mpccdist2 details:

Keywords: analytic solution

Global classification: nonlinear-quadratic

Functional: convex quadratic

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

Design constraints:

State constraints:

Mixed constraints:



Submitted on 2014-04-23 by Thomas Betz. Published on 2014-05-27


mpccdist2 description:


Introduction

We consider the optimal control of static elastoplasticity with linear kinematic hardening. This leads to an optimal control problem governed by an elliptic variational inequality of first kind in mixed form or, equivalently, an MPCC in function space.
The objective functional tracks the state both on the entire domain and additionally on a submanifold of the domain. The control cost is taken into account by a tracking type term as well. There are no constraints on the control, which acts in a distributed way on the domain. The Dirichlet boundary is the entire boundary of the domain.
A locally optimal control is known, whose corresponding state has a bi-active set with a positive measure.

The problem and its solution are taken from [Betz et al.2014, Section 6.1].

Variables & Notation

Unknowns

f L2Ω; 2 control variable σ,χ,u,λ L2Ω; sym2×22 × H 01Ω; 2 × L2(Ω)state variable

The state is composed of the stress σ, back stress χ, displacement u and plastic multiplier λ.

Given Data

Ω = {x 2 : x < 1} computational domain B = {x 2: x < 12}subdomain R = Ω B subdomain k1 = 1.0 hardening constant E = 1.0 elasticity modulus ν = 0.0 Poisson number μL = 0.5 Lamé constant λL = 0.0 Lamé constant σ0 > 0 yield stress (arbitrary) α > 0 control cost parameter (arbitrary)

desired displacement in the domain:

uΩ(x) = Ux2 + 8x12 + 4x22 + 4x1x2Ux2 + 6Ux2 Ux2 + 4x12 + 8x22 + 4x1x2Ux2 + 6Ux2 ,x B Ux2 + 6x12 + 2x22 + 4x1x2Ux2 + 4Ux2 Ux2 + 2x12 + 6x22 + 4x1x2Ux2 + 4Ux2 , x R

with

U(t) = σ0t2 + 3 2σ0t 13 16σ0,t < 1 4 σ0t σ0, t 1 4 U(t) = 2σ0t + 3 2σ0,t < 1 4 0.5σ0t12, t 1 4 U(t) = 2σ0, t < 1 4 0.25σ0t32,t 1 4

desired displacement on the submanifold B:

uB(x) = σ0 σ0

desired control:

fΩ(x) = 2 αUx2 4x12 + 2x22 + 2x1x2Ux2 3Ux2 2 αUx2 2x12 + 4x22 + 2x1x2Ux2 3Ux2

Problem Description

MinimizeJ(u,f) := 1 2uuΩL2(Ω;2)2 + 1 2uuBL2(B;2)2 + α 2 f fΩL2(Ω;2)2 s.t. 1σ 𝜀(u) + λσD + χD = 0in L2Ω; sym2×2, 1χ + λσD + χD = 0in L2Ω; sym2×2, σ,𝜀(v)L2Ω;sym2×2 + (f,v)L2(Ω;2) = 0v H01Ω; 2, 0 λ ϕ(σ,χ) 0a.e. in Ω.

With the given Lamé coefficients, the inverse elasticity and hardening tensors read

1σ = 1 2μLσ λL 2μL(2μL + 2λL)trace(σ)I = σ, 1χ = 1 k1χ = χ,

where I : sym2×2 sym2×2 is the identity mapping and

𝜀(u) = 1 2u + (u)the linearized strain, σD = σ trace(σ)Ithe deviatoric part of σ.

The yield function is given by

ϕ(σ,χ) = σD + χDF 2 σ02 2 ,

where F denotes the Frobenius norm of a matrix. The corresponding inner product trace(AB) is used in the calculation of (,)L2(Ω;sym2×2).

Optimality System

According to [Betz and Meyer2015, Theorem 4.6] (Theorem 4.4 in the Preprint) the following conditions are sufficient for local optimality of the control f with associated state σ,χ,u,λ.
There exist adjoint variables ζ,ψ,w LΩ; sym2×22 × H01Ω; 2 and multipliers (μ,𝜃) L(Ω) × L(Ω) satisfying

