Functional a posteriori error estimates for incremental models in elasto-plasticity

Cent. Eur. J. Math. • 7(3) • 2009 • 506-519 DOI: 10.2478/s11533-009-0035-2

Central European Journal of Mathematics

Functional a posteriori error estimates for incremental models in elasto-plasticity Research Article

Sergey I. Repin1∗ , Jan Valdman2† 1 V.A. Steklov Institute of Mathematics in St.-Petersburg, St.-Petersburg, Russia 2 Department of Mathematics, University of Bergen, Bergen, Norway

Received 12 December 2008; accepted 26 May 2009

Abstract: We consider incremental problem arising in elasto-plastic models with isotropic hardening. Our goal is to derive computable and guaranteed bounds of the difference between the exact solution and any function in the admissible (energy) class of the problem considered. Such estimates are obtained by an advanced version of the variational approach earlier used for linear boundary-value problems and nonlinear variational problems with convex functionals [24, 30]. They do no contain mesh-dependent constants and are valid for any conforming approximations regardless of the method used for their derivation. It is shown that the structure of error majorant reflects properties of the exact solution so that the majorant vanishes only if an approximate solution coincides with the exact one. Moreover, it possesses necessary continuity properties, so that any sequence of approximations converging to the exact solution in the energy space generates a sequence of positive numbers (explicitly computable by the majorant functional) that tends to zero. MSC:

65N30, 74C05

Keywords: A posteriori error estimates • Elasto-plastic problem with isotropic hardening • Variational inequalities © Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1.

Introduction

Incremental models in the theory of elasto-plasticity are among the most widely used in the numerical analysis of processes that include plasticity phenomenon. These typically include memory effect and exhibit hysteresis [12, 22] behavior which are described by time-dependent variational inequalities. If an implicit Euler scheme is used, then the evolutionary variational inequality is approximated by a sequence of stationary variational inequalities of the second kind [3, 20] in which the unknown functions are displacement u and plastic strain p. Each of these inequalities is equivalent to a minimization problem with a convex but non-smooth energy functional, J(u, p) → min. The minimization ∗ †

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E-mail: [email protected] E-mail: [email protected]

S.I. Repin, J. Valdman

problem is solved by iterative methods like a classical return mapping algorithm [35], inexact Newton methods [9] or SQP method [37] among many others. The main focus here is not to develop new methods for solving the minimization problems, but to deduce a guaranteed estimates of the difference between exact and numerical solutions. First results in the error analysis of elasto-plastic problems were obtained within the framework of a priori conception (see, e.g., [20, 35, 37] and the references cited therein). In this analysis, the case of perfect plasticity was the most difficult because for such type problems the initial variational problem (for displacements) does not possess coercivity on a suitable reflexive space. In spite of these difficulties, qualified priori rate convergence estimates (for a perfectly elasto-plastic model) were derived in terms of the dual problem, i.e., for approximations of the corresponding field of stresses (see [26, 31]). Those estimates, were 1 based upon local regularity (W2,loc (Ω, Rd×d )) of the stress tensor established in [6, 34]. In [27] a somewhat different error estimation method was used. It was shown that approximations of the stress field associated with the perfectly elasto-plastic model can be approximated with the help of a sequence of approximate solutions (computed in terms of displacements) generated by elasto-plastic models with hardening provided that the hardening and mesh parameters are properly related. Also, qualified a priori rate convergence estimates for stresses were established. However, a priori error estimates provide only a very general knowledge on the behavior of computational errors and, in general, are unable to give an adequate presentation on the computational error. For this purpose, we need to have an a posteriori error estimate, which is intended to (a) give an indication of the overall accuracy of an approximate solution and (b) serve as an error indicator that show regions with excessively high errors (typically a new finite dimensional space constructed on the basis of this information has extra trial functions in each regions). There exist various approaches to the construction of a posteriori error estimates (a discussion of them can be found e. g., in the books [1, 4, 5, 24] or in the recent monograph [30]. For applications to elasto-plasticity, we refer to the works [14, 25] and to the literature cited in these publications. In this paper, we apply the framework introduced in the book [24], where the estimates are derived by the analysis of the variational problem and its dual counterpart. A computable upper bound of the error is obtained on a purely functional level without exploitation of specific properties of the approximation or the method used for its computation. Estimates of such a type are often called “functional a posteriori estimates”. One of the first publications presenting this method was [33], where a posteriori estimates were derived for a deformation plasticity model with hardening. Recently, the method was applied to the Ramberg-Osgood model (sometimes also called Norton-Hoff) in the theory of nonlinear solid media [8], to nonlinear viscous flow problems [7, 16] and to problems with nonlinear boundary conditions [32].

2.

