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On the equivalent static load method for flexible multibody systems described with a nonlinear finite element formalism Tromme, Emmanuel ; Sonneville, Valentin ; Bruls, Olivier et al in International Journal for Numerical Methods in Engineering (2016) The equivalent static load (ESL) method is a powerful approach to solve dynamic response structural optimization problems. The method transforms the dynamic response optimization into a static response ... [more ▼] The equivalent static load (ESL) method is a powerful approach to solve dynamic response structural optimization problems. The method transforms the dynamic response optimization into a static response optimization under multiple load cases. The ESL cases are defined based on the transient analysis response whereupon all the standard techniques of static response optimization can be used. In the last decade, the ESL method has been applied to perform the structural optimization of flexible components of mechanical systems modeled as multibody systems (MBS). The ESL evaluation strongly depends on the adopted formulation to describe the MBS and has been initially derived based on a floating frame of reference formulation. In this paper, we propose a method to derive the ESL adapted to a nonlinear finite element approach based on a Lie group formalism for two main reasons. Firstly, the finite element approach is completely general to analyze complex MBS and is suitable to perform more advanced optimization problems like topology optimization. Secondly, the selected Lie group formalism leads to a formulation of the equations of motion in the local frame, that turns out to be a strong practical advantage for the ESL evaluation. Examples are provided to validate the proposed method. [less ▲] Detailed reference viewed: 13 (2 ULg)Flexible multibody system modelling: A geometric local frame approach on SE(3) Sonneville, Valentin Scientific conference (2015, October 30) In order to improve computational costs in the frame of flexible multibody system modelling, we propose a geometrically exact framework relying on a local frame description. Thanks to this framework ... [more ▼] In order to improve computational costs in the frame of flexible multibody system modelling, we propose a geometrically exact framework relying on a local frame description. Thanks to this framework, geometric non-linearities are strongly reduced for two reasons. Firstly, the equations of motion can be solved without introducing a global parametrization of the motion, which also leads to a naturally singularity-free description of large rotations. Secondly, the equations of motion are expressed in a local frame, both at position and rotation level, leading to intrinsic equations. Accordingly, these equations are insensitive to large amplitude motions, such that the geometric non-linearities are naturally filtered. The mathematical developments are carried out in a Lie group setting which, albeit more abstract than classical treatments, provides generic and powerful well established tools. So far, several elements have been formulated in this framework: rigid body, kinematic joints and a flexible beam finite element. A shell finite element and a super-element are also under development. Some interesting numerical advantages are observed following the reduction of geometric non-linearities, e.g., the iteration matrix depends on relative motion within the elements only and is thus invariant under rigid body motions. [less ▲] Detailed reference viewed: 77 (15 ULg)Optimal design of flexible mechanisms using the Equivalent Static Load method and a Lie group formalism Tromme, Emmanuel ; Sonneville, Valentin ; Bruls, Olivier et al Conference (2015, July 02) Detailed reference viewed: 76 (12 ULg)Exploiting frame-invariant operators for the efficient numerical simulation of flexible multibody systems Sonneville, Valentin ; ; Bruls, Olivier Conference (2015, July) Detailed reference viewed: 14 (3 ULg)A geometrically exact shell finite element with an almost constant tangent stiffness matrix Sonneville, Valentin ; ; Bruls, Olivier Conference (2015, July) Detailed reference viewed: 14 (1 ULg)Trajectory optimization for 3D robots with elastic links Lismonde, Arthur ; Sonneville, Valentin ; Bruls, Olivier Scientific conference (2015, June 29) Detailed reference viewed: 33 (3 ULg)A geometric local frame approach for flexible multibody systems Sonneville, Valentin Doctoral thesis (2015) The notion of frame is ubiquitous in the kinematic description of flexible multibody models. In this work, a differential-geometric framework is selected to describe frame operations in a rigorous and ... [more ▼] The notion of frame