  1. the optimality system 1ζ 𝜀(w) + λζD + ψD + 𝜃σD + χD = 0in L2Ω; sym2×2 1ψ + λζD + ψD + 𝜃σD + χD = 0in L2Ω; sym2×2 ζ,𝜀(v)L2Ω;sym2×2 + (uuΩ,v)L2(Ω;2) + (uuB,v)L2(B;2) = 0v H01Ω; 2 w + α(f fΩ) = 0in L2Ω; 2 σD + χD : ζD + ψD μ = 0a.e. in Ω μλ = 0a.e. in Ω 𝜃ϕ(σ,χ) = 0a.e. in Ω μ 0a.e. in Ω 𝜃 0a.e. in Ω
  2. the second-order condition
    There exists κ > 0 such that
    σ,χ,u,λ,f2σ,χ,u,λ,f,ζ,ψ,w,μ,𝜃σ,χ,u,λ,h2 κh U2

    holds for all h L2Ω; 2 and σ,χ,u,λ solving

    1σ𝜀(u) + λ(σ)D + (χ)D + λσD + χD = 0in L2Ω; sym2×2 1χ + λ(σ)D + (χ)D + λσD + χD = 0in L2Ω; sym2×2 σ,𝜀(v) L2Ω;sym2×2 + (h,v)L2(Ω;2) = 0v H01Ω; 2 λσD + χD : (σ)D + (χ)D = 0a.e. in 𝒜 s 0 λσD + χD : (σ)D + (χ)D 0a.e. in  0 = λσD + χD : (σ)D + (χ)D a.e. in .

    The sets are defined as follows,

    𝒜s := {x Ω : λ > 0} strongly active set, := {x Ω : ϕ(σ,χ) = λ = 0}bi-active set, := {x Ω : ϕ(σ,χ) < 0} inactive (elastic) set,

    and the Lagrangian is defined by

    (σ,χ,u,λ,f,ζ,ψ,w,μ,𝜃) = J(u,f) + σ 𝜀(u) + λσD + χD,ζ L2Ω;sym2×2 + χ + λσD + χD,ψ L2Ω;sym2×2 σ,𝜀(w)L2Ω;sym2×2 + (f,w)L2Ω;2 (λ,μ)L2(Ω) + (ϕ(σ,χ),𝜃)L2(Ω).

    Consequently σ,χ,u,λ,f2σ,χ,u,λ,f,ζ,ψ,w,μ,𝜃σ,χ,u,λ,h2 is given as follows:

    σ,χ,u,λ,f2σ,χ,u,λ,f,ζ,ψ,w,μ,𝜃σ,χ,u,λ,h2 = (u,u) L2(Ω;2) + (u,u) L2(B;2) + α(h,h)L2(Ω;2) + 2λ(σ)D + (χ)D,ζD + ψD L2(Ω;sym2×2) + (σ)D + (χ)D F 2,𝜃 L2(Ω).

Supplementary Material

Locally optimal control, state, adjoint state and multipliers are known analytically:

u = U x2 U x2 displacement, σ = 𝜀(u) = U(x2) 2x1 x1 + x2x 1 + x2 2x2 stress, χ = 0 back stress, λ = 0 plastic multiplier, f = div(𝜀(u)) = 3U(x2) + U(x2)(4x12 + 2x1x2 + 2x22) 3U(x2) + U(x2)(2x12 + 2x1x2 + 4x22) control (right hand side force), ζ = 2𝜀(u), x B 𝜀(u) + 𝜀(u)S,x R adjoint stress, ψ = 0, x B 𝜀(u)D,x R adjoint back stress, w = 2u adjoint displacement, μ = 2𝜀(u)DF 2,x B 0, x R multiplier, 𝜃 = 0,x B 1, x R multiplier.

The magnitude of the optimal state and control for the values α = 1 and σ0 = 2 as given in Betz et al. [2014] are depicted in Figure 0.1. The reference value for the objective,

J 156.44873847926598083

corresponding to these values of α and σ0 is given in Betz et al. [2014].


PICPIC

Figure 0.1: Analytical values of the optimal state |u| (left) and control |f| (right) for parameters α = 1 and σ0 = 2. Figure courtesy of K. Rosin

Revision History

References

   T. Betz and C. Meyer. Second-order sufficient optimality conditions for optimal control of static elastoplasticity with hardening. ESAIM: Control, Optimisation and Calculus of Variations, 21(1):271–300, 2015. doi: 10.1051/cocv/2014024.

   T. Betz, C. Meyer, A. Rademacher, and K. Rosin. Adaptive optimal control of elastoplastic contact problems. Technical report, Fakultät für Mathematik, TU Dortmund, May 2014. URL http://www.mathematik.tu-dortmund.de/papers/BetzMeyerRademacherRosin2014.pdf. Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 496.