Juan C. García Orden

Dynamic analysis of rigid and deformable multibody systems with penalty methods and energy–momentum schemes

Abstract A multibody formulation for the nonlinear dynamics of mechanical systems composed of both rigid and deformable bodies is proposed in this work, focusing on its conservation properties for basic magnitudes such as total energy and momentum. The approach is based on the use of dependent variables (cartesian coordinates of selected points) and the enforcement of the constraints through the penalty method. This choice has the advantage of providing a simple overall structure that allows the inclusion of both rigid bodies (discrete model) and elastic bodies (continuum model discretised with the finite element method) under the same framework, in order to build a single set of ordinary differential equations. The elastic bodies are represented by general hyperelastic models and may undergo large displacements, rotations and strains. An energy–momentum time integration method has been employed, achieving remarkable stability and robustness with exact conservation of total energy. This approach effectively overcomes drawbacks associated with penalty formulations in other time integration algorithms. This important result in fact proves to be the main conclusion of this work. Some representative numerical simulations are presented for mechanical systems comprised of rigid and deformable bodies.

Conserving Properties in Constrained Dynamics of Flexible Multibody Systems

The context of this work is the non-linear dynamics ofmultibody systems (MBS). The approach followed for parametrisation ofrigid bodies is the use of inertial coordinates, forming a dependent setof parameters. This approach mixes naturally with nodal coordinates in adisplacement-based finite element discretisation of flexible bodies,allowing an efficient simulation for MBS dynamics. An energy-momentumtime integration algorithm is developed within the context of MBSconstraints enforced through penalty methods. The approach follows theconcept of a discrete derivative for Hamiltonian systems proposed byGonzalez, achieving exact preservation of energy and momentum. Thealgorithm displays considerable stability, overcoming the traditionaldrawback of the penalty method, namely numerical ill-conditioning thatleads to stiff equation systems. Additionally, excellent performance isachieved in long-term simulations with rather large time-steps.

Quadratic and Higher-Order Constraints in Energy-Conserving Formulations of Flexible Multibody Systems

The treatment of constraints is considered here within the framework ofenergy-momentum conserving formulations for flexible multibody systems.Constraint equations of various types are an inherent component of multibodysystems, their treatment being one of the key performance features ofmathematical formulations and numerical solution schemes.Here we employ rotation-free inertial Cartesian coordinates of points tocharacterise such systems, producing a formulation which easily couples rigidbody dynamics with nonlinear finite element techniques for the flexiblebodies. This gives rise to additional internal constraints in rigid bodies topreserve distances. Constraints are enforced via a penalty method, which givesrise to a simple yet powerful formulation. Energy-momentum time integrationschemes enable robust long term simulations for highly nonlinear dynamicproblems.The main contribution of this paper focuses on the integration of constraintequations within energy-momentum conserving numerical schemes. It is shownthat the solution for constraints which may be expressed directly in terms ofquadratic invariants is fairly straightforward. Higher-order constraints mayalso be solved, however in this case for exact conservation an iterativeprocedure is needed in the integration scheme. This approach, together withsome simplified alternatives, is discussed.Representative numerical simulations are presented, comparing the performanceof various integration procedures in long-term simulations of practicalmultibody systems.

Energy Considerations for the Stabilization of Constrained Mechanical Systems with Velocity Projection

There are many difficulties involved in the numerical integration of index-3 Differential Algebraic Equations (DAEs), mainly related to stability, in the context of mechanical systems. An integrator that exactly enforces the constraint at position level may produce a discrete solution that departs from the velocity and/or acceleration constraint manifolds (invariants). This behavior affects the stability of the numerical scheme, resulting in the use of stabilization techniques based on enforcing the invariants. A coordinate projection is a post-stabilization technique where the solution obtained by a suitable DAE integrator is forced back to the invariant manifolds. This paper analyzes the energy balance of a velocity projection, providing an alternative interpretation of its effect on the stability and a practical criterion for the projection matrix selection.

A conservative augmented Lagrangian algorithm for the dynamics of constrained mechanical systems

Abstract The motion of many practical mechanical systems is often constrained. An important example is the dynamics of multibody systems, where the numerical solution of this type of systems faces several difficulties. A strategy to solve this type of problem is the augmented Lagrange formulation, which allows the use of numerical integrators for ODEs, combined with an update scheme for the algebraic variables, accomplishing exact fulfillment of the constraints. This work focuses on the design of a conservative version of this augmented Lagrangian formulation for holonomic constraints, proposing a numerical procedure that exhibits excellent stability.