is ubiquitous in the kinematic description of flexible multibody models. In this work, a differential-geometric framework is selected to describe frame operations in a rigorous and systematic way. A frame transformation is thus seen as an element of the special Euclidean group $SE(3)$, which is represented by a four by four transformation matrix, and frame operations, such as spatial interpolation or time integration, rely on non-linear but analytical expressions in which translation and rotation contributions are inherently coupled. Based on this formalism, this thesis develops geometrically exact formulations of many classical components used in flexible multibody system modelling, which includes the formulation of a rigid body, several kinematic joints, a flexible beam, a flexible shell and a superelement. As opposed to most popular techniques in the literature, a local frame representation of the equations of motion is adopted in this work. This means that the unknown kinematic variables such as the motion increments, the velocities and the accelerations, as well as the generalized forces are all expressed in a local frame attached to the body. After spatial semi-discretization, the equations of motion of a multibody system take the form of differential-algebraic equations on a Lie group which can be conveniently solved in a global parametrization-free approach using a Lie group integration scheme. This thesis presents numerous arguments to recommend this framework for the development of efficient codes for the numerical simulation of flexible multibody systems. On the one hand, the proposed framework leads to novel and interesting theoretical aspects. For instance, it features a naturally singularity-free description of large rotations and it leads to inherently shear-locking free beam and shell finite elements. On the other hand, the formulation leads to unprecedented computational properties. The geometric non-linearities are naturally filtered out of the equilibrium equations such that non-linearities are significantly reduced, as compared to classical formulations. In particular, the iteration matrix, which is used in implicit integration schemes, is insensitive to overall large amplitude motions and is only affected by local relative transformations, such as deformations in flexible elements and relative motions in kinematic joints. This property can be exploited to strongly reduce computational costs, as compared to classical formulations. [less ▲] Detailed reference viewed: 151 (34 ULg)Nonlinear analysis of tape springs: Comparison of two geometrically exact finite element formulations Dewalque, Florence ; Sonneville, Valentin ; Bruls, Olivier Conference (2014, July 22) Detailed reference viewed: 72 (29 ULg)A formulation on the special Euclidean group for dynamic analysis of multibody systems Sonneville, Valentin ; Bruls, Olivier in Journal of Computational and Nonlinear Dynamics (2014), 9(4), 041002 Detailed reference viewed: 64 (17 ULg)FORMULATION OF A NON-LINEAR SHELL FINITE ELEMENT ON THE LIE GROUP SE(3) Sonneville, Valentin ; Bruls, Olivier Conference (2014, July) Detailed reference viewed: 39 (11 ULg)Model reduction of geometrically exact structures formulated on the Lie group SE(3) Sonneville, Valentin ; Bruls, Olivier Conference (2014, July) Detailed reference viewed: 39 (6 ULg)Geometric interpretation of a non-linear beam finite element on the Lie group SE(3) Sonneville, Valentin ; ; Bruls, Olivier in Archive of Mechanical Engineering (2014), 61(2), 305-329 Detailed reference viewed: 64 (12 ULg)Sensitivity analysis for multibody systems formulated on a Lie group Sonneville, Valentin ; Bruls, Olivier in Multibody System Dynamics (2014), 31 A direct differentiation method and an adjoint variable method are proposed for the efficient semi-analytical evaluation of the sensitivities of multibody systems formulated in a matrix Lie group ... [more ▼] A direct differentiation method and an adjoint variable method are proposed for the efficient semi-analytical evaluation of the sensitivities of multibody systems formulated in a matrix Lie group framework. These methods rely on the linearization of the equations of motion and/or of the time integration procedure. The simpler structure of the equations of motion in the Lie group formalism appears as an advantage for that purpose. Lie bracket contributions and the non-linearity of the exponential map need to be taken into account in the sensitivity algorithms. Nevertheless, essential characteristics of formulations of the direct differentiation method and the adjoint variable method on linear spaces are recovered. Some implementation issues are discussed and two relevant examples illustrate the properties of these