Time-adaptive space-time integrator

In many industrial applications of structural dynamics one is interested in the determination of the structural response of a mechanical system at a specific time or during a fixed period. A particular task are drop test simulations, in which nonlinear materials and constraints arising in domain decomposition methods and impact situations with particular consideration of the geomtrical complexity (i.e. contact detection) must be considered. The solution method to be selected must be able to (1) efficiently express and solve the governing equations, (2) to generate the constraints reliably and fast, and (3) to exhibit good long time stability while maintaining sufficient accuracy. Furthermore, the applied partial algorithms should be self adaptive in the sense that the amount of user input and, thus, the required knowledge about the structure of the considered mechanical problem is minimized. In previous work, a stabilization method for explicit integrators based on modal reductions has been developed and successfully applied in the context of industrial application. The proposed project shall provide a systematic approach to the development of efficient integration method which are capable of handling collisions in such a way that the energy balance will be maintained and still time steps do not become too small. Therefore, on the one hand the research project shall contain the spatial and temporal formulation of the contact problem. Recently introduced distance fields promise a robust and efficient determination of contact constraints, although the methodology has been applied to frictionless penalty contact of simplicial domains only and must be appropriately extended to friction, general finite element meshes and Lagrange contact. Within the context of time integrators, there has been much research interest during the last two decades on integrators which systematically preserve the invariants (energy, momentum, symplecticity) while enforcing constraints. Beyond that, there exist a few approaches to discrete impact laws. A specific group, the variational integrators, shall be used as a basis to derive an explicit integrator. It is attempted to derive smooth continuity conditions between time steps to increasy the accuracy at almost constant numerical efficiency. Time adaptation shall be examined as well. Concluding, the numerical efficiency shall be improved by application of local time integration, i.e. each part of the system will be separtely integrated utilizing individual time step lengths. Preconditions are coupling conditions between these parts, which may be expressed either weakly or continuously in the spacetime domain. The contact formulation must be adapted to this approach. The resulting integration method shall be implemented for two finite element types and will be tested using an industrial example.

In many engineering applications people are interested in quantifying the distribution of displacements, deformations, stress, strains and material failure within a structure over time. Most of these tasks are performed by simulation. While the computing power of modern workstations is steadily growing, the complexity of the numerical models is growing at the same time. This complexity arises from considering nonlinearities (for example, from material laws), discontinuous effects (for example, from impacts between bodies) and desired high accuracies in certain domains leading to locally very fine discretizations. When considering families of numerical methods, in particular the numerical efficiency of explicit time integrators suffers when locally fine meshes are used. This is because the time domain is subdivided into a finite number of time steps. For each step a system of equations must be solved. The number of required time steps may be increased drastically if only a few very small finite elements are in the system because the upper critical time step size limit is dependent on the size and stiffness of the stiffest oscillator in a system. An approach of numerical methods to circumvent this problem are asynchronous variational integrators. They associate an individual time step size to each spatial point and, thus, every point is associated with a time step size which is the maximum to ensure stability. Larger (and softer) elements can then be integrated with a much larger time step compared with the standard approach. Within the project, the following advances were made in the context of asynchronous integrators and explicit time stepping: A finite element formulation based on continuous assumed deformation gradients was developed, termed Continuous Assumed Gradient elements. Those increase the accuracy of low-order finite elements while not increasing the number of degrees of freedom and increase the computational effort insignificantly. In particular for explicit simulations, a first order tetrahedral element was developed which drastically increases accuracy and even improves numerical efficiency when used in explicit dynamics. For the new finite element formulation a new strategy was developed to estimate stability of the asynchronous time integrator. Aside spatially varying time steps, temporally varying time steps were studies. It was found that most common formulations destabilize the asynchronous algorithm. For impact/collision dynamics a new strategy was implemented for local contact search based on discrete distance fields. An asynchronous collision integrator was developed which can be efficiently used in conjunction with friction and nodal restraint conditions. All of the mentioned approaches were shown to improve either accuracy, or numerical efficiency, or both compared with standard approaches.