Design sensitivity analysis for rate-independent elastoplasticity

Computer Methods in Applied Mechanics and Engineering 107 (1993) 393-431 North-Holland

CMA 407

Design sensitivity analysis for rate-independent elastoplasticity

Creto August0 Vidal Department of Civil Engineering, University of Illinois at Urbana-Champaign, IL, USA

Robert Bruce Haber Department of Theoretical and Applied Mechanics Center for Supercomputing Research

and Development, University of Illinois at Urbana-Champaign, IL, USA

Received 5 June 1992

A new incremental, direct differentiation method for design sensitivity analysis of structures with rate-independent elastoplastic behavior is presented. We formulate analytical sensitivity expressions that are consistent with numerical algorithms for elastoplasticity that use implicit methods to integrate the constitutive equations and return mappings to enforce the consistency conditions. The sensitivity expressions can be evaluated with only a modest increase in computational expense beyond the cost of simulation. Combined with the inherent advantages of implicit integration strategies, this represents a significant improvement over previous sensitivity formulations for history-dependent materials. First- order sensitivity expressions involving the complete set of design variables, including shape design variables, are derived for a generic response functional. The reduced form of the consistent tangent stiffness matrix obtained at the end of each time or load step in the finite element procedure is used to update the response sensitivities for that time step. No iterations are needed in the sensitivity computations. A numerical example demonstrates the accuracy and efficiency of the new sensitivity analysis method for an elastoplastic analysis problem. Explicit sensitivities from the new method are confirmed by finite difference estimates.

1. Introduction

1.1. Background

Design sensitivity analysis involves the determination of the rate of change with respect to a set of design parameters of one or more performance measures which are often expressed as functionals involving a combination of design and response fields. Sensitivity analysis plays an important role in structural optimization, reliability analysis, inverse problems, and parameter identification problems. It can also be useful in guiding manual redesign procedures. The central problem in sensitivity analysis is the determination of the implicit variations in the response fields generated by a specified design variation. In general, there are three classes of methods to solve this problem: the finite difference method, the adjoint variable method and

Correspondence to: Professor Robert Haber, Department of Theoretical and Applied Mechanics, University of Illinois, 216 Talbot Laboratory, 104 South Wright St., Urbana, IL 61801-2935, USA.

00457825/93/$06.00 @ 1993 Elsevier Science Publishers B.V. All rights reserved

394 C. A. Vidal, R. B. Haber, Design sensitivity analysis

the direct differentiation method [l]. Finite difference sensitivity analysis methods are simple to implement, but they can be computationally expensive and deficient in terms of accuracy and reliability [2,3]. For this reason, the adjoint variable and direct differentiation methods are generally preferred despite their relative complexity.

Design sensitivity analysis of nonlinear systems involving history-dependent material response has been the focus of recent study. In these problems, the state of stress at a given time depends on the full history of the state variables. History-dependent behavior appears in many practical problems, including metal structures which exhibit plastic and/or creep response at elevated temperatures; manufacturing processes, such as rolling, extrusion and drawing; reinforced concrete structures which exhibit creep behavior and composite material systems subjected to various forms of damage or creep. Thus there is considerable interest in extending existing sensitivity analysis methods to handle this class of problems.

One consequence of history-dependent behavior is that the response sensitivity at a given time and position depends on both the response and the response sensitivities at all previous times and locations in the structure. Ryu et al. [4] were among the first to point this out. Incremental, step-by-step versions of the direct-differentiation method are natural choices for sensitivity analysis of history-dependent problems, since these methods generate sensitivity information for the complete response field at each step. Adjoint sensitivity formulations are not well-suited for such forward-marching, step-by-step methods because each adjoint solution yields the sensitivity of only a single performance functional, rather than the sensitivities of the full response fields [5].

On the other hand, adjoint sensitivity analysis procedures involving a transient, terminal- value problem are feasible for history-dependent problems. In this case, the sensitivity analysis cannot be carried out simultaneously with the forward solution since the definition of the adjoint terminal-value problem depends on the solution of the original initial-value problem. This substantially increases the computational expense and complexity. Thus, the direct differentiation method is usually the method of choice for history-dependent problems.

The main challenge in formulating the direct differentiation method is the derivation of the incremental stress sensitivities as a function of the incremental response and the total response sensitivities evaluated at the beginning of the time step. Several methods have been proposed over the past few years. Wu and Arora suggested that a semi-analytical approach could be used in combination with the direct differentiation method [6]. In this approach, the response sensitivities for which analytical expressions are difficult to derive are computed by the finite difference method. More recently, Tsay and co-workers [5,7] and Mukherjee and co-workers [8,9] have presented formulations of incremental direct differentiation methods based on finite element and boundary element models. Mukherjee and co-workers used explicit methods to integrate the constitutive equations. This approach can be expensive since it requires the use of extremely small time steps to ensure accuracy. Tsay and co-workers present only analytical solutions and do not address the issue of numerical integration.

In this paper, we explore sensitivity formulations for numerical models of history-dependent behavior in which implicit integration schemes are used to integrate the constitutive equations. We demonstrate that when consistency between the sensitivity and the simulation model is maintained, the direct differentiation method leads to a very accurate and efficient sensitivity formulation with only a modest increase in computational expense beyond the cost of the underlying simulation. This paper extends our previous work [lo-121 on the sensitivity

C.A. Vidal, R.B. Haber,

analysis of systems with creep behavior to linearization of the incremental constitutive formulation.

Design sensitivity analysis 395

time-independent elastoplasticity. Consistent model is the key to achieving an effective

1.2. Exact linearization and the consistent tangent operator

Tangent operators appear in the linear forms of the incremental constitutive equations for nonlinear materials. They are commonly used in iterative solution methods, such as the Newton-Raphson method, but they also play a role in sensitivity formulations. The definition of each tangent operator depends on a corresponding form of the incremental constitutive model: the continuum model in the case of the continuum tangent operator and the algorithmic model in the case of the consistent tangent operator.

The incremental constitutive relations for an arbitrary hypoelastic material can be written as

I E

Tc = nT + “&

C,(s) d& >

where rc is the stress predicted by the constitutive equations for a given strain path between ‘e and E and C,(s) is the continuum tangent operator that depends on a number of state variables contained in the vector s. That is, the continuum tangent operator is defined as

C,(s)= 2 , s

so that

dr, = C,(s) de . (3)

In elastoplasticity, C, corresponds to the familiar elastoplastic combining the elastic stress-strain relation, the flow rule and

operator C,, obtained by the consistency condition.

Equation (1) defines the continuum incremental constitutive model, expanded about configuration n. The continuum tangent is the slope of the curve for the one-dimensional continuum constitutive model shown in Fig. 1.

(2)

In practice, it is generally not possible to integrate (1) exactly, so a numerical integration algorithm, such as the generalized midpoint rule [13], is substituted for the continuum integration operator. This leads to an incremental constitutive relation of the form

T, = “7 + FJ’b, As) , (4)

where TV is the stress according to the numerical integration algorithm and F, is a nonlinear mapping of “s and the strain increment, AE = E - ‘e , into an incremental stress (a specific example is presented in Section 2 for elastoplasticity). This defines the algorithmic incremental constitutive model, also shown in Fig. 1. The tangent operator 6 corresponding to the algorithmic constitutive relation is called the consistent tangent operator, since it is consistent with the constitutive model that is actually used in computation. It maps de, an infinitesimal variation of the strain increment about the current value AE”, into a stress variation:

396 C. A. Vidal, R. B. Haber, Design sensitivity analyst

algorithmic incnmenml

constitutive model from

__------“~:~:\*

TC(“S) = ZywIJ

-I

/

continuum incremental

constitutive mode4 from

configuration n

.---

LIE 1 Li

Fig. 1. Comparison of continuum and consistent tangent operators.

so that

dr, = c(“s, AZ) de . (6)

In Fig. 1, the consistent tangent is the slope of the curve representing the one-dimensional algorithmic constitutive model. Note that the consistent tangent operator depends on the state at configuration n and the current value of the incremental strain.

A properly formulated numerical integration procedure should be asymptotically exact as ha-+ 0. Therefore, the algorithmic and continuum incremental constitutive models are in close agreement for small values of AS, but the curves diverge for larger values. The result is that the initial continuum and consistent tangent operators, corresponding to configuration 12 at the beginning of the step, are identical;

IIowever, the two tangent operators are not in agreement for a nonzero strain increment.

C. A. Vidal, R. B. Haber, Design sensitivity analysis 397

That is, if s”, is the state associated with the strain E” = ‘e + AE” according to the algorithmic constitutive model (see Fig. l), then

C,(iJ f c(“s, AE”) . G-4)

In particular, the terminal continuum and consistent tangent operators at the end of a step are distinct. This fact has important implications in the formulation of efficient nonlinear solution procedures and reliable sensitivity analysis methods for problems with history-dependent material models.

The Newton-Raphson method is often used to solve nonlinear equations, such as those that describe the equilibrium of systems with history-dependent response. It is an iterative method that solves a nonlinear problem as a sequence of linearized problems. Its quadratic rate of asymptotic convergence is one of its key advantages. However, this rate can be preserved only if the tangent moduli that appear in the linearized problem are obtained by consistent linearization of the response function resulting from the integration algorithm [14]. The use of the consistent tangent operator corresponds to a consistent linearization of the nonlinear problem and, therefore, preserves the quadratic rate of convergence of the Newton-Raphson method. On the other hand, use of an inconsistent continuum tangent destroys the quadratic rate of convergence. This problem was reported by Nagtegaal in 1982 [15]. Simo and Taylor [16] demonstrated the advantages of using the consistent tangent operator over the continuum tangent operator, especially when large time steps are used.

