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See detailElastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework
Wu, Ling ULg; Becker, Gauthier ULg; Noels, Ludovic ULg

in Computer Methods in Applied Mechanics & Engineering (2014), 279

One current challenge related to computational fracture mechanics is the modeling of ductile fracture and in particular the damage to crack transition. On the one hand, continuum damage models, especially ... [more ▼]

One current challenge related to computational fracture mechanics is the modeling of ductile fracture and in particular the damage to crack transition. On the one hand, continuum damage models, especially in their non-local formulation which avoids the loss of solution uniqueness, can capture the material degradation process up to the localization of the damage, but are unable to represent a discontinuity in the structure. On the other hand cohesive zone methods can represent the process zone at the crack tip governing the crack propagation, but cannot account for the diffuse material damaging process. In this paper we propose to combine, in a small deformations setting, a non-local elastic damage model with a cohesive zone model. This combination is formulated within a discontinuous Galerkin nite element discretization. Indeed this DG weak formulation can easily be developed in a non-local implicit form and naturally embeds interface elements that can be used to integrate the traction separation law of the cohesive zone model. The method remains thus consistent and computationally e cient as compared to other cohesive element approaches. The effects of the damage to crack transition and of the mesh discretization are respectively studied on the compact tension specimen and on the double-notched specimen, demonstrating the efficiency and accuracy of the method. [less ▲]

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See detailEvent-driven integration of linear structural dynamics models under unilateral elastic constraints
Depouhon, Alexandre ULg; Detournay, Emmanuel; Denoël, Vincent ULg

in Computer Methods in Applied Mechanics & Engineering (2014), 276

This paper proposes an algorithm for the numerical simulation of linear structural dynamics problems under unilateral elastic constraints, i.e., constraints with a linear force/displacement characteristic ... [more ▼]

This paper proposes an algorithm for the numerical simulation of linear structural dynamics problems under unilateral elastic constraints, i.e., constraints with a linear force/displacement characteristic whenever active. The presented procedure relies on an event-driven strategy for the handling of the contact constraints, in combination with one-step schemes dedicated to the time integration of the second-order equations of motion. Efficiency of the procedure follows from the use of cubic Hermite interpolation to continuously extend the normal gap functions that reflect the openings of the contact interfaces. Robustness follows from the proper handling of complex numerical situations, e.g., numerical grazing or discontinuity sticking, through appropriate algorithm structure and numerical implementation. And, integration stability is guaranteed by the very nature of the algorithm and that of the one-step integration scheme. Following a detailed coverage of the integration procedure and the countermeasures to the expected numerical difficulties, three application examples are treated for illustration purposes. A MATLAB implementation of the procedure is provided online; download and usage information are given in the Appendix. [less ▲]

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See detailGeometrically exact beam finite element formulated on the special Euclidean group SE(3)
Sonneville, Valentin ULg; Cardona, Alberto; Bruls, Olivier ULg

in Computer Methods in Applied Mechanics & Engineering (2014), 268

This paper describes a dynamic formulation of a straight beam finite element in the setting of the special Euclidean group SE(3). First, the static and dynamic equilibrium equations are derived in this ... [more ▼]

This paper describes a dynamic formulation of a straight beam finite element in the setting of the special Euclidean group SE(3). First, the static and dynamic equilibrium equations are derived in this framework from variational principles. Then, a non-linear interpolation formula using the exponential map is introduced. It is shown that this framework leads to a natural coupling in the interpolation of the position and rotation variables. Next, the discretized internal and inertia forces are developed. The semi-discrete equations of motion take the form of a second-order ordinary differential equation on a Lie group, which is solved using a Lie group time integration scheme. It is remarkable that no parameterization of the nodal variables needs to be introduced and that the proposed Lie group framework leads to a compact and easy-to-implement formulation. Some important numerical and theoretical aspects leading to a computationally efficient strategy are highlighted and discussed. For instance, the formulation leads to invariant tangent stiffness and mass matrices under rigid body motions and a locking free element. The proposed formulation is successfully tested in several numerical static and dynamic examples. [less ▲]

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See detailMultiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation
Nguyen, Van Dung ULg; Becker, Gauthier ULg; Noels, Ludovic ULg

in Computer Methods in Applied Mechanics & Engineering (2013), 260

When considering problems of dimensions close to the characteristic length of the material, the size e ects can not be neglected and the classical (so–called first–order) multiscale computational ... [more ▼]

