References of "Cardona, Alberto"
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See detailError analysis of generalized-alpha Lie group time integration methods for constrained mechanical systems
Arnold, Martin; Bruls, Olivier ULg; Cardona, Alberto

in Numerische Mathematik (in press)

Generalized-alpha methods are very popular in structural dynamics. They are methods of Newmark type and combine favourable stability properties with second order convergence for unconstrained second order ... [more ▼]

Generalized-alpha methods are very popular in structural dynamics. They are methods of Newmark type and combine favourable stability properties with second order convergence for unconstrained second order systems in linear spaces. Recently, they were extended to constrained systems in flexible multibody dynamics that have a configuration space with Lie group structure. In the present paper, the convergence of these Lie group methods is analysed by a coupled one-step error recursion for differential and algebraic solution components. It is shown that spurious oscillations in the transient phase result from order reduction that may be avoided by a perturbation of starting values or by index reduction. Numerical tests for a benchmark problem from the literature illustrate the results of the theoretical investigations. [less ▲]

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See detailGeometric interpretation of a non-linear beam finite element on the Lie group SE(3)
Sonneville, Valentin ULg; Cardona, Alberto; Bruls, Olivier ULg

in Archive of Mechanical Engineering (in press)

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See detailA mortar method combined with an augmented Lagrangian approach for treatment of mechanical contact problems
Cavalieri, Federico J.; Bruls, Olivier ULg; Cardona, Alberto

in Terze, Zdravko (Ed.) Multibody Dynamics: Computational Methods and Applications (2015)

This work presents a mixed penalty-duality formulation from an augmented Lagrangian approach for treating the contact inequality constraints. The augmented Lagrangian approach allows to regularize the non ... [more ▼]

This work presents a mixed penalty-duality formulation from an augmented Lagrangian approach for treating the contact inequality constraints. The augmented Lagrangian approach allows to regularize the non differentiable contact terms and gives a C1 differentiable saddle-point functional. The relative displacement of two contacting bodies is described in the framework of the Finite Element Method (FEM) using the mortar method, which gives a smooth representation of the contact forces across the bodies interface. To study the robustness and performance of the proposed algorithm, validation numerical examples with finite deformations and large slip motion are presented. [less ▲]

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See detailOrder reduction in time integration caused by velocity projection
Arnold, Martin; Cardona, Alberto; Bruls, Olivier ULg

in Proceedings of the 3rd Joint International Conference on Multibody System Dynamics and the 7th Asian Conference on Multibody Dynamics (2014, July)

Holonomic constraints restrict the configuration of a multibody system to a subset of the configuration space. They imply so called hidden constraints at the level of velocity coordinates that may ... [more ▼]

Holonomic constraints restrict the configuration of a multibody system to a subset of the configuration space. They imply so called hidden constraints at the level of velocity coordinates that may formally be obtained from time derivatives of the original holonomic constraints. A numerical solution that satisfies hidden constraints as well as the original constraint equations may be obtained considering both types of constraints simultaneously in each time step (stabilized index-2 formulation) or using projection techniques. Both approaches are well established in the time integration of differential-algebraic equations. Recently, we have introduced a generalized- alpha Lie group time integration method for the stabilized index-2 formulation that achieves second order convergence for all solution components. In the present paper, we show that a separate velocity projection would be less favourable since it may result in an order reduction and in large transient errors after each projection step. This undesired numerical behaviour is analysed by a one-step error recursion that considers the coupled error propagation in differential and algebraic solution components. This one-step error recursion has been used before to prove second order convergence for the application of generalized-alpha methods to constrained systems. As a technical detail, we discuss the extension of these results from symmetric, positive definite mass matrices to the rank deficient case. [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 detailFormulation of a geometrically exact beam finite element on the Lie group SE(3)
Sonneville, Valentin ULg; Cardona, Alberto; Bruls, Olivier ULg

Conference (2013, July)

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See detailAn augmented Lagrangian and reduced index formulation for the analysis of multibody systems with impacts
Bruls, Olivier ULg; Acary, Vincent; Cavalieri, Federico J. et al

Conference (2013, July)

This paper presents a formalism for the simulation of nonsmooth dynamic systems with unilateral constraints, which integrates the contributions of impacts with first order accuracy in a fully implicit way ... [more ▼]