Minimization problem and variational inequality

We consider the first time-step problem for the elasto-plasticity model with isotropic hardening and von Mises yield criterion. It can be represented in a variational form as an energy minimization problem (see [3], Definition 3.3)

J(v, q) :=

1 2

Z


C(ε(v) − q) : (ε(v) − q) + σy2 H 2 |q|2 dx +

Ω

Z


σy |q| − fv dx → min,

(1)

Ω

for an unknown displacement v and a plastic strain q. Here, Ω ⊂ Rd is a bounded connected domain with Lipschitz boundary Γ. In (1), ε(·) denotes the linearized Green-St. Venant strain tensor defined as

ε(v) :=


1 ∇v + (∇v)T , 2

(2)

where ∇ denotes a vector gradient operator. J(v, q) is minimized over  v ∈ V0 + u0 := w + u0 | w ∈ H01 (Ω; Rd ) , and  q ∈ Q0 := q ∈ Q = L2 (Ω; Rd×d sym ) | tr q = 0 a. e. in Ω ,

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Functional a posteriori error estimates for incremental models in elasto-plasticity

where the function u0 ∈ H 1 (Ω; Rd ) defines a nonhomogeneous Dirichlet boundary condition on Γ (which is understood in the sense of traces) and tr denotes the trace operator defined by the relation tr A = A : I for all A ∈ Rd×d with I ∈ Rd×d meaning the identity matrix. The positive scalar constants H and σy represent the modulus of hardening and yield stress, respectively, and d×d C ∈ L(Rd×d sym , Rsym ) denotes the fourth-order elastic stiffness tensor which satisfies the relation (for known positive constants c1 ≤ c2 ) c1 |q|2 ≤ Cq : q ≤ c2 |q|2

(3)

for all q ∈ Rd×d sym . Finally, the vector f ∈ L2 (Ω; Rd ) expresses external forces acting on an elastoplastic continuum located in the domain Ω. For more details on mechanical aspects of this mathematical model and its possible generalization please refer to [13, 20].

Theorem 2.1. There exists a pair (u, p) ∈ (V0 + u0 ) × Q0 that solves (1). It satisfies the variational inequality a(u, p; v − u, q − p) + Ψ(q) − Ψ(p) − l(v − u) ≥ 0,

(4)

where Z a(u, p; v, q) :=


C(ε(u) − p) : (ε(v) − q) + σy2 H 2 p : q dx,

Ω

Z σy |q| dx,

Ψ(q) := Ω

Z l(v) :=

fv dx Ω

for all (v, q) ∈ V0 × Q0 .

Proof.

Existence of (u, p) ∈ (V0 + u0 ) × Q0 follows from known results in the calculus of variations. Indeed, the functional J(v, q) is convex and coercive on (V0 + u0 ) × Q0 , which is a convex closed subset of the product space V × Q, where V = H 1 (Ω, Rd ) and Q are reflexive spaces. Due to the assumption (3), the ellipticity and boundedness of the bilinear form a(u, p; v, q) are easily proved. Then, the equivalence of the variational inequality (4) and (1) follows from the Lions-Stampacchia Theorem [23].

Remark 2.1. By the variation of the energy functional J(v, q), it is simple to show that the minimizer (u, p) must satisfy the following relations almost everywhere in Ω: σ := C(ε(u) − p),

(5)

div σ + f = 0,

(6)

σ D = σy2 H 2 p + σy λ,  Λ if p = 0, λ∈ p otherwise, |p|

(7) (8)

where σ is the stress tensor associated with the exact solution. These relations have a clear physical meaning: (5) expresses an exact stress tensor σ as an additive decomposition of a linearized elastic strain ε(v) and a plastic strain p combined with a Hook’s law. (6) formulates an equilibrium of internal and external forces in the quasistatic case. (7) and (8) formulate a plasticity flow law in the case of von Mises yield function. See [3] for more details on the mechanical model and its mathematical aspects.

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3.

Basic estimate of the deviation from exact solution

We formulate an estimate allowing a posteriori analysis of (4). An abstract form of this estimate was originally stated in [29], here it is adapted to our particular nonlinear problem. Another adaptation was applied to problems with nonlinear boundary conditions [32].

Theorem 3.1. For any (v, q) ∈ (V0 + u0 ) × Q0 , the estimate 1 ||| (u − v), (p − q) |||2 ≤ J(v, q) − J(u, p) 2

(9)

holds, where the norm in the left hand side is defined by the relation ||| (u − v), (p − q) |||2 : = a(u − v, p − q; u − v, p − q) = kC(ε(u − v) − p +

q)k2C−1 ;Ω

+

(10) σy2 H 2

kp −

qk2Ω

and kκk2C−1 ;Ω :=

Proof.

Z

C−1 κ : κ dx.

