‐linear equations forming the core of constitutive model integration is proposed. Specifically, the trust‐region method that has been developed in the numerical optimization community is successfully modified for use in implicit integration of elastic‐plastic models. Although attention here is restricted to these rate‐independent formulations, the proposed approach holds substantial promise for adoption with models incorporating complex physics, multiple inelastic mechanisms, and/or multiphysics. As a first step, the non‐quadratic Hosford yield surface is used as a representative case to investigate computationally challenging constitutive models. The theory and implementation are presented, discussed, and compared with other common integration schemes. Multiple boundary value problems are studied and used to verify the proposed algorithm and demonstrate the capabilities of this approach over more common methodologies. Robustness and speed are then investigated and compared with existing algorithms. Through these efforts, it is shown that the utilization of a trust‐region approach leads to superior performance versus a traditional closest‐point projection Newton–Raphson method and comparable speed and robustness to a line search augmented scheme. Copyright © 2017 John Wiley & Sons, Ltd.     

Second‐order optimal control algorithm for complex systems

The solution to large‐scale optimal control problems, characterized by complex dynamics and extended time periods, is often computationally demanding. We present a solution algorithm with favourable local convergence properties as a way to reduce simulation times. This method is based on using a trapezoidal direct collocation to convert the differential equations into algebraic constraints. The resulting constrained minimization problem is then solved with an augmented Lagrangian formulation to accommodate both equality and inequality constraints. In contrast to the prevalent optimal control software implementations, we calculate analytical first and second derivatives. We then apply a generalized Newton method to the augmented Lagrangian formulation, solving for all unknowns simultaneously. The computational costs of the Hessian formation and matrix solution remain manageable as the system size increases due to the sparsity of all tensor quantities. Likewise, the total iterations for convergence scale well due to the local quadratic convergence of the generalized Newton method. We demonstrate the method with an inverted pendulum problem and a neuromuscular control problem with complex dynamics and 18 forcing functions. The optimal control solutions are successfully found. In both examples, we obtain quadratic convergence rates in the neighbourhood of the solution. Copyright © 2002 John Wiley & Sons, Ltd.     

Post buckling analysis of shear deformable shallow shells by the boundary element method

This paper presents four boundary element formulations for post buckling analysis of shear deformable shallow shells. The main differences between the formulations rely on the way non‐linear terms are treated and on the number of degrees of freedom in the domain. Boundary integral equations are obtained by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Four different sets of non‐linear integral equations are presented. Some domain integrals are treated directly with domain discretization whereas others are dealt indirectly with the dual reciprocity method. Each set of non‐linear boundary integral equations are solved using an incremental approach, where loads and prescribed boundary conditions are applied in small but finite increments. The resulting systems of equations are solved using a purely incremental technique and the Newton–Raphson technique with the Arc length method. Finally, the effect of imperfections (obtained from a linear buckling analysis) on the post‐buckling behaviour of axially compressed shallow shells is investigated. Results of several benchmark examples are compared with the published work and good agreement is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

A computational stream function method for two‐dimensional incompressible viscous flows

This work concerns the development of a numerical method based on the stream function formulation of the Navier–Stokes equations to simulate two‐dimensional—plane or axisymmetric—viscous flows. The main features of the proposed method are: the use of the high order finite‐difference compact method for the discretization of the stream function equation, the implicit pseudo‐transient Newton–Krylov‐multigrid matrix free method for the stationary stream function equation and the fourth order Runge–Kutta method for the integration of non‐stationary flows. Copyright © 2005 John Wiley & Sons, Ltd.     

A GPU‐based preconditioned Newton‐Krylov solver for flexible multibody dynamics

This paper describes an approach to numerically approximate the time evolution of multibody systems with flexible (compliant) components. Its salient attribute is that at each time step, both the formulation of the system equations of motion and their numerical solution are carried out using parallel computing on graphics processing unit cards. The equations of motion are obtained using the absolute nodal coordinate formulation, yet any other multibody dynamics formalism would fit equally well the overall solution strategy outlined herein. The implicit numerical integration method adopted relies on a Newton–Krylov methodology and a parallel direct sparse solver to precondition the underlying linear system. The proposed approach, implemented in a software infrastructure available under an open‐source BSD‐3 license, leads to improvements in overall simulation times of up to one order of magnitude when compared with matrix‐free parallel solution approaches that do not use preconditioning. Copyright © 2015 John Wiley & Sons, Ltd.     

A new predictor–corrector approach for the numerical integration of coupled electromechanical equations

In this paper, a new approach for the numerical solution of coupled electromechanical problems is presented. The structure of the considered problem consists of the low‐frequency integral formulation of the Maxwell equations coupled with Newton–Euler rigid‐body dynamic equations. Two different integration schemes based on the predictor–corrector approach are presented and discussed. In the first method, the electrical equation is integrated with an implicit single‐step time marching algorithm, while the mechanical dynamics is studied by a predictor–corrector scheme. The predictor uses the forward Euler method, while the corrector is based on the trapezoidal rule. The second method is based on the use of two interleaved predictor–corrector schemes: one for the electrical equations and the other for the mechanical ones. Both the presented methods have been validated by comparison with experimental data (when available) and with results obtained by other numerical formulations; in problems characterized by low speeds, both schemes produce accurate results, with similar computation times. When high speeds are involved, the first scheme needs shorter time steps (i.e., longer computation times) in order to achieve the same accuracy of the second one. A brief discussion on extending the algorithm for simulating deformable bodies is also presented. An example of application to a two‐degree‐of‐freedom levitating device based on permanent magnets is finally reported. Copyright © 2015 John Wiley & Sons, Ltd.     

Globalization technique for projected Newton–Krylov methods

Large‐scale systems of nonlinear equations appear in many applications. In various applications, the solution of the nonlinear equations should also be in a certain interval. A typical application is a discretized system of reaction diffusion equations. It is well known that chemical species should be positive otherwise the solution is not physical and in general blow up occurs. Recently, a projected Newton method has been developed, which can be used to solve this type of problems. A drawback is that the projected Newton method is not globally convergent. This motivates us to develop a new feasible projected Newton–Krylov algorithm for solving a constrained system of nonlinear equations. Combined with a projected gradient direction, our feasible projected Newton–Krylov algorithm circumvents the non‐descent drawback of search directions which appear in the classical projected Newton methods. Global and local superlinear convergence of our approach is established under some standard assumptions. Numerical experiments are used to illustrate that the new projected Newton method is globally convergent and is a significate complementarity for Newton–Krylov algorithms known in the literature. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by 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.