When considering problems of dimensions close to the characteristic length of the material, the size e ects can not be neglected and the classical (so–called first–order) multiscale computational homogenization scheme (FMCH) looses accuracy, motivating the use of a second–order multiscale computational homogenization (SMCH) scheme. This second–order scheme uses the classical continuum at the micro–scale while considering second–order continuum at the macro–scale. Although the theoretical background of the second–order continuum is increasing, the implementation into a finite element code is not straightforward because of the lack of high–order continuity of the shape functions. In this work, we propose a SMCH scheme relying on the discontinuous Galerkin (DG) method at the macro–scale, which simplifies the implementation of the method. Indeed, the DG method is a generalization of weak formulations allowing for inter-element discontinuities either at the C0 level or at the C1 level, and it can thus be used to constrain weakly the C1 continuity at the macro–scale. The C0 continuity can be either weakly constrained by using the DG method or strongly constrained by using usual C0 displacement–based finite elements. Therefore, two formulations can be used at the macro–scale: (i) the full–discontinuous Galerkin formulation (FDG) with weak C0 and C1 continuity enforcements, and (ii) the enriched discontinuous Galerkin formulation (EDG) with high–order term enrichment into the conventional C0 finite element framework. The micro–problem is formulated in terms of standard equilibrium and periodic boundary conditions. A parallel implementation in three dimensions for non–linear finite deformation problems is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward and e cient way. [less ▲]

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See detailA multiscale mean-field homogenization method for fiber-reinforced composites with gradient-enhanced damage models
Wu, Ling ULg; Noels, Ludovic ULg; Adam, L et al

in Computer Methods in Applied Mechanics & Engineering (2012), 233-236

In this work, a gradient-enhanced homogenization procedure is proposed for fiber reinforced materials. In this approach, the fiber is assumed to remain linear elastic while the matrix material is modeled ... [more ▼]

In this work, a gradient-enhanced homogenization procedure is proposed for fiber reinforced materials. In this approach, the fiber is assumed to remain linear elastic while the matrix material is modeled as elasto-plastic coupled with a damage law described by a non-local constitutive model. Toward this end, the mean-field homogenization is based on the knowledge of the macroscopic deformation tensors, internal variables and their gradients, which are applied to a micro-structural representative volume element (RVE). The macro-stress is then obtained from a homogenization procedure. The methodology holds for 2-phase composites with moderate fiber volume ratios, and for which, at the RVE size, the matrix can be considered as homogeneous isotropic and the ellipsoidal fibers can be considered as homogeneous transversely isotropic. Under these assumptions, the method is successfully applied to simulate the damage process occurring in unidirectional carbon-fiber reinforced epoxy composites submitted to different loading conditions. [less ▲]

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See detailA one Field Full Discontinuous Galerkin Method for Kirchhoff-Love Shells Applied to Fracture Mechanics
Becker, Gauthier ULg; Geuzaine, Christophe ULg; Noels, Ludovic ULg

in Computer Methods in Applied Mechanics & Engineering (2011), 200(45-46), 3223-3241

In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the Finite Element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive ... [more ▼]

In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the Finite Element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive elements. It is well known that, on one the hand, if these elements are present before fracture there is a modification of the structure stiffness, and that, on the other hand, their insertion during the simulation requires very complex implementation, especially with parallel codes. These drawbacks can be avoided by combining the cohesive method with the use of a discontinuous Galerkin formulation. In such a formulation, all the elements are discontinuous and the continuity is weakly ensured in a stable and consistent way by inserting extra terms on the boundary of elements. The recourse to interface elements allows to substitute them by cohesive elements at the onset of fracture. The purpose of this paper is to develop this formulation for Kirchhoff-Love plates and shells. It is achieved by the establishment of a full DG formulation of shell combined with a cohesive model, which is adapted to the special thickness discretization of shell formulation. In fact, this cohesive model is applied on resulting reduced stresses which are the basis of thin structures formulations. Finally, numerical examples demonstrate the efficiency of the method. [less ▲]