1.3. Outline

The remainder of this paper is divided into four sections as follows. Section 2 presents a formulation based on the consistent tangent operator for rate-independent elastoplastic behavior and Section 3 presents the corresponding sensitivity formulation. A numerical example is presented in Section 4 and conclusions are presented in Section 5.

2. A numerical formulation for elastoplastic analysis

This section presents continuum and discrete formulations of the initial boundary-value problem for elastoplastic analysis. The continuum variational formulation provides the basis for an incremental finite element formulation. Finally, the consistent tangent operator and a solution algorithm, based on the incremental finite element formulation, are introduced.

2.1. The continuum initial-boundary-value problem for elastoplasticity

2.1 .l. Strong form of the continuum problem Here we limit our attention to elastoplastic systems undergoing infinitesimal deformation.

Therefore, the total strain can be expressed by an additive decomposition into elastic and plastic components [17];

& = E, + Ep .

We assume that the loading process is quasi-static, so inertia effects are neglected.


Suppose we are given an open domain V with a boundary A comprised of two disjoint, open subsets, A, and A, (A = A,, U A,). We seek a solution on V over the time interval Z = [0, T] that satisfies the following equations: - Equilibrium condition

V*7+b=O inVXZ.

- Strain-displacement relations

E= $(Vu+(Vu)‘) inVXZ.

- Constitutive relations and initial conditions - elastic stress-strain relation

~=C:(E-&~) inVXZ;

(10)

(11)

- flow rule

8, = );f(r, q) in V X Z ;

- hardening law

4 = Ah(r, q) in V X Z ;

- Kuhn-Tucker complementarity

h>O, Y(r, 4)sO 9

- consistency condition

liF(r, q)=O inVXZ;

( 124

ww

conditions

iY(T,q)=O inVXZ;

- evolution of the plastic strains and the internal variables

Ep = OEp + I

f

0 a+,(~, q) dt in V x Z ;

q=Oq+ I

*

0 4(~, q) dt in V x Z ;

- Cauchy’s relation

t=T*n onAxZ.

- Boundary conditions

u=li onA,xZ, t=t onA,xZ.

Wf)

(133)

(13)

(Ida, b)


u is the displacement, b represents the body force, t represents the surface traction and n is the unit vector normal to the surface A. Denoting by S the space of second-order symmetric tensors, E E S is the total strain, E,, E S is the plastic strain (the part of the total strain due to the plastic flow) and 7 E S is the stress. q is an array in R” comprised of a set of m internal variables that describe the internal dissipation mechanisms in a phenomenological plasticity model [17-191. Y is the yield function which delimits the elastic domain E,,

C is the elastic constitutive tensor and f is the plastic flow vector. In associative plasticity, f is defined as the gradient of the yield function with respect to the stress components [20]. h E R” is an array of hardening functions corresponding to the- internal variables q. The boundary conditions for displacement and surface traction are specified by the prescribed functions U and ?, and the initial conditions for the plastic strain and for the internal variables are specified by the functions ‘cP and ‘q, respectively.

2.1.2. Variational form of the continuum problem Next we present a variational form of the continuum initial-boundary-value problem that

provides a suitable basis for a finite element stiffness formulation. The solution of the variational problem involves a displacement field that satisfies the equilibrium equation (10) and the traction boundary condition (14b) in a weak sense over V, tit E I, subject to strict enforcement of (ll)-(14a). Using the function spaces

u = {u E H’(V): UIA, = ii} ) i, = {ii E zP(V): pI,” = 0) ) (16a, b)

where ( )IA is the trace of ( ) on A,, we state the variational form of the initial boundary-value problem as follows.

Find u(x, t) such that (ll)-(14a) are satisfied, u(., t) E u, Vt E I and

ti*tdA V@,t)EGxZ, I

where b is given by

b = pii + (Vii)‘] .

(17)

(18)

2.2. Numerical model

In this subsection we present a numerical model for the elastoplasticity problem based on the above continuum model. First, incremental forms are defined over a sequence of discrete time intervals that span [0, T]. Then numerical approximations for the time integrals in the constitutive model are introduced. Finite element discretization of the spatial domain and the response field completes the numerical model.

2.2.1. A temporal partition and incremental forms Let Z = [0, T] C R be the closed time interval of the simulation. We partition Z into


subintervals, Zj = [tj_l, ti] of length Atj = tj - tj_l, j = 1, m, such that 0 = t, < t, < - * - < t, = T. This partition provides a basis for incremental integration strategies in which we seek solutions of the evolution equations at the discrete times tj, j = 0, m. In a typical increment, the state at time tj_ 1 is known and we use an integration procedure to determine the solution at time tj = tj_l + Atj. From here on, we drop the subscripts to simplify the notation, replacing [tj-,) tj] by [t, t + At].

We now formulate the incremental constitutive relations. These describe the change in state at a given material point over a time step [t, t + At] induced by a displacement increment Au. Let ‘s = (‘7, ‘85, ‘ep, ‘q), the state at the given material point at the beginning of the time step, and the incremental displacement field Au be given. Now suppose we wish to find the state at the end of the time step ‘+*(s, such that (12a)-(12g) are satisfied. We use the following expansions to generate the incremental constitutive relations:

‘+*‘u = ‘u + Au , (19)

r+Are = ‘6 + be = % + $[V(Au) + (V(Au))‘] , (20)

I r+At

I+Atcp = lep + f 8, dt ,

t+Atq = fq + I t+At

4 dt , I

(21)

(22)

‘+AtT = tT + C: [ libldpdt]. AE - (23)

In an incremental stiffness formulation, we require expressions for the solution variables at time t + At in terms of the displacement increment Au. Such expressions are apparent in (19) and (20), but not in (21)-(23). We use the generalized midpoint rule to introduce numerical approximations for the time integrals in (21), (22) and (23) to obtain the desired form. These approximations generate the algorithmic constitutive model.

2.2.2 The generalized midpoint rule and the algorithm constitutive model The integral of a function c(t) over the subinterval [t, t + At] using the generalized midpoint

rule is written as

s t+At

c(5)dc=c(t+cuAt)At, Os~~al, (24) I

where (Y is an integration parameter. Particular choices of cr reproduce some of the familiar integration methods. For example (Y = 0 generates the forward Euler method, cy = 1 corresponds to the backward Euler method and (Y = 0.5 gives the standard midpoint rule.

Applying the generalized midpoint rule to (21), (22) and (23), we have

r+At E P =kp + d,(h*, T*, q”) At, (25)


‘+*‘q = ‘q + t&i*, T*, q*) At,

r+Ar T=~T+C:[AE-+~*,T*,~*)A~], (27)

where h *, T* and q* are approximations of the plastic strain-rate multiplier, of the stress tensor and of the internal variables at the interior point t + (Y At of the time interval [t, t + At]. The approximations T* and q* are determined by the following linear interpolations:

T*(‘T, r+ArT) = (1 - Ct)'T + (Y'+*'T ,

q*(k r+Arq) = (1 - a)‘q + ar+Arq )

(28)

whereas the approximation of the plastic strain-rate multiplier A* is determined from the discrete enforcement of the consistency condition, as discussed below. The notation (*)* denotes

(. >* = (1 - cy)‘(‘) + (Y’+*‘(*) . (30)

In the continuum evolution problem, the consistency condition (12e) constrains the state variables to remain within the closure of the elastic region E, throughout the time interval [t, t + At]. The introduction of the generalized midpoint rule transforms the constrained evolution problem into a constrained discrete algebraic problem for the state variables at t + At. There are enough degrees of freedom in this discrete problem to enforce the consistency condition at only one point in each time step. The choice of the time at which to enforce the consistency condition should be guided by considerations of accuracy, simplicity of implementation and efficiency. In this work, we enforce consistency at time t + a At since this ensures symmetry for an important class of plasticity models, as explained in Section 2.3 [13,21]. This simplifies the implementation and allows for more efficient solution algorithms. Denoting h = h * At, the discrete versions of the Kuhn-Tucker complementarity conditions are written as

/GO, Y(T*, q*)so, hY(~*,q*)=0 inVXZ. (31)

The algorithmic constitutive model described by (25)-(31) can be solved by Newton- Raphson iterations. However, a robust integration scheme for this set of nonlinear equations is provided by the return mapping algorithms [21,22]. These are based on a split elastic-plastic operator. The solution is reached in two steps. In the first step, called the elastic predictor step (see Section 2.4.1.1), a trial state is predicted according to the elastic incremental stress-strain relations. If the trial midpoint state falls outside the closure of the elastic region E,, or equivalently, if the trial state at the end of the interval falls outside the closure of, the region E: (see Fig. 2),

E:= ((7, q) E s X R” 1 (T*, q*) E E,} ,

then the plastic corrector step (see Section 2.4.1.2) restores consistency at the midpoint. This


t+At,trial

Fig. 2. Closest-point-projection of the trial-state onto the yield surface.

procedure defines the algorithmic constitutive model, a nonlinear mapping from the state variables at time t and the strain increment to the state variables at time t + At:

(33)

The total strain increment AE is defined in terms of the displacement increment by (20). The nonlinear stress-update mapping is represented as

F, :(‘s, A++ t+Afr . (34)

This is our final numerical representation of history-dependent elastoplastic behavior. It defines our stress update procedure and generates the consistent tangent stiffness operator. It also plays a key role in the formulation of consistent design sensitivity expressions, as explained in Section 3.

