On the superlinear convergence in computational elasto-plasticity ...

Karlsruhe Institute of Technology

On the superlinear convergence in computational elasto-plasticity Application to non-associated models in soil mechanics Christian Wieners (joint work with M. Sauter) Institut für Angewandte und Numerische Mathematik

KIT – University of the State of Baden-Württemberg and National Research Center of the Helmholtz Association

www.kit.edu

Overview

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1. Representative Models in Soil Mechanics Drucker-Prager — Cam-clay 2. Generalized Newton Methods for Incremental Plasticity Semismooth Netwon methods — active set strategies 3. Nonsmooth Convergence Analysis Semismooth functions — generalized implicit function theorem 4. Applications and Convergence Results Drucker-Prager — Cam-clay

1

Quasi-static Infinitesimal Plasticity

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Let Ω ⊂ R3 and ∂ Ω = ΓD ∪ ΓN . We consider for t ∈ [0, T ] σ (x, t) = b(x, t) in Ω , − divσ

σ (x, t)n(x) = t N (x, t) on ΓN .

Balance of angular momentum implies σ (x, t) ∈ S := {ττ ∈ Rd,d : τ = τ T }. We only consider the small strain setting, i.e., the stress-strain relation is given by σ = C[εε (u) −εε p ] . The plastic evolution is described by the plastic strain ε p and internal variables η. σ , η) ∈ S = S × Rm is said to be admissible if The generalized stress Σ = (σ ˆ ∈ S : fi (Σ) ˆ ≤ 0 for i = 1, . . . , p} Σ ∈ K := {Σ for given yield functions fi : S → R. The plastic strain rate ε˙ p is given by p

ε˙ p =

∑ λi r i (Σ) ,

i=1

with prescribed plastic flow directions r i : S → S (possibly multi-valued) and the consistency parameters λi ≥ 0 determined by the complementarity conditions 0 = λi fi (Σ) ,

λi ≥ 0 ,

fi (Σ) ≤ 0 .

The equations of the internal variables η depend on the model. 2

The Drucker-Prager Model

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The classical Drucker-Prager model is defined by the single yield function σ ) = | dev(σ σ )| + k0 (tan φ 13 tr(σ σ ) − c) f (σ c tan φ 1. k0 > 0 is a shape factor of the cone, c ≥ 0 is related to the cohesion, and φ > 0 is the angle of friction.

defines the admissible set K , which is a cone with apex σ apex = The non-associated flow rule is based on the plastic potential σ ) − c) , σ ) = | dev(σ σ )| + k0 (tan ψ 31 tr(σ g(σ

where ψ ∈ [0, φ ] is the dilatency. The incremental flow rule is then given as  σ ) = s ∈ S : g(ττ ) ≥ g(σ σ ) + s : (ττ − s) . ε p = ε old s ∈ ∂ g(σ p + 4λ s , σ ) = {Dg(σ σ )} for dev(σ σ ) 6= 0. Note that in this case ∂ g(σ Since f is not differentiable, we also consider a smoothed variant with θ > 0 q σ )|2 + θ 2 + k0 (tan φ 31 tr(σ σ ) − c) , σ ) = | dev(σ fθ (σ q σ ) = | dev(σ σ )|2 + θ 2 + k0 (tan ψ 31 tr(σ σ ) − c) gθ (σ (cf. Krabbenhoft et al. 2006 or Miehe-Lambrecht 1999). 3

The Drucker-Prager Model

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4

The smoothed Drucker-Prager Model

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5

The Modified Cam-clay Model

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Cam-clay plasticity and its variants (see, e.g., Roscoe et al. 1968) are fundamental models in critical state soil mechanics. Here, we consider the modification as in Borja-Lee 1990 or Zouain-Filho-Borges-Costa 2007. σ , η) ∈ S × R includes a material strength parameter The generalized stress Σ = (σ η > 0 (related to the pre-consolidation pressure). The yield function is given as   2 σ )|2 + M3 tr(σ σ ) 31 tr(σ σ ) + 2η . f (Σ) = 23 | dev(σ For the plastic strain rate we assume normality, and for the evolution of the strength parameter, a non-associated evolution law is proposed: σ , η) , ε˙ p = λ Dσ f (σ η˙ = −k η tr(ε˙ p ) . The parameter k related to the virgin compression and the swell-recompression index.

