A Generalized Finite Element Method with Global

A Generalized Finite Element Method with Global-Local Enrichment Functions for Confined Plasticity Problems D.-J. Kima∗, C.A. Duarteb , S.P. Proencac a Department

of Architectural Engr., Kyung Hee University,

Engineering Building, 1 Sochon-Dong Kihung-Gu, Yongin, Kyunggi-Do, Korea 446-701 b Department

of Civil and Environmental Engr., University of Illinois at Urbana-Champaign,

Newmark Laboratory, 205 North Mathews Avenue, Urbana, Illinois 61801, USA c

Structural Engineering Department, School of Engineering at São Carlos - University of São Paulo, Av.Trabalhador São-carlense, 400 São Carlos, São Paulo, CEP 13566-590 , Brazil

Abstract The main feature of partition of unity methods such as the generalized or extended finite element method is their ability of utilizing a priori knowledge about the solution of a problem in the form of enrichment functions. However, analytical derivation of enrichment functions with good approximation properties is mostly limited to two-dimensional linear problems. This paper presents a procedure to numerically generate proper enrichment functions for three-dimensional problems with confined plasticity where plastic evolution is gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with the local solutions through the partition of unity method framework. This approach can produce accurate nonlinear solutions with a reduced computational cost compared to standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing localized nonlinear behavior. Several three-dimensional nonlinear problems based on the rate-independent J2 plasticity theory with isotropic hardening are solved using the proposed procedure to demonstrate its robustness, accuracy and computational efficiency.

1

Introduction

In many practical engineering applications, it is common to analyze complex structures which show mostly a linear elastic behavior, but exhibit confined plasticity in some small critical regions. The analysis of these types of problems using commercial finite element packages generally requires that loading history be divided into several load increments. The nonlinear equilibrium equations are then iteratively solved by a Newton-Raphson method at each load increment step. However, if the size of the problem is large or the loading history is complex, the computational cost may become too high, and thus a simplified ∗ Corresponding

author. E-mail: [email protected]

1

linear elastic analysis is often preferred in practice even though it produces less accurate results. Highperformance parallel machines may be utilized to facilitate this type of analysis, but the development of more efficient numerical methodologies that can take advantage of the small size of the nonlinear regions is also required to further reduce the computational cost. The Generalized or eXtended finite element methods (G/XFEM) [2, 3, 5, 8, 18, 19, 27, 29] is able to utilize a priori knowledge about the solution of a problem in the form of enrichment functions. Linear combination of partition of unity shape functions can reproduce exactly any enrichment function and thus their approximation properties are preserved. Due to this reproducing property, the GFEM has been applied to many classes of problems where a priori knowledge about the solution exists such as modeling of cracks [20], inclusions [28] and microstructures [26]. Early works on elasto-plastic and damage analysis of solids by Barros et al. [4] and Torres et al. [31] showed the viability of the GFEM applied to nonlinear analysis by utilizing only polynomial enrichment functions. Enrichment functions for nonlinear fracture analysis based on the Hutchinson-Rice-Rosengren (HRR) elastic-plastic fields were proposed by several researchers [13, 22, 23, 30]. However, they are not general enough to be applied for elastic-plastic problems with other nonlinear constitutive models and not readily applicable to three-dimensional problems. Problems involving confined plasticity like those considered here, can also be dealt with using, for example, the two-scale XFEM proposed in [21] and based on a mixed three-field weak formulation, or model reduction techniques like those proposed in [14]. In this paper, we propose a generalized finite element method with global-local enrichment functions (GFEM gl ) to generate proper enrichment functions for problems with confined plasticity where plastic evolution is gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with local solutions through the partition of unity framework. The methodology does not involve Lagrange multipliers, point constraints or any other modification on the variational principle used with standard finite element methods. This approach can produce accurate nonlinear solutions with a reduced computational cost compared to standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing local nonlinear behavior. The outline of this paper is as follows. In Section 2, a brief review of elasto-plastic analysis based on the rate-independent J2 flow theory and the isotropic hardening model is given. Section 3 presents a brief review of the GFEM and the weak formulation of a GFEM gl suited to elasto-plastic analysis with localized nonlinearities. Several three-dimensional nonlinear problems based on the rate-independent J2 plasticity theory with isotropic hardening are solved in Section 4 using the proposed procedure to demonstrate its robustness, accuracy and computational efficiency. The summary and conclusions are provided in Section 5.

2

Elasto-Plastic Problem Formulation

This section reviews some relevant concepts and formulation for the elasto-plastic analysis performed in this paper. It is based on the classical rate-independent J2 flow theory for small strains with isotropic hardening. ¯ G = ΩG ∪ ∂ ΩG in ℜ3 . The The strong form of the equilibrium equation is defined on a domain Ω 2

boundary is decomposed as ∂ ΩG = ∂ ΩuG ∪ ∂ ΩσG with ∂ ΩuG ∩ ∂ ΩσG = 0. / The equilibrium equations are given by in ΩG , (1) ∇ · σ +bb = 0 where σ is the second-order stress tensor and b is a body force vector. The following boundary conditions are prescribed on ∂ ΩG

σ · n = t¯ on ∂ ΩσG ,

u = u¯ on ∂ ΩuG

(2)

where n is the outward unit normal vector to ∂ ΩσG and t¯ and u¯ are prescribed tractions and displacements, respectively. The total strain tensor is expressed as the symmetric part of the displacement gradient tensor 1 ε = (∇uu + (∇uu)T ), 2

(3)

and is based on the additive decomposition into elastic and plastic parts as

ε = ε e + ε p.

