Pierre Ladeveze

• Eng., Ph.D and D.Sc. degrees
• Professor Emeritus at Ecole normale supérieure de Cachan

We provide first the functional analysis background required for reducedorder modeling and present the underlying concepts of reduced basis model reduction. The projection-based model reduction framework under affinity assumptions, offline-online decomposition, and error estimation are introduced. Several tools for geometry parameterizations such a...

The so-called “reduced” models have always been very popular and often essential in engineering to analyze the behavior of structures and materials, especially in dynamics. They highlight the relevant information and lead, moreover, to less expensive and more robust calculations. In addition to conventional reduction methods, a generation of reduct...

High fidelity structural problems that involve nonlinear material behaviour, when subjected to cyclic loading, usually demand infeasible computational resources; this demonstrates the need for efficient model order reduction (MOR) techniques in order to shrink these demands to fit into the available means. The solution of cyclic damage problems in...

This paper introduces a new vision of data-driven structure computation taking advantage of Material Science, especially for highly nonlinear and time-dependent material behaviours. Technical solutions are also derived, in order to build internal hidden variables defining the so-called “Experimental Constitutive Manifold”.

The goal of this paper is to introduce a model order reduction method for high-cycle fatigue simulations using a kinetic damage model, i.e. a constitutive model in which the damage evolution law is defined as a rate form for the damage variable. In the framework of continuum mechanics, high-cycle fatigue simulation involves a two-scale damage model...

The LATIN-PGD method is a fast non-linear and non incremental solver generally applied in solid mechanics that use on their formulation a model reduction technique called Proper Generalize Decomposition. To date, the method has only been applied for solving the response of structures under quasistatic conditions, leaving aside the inertial effects....

The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive inc...

The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive inc...

The objective of this article is to introduce a new method including model order reduction for the life prediction of structures subjected to cycling damage. Contrary to classical incremental schemes for damage computation, a non-incremental technique, the LATIN method, is used herein as a solution framework. This approach allows to introduce a PGD...

To establish a reliable virtual testing strategy for composite design, the damage mesomodel for laminated composites has been developed at LMT-Cachan since the 1980s. Nevertheless, the predictions using the mesomodel could be too conservative in particular situations such as composite structures involving extensive splitting. Indeed, the predicted...

One of the challenges of fatigue simulation using continuum damage mechanics framework over the years has been reduction of numerical cost while maintaining acceptable accuracy. The extremely high numerical expense is due to the temporal part of the quantities of interest which must reflect the state of a structure that is subjected to exorbitant n...

Model reduction techniques such as Proper Generalized Decomposition (PGD) are decision-making tools that are about to revolutionize many domains. Unfortunately, their computation is still problematic for problems involving many parameters, for which one has to face the “curse of dimensionality”. An answer to this challenge is given in solid mechani...

Reduced models and especially those based on Proper Generalized Decomposition (PGD) are decision-making tools which are about to revolutionize many domains. Unfortunately, their calculation remains problematic for problems involving many parameters, for which one can invoke the “curse of dimensionality”. The paper starts with the state-of-the-art f...

The simulation of mechanical responses of structures subjected to cyclic loadings for a large number of cycles remains a challenge. The goal herein is to develop an innovative computational scheme for fatigue computations involving non-linear mechanical behaviour of materials, described by internal variables. The focus is on the Large Time Incremen...

This contribution focuses on the use of a new method to reduce the computational demands of fatigue damage computations using continuum damage mechanics. The LArge Time INcrement (LATIN) method incorporates a model order reduction approach namely the proper generalised decomposition (PGD). LATIN has been extended to tackle damage problems. (© 2017...

The use of constitutive equations calibrated from data has been implemented into standard numerical solvers for successfully addressing a variety problems encountered in simulation-based engineering sciences (SBES). However, the complexity remains constantly increasing due to the need of increasingly detailed models as well as the use of engineered...

Different numerical tests for fatigue damage in metals using non-incremental approach

Different numerical tests for fatigue damage in metals using non-incremental approach

Non incremental approach for simulating fatigue damage in metals.

Medium frequency heterogeneous Helmholtz problem gives rise to severe numerical challenge to Finite Element Method (FEM). An extended Variational Theory of Complex Rays (VTCR) is developed for solving this problem. Since VTCR is kind of Trefftz method, analytic solutions of governing equation need to be known a priori. But this requirement limits t...