This paper presents a formalism for the simulation of nonsmooth dynamic systems with unilateral constraints, which integrates the contributions of impacts with first order accuracy in a fully implicit way, while the other contributions are integrated with second-order accuracy. Here, the smooth contribution of the reaction forces associated with bilateral and unilateral constraints is also integrated with second-order accuracy. Therefore, in free flight phases, the algorithm is strictly equivalent to a classical second-order scheme. Moreover, following the idea of Gear, Gupta and Leimkuhler, bilateral and unilateral constraints are imposed both at position and velocity levels. This means that the constraints are enforced exactly and no penetration is allowed. In the absence of unilateral constraints, the method leads to a formulation of the equation of motion as an index-2 differential-algebraic equation. Finally, the method is based on a reformulation of the LCP as an augmented Lagrangian equation with a suitable activation / deactivation criterion for the unilateral constraints. The resulting system of nonsmooth equations is solved using a monolithic semi-smooth Newton algorithm. [less ▲]

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See detailA mortar algorithm combined with an Augmented Lagrangian approach for treatment of frictional contact problems
Cavalieri, Federico J.; Bruls, Olivier ULg; Cardona, Alberto

Conference (2013, July)

Contact mechanics is present in a wide range of mechanical engineering applications, and numerous works have been dedicated to the numerical solution of contact problems. Mathematically, the frictional ... [more ▼]

Contact mechanics is present in a wide range of mechanical engineering applications, and numerous works have been dedicated to the numerical solution of contact problems. Mathematically, the frictional contact problem can be seen as defined by a variational inequality. The solution corresponds to the minimum of the total energy of the system, subjected to inequalities constraints associated to the normal and tangential components of the traction and distance vectors, respectively. Nonlinear contact mechanics can be related to nonlinear optimization problems formulated by using the method of Lagrange multipliers, resulting in a saddle point system to be solved at each iteration. The method of Lagrange multipliers is very popular in contact mechanics because it overcomes the ill-conditioning inconvenience of the penalty methods. However, the size of the global matrix increases due to the additional unknowns (the Lagrange multipliers) and zero entries appear on the main diagonal of the stiffness matrix. These drawbacks can be avoided by using an augmented Lagrangian method, which consists in a combination of both the penalty and the Lagrange multipliers techniques. In this work, a mixed penalty-duality formulation based on an augmented Lagrangian approach for treating the contact and the friction inequality constraints is presented. The augmented Lagrangian approach allows to regularize the non differentiable contact terms, giving a C1 differentiable saddle-point function. The relative displacement of the contacting bodies is described in the framework of the Finite Element Method (FEM) using the mortar method, which gives a smooth representation of the contact forces across the bodies interface. [less ▲]

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See detailSpurious oscillations in generalized-alpha time integration methods
Arnold, Martin; Bruls, Olivier ULg; Cardona, Alberto

Conference (2012, March)

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See detailLie group generalized-alpha time integration of constrained flexible multibody systems
Bruls, Olivier ULg; Cardona, Alberto; Arnold, Martin

in Mechanism & Machine Theory (2012), 48

This paper studies a Lie group extension of the generalized-alpha time integration method for the simulation of flexible multibody systems. The equations of motion are formulated as an index-3 ... [more ▼]

This paper studies a Lie group extension of the generalized-alpha time integration method for the simulation of flexible multibody systems. The equations of motion are formulated as an index-3 differential-algebraic equation (DAE) on a Lie group, with the advantage that rotation variables can be taken into account without the need of introducing any parameterization. The proposed integrator is designed to solve this equation directly on the Lie group without index reduction. The convergence of the method for DAEs is studied in detail and global second-order accuracy is proven for all solution components, i.e. for nodal translations, rotations and Lagrange multipliers. The convergence properties are confirmed by three benchmarks of rigid and flexible systems with large rotation amplitudes. The Lie group method is compared with a more classical updated Lagrangian method which is also formulated in a Lie group setting. The remarkable simplicity of the new algorithm opens interesting perspectives for real-time applications, model-based control and optimization of multibody systems. [less ▲]

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See detailModelling of frictional unilateral contact in automotive differentials
Virlez, Geoffrey ULg; Cardona, Alberto; Bruls, Olivier ULg et al

Conference (2011, November 14)

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See detailImproved stability and transient behaviour of generalized-alpha time integrators for constrained flexible systems
Arnold, Martin; Bruls, Olivier ULg; Cardona, Alberto

Conference (2011, November)

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See detailTwo Lie Group Formulations for Dynamic Multibody Systems with Large Rotations
Bruls, Olivier ULg; Arnold, Martin; Cardona, Alberto

in Proceedings of the ASME 2011 International Design Engineering Technical Conferences (2011, August)

