Nonsmooth Multi-Level Optimization Algorithms for Energetic Formulations of Finite-Strain Elastoplasticity

有限应变弹塑性的能量公式是速率无关系统的一般理论的一个实例。他们推广了小应变弹塑性的原始公式，其中变量是位移、塑性应变和可能的硬化变量。由于它们不涉及导数，非光滑现象可以用一种特别优雅的方式建模。在能量形式中，时间离散弹塑性问题是最小化问题的序列，这使得它们适合于优化算法。增量最小化问题结合了各种困难：它们是高度非线性、非凸和非光滑的，一些自变量在李群中取值，模拟了塑性变形的不可压缩性。从积极的方面来看，离散化后的非光滑项是块可分的，即它们可以写成具有小的不相交自变量集的非光滑函数的和。这一事实可以通过优化算法加以利用。在这个项目中，我们计划为有限应变弹塑性的能量公式开发有效的优化求解器。在特定问题结构的激励下，我们将使用近似牛顿方法，它将给定的非凸非光滑问题简化为凸序列，但仍然是非光滑子问题。然后，利用非光滑多重网格方法对这些子问题进行有效求解。这克服了牛顿近似解的一个众所周知的局限性，它通常缺乏子问题的有效解。本文研究了代数集和函数空间中的新的近端牛顿算法。我们将研究强制塑性变形不可压缩性的两种替代方法。一方面，我们将它们视为矩阵向量空间的元素，并将它们置于非线性等式约束之下。对于这个公式，我们将构造非光滑复合阶跃方法。作为一种补充方法，我们将把多水平近牛顿方法推广到流形上的优化问题的设置。将在一系列基准中比较这些方法的相对优点和缺点。

Energetic formulations of finite-strain elastoplasticity are an instance of the general theory of rate-independent systems. They generalize the primal formulation of small-strain elastoplasticity, where the variables are the displacements, plastic strain, and possibly hardening variables. As they do not involve derivatives, nonsmooth phenomena can be modeled in a particularly elegant way.In the energetic formulation, time-discrete elastoplastic problems are sequences of minimization problems, which makes them amenable to optimization algorithms. The increment minimization problems combine various difficulties: They are highly nonlinear, nonconvex and nonsmooth, and some of the independent variables take values in a Lie group, modelling incompressibility of plastic deformation.On the positive side, after discretization the nonsmooth terms are block-separable, i.e., they can be written as sums of nonsmooth functions with small disjoint sets of independent variables. This fact can be exploited by optimization algorithms.In this project we plan to develop efficient optimization solvers for energetic formulations of finite-strain elastoplasticity. Motivated by the specific problem structure we will use proximal Newton methods, which reduce the given nonconvex nonsmooth problems to sequences of convex, but still nonsmooth subproblems. Then, these subproblems are solved efficiently with the help of a nonsmooth multigrid method. This overcomes a well-known limitation of proximal Newton solvers, which typically lack efficient solvers for the subproblems. We study the new proximal Newton algorithms both in an algebraic setting and in function spaces.We will investigate two alternative approaches for enforcing incompressibility of plastic deformation. On the one hand, we will consider them as elements of the vector space of matrices and subject them to a nonlinear equality constraint. For this formulation we will construct nonsmooth composite step methods. As a complementary approach, we will generalize the multilevel proximal Newton methods to the setting of optimization problems posed on manifolds. The relative merits and shortcomings of these approaches will be compared in a series of benchmarks.