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See detailFinite element analysis on implicitly defined domains: An accurate representation based on arbitrary parametric surfaces
Moumnassi, Mohammed; Belouettar, Salim; Béchet, Eric ULg et al

in Computer Methods in Applied Mechanics & Engineering (2011), 200(5-8), 774-796

In this paper, we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis. The novel contributions of this paper are threefold ... [more ▼]

In this paper, we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis. The novel contributions of this paper are threefold: (1) describe and validate a method to represent arbitrary parametric surfaces implicitly; (2) represent arbitrary solids implicitly, including sharp features using level sets and boolean operations; (3) impose arbitrary Dirichlet and Neumann boundary conditions on the resulting implicitly defined boundaries. The methods proposed do not require local refinement of the finite element mesh in regions of high curvature, ensure the independence of the domain’s volume on the mesh, do not rely on boundary regularization, and are well suited to methods based on fixed grids such as the extended finite element method (XFEM). Numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries. This approach is one step in the desired direction of tying numerical simulations to computer aided design (CAD), similarly to the isogeometric analysis paradigm. [less ▲]

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See detailA scalable 3D fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, Cohesive Element Method
Radovitzky, Raúl; Seagraves, Andrew; Tupek, Mike et al

in Computer Methods in Applied Mechanics & Engineering (2011), 200(1-4), 326-344

A scalable algorithm for modeling dynamic fracture and fragmentation of solids in three dimensions is presented. The method is based on a combination of a discon- tinuous Galerkin (DG) formulation of the ... [more ▼]

A scalable algorithm for modeling dynamic fracture and fragmentation of solids in three dimensions is presented. The method is based on a combination of a discon- tinuous Galerkin (DG) formulation of the continuum problem and Cohesive Zone Models (CZM) of fracture. Prior to fracture, the flux and stabilization terms aris- ing from the DG formulation at interelement boundaries are enforced via interface elements, much like in the conventional intrinsic cohesive element approach, albeit in a way that guarantees consistency and stability. Upon the onset of fracture, the traction-separation law (TSL) governing the fracture process becomes operative without the need to insert a new cohesive element. Upon crack closure, the rein- statement of the DG terms guarantee the proper description of compressive waves across closed crack surfaces. The main advantage of the method is that it avoids the need to propagate topo- logical changes in the mesh as cracks and fragments develop, which enables the indistinctive treatment of crack propagation across processor boundaries and, thus, the scalability in parallel computations. Another advantage of the method is that it preserves consistency and stability in the uncracked interfaces, thus avoiding issues with wave propagation typical of intrinsic cohesive element approaches. A simple problem of wave propagation in a bar leading to spall at its center is used to show that the method does not affect wave characteristics and as a consequence properly captures the spall process. We also demonstrate the ability of the method to capture intricate patterns of radial and conical cracks arising in the impact of ceramic plates which propagate in the mesh impassive to the presence of processor boundaries. [less ▲]

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See detailProbabilistic equivalence and stochastic model reduction in multiscale analysis
Arnst, Maarten ULg; Ghanem, Roger

in Computer Methods in Applied Mechanics & Engineering (2008), 197(43-44), 3584-3592

This paper presents a probabilistic upscaling of mechanics models. A reduced-order probabilistic model is constructed as a coarse-scale representation of a specified fine-scale model whose probabilistic ... [more ▼]

This paper presents a probabilistic upscaling of mechanics models. A reduced-order probabilistic model is constructed as a coarse-scale representation of a specified fine-scale model whose probabilistic structure can be accurately determined. Equivalence of the fine- and coarse-scale representations is identified such that a reduction in the requisite degrees of freedom can be achieved while accuracy in certain quantities of interest is maintained. A significant stochastic model reduction can a priori be expected if a separation of spatial and temporal scales exists between the fine- and coarse-scale representations. The upscaling of probabilistic models is subsequently formulated as an optimization problem suitable for practical computations. An illustration in stochastic structural dynamics is provided to demonstrate the proposed framework. [less ▲]

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See detailA smoothed finite element method for plate analysis
Nguyen-Xuan, Hung; Rabczuk, Timon; Bordas, Stéphane et al

in Computer Methods in Applied Mechanics & Engineering (2008), 197(13-16), 1184-1203

A quadrilateral element with smoothed curvatures for Mindlin-Reissner plates is proposed. The curvature at each point is obtained by a non-local approximation via a smoothing function. The bending ... [more ▼]