This paper studies the formulation of the dynamics of multibody systems with large rotation variables and kinematic constraints as differential-algebraic equations on a matrix Lie group. Those equations ... [more ▼]

This paper studies the formulation of the dynamics of multibody systems with large rotation variables and kinematic constraints as differential-algebraic equations on a matrix Lie group. Those equations can then be solved using a Lie group time integration method proposed in a previous work. The general structure of the equations of motion are derived from Hamilton principle in a general and unifying framework. Then, in the case of rigid body dynamics, two particular formulations are developed and compared from the viewpoint of the structure of the equations of motion, of the accuracy of the numerical solution obtained by time integration, and of the computational cost of the iteration matrix involved in the Newton iterations at each time step. In the first formulation, the equations of motion are described on a Lie group defined as the Cartesian product of the group of translations R^3 (the Euclidean space) and the group of rotations SO(3) (the special group of 3 by 3 proper orthogonal transformations). In the second formulation, the equations of motion are described on the group of Euclidean transformations SE(3) (the group of 4 by 4 homogeneous transformations). Both formulations lead to a second-order accurate numerical solution. For an academic example, we show that the formulation on SE(3) offers the advantage of an almost constant iteration matrix. [less ▲]

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See detailConvergence analysis of generalized-alpha Lie group integrators for constrained systems
Arnold, Martin; Bruls, Olivier ULg; Cardona, Alberto

in Proceedings of Multibody Dynamics ECCOMAS Conference (2011, July)

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See detailConvergence analysis for a generalized-alpha Lie group time integrator
Arnold, Martin; Bruls, Olivier ULg; Cardona, Alberto

Conference (2011, January)

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See detailLie group integrators for the numerical solution of DAE’s in flexible multibody dynamics
Cardona, Alberto; Bruls, Olivier ULg; Arnold, Martin

Conference (2010, November)

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See detailNumerical solution of DAEs in flexible multibody dynamics using Lie group time integrators
Bruls, Olivier ULg; Cardona, Alberto; Arnold, Martin

in Proceedings of the First Joint International Conference on Multibody System Dynamics (2010, May)

This paper studies a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parameterization problem and, as an ... [more ▼]

This paper studies a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parameterization problem and, as an extension of the classical generalized-alpha method for dynamic systems, it can deal with constrained equations of motion. Here, second-order accuracy of the Lie group method is demonstrated for constrained problems. The convergence analysis explicitly accounts for the nonlinear geometric structure of the Lie group. The performance is illustrated on two critical benchmarks of rigid and flexible systems with large rotation amplitudes. Second-order accuracy is evidenced in both of them. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control and optimization of multibody systems. [less ▲]

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See detailTime integration of finite rotations in flexible multibody dynamics using Lie group integrators
Bruls, Olivier ULg; Cardona, Alberto

Conference (2010, May)

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See detailOn the use of Lie group time integrators in multibody dynamics
Bruls, Olivier ULg; Cardona, Alberto

in Journal of Computational and Nonlinear Dynamics (2010), 5(3), 031002

This paper proposes a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parameterization problem. As an ... [more ▼]

This paper proposes a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parameterization problem. As an extension of the classical generalized-alpha method for dynamic systems, it can deal with constrained equations of motion. Second-order accuracy is demonstrated in the unconstrained case. The performance is illustrated on several critical benchmarks of rigid body systems with high rotation speeds and second order accuracy is evidenced in all of them, even for constrained cases. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control and optimization of multibody systems. [less ▲]

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See detailLie group vs. classical time integrators in multibody dynamics: Formulations and numerical benchmarks
Bruls, Olivier ULg; Cardona, Alberto

Conference (2009, September)

The dynamics of flexible multibody systems with large rotations is often described using large sets of index-3 differential-algebraic equations. In this context, the Lie group structure of the dynamic ... [more ▼]

The dynamics of flexible multibody systems with large rotations is often described using large sets of index-3 differential-algebraic equations. In this context, the Lie group structure of the dynamic system may be exploited in order to provide an elegant solution to the rotation parameterization problem. The talk discusses an original Lie-group extension of the classical generalized-alpha method, which can be used to solve index-3 differential-algebraic equations in multibody dynamics. Second-order accuracy is demonstrated at least in the unconstrained case and the performance is illustrated on several critical benchmarks with high rotational speeds. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control and optimization of multibody systems. [less ▲]

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