| In this work, issues with regard to the design of time integrators for nonlinear transient problems are addressed. Two subjects are covered: nonlinear stable time integrators for conservative/dissipative dynamical systems; and modeling viscoelastic torsional response of biphasic material such as human cartilage.; In the first case, the basic theories of dynamical systems are discussed with emphasis on the conservation principles, which an energy consistent algorithm should obey in a discrete sense. Based on the algorithmic framework presented, a new solution strategy is proposed that applies to general hyperelastic material models in elastodynamics. Treating the weak form and the energy conservation equality as a coupled system, this new algorithm achieves quadratic rate of convergence without increasing the problem size, through a static condensation process. Following this algorithm for elastodynamics, algorithms that apply to dissipative dynamical systems are proposed, while such algorithms will not necessarily introduce numerical damping into the simulation. Two key implementation components are discussed in detail: energy consistency of plasticity models; and the construction of an algorithmic stress tensor that satisfies the reduced dissipation inequality automatically. Two types of plasticity models—an additive model and a multiplicative model—are adopted in the algorithms. Both model can be integrated using the classical strain driven return mapping scheme and amenable to consistent linearization.; Subsequently, an algorithm that predicts the viscoelastic behavior of biphasic materials is presented. Since the torsional response is of particular interest, the algorithm is formulated in a cylindrical coordinated system. This complicates the linearization of the weak form, because covariant differentiation is involved and gives rise to additional geometric terms that do not exist in a Cartesian formulation.; Finally, numerical examples are given to illustrate the performance of the algorithms presented and are compared to results from traditional algorithms and/or experiments. |