On the Stabilizing Properties of Energy-Momentum Integrators and Coordinate Projections for Constrained Mechanical Systems

Several considerations are important if we try to carry out fast and precise simulations in multibody dynamics: the choice of modeling coordinates, the choice of dynamical formulations and the numerical integration scheme along with the numerical implementation. All these matters are very important in order to decide whether a specific method is good or not for a particular purpose.

Controllable velocity projection for constraint stabilization in multibody dynamics

The direct numerical solution of the index-3 algebraic-differential equation system (DAE) associated with the constrained dynamics of a multibody system poses several computational difficulties mainly related to stability. Specially in long simulations, the instability is related to the drift of the solution from the velocity constraint manifold. One of the techniques proposed in the literature to overcome this problem is the velocity projection, which is a post-stabilization method that brings the solution back to the invariant manifold. This projection introduce a nonnegative artificial dissipation of energy when performed with the system mass matrix, which can affect the long term quality of the solution. This paper proposes a novel controllable velocity projection procedure capable of meeting certain requirements in the projected solution; namely, a maximum constraint value or a maximum dissipated energy. Several issues related to the efficiency of the implementation are discussed and two numerical simulations are presented. These simulations illustrate the performance of the proposed methodology and provide interesting insights about the relevance of the accuracy of the projection in the stabilization effect.

Galerkin meshfree methods applied to the nonlinear dynamics of flexible multibody systems

Meshfree Galerkin methods have been developed recently for the simulation of complex mechanical problems involving large strains of structures, crack propagation, or high velocity impact dynamics. At the present time, the application of these methods to multibody dynamics has not been made despite their great advantage in some situations over standard finite element techniques.We adopt in this paper a geometric nonlinear formulation embedded in a multibody framework. The proposed approach allows the implementation of flexible solid bodies with minor changes in a multibody code. The flexibility is formulated by a Galerkin weak form. Among all the meshfree discretization methods, radial basis shape functions have been identified has the best ones for this kind of approach. The formulation is suited for both 2D and 3D problems, including static and dynamic analysis.

Energy and symmetry-preserving formulation of nonlinear constraints and potential forces in multibody dynamics

Practical multibody models are typically composed by a set of bodies (rigid or deformable) linked by joints, represented by constraint equations, and in many cases are subject to potential forces. Thus, a proper formulation of these constraints and forces is an essential aspect in the numerical analysis of their dynamics. On the other hand, geometric integrators are particular time-stepping schemes that have been successfully employed during the last decades in many applications, including multibody systems. One category is the so-called energy-momentum (EM) schemes that exhibit excellent stability and physical accuracy, conserving the discrete energy in conservative systems and preserving possible symmetries related with the conservation of linear and angular momenta. The use of the discrete derivative concept greatly systematizes their formulation, but its particular expression is not unique. This paper discusses the properties of several discrete derivative formulas proposed in the literature for models possessing finite-dimensional and linear configuration spaces, showing that not all of them lead to proper EM schemes. Some formulas, when applied to constraints or potentials endowed with certain symmetries (associated with the conservation of linear and angular momenta, found in many common practical systems), may produce numerical results that conserves the energy but violates the symmetries, introducing numerical instabilities and/or producing unphysical motions. This fact, surprisingly overlooked by many authors, is carefully analyzed and illustrated with several numerical experiments related with the dynamics of representative multibody systems.

APPLICATION CRITERIA FOR CONSERVING INTEGRATORS AND PROJECTION METHODS IN MULTIBODY DYNAMICS

This work presents the application to the dynamics of multibody systems of two methods based on augmented Lagrangian techniques, compares them, and gives some criteria for its use in realistic problems. The methods are an augmented Lagrangian method with orthogonal projections of velocities and accelerations, and an augmented Lagrangian energy conserving method. Both methods were presented by the authors in a very recent work, but it was not complete since the testing and the comparison of the methods was done by simulating a simple and academic example, and that was not sufficient to draw conclusions in terms of efficiency. For this work, the whole model of a vehicle has been simulated through both formulations, and their performance compared for such a large and realistic problem.Copyright