Fractional step methods for thermo‐mechanical damage analyses at transient elevated temperatures

In the interest of computational efficiency this paper describes the implementation of a coupled thermo‐damage constitutive model into a coupled time‐stepping analysis using fractional step methods. To begin it is demonstrated that a thermo‐damage model can be presented in a thermodynamic framework with the evolution equations satisfying the first and second laws of thermodynamics. The equations of evolution are partitioned in two ways, thus defining two fractional step methods: an isothermal method and an isentropic method. When implemented into a time‐stepping algorithm the isentropic method maintains a precise energy balance for the entire analysis where as the isothermal method can only provide an energy balance at the end of each thermal time step. In addition, a stability analysis shows that the isentropic analysis is unconditionally stable while a isothermal analysis is at best conditionally stable. Simulations of thermal fracture in a restrained specimen under heat show stable growth of damage to failure. Copyright © 2000 John Wiley & Sons, Ltd.     

An efficient 3D implicit approach for the thermomechanical simulation of elastic–viscoplastic materials submitted to high strain rate and damage

This paper aims at presenting a general consistent numerical formulation able to take into account, in a coupled way, strain rate, thermal and damage effects on the behavior of materials submitted to quasistatic or dynamic loading conditions in a large deformation context. The main features of this algorithmic treatment are as follows:

• A unified treatment for the analysis and implicit time integration of thermo‐elasto‐viscoplastic constitutive equations including damage that depends on the strain rate for dynamic loading conditions. This formalism enables us to use dynamic thermomechanically coupled damage laws in an implicit framework.
• An implicit framework developed for time integration of the equations of motion. An efficient staggered solution procedure has been elaborated and implemented so that the inertia and heat conduction effects can be properly treated.
• An operator split‐based implementation, accompanied by a unified method to analytically evaluate the consistent tangent operator for the (implicit) coupled damage–thermo‐elasto‐viscoplastic problem.
• The possibility to couple any hardening law, including rate‐dependent models, with any damage model that fits into the present framework.

All the developments have been considered in the framework of an implicit finite element code adapted to large strain problems. The numerical model will be illustrated by several applications issued from the impact and metal‐forming domains. All these physical phenomena have been included into an oriented object finite element code (implemented at LTAS‐MN ²L, University of Liège, Belgium) named Metafor.Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive integration in elasto‐plastic boundary element analysis

Abstract

This paper describes an efficient adaptive integration technique for both internal cell integration and boundary element integration. The adaptive algorithm can cope with the common situation where the sizes of adjacent cells and boundary elements are significantly different. Various cases are examined numerically and some numerical applications demonstrate the effectiveness of this method.     

Design of order‐preserving algorithms for transient first‐order systems with controllable numerical dissipation

Using a new design procedure termed as Algorithms by Design, which we have successfully introduced in our previous efforts for second‐order systems, alternatively, we advance in this exposition, the design and development of a computational framework that permits order‐preserving second‐order time accurate, unconditionally stable, zero‐order overshooting behavior, and features with controllable numerical dissipation and dispersion via a family of algorithms for effectively solving transient first‐order systems. The key feature is the incorporation of a spurious root to introduce controllable numerical dissipation while preserving second‐order accuracy (order‐preserving feature) resulting in a two‐root system, namely, the principal root (ρ_1∞) and a spurious root (ρ_2∞). In contrast to the classical Trapezoidal family of algorithms which are the most popular, the present framework has the same order of computational complexity, but a higher payoff that is a significant advance to the field for tackling a wide class of applications dealing with first‐order transient systems. We also present the special case with selection of ρ_1∞ = 1 and any ρ_2∞ leading to the design of a family of generalized single‐step single‐solve [GS4‐1] algorithms recovering the Crank–Nicolson method at one end (ρ_2∞ = 1) and the Midpoint Rule at the other end (ρ_2∞ = 0) and anything in between, all of which have spectral radius features resembling that of the Crank–Nicolson method. More interestingly, with the particular choice of ρ_1∞ = ρ_2∞ = 0, the developed framework additionally inherits L‐stable features. We illustrate the successful design of the developed GS4‐1 framework using two simple illustrative numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

A novel non‐linearly explicit second‐order accurate L‐stable methodology for finite deformation: hypoelastic/hypoelasto‐plastic structural dynamics problems

A novel non‐linearly explicit second‐order accurate L‐stable computational methodology for integrating the non‐linear equations of motion without non‐linear iterations during each time step, and the underlying implementation procedure is described. Emphasis is placed on illustrative non‐linear structural dynamics problems employing both total/updated Lagrangian formulations to handle finite deformation hypoelasticity/hypoelasto‐plasticity models in conjunction with a new explicit exact integration procedure for a particular rate form constitutive equation. Illustrative numerical examples are shown to demonstrate the robustness of the overall developments for non‐linear structural dynamics applications. Copyright © 2004 John Wiley & Sons, Ltd.     