Ω

The direct calculation shows 1 1 a(v, q; v, q) − a(u, p; u, p) + Ψ(q) − Ψ(p) − l(v) + l(u) 2 2 1 = a(u − v, p − q; u − v, p − q) + a(u, p; v − u, q − p) + Ψ(q) − Ψ(p) − l(v − u). 2

J(v, q) − J(u, p) =

In view of (4), we obtain (9).

Remark 3.1. Since σy2 H 2 > 0, the term σy2 H 2 kp − qk2Ω can be dropped to obtain a weaker estimate (formulated as (28) in [14]) 1 kσ − τk2C−1 ;Ω ≤ J(v, q) − J(u, p), 2 where σ := C(ε(u) − p) and τ := C(ε(v) − q) have meaning of an exact stress σ and an approximate stress τ. In the following, we will bound the difference J(v, q) − J(u, p) in (9) from above by a directly computable and physically meaningful term, which does not involve the exact solution (u, p).

4.

Perturbed problem and Lagrangian

The value of J(u, p) is unknown in the estimate (9). To use this estimate, we need to find a computable lower bound of this quantity. For this purpose, we introduce a ”perturbed” functional

Jλ (v, q) :=

1 a(v, q; v, q) + 2

Z σy λ : q dx − l(v), Ω

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Functional a posteriori error estimates for incremental models in elasto-plasticity

where the multiplier λ belongs to the set  Λ := λ ∈ L∞ (Ω, Rd×d ) | |λ| ≤ 1, tr λ = 0 a. e. in Ω . The relation of the original and the perturbed problem is given by sup Jλ (v, q) = J(v, q)

(11)

λ∈Λ

for all (v, q) ∈ (V0 + u0 ) × Q0 . Further we define the respective ”perturbed” Lagrangian Z Lλ (v, q; τ, ξ) : = Ω

C−1 τ : τ |ξ|2 τ : (ε(v) − q) − +ξ :q− 2 2 2 2σy H

!

Z dx +


σy λ : q − fv dx,

(12)

Ω

where new tensor-valued functions τ ∈ Q, ξ ∈ Q0 . Since Z  τ : (ε(v) − q) −

sup τ∈Q

C−1 τ : τ 2

 dx =

1 2

Ω

Z C(ε(v) − q) : (ε(v) − q) dx, Ω

|ξ|2 ξ :q− 2 2 2σy H

Z sup ξ∈Q0

Ω

! dx =

1 2

Z

σy2 H 2 |q|2 dx,

Ω

it is easy to see sup Lλ (v, q; τ, ξ) = Jλ (v, q)

(13)

τ∈Q ξ∈Q0

for all (v, q) ∈ (V0 + u0 ) × Q0 . Thus, the combination of (11) and (13) provides an estimate J(u, p) =

inf

v∈V0 +u0 q∈Q0

J(v, q) ≥

inf

v∈V0 +u0 q∈Q0

Jλ (v, q) =

inf

≥ sup

inf

≥

Lλ (v, q; τ, ξ).

τ∈Q v∈V0 +u0 q∈Q0 ξ∈Q0

inf

v∈V0 +u0 q∈Q0

sup Lλ (v, q; τ, ξ)

v∈V0 +u0 τ∈Q q∈Q0 ξ∈Q0

Lλ (v, q; τ, ξ)

Here, we used an estimate for a functional defined on the elements of two nonempty sets X and Y inf sup f(x, y) ≥ sup inf L(x, y)

x∈X y∈Y

y∈Y x∈X

which holds regardless of the structure of L(x, y), see [15], chapter 6, section 1, proposition 1.1. Substitution of the last estimate in (9) yields the inequality 1 ||| (u − v), (p − q) |||2 ≤ J(v, q) − 2

inf

v∈V0 +u0 q∈Q0

Lλ (v, q; τ, ξ) =: M(v, q; τ, ξ, λ).

valid for all τ ∈ Q, ξ ∈ Q0 . The right hand side of the last estimate defines an error majorant M(v, q; τ, ξ, λ). Its explicit form of is derived in the next section.

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S.I. Repin, J. Valdman

5.

Derivation of the error majorant

The infimum of Lagrangian (12) can be rewritten in the form inf

v∈V0 +u0 q∈Q0

1 = − 2

Lλ (v, q; τ, ξ) = inf Lλ (w + u0 , q; τ, ξ) w∈V0 q∈Q0

|ξ|2 C τ:τ+ 2 2 σy H

Z

!

−1

Ω

Z (τ : ε(u0 ) − fu0 ) dx

dx + Ω

Z

Z (τ : ε(w) − fw) dx + inf

+ inf

w∈V0

(ξ + σy λ − τ) : q dx

q∈Q0

Ω

(14)

Ω

Note it holds (

Z

0 if div τ + f = 0 a. e. in Ω, −∞ otherwise, Ω ( Z 0 if τ D = ξ + σy λ a. e. in Ω, inf (ξ + σy λ − τ) : q dx = q∈Q0 −∞ otherwise, (τ : ε(w) − fw) dx =

inf

w∈V0

Ω

where ·D denotes a deviatoric operator, i. e., AD = A −

inf

v∈V0 +u0 q∈Q0

Lλ (v, q; τ, ξ) =

tr A I, d

for all A ∈ Rd×d . Hence, we arrive at the following result.

