Asynchronous variational integration of structural collision dynamics: Numerical methods in structural mechanics,Used

From the Inside FlapThe growing power of modern workstations enables engineers to simulate more and more complex mechanical models by computers. In particular, nonlinear problems from structural dynamics are computationally intensive. Hence, there is ongoing demand in the development of new and improvement of existing algorithms. The present thesis deals with the simulation of the dynamics of flexible bodies subject to material and geometrical nonlinearities, as well as discontinuous phenomena arising from collisions.The equation of motion is discretized following the principle of variational integration, by what conservation laws of the continuous problem are valid in the discrete model. Existing approaches are presented. By combination of different procedures a mollified implicitexplicit algorithm is developed. It allows larger critical time steps and is particular suited for problems with nondominant nonlinearities. The presentation of variational integrators includes the temporal discretization of holonomic and unilateral constraints.The spatial discretization is performed by a modified finite element method. The accuracy of isoparametric elements is increased by enforcing stress continuity locally. This happens by the assumption of a continuously interpolated deformation gradient. The stability of the formulation is discussed in detail.For the temporal discretization an asynchronous strategy is employed. The equation of motion is integrated explicitly, whereby some critical time step length must not be exceeded. Asynchronous methods apply individual time steps to each spatial domain. Substructures with softer material behaviour or larger finite elements can, therefore, be integrated by a larger time step. The thesis develops strategies to estimate the local time step size for the new element formulation and to efficiently treat nodal restraint conditions. It studies, how temporallyadaptive step sizes influence stability and accuracy.Furthermore, this work presents procedures for spatial discretization and detection of collision problems. In particular, the concept of distance fields is enhanced in this respect. The contact conditions from impenetrability and friction are enforced by discontinuous velocity changes in a spatially asynchronous and temporally adaptive manner.Contents1 Introduction2 Variational mechanics2.1 Introduction2.2 Preliminaries2.3 Principle of Hamilton2.4 Preserved quantities2.5 Example3 Variational integrators3.1 Introduction3.2 Geometric integrators3.3 Discrete EulerLagrange equation3.4 Preserved discrete quantities3.5 Error analysis3.6 Linear stability analysis3.7 Example integrators3.8 Constraints3.9 Examples4 Continuous assumed gradient method4.1 Introduction4.2 Fundamentals of continuum mechanics4.3 Finite element interpolation of the continuum4.4 Assumed gradient field4.5 Regular mesh generation4.6 Nodal integration4.7 Smoothed Finite Element Method4.8 Stable interpolation schemes4.9 Implementation4.10 Error analysis4.11 Examples5 Asynchronous variational integration5.1 Introduction5.2 Asynchronous Euler scheme5.3 Discretization of the spacetime integral5.4 Nodal restraints5.5 Estimating the time step length5.6 Example6 Variable step size integration6.1 Introduction6.2 Explicit symplectic energy momentum integration6.3 Time transformations6.4 Variational kick and drift operators6.5 Asynchronous variable time steps6.6 Example time step functions6.7 Time step selection and solution6.8 Examples7 Collision dynamics7.1 Introduction7.2 Contact mechanics7.3 Distance field7.4 Asynchronous collisions7.5 Examples8 SummaryAppendixA Verification of CAG elementsB Gradient interpolating functions of CAG elementsC Variations on the contact interfaceD Spatial discretization of the contact boundaryE Collision detectionProduct DescriptionThe growing power of modern workstations enables engineers to simulate more and more