We define an a posteriori verification procedure that enables to control and certify PGD-based model reduction techniques applied to parametrized linear elliptic or parabolic problems. Using the concept of constitutive relation error, it provides guaranteed and fully computable global/goal-oriented error estimates taking both discretization and PGD...

The paper deals with the use of model order reduction within a posteriori error estimation procedures in the context of the Finite Element Method. More specifically, it focuses on the Constitutive Relation Error (CRE) concept which has been widely used over the last 40 years for FEM verification of Computational Mechanics models. A technical key-po...

In the past years, a numerical technique method called Variational Theory of Complex Rays (VTCR) has been developed for vibration problems in medium frequency. It is a Trefftz Discontinuous Galerkin method which uses plane wave functions as shape functions. However this method is only well developed in homogeneous case. In this paper, VTCR is exten...

A new approximation technique, called Reference Point Method (RPM), is proposed in order to reduce the computational complexity of algebraic operations for constructing reduced-order models in the case of time dependent and/or parametrized nonlinear partial differential equations. Even though model reduction techniques enableone to decrease the dim...

The use of constitutive equations calibrated from data collected from adequate testing has been implemented successfully into standard solvers for successfully addressing a variety of problems encountered in SBES (simulation based engineering sciences). However, the complexity remains constantly increasing due to the more and more fine models being...

The chapter deals with a crucial question for the design of composite structures: how can one predict the evolution of damage up to and including final fracture? Virtual testing, whose goal is to drastically reduce the huge number of industrial tests involved in current characterization procedures, constitutes one of today’s main industrial challen...

The capabilities and limits of current model reduction methods are examined in the case of solid mechanics problems involving significant nonlinearities—such as (visco)plasticity, damage, contact with friction, …—and parameters. Particular emphasis will be put on the PGD method (Proper Generalized Decomposition) and its last developments. These red...

Nowadays, the interest of aerospace and automotive industries on virtual testing of medium-frequency vibrational behavior of shallow shell structures is growing. The development of software capable of predicting the vibrational response in such frequency range is still an open question because classical methods (i.e., FEM, SEA) are not fully suitab...

In this paper, we consider the Weak Trefftz Discontinuous Galerkin (WTDG) method, which enables one to use at the same time the Finite Element Method (FEM) or Variational Theory of Complex Rays (VTCR) discretizations (polynoms and waves), for vibration problems. It has already been developed such that the FEM and the VTCR can be used in different a...

The third generation of composite materials possesses a functional interphase between plies, which is reinforced by thermoplastic particles. Understanding the mechanism of matrix crack propagation through the interphase is an important prerequisite to the development of a robust code for the computation of damage resistance of cross ply laminates p...

LATIN-PGD approach for cyclic viscoplasticity involving large number of cycles

The present paper is a review of the main activities carried out within the context of the COMPTINN` program, a joint research project founded by a FUI program (Fonds Unifi{\acute{e}}sInterminist{\acute{e}}riels) in which four research teams focused on the thermo-oxidation behaviour of HTS-TACTIX carbon-epoxy composite at `high` temperatures (120^{...

This article deals with the efficient construction of approximations of fields and quantities of interest used in geometric optimisation of complex shapes that can be encountered in engineering structures. The strategy, which is developed herein, is based on the construction of virtual charts that allow, once computed offline, to optimise the struc...

This paper is a revisit of the work [Ladevèze and Pelle, Int. J. Numer. Methods Engrg. 28 (1989)] where the goal here is to acquire guaranteed, accurate and computable bounds of eigenfrequencies through post-processing of conventional finite element results. To this end, a new theoretical quotient is introduced and thereafter, a practical way to de...

This chapter reviews the Constitutive Relation Error method as a general verification tool which is very suitable to compute strict and effective error bounds for linear and more generally convex Structural Mechanics problems. The review is focused on the basic features of the method and the most recent developments.

Recently, interest of aerospace and automotive industries on medium-frequency vibrational behavior of composite shell structures has grown due to their high specific stiffness and fatigue resistance. Conventional methods such as the finite element method and the statistical energy analysis are not suitable for the medium-frequency bandwidth. Conver...