22.3. Finite element discretization In this subsection we introduce a finite element discretization of the spatial domain V to

obtain a discrete counterpart of the variational problem of Section 2.1.2. We follow the standard finite element stiffness procedure and establish a partition of the domain V consisting of n,, nonoverlapping elements. The domain, the prescribed-displacement boundary and the prescribed-traction boundary of element e are denoted V,, (A,), and (A,),, respectively. (For

C.A. Vidal, R.B. Haber, Design sensitivity analysis 403

many elements, (A,), and/or (A,), represent the null set.) We introduce a set of smooth basis functions {N’,} to define a finite-dimensional function space ue over each element e.

where U; is a vector-valued parameter at node 5 in element e and 5 ranges from 1 to the number of nodes in element e. Next, we introduce the finite-dimensional function spaces,

u* = {u E u: t&e E u”} , Gk = {ti E 6: Z.&e E v”} , (36a, b)

where ( )IvE is the restriction of ( ) to V’. We also define the space _vh of global nodal displacement vectors and the space c* of global weighting parameter vectors.

v* = {U E RP: N”U E vkjVeVelements e} ,

5’~ {i?E RP: Ne~EOk~VeVelementse}.

(374

w-3

We apply the temporal partition of Section 2.2.1, the algorithmic constitutive model, and the finite element discretization to the continuum variational problem (see (17)) to obtain the discrete problem. The numerical solution of the history-dependent problem begins with the given state OS at time t, = 0 and solves sequentially for the states at times t,, f,, . . . , t, = T. The typical incremental variational problem is stated as follows.

Given the state $, find ‘+” U E_v* such that (ll), (13), (19), (20), (25)-(27), (31) and

? 1” (B’)‘(‘+‘,) dV= 2 j- (,=)‘(‘+‘,) dV + 2 j-A (,,),(,+A,,) dA e=l e e-l “e e=l 1,

(38)

are satisfied.

2.3. The consistent tangent o~eratur

The rate of change of z’A’r with respect to changes in As plays an important role in the formulation of the tangent stiffness equations and the sensitivity expressions. Suppose A; and r+Atg represent the current estimate of the strain increment and the corresponding estimate of the stress at time t + At, and further suppose that we wish to determine the stress at a new value of the strain increment, AE = AE” + d(Aa). Applying a Taylor’s expansion about Ai? to the nonlinear mapping F,(‘s, A&), we find

f+Af7 = F,(‘s, AE) = F,($, AE) + c: de + H.O.T. , (3%

where the term c =VbaFrjAc=Aii’ is the consistent tangent operator [14,161. It determines a linear approximation for tiAc~ in terms of ds, and is evaluated by imp~citly differentiating the algorithmic constitutive model ((25)-(31)) with respect to A&. The derivation presented below follows that presented for associative plasticity by Simo and Hughes 1211.

404 C. A. Vidal, R.B. Haber, Design sensitivity analysis

For convenience, we rewrite (25)-(27), using the relations (12b) and (12c), as

r+At

5-J = kp + hf(7*, q”) inV,

t+Arq = lq + hh(T*, q”) in V ,

r+At~ = ‘T + C: [As - hf(r*, q*)] in V ,

and the Kuhn-Tucker complementarity conditions,

LO, Y(T”, q*) s 0, iY(T*, q*) = 0.

(40)

(41)

(42)

(43a, b, 4

Assuming plastic loading so that h > 0 and using the notation

the differentiation of (41), (42) and (43b) yields

V,,(t+Atq) = h*@VV,,i + i(V,h)*: V,+a,,r*: c + i&h)* ++a,q* -VA,(t+Afq) ,

(44)

c = c - c: (f * @V,,A) - c: ii(v v,+a,,7*: C]

- c: [h&f)* *vr+a,q* *Vbe(*+Arq)] )

(v,Y)*: Vl+M*7”: c + (v,Y)* -vt+a,,q* -VAE(l+Atq) = 0 )

(45)

(46)

where V, ]( ) is the partial Fr&het derivative of ( ) with respect to [ 3. Equations (44)-(46) are solved for VAEh, VAE(‘+A’q) and c. Thus,

I’& $ [(V,Y)*: &+ ia(O,Y)*+(V,h)*: i;], (47)

E=&+‘: f*@((V,Y)*:e+ia(V,Y)*&(V,h)*:~)

+ id: (VJ)* . e”. h* ‘23 ((V,Y)*: t + ha(V,Y)* . $. (V,h)*: e)] , (48)

V,,(‘+Atq) = iff& * (V,h)*: 2 + f [$. h* @((V,Y)*: t + ia(VqY)* * & * (V,h)*: e)

- ir$ . (V,h)*: Q.f*: (v,Y)*: c + ha(v*Y)* * jfj * (v,h)*: f?)] ) (49) where

6 = [I, - ,icr(V,h)*]-’ , (50)

c” = [P + ia(vJ) ) (51)

C.A. Vidal, R. B. Haber, Design sensitivity analysis 405

$ = [I, - ha(V,h)* + ( /L)“(v7h)*: E: (v,f)*]-’ ) (52)

c = [c-l + ia(V,f)” + (ia)*( - 6 * (V,h)*]-’ ) (53)

y = ((v,Y)*: e + /ia( * fj * (V,h)*: E): f*

- ((v,Y)* - $ - Aa(V,Y)*: c: (v,f)* * fj) - h* . (54)

Z, is the identity map on R” and Z is the fourth-order identity tensor. If associative plasticity is assumed, the flow and the hardening rules can be written as [21]

i, = if(T, q) = iVTY(T, q) ,

4 = hh(T, q) = hD 'VIY(T, q) )

(55)

(56)

where D is the array of generalized plastic moduli. Thus,

VJ =r&Y =V,(V,Y) ) (57)

V7h=DV,,Y=DV,(V,Y). (58)

We see that for yield functions of the form

Y(7, q) = E(7) + F(q) (5%

or for those where the coupling between 7 and q is linear, the terms in (57) and (58) vanish. Under such conditions, e = 6 and $ = 6 (see (50)-(53)). The expression for y is also simplified in these circumstances. Under these simplifications, the consistent tangent operator takes on a symmetric form,

c = e - $ [c”: (v,Y)*@(v,Y)*: c”] . (60)

2.4. A solution algorithm for the incremental problem

In a typical time step, the solution at time t is available and we seek increments of the displacements, the strains, the plastic strains, the internal variables and the stresses that simultaneously satisfy the algorithmic constitutive model ((25)-(31))) the relations (11) and (13), and the variational equilibrium equation (38) at time t + At.

Two-level nested iterations are required to solve the incremental problem. The outer equilibrium iterations seek displacement increments that satisfy (38) at time t + At; while the inner constitutive iterations enforce the algorithmic constitutive model for the current estimate of the displacement increments. The constitutive iterations are the basis of the stress update algorithm, described in Section 2.4.1. The consistent tangent stiffness formulation is the basis of the equilibrium iterations described in Section 2.4.2.


2.4.1. Stress update algorithm : constitutive iterations Suppose that ‘s has been determined by the solution for the previous time step and that we

are given an estimate of the incremental strain As@) incremental displacement AU@)

corresponding to the estimated in the kth equilibrium iteration of the current time step. Then

the nonlinear algorithmic constitutive equations, (25)-(31), can be solved by the Newton- Raphson method for the corresponding stress ‘fArr(k! The solution is achieved in two steps. In the first step, the elastic predictor step, we assume a purely elastic process with zero plastic flow to determine a fictitious intermediate state called the trial state. In the second step, the plastic corrector step, the state variables evolve from the trial state with nonzero plastic flow until the consistency condition is satisfied. This latter process corresponds to a projection of the trial state back to the yield surface. Several algorithms are available for the plastic correction step [21-231. Here we adopt the closest point projection method (see Fig. 2) because it leads to analytical expressions for the consistent tangent operator and because it is an unconditionally stable method [21].

2.4.1.1. The elastic predictor step to the trial state. There is no dissipation mechanism during the loading process of the elastic predictor step. This is equivalent to setting $, = 0 and 4 = 0 for the evolution problem in the interval [t, t + At]. Therefore, the solution for the trial state is obtained directly from the incremental displacement estimate Au’k’_as

t+Are(k) = 'E + A#‘) = ‘E + 4 [V(Azbk’) + (V(A#)))‘] , (614

r+AfE(k,O) = r+Ar (k,O) = P *Ep , 4 b 9 ‘+ArT(k*o) = ; + C: Acck’ , W, c, d)

where a superscript (k, 0) indicates the trial solution for the kth equilibrium iteration. The trial state is accepted as the final state if ((r*)‘kPO’, ( q*)(kyO)) corresponds to a point in the closure of the elastic region, that is Y((T*)@’ O), ( q*)(kyO’) S 0, where ( q*)(k90) = ‘q. However, if ((T*)‘~,“, ( q*)(kTO)) corresponds to a point outside the elastic region ( Y((T*)(~*‘), ( q*)(kpO)) >

0), the Kuhn-Tucker conditions are violated and consistency is restored using the return mapping of the plastic corrector step.