(4)

The equations for the classical rate-independent J2 flow theory for small strains with isotropic hardening are summarized in Table 1 by following [24]. The stress tensor σ depends only on the elastic strain tensor ε e and is given by 1 e σ = C : ε = κ[11 ⊗ 1 ] + 2µ I + 1 ⊗ 1 : ε e, (5) 3

where κ denotes the bulk modulus, µ represents the shear modulus, I is the fourth-order unit tensor and 1 the second-order unit tensor. Admissible stress states are defined by the von Mises yield condition expressed by r 2 K(α ) ≤ 0, f (σ , α ) = ||dev[σ ]|| − (6) 3

where || · || denotes the Euclidean norm of a tensor, dev[·] the deviatoric part of a tensor and α is an internal state variable. We use an isotropic hardening modulus function given by K(α ) = σyield + hα ,

(7)

where σyield is the initial yield stress and h is a plastic hardening modulus. A summary of the constitutive model is provided in Table 1. The parameter γ represents the plastic flow rate that satisfies the consistency and Kuhn-Tucker conditions presented in the table. Refer to [24] for more details about the consistency and Kuhn-Tucker conditions. The rate-independent elasto-plastic boundary value problem described above exhibits a nonlinear behavior due to the nonlinear constitutive relationship depending on strain history. In this work, a predictorcorrector scheme is adopted to impose the constitutive model, whereas an incremental Newton-Raphson procedure is used to iteratively verify a linearized form of the equilibrium condition. In the context of the incremental form of the problem, each incremental step can be defined by a pseudo time step tn , where n denotes an index for each step. We employ the radial return mapping algorithm (RRA) [25] that can deliver

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Table 1: Classical J2 flow theory with isotropic hardening. 1. Elastic stress-strain relationship: 2. Elastic domain in stress space: 3. Flow rule and hardening law:

4. Kuhn-Tucker complementarity conditions: 5. Consistency condition:

σ = C : (ε − ε p ) IE σ = {((σ , α )| f (σ , α ) ≤ 0} dev[σ ] ε˙ p = γ ||dev[ r σ ]|| 2 α˙ = γ 3 γ ≤ 0, f (σ , α ) ≤ 0, γ f (σ , α ) = 0 γ f˙(σ , α ) = 0

σ n+1 consistent with the yield condition for any given strain increment ∆ε = ε n+1 − ε n . The consistent tangent operator (CTO) is constructed to relate incremental strains and stresses. The CTO was first proposed by Simo and Taylor in the FEM context [25] and also applied to the BEM [7]. Further details on the elasto-plastic analysis can be found in, e.g., [24].

3

A Generalized Finite Element Method with Global-Local Enrichment Functions for Confined Plasticity Problems

This section describes the basic concepts of the generalized finite element method (GFEM) and a procedure to construct the so-called global-local enrichment functions for confined plasticity analysis. The main features of these functions and their possible improvements are also discussed.

3.1 Generalized Finite Element Approximations The construction of the generalized finite element approximations is briefly reviewed in this section. Further details can be found in many references, for example, [3, 10, 15, 19, 27]. The generalized FEM is an instance of the so-called partition of unity method [3, 11, 19]. The generalized finite element shape functions, φα i , in this class of methods are built from the product of a partition of unity, ϕα , and enrichment functions, Lα i

φα i := ϕα Lα i

i ∈ I (α )

(no sum on α ),

(8)

where ϕα , α = 1, . . . , N, N being the number of functions, constitutes a partition of unity, i.e., a set of functions defined in a domain Ω with the property that ∑αN =1 ϕα (xx) = 1 for all x in Ω. The index set of the enrichment functions at a vertex node x α is denoted by I (α ). The support of ϕα , {xx : ϕα (xx) 6= 0}, is denoted by ωα . In the generalized finite element method, the partition of unity is, in general, provided by linear Lagrangian finite element shape functions. The support ωα of ϕα is then given by the union of the finite elements sharing a vertex node x α . The resulting shape functions are called generalized finite element shape functions. The main advantage of the generalized FEM is that it has a great freedom in selecting enrichment 4

functions Lα i . As an example, Heaviside (or Step) and Westergaard (or Branch) functions can be used as enrichment functions to accurately describe the discontinuity and singularity in the interior of a finite element caused by the presence of a crack [6, 8, 18, 29]. Enrichment functions that are solutions of local boundary value problems can be used as well [9]. These so-called global-local enrichment functions are very effective when solving a problem where limited a priori information about its solution is available. A partition of unity-based approximation of a scalar field u(xx) defined on a domain Ω ⊂ R I n , n = 1, 2, 3, can be written, using shape functions (8), as N

uh (xx) =

∑ ∑

α =1 i∈I (α )

u α i φα i (xx) =

N

∑ ϕα (xx)uhα (xx),

(9)

α =1

where u α i , α = 1, . . . , N, i ∈ I (α ), are nodal degrees of freedom and uhα (xx) := ∑i∈I (α ) u α i Lα i (xx), denotes a local approximation of the field u(xx) defined on ωα and belonging to the local space

χα (ωα ) = span{Lα i (xx)}i∈I (α )

(10)

where Lα i , i ∈ I (α ), are basis or enrichment functions and Lα 1 = 1. The GFEM function (9) can be used with a Galerkin method to find an approximate solution to a boundary value problem, following the same steps as in the standard FEM. This procedure leads to a system of equations for the unknown degrees of freedom u α i .