In the incremental approach we solve the differential equation exactly, i.e.,
η = η old exp − k tr(εε p −εε old p ) . The exact integration guarantees η > 0. 6

Incremental Plasticity

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Our objective is to construct and to analyze generalized stress response functions R n = (R n , E n ) : S → S ,

σ tr ) . Σn = R n (σ

By means of the stress response function, the incremental elasto-plasticity problem and η n−1 find σ n and u n such that reads as follows: depending on ε n−1 p σ n (x) = b(x, tn ) , − divσ σn

σ (x) = R

n

C[εε (u

x ∈ Ω, n

(x)) −εε n−1 p (x)]




x ∈ Ω,

,

u n (x) = u D (x, tn ) ,

x ∈ ΓD ,

σ n (x)n(x) = t N (x, tn ) ,

x ∈ ΓN .

The incremental solution then defines the plastic strain and history variables σ n (x)] , ε np (x) = ε (u n (x)) − C−1 [σ n

The FE

Ω

Rn

n

x ∈ Ω,

η (x) = E C[εε (u (x)) −εε n−1 p (x)] , n discretization determines u with u n |ΓD = u D (tn ) Z

n




x ∈ Ω.

and Z Z
C[εε (u n ) −εε n−1 b(tn ) · w dx + p ] : ε (w ) dx = Ω

ΓN

t N (tn ) · w da

for all test functions with w |ΓD = 0. In the following we write u = u n , u old = u n−1 . 7

Implicit Characterization of Response Functions

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σ , η) ∈ S We determine simultaneously the generalized stress Σ = (σ p and the consistency parameter 4λ ∈ R≥0 . Thus, we define the space T = S × Rp , and we assume that a function G : T × S → S exists, such that
σ tr = 0 . G (Σ, 4λ ),σ Drucker-Prager Cam-Clay


σ , 4λ ),σ σ tr = C−1 [σ σ −σ σ tr ] + 4λi Dgθ (Σ) G (σ " # σ −σ σtr ] + 4λ Dσ f (σ σ , η)  C−1 [σ

σ , η), 4λ ),σ σ tr = G ((σ σ −σ σ tr ] η − η old exp k tr C−1 [σ

For α > 0 we define Φi (f , λ ) = max{0, λi + αfi } − λi and   σ tr ) G((Σ, 4λ ),σ σ tr ) = . T : T ×S → T , T ((Σ, 4λ ),σ Φ(f (Σ), 4λ ) For the ncp-function holds Φi (f , λ ) = 0 if and only if 0 = λi fi (Σ), λi ≥ 0, fi (Σ) ≤ 0. p

For a given trial stress σ tr , the solution (Σ∗ , 4λ ∗ ) ∈ K × R≥0 ⊂ T of σ tr ) = 0 T ((Σ∗ , 4λ ∗ ),σ σ ∗ , η ∗ ) = (R(σ σ tr ), E(σ σ tr )). defines the response Σ∗ = (σ 8

An Active Set Method

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 The active index set A (Σ, 4λ ) = i ∈ {1, . . . , p} : 4λi + αfi (Σ) > 0 and its complement I (Σ, 4λ ) defines  
σ tr ) D4λ G((Σ, 4λ ),σ σ tr ) D G((Σ, 4λ ),σ σ tr = Σ , T (Σ, 4λ ),σ αDf (Σ)A (Σ,4λ ) − idI (Σ,4λ ) where AJ is defined row-wise via (AJ )i = Ai if i ∈ J and (AJ )i = 0 otherwise. (AS0) (AS1)

Choose (Σ0 , 4λ 0 ) ∈ T , ε ≥ 0, α > 0 and set k := 1. σ tr ) ≤ ε, If T ((Σk −1 , 4λ k −1 ),σ

(AS2)

set (Σ∗ , 4λ ∗ ) = (Σk −1 , 4λ k −1 ) and T∗ = Tk −1 , STOP.
σ tr . Set Tk = T (Σk −1 , 4λ k −1 ),σ

(AS3)

σ tr ). Solve Tk |δ Σk , δ 4λ k ] = −T ((Σk −1 , 4λ k −1 ),σ

(AS4)

Set (Σk , 4λ k ) = (Σk −1 , 4λ k −1 ) + (δ Σk , δ 4λ k ). Set k := k + 1 and go to (AS1).