2.4.1.2. The plastic corrector step (closest point projection method). In general, the plastic corrector step involves a nonlinear problem that must be solved iteratively. Starting from the trial state, we seek converged fields ‘+A’r(k’, r+Are~) and r+Arq(k) such that

t+AtT(k) _ r+AfT(k,O) - - c: /Pk’f((7*yk), ( q*fk’) ) (62)

‘+Q;) = t+A$r.O) + $k’f((7*)‘k’, ( q*)‘k’) ,

r+AtqW = r+A~qW) + $k’h((7*)‘k), (q*)‘k’) ,

(63)

w Y((T*yk), ( q*yk)) = 0 . (65)

~~~ r+Ai(k,i) r+AtE(k,Qand t+Af 4

(kj) ,i=1,2 ,..., be solution estimates from the ith Newton- Raphson iteraiion 0; the plastic corrector step within the kth equilibrium iteration. Then, the


residual expressions for the plastic strains and the internal variables are written as

R&i) = &P

r+A$(k,i) _ r+Arc(k,O)

P P

_ h(k.i)f((T*)(k.i), (4*)(k.i)) ,

R(k.i) _ t+Ab(k.i) _ r+At4(k4) _ h(k.i)q(7*)(k.i), (4*)(k.i))

4 - ,

R(yk,i) = y((T*)(k.i), (4*)(k.i)) .

(66)

(67)

(6%

We seek iteration corrections, d7 W.9 dEfTi), dq(kd , and dhck”), that define an improved solution estimate,

t+AtT(k,i+l) = f+At7(k,i) + dT(k,i) , (6%

r+Arer.i+l) = t+A$F,i) + daff’) ,

t+Ar4(k,i+l) = t+Ar4(k,i) + dq(k,i) ,

h(k,i+l) = hW,i) + dh(k,i) ,

(70)

(71)

(72)

so that linearized estimates of the residuals vanish. The residual estimates are obtained from truncated Taylor’s expansions as

R&i+l) = R(k,i) + V EP &P

r+Ar, R(k.i). ds(k,i) + V

P &p - P t+Ar,

R(k.i). dr(k.i) =P *

+ VlialqR;d) . dq(k,‘) + V,Rr+‘) d,$k.‘) ,

P (7%

R(k,i+l) = R(kd + V Q 4

r+At, R(k,i). &k,‘) + V,+A,qR~y’) . dq(k.‘) + V,R;“’ ,jh(k.i) ,

cl * (74)

R(k.i+l) = R(kJ) + v Y Y r+Af, y .

R(k.i). d7(kvi) + V,+ArqR;.i) . &fk,i) . (75)

From (62), (63), (69) and (70), we write

de@,‘) = -C-l: d7(k.i) P

(76)

and substitute (76) into (73) to obtain

R(k.i+l) = R(k,i)

&P &P - (V,+A,pRti’: C-’ -V~+A,,R~‘~)): dr(k*i)

+ Vt+atqREp (k-i) &+k” + V,RI((,i) &k.‘) . P

(77)

Solving for dhcksi), dqck”’ and d7(k,i) such that the residual estimates in (77), (74) and (75) vanish simultaneously, we find

+ ((V,Y)*: C + &(V,Y)* . $a (I’$)*: C): Rt'i'

- ((V,Y)* * $ - hcy(V,Y)*: C: (V, f)* * 4). Rb"."] , (78)


drl (kif _ - -0. R, + ,(a$ - (v,h)*: e: R:,~’ + $ - (h* - hLY(V,h)*: & f*) dh(k.i) ,

(79)

Now, we use (76) to compute de:,” and compute new estimates of the state variables using (69)-(72) and new residuals using (66)) (67) and (68). The iterations are repeated for a fixed value of the estimated incremental strain Asck), until all three residuals R, , R, and R, vanish within specified tolerances, TOLep, TOL, and TOL,.

Q

2.4.2. Consistent tangent stiffness formulation: equilibrium iterations Let r+A4y(k) be the converged stresses generated by the constitutive iterations for the

estimate of the incremental strains in the kth equilibrium iteration Aeck). In general, these stresses will not satisfy equilib~um, so it is necessary to find an improved estimate of the incremental strain, Ae(kilf = Asfkf + ds, that generates a new stress t+A’~(k+l) to reduce the equilibrium error. The iterational strain corrections de are determined using the consistent tangent operator of (48) in a linearized form of the nonlinear algorithmic constitutive model,

f+A;(k+l) r t+A;(k) + C(k): de a (81)

Substituting (81) into the equilibrium equation (38), and letting

d& = B' dU’ (82)

be the iterational strain correction within element e, we find

5 e=l

{[j-- (Be)ti?k)Be dv] dU’} = 5 i (iv”)% dV P e=l e

+ =gl jA (W)?dA - c/$l j-- (Bc)t~(k’ dV , f e P

(83)

The term in square brackets is the element consistent tangent stiffness matrix. Equation (83) is a set of linear equations for the iterative displacement corrections dU”. Once these corrections are computed, the iterative strain corrections are computed from (82) and a new set of constitutive iterations is initiated.

2.4.3. Iterative solution procedure Here we summarize the two-level, iterative solution procedure.

(1) Begin the new load step; set r+A’~(o’, r+Are’o’, “‘$r) and ‘+A’r(o) to *u, %, %P and ‘7. (2) Start the equilib~~ iterations; set k = 1.

(3) Check the convergence of the previous equilibrium iteration. - If (38) is satisfied within a specified tolerance, go to Step (13).

C.A. Vidal, R. B. Haber, Design sensitivity analysis 409

(4) Solve (83) for the iterative displacement correction. (5) Update:

- the displacements, (U’)‘“’ = (Ue)(k-l) + dU’; the total strain increment, Asck) = ‘+Are(k) - ‘E.

(6) Compute the trial state by the elastic predictor (61a-d) and check the yield condition. - If Y((r*)(k,O), ( q*)(k.O’) < 0, then the trial state is elastic. Go to Step (11).

(7) Start the return mapping iterations; set i = 0. (8) Check the pointwise residuals R[,", RF.” and R(ykyi) using (66)-(68).

if IIRckqi)lI GTOL, , ((R(~*~)([ sTOL and IR~~“I STOL,; go to Step (11). (9) Cornput: the new s&e v&iable estimites using (69)-(72).

(10) Set i = i + 1; go to Step (8). (11) End the return mapping iterations. (12) Set k = k + 1; go to Step (3).

(13) End the equilibrium iterations; continue with the next load step.

3. Sensitivity formulation

This section presents the derivation of a sensitivity formulation that is consistent with the numerical model developed in Section 2. We wish to determine the variation of the response at the end of a typical load step (time t + At) with respect to a specified design variation. We assume that the response at the beginning and at the end of the step (times t and t + At) and the response sensitivity for time t have already been computed. First, we formulate the shape design sensitivity analysis problem. Then, the direct-differentiation method is applied to the variational equilibrium equation and the algorithmic constitutive model to obtain a system of equations for the response sensitivity. The variational equilibrium equation is written in the reference geometry of the domain parameterization method to allow for the treatment of shape variations. Finally, finite element forms of the sensitivity equations are presented and the accuracy and efficiency of the method are discussed.

3.1. Overview of the sensitivity analysis problem

3.1.1. The design and response descriptions We describe a physical system through a set of design specifications that is comprised of two

subsets: the subset that defines the geometric model and the set that ascribes physical attributes to every location in the gee_metric model. We represent the full set of attribute functions by a vector-valued function @(x, t), where x E V and t E I.

In design sensitivity analysis, we are interested in studying the effects of both geometric and nongeometric design variations on a given performance functional. The domain parameterization (or reference volume) method [24-271, depicted in Fig. 3, is used here to treat both types of variations in a common framework. The fixed reference domain, 0 C R” (s = 1,2 or 3) with boundary r, provides an invariant geometric base for the sensitivity formulation. The reference position vector, r, is the independent spatial variable for the formulation, and !P : L? + Rd is an invertible mapping defined so that the domain V C Rd (d = 1,2 or 3; d 2 s) is the image of 0 under ly, and x is the value of !P at r.

410 C.A. Vidai, R. B. Haber, Design sensitivity analysis

deformed configuradon

reference configuration

reference coordinate frame

Fig. 3. Domain parameterization model for describing shape variations.

v= !Py(q ) x = P(r) . (84)

In the following, all expressions are written in terms of the reference domain L!. Thus, any function Z”(X) on V is replaced by z(r) on 0, such that z = z”o ?P. In particular, the set of attribute functions is written in referential form as @(r, t) .

The Jacobian J of the mapping ?P plays an important role in the transformation of function derivatives and integral expressions.

J-’ =V,('F') , (86)

vx( ) =v,( )v,(V-‘) =V,( )a J-' , (87)

dV=JdO, (88)

CA. Vidal, R.B. Haber, Design sensitivity analysis 411

dA=KdT. (89)

J is the determinant of J and K is a metric that relates corresponding differential areas on A and r. If q is the outward unit normal to r, then [28]

K = J[(J-' * q). (J-” * r))]“* - (90)

It is convenient to restrict the attribute functions to a linear combination of P vector-valued basis functions, +?(r, t), where the combination coefficients are the elements of a vector of discrete design parameters, 0 E Rp.

Similarly, the components of the geometry mapping 4v are restricted to a linear combination of Md vectors of geometry basis functions $(I-, t), where the combination coefficients are the elements of a vector of discrete geometry design parameters, X E RMd.

x = P(r, t) = 2 $&Jr, t)Xc . I=1

C-92)