A continuum sensitivity method for finite thermo‐inelastic deformations with applications to the design of hot forming processes

A computational framework is presented to evaluate the shape as well as non‐shape (parameter) sensitivity of finite thermo‐inelastic deformations using the continuum sensitivity method (CSM). Weak sensitivity equations are developed for the large thermo‐mechanical deformation of hyperelastic thermo‐viscoplastic materials that are consistent with the kinematic, constitutive, contact and thermal analyses used in the solution of the direct deformation problem. The sensitivities are defined in a rigorous sense and the sensitivity analysis is performed in an infinite‐dimensional continuum framework. The effects of perturbation in the preform, die surface, or other process parameters are carefully considered in the CSM development for the computation of the die temperature sensitivity fields. The direct deformation and sensitivity deformation problems are solved using the finite element method. The results of the continuum sensitivity analysis are validated extensively by a comparison with those obtained by finite difference approximations (i.e. using the solution of a deformation problem with perturbed design variables). The effectiveness of the method is demonstrated with a number of applications in the design optimization of metal forming processes. Copyright © 2002 John Wiley & Sons, Ltd.     

An energy–momentum consistent method for transient simulations with mixed finite elements developed in the framework of geometrically exact shells

In this paper, a mixed variational formulation for the development of energy–momentum consistent (EMC) time‐stepping schemes is proposed. The approach accommodates mixed finite elements based on a Hu–Washizu‐type variational formulation in terms of displacements, Green–Lagrangian strains, and conjugated stresses. The proposed discretization in time of the mixed variational formulation under consideration yields an EMC scheme in a natural way. The newly developed methodology is applied to a high‐performance mixed shell finite element. The previously observed robustness of the mixed finite element formulation in equilibrium iterations extends to the transient regime because of the EMC discretization in time. Copyright © 2016 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.     

Quadrilateral elements for the solution of elasto‐plastic finite strain problems

In this paper two plane strain quadrilateral elements with two and four variables, are proposed. These elements are applied to the analysis of finite strain elasto‐plastic problems. The elements are based on the enhanced strain and B‐bar methodologies and possess a stabilizing term. The pressure and dilatation fields are assumed to be constant in each element's domain and the deformation gradient is enriched with additional variables, as in the enhanced strain methodology. The formulation is deduced from a four‐field functional, based on the imposition of two constraints: annulment of the enhanced part of the deformation gradient and the relation between the assumed dilatation and the deformation gradient determinant. The discretized form of equilibrium is presented, and the analytical linearization is deduced, to ensure the asymptotically quadratic rate of convergence in the Newton–Raphson method. The proposed formulation for the enhanced terms is carried out in the isoparametric domain and does not need the usually adopted procedure of evaluating the Jacobian matrix in the centre of the element. The elements are very effective for the particular class of problems analysed and do not present any locking or instability tendencies, as illustrated by various representative examples. Copyright © 2001 John Wiley & Sons, Ltd.     

A monolithic computational approach to thermo‐structure interaction

In the present work, a monolithic solution approach for thermo‐structure interaction problems motivated by the challenging application of the behaviour of rocket nozzles is proposed. Structural and thermal fields are independently discretised via finite elements. The resulting system of equations is solved via a monolithic thermo‐structure interaction scheme, which is constructed by a block Gauss–Seidel preconditioner in combination with algebraic multigrid methods. The proposed method is tested for four numerical examples, the second Danilovskaya problem, a simplified rocket nozzle configuration, an internally loaded hollow sphere, and a fully three‐dimensional nozzle configuration of a subscale thrust chamber. Good agreement of the numerical results with results from the literature is observed. Furthermore, it is shown that the monolithic solution algorithm can handle the complete range of the parameter spectrum, whereas partitioned algorithms are limited to a certain parameter range only. Moreover, the monolithic algorithm exhibits improved efficiency and robustness compared to partitioned algorithms. Copyright © 2013 John Wiley & Sons, Ltd.     

Energy‐momentum consistent algorithms for dynamic thermomechanical problems—Application to mortar domain decomposition problems

An energy‐momentum consistent integrator for non‐linear thermoelastodynamics is newly developed and extended to domain decomposition problems. The energy‐momentum scheme is based on the first law of thermodynamics for strongly coupled, non‐linear thermoelastic problems. In contrast to staggered algorithms, a monolithic approach, which solves the mechanical as well as the thermal part simultaneously, is introduced. The approach is thermodynamically consistent in the sense that the first law of thermodynamics is fulfilled. Furthermore, a domain decomposition method for the thermoelastic system is developed based on previous developments in the context of the mortar method. The excellent performance of the new approach is illustrated in representative numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

A0 – Stable family of single step methods for semidiscretized parabolic problems

A general family of single-step methods for use in first-order ODEs is presented. This family of methods, called SS3-methods, embraces generalized trapezoidal methods, SS21-methods, α-methods and many other well-known methods, thus providing a good basis for comparison of these methods. Corresponding parameter values of SS3 methods for these methods are presented. The analysis of SS3-methods is restricted to linear and symmetric systems. Stability, convergence, accuracy, numerical dissipation and overshoot of the methods are considered. The conditions for algorithm parameters to exhibit good characteristics in these respects are given. It is shown that the optimal subfamily of SS3-methods coincides with α-methods.     