3.2 Global-Local Enrichment Functions for Confined Plasticity Analysis In this section, we present a global-local procedure to build enrichment functions for a class of problems where nonlinear behavior is locally exhibited. In particular, we focus on three-dimensional elasto-plastic problems where plastic strains are confined to small regions of the analysis domain. The global-local procedure to build enrichment functions described in this section is conceptually similar to the one introduced in [17] (denoted by GFEM gl ), but it requires modifications to be suitable for the analysis of elasto-plastic problems with localized plastic regions. The main concept of the approach is to numerically build functions that can accurately describe localized nonlinear behavior and use them to enrich the global solution space. The procedure comprises three main steps as illustrated in Figure 1. In the first step, the global problem subjected to the boundary conditions at the final pseudo time step is solved assuming a linear elastic constitutive model. This step is illustrated in Figure 1(a). Next, a local problem containing a small critical region exhibiting nonlinear behavior is constructed and solved using the global solution computed at the previous step as boundary conditions. A Newton-Rhapson iterative algorithm is used to solve this nonlinear local problem as illustrated Figure 1(b). The local solution is able to approximate the localized nonlinear features of the problem, in particular, at the final pseudo time step. At the third step, the global problem is enriched with the nonlinear local solution and solved incrementally using a nonlinear constitutive model as illustrated in Figure 1(c). Details of the three steps are discussed with precise definition of the weak formulation in the following subsections. The following notational conventions are adopted to derive the formulations of the nonlinear GFEM gl . Subscripts G and L represent global and local problems, respectively. An additional subscript lin indicates a solution obtained using a linear elastic constitutive law. If a subscript lin is not used, it represents a 5

Figure 1: Algorithm of the elasto-plastic GFEM gl . case where an elasto-plastic constitutive relationship is used. A superscript n is an index to represent each pseudo time step t n , n = 1, ..., nmax where nmax is the maximum number of pseudo time steps.

3.3 Linear Initial Global Problem max The generalized FEM approximation u nG,lin is the solution of the problem defined by (1) and (2) at the final n max pseudo time step t and obtained by assuming a linear elastic constitutive relationship represented only by the bulk and shear moduli (κ and µ ) in (5). It is the solution of the following problem: max Find u nG,lin ∈ X G,lin (ΩG ) ⊂ H 1 (ΩG ) such that, ∀ v G,lin ∈ X G,lin (ΩG )

Z

Z

ΩG

max σ (uunG,lin ) : ε (vvG,lin )dxx + η

∂ ΩσG

t¯nmax · vG,lin dss + η

Z

∂ ΩuG

Z

∂ ΩuG

max u nG,lin · v G,lin dss =

u¯ nmax · vG,lin dss,

(11)

where X G,lin (ΩG ) is a discretization of H 1 (ΩG ), a Hilbert space defined on ΩG , built with polynomial generalized FEM shape functions ) ( X G,lin (ΩG ) =

u hp =

NG DL

∑ ∑ ϕα (xx) u α i Lα i (xx)

,

(12)

α =1 i=1

where u α i , α = 1, . . . , NG , i = 1, . . . , DL , are nodal degrees of freedom and DL is the dimension of a set of polynomial enrichment functions, Lα i (xx) defined at each node α . Details on the construction of polynomial GFEM shape functions can be found, e.g., in [8, 19]. The parameter η in (11) is a penalty parameter. The mesh needed to solve problem (11) is typically coarse and quasi uniform. This problem corresponds to step (a) shown in Figure 1 and is denoted as the linear initial global problem for convenience.

6

3.4 Nonlinear Local Problem Let ΩL denote a subdomain of ΩG that contains local regions exhibiting nonlinear behavior. The following local problem is solved on ΩL for the final pseudo time step t nmax by assuming the nonlinear constitutive max relationship described by equations (5), (6) and (7) and Table 1 after the linear global solution u nG,lin is computed as described above. Find unLmax ∈ X L (ΩL ) ⊂ H 1 (ΩL ) such that, ∀ vL ∈ X L (ΩL ) Z

ΩL

κ η

Z

σ (uunLmax )

: ε (vvL )dxx + η

∂ ΩL \(∂ ΩL ∩∂ ΩG )

Z

∂ ΩL ∩∂ ΩuG

Z

∂ ΩL ∩∂ ΩuG

u nLmax · v L dss =

u¯ nmax · v L dss +

Z

Z

u nLmax · v L dss +

∂ ΩL ∩∂ ΩσG

∂ ΩL \(∂ ΩL ∩∂ ΩG )

t¯nmax · v L dss + max max (tt (uunG,lin ) + κ u nG,lin ) · v L dss,

(13)

where X L (ΩL ) is a discretization of H 1 (ΩL ), a Hilbert space defined on ΩL , built with polynomial generalized FEM shape functions. It is the analogous to space X G,lin (ΩG ) but defined on ΩL and based on a finer mesh and higher order polynomial enrichment functions than those used in X G,lin (ΩG ). The parameter η is a penalty parameter. In all numerical examples presented in Section 4, local nonlinear behavior is fully confined to the interior of ΩL . This is guaranteed by referring to the von Mises stress distribution of the linear initial global solution and selecting the size of the local domain ΩL such that it can fully contain the region of the global domain where von Mises stress is greater than the initial yield stress. A more detailed description of this procedure max is provided in the numerical examples of Section 4. The traction vector, t (uunG,lin ), that appears in the integral over ∂ ΩL \(∂ ΩL ∩ ∂ ΩG ) can be computed as presented in [17], max max max C : ε (uunG,lin t (uunG,lin ) = nˆ · σ (uunG,lin ) = nˆ · (C )),

(14)

where nˆ is the outward unit normal vector to ∂ ΩL . In this case, the spring stiffness, κ , can be selected by following the approach for the linear elastic case introduced in [17] as follows : E √ , κ = nd V0 J

(15)

where E is the Young’s modulus, nd is the number of spacial dimensions of the problem, V0 is the volume of the master element used and J is the Jacobian of the global element across the local boundary where the spring boundary condition is imposed. max The key aspect of this problem is that the solution of the linear initial global problem, u nG,lin , is used as boundary condition on ∂ ΩL \(∂ ΩL ∩ ∂ ΩG ) and the solution of this problem is able to describe local nonlinear behavior on ∂ ΩG at the final pseudo time step. Furthermore, this procedure can be fully parallelized and high scalability can be achieved as demonstrated in [16]. This problem is hp-adapted to obtain accurate solutions and is illustrated in step (b) of Figure 1. We denote this problem as the nonlinear local problem for convenience.

