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## Estimation of the validity domain of hyper-reduction approximations in generalized standard elastoviscoplasticity

### Advanced Modeling and Simulation in Engineering Sciences (2015-05-19) 2: 1-19 , May 19, 2015

### Background

We propose an a posteriori estimator of the error of hyper-reduced predictions for elastoviscoplastic problems. For a given fixed mesh, this error estimator aims to forecast the validity domain in the parameter space, of hyper-reduction approximations. This error estimator evaluates if the simulation outputs generated by the hyper-reduced model represent a convenient approximation of the outputs that the finite element simulation would have predicted. We do not account for the approximation error related to the finite element approximation upon which the hyper-reduction approximation is introduced.

### Methods

We restrict our attention to generalized standard materials. Upon use of incremental variational principles, we propose an error in constitutive relation. This error is split into three terms including a tailored norm of the hyper-reduction approximation error. This error norm is defined by using the convexity of an incremental potential introduced to state the constitutive equations. The second term of the a posteriori error is related to the stress recovery technique that generates stresses fulfilling the finite element equilibrium equations. The last term is a coupling term between the hyper-reduction approximation error at each time step and the errors committed before this time step. Unfortunately, this last term prevents error certification. In this paper, we restrict our attention to outputs extracted by a Lipschitz function of the displacements.

### Results

In the proposed numerical examples, we show very good preliminary results in predicting the validity domain of hyper-reduction approximations. The average computational time of the predictions obtained by hyper reduction, is accelerated by a factor of 6 compared to that of finite element simulations. This speed-up incorporates the computational time devoted to the error estimation.

### Conclusions

The numerical implementation of the proposed error estimator is straightforward. It does not require the computation of the incremental potential. In the numerical results, the estimated validity domain of hyper-reduced approximations is inside the reference validity domain. This paper is a first attempt for a posteriori error estimation of hyper-reduction approximations.

## Coupling local and non-local damage evolutions with the Thick Level Set model

### Advanced Modeling and Simulation in Engineering Sciences (2014-10-15) 1: 1-21 , October 15, 2014

### Background

The Thick Level Set model (TLS) is a recent method to delocalize local constitutive models suffering spurious localization. It has two major advantages compared to other delocalization methods. The first one is that the transition from localization to fracture is taken into account in the model. The second one is that the delocalization only acts when and where needed. In other words, the TLS has no effect when the local model is stable. The former advantage was already detailed in several papers (IJNME 86:358-380, 2011, CMAME 233:11-27, 2012, IJF 174:49-60, 2012). This paper concentrates on the latter advantage.

### Methods

The TLS delocalization approach is formulated as a bound on the damage gradient. The non-local zone is defined as the zone where the bound is met whereas the local zone is defined as the zone where it is not met. The boundary (localization front) between the local and non-local zone is the main unknown in the problem.

### Results

Based on the new model, a 1D pull-out test is solved both analytically and numerically. Different regimes are observed in the solution as the loading progresses: fully elastic, local damage, coupled local/non-local damage and, finally, purely non-local damage.

### Conclusions

The new model introduces delocalization as an inequality allowing local damage to develop in zones whereas non-local damage may develop in other zones. This reduces dramatically the cost of implementation of such models compared to fully non-local models.

## Static fracture and modal analysis simulation of a gas turbine compressor blade and bladed disk system

### Advanced Modeling and Simulation in Engineering Sciences (2016-11-24) 3: 1-23 , November 24, 2016

This paper presents a methodology for conducting a 3-D static fracture analysis with applications to a gas turbine compressor blade. An open crack model is considered in the study and crack-tip driving parameters are estimated by using 3-D singular crack-tip elements in ANSYS $$\circledR $$ . The static fracture analysis is verified with a special purpose fracture code (FRANC3D). Once the crack front is perfectly defined and validated, a free vibration study is conducted by analyzing the natural frequencies and modeshapes for both a single blade and bladed disk system. Taking advantage of high performance computing resources, a high fidelity finite element model is considered in the parametric investigation. In the fracture simulation, the influence of the size of a single edged crack as well as the rotational velocity on fracture parameters (stress intensity factors and J-Integral) are evaluated. Results demonstrate that for the applied loading condition, a mixed mode crack propagation is expected. In the modal analysis study, increasing the depth of the crack leads to a decrease in the natural frequencies of both the single blade and bladed disk system, while increasing the rotational velocity increases the natural frequencies. The presence of a crack also leads to mode localization for all mode families, a phenomenon that cannot be captured by a single blade analysis.