 R  −1 C τ:τ  +  2 − Ω

  

|ξ|2 2σy2 H 2

 − τ : ε(u0 ) + fu0 dx (15)

if (τ, ξ) ∈ Qfλ , −∞ otherwise,

where  Qfλ := (τ, ξ) ∈ Q × Q0 | div τ + f = 0, τ D = ξ + σy λ a. e. in Ω .

(16)

The combination of (1) and (15) yields an explicit form of the error majorant estimate under the assumption (τ, ξ) ∈ Qfλ , 1 2

Z


C(ε(v) − q) : (ε(v) − q) + σy2 H 2 |q|2 dx +

Ω

+

1 2


σy |q| − fv dx

Ω

Z

C−1 τ : τ +

Ω

1 = 2

Z

Z

2

|ξ| σy2 H 2

!

Z dxĺ −

(τ : ε(u0 ) − fu0 ) dx Ω

−1

−1

C(ε(v) − q − C τ) : (ε(v) − q − C τ) +

σy2 H 2 (q

Ω

Z

Z σy |q| dx −

+ Ω

ξ − 2 2 )2 σy H

! dx

Z (q : τ − ξ : q) dx +

Ω

(τ : ε(v − u0 ) − f(v − u0 )) dx. Ω

After the simplification of the last integral terms due to the constrain (τ, ξ) ∈ Qfλ , Z

Z q : τ − ξ : q dx =

Ω

σy λ : q dx

for all q ∈ Q0 ,

Ω

Z τ : ε(v − u0 ) − f(v − u0 ) dx = 0, Ω

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Functional a posteriori error estimates for incremental models in elasto-plasticity

we deduce an explicit form of the error majorant

M(v, q; τ, ξ, λ) =

1 2

Z

C(ε(v) − q − C−1 τ) : (ε(v) − q − C−1 τ) dx

Ω

+

1 2

Z

σy2 H 2

Ω

q−

1 ξ σy2 H 2

!2

Z (σy |q| − σy λ : q) dx.

dx +

(17)

Ω

Summarizing the above considerations, we formulate the following result.

Theorem 5.1. The majorant (17) represents a guaranteed upper bound of the combined error norm 1 ĺ ||| (u − v), (p − q) |||2 ≤ M(v, q; τ, ξ, λ). 2

(18)

It is valid for any (v, q) ∈ (V0 + u0 ) × Q0 , λ ∈ Λ, and (τ, ξ) ∈ Qfλ .

Remark 5.1. The majorant (17) was derived by purely functional analysis of the problem in question. It does not involve meshdependent constants and is valid for any admissible (conforming) approximations from the respective functional classes associated with the primal variational problem. For this reason, error majorants (or a posteriori error estimates) of this type are called functional.

Remark 5.2. In order to get the upper bound as sharp as possible, we should minimize the right hand side with respect to free functions and use the estimate 1 ||| (u − v), (p − q) |||2 ≤ 2

M(v, q; τ, ξ, λ)

inf

(τ,ξ)∈Qf

(19)

λ

This estimate is also valid for any (v, q) ∈ (V0 + u0 ) × Q0 , λ ∈ Λ.

Remark 5.3. It is easy to see that the functional error majorant M(v, q; τ, ξ, λ) defined in (17) reflects natural conditions (5)–(8). Indeed, it attains the zero value if and only if the following conditions hold almost everywhere in Ω: τ = C(ε(v) − q),

(20)

div τ + f = 0,

(21)

λ : q = |q|, τ D = ξ + σy λ, ξ=

σy2 H 2 q.

λ ∈ Λ,

(22) (23) (24)

These conditions mean that (v, q) and τ satisfy (5)–(8); in other words they must be equal to the solution (u, p) of the minimization problem (1) and the exact stress tensor σ , respectively.

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6.

Modification of the majorant

For practical computation, an approximated displacement v ∈ (V0 + u0 ) and q ∈ Q0 are computed numerically, e. g., by the finite element method. The error of such approximation is bounded in the combined norm from above by a functional majorant M(v, q; τ, ξ, λ) defined in (17). To obtain a finite and therefore meaningful value of the functional majorant, free parameters must satisfy the conditions (τ, ξ) ∈ Qfλ and λ ∈ Λ. It is known that the equilibrium constrain div τ + f = 0 or its equivalent weak formulation Z (−τ : ε(w) + fw) dx = 0 for all w ∈ V0

(25)

Ω

is difficult to satisfy. Therefore, an upper bound of M(v, q; τ, ξ, λ) is derived here. It does not contain the parameter ξ as well as it remains independent of the equilibrium constrain, which is transformed into a penalty term. It turns out useful to split the majorant (17) in three parts: 1 M1 (v, q; τ) := 2

Z

C(ε(v) − q − C−1 τ) : (ε(v) − q − C−1 τ) dx,

(26)

Ω

M2 (q; ξ) :=

1 2 Z

M3 (q; λ) :=

Z

σy2 H 2

q−

Ω

1 ξ σy2 H 2

!2 dx,

(27)

(σy |q| − σy λ : q) dx.