The integer scalar d is the dimensionality of the geometry (d = 1,2 or 3) and M is the number of geometry control points. In finite element applications it is natural to choose node coordinates or mesh generation control points as the geometry design parameters. Thus, a design is specified by an ordered pair of discrete parameter arrays, (0, X) E RP X RMd, and the design basis functions, (pY and +QC.

3.1.2. Representation of a general performance functional Let the functions U, E and T on 0 x [0, T] represent the displacement, strain and stress

response fields, and let the function f on F x [0, T] represent the surface tractions. These response fields are implicit functions of the design, determined by the governing equations, the initial conditions and the boundary conditions.

Now we construct a general pe~o~ance functional that can be used to characterize the performance of the structure over the time interval of interest.

T

II= I [I 0 n f((u, E, T, CD, Ilr, t)J da +

I r, g(u, Qi, 0, t)K dT

+ @t, Qi, !P, t)/c dr I

dt . (93)

Equation (93) provides a framework for defining performance measures as integral weighted averages of the response and the design fields on fl x [0, T]. These can be used to represent objective unctions and constraint functions in optimization problems, or error measures in inverse and identi~cation problems.

Weighting functions with local support can be incorporated in f, $ and h to restrict the


performance measure in space and/or time. For example, we might be interested in the instantaneous behavior of the structure at time t”E [0, T]. Then we choose f(u, E, r, @, Y?‘, t) = f(u, E, T, Qi, Y)S(t - 0, g(u, a, P, t) = g(u, Qi, P)S(t - t‘> and &t, Qi, !P, t) = h(t, @, lu)s(t - 9 to define the instantaneous performance measure,

n R ?= I [f R f(U, E, T, @, e)J da -i- g(il, @, ‘;u)K dT + h(t, qZ, !P)K dr 1 . i (94)

In the following, we restrict our attention to the instantaneous measure (94), and drop the notation Ir for convenience, If we wish to employ a space-time performance measure, an approximation of (93) can be obtained by evaluating (94) at a number of discrete times and employing numerical quadrature.

3.1.3, Design sensitivities of the performance functional The variations of the mapping Ur, denoted by Slu, describe shape design variations. They

induce the corresponding Jacobian variations [27]

SJ = 6(V, !I’) = Vr(S’P) , (93

&(J-‘) = -J-l . jjJ. J-’ , WI

The variations of J and K follow directly. Based on (91) and (92), we describe a design variation (S@, MP) in terms of variations of

the discrete design parameter array (80,6X) E RP X RMd, where

Q@(C t) = f: &(C t) a@? ; W(r) = fg Jis(r) 8X< . y=l &.=I

We represent the design variation in a given perturbation direction B = (6,x) as

(97)

(se, SX) = 87jE) (98)

where 6~ is the variation of a parameter that scales the design change. The notation S=(e) represents the variation of (*) due to the design variation 8~2 and e,(s) denotes the directional design sensitivity of (e). That is, S=(e) = 67 a=(*). Response variations 6&r, t),

8&t, t), 6,~(r, t) and S&r, t) are associated with the design variation 6~ B through the implicit relations defined by the system equations. Utilizing the referential form of the performance functional,

II= I n f(u, e, I, a, v)J dL? + I, g(u, 8, p)K dr + jr h(t, @, *)K dr, (99) I U

we write the variation 6,ll as the sum of two parts that depend, respectively, on the explicit design variation, (6,Qz, SE Y), and the implicit response variations, &u, &E, $=r and 6,t:

C.A. Vidal, R.B. Haber, Design

6,17 = 6r)(D + R) .

The explicit sensitivity D and the implicit sensitivity

sensitivity analysis

R are expressed as

+ V,,, f - a, ‘P)J] dR

+ I r MC a, WY> a, K + (V,h . a=@ + V,,,h * a, q)K] dT , u

R= ~(vuf~aEu+vEf:a,E+vTf:aa7)Jd~ I

+ I r, V,,@,uKdr+ I r”

V,h+tKdr. (102)

413

W)

W)

D can be evaluated directly, using (95)-(98), once the response for the current design has been determined. Using (97) and (98)) we express aI@ and a, W as

R is the part of the design sensitivity associated with the implicit variations of the referential response fields, 6,~ = [V,u* 6 +V,u*%] ST, 8=e =[V,E* @+Vxa-X] 677, &T=[V~T- 6 + VXr . $]6q and 8& = [V,t . 6 + Vxt. _%I 87. The terms in brackets are the directional design sensitivities, a=u, a,-&, a,7 and a&. The central issue in design sensitivity analysis is the evaluation (in the finite difference and direct differentiation methods) or the elimination (in the adjoint variable method) of the response variations appearing in (102).

3.2. The incremental direct differentiation method

3.2.1. Implicit differentiation of the variational equation of equilibrium and the algorithmic constitutive equations

We rewrite the variational equation (17) and the strain-displacement relations in the reference configuration as

I R 6: T(u)JdO = n li.bJdO +

I I rci*&dr V(h,t)E6hXz, (105) I

E = $[V,u . J-’ + (V,u *J-‘)‘I , (106)

414 CA. Vidal, R.B. Haber, Design sensitivity analysis

2 = pi *J-l + (V,ci P)‘] * (107)

The direct differentiation formulation begins with the implicit differentiation of the variational equation (105) at time t + At.

I D V,ti -J-l: c~~(“~~T).I d0 =

I n h - P&d ‘+Arb)J + ‘+‘% a, J] da

+ I r,

2 - M t+A’i)K + t+Af+c]dr- V,&a,J-‘: “AtrJdO

- I nVji. J-‘:t+At~a,JdL3 VriE6u. (108)

Implicitly differentiating (28), (29), (40), (41), (42), (43b) and the incremental strain- displacement relation (obtained from (106)), we have

a,( f+AfT) = a&) + a&: [A& - hf*] + c: [a,(AE) - a&f * - h a,f *] , (109)

al(f+A’q) = a,(&) + a,ih* + i a,h* , (110)

a,( ‘+A’~p) = a&J + ahif * + i a,f * , (111)

(V,Y)* a&*) + (v,Y)* a,( q*) = 0 ,

a,f * = a:f * + (v,f >* a,(T*) + ($f )* a&*) ,

(112)

(113)

aph* = a;h* + (V,h)* a&*) + (V,h)* a&*) )

a&*) = (1 - a) a&) + or ae’(f+A;) ,

(114)

(115)

a&q*) = (I- (Y) a,(b) + (Y a=(‘+“q),

a,(A&) = ; {V,@,(Au)) * J-l + [V,(a,(AU)) . J-lit

+ V,(AU) . a,J-’ + [V,(Au) . a, J-l]‘} .

(116)

(117)

The notations tlgf * and agh* denote the explicit parts of the design sensitivities off and h with 7* and q* fixed. Combining (109)-(117) and accounting for the symmetry of C, we have

aI(f+A%) = a&) + E: c? a,c: (AE - if *)

+ 2: [V&(Au) *J-l + V,(Au) . a&‘]

- 2: [agf* + (V, f)*: a& + (1 - a)(Vgf)* ma,(rq)]

-a&!: f*-L&(Vqf)*e a,(““&), (118)


az(t+Atq) = ap(tq) + a,@. h* + &tfj. (V,h)*: a,(“A’7)

+ ifj. [a;h* + (V,h)* . a,(b) + (1 - a)(V,h)*: a,(‘~)] . (119)

Combining (118) and (119), we obtain expressions for a,( ‘+“r) and dS(f+Afq) in terms of a,A.

as(""'~)=a,('~)+ {&[c-':a&: (AE - hf*)+V,(Au). a&']

- he: [(VJ)* + /ia(V (j. (V,h)*]: a&)

- A& [(VJ)* + hcu(V,f)* . fi. (V,h)*] . a,(b)

-A&[a;f*+ hcu(v$)* +aph*]>

+ c: [V&(Au)) . J-l]- & [f* + k&f)*. 6. h*] a,i ,

a,("+Aqq) = a&) + hcy& - (V,h)*: & [V,(a,(Au)) T1]

+ { ia& - (V,h)*: e: [C-l: a&: (he - if*) + V,(Au). a&-‘1

+ h&s [(V,h)* - icr(V,h)*: 6: (V,f)*] * ds(‘q)

+ h& - [(V,h)* - icx(V,h)*: 6: (V,.f)*]: a=(k)

+ h$ * [a;h* - k(V,h)*: E: apf*]}

+ $. [h* - ha(V,h)*: 6: f*] a,h .

(120)

(121)

The terms between braces in (120) and (121) are now denoted by a;“(Ar) and ai (Aq), respectively. Substituting (120) and (121) into (112), we obtain an expression for a,A.

a,i= $ KV)*:a,W+ (v,y)* .a&~)]

+ $ [(v,Y)*: a;(AT) + (v*Y)* * a;(Aq)]}

+ $ [(V,Y)*: (? + k(p,Y)* - &- (V,h)*: 61: [V,(a,(Au)).J-‘1 . (122)

Denoting the term between braces by a; h and substituting (122) into (120), we obtain

as(‘+AfT) = a&T) + a;(AT) -I 6: [V,(a,(Au)) . J-‘1

- & [f* + ha(V* f)* - (fj - h*] dgi . (123)

The only implicit term appearing on the right-hand side of (123) is a,(Au). Thus, determining

416 C.A. Vidal, R. B. Haber, Design sensitivity analysis

the design sensitivities of the incremental displacements is sufficient to find the sensitivities of all the other response variables at time t + At.

3.2.2. Computation of the design sensitivities of the incremental displacements The sensitivities of the incremental displacements are the solution of the following

variational problem which is obtained by substituting (123) into (108). Find dE(Au) E H’(0) such that a&Au) = a&Au) on r, and

I n Vii. J-‘: c: V&(Au)) . J-‘J d0 =

I n 22 * [a,( ‘+Atb) J + ‘+‘% a, J] da + lr li - [a,(““‘i)~ + ‘+Atia,~] dr ,

- I n v,ri -J-l: {a&) + I; - & [f* + iu(Vqf)* - (j - h*] &$}J d0

- I yii * a,J -I: r+AtrJ d0 - R

-‘:‘+“‘~a,Jdfi tlliEG,, . (124)

The finite element discretization of (124) yields the following equations for a&AU):

$ [I (B’)‘@J da] &(AU’) e=l

= 2 [I (B’)‘& (f * + k&f)*. 4. h*) a;/iJ dL? e=l

1 - z (I [(a,B’)t(t+A’~)J + (Be)t(‘+At~) a=J + (B’)‘(a&) + ap(AT))J] dR)

e=l

+ $ {l[(N’)‘a=(““‘b)J + (Ne)t(t+Afb) a= J] da} .C=l

‘+“t)~ + (,e)t(*+A’,) &K] dl‘) (125)