## Separation of particles in non-Newtonian fluids flowing in T-shaped microchannels

### Advanced Modeling and Simulation in Engineering Sciences (2015-06-16) 2: 1-23 , June 16, 2015

### Background

The flow of suspensions through bifurcations is encountered in several applications. It is known that the partitioning of particles at a bifurcation is different from the partitioning of the suspending fluid, which allows particle separation and fractionation. Previous works have mainly investigated the dynamics of particles suspended in Newtonian liquids.

### Methods

In this work, we study through 2D direct numerical simulations the partitioning of particles suspended in non-Newtonian fluids flowing in a T-junction. We adopt a flow configuration such that the two outlets are orthogonal, and their flow rates can be tuned. A fictitious domain method combined with a grid deformation procedure is used. The effect of fluid rheology on the partitioning of particles between the two outlets is investigated by selecting different constitutive equations to model the suspending liquid. Specifically, an inelastic shear-thinning (Bird-Carreau) and a viscoelastic shear-thinning (Giesekus) models have been chosen; the results are also compared with the case of a Newtonian suspending liquid.

### Results

Simulations are carried out by varying the confinement, the inlet flow rate and the relative weight of the two outlet flow rates. For each condition, the fluxes of particles through the two outflow channels are computed. The results show that shear-thinning does not have a relevant effect as compared to the equivalent Newtonian case, i.e., with the same choice of the relative outlet flow rates. On the other hand, fluid elasticity strongly alters the fraction of particles exiting the two outlets as compared to the inlet. Such effect is more pronounced for larger particles and inlet flow rates.

### Conclusions

The results illustrated here show the feasibility to efficiently separate/fractionate particles by size, through the use of viscoelastic suspending liquids.

## An upwind least square formulation for free surfaces calculation of viscoplastic steady-state metal forming problems

### Advanced Modeling and Simulation in Engineering Sciences (2015-07-14) 2: 1-27 , July 14, 2015

Despite using very large parallel computers, numerical simulation of some forming processes such as multi-pass rolling, extrusion or wire drawing, need long computation time due to the very large number of time steps required to model the steady regime of the process. The direct calculation of the steady-state, whenever possible, allows reducing by 10–20 the computational effort. However, removing time from the equations introduces another unknown, the steady final shape of the domain. Among possible ways to solve this coupled multi-fields problem, this paper selects a staggered fixed-point algorithm that alternates computation of mechanical fields on a prescribed domain shape with corrections of the domain shape derived from the velocity field and the stationary condition *v.n* = 0. It focuses on the resolution of the second step in the frame of unstructured 3D meshes, parallel computing with domain partitioning, and complex shapes with strong contact restraints. To insure these constraints a global finite elements formulation is used. The weak formulation based on a Galerkin method of the v.n = 0 equation is found to diverge in severe tests cases. The least squares formulation experiences problems in the presence of contact restraints, upwinding being shown necessary. A new upwind least squares formulation is proposed and evaluated first on analytical solutions. Contact being a key issue in forming processes, and even more with steady formulations, a special emphasis is given to the coupling of contact equations between the two problems of the staggered algorithm, the thermo-mechanical and free surface problems. The new formulation and algorithm is finally applied to two complex actual metal forming problems of rolling. Its accuracy and robustness with respect to the shape initialization of the staggered algorithm is discussed, and its efficiency is compared to non-steady simulations.

## A micromechanics-based interface mesomodel for virtual testing of laminated composites

### Advanced Modeling and Simulation in Engineering Sciences (2014-01-29) 1: 1-16 , January 29, 2014

### Background

The prediction of the behavior of laminated composite structures up to final fracture continues to be a challenge today. Indeed, failure may occur due to the interaction of small-scale degradations, such as transverse intraply cracks and interface delamination, which are difficult to account for in calculations on the structure’s scale.