(28)

Ω

We exclude ξ by setting ξ = τ D − σy λ according to (16) to rewrite 1 M2 (q; ξ) = 2

Z

1 = 2

Z

Ω

Ω

σy2 H 2

!2 1 D q − 2 2 (τ − σy λ) dx σy H

1 ¯ 2 (q; τ, λ), (τ D − ζ)2 dx =: M σy2 H 2

where ζ := σy2 H 2 q + σy λ and obtain a simplified error majorant independent of ξ ¯ ¯ 2 (q; τ, λ) + M3 (q; λ), M(v, q; τ, λ) := M1 (v, q; τ) + M which is defined on τ ∈ Qf := {τ ∈ Q | div τ + f = 0 a. e. in Ω} . Let us decompose τ = τ − τˆ + τˆ, where  τˆ ∈ Qdiv := τ ∈ Q | div τ ∈ L2 (Ω, Rd ) , i. e., τˆ does not have to satisfy the equilibrium condition (25). Then, we obtain M1 (v, q; τ) ≤

1 (1 + β) 2

Z

C(ε(v) − q − C−1 τˆ) : (ε(v) − q − C−1 τˆ) dx

Ω

Z 1 1 + (1 + ) C−1 (τ − τˆ) : (τ − τˆ) dx, 2 β Ω Z Z 1 1 1 1 1 D 2 ¯ 2 (q; τ, λ) ≤ (1 + δ) M (ˆ τ − ζ) dx + (1 + ) (τ D − τˆD )2 dx 2 σy2 H 2 2 δ σy2 H 2 Ω

Ω

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Functional a posteriori error estimates for incremental models in elasto-plasticity

which is valid for τˆ ∈ Qdiv , τ ∈ Qf and for all β, δ > 0. Here, we used the inequality (a + b)2 ≤ (1 + β)a2 + (1 +

1 2 )b β

¯ 2 (q; τ, λ) estimate fulfills valid for all β > 0, a, b ∈ R and its modification for β = δ. Since the last integral in the M Z

(τ D − τˆD )2 dx ≤

Ω

Z

(τ − τˆ)2 dx ≤ c2

Ω

Z

C−1 (τ − τˆ) : (τ − τˆ) dx,

Ω

¯ 2 (q; τ, λ) to obtain an ξ-independent estimate we combine available bounds on M1 (v, q; τ) and M 1 ¯ M(v, q; τ, λ, β, δ) ≤ (1 + β) 2

Z

C(ε(v) − q − C−1 τˆ) : (ε(v) − q − C−1 τˆ) dx

Ω

Z Z 1 1 D 2 + (1 + δ) (ˆ τ − ζ) dx + (σy |q| − σy λ : q) dx 2 σy2 H 2 Ω Ω " #Z 1 1 c2 1 + (1 + ) + 2 2 (1 + ) C−1 (τ − τˆ) : (τ − τˆ) dx 2 β σy H δ

(29)

Ω

¯ =: M(v, q; τ, λ, β, δ, τˆ). valid for τˆ ∈ Qdiv .

Lemma 6.1. Let τˆ ∈ Qdiv . Then, it holds inf

τ∈Qf

1 2

Z

C−1 (τ − τˆ) : (τ − τˆ) dx ≤

1 2 C kdiv τˆ + fk2 , 2

Ω

where C > 0 satisfies the inequality kwk ≤ C kε(w)kC

Proof.

for all w ∈ V0 .

The direct calculation reveals Z 1 I(ˆ τ ) := inf C−1 (τ − τˆ) : (τ − τˆ) dx τ∈Qf 2 Ω Z
1 C−1 (τ − τˆ) : (τ − τˆ) + τ : ε(w) − fw dx = inf sup τ∈Q w∈V 2 0 Ω

the interchange of operators follows e. g., from [15], Theorem 4.1.  Z  1 −1 = sup inf C (τ − τˆ) : (τ − τˆ) + τ : ε(w) − fw dx 2 w∈V0 τ∈Q Ω

=

≤ ≤ ≤

514

the infimum is attained in the argument τ = τˆ − Cε(w)   Z 1 2 sup − kε(w)kC − (div τˆ + f)w dx  2 w∈V0 Ω   1 2 sup − kε(w)kC + kdiv τˆ + fk kwk 2 w∈V0   1 sup − kε(w)k2C + C kdiv τˆ + fk kε(w)kC 2 w∈V0   1 1 sup − t 2 + C kdiv τˆ + fk t = C 2 kdiv τˆ + fk2 , 2 2 t≥0

