Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators

Formulation and performance of variational integrators for rotating bodies

Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere. In the first case, an integration algorithm is presented for Euler’s equations of the free rigid body, following the ideas of Marsden et al. (Nonlinearity 12:1647–1662, 1999). In the second example, a variational time integrator is formulated for the rigid dumbbell. Both methods are formulated directly on their nonlinear configuration spaces, without using Lagrange multipliers. They are one-step, second order methods which show exact conservation of a discrete angular momentum which is identified in each case. Numerical examples illustrate their properties and compare them with existing integrators of the literature

Phenomenological modelling of damage in polymer blends

To describe the constitutive behaviour of a certain class of polymer blends an elastoperfectly-viscoplastic and creep damageable material characterization is proposed. For a composite of 80 % Polystyrene and 20 % Ethylene Propylene Diene Monomer rubber (PSIEPDM) the specific parameters are determined from tensile tests in a particular range of strain velocities. To investigate the applicability of the model, the results of a finite element analysis for a laterally loaded thin plate (plane stress) with a circular hole are compared to measurements. Numerically calculated values are in reasonable agreement with reality; discrepancies can be ascribed to noise in experimental data. The finite element approach is evaluated with respect to the occurrence of mesh-dependence. Mesh-refinement shows convergence of solutions, attributable to the stabilizing influence of the viscous contribution in the constitutive equations

A Nonlocal Elasto-Plastic Model for Structured Soils at Large Strains for the Particle Finite Element Method

This work presents a robust and mesh-independent implementation of an elasto-plastic constitutive model at large strains, appropriate for structured soils, into a Particle Finite Element code specially developed for geotechnical simulations. The constitutive response of structured soils is characterized by softening and, thus, leading to strain localization. Strain localization poses two numerical challenges: mesh dependence of the solution and computability of the solution. The former is mitigated by employing a non-local integral type regularization whereas an Implicit-Explicit integration scheme is used to enhance the computability. The good performance of these techniques is highlighted in the simulation of the cone penetration test in undrained conditions.Peer ReviewedPostprint (published version

Combined sticking: a new approach for finite-amplitude Coulomb frictional contact

Engineering-level accuracy of discretization methods for frictional contact originates from precise representation of discontinuous frictional and normal interaction laws and precise discrete contact techniques. In terms of discontinuous behavior in the quasi-static case, two themes are of concern: the normal interaction (i.e. impact) and the jumps in tangential directions arising from high frictional values. In terms of normal behavior, we use a smoothed complementarity relation. For the tangential behavior, we propose a simple and effective algorithm, which is based a stick predictor followed by corrections to the tangential velocity. This allows problems with impact and stick-slip behavior to be solved with an implicit code based on Newton–Raphson iterations. Three worked examples are shown with comparisons with published results. An extension to node-to-face form in 3D is also presented