7

3.5 Global-Local Enrichment Functions for Nonlinear Global Problem The linear initial global and nonlinear local problems described in Sections 3.3 and 3.4 are preliminary stages to provide nonlinear global-local enrichment functions, u nLmax , to the nonlinear global problem. After this step, nonlinear iterations are performed to obtain the solution of the global problem at each pseudo time step. The nonlinear global problem enriched with u nLmax is solved to obtain its solution u nG as illustrated in Figure 1(c). The global solution space (12) is augmented with the following global-local GFEM shape functions φ α (xx) = ϕα (xx)uunLmax (xx) (16) The weak formulation of the nonlinear global problem at t n enriched with u nLmax is given by Find u nG ∈ X G (ΩG ) ⊂ H 1 (ΩG ) such that, ∀ v G ∈ X G (ΩG ) Z

Z

ΩG

σ (uunG ) : ε (vvG )dxx + η

∂ ΩσG

t¯n · v G dss + η

Z

∂ ΩuG

Z

∂ ΩuG

u nG · v G dss =

u¯ n · v G dss,

(17)

where X G (ΩG ) is the space X G,lin (ΩG ) augmented with GFEM functions (16) and given by               NG DL gl hp X G (ΩG ) = u = ∑ ∑ ϕα (xx) u α i Lα i (xx) + ∑ ϕβ (xx)uuβ (xx) ,     α =1 i=1 β ∈Igl   {z } |   {z } |     global approx.

(18)

local approx.

where Igl is the index set of nodes enriched with function u nLmax and 

 nmax x u gl β 1 uL1 (x )  gl  u βgl (xx) =  u β 2 unL2max (xx)  , nmax x u gl β 3 uL3 (x ) nmax x where u gl β j , β ∈ Igl , j = 1, 2, 3, are nodal degrees of freedom. uL j (x ), j = 1, 2, 3, are the Cartesian nmax components of displacement vector u L .

This problem is denoted as the nonlinear enriched global problem. It is solved on the same coarse global mesh at each pseudo time step t n . Nonlinear global-local enrichment functions add only three degrees of freedom to each node β ∈ Igl of the coarse global mesh when solving a three-dimensional elasto-plastic problem regardless of the number of degrees of freedom of the local problem. Thus, highly adapted local discretizations can be used in the local problem to capture fine-scale features of the solution since the level of local mesh refinement/enrichment does not affect the size of the global problem. In the proposed GFEM gl , the nonlinear local solution customized for the final pseudo time step nmax is used as enrichment functions for the nonlinear global problem at every pseudo time step. Only the nonlinear local solution is used in the global problem as enrichment. State variables from the local problem are not

8

transfered or shared with the global problem. Each problem has its own set of state variables stored at their integration points as in the standard FEM. This approach can be justified if the plastic evolution in the region enriched with global-local functions is gradual, as in the analysis of problems with hardening type material models and monotonic loading. A more general algorithm, in which global local enrichment functions used in the global problem are updated at several pseudo time steps is also conceivable. However, such an approach may be more computationally demanding than the approach introduced in this paper, which shows high accuracy for hardening type confined plasticity analysis, as demonstrated in Section 4. Consequently, the exploration of the more general algorithm will be our future research topic and used to analyze other types of nonlinear problems where plastic evolution is localized but not gradual.

4

Numerical Experiments

In this section, we perform numerical experiments to investigate the effectiveness of the proposed GFEM gl by focusing on three-dimensional elasto-plastic mechanics problems with confined plasticity. Numerical examples are analyzed using both of the hp-GFEM and the proposed GFEM gl and their solutions are compared to evaluate the accuracy and computational efficiency of the GFEM gl . Here, the hp-GFEM corresponds to the case in which the global mesh is directly refined and enriched only with the polynomial enrichment functions described in [8] along with the implementation of the elasto-plastic material model introduced in Section 2. In all of the numerical examples of this section, the relative norm of the residual with respect to its initial value at the beginning of each pseudo time step is used as a tolerance criterion for the convergence of Newton-Rhapson iterations discussed in Section 2.

4.1 Bi-Material Bar Example The first example is a bar with two material interfaces. It is discretized with a uniform mesh of 12 ∗ (1 × 7 × 1) tetrahedron elements as shown in Figure 2. The mesh has a total of 84 elements. Only the element faces at the boundary of the domain are shown in the figure. A layer of material in the middle of the bar has a smaller initial yield stress than the material in the rest of the bar as indicated in the figure. Thus, yielding of material occurs only in that area as the external load increases and plastic strain is confined in a small region of the domain. The following geometric parameters are used in this example: Bar dimensions h = 7.0, a = 1.0, b = 1.0; length of the middle layer M = 1.0. A constant traction ty = 7.0 is applied at the top and bottom of the domain and six point constraints are provided as illustrated in the figure in order to prevent rigid body motions. Table 2: Material parameters of the isotropic hardening model used in the bi-material bar example. Material parameters Young’s modulus (E) Poisson’s ratio (ν ) Initial yield stress (σyield ) Plastic modulus (h)

Middle layer 4.0 0.0 4.0 0.2

Outer layers 4.0 0.0 40.0 1.0

A linear isotropic hardening model is used to define material properties of the model and the material 9