### Methods

Here, in order to model the interaction of intralaminar and interlaminar degradations, we develop a new and relatively simple micromechanics-based interface mesomodel which differs from classical cohesive interface models, since it includes the coupling between transverse intraply cracks and interface delamination.

### Results

The new interface model was implemented in a finite element code and used in the simulation of tensile tests on unnotched and holed specimens. Simulations with a classical cohesive interface model (not including coupling) were also carried out.

### Conclusions

The simulations highlight the need for introducing intra-/interlaminar’s behavior coupling in order to accurately predict the damage evolution and failure stress and mode.

## High order cut finite element methods for the Stokes problem

### Advanced Modeling and Simulation in Engineering Sciences (2015-09-25) 2: 1-23 , September 25, 2015

We develop a high order cut finite element method for the Stokes problem based on general inf-sup stable finite element spaces. We focus in particular on composite meshes consisting of one mesh that overlaps another. The method is based on a Nitsche formulation of the interface condition together with a stabilization term. Starting from inf-sup stable spaces on the two meshes, we prove that the resulting composite method is indeed inf-sup stable and as a consequence optimal *a priori* error estimates hold.

## General treatment of essential boundary conditions in reduced order models for non-linear problems

### Advanced Modeling and Simulation in Engineering Sciences (2016-03-08) 3: 1-14 , March 08, 2016

Inhomogeneous essential boundary conditions must be carefully treated in the formulation of Reduced Order Models (ROMs) for non-linear problems. In order to investigate this issue, two methods are analysed: one in which the boundary conditions are imposed in an strong way, and a second one in which a weak imposition of boundary conditions is made. The ideas presented in this work apply to the big realm of a posteriori ROMs. Nevertheless, an a posteriori hyper-reduction method is specifically considered in order to deal with the cost associated to the non-linearity of the problems. Applications to nonlinear transient heat conduction problems with temperature dependent thermophysical properties and time dependent essential boundary conditions are studied. However, the strategies introduced in this work are of general application.

## ASTER Shell: a simple concept to significantly increase the plastic buckling strength of short cylinders subjected to combined external pressure and axial compression

### Advanced Modeling and Simulation in Engineering Sciences (2015-10-13) 2: 1-27 , October 13, 2015

This paper proposes a new type of shell, similar to a cylindrical shell, which has significantly higher buckling strength when subjected to an arbitrary combination of uniform external pressure and axial compression. The underlying principle consists in a slight modification of the perfect cylinder in order to counteract the natural deformations which get larger and larger and lead to the collapse of the structure. Such shells are called ASTER shells. The concept has been validated through experiments, then analyzed numerically in order to explain what was observed and to propose avenues for improvements. The shells were made of electrodeposited nickel. The material was characterized. The chosen specimens were carefully measured to characterize their thickness and initial imperfections, then tested under the various types of loading. Then they were analyzed using finite elements. Thus, we were able to compare the finite element predictions with the experimental results. This comparison shows that plasticity has a decisive influence on the critical load and that linear elastic dimensioning leads to a serious overestimation of the experimental critical load. Contrary to perfect cylindrical shells, this type of shell is not significantly affected by geometric imperfections: this is another advantage of this type of design. Finally, we propose a numerical analysis in order to optimize the choice of the shape and propose shapes which resist buckling much better than a smooth cylinder when subjected to uniform external pressure, axial compression or a combination of both.

## Recursive POD expansion for reaction-diffusion equation

### Advanced Modeling and Simulation in Engineering Sciences (2016-03-09) 3: 1-22 , March 09, 2016

This paper focuses on the low-dimensional representation of multivariate functions. We study a recursive POD representation, based upon the use of the power iterate algorithm to recursively expand the modes retained in the previous step. We obtain general error estimates for the truncated expansion, and prove that the recursive POD representation provides a quasi-optimal approximation in $$L^2$$ norm. We also prove an exponential rate of convergence, when applied to the solution of the reaction-diffusion partial differential equation. Some relevant numerical experiments show that the recursive POD is computationally more accurate than the Proper Generalized Decomposition for multivariate functions. We also recover the theoretical exponential convergence rate for the solution of the reaction-diffusion equation.