We consider the deformation of a geological structure with non-intersecting faults that can be represented by a layered system of viscoelastic bodies satisfying rate- and state-depending friction conditions along the common interfaces. We derive a mathematical model that contains classical Dieterich- and Ruina-type friction as special cases and accounts for possibly large tangential displacements. Semi-discretization in time by a Newmark scheme leads to a coupled system of non-smooth, convex minimization problems for rate and state to be solved in each time step. Additional spatial discretization by a mortar method and piecewise constant finite elements allows for the decoupling of rate and state by a fixed point iteration and efficient algebraic solution of the rate problem by truncated non-smooth Newton methods. Numerical experiments with a spring slider and a layered multiscale system illustrate the behavior of our model as well as the efficiency and reliability of the numerical solver.
翻译:我们认为一个地质结构的变形,其非交叉断层可由一个能满足共同界面上速度和国家摩擦条件的多层粘合体系统来代表。我们得出一个数学模型,其中包括传统的迪特里奇式摩擦和鲁伊纳型摩擦,作为特殊案例,并说明了可能发生的大相向迁移。一个Newmark 方案在时间上的半分化导致一个结合的系统,即每个步骤要解决的速率和状态的非摩擦、最小化问题和分级问题。通过一个迫击炮法和条形常数不变的固定固定固定不变元素使速率和状态脱钩,通过固定点迭代和高效的代数法解决速率问题。与一个弹簧滑块和一个分层的多尺度系统进行的数值实验显示了我们模型的行为以及数字求解器的效率和可靠性。