Equation (125) is a system of linear equations that can be solved directly for the incremental displacement sensitivities. Once the displacement sensitivities are obtained, we obtain a&A&), a& d,(““‘q), a&““%) and aB(‘+AtEP) using (117), (121)-(123) and (ill)-(116). The matrix on the left-hand side of (125) is the consistent tangent stiffness matrix obtained in the elastoplastic stiffness formulation (see (83)). The right-hand side is the pseudo-load vector corresponding to a design direction 3.

The sensitivity computations are relatively inexpensive, since no iterations are required in the sensitivity analysis. Furthermore, the reduced form of the consistent tangent stiffness matrix formed during Step (4) of the iterative solution scheme can be stored. Therefore, the


only additional operations required to obtain a&AU) for each design perturbation direction B are (1) the assembly of the pseudo-load vector for 3, (2) reduction of the pseudo-load vector using the reduced form of the consistent tangent stiffness matrix retained from the final equilibrium iteration of the load step, and (3) back-substitution to solve for a,(AU).

4. Numerical example

The problem depicted in Fig. 4 demonstrates the use of the consistent sensitivity analysis formulation for elastoplastic problems developed in the previous section. A thick-walled hollow cylinder subjected to an internal pressure is analyzed for cyclic loading. Plane strain conditions are assumed in the axial direction. The cylinder dimensions, material properties and yield criterion are as follows: internal radius, ri = 100 mm; external radius, re = 200 mm; elastic modulus, E = 2.1 x lo4 dN/mm*; Poisson’s ratio, u = 0.3; uniaxial yield stress,

/ / /’ ‘i

/ / / /

re

a) Geometry and loading

\ \ \ 41 nodes @ 2.5mm

\

internal surface external surface

b) Node arrangement in the radial direction

Fig. 4. Thick-walled hollow cylinder under internal pressure loading.


a,, = 24.0 dN/mm*; Von Mises yield criterion, Y(r) = J2 - ~$3 = 0 (where J2 = 4s: S and S is the deviatoric stress tensor). Due to the symmetry of the problem, only the first quadrant of the cylinder is modeled. The finite element mesh consists of 600 eight-node quadrilateral elements (20 elements in the radial direction and 30 elements in the tangential direction).

The cyclic load history is summarized as follows. The pressure is increased from zero to pO, a load level at which some plastic deformation occurs, during the initial loading phase. Then the internal pressure is decreased to zero during the unloading phase. Finally, the load is increased during the reloading phase until a pressure p1 > p,, is achieved. The maximum internal pressure in the initial loading process is p,, = 16 dN/mm*, and the maximum internal pressure in the reloading process is p1 = 18 dN/mm*.

Three load incrementation schemes were tested to study the effects of load-step size on the accuracy of the simulation and the sensitivity analysis. The load steps used for each scheme are shown in Fig. 5. Three sensitivity analysis methods were tested for each load increment scheme. The central finite difference sensitivity analysis method provides reference solutions

-----------

i 1 9 17 20 21 22 27 35 43 ’

load step number

1 5 9 11 12 13 16 20 24

load step number Scheme 1 +____________________--_-_-_______.__ 18.

1 3 5 7 8 9 11 13 15

load step number

Fig. 5. Load increment schemes.


for evaluating the other two methods. Careful convergence studies were performed to find a favorable step length for the design perturbations in the finite difference calculations. The second method is based on the consistent direct-differentiation formulation developed in this work, and the third method is based on an inconsistent direct-differentiation formulation that uses the initial continuum tangent operator from the beginning of each load step to compute the sensitivities of the incremental displacements. This method was included to demonstrate the importance of using the consistent tangent operator in sensitivity analysis of implicit algorithmic constitutive models. The three load schemes provide a test of the extent to which the accuracy of the sensitivity computations depends on the size of the load increments.

The computed response data for the three load schemes are compared in Fig. 6 and Table 1. There is excellent agreement between the three simulations. This indicates that the larger load steps used in the first load scheme do not affect the accuracy of the implicit integration strategy, and that the subincrementation in schemes 2 and 3 are wasteful of computational resources.

Figures 7 and 8 compare the sensitivities of the radial displacement U, of the interior surface with respect to changes in ri obtained with the three sensitivity methods for load scheme 1. The results obtained with the consistent method agree almost perfectly with those obtained by the central finite difference method throughout the load history. On the other hand, the sensitivities obtained by the inconsistent method register maximum errors of 9.4%, 91.7% and 36.4% at the end of the first loading phase, the unloading phase and the reloading phase,

16.0

8.0

- Scheme 3

0.0

0.00 0.04 0.08 0.12 0.16 0.20

Interior surface radial displacement (mm)

0.24 0.28

Fig. 6. Comparison of the interior surface displacement histories obtained with three load increment schemes.


Table 1 Radial displacement u, of the interior surface for the three load increment schemes

Scheme 1 Scheme 2 Scheme 3

Step Step U, Step

1 0.07263474 1 0.07263474 1 0.07263474

2 0.09079343 3 0.09079343 5 0.09079343

3 0.11111684 5 0.11111564 9 0.11111420

4 0.13982541 7 0.13981205 13 0.13980223

5 0.18353942 9 0.18350161 17 0.18347533

6 0.17446007 10 0.17442227 19 0.17439599

7 0.14722204 11 0.14718424 20 0.14715796

8 0.11090467 12 0.11086686 21 0.11084058

9 0.03826992 13 0.03823211 22 0.03820583

10 0.11090467 14 0.11086686 23 0.11084058

11 0.12906336 16 0.12902555 27 0.12899927

12 0.14722204 18 0.14718424 31 0.14715796

13 0.16538073 20 0.16534292 35 0.16531664

14 0.18353942 22 0.18350161 39 0.18347533

15 0.26305091 24 0.26295719 43 0.26288544

20.0

16.0

_ Finite difference sensitivity and conwtent sensitlvlty

- -0 - - Inconsistent sensitivity

0.00 0.05 0.10 0.15 0.20 0.25 0.30 Vr,U, ( x 10-q

I I, I I u I, I n I I, I II 11 11 1 ’

Detail of boxed region

Fig. 7. Sensitivity of the radial displacement U, of the interior surface with respect to ri for load scheme 1.

Fig. 8. Sensitivity percentage of the


I10

100

90

80

70

60

50

40

30

20

10 consistent method lj

---e-e lnconsisrent melhod ’

1 3 5 7 9 .ll 13 15 Load step

/ of the radial displacement of the interior surface with respect to ri for finite difference sensitivity results.

20.0

16.0

“T

g 12.0

E !? ; 8.0

aii z _ Finite difference sensitivity and

4.0 sensttwity consistent - -D . - Inconsistent sensitivity

0.0 v- 0.00 0.05 0.10 0.15 0.20 0.25 0.30

load scheme 1 as a

Fig. 9. Sensitivity of the radial displa~ment U, of the interior surface with respect to ri for load scheme 2.


respectively. Figures 9 and 10 show sensitivity results for load scheme 2. Again, the results obtained with the consistent method are almost exactly the same as those obtained with the central finite difference method. However, the sensitivities obtained with the inconsistent method follow approximately the same pattern as in load scheme 1, but smaller errors are observed. The error increases to a maximum of 8% in the initial loading phase. The relative error increases during the unloading process to a maximum of 81% at p = 0. The error decreases during the early stages of the reloading process to a minimum of 5.6%, but then grows to a maximum of 22.4% at p = pl. Further improvement is observed for the inconsistent method in load scheme 3 due to the reduction of the load-step lengths. Once more, the results for the consistent and the central finite difference methods agree almost exactly. For the inconsistent method, Figs. 11 and 12 show that the pattern is essentially the same as before, but now a maximum error of 6% is observed during the initial loading phase. The percentage error increases during the unloading process, reaching a maximum of 44.6% when the load is completely removed. During the reloading process, the error first decreases to a minimum of 3% at the load level p = po, but then grows to a maximum of 12.6% at the final load level

P=Pl* Figure 13 shows sensitivity results obtained with the consistent method for the three load

schemes. Since the underlying simulation models are in close agreement (see Fig. 6), the sensitivity predictions of the consistent formulation for the three load schemes are almost the same. However, Fig. 14 shows that the inconsistent method generates significantly different sensitivity results for the three load schemes. These differences are attributed to the discrepancies between the inconsistent formulation and the analytical sensitivity of the