A quadrilateral element with smoothed curvatures for Mindlin-Reissner plates is proposed. The curvature at each point is obtained by a non-local approximation via a smoothing function. The bending stiffness matrix is calculated by a boundary integral along the boundaries of the smoothing elements (smoothing cells). Numerical results show that the proposed element is robust, computational inexpensive and simultaneously very accurate and free of locking, even for very thin plates. The most promising feature of our elements is their insensitivity to mesh distortion. (c) 2007 Elsevier B.V. All rights reserved. [less ▲]

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See detailInversion of probabilistic structural models using measured transfer functions
Arnst, Maarten ULg; Clouteau, Didier; Bonnet, Marc

in Computer Methods in Applied Mechanics & Engineering (2008), 197(6-8), 589-608

This paper addresses the inversion of probabilistic models for the dynamical behaviour of structures using experimental data sets of measured frequency-domain transfer functions. The inversion is ... [more ▼]

This paper addresses the inversion of probabilistic models for the dynamical behaviour of structures using experimental data sets of measured frequency-domain transfer functions. The inversion is formulated as the minimization, with respect to the unknown parameters to be identified, of an objective function that measures a distance between the data and the model. Two such distances are proposed, based on either the loglikelihood function, or the relative entropy. As a comprehensive example, a probabilistic model for the dynamical behaviour of a slender beam is inverted using simulated data. The methodology is then applied to a civil and environmental engineering case history involving the identification of a probabilistic model for ground-borne vibrations from real experimental data. [less ▲]

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See detailMicrobeam pull-in voltage topology optimization including material deposition constraint
Lemaire, Etienne ULg; Rochus, Véronique ULg; Golinval, Jean-Claude ULg et al

in Computer Methods in Applied Mechanics & Engineering (2008), 197

Because of the strong coupling between mechanical and electrical phenomena existing in electromechanical microdevices, some of them experience, above a given driving voltage, an unstable behavior called ... [more ▼]

Because of the strong coupling between mechanical and electrical phenomena existing in electromechanical microdevices, some of them experience, above a given driving voltage, an unstable behavior called pull-in effect. The present paper investigates the application of topology optimization to electromechanical microdevices for the purpose of delaying this unstable behavior by maximizing their pull-in voltage. Within the framework of this preliminary study, the pull-in voltage maximization procedure is developed on the basis of electromechanical microbeams reinforcement topology design problem. The proposed sensitivity analysis requires only the knowledge of the microdevice pull-in state and of the first eigenmode of the tangent stiffness matrix. As the pull-in point research is a highly non-linear problem, the analysis is based on a monolithic finite element formulation combined with a normal flow algorithm (homotopy method). An application of the developed method is proposed and the result is compared to the one obtained using a linear compliance optimization. Moreover, as the results provided by the developed method do not comply with manufacturing constraints, a deposition process constraint is added to the optimization problem and its effect on the final design is also tested. [less ▲]

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See detailA new discontinuous Galerkin method for Kirchhoff-Love shells
Noels, Ludovic ULg; Radovitzky, Raúl

in Computer Methods in Applied Mechanics & Engineering (2008), 197

Discontinuous Galerkin methods (DG) have particular appeal in problems involving high-order derivatives since they provide a means of weakly enforcing the continuity of the unknown-field derivatives. This ... [more ▼]

Discontinuous Galerkin methods (DG) have particular appeal in problems involving high-order derivatives since they provide a means of weakly enforcing the continuity of the unknown-field derivatives. This paper proposes a new discontinuous Galerkin method for Kirchhoff–Love shells considering only the membrane and bending response. The proposed one-field method utilizes the weak enforcement in such a way that the displacements are the only unknowns, while the rotation continuity is weakly enforced. This work presents the formulation of the new discontinuous Galerkin method for linear elastic shells, demonstrates the consistency and stability of the proposed framework, and establishes the method’s convergence rate. After a description of the formulation implementation into a finite-element code, these properties are demonstrated on numerical applications. [less ▲]

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See detailOn the numerical damping of time integrators for coupled mechatronic systems
Bruls, Olivier ULg; Golinval, Jean-Claude ULg

in Computer Methods in Applied Mechanics & Engineering (2008), 197(6-8), 577-588

The generalized-alpha time integrator is considered for the simulation of mechatronic systems. In this context, the fundamental concept of numerical damping is analysed for coupled sets of first and ... [more ▼]