σ ∗ , η ∗ ) = (R(σ σ tr ), E(σ σ tr )) and the Jacobian This computes the response Σ∗ = (σ  ∗ S E∗  = −(T∗ )−1 C−1 Λ∗ 9

A Generalized Newton Method

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If the response function R is Lipschitz continuous, it is differentiable in a dense the set ΘR , and the set-valued B(ouligand)-subdifferential of R is given as  σ tr ) = S ∈ Lin(S, S) : S = ∂ B R(σ lim DR(θ ) . σ tr ,θ ∈ΘR θ →σ

 σ tr ) = conv ∂ B R(σ σ tr ) . Clarke’s subdifferential is its convex hull ∂ R(σ σ tr + θ ) R is semismooth, if for any S ∈ ∂ R(σ R(σ σ tr + θ ) − R(σ σ tr ) − S[θ ] = o(|θ |) as θ → 0 . (GN0)

Choose u 0h ∈ X h (u D ), ε ≥ 0, and set k := 1.

(GN1)

k −1 = R(σ σ ktr−1 ), Compute σ ktr−1 = C[εε (u k −1 ) − ε old p ], the stress response σ and the residual Z

rk −1 (w h ) =

(GN2)

Z

(GN4)

σ k −1 : ε (w h ) dx − `(w h ) ,

Ω If krk −1 k ≤ ε, set u ∗h = u kh −1 , STOP. σ ktr−1 ) and compute Choose S ∈ ∂ R(σ

Set

wh ∈ Xh .

δ u kh ∈ X h solving the linearization

S[C[εε (δ u kh )]] : ε (w h ) dx = −rk −1 (w h ) , w h ∈ X h .

Ω u kh = u kh −1 + δ u kh

and k := k + 1. Go to (GN1).

10

Results from Non-smooth Analysis

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For T : RN × RM → RN we define  ∂xB T (x, y ) = Ax ∈ RN,N : Ay exists s.t. [Ax Ay ] ∈ ∂ B T (x, y ) .

Theorem (Gowda 04, Kanzow-Heusinger 08) Let T be semismooth in a neighborhood of a point (x ∗ , y ∗ ) satisfying T (x ∗ , y ∗ ) = 0, and let all matrices ∂xB T (x ∗ , y ∗ ) be non-singular. Then, there exists an open neighborhood U(y ∗ ) of y ∗ and a function Y : U(y ∗ ) → RN which is locally Lipschitz and semismooth such that Y (y ∗ ) = x ∗ and T (Y (y ), y ) = 0 for all y ∈ U(y ∗ ). Moreover, if [Ax Ay ] ∈ ∂ B T (Y (y ), y ), we have B −A−1 x Ay ∈ ∂ Y (y ) ⊂ ∂ Y (y ) .

A generalized Newton iteration to compute a root x ∗ of a Lipschitz continuous function F : RN → RN is defined by x k = x k −1 − Jk−1 F (x k −1 ) with Jk ∈ ∂ F (x k −1 ).

Theorem (Mifflin 77, Clarke 83, Qi-Sun 93 ...) Let x ∗ be a solution of F (x ∗ ) = 0. Assume that F is semismooth at x ∗ , and that all matrices in ∂ F (x ∗ ) are regular. Then, provided that |x 0 − x ∗ | is small enough, the Newton-iteration is well-defined and converges superlinearly to the solution x ∗ . 11

Semismooth Response for the Drucker-Prager Model

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The non-associated deviation between flow rule and plastic potential σ) f (σ

=

σ )| + k0 (tan φ 31 tr(σ σ ) − c) | dev(σ

σ) g(σ

=

σ )| + k0 (tan ψ 13 tr(σ σ ) − c) | dev(σ

is measured by G[εε ] = dev(εε ) +

tan ψ 1 tr(εε )1 and F = C ◦ G. tan φ 3

The plastic response R is the orthogonal projection onto the admissible set −1 σ ∈ S : f (σ σ ) ≤ 0} w.r.t. the inner product K = {σ induced by   F . tan φ σ ) := κ tan φ tan ψ | dev(σ σ )| − 2µ 3 tr(σ σ ) − c , we obtain With k ◦ (σ   σ tr ) ≤ 0 , f (σ  σ tr ◦ σ tr ) ≤ 0 , k (σ σ tr ) = σ apex R(σ  σ tr ) f (σ  σ tr − σ tr )] else F[Df (σ 2µ+tan φ tan ψκ

for the plastic response, and the consistent tangent is given by    I σ tr ) = 0 S(σ     1 I − σ tr )] ⊗ Df (σ σ tr ) + f (σ σ tr )F ◦ D 2 f (σ σ tr ) F[Df (σ 2µ+κ tan φ tan ψ

σ tr ) ≤ 0 , f (σ σ tr ) ≤ 0 , k ◦ (σ else . 12

Algorithmic Response for the Cam-clay Model

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σ , η, λ ) with a vector z ∈ R8 , and for given σ tr we solve We identify the triple (σ σ tr ) = 0 , T1 (z,σ

...