methods. [less ▲] Detailed reference viewed: 112 (31 ULg)Geometrically exact beam finite element formulated on the special Euclidean group SE(3) Sonneville, Valentin ; ; Bruls, Olivier 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 ▲] Detailed reference viewed: 82 (26 ULg)Contact model between superelements in dynamic multibody systems Virlez, Geoffrey ; Bruls, Olivier ; Sonneville, Valentin et al in Proceedings of ASME2013 International Design Engineering Technical Conference & Computers and Information in Engineering Conference IDETC/CIE 2013 (2013, August) In this paper, a new contact formulation defined between flexible bodies modeled as superelements is investigated. Unlike rigid contact models, this approach enables to study the deformation and vibration ... [more ▼] In this paper, a new contact formulation defined between flexible bodies modeled as superelements is investigated. Unlike rigid contact models, this approach enables to study the deformation and vibration phenomena induced by hard contacts. Compared with full-scale finite element models of flexible bodies, the proposed method is computationally more efficient, especially in case of a large number of bodies and contact conditions. The compliance of each body is described using a reduced-order elastic model which is defined in a corotational frame that follows the gross motion of the body. The basis used to reduce the initial finite element model relies on the Craig-Bampton method which uses both static boundary modes and internal vibration modes. The formulation of the contact condition couples all degrees of freedom of the reduced model in a nonlinear way. The relevance of the approach is demonstrated by simulation results first on a simple example, and then on a gear pair model. [less ▲] Detailed reference viewed: 156 (28 ULg)A FEW GOOD REASONS TO CONSIDER A BEAM FINITE ELEMENT FORMULATION ON THE LIE GROUP SE(3) Sonneville, Valentin ; Bruls, Olivier in Proceedings of the ASME 2013 International Design Engineering Technical Conference & Computers and Information in Engineering Conference IDETC/CIE 2013 (2013, August) Based on an original interpolation method we develop a beam finite element formulation on the Lie group SE(3) which relies on a mathematically rigorous framework and provides compact and generic notations ... [more ▼] Based on an original interpolation method we develop a beam finite element formulation on the Lie group SE(3) which relies on a mathematically rigorous framework and provides compact and generic notations. We work out the beam kinematics in the SE(3) context, the beam deformation measure and obtain the expression of the internal forces using the virtual work principle. The proposed formulation exhibits important features from both the theoretical and numerical points of view. The approach leads to a natural coupling of position and rotation variables and thus differs from classical Timoshenko/Cosserat formulations. We highlight several important properties such as a constant deformation measure over the element, an invariant tangent stiffness matrix under of rigid motions or the absence of shear locking. [less ▲] Detailed reference viewed: 90 (13 ULg)Formulation of a geometrically exact beam finite element on the Lie group SE(3) Sonneville, Valentin ; ; Bruls, Olivier Conference (2013, July) Detailed reference viewed: 29 (9 ULg)Formulation of Kinematic Joints and Rigidity Constraints in Multibody Dynamics using a Lie Group Approach Sonneville, Valentin ; Bruls, Olivier in Proceedings of the 2nd Joint International Conference on Multibody System Dynamics (IMSD) (2012, May) The matrix Lie group approach allows to formulate and solve the equations of motion of a multibody system in a parametrization-free framework. The kinematic joints and the rigidity constraints should also ... [more ▼] The matrix Lie group approach allows to formulate and solve the equations of motion of a multibody system in a parametrization-free framework. The kinematic joints and the rigidity constraints should also be formulated as constraint equations on the Lie group. Working on the Special Euclidean group SE(3), we introduce a method to obtain appropriate vectorial constraint equations in terms of mixed coordinates. Moreover, we present an absolute coordinates formulation, based on an relative coordinates elimination method, so that the minimum number of constraint equations necessary to describe the joints is used. [less ▲] Detailed reference viewed: 155 (22 ULg)Sensitivity analysis for flexible multibody systems formulated on a Lie group Bruls, Olivier ; Sonneville, Valentin Conference (2012, February) Detailed reference viewed: 38 (14 ULg) |
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