ty

layer with smaller initial yield stress

M

h

y

z

ty a

x

b

Figure 2: Description of a bi-material bar example. parameters are listed in Table 2. The tolerance for Newton-Rhapson convergence is taken as 10−4 . Fourteen uniform pseudo time steps are used for incremental analysis, which means that the load increment at each pseudo time step is 0.5. As a measure for the accuracy of solutions, we use the y-displacement measured at the center of the top face of the bar, which is equal to the maximum y displacement. For verification of correctness of the elasto-plastic implementation, the results of this example obtained by the hp-GFEM are compared with those of ANSYS [1], which is a widely used commercial finite element analysis package. Figure 3 shows the load-displacement curves computed by the hp-GFEM and ANSYS with quadratic approximations on the coarse mesh shown in Figure 2. It indicates that the two curves obtained by the two methods match exactly. Their behavior also coincides with the physical interpretation of the problem. Since the sectional area of the bar is 1.0, the external force is the same as the magnitude of the traction. Therefore, the middle layer must start to yield at the beginning of the ninth pseudo time step (or at the end of the eighth pseudo time step) at which the von Mises stress of the domain reaches the initial yield stress, which is equal to 4.0. The curves in the figure exactly follow this description. They initially show a linear elastic behavior and the slope is equal to the value of the Young’s modulus (E) divided by the bar length (h) (= 4.0/7.0). It then drops after reaching the initial yield stress. Due to the simple geometry of the domain and bilinear isotropic hardening rule used in this model, the load-displacement history also shows an almost bilinear behavior. This confirms that the implementation of the elasto-plastic material model is correctly done. The procedure to analyze the bi-material bar example using the GFEM gl described in Section 3 is illustrated in Figure 4. In the first step, the global problem is solved on a coarse mesh by assuming a linear elastic material model as illustrated in Figure 4(a). The second step involves the creation and solution of 10

8 7 6

External load

5 4 3 2 Linear solution

1

Hp-GFEM ANSYS

0

0

2

4

6

8

10

12

14

16

18

20

22

24

26

Maximum y displacement

Figure 3: Nonlinear load-displacement curves for the verification of the elasto-plastic implementation.

Provide BCs

(a) Linear initial global problem

Provide Enrichment

(b) Nonlinear local problem

(c) Nonlinear enriched global problem

Figure 4: Nonlinear solution of the bi-material bar example using the proposed GFEM gl . the nonlinear local problem and is described in Figure 4(b). A local domain is extracted from the coarse global mesh. It fully contains the region with plastic strains as discussed in Section 3.4. The solution of the coarse global problem is used to prescribed boundary conditions for the local problem in terms of spring boundary conditions. The local mesh is refined to obtain an accurate nonlinear solution. The spring stiffness is selected using Equation (15) which gives κ = 9.158. By assuming the linear isotropic hardening models given in Table 2, the local problem is solved for the final pseudo time step through Newton-Rhapson iterations. Finally, the computed nonlinear local solution is used to enrich the coarse global problem. Only eight global nodes indicated by spheres in Figure 4(c) are enriched. Like in the local problem, the nonlinear 11

constitutive model is adopted and the enriched global problem is solved by incremental analysis. Table 3: Solutions of the bi-material bar example computed by the hp-GFEM and GFEM gl . Time step 1 2 3 4 5 6 7 8 9 10 11 12 13 14

External load 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

Max. y-displacement hp-GFEM GFEM gl Ref. 0.875 0.875 0.875 1.750 1.750 1.750 2.625 2.625 2.625 3.500 3.500 3.500 4.375 4.375 4.375 5.250 5.250 5.250 6.125 6.125 6.125 7.000 7.000 7.000 9.618 9.616 9.646 12.366 12.346 12.425 15.182 15.120 15.271 18.037 17.914 18.160 20.916 20.717 21.092 23.811 23.526 24.073

8 7 6

External load


Hp-GFEM

1

GFEM

gl

Ref.

0

0

2

4

6

8

10

12

14

16

18

20

22

24

26


Figure 5: Comparison of nonlinear load-displacement curves computed by hp-GFEM and GFEM gl . Table 3 lists the maximum y-displacement at each pseudo time step computed using both of the hpGFEM and GFEM gl described above. Quadratic shape functions are used in both of the global and local domains to obtain the hp-GFEM and GFEM gl solutions. In both analyses, the same level of mesh refinement is performed in the middle layer where yielding of material occurs as external load increases. The reference solution is included in the table to evaluate the accuracy of solutions computed by the two approaches. It 12

(a) Hp-GFEM

(b) GFEM

gl

(c) Reference

Figure 6: Plastic strain distributions at the final pseudo time step. is obtained using the hp-GFEM with cubic approximation on a highly refined mesh over the whole the domain. The results in the table are plotted in Figure 5. It can be noted from the results that the GFEM gl solution is close to that of the hp-GFEM at all pseudo time steps, which confirms that highly accurate solutions can be obtained by the proposed GFEM gl using coarse global meshes. Figure 6 shows the global mesh discretizations and plastic strain distributions at the final pseudo time step for the three cases discussed above. The plastic strain distributions of the hp-GFEM and GFEM gl solutions are quite similar as expected from the results of the table and the plot shown in Figure 5. Figure 7 shows the performance of the proposed generalized finite element method for non-proportional loading. Smaller pseudo time step intervals are used around the kink in the load-displacement curve where plasticity starts to evolve in this analysis. It can be noted from the results in the figure that the same load-displacement curve is obtained as in the case of the uniform load increment and the proposed method performs well with non-proportional loading.

4.2 L-Shaped Domain As a second example, we analyze an L-shaped domain subjected to a constant vertical traction as illustrated in Figure 8. Under the given loading condition, yielding of material occurs around the re-entrant corner of the domain as the external load increases. The length L indicated in the figure is taken as 50.0 and the thickness of the domain b is 10.0. A uniform vertical traction ty = 10.0 is applied at the free end on the right side of the domain and the top of the domain is fixed. A linear isotropic hardening model is used to define material properties of the model and the material parameters are given as follows: Young’s modulus E = 40.0; Poisson’s ratio ν = 0.0; Initial yield stress σyield = 120.0; Plastic modulus h = 10.0. Ten uniform pseudo time steps are used for incremental analysis. Thus the load increment at each pseudo time step is 0.8. The tolerance limit for the convergence of NewtonRhapson incremental analysis is 10−4 . As a measure to compare the quality of the solutions computed by 13

8 7 6

External load


Hp-GFEM

1 0

GFEM

0

2

4

6

8

10

12

14

16

18

20

gl

22

24

26


Figure 7: Performance of the proposed generalized finite element method for non-proportional loading. b L

y L x

z L

L

ty

Figure 8: Description of the L-shaped domain example. the hp-GFEM and GFEM gl , E εyy /σyield is used. The total strain in y direction, εyy , is measured at point (50.0, 50.1, 5.0) to capture localized nonlinear behavior near the re-entrant corner of the L-shaped domain. These coordinates are calculated by assuming that the origin of the coordinate system is located at the back of the lower left corner of the domain. The GFEM gl solution is computed following the same steps described in the previous section as illus-

14

Provide BCs


Provide Enrichment



Figure 9: Nonlinear solution of the L-shaped domain example using the proposed GFEM gl . The mesh refinement performed in the local problem corresponds to Level 3 indicated in Table 4.