A comparative study of stress update algorithms for rate‐independent and rate‐dependent crystal plasticity

The paper presents a comparative discussion of stress update algorithms for single‐crystal plasticity at small strains. The key result is a new unified fully implicit multisurface‐type return algorithm for both the rate‐independent and the rate‐dependent setting, endowed with three alternative approaches to the regularization of possible redundant slip activities. The fundamental problem of the rate‐independent theory is the possible ill condition due to linear‐dependent active slip systems. We discuss three possible algorithmic approaches to deal with this problem. This includes the use of alternative generalized inverses of the Jacobian of the currently active yield criterion functions as well as a new diagonal shift regularization technique, motivated by a limit of the rate‐dependent theory. Analytical investigations and numerical experiments show that all three approaches result in similar physically acceptable predictions of the active slip of rate‐independent single‐crystal plasticity, while the new proposed diagonal shift method is the most simple and efficient concept. Copyright © 2001 John Wiley & Sons, Ltd.     

A computational approach for multi‐scale analysis of heterogeneous elasto‐plastic media subjected to short duration loads

The asymptotic expansion homogenization (AEH) approach has found wide acceptance for the study of heterogeneous structures due to its ability to account for multi‐scale features. The emphasis of the present study is to develop consistent AEH numerical formulations to address elasto‐plastic material response of structures subjected to short‐duration transient loading. A second‐order accurate velocity‐based explicit time integration method, in conjunction with the AEH approach, is currently developed that accounts for large deformation non‐linear material response. The approach is verified under degenerate homogeneous conditions using existing experimental data in the literature and its ability to account for heterogeneous conditions is demonstrated for a number of test problems. Copyright © 2004 John Wiley & Sons, Ltd.     

On numerical analyses in the presence of unstable saturated porous materials

The numerical analysis of the dynamic evolution problem concerning an elastic–plastic saturated porous media in the presence of softening (or non‐associativity) is considered in the framework of the Biot formulation extended to take into account plastic phenomena. The finite step boundary value problem, obtained by discretization in time of the continuous initial boundary value problem, is studied and the issue of its ill‐posedness is particularly addressed. The conditions for the loss of ellipticity are established for the linearized problem solved at each iteration when using the Newton–Raphson scheme. In particular, the roles of the algorithmic properties on this loss of ellipticity are derived in detail. The integration scheme of the balance of mass equation plays a major role and it is shown that the fluid flow (Darcy's law) does indeed introduce a length scale but in addition to being dependent on the integration time step, it is found to be insufficient for regularization. To illustrate and corroborate the obtained results, a one‐dimensional example (exhibiting all the features of the three‐dimensional situation) is considered and the corresponding linearized finite step problem is solved in closed form. Copyright © 2002 John Wiley & Sons, Ltd.     

A new hybrid velocity integration method applied to elastic wave propagation

We present a novel space–time Galerkin method for solutions of second‐order time‐dependent problems. By introducing the displacement–velocity relationship implicitly, the governing set of equations is reformulated into a first‐order single field problem with the unknowns in the velocity field. The resulting equation is in turn solved by a time‐discontinuous Galerkin approach (Int. J. Numer. Anal. Meth. Geomech. 2006; 30 :1113–1134), in which the continuity between time intervals is weakly enforced by a special upwind flux treatment. After solving the equation for the unknown velocities, the displacement field quantities are computed a posteriori in a post‐processing step. Various numerical examples demonstrate the efficiency and reliability of the proposed method. Convergence studies with respect to the h‐ and p‐refinement and different discretization techniques are given. Copyright © 2007 John Wiley & Sons, Ltd.     

Elasto‐plasticity revisited: numerical analysis via reproducing kernel particle method and parametric quadratic programming

Aiming to simplify the solution process of elasto‐plastic problems, this paper proposes a reproducing kernel particle algorithm based on principles of parametric quadratic programming for elasto‐plasticity. The parametric quadratic programming theory is useful and effective for the assessment of certain features of structural elasto‐plastic behaviour and can also be exploited for numerical iteration. Examples are presented to illustrate the essential aspects of the behaviour of the model proposed and the flexibility of the coupled parametric quadratic programming formulations with the reproducing kernel particle method. Copyright © 2002 John Wiley & Sons, Ltd.