The generalized-alpha time integrator is considered for the simulation of mechatronic systems. In this context, the fundamental concept of numerical damping is analysed for coupled sets of first and second-order differential-algebraic equations. First, it appears that the algebraic variables do not influence the spectral properties of the dynamic variables. Second, we demonstrate that the coupling between the dynamic variables does not influence the high-frequency spectral response, so that the numerical damping can be determined as usual from elementary characteristic polynomials. Those results are exploited to assess the stability properties of the scheme and to select an algorithm with optimal damping properties. [less ▲]

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See detailX-FEM explicit dynamics for constant strain elements to alleviate mesh constraints on internal or external boundaries
Rozycki, P.; Moes, N.; Béchet, Eric ULg et al

in Computer Methods in Applied Mechanics & Engineering (2008), 197(5), 349-363

This paper deals with the use of the extended Finite Element Method (X-FEM) for rapid dynamic problems. To solve the equations of motion, a common technique is the explicit direct integration with a ... [more ▼]

This paper deals with the use of the extended Finite Element Method (X-FEM) for rapid dynamic problems. To solve the equations of motion, a common technique is the explicit direct integration with a Newmark scheme. Since this temporal scheme is only conditionally stable, the critical time step must be determined. It is generally induced by mesh constraints. The idea of the paper is to weaken constraints on mesh generation algorithms so that the critical time step is as large as possible. Using the X-FEM one allows a non-conformity between mesh and discontinuities such as cracks, holes or interfaces. In a first part, we present a summary about direct integration schemes and about the eXtended Finite Element Method. Then, we focus on the theoretical description of a ID X-FEM finite element and its generalization to 2D and 3D finite elements. Then, dynamic numerical simulations are shown. They concern structures under impact with holes or external boundaries not exactly matched by the mesh. Comparisons are made with numerical results coming from the ABAQUS software. It shows that developments are satisfactory. We conclude with some outlooks concerning this work. (c) 2007 Elsevier B.V. All rights reserved. [less ▲]

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See detailA first-order energy-dissipative momentum-conserving scheme for elasto-plasticity using the variational updates formulation
Noels, Ludovic ULg; Stainier, Laurent ULg; Ponthot, Jean-Philippe ULg

in Computer Methods in Applied Mechanics & Engineering (2008), 197(6-8), 706726

In a previous paper [L. Noels, L. Stainier, J.-P. Ponthot, An energy momentum conserving algorithm using the variational formulation of visco-plastic updates, Int. J. Numer. Methods Engrg. 65 (2006) 904 ... [more ▼]

In a previous paper [L. Noels, L. Stainier, J.-P. Ponthot, An energy momentum conserving algorithm using the variational formulation of visco-plastic updates, Int. J. Numer. Methods Engrg. 65 (2006) 904-942] the authors demonstrated the efficiency of the variational formulation of elasto-plastic updates to develop energy-momentum conserving time integration algorithms. Indeed, within such a framework, the stress tensor always derives from an incremental potential, even when plastic behavior is considered. Therefore the verification of the conservation of energy in the non-linear range can easily be demonstrated: the sum of the reversible stored energy and irreversible dissipated energy exactly corresponds to the work of the external forces applied to the structure. Although this formulation was shown to be accurate and robust, the introduction of numerical dissipation for high-frequency numerical modes can be necessary to simulate complex phenomena. In this work, we propose a modification of the variational updates framework to introduce this numerical property, leading to a new energy-dissipative momentum-conserving time-integration algorithm for elasto-plasticity. (c) 2007 Elsevier B.V. All rights reserved. [less ▲]

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See detailFinite element study of the effect of some local defects on the risk of transverse cracking in continuous casting of steel slabs
Pascon, Frédéric ULg; Habraken, Anne ULg

in Computer Methods in Applied Mechanics & Engineering (2007), 196

This paper introduces a numerical 2.5D model of continuous casting of steel slabs. This model is based on the finite element method and it has been applied to the study of some local defects in a ... [more ▼]