(a)

(b)

(c)

Figure 10: Selection of the local domain based on von Mises stress distribution in the initial global problem p (a). Plastic strain component εyy in the local (b) and enriched global (c) problems at the final pseudo time step are also shown. trated in Figure 9. The linear initial global problem is solved on the coarse global mesh shown in Figure 9(a) and the local domain is created around the re-entrant corner of the domain where plastic strain is concentrated. The size of the local domain is determined from the von Mises stress distribution of the initial global solution such that it can contain the region where von Mises stress of the initial global solution is greater than the initial yield stress as shown in Figure 10(a). Figure 10(b) illustrates the plastic strain distribution of the nonlinear local solution, which clearly indicates that the plastic strain is fully confined inside the local domain. Since the global mesh is not refined enough to accurately capture the nonlinear behavior around the re-entrant corner of the domain, a mesh refinement of the local domain is performed to obtain accurate solutions as illustrated in Figure 9(b). The solution of the linear initial global problem is applied 15

in terms of spring boundary conditions. The spring stiffness is selected using Equation (15) which gives κ = 7.268. As Figure 9(c) indicates, the solution of the nonlinear local problem computed for the final pseudo time step is used to enrich 16 global nodes around the re-entrant corner and the nonlinear global problem is solved by incremental analysis. Figure 10(c) displays the plastic strain distribution of the nonlinear global solution computed with the local solution enrichment shown in Figure 10(b). Figure 11 plots E εyy /σyield of the enriched global solution for several spring stiffness values. It indicates that the quality of the nonlinear global solution is not much affected by the choice of spring stiffness and the use of spring boundary conditions for the nonlinear local problem is able to produce quite robust results. 4.2

4.1

E*

yy

/

yield

4.0

3.9

3.8

3.7 0

100

200

300

400

500

600

700

800

900

1000

Spring stiffness

Figure 11: Sensitivity analysis of the spring stiffness used for boundary conditions on the local problem. To investigate the effectiveness of the proposed GFEM gl , three different levels of mesh refinement are performed in the local problem and the nonlinear global solutions computed with these local enrichment functions are compared with hp-GFEM solutions obtained by performing the same level of mesh refinement directly performed on the global mesh. Table 4 lists the number of degrees of freedom required by the hp-GFEM and GFEM gl analyses with quadratic approximation for the three different levels of mesh refinement. It can be noted from the table that only 48 more degrees of freedom are added to the nonlinear global problem by local solution enrichments regardless of the increase in the size of the nonlinear local problem. In contrast, the size of the problem increases with mesh refinement in the hp-GFEM analyses. Table 4: Number of degrees of freedom required for hp-GFEM and GFEM gl analyses with three different levels of mesh refinement. Level of uniform mesh refinement 1 2 3

hp-GFEM Initial global 2,676 3,828 6,948

2,304

16

GFEM gl Local Enriched global 852 2,004 2,352 5,124

The nonlinear solutions computed by the two methods for the three levels of mesh refinement are listed in Table 5. The reference solution is provided by the hp-GFEM with cubic approximation and Level 3 mesh refinement. These results are plotted in Figure 4.2. It indicates that the solutions of the GFEM gl are more accurate than the hp-GFEM solutions for all three levels of refinement cases. For example, the ratios between the GFEM gl solution and the reference value at the final pseudo time step are 78.2, 82.9 and 94.2 (%) for mesh refinement levels 1, 2 and 3, respectively, while the corresponding values for the hp-GFEM analyses are 59.0, 64.1 and 79.6 (%). The last row of the table provides the results that are obtained by using only one pseudo time step in the (enriched) global problem (one-shot analysis). This kind of analysis can be performed if the whole load-displacement history is not required. The values are almost identical to the results at the tenth pseudo time step of the table and the small differences between them seem to be caused by the accumulation of errors in residuals at each increment step. Figure 4.2 shows the von Mises stress distributions for Level 3 mesh refinement at the final pseudo time step computed by the hp-GFEM and GFEM gl . The quality of the GFEM gl solution is comparable to or even better than that of the hp-GFEM solution even though a coarse global mesh is used for the GFEM gl analysis. Table 5: Solutions of the L-shaped domain example computed by the hp-GFEM and GFEM gl . Time ty /σyield step 1 0.0083 2 0.0167 0.0250 3 4 0.0333 0.0417 5 6 0.0500 0.0583 7 0.0667 8 9 0.0750 10 0.0833 One-shot

E εyy /σyield Level 1 0.206 0.413 0.619 0.826 1.032 1.259 1.527 1.816 2.137 2.480 2.478

hp-GFEM Level 2 0.222 0.444 0.666 0.888 1.110 1.371 1.669 1.992 2.337 2.695 2.693

Level 3 0.266 0.531 0.797 1.066 1.372 1.736 2.126 2.537 2.960 3.349 3.339

GFEM gl Level 1 Level 2 Level 3 0.259 0.275 0.301 0.518 0.551 0.603 0.777 0.826 0.904 1.036 1.104 1.244 1.334 1.431 1.622 1.675 1.800 2.035 2.043 2.197 2.479 2.440 2.607 2.940 2.855 3.037 3.436 3.290 3.486 3.963 3.293 3.482 3.966

