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.

有限应变弹性性的能量公式是速率独立系统的一般理论的一个实例。它们概括了小型弹性塑性性的原始公式，其中变量是位移，塑性应变，可能是硬化变量。由于它们不涉及衍生物，因此可以以特别优雅的方式对非平滑现象进行建模。在充满活力的表述中，时间散射的弹性塑料问题是最小化问题的序列，这使得它们可以与优化算法相吻合。增量最小化问题结合了各种困难：它们是高度非线性的，非凸和非平滑的，并且某些自变量在谎言组中取值，对塑性变形的不可压缩性进行建模。在离散的一面，在离散的一面不可压缩的一面是非这个光的术语是块分别的术语，即，它们可以用少量的nonsoth函数编写，它们的变化是不合时宜的。可以通过优化算法来利用这一事实。在该项目中，我们计划为有限型弹性性的能量配方开发有效的优化求解器。 由特定的问题结构激发，我们将使用近端牛顿方法，从而将给定的非凸态问题减少到凸的序列，但仍然不太平滑的子问题。然后，借助于非平滑的多机方法，有效地解决了这些子问题。这克服了近端牛顿求解器的众所周知的局限性，通常缺乏子问题的有效求解器。我们在代数环境和功能空间中研究了新的牛顿近端算法。我们将研究两种替代方法，以实现塑性变形的不可压缩性。一方面，我们将它们视为矩阵的向量空间的要素，并将其予以非线性平等约束。 对于此公式，我们将构建非平滑复合步骤方法。作为一种互补方法，我们将概括多级牛顿近端方法来设置在歧管上提出的优化问题。这些方法的相对优点和缺点将在一系列基准测试中进行比较。