(30)

S.I. Repin, J. Valdman

where the constant C > 0 comes from (30). The existence of such constant follows from the Korn’s and Friedrichs’ inequalities. Lemma 6.1 allows for the reformulation of (30) in ¯ ¯ ˆ M(v, q; τ, λ, β, δ) ≤ inf M(v, q; τ, λ, β, δ, τˆ) =: M(v, q; τˆ, λ, β, δ), τ∈Qf

ˆ where an non-equilibrated majorant M(v, q; τˆ, λ, β, δ) is defined as 1 ˆ M(v, q; τˆ, λ, β, δ) := (1 + β) 2

Z

C(ε(v) − q − C−1 τˆ) : (ε(v) − q − C−1 τˆ) dx

Ω

Z Z 1 1 D 2 + (1 + δ) (ˆ τ − ζ) dx + (σy |q| − σy λ : q) dx 2 σy2 H 2 Ω Ω " # 1 1 c2 1 (1 + ) + 2 2 (1 + ) C 2 kdiv τˆ + fk2 . + 2 β σy H δ

(31)

It is clear that the majorant vanishes if and only if (v, q) and τˆ satisfy (5)–(8), i. e., if these functions coincide with exact solutions. Hence, we arrive at the following result:

Theorem 6.1. The majorant (31) represents a guaranteed upper bound of the combined error norm 1 ˆ ||| (u − v), (p − q) |||2 ≤ M(v, q; τˆ, λ, β, δ). 2

(32)

It is valid for any (v, q) ∈ (V0 + u0 ) × Q0 , λ ∈ Λ, τˆ ∈ Qdiv , β, δ > 0. The majorant vanishes if and only if (v, q) coincides with the solution (u, p) of the minimization problem (1) and τˆ coincides with the exact stress σ = C(ε(u) − p).

ˆ over λ). Remark 6.1 (minimization of M Since λ ∈ Λ enters only integral terms 1 (1 + δ) 2

Z Ω

1 (ˆ τ D − ζ)2 dx + σy2 H 2

Z (σy |q| − σy λ : q) dx Ω

ˆ with respect to λ provides of (31), recall ζ = σy2 H 2 q + σy λ. If the constrain λ ∈ Λ is not active, then the variation of M the relation (1 + δ)

1 ∂ζ (ζ − τˆD ) + σy q = 0, σy2 H 2 ∂λ

which implies

λ = λ0 :=

1 D δσy H 2 q τˆ − . σy 1+δ

ˆ is attained for λ = λ1 := If |λ0 | > 1, then minimal value of M

λ0 . |λ0 |

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Functional a posteriori error estimates for incremental models in elasto-plasticity

7.

Analysis of the majorant

From Theorem 6.1 it follows that the quantity ˆ ρ(v, q) := inf M(v, q; τˆ, λ, β, δ)

(33)

τˆ∈Qdiv λ∈Λ

defined the guaranteed upper bound of the error in the combined error norm. Consider a theoretical choice depending on the exact solution (u, p) ( λ = λ(q) ∈

Λ q |q|

if q = 0, otherwise,

(34)

σ = C(ε(u) − p).

(35)

ˆ Hence, the third and fourth integrals in (31) vanish and the majorant M(v, q; σ , λ(q)) simplifies by taking δ = β = 0 in Z

1 C(ε(v) − q − C σ ) : (ε(v) − q − C σ ) dx + 2 −1

−1

Ω

Z

1 (σ D − σy2 H 2 q − σy λ(q))2 dx. σy2 H 2

Ω

The last integral is bounded as 1 2

2 Z  2 2 Z Z σy H (p − q) + σy (λ(p) − λ(q)) (λ(p) − λ(q))2 2 2 2 dx ≤ σ H (p − q) dx + dx. y σy2 H 2 H2 Ω

Ω

Ω

With respect to the definition (10) of the combined norm, we obtain 1 ˆ M(v, q; σ , λ(q)) ≤ ||| (u − v), (p − q) |||2 +ρ1 (p − q), 2

(36)

where a new quantity Z ρ1 (p − q) :=

(λ(p) − λ(q))2 dx H2

(37)

Ω

ˆ is introduced. The value of ρ1 (p − q) measures, how the majorant value M(v, q; σ , λ(q)) overestimates the error in the combined norm. In the following, this value will be bounded by a simpler expression. Let us decompose the domain Ω in its elastic and plastic parts in dependence of p or q as p

p

Ω := Ωela ∪ Ωpla ,

q

q

Ω := Ωela ∪ Ωpla ,

where p

Ωpla := Ω \ Ωela ,

q

Ωpla := Ω \ Ωela ,

Ωela := {x ∈ Ω | p(x) = 0} , Ωela := {x ∈ Ω | q(x) = 0} ,

p

p

q

q

and let us define q

p

ω1 := Ωela ∩ Ωela ,

516

q

p

ω2 := Ωpla ∩ Ωpla ,

ω3 := Ω \ {ω1 ∪ ω2 }.