This paper investigates local phenomena that govern the damage behaviour of laminated composites of third generation that possess a functional interphase between plies reinforced by thermoplastic particles. The effects of interphase parameters (thickness and mod-ulus) on the propagation of the matrix transverse crack and on the micro-delamination p...

The use of polymer matrix composites at relatively high temperatures (150-300°C) requires accounting for the oxidation of the polymer and its coupling with mechanical degradation. A vast body of literature investigates these phenomena at very small scales, from the chemistry of the polymer to the mechanical behavior at the fiber's scale. The aim of...

First, the effectivity of classical Proper Generalized Decomposition (PGD) computational methods is analyzed on a one dimensional transient diffusion benchmark problem, with a moving load. Classical PGD methods refer to Galerkin, Petrov–Galerkin and Minimum Residual formulations. A new and promising PGD computational method based on the Constitutiv...

The calculation of vibrational responses of complex systems on frequency bands appears to be more and more important in engineering simulation. This is particularly true in the medium frequency regime where the solution is very sensitive to the frequency. In this work, we propose a new path to determine the frequency response of a system at many fr...

This paper introduces a review of the main activities carried out within the context of the 'COMPTINN' research program, for the activities focusing on the durability and thermo-oxidation behaviour of organic matrix composite materials at 'high' temperatures (120°C-180°C). The scientific aim of 'COMPTINN' research program was to better identify, wi...

The evolution of damage in a woven composite results from the interaction and competition among several elementary mechanisms, which are greatly influenced by the microstructure. Thus, in order to develop a model as close as possible to the underlying physics, it is necessary to observe, understand and describe each mechanism on the most appropriat...

Ceramic matrix composites (CMC) are good candidates for structural applications at high temperatures in oxidizing environments. A main question is then to be able to predict, that is, to compute the lifetime of engineering structures. Multiphysic models are needed to reproduce couplings between the mechanical and environmental fields.The chapter is...

We introduce a goal-oriented procedure for the updating of mechanical models. It exploits as usual information coming from experimental data, but these are post-processed in a specific way in order to firstly update model parameters which are the most influent for the prediction of a given quantity of interest. The objective is thus to perform a pa...

The Variational Theory of Complex Rays (VTCR) was developed in order to calculate the vibrational response of structures in the medium-frequency range. It leads to a numerical approximation of this response through the resolution of a small system of equations which, contrary to element-based methods, does not result from a refined spatial discreti...

In this paper, the wave approach called the Variational Theory of Complex Rays (VTCR), which was developed for medium-frequency acoustics and vibrations, is revisited as a discontinuous Galerkin method. Extensions leading to a weak Trefftz constraint are introduced. This weak Trefftz discontinuous Galerkin approach enables hybrid FEM/VTCR strategie...

When dealing with high-fidelity models, the number of degrees of freedom can lead to sys-tems so large that direct techniques are unsuitable. Reduced order modeling (ROM) seeks to reduce the computational complexity and computational time of large-scale systems by approximations of much lower dimension. Several techniques have been developed in ord...

WCCM XI - 11th World Congress on Computational Mechanics, BARCELONE, ESPAGNE, 20-/07/2014 - 25/07/2014

In this paper, we introduce a goal-oriented procedure for the updating of mechanical models. It is based as usual on information coming from measurement data, but these data are post-processed in a convenient way in order to firstly update model parameters which are the most influent for the prediction of a given quantity of interest. The objective...

The calculation of the acoustic response of systems in frequency bands is becoming increasingly important in simulation-based engineering design. This is particularly true in medium-frequency bands, where the response is very sensitive to the frequency. Some standard techniques for addressing these problems present a frequency dependent formulation...

Woven composite materials are receiving particular attention in a wide range of specialized aeronautical applications. Reliable numerical prediction tools based on computational modeling are required to quan-titatively characterize the role of the microstructure and damage mechanisms at the mesoscale. In this paper, such a computational strategy is...

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....

Today, the prediction of vibrational responses of industrial systems has become more and more important. To do this, it is of great interest to use numerical strategies able to take into account the complexity of these systems, which have then to be efficient in different frequency regimes. The finite element method is very useful in the low freque...

The evolution of damage in a woven composite results from the interaction and competition among several elementary mechanisms, which are greatly influenced by the microstructure. Thus, in order to develop a model as close as possible to the physics, it is necessary to observe, understand and describe each mechanism on the most appropriate scale. Th...