Ref. 0.309 0.617 0.926 1.283 1.714 2.177 2.663 3.163 3.643 4.207

Table 6 presents the numbers of Newton-Rhapson iterations required at each pseudo time step for the three mesh refinement cases in the hp-GFEM and GFEM gl analyses to evaluate the computational costs required by the two approaches. The last row of the table provides the numbers of the iterations required for the one-shot analysis discussed above. It is expected that the hp-GFEM has a computational cost close to the standard FEM when solving confined plasticity problems since both methodologies utilize basically the same polynomial solution space and share the same weakness that the size of a problem is directly proportional to the level of local p or h extension. In contrast, in the proposed GFEM gl , p or h extension performed in the local problem does not impact the size of the global problem, thus computationally intensive nonlinear iterations can be performed on a coarse global mesh. It can be noted from the table that the total number of iterations required for the GFEM gl analyses is only slightly greater than that of the hp-GFEM analysis. This seems to be caused by the fact that the size of the plastic zone is slightly larger in the GFEM gl analysis than in the hp-GFEM analysis as indicated by 17

0.10

0.09

0.08

0.07

0.05

/

yield

0.06

t

y

Hp-GFEM (Level 1)

0.04

GFEM

gl

(Level 1)

Hp-GFEM (Level 2) 0.03

GFEM

gl

(Level 2)

Hp-GFEM (Level 3) 0.02

GFEM

gl

(Level 3)

Ref. 0.01

0.00 0.0

0.5

1.0

1.5

2.0

E*

yy

2.5

/

3.0

3.5

4.0

4.5

yield

Figure 12: Comparison of nonlinear load-displacement curves computed by hp-GFEM and GFEM gl .

(b) GFEM

(a) Hp-GFEM

gl

Figure 13: Von Mises stress distributions of the L-shaped domain example for the final pseudo time step computed by the hp-GFEM and GFEM gl . the results in the table and plot of Figure 4.2, which apparently results in more nonlinear iterations in the former than in the latter. Considering that the size of the global problem in the hp-GFEM analysis (6,948 DOFs) is almost three times larger than that of the GFEM gl analysis (2,352 DOFs) and the only additional cost of the GFEM gl analysis compared with the hp-GFEM analysis is the solution of the nonlinear local problem (5,124 DOFs for Level 3 mesh refinement), it can be concluded that the proposed GFEM gl can produce more accurate solutions than the hp-GFEM while it requires less computational cost. This will become even more clear when analyzing problems containing multiple localized nonlinear features.

18

Table 6: Numbers of Newton iterations required for the hp-GFEM and GFEM gl analyses of the L-shaped domain example.

Time step 1 2 3 4 5 6 7 8 9 10 One-shot

Number of Newton iterations hp-GFEM GFEM gl (G) GFEM gl (L) Level 1 Level 2 Level 3 Level 1 Level 2 Level 3 Level 1 Level 2 Level 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 2 3 1 1 3 3 3 3 3 3 3 3 3 3 3 4 4 3 3 3 3 3 4 3 3 4 3 4 4 4 4 4 4 4 4 3 4 6 4 4 4 4 4 6 4 4 4 3 3 3

4.3 Plate with a Hole In this section, we analyze a plate with a hole subjected to tension. Only half of the domain is modeled due to symmetry of geometry and boundary conditions are shown in Figure 14. P-extensions on a moderately refined mesh around the hole are very effective to solve this problem [12]. The mesh needs to be refined due to loss of solution smoothness at the elastic/elastic-plastic interfaces. Furthermore, the mesh must also be able to describe the circular geometry of the hole with sufficient accuracy. We investigate the effectiveness of the proposed GFEM gl in regard to this approach. Another purpose of analyzing this problem is to show that the GFEM gl is able to handle cases with relatively large plastic zones. The following parameters are assumed in this example: Plate dimensions h = 0.036, a = 0.010, b = 0.001; Radius of the hole r = 0.005. A constant traction ty = 15.0 is applied at the top and bottom of the domain and three point constraints are provided to prevent rigid body translations and rotations in addition to the symmetry boundary conditions as indicated in the figure. A linear isotropic hardening model is used to define material properties of the model. The material parameters are given as follows: Young’s modulus E = 7000.0; Poisson’s ratio ν = 0.2; Initial yield stress σyield = 24.3; Plastic modulus h = 1500.0. Twenty five uniform pseudo time steps are used in the incremental analysis, thus the load increment at each pseudo time step is 0.4. The tolerance for Newton-Rhapson convergence is taken as 10−3 . As a measure to compare the quality of the solutions computed by the hpGFEM and GFEM gl , E εyy /σyield and the absolute value of E εxx /σyield (—E εxx /σyield —) are used. The total strains in x and y directions, εxx and εyy , are measured at point A in Figure 14. This point is located at its symmetry line in x-direction and at z = b/2. The procedure to obtain the GFEM gl solution for this example is basically the same as before and is illustrated in Figure 15. The linear initial global problem is solved on the global mesh shown in Figure 15(a). The local domain is created in the region near the hole where plastic strains occur. The local domain

