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

有限应变弹塑性能量公式的非光滑多级优化算法

• 批准号: 423764152
• 负责人: Professor Dr. Oliver Sander
• 金额: --
• 依托单位: Institut für Numerische Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/423764152?language=en
• 关键词: Nonsmooth; Multi; Level; Optimization; Algorithms

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.

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例子。他们推广了小应变弹塑性的原始公式，其中的变量是位移，塑性应变，并可能硬化变量。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来建模。在能量公式中，时间离散的弹塑性问题是一系列最小化问题，这使得它们适合于优化算法。增量最小化问题联合收割机结合了各种困难：它们是高度非线性的、非凸的和非光滑的，并且一些独立变量在李群中取值，模拟塑性变形的不可压缩性。在积极的方面，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。这个事实可以利用优化算法。在这个项目中，我们计划开发有效的优化求解有限应变弹塑性的能量公式。 出于特定的问题结构，我们将使用近端牛顿方法，减少给定的非凸非光滑问题的序列凸，但仍然非光滑的子问题。然后，这些子问题的有效解决的帮助下，非光滑多重网格方法。这克服了近端牛顿求解器的一个众所周知的限制，该限制通常缺乏有效的子问题求解器。我们在代数和函数空间中研究了新的近似牛顿算法，并研究了两种增强塑性变形不可压缩性的方法。一方面，我们将把它们看作矩阵的向量空间的元素，并使它们受到非线性等式约束。 对于这个公式，我们将构造非光滑复合步长方法。作为一种补充方法，我们将推广的多层近似牛顿方法设置的流形上提出的优化问题。这些方法的相对优点和缺点将在一系列基准中进行比较。