S.I. Repin, J. Valdman

Then, ω1 represents a part of domain Ω, where both exact and approximate plastic strains show elasticity behavior. It this subdomain, we can choose any λ(p) = λ(q) ∈ Λ to obtain Z

(λ(p) − λ(q))2 dx = 0.

ω1

In the subdomain ω3 , where the exact and approximate plastic strains indicate different behaviors (one indicates elastic behavior, whereas the other one plastic behavior), we can choose one parameter (either λ(p) from (8) or λ(q) from (34)) to ensure Z (λ(p) − λ(q))2 dx = 0.

ω3

In ω2 , both exact and approximate plastic strains show plasticity behavior and Z

2 2 Z  p q p |q| − |p| q − p − dx = − dx |p| |q| |p| |q| |q| ω2 ω2  Z  Z (|q| − |p|)2 (q − p)2 (q − p)2 ≤ 2 + dx ≤ 4 dx 2 2 |q| |q| |q|2 ω2 ω2 Z (q − p)2 ≤ 4 dx q |q|2 Ωpla

(λ(p) − λ(q))2 dx =

ω2

Z



which results in

Theorem 7.1. ˆ The majorant value M(v, q; σ , λ(q)) in the case of the exact stress (35) and λ(q) from (34) satisfies 1 ˆ M(v, q; σ , λ(q)) ≤ ||| (u − v), (p − q) |||2 +ρ2 (p − q), 2

(38)

where the overestimation functional ρ2 (p − q) is defined as Z ρ2 (p − q) := 4

q

Ωpla

(q − p)2 dx. |q|2

(39)

In forthcoming works, we plan to extend obtained functional a posteriori error estimates to other hardening problems such as kinematic hardening and multi-yield hardening models [10, 11]. We are also interested in practical computations of these estimates combining our experience in computational elastoplasticity [14, 18, 19, 21] and fast computations of functional estimates for linear problems [36]. Since the functional majorant is typically a nonlinear functional, its efficient numerical minimization creates a main challenge.

Acknowledgements The first author was supported from Johann Radon Institute for Computational and Applied Mathematics (RICAM) during his stay in September 2007. The second author acknowledges the support of Austrian Science Fund ’Fonds zur Förderung der wissenschaftlichen Forschung (FWF)’ for his support under grant SFB F013/F1306 in Linz, Austria as well as the support of YFF project ’Modelling Transport in Porous Media over Multiple Scales’ at the University Bergen.

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Functional a posteriori error estimates for incremental models in elasto-plasticity