19

b ty

h/2

r A

O

h/2

y

ty

a

z

x

Figure 14: Description of a plate with a hole example. fully contains the region where the von Mises stress in the initial global solution is greater than the initial yield stress. The linear initial global solution is applied as spring boundary conditions with spring stiffness κ = 107 . The size of global elements are not constant as shown in Figure 15. Thus, this is representative value for κ selected using Equation (15) and used for elements located at the boundary of the local domain, since our current implementation uses a single value for the spring stiffness. The local domain is illustrated in Figure 15(b). The nonlinear solution of the local problem at the final pseudo time step is used to enrich global nodes around the hole as illustrated in Figure 15(c). The enriched global problem adopts the same nonlinear material law used in the local problem. A non-uniform p-enrichment strategy is performed in both the hp-GFEM and in the enriched global problem of the GFEM gl . In the GFEM gl analysis, a quadratic approximation is used in the local problem shown in Figure 15(b). Linear polynomial enrichments are employed in all nodes of the enriched global problem in addition to global-local enrichment functions used at the nodes indicated by spheres in Figure 15(c). In the hp-GFEM analysis, quadratic polynomial enrichments are used at the nodes with the globallocal enrichment functions in the GFEM gl analysis. Linear polynomial enrichments are used elsewhere. Table 7 lists the number of degrees of freedom required by the hp-GFEM and GFEM gl discretizations. The number of degrees of freedom for the hp-GFEM discretization with uniform linear p-enrichment in the whole domain is also included in the table. Abbreviations Gp and Lp in the table represent the polynomial orders of global and local (around hole) polynomial enrichments, respectively. Figures 16(a) and (b) plot E εyy /σyield and |E εx /σyield | versus ty /σyield curves for the three discretizations described above, respectively. For comparison purpose, a reference solution computed by the hp-GFEM

20

Provide BCs


Provide Enrichment



Figure 15: Nonlinear solution of the plate with a hole example using the proposed GFEM gl . Table 7: Number of degrees of freedom required for hp-GFEM and GFEM gl discretizations. Order of polynomial enrichment Uniform (Gp = 1) Non-uniform (Gp = 1; Lp = 2)

hp-GFEM Initial global 2,122 5,595

2,122

GFEM gl Local Enriched global 4,704

3,108

with a cubic approximation in the whole global mesh is also included. The figure shows that in the entire range of the plot, the GFEM gl solution is closer to the reference solution than that of the hp-GFEM. All discretizations show great improvement from the hp-GFEM solution computed with linear approximation. This indicates that the GFEM gl can produce highly accurate solutions and the local p-extension is a very effective strategy to handle this class of problems. This behavior can also be observed from the results p in the region around the hole at several in Figure 17, which displays the development of plastic strain εyy pseudo time steps computed by the two methods. It can be noted from the figure that the plastic distributions of the GFEM gl and hp-GFEM solutions are quite similar to each other at all the steps shown.

21

0.6

0.6

0.5

0.5

0.4

0.4

/ y

0.3

t

/

0.3

t

y

yield

0.7

yield

0.7

0.2

0.2

Hp-GFEM (p = 1)

Hp-GFEM (p = 1)

Hp-GFEM (Gp = 1; Lp = 2) GFEM 0.1

gl

Hp-GFEM (Gp = 1; Lp = 2)

(Gp = 1; Lp = 2)

GFEM 0.1

Ref.

0.0

gl

(Gp = 1; Lp = 2)

Ref.

0.0 0.0

1.0

2.0

3.0

E*

4.0

yy

/

5.0

6.0

7.0

0.0

0.5

1.0

1.5

|E*

yield

(a) E εyy /σyield

xx

/

2.0

yield

2.5

3.0

|

(b) |E εxx /σyield |

Figure 16: Comparison of nonlinear load-displacement curves computed by hp-GFEM and GFEM gl .

GFEM

gl

Hp-GFEM

(a) time step = 15

(b) time step = 20

(c) time step = 25

Figure 17: Plastic strain distributions of the plate with a hole example at several pseudo time steps computed by the hp-GFEM and GFEM gl .

5

Summary and Concluding Remarks

In this paper, we proposed a generalized finite element method with global-local enrichment functions to generate proper enrichment functions for problems with confined plasticity where plastic evolution is

22

gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with local solutions through the partition of unity framework used in the generalized finite element method. This approach can produce accurate nonlinear solutions with a reduced computational cost compared with standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing local nonlinear behavior. The effectiveness of the method was investigated in terms of accuracy and problem size through several three-dimensional examples exhibiting confined plasticity. The main conclusions of this paper are as follows. • The GFEM gl allows the numerical construction of enrichment functions for problems exhibiting localized nonlinear behavior where no or limited a-priori knowledge about their solutions is available. Thus it can bring the benefits of the generalized FEM to a broader class of problems; • The numerical examples analyzed in this paper show that the GFEM gl can provide nonlinear solutions with comparable to or even higher accuracy than those obtained by the hp-GFEM with reduced problem size; • The use of spring boundary conditions for the local problem is able to produce accurate solutions and is not sensitive to the choice of spring stiffness. A more general algorithm, in which global local enrichment functions used in the global problem are updated at several pseudo time steps is also conceivable. We are currently exploring variations of the algorithm proposed here with the goal of solving nonlinear problems where plastic evolution is not gradual. The following topics may also to be considered when extending the proposed methodology to other classes of non-linear problems: • Creation of optimal enrichment functions for the analysis of problems exhibiting global softening response (e.g. shear bands). In this class of problems, plastic evolution is not gradual and a single enrichment function most likely will not be able to produce accurate nonlinear global solutions at all pseudo time steps. Thus, the enrichment functions may need to be updated in order to achieve accurate solutions. Proper error indicators could be used to trigger this update; • Computation of solution residuals with updated shape functions: If the shape functions are updated at all (or at several) pseudo time steps as suggested above, the computation of the nonlinear solution at integration points can not be done as in the standard FEM. The solution vectors from different load steps can not be used in the computations since they have coefficients of different shape functions. Therefore, proper modifications to standard algorithms used to update the nonlinear solution needs to be developed; • Computational cost of the more general algorithm: It is clear that a more general algorithm will lead higher computational cost than the one introduced in this paper even though it is able to produce more accurate solutions for nonlinear problems where plastic evolution is not gradual. Thus, it is required to perform its cost analysis and to demonstrate its computational efficiency over the hp-GFEM or the standard FEM.

Acknowledgments: The first author acknowledges the support by a grant from the Kyung Hee University in 2010. (KHU-20101836) 23

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