σ tr ) = 0 . T8 (z,σ

This is done with a black box iteration, where the Jacobian T in (AS) and thus the consistent tangent S in (GN) is approximated by symmetric finite differences, i.e.,  1  σ tr ) − Ti (z − δ ej ,σ σ tr ) T= Ti (z + δ ej ,σ ∈ R8×8 . 2δ i,j=1,...,8 Here, we choose δ = 5 · 10−7 . void Constraint (const Tensor& T, double eta_old, const Tensor& S, double eta, double Lambda, SmallVector& c) { Tensor DS = S; DS -= T; DS = StressStrain(DS); c[6] = eta - eta_old * exp(CamClay_k * trace(DS)); DS += Lambda * CamClayFlowDirection_S(S,eta); c[0] = DS[0][0]; c[1] = DS[1][1]; c[2] = DS[2][2]; c[3] = DS[0][1]; c[4] = DS[0][2]; c[5] = DS[1][2]; c[7] = CamClayNCP(S,eta,Lambda); } 13

A Strip Footing with the Drucker-Prager Model

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Shear modulus µ = 5.5 [MPa] Bulk modulus κ = 12.07 [MPa] Cohesion c = 0.01 [MPa] Friction angle φ = 30◦ Dilatancy angle ψ = 15◦ Scaling factor k0 = 0.7 Smoothing parameter θ = 0.0001

Convergence history of the (outer) generalized Newton iteration for Drucker-Prager and smoothed Drucker-Prager elasto-plasticity.

k 0 1 2 3 4 5 6 7 8

t20 4.4e-05 2.8e-05 1.0e-05 3.8e-07 1.1e-08 1.8e-14

Drucker-Prager t40 t60 8.2e-05 2.8e-04 8.3e-06 1.4e-04 4.4e-07 5.0e-06 1.5e-08 1.0e-06 4.6e-14 3.4e-07 1.5e-07 6.9e-08 2.2e-08 1.5e-11

Smoothed Drucker-Prager t20 t40 t60 4.4e-05 8.2e-05 2.8e-04 4.4e-05 8.2e-06 1.4e-04 2.8e-06 4.3e-07 5.1e-06 9.5e-08 2.1e-08 1.0e-06 1.8e-09 7.8e-14 3.3e-07 2.4e-15 1.5e-07 5.9e-08 1.1e-08 4.7e-13 14

Configuration for a Slope Failure Problem

2.5

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Lg(t) Lt(t)

Load functions

2

1.5

1

0.5

0 0

0.5

1

1.5

2

2.5

3

3.5

Time

15

A Slope Failure Problem with the Drucker-Prager Model Newton convergence at time step 29 and accumulated plastic strain at time t = 2.9.

k 0 1 2 3 4 5 6 7 8

826 995 2.7e-04 5.7e-05 3.7e-06 4.9e-07 4.8e-10

6 452 451 1.9e-04 5.3e-05 1.5e-05 1.7e-06 2.2e-07 7.8e-09 6.1e-12

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50 973 123 1.1e-04 6.2e-05 3.4e-05 6.9e-06 5.3e-07 1.1e-07 2.6e-08 3.1e-09 5.5e-11

16

A Slope Failure Problem with the Cam-clay Model kηk∞

k

3 792 d.o.f. 14 943 d.o.f. 108 603 d.o.f. 826 995 d.o.f. 6 452 451 d.o.f.

7

0.0495

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6 5

0.049

4

0.0485

3 2

level 0 level 1 level 2 level 3 level 4

0.048 0

0.5

1

1

1.5

2

2.5

3

0

t

Evolution of the material strength parameter η

0

5

10

15

20

25

30

n

Number of Newton steps k in time step n

Distribution of the mean stress σm for the Cam-clay model on refinement level 4 at time t = 2.9 (and surface mesh on refinement level 2). M. Sauter and C. Wieners: On the superlinear convergence in computational elastoplasticity. Comput. Methods Appl. Mech. Engrg. 200 (2011) 3646-3658 17