Mechanics continues to supply numerous science and engineering problems which remain inaccessible to standard FE codes. Not all these problems are exotic, and many are indeed practical problems. A significant number of these engineering challenges are related to the today’s growing interest in physics-based material models described on a scale smal...

The papers in this volume start with a description of the construction of reduced models through a review of Proper Orthogonal Decomposition (POD) and reduced basis models, including their mathematical foundations and some challenging applications, then followed by a description of a new generation of simulation strategies based on the use of separ...

This paper deals with the offline resolution of nonlinear multiscale problems leading to a PGD-reduced model from which one can derive microinformation which is suitable for design. Here, we focus on an improvement to the Proper Generalized Decomposition (PGD) technique which is used for the calculation of the homogenized operator and which plays a...

The accurate prediction of damage and failure in laminated composites is still a major issue in many structural applications. The paper provides a detailed description of a new damage mesomodel and examines its application to solve material and structural problems for the Test Cases proposed in the Third World-Wide Failure Exercise (WWFE-III). The...

This work concerns the Proper Generalized Decomposition (PGD), an a priori model reduction tech-nique used to solve problems, eventually nonlinear, defined over the time-space domain. PGD seeks the solution of a problem in a reduced-order basis generated by a dedicated algorithm. This is the LATIN method, an iterative strategy which generates the a...

In this work, we propose new bounding techniques that enable to derive accurate and strict error bounds on outputs of interest computed from numerical approximation methods such as the finite element method. These techniques are based on Saint-Venant's principle and exploit specific homotheticity properties in order to improve the quality of the bo...

This paper describes the formulation and numerical implementation of a family of anisotropic and unilateral damage models for the prediction of damage and final rupture in engineering structures. The damage can be load oriented, microstructure oriented, or (for the first time within this modeling framework) softening. The local equations are solved...

The Variational Theory of Complex Rays (VTRC) is an approach to the simulation of mid-frequency phenomena whose wavelengths are relatively small compared to the dimensions of the domain. This is a wave-based computational technique which involves a nonclassical variational formulation. This paper focuses on the development of the approach for 3D en...

Recently, a new type of numerical technique called the Variational Theory of Complex Rays (VTCR) was proposed for the efficient solution of the acoustic Helmholtz equation in the medium frequency regime. However this method was fully developed only for bounded domains. In this paper, the VTCR is extended to exterior Helmholtz problems in order to e...

The paper deals with the accuracy of guaranteed error bounds on outputs of interest computed from approximate methods such as the finite element method. A considerable improvement is introduced for linear problems thanks to new bounding techniques based on Saint-Venant's principle. The main breakthrough of these optimized bounding techniques is the...

A new multiscale computational strategy was recently proposed for the analysis of structures described both on a fine space scale and a fine time scale. This strategy, which involves homogenization in space as well as in time, could replace in several domains of application the standard homogenization techniques, which are generally limited to the...

This paper proposes an extension of the variational theory of complex rays (VTCR) to three-dimensional linear acoustics, The VTCR is a Trefftz-type approach designed for mid-frequency range problems and has been previously investigated for structural dynamics and 2D acoustics. The proposed 3D formulation is based on a discretization of the amplitud...

Laminated composites may undergo significant degradations prior to their ultimate failure. Even though these degradations are associated with length scales which are much smaller than the structural scale, their interactions may greatly influence global failure. Various approaches have been proposed to seek the best compromise between a detailed ph...

This paper introduces a new computational technique for deriving guaranteed upper bounds of the error in calculated outputs of interest for plasticity problems under quasi-static conditions. This approach involves the resolution of some nonstandard additional problems, for which resolution techniques are given. All sources of error (spatial and tem...

This paper introduces a discontinuous method for the efficient determination of an approximate numerical solution of the two-dimensional advection-diffusion equation. Using the VTCR methodology (see [1]), this method involves free-space solutions of the governing partial differential equation. For the advection-diffusion equation with constant coef...

In this paper, a verification approach is introduced to build guaranteed PGD-reduced models for linear elliptic or parabolic problems depending on parameters. It is based on the concept of constitutive relation error and provides for strict bounds on both global error and error on outputs of interest defined with respect to the reference multi-para...