Anthony Gravouil

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

An algorithm to solve transient structural non-linear problems for non-matching time-space domains

Abstract This paper presents a new transient algorithm to couple sub-domains having non-matching time and space interfaces. The time integrators in each of the sub-domains are of the Newmark family but can be different. The method is also developed for non-linear explicit transient computations for which the time steps are not constant. A number of examples are presented which illustrate the interest, quality and the efficiency of the proposed method.

Extended finite element method for numerical simulation of 3D fatigue crack growth

Abstract This paper is devoted to the presentation of the Extended Finite Element Method (X-FEM) compared to usual Finite Element Method for simulation of 3D fatigue crack propagation under complex loading. This type of situation is very typical of crack propagation in tribological environment. The X-FEM method will be presented in a first part, and two techniques allowing the discretization of cracks will be explained. The interaction integral concept, which allows to compute with a very good accuracy the stress intensity factors KI, KII, KIII, in complex situation will be presented. Once these tools are available, the method can be used very efficiently to simulate crack propagation without remeshing. This property is essential because it guarantees exact energy conservation when the crack is updated. The precise knowledge of individual stress intensity factors KI ,KII and KIII allows deciding the direction of the crack growth (using any crack propagation law). This tool can hence be used to identify the propagation laws in complex cases using comparison between experimental and numerical studies. One example of stress intensity factor computation using a 3D real crack measured by tomography will be shown. 3D crack propagation simulation will be presented. Very recent results will be given for small scale yielding and crack closure situations, which permit to explain, for instance, slowed down crack propagation due to overload.

X-Ray Micro-Tomography Coupled to the Extended Finite Element Method to Investigate Microstructurally Short Fatigue Cracks

In this paper we will present how it is possible to couple a 3D experimental technique with a 3D numerical method in order to calculate the stress intensity factors along the crack front taking into account the real shape of the crack. This approach is used to characterize microstructurally short fatigue cracks that exhibit a rather complicated 3D shape. The values of the stress intensity factors are calculated along the crack front at different stages of crack propagation and it can be seen that the crack shape irregularities introduce rather important fluctuations of the values of KI, KII and KIII along the crack front. The values of KI obtained taking into account the real shape of the crack are significantly different from the ones calculated using an approach based on a shape assumption

Space–time POD based computational vademecums for parametric studies: application to thermo-mechanical problems

Standard numerical simulations for optimization or inverse identification of welding processes remain costly and difficult due to their multi-parametric aspect and inherent complexity. The aim of this paper is to propose a non-intrusive strategy for building computational vademecums dedicated to real-time simulations of nonlinear thermo-mechanical problems. There is in essence, a set of precomputed space–time parametric solutions (snapshots), selected by an appropriate approach in the parameter space and stored in memory as quasi-optimal reduced bases (RBs) provided by the proper orthogonal decomposition method. Once the RBs are obtained, the computational vademecums can be used online and provide real-time space–time transient nonlinear thermo-mechanical solutions for any desired parameter value. The contributions of the paper consist in a space–time RBs interpolation approach with the Grassmann manifolds method, and a localized multigrid selection method that allows an automatic selection of snapshots in the parameter areas of interest for a given level of accuracy. As application, the welding simulation is considered with a transient non-linear thermo-mechanical model using the finite element method. It is shown that the moving frame allows an optimal design of the RBs. A good efficiency of the proposed approach is demonstrated. Computational vademecums can be used for optimization or inverse identification problems of welding.

Automatic refinement and efficient solver for non linear dynamic structural problems

This paper presents an adaptive strategy dedicated to non-linear transient dynamic problems. The spatial mesh is optimized to ensure the accuracy of the solution. Beginning from a coarse mesh, an error indicator is used to estimate the discretization error and new elements are created where the prescribed accuracy is not reached. A localized multigrid solver is used and the strategy is applied recursively until the local mesh size ensures that the discretization error is less than the prescribed accuracy. The spatial mesh is recreated at each time step.

The Extended Finite Element and Level Set Methods for Non-Planar 3D Crack Growth

A methodology for treating non-planar three-dimensional cracks with geometries that are independent of the mesh is described. The method is based on the extended finite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The crack geometry is described by two signed distance functions (level sets), which in turn can be defined by nodal values. Consequently, no explicit representation of the crack is needed. A Hamilton-Jacobi equation is used to update the level sets as the crack grows. Numerical experiments show the robustness of the method in treating cracks with significant changes in topology. The method is readily extendable to inelastic fracture problems.

A time variational method to couple heterogeneous time integrators

This paper is devoted to a brief presentation of recent research results upon structural mechanics code coupling in transient analysis. The domain is supposed to be decomposed into a series of sub domains which are treated independently with their own time integration scheme and or their own code. The paper gives a general method which allows to couple these subdomains. The proposed method is rather general and based upon a weak vision of dynamic equilibrium equation. This new vision allows to design a coupling strategy which ensure by design that no energy is introduced or dissipated in the interfaces between the sub domains. The proposed coupling method hence does not perturb the quality of the time integrators of each sub domain. This also allows to develop a general code coupler for transient dynamics. Two examples are given to illustrate the paper.

Application of X-FEM to 3D Real Cracks and Elastic-Plastic Fatigue Crack Growth

In a general point of view, X-FEM plus level set representation of the interfaces reveals to be of great interest in the aim to couple experimental data with numerical simulations. This can be highly illustrated in the case of 3D fatigue crack growth simulations where the initial 3D “real crack” is extracted from tomo-graphic images. The experimentally observed fatigue crack propagation is compared to numerical simulations. Good agreement is found when a linear variation of closure stress along the crack front is taken into account in the “3D crack propagation law” used for the simulation. Furthermore, in order to take into account plasticity during fatigue crack growth, one develops an augmented Lagrangian formulation in the X-FEM framework that is able to deal with elastic-plastic crack growth with treatment of contact and friction. Numerical issues such as contact treatment and numerical integration are addressed, and finally numerical examples are shown to validate the method.

2D X-FEM Simulation of Dynamic Brittle Crack Propagation

The application of X-FEM technique to the prediction of two dimensional dynamic brittle crack growth is presented in this paper. The method is known to guarantee exact energy conservation in case of crack propagation and it is applied to the simulation of one dynamic crack propagation experiment submitted to a mixed mode loading and showing stop and restart of a crack.