This paper introduces a numerical 2.5D model of continuous casting of steel slabs. This model is based on the finite element method and it has been applied to the study of some local defects in a continuous caster, such as partial blockage of nozzles (leading to a local reduction of secondary cooling rate), locking or misalignment of rolls. The purpose of the study was the evaluation of the effect of such defects on the risk of transverse cracking during bending and unbending operations. To do so, the simulation at macro-scale of the complete process has been first performed in standard conditions to get reference values and then each defect has been introduced. Defining two indexes (indicators) of the risk of transverse cracking, it has been possible to classify the defects in terms of risk increase, helping steel producers to focus on the most critical problems. [less ▲]

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See detailA cascade optimization methodology for automatic parameter identification and shape/process optimization in metal forming simulation
Ponthot, Jean-Philippe ULg; Kleinermann, Jean-Pascal

in Computer Methods in Applied Mechanics & Engineering (2006), 195(41-43), 5472-5508

Computer simulations of metal forming processes using the finite element method (FEM) are, today, well established. This form of simulation uses an increasing number of sophisticated geometrical and ... [more ▼]

Computer simulations of metal forming processes using the finite element method (FEM) are, today, well established. This form of simulation uses an increasing number of sophisticated geometrical and material models, relying on a certain number of input data, which are not always readily available. The aim of inverse problems, which will be considered here, is to determine one or more of the input data relating to these forming process simulations, thereby leading to a desired result. In this paper, we will focus on two categories of such inverse problems. The first category consists of parameter identification inverse problems. These involve evaluating the material parameters for material constitutive models that would lead to the most accurate results with respect to physical experiments, i.e. minimizing the difference between experimental results and FEM simulations. The second category consists of shape/process optimization inverse problems. These involve determining the initial geometry of the specimen and/or the shape of the forming tools, as well as some parameters of the process itself, in order to provide the desired final geometry after the forming process. These two categories of inverse problems can be formulated as optimization problems in a similar way, i.e. by using identical optimization algorithms. In this paper, we intend firstly to solve these two types of optimization problems by using different non-linear gradient based optimization methods and secondly to compare their efficiency and robustness in a variety of numerical applications. (c) 2005 Elsevier B.V. All rights reserved. [less ▲]

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See detailEnergy conserving balance of explicit time steps to combine implicit and explicit algorithms in structural dynamics
Noels, Ludovic ULg; Stainier, Laurent ULg; Ponthot, Jean-Philippe ULg

in Computer Methods in Applied Mechanics & Engineering (2006), 195(19-22), 21692192

Recent developments have proved the advantage of combining both time implicit and time explicit integration algorithms in structural dynamics. A major problem is to define the initial conditions for the ... [more ▼]

Recent developments have proved the advantage of combining both time implicit and time explicit integration algorithms in structural dynamics. A major problem is to define the initial conditions for the implicit simulation on the basis of a solution obtained from an unbalanced explicit resolution. The unbalanced nature of the explicit algorithm leads to oscillations in the fields of interest. Therefore, the values obtained after an explicit computation cannot be used directly as initial conditions for an implicit simulation. In this paper, we develop such initial values that lead to a stable (no numerical creation of energy) and energy-conserving transition. (c) 2005 Elsevier B.V. All rights reserved. [less ▲]

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See detailOn the exploitation of chaos to build reduced-order models
Kerschen, Gaëtan ULg; Feeny, B. F.; Golinval, Jean-Claude ULg

in Computer Methods in Applied Mechanics & Engineering (2003), 192(13-14), 1785-1795

The present study focuses on the model reduction of non-linear systems. The proper orthogonal decomposition is exploited to compute eigenmodes from time series of displacement. These eigenmodes, called ... [more ▼]

The present study focuses on the model reduction of non-linear systems. The proper orthogonal decomposition is exploited to compute eigenmodes from time series of displacement. These eigenmodes, called the proper orthogonal modes, are optimal with respect to energy content and are used to build a low-dimensional model of the non-linear system. For this purpose, the proper orthogonal modes obtained from a chaotic orbit are considered. Indeed, such an orbit is assumed to cover a portion of the phase space of higher dimension, and hence of greater measure. This higher dimensional data is further assumed to contain more information about the system dynamics than data of a lower-dimensional periodic orbit. In an example, it is shown that the modes for this particular behaviour are more representative of the system dynamics than any other set of modes extracted from a non-chaotic response. This is applied to a buckled beam with two permanent magnets and the reduced-order model is validated using both qualitative and quantitative comparisons. (C) 2003 Elsevier Science B.V. All rights reserved. [less ▲]

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