References

[1] Ainsworth M., Oden J.T., A posteriori error estimation in finite element analysis, Pure and Applied Mathematics, A Wiley-Interscience Series of Texts, Monographs, and Tracts, Wiley and Sons, New York, 2000 [2] Alberty J., Carstensen C., Numerical analysis of time-depending primal elastoplasticity with hardening, SIAM J. Numer. Anal., 2000, 37, 1271–1294 [3] Alberty J., Carstensen C., Zarrabi D., Adaptive numerical analysis in primal elastoplasticity with hardening, Comput. Methods Appl. Mech. Engrg., 1999, 171, 175–204 [4] Babuška I., Strouboulis T., The finite element method and its reliability, Oxford University Press, New York, 2001 [5] Bangerth W., Rannacher R., Adaptive finite element methods for differential equations, Birkhäuser, Berlin, 2003 [6] Bensoussan A., Frehse J., Asymptotic behaviour of Norton-Hoff’s law in plasticity theory and H 1 regularity, In: Lions J.L. (Ed.) et al., Boundary value problems for partial differential equations and applications, Dedicated to Enrico Magenes on the occasion of his 70th birthday, Paris: Masson. Res. Notes Appl. Math., 1993, 29, 3–25 [7] Bildhauer M., Fuchs M., Repin S., A posteriori error estimates for stationary slow flows of power–law fluids, Journal of Non-Newtonian Fluid Mechanics, 2007, 142, 112–122 [8] Bildhauer M., Fuchs M., Repin S., A functional type a posteriori error analysis for Ramberg–Osgood Model, ZAMM Z. Angew. Math. Mech., 2007, 87(11–12), 860–876 [9] Blaheta R., Numerical methods in elasto-plasticity, Comput. Methods Appl. Mech. Engrg., 1997, 147, 167–185 [10] Brokate M., Carstensen C., Valdman J., A quasi-static boundary value problem in multi-surface elastoplasticity, I, Analysis, Math. Methods Appl. Sci., 2004, 27, 1697–1710 [11] Brokate M., Carstensen C., Valdman J., A quasi-static boundary value problem in multi-surface elastoplasticity, II, Numerical solution, Math. Methods Appl. Sci., 2005, 28, 881–901 [12] Brokate M., Sprekels J., Hysteresis and phase transitions, Springer, New York, 1996 [13] Carstensen C., Numerical analysis of the primal problem of elastoplasticity with hardening, Numer. Math., 1999, 82(4), 577–597 [14] Carstensen C., Orlando A., Valdman J., A convergent adaptive finite element method for the primal problem of elastoplasticity, Internat. J. Numer. Methods Engrg., 2006, 67(13), 1851–1887 [15] Ekeland I., Teman R., Convex analysis and variational problems, North–Holland, Oxford, 1976 [16] Fuchs M., Repin S., Estimates for the deviation from the exact solutions of variational problems modeling certain classes of generalized Newtonian fluids, Math. Methods Appl. Sci., 2006, 29, 2225–2244 [17] Glowinski R., Lions J.L., Tremolieres R., Analyse numerique des inequations variationnelles, Dunod, Paris, 1976 (in French) [18] Gruber P., Valdman J., Implementation of an elastoplastic solver based on the Moreau˛Yosida theorem, Math. Comput. Simulation, 2007, 76(1–3), 73–81 [19] Gruber P., Valdman J., Solution of one-time-step problems in elastoplasticity by a slant Newton method, SIAM J. Sci. Comput., 2009, 31(2), 1558–1580 [20] Han W., Reddy B.D., Computational plasticity: the variational basis and numerical analysis, Comput. Methods Appl. Mech. Engrg., 1995, 283–400 [21] Hofinger A., Valdman J., Numerical solution of the two-yield elastoplastic minimization problem, Computing, 2007, 81, 35–52 [22] Krejˇcí P., Hysteresis, convexity and dissipation in hyperbolic equations, GAKUTO Internat. Ser. Math. Sci. Appl., Vol. 8, Gakkotosho, Tokyo, 1996 [23] Lions J.L., Stampacchia G., Variational inequalities, Comm. Pure Appl. Math., 1967, XX(3), 493–519 [24] Neittaanmäki P., Repin S., Reliable methods for computer simulation, Error control and a posteriori estimates, Elsevier, New York, 2004 [25] Rannacher R., Suttmeier F.T., A posteriori error estimation and mesh adaptation for finite element models in elastoplasticity, Comput. Methods Appl. Mech. Engrg., 1999, 176, 333–361 [26] Repin S., A priori error estimates of variational-difference methods for Hencky plasticity problems, Zap. Nauchn. Semin. POMI, 1995, 221, 226–234 (in Russian), English translation: J. Math. Sci., New York, 1997, 87(2), 3421–3427 [27] Repin S.I., Errors of finite element methods for perfectly elasto-plastic problems, Math. Models Meth. Appl. Sci.,

518

S.I. Repin, J. Valdman

1996, 6(5), 587–604 [28] Repin S., A posteriori estimates for approximate solutions of variational problems with strongly convex functionals, Problems of Mathematical Analysis, 1997, 17, 199–226 (in Russian), English translation: J. Math. Sci., 1999, 97(4), 4311–4328 [29] Repin S., A posteriori error estimation for variational problems with uniformly convex functionals, Math. Comp., 2000, 69(230), 481–500 [30] Repin S., A posteriori estimates for partial differential equations, Walter de Gruyter Verlag, Berlin, 2008 [31] Repin S.I., Seregin G.A., Error estimates for stresses in the finite element analysis of the two-dimensional elastoplastic problems, Internat. J. Engrg. Sci., 1995, 33(2), 255–268 [32] Repin S., Valdman J., Functional a posteriori error estimates for problems with nonlinear boundary conditions, J. Numer. Math., 2008, 16(1), 51–81 [33] Repin S.I., Xanthis L.S., A posteriori error estimation for elasto-plastic problems based on duality theory, Comput. Methods Appl. Mech. Engrg., 1996, 138, 317–339 [34] Seregin G., On the regularity of weak solutions of variational problems of plasticity theory, Algebra i Analiz, 1990, 2(2), 121–140 (in Russian), English translation: Leningrad Mathematical Journal, 1991, 2(2), 321–338 [35] Simo J.C., Hughes T.J.R., Computational inelasticity, Springer-Verlag New York, 1998 [36] Valdman J., Minimization of functional majorant in a posteriori error analysis based on H(div) multigridpreconditioned CG method, Advances in Numerical Analysis, to appear [37] Wieners Ch., Nonlinear solution methods for infinitesimal perfect plasticity, ZAMM Z. Angew. Math. Mech., 2007, 87(8–9), 643–660

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