------ Inconsistent method 10 ““““““““““”

1 4 7 10 13 16 19 22 24

Load step

Fig. 10. Sensitivity of the radial displacement of the interior surface with respect to ri for load scheme 2 as a percentage of the finite difference sensitivity results.


16.0

*E a ?j 12.0

i

k z

8.0

E S 9 Finite difference sysitivity and

4.0 _

comstent sensltivlty - -0 - - Inconsistent sensitivity

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Vr,U, (X 10-q

Detail of boxed region

Fig. 11. Sensitivity of the radial displacement U, of the interior surface with respect to ri for load scheme 3.

numerical model. The significant errors in the results for schemes 2 and 3 indicate that even the highly subdivided load steps of scheme 3 do not ensure reliable sensitivity predictions with the inconsistent method. Quantitative comparisons of the consistent and inconsistent sensitivity results with the central finite difference results are presented in Tables 2,3 and 4 for the final load step (p = pl). The results obtained with the consistent sensitivity method exhibit seven figures of agreement with the results obtained with the central finite difference method for all the nodes in the mesh and for all three load schemes. On the other hand, the sensitivities obtained with the inconsistent sensitivity method show maximum relative errors of 36.35%) 22.44% and 12.63% for load schemes 1,2 and 3, respectively. Table 5 shows a comparison of CPU times on a CRAY YMP for the simulation and the three sensitivity analysis methods. The central finite difference times do not include the cost of the search for a suitable finite difference increment.

The following observations summarize the findings of this numerical study. (1) The larger load increments of load scheme 1 do not significantly affect the accuracy of the

simulation. Therefore, there is little to be gained (as far as the simulation is concerned) for the significant extra costs involved in using the smaller load steps of schemes two and


110 ‘,,‘, ,, ,, ,, 1. ,, (, ., ., ,, ,, (. *.

coasistentmethod _

------* Inconsistent method - 100 +-.

J

_---3

\

*- t-

t

4

504 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43

Load step

Fig. 12. Sensitivity of the radial displacement of the interior surface with respect to r, for load scheme 3 as a percentage of the finite difference sensitivity results.

Fig. 13. Sensitivity of the radial displacement u, of the interior surface with respect to rr for the consistent sensitivity analysis method.

20.0

16.0

% $

g 12.0

c e

; 8.0

E is? 9

4.0

0.0


0.00 0.05 0.10 0.15 0.20 0.25 0.30

Vr,Ur (X 10-q

Fig. 14. Sensitivity of the radial displacement u, of the interior surface with respect to r, for the inconsistent sensitivity analysis method.

(2)

(3)

(4)

(5)

three. This favorable behavior is a benefit of using an implicit integration method in the algorithmic constitutive model. The consistent sensitivity formulation produces extremely accurate results in all cases tested. There is precise agreement between the reference central finite difference sensitivity results and the consistent sensitivity results for each load scheme. Moreover, there is good agreement between the consistent sensitivity predictions obtained with the three load schemes (see Tables 2-4), since the underlying simulations are in reasonably close agreement for the three loading strategies. The inconsistent sensitivity formulation produces significant error in all three cases tested. The sensitivity error tends to accumulate during the course of the analysis. Although the magnitude of the sensitivity error decreases as the load step size is decreased, we did not obtain acceptable accuracy in any of the three cases for the inconsistent method. Thus, extremely small time steps would be required with the inconsistent method to ensure reliable sensiti~ty results, even though the implicit integration scheme produces reliable simulation results for larger steps. The cost of the finite difference sensitivity analysis is an order of magnitude larger than the consistent formulation for a given loading scheme. The consistent sensitivity analysis method increases the computational cost by only 16% over the cost of the simulation in this example (see Table 5), so it is feasible for practical applications. Although the costs of the consistent and inconsistent methods are roughly the same for a given load scheme, the costs required to achieve comparable accuracy in the sensitivities are quite different. Clearly, load scheme 1 is sufficient to obtain an accurate response simulation. The incremental cost to obtain sensitivity information with a terminal accuracy of 12.6% by the inconsistent method is 490 CPU seconds (load scheme 3) which is double the cost of the simulation for load scheme 1, compared to an incremental cost of 40 CPU

426

Table 2

C. A. Vidal, R.B. Haber, Design sensitivity analysis

Sensitivities of the nodal radial displacements u, with respect to ri for load scheme 1 and load level p1

Node Central finite difference

(Ari = 0.01 mm)

Consistent method Inconsistent method

Sensitivity % difference Sensitivity % difference (x 10-q (x 10-q

1 0.28137386 0.28137386 o.ooooo 2 0.28986067 0.28986067 o.ooooo 3 0.29712831 0.29712831 0.00000 4 0.30330616 0.30330616 0.00000 5 0.30851176 0.30851176 0.00000 6 0.31284504 0.31284504 0.00000 7 0.31639494 0.31639494 0.00000 8 0.31924183 0.31924183 0.00000 9 0.32145721 0.32145721 o.ooooo

10 0.32309762 0.32309762 0.00000 11 0.32422330 0.32422330 0.00000 12 0.32488574 0.32488574 0.00000 13 0.32513114 0.32513114 0.00000 14 0.32500180 0.32500180 o.ooooo 15 0.32453592 0.32453592 0.00000 16 0.32376559 0.32376559 0.00000

17 0.32268622 0.32268622 o.ooooo 18 0.32135397 0.32135397 0.00000 19 0.3197970s 0.3197970s 0.00000 20 0.31803956 0.31803956 0.00000 21 0.31610364 0.31610364 o.ooooo 22 0.31400992 0.31400992 0.00000 23 0.31177747 0.31177747 0.00000 24 0.30942412 0.30942412 0.00000 25 0.30696639 0.30696639 0.00000 26 0.30653478 0.30653478 o.ooooo 27 0.30605394 0.30605394 o.ooooo 28 0.30552955 0.30552955 0.00000 29 0.30496667 0.30496667 0.00000 30 0.30436996 0.30436996 0.00000 31 0.30374358 0.30374358 0.00000 32 0.30309136 0.30309136 0.00000 33 0.30241674 0.30241674 o.ooooo 34 0.30172286 0.30172286 0.00000 3s 0.30101256 0.30101256 0.00000 36 0.30028845 0.30028845 0.00000 37 0.29955285 0.29955285 0.00000 38 0.29880792 0.29880792 0.00000 39 0.29805558 0.29805558 o.ooooo 40 0.29729762 0.29729762 o.ooooo 41 0.29653562 0.29653562 0.00000

0.17909836 -36.34861 0.18939947 -34.65844 0.19848093 -33.20026 0.20646758 -31.92767 0.21346841 -30.80704 0.21958563 -29.81010 0.22490575 -28.91614 0.22950656 -28.10887 0.23345733 -27.37530 0.23682141 -26.70283 0.23965653 -26.08288 0.24201157 -25.50871 0.24393037 -24.97477 0.24545314 -24.47638 0.24661618 -24.00959 0.24746998 -23.56508 0.24825478 -23.06620 0.24880699 -22.57541 0.24913118 -22.09710 0.24924629 -21.63041 0.24916946 -21.17476 0.24891649 -20.72974 0.24850175 -20.29515 0.24793856 - 19.87097 0.24723908 - 19.45728 0.24734885 - 19.30806 0.24738722 - 19.16875 0.24736086 - 19.03865 0.24727578 -18.91711 0.24713749 - 18.80359 0.24695101 - 18.69754 0.24672094 -18.59849 0.24645145 -18.50602 0.24614639 - 18.41971 0.24580927 - 18.33920 0.24544331 -18.26415 0.24505143 -18.19426 0.24463635 -18.12923 0.24420053 - 18.06879 0.24374626 - 18.01271 0.24327560 - 17.96075


Table 3 Sensitivities of the nodal radial displacements u, with respect to ri for load scheme 2 and load level p1

Node Central finite Consistent method Inconsistent method

difference

(Ari = 0.01 mni) Sensitivity % difference Sensitivity % difference

(x lo-*) (x 10-q

1 0.28099253 0.28099252 o.ooooo 2 0.28948420 0.28948420 O.OOOOO 3 0.29675721 0.29675721 O.OOOOO 4 0.30293951 0.30293951 o.ooooo 5 0.30814816 0.30814816 o.ooooo 6 0.31248472 0.31248472 O.OOOOO 7 0.31603867 0.31603867 O.OOOOO 8 0.31888839 0.31888839 0.00000 9 0.32110603 0.32110603 0.00000

10 0.32275213 0.32275213 0.00000 11 0.32388557 0.32388557 O.OOOOO 12 0.32455775 0.32455775 O.OOOOO 13 0.32480765 0.32480765 O.OOOOO 14 0.32468228 0.32458227 o.ooooo 15 0.32422024 0.32422023 O.OOOOO 16 0.32345515 0.32345514 040000 17 0.32239985 0.32239985 O.OOOOO 18 0.32109782 0.32109782 o.ooooo 19 0.31957685 0.31957684 o.ooooo 20 0.31785848 0.31785848 030000 21 0.31592827 0.31592826 o.ooooo 22 0.31383762 0.31383762 O.OOOOO 23 0.31160814 0.31160814 O.OOOOO 24 0.30925766 0.30925766 O.OOOOO 25 0.30680271 0.30680271 O.OOOOO 26 0.30637245 0.30637245 0.00000 27 0.30589293 0.30589293 O.OOOOO 28 0.30536979 0.30536979 o.ooooo 29 0.30480812 0.30480812 O.OOOOO 30 0.30421257 0.30421257 o.ooooo 31 0.30358731 0.30358731 o.ooooo 32 0.30293617 0.30293617 O.OOOOO 33 0.30226258 0.30226258 o.ooooo 34 0.30156970 0.30156970 O.OOOOO 35 0.30086036 0.30086036 O.OOOOO 36 0.30013716 0.30013716 O.OOOOO 37 0.29940245 0.29940245 O.OOOOO 38 0.29865837 0.29865837 O.OOOOO 39 0.29790686 0.29790686 O.OOOOO 40 0.29714969 0.29714969 O.OOOOO 41 0.29638845 0.29638845 o.ooooo

0.21793277 -22.44179 0.22783876 -21.29492 0.23646722 -20.31627 0.24395051 - 19.47220 0.25040739 -18.73799 0.25594294 -18.09425 0.26064906 -17.52621 0.26460653 - 17.02221 0.26788987 - 16.57277 0.27045236 -16.17023 0.27268432 - 15.80844 0.27430886 -15.48226 0.27548029 -15.18664 0.27624356 -14.91881 0.27663791 -14.67593 0.27669918 - 14.45516 0.27645763 - 14.25007 0.27594250 - 14.06279 0.27518059 -13.89220 0.27424446 - 13.72121 0.27374116 -13.35338 0.27311626 -12.97530 0.27233528 -12.60328 0.27141161 -12.23771 0.27035742 -11.87906 0.27025751 -11.78792 0.27009480 -11.70283 0.26987556 -11.62336 0.26960544 -11.54913 0.26928962 -11.47979 0.26893278 -11.41501 0.26853923 -11.35452 0.26811285 -11.29803 0.26765723 -11.24532 0.26717560 -11.19614 0.26667095 -11.15031 0.26614597 -11.10762 0.26560317 -11.06790 0.26504479 -11.03099 0.26447293 -10.99673 0.26388946 - 10.96500


Table 4 Sensitivities of the nodal radial displacements u, with respect to r, for load scheme 3 and load level p,

Node Central finite Consistent method Inconsistent method difference

(Ari = 0.01 mm) Sensitivity % difference Sensitivity % difference (x 10-q (x 10-q

1 0.28072226 0.28072225 o.ooooo 0.24527919 - 12.62567 2 0.28921761 0.28921761 o.ooooo 0.25459934 -11.96963 3 0.29649318 0.29649317 o.ooooo 0.26266338 -11.40998 4 0.30267824 0.30267823 o.ooooo 0.26960283 - 10.92758 5 0.30788896 0.30788896 o.ooooo 0.27553571 - 10.50809 6 0.31222768 0.31222768 0.00000 0.28056622 -10.14050 7 0.31578344 0.31578344 0.00000 0.28478500 -9.81636 8 0.31863574 0.31863574 o.ooooo 0.28827322 -9.52891 9 0.32085531 0.32085531 0.00000 0.29110310 -9.27278

10 0.32250517 0.32250516 o.ooooo 0.29333920 -9.04357 11 0.32364365 0.32364364 o.ooooo 0.29504046 -8.83787 12 0.32431843 0.32431843 0.00000 0.29625671 -8.65252 13 0.32457507 0.32457507 0.00000 0.29703460 -8.48508 14 0.32445796 0.32445796 o.ooooo 0.29741836 -8.33378 15 0.32399936 0.32399936 0.00000 0.29744288 -8.19646 16 0.32323815 0.32323815 0.00000 0.29714578 -8.07218 17 0.32219701 0.32219700 o.ooooo 0.29655256 -7.95925 18 0.32091147 0.32091147 o.ooooo 0.29569718 -7.85709 19 0.31939517 0.31939517 0.00000 0.29459862 -7.76360 20 0.31768218 0.31768218 O.OOOOO 0.29328767 -7.67890 21 0.31577713 0.31577713 O.OOOOO 0.29178845 -7.59671 22 0.31371430 0.31371429 o.ooooo 0.29016331 -7.50715 23 0.31148865 0.31148865 0.00000 0.28898313 -7.22515 24 0.30914013 0.30914013 o.ooooo 0.28771352 -6.93103 25 0.30668708 0.30668708 0.00000 0.28631976 -6.64108 26 0.30625774 0.30625774 0.00000 0.28607504 -6.59010 27 0.30577910 0.30577909 o.ooooo 0.28577345 -6.54252 28 0.30525681 0.30525680 o.ooooo 0.28542100 -6.49807 29 0.30469596 0.30469596 o.ooooo 0.28502308 -6.45656 30 0.30410120 0.30410119 o.ooooo 0.28458464 -6.41778 31 0.30347670 0.30347670 0.00000 0.28411015 -6.38156 32 0.30282629 0.30282628 0.00000 0.28360369 -6.34773 33 0.30215340 0.30215340 O.OOOOO 0.28306896 -6.31614 34 0.30146120 0.30146120 o.ooooo 0.28250935 -6.28666 35 0.30075252 0.30075251 o.ooooo 0.28192792 -6.25917 36 0.30002995 0.30002995 o.ooooo 0.28132748 -6.23353 37 0.29929585 0.29929585 o.ooooo 0.28071059 -6.20966 38 0.29855235 0.29855235 0.00000 0.28007958 -6.18745 39 0.29780141 0.29780140 O.OOOOO 0.27943657 -6.16681 40 0.29704477 0.29704477 o.ooooo 0.27878349 -6.14765 41 0.29628406 0.29628406 o.ooooo 0.27812213 -6.12990

C. A. Vidal, R. 3. Haber, Design sensitivity analysis 429

Table 5 CPU times for the three sensitivity methods for each load increment scheme

Sensitivity Total cost of Total cost of method simulation sensitivity

computations (CPU seconds) (CPU seconds)

Total cost

(CPU seconds)

Sensitivity x 100o/ Simulation

*

Load scheme 1 Central finite difference Consistent method Inconsistent method



0.23E -t 03

0.23E + 03

0.23E + 03

0.35E + 03

0.35E + 03

0.35E + 03

0.61E + 03

0.61E + 03

OAlE + 03

OSOE + 03

0.04E + 03

0.04E + 03

0.78E + 03 l.l3E+03 223

0.06E + 03 0.41E f 03 17

O.06E + 03 0.41E + 03 17

1.36E+O3

O.llE + 03

O.llE + 03

0.73E + 03 217

0.27E f 03 17

0.27E + 03 17

1.97E f 0.3 223

0.72E + 03 18

0.72E + 03 18

seconds for nearly exact results by the consistent method. The cost of the inconsistent scheme would be even greater if more precision were required or if the sensitivities of the residual stresses after unloading were required. The cost of the extreme load-step subdivision required to achieve accuracy in the inconsistent method makes it unsuitable for practical applications.

5. Summary and conclusions

There is a clear need for robust design sensitivity fo~ulations which provide accurate sensitivities at reasonable cost. These requirements are particularly important for history- dependent problems, where the simulation costs are relatively high. A promising new design sensitivity analysis formulation for history-dependent response has been proposed in this work. In particular, for the first time we treat algorithmic constitutive models based on implicit integration strategies and return mappings associated with elastoplastic consistency conditions. We have emphasized the necessity of maintaining consistency between the sensitivity formulation and the underlying simulation model if accurate sensitivities are to be obtained. The new method fully satisfies this requirement and the numerical example


demonstrates the highly accurate results it can produce. The proposed sensitivity analysis method is also very efficient. In contrast to previous formulations, the consistent formulation does not require iterations in the sensitivity calculations and there is no need to subdivide load or time steps to improve the accuracy of the sensitivity results. In fact, there is nothing to be gained by these procedures in the proposed method, since the new sensitivity formulation generates analytical design gradients of the simulation model. Thus, the choice of the time or load step scheme can be based solely on the accuracy requirements of the simulation algorithm.

The proposed sensitivity formulation involves nothing more than the consistent application of the direct differentiation method to the incremental simulation model. Consistent with the notion of an incremental stress update procedure [14], there is no need (or reason) to perform sensitivity calculations for the unconverged configurations of the equilibrium iterations. Thus, only one sensitivity calculation is required per load or time step.

It turns out that a linearization of the incremental algorithmic constitutive model, expanded about the converged configuration achieved at the end of the equilibrium iterations for a given load or time step, is required in the direct differentiation sensitivity equations. If an exact linearization is used, then the identical consistent tangent operator appears in both the incremental sensitivity formulation and the simulation formulation. The numerical examples demonstrate that consistent linearization is essential to the accuracy and reliability of the incremental sensitivity formulation. It has the added benefit of computational efficiency, since it ensures that the system matrix appearing in the incremental sensitivity equations is identical to the consistent tangent stiffness matrix from the final equilibrium iteration of the time step. Thus, the reduced form of this matrix can be retained from the final equilibrium iteration of a load step for immediate use in the sensitivity analysis. Since the proposed sensitivity analysis method requires only a modest increase in computational expense over the cost of simulation, it should be feasible to apply it to practical analysis and design problems.

The proposed formulation could be used in a number of practical applications in structural and machine design as well as in a broad range of new research initiatives involving design optimization, identification problems and other inverse problems with path-dependent materials. Explicit integration techniques are often too expensive for these applications. In such cases, implicit integration methods, such as those presented here, are required for practical simulation and sensitivity analysis.

Acknowledgment

The authors would like to thank the Conselho National de Desenvolvimento Cientifico e Tecnologico of Brazil, the National Center for Supercomputing Applications, AT&T, the Center for Supercomputing Research and Development, CRAY Research Inc., and the National Science Foundation for their financial support.

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Documents

Design sensitivity analysis for rate-independent elastoplasticity