COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. O˜ nate and E. Dvorkin (Eds.) c
CIMNE, Barcelona, Spain 1998
A DIRECTIONAL ERROR ESTIMATOR FOR ADAPTIVE FINITE ELEMENT ANALYSIS Lavinia Borges∗ , Ra´ ul Feij´ oo†, Claudio Padra†† and Nestor Zouain∗ ∗ COPPE/EE
- Mechanical Engineering Department Federal University of Rio de Janeiro Cx. Postal 68503 Rio de Janeiro,RJ,Brasil Cep: 21945.970 e-mail:
[email protected] and
[email protected] †Laboratrio Nacional de Computa¸c˜ao cient´ıfica (LNCC/CNPq) Lauro Muller 455 -Botafogo, Rio de Janeiro - RJ - Brasil Cep: 22290.160 E-mail:
[email protected] web page:www.lncc.br/˜tacsom ††Instituto Balseiro (CAB) Centro At´omico de Bariloche, 8400, Bariloche, Argentina. E-mail:
[email protected]
Key words: Finite Elements, Mesh Generation, Error estimator, Adaptive Analysis, Limit Analysis Abstract. We present an error estimator based on first- and second-order derivatives recovery for finite element adaptive analysis. At first, we briefly discuss the abstract framework of the adopted error estimation techniques. Some possibilities of derivatives recovery are considered, including the proposal of a directional error estimator. Using the directional error estimator proposed, an adaptive finite element analysis is performed which gives an adapted mesh where the estimated error is uniformly distributed over the domain. The advantages of adapting meshes are well known, but we place particular emphasis on the anisotropic mesh adaptation process generated by the directional error estimator. This mesh adaptation process gives improved results in localizing regions of rapid or abrupt variations of the variables, whose location is not known a priori. We apply the above abstract formulation to analyze the behaviour of the recovery technique and the proposed adaptive process for some particular functions. Finally, we apply the procedure to some finite element models for limit analysis.
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
1 INTRODUCTION The main objective of this paper it to present an error estimator based on firstand second-order derivatives recovery for finite element adaptive analysis, including the proposal of a directional error estimator. Using the proposed directional error estimator, we perform an adaptive finite element analysis, which gives an adapted mesh where the estimated error is uniformly distributed over the domain. The advantages of adapting meshes are well known, but we place particular emphasis on the anisotropic mesh adaptation process, generated by a directional error estimator based on the recovering of second derivatives of the finite element solution. The goal of this approach is to achieve a mesh-adaptive strategy accounting for mesh size refinement, as well as redefinition of the oriented element stretching. This way, along the adaptation process, the mesh turns aligned with the direction of maximum curvature of the function. This mesh adaptation process gives improved results in localizing regions of rapid or abrupt variations of the variables, whose location is not known a priori1,2,3,4 . So, we can obtain an accurate representation of shocks, boundary layers, wakes and other discontinuities. In the light of the abstract theory, it is presented an adaptive mesh-refinement process for some classical examples in limit analysis. A similar approach is adopted in applications of Computacional Fluid Dynamics problems by Regina C. de Almeida et al. in the paper “Adaptive Finite element Computacional Fluid Dynamics using an Anisotropic Error Estimator”, also presented in this event. Limit analysis deals with the direct computation of the load producing plastic collapse of a body - a phenomenon where, under constant stresses, kinematically admissible plastic strain rates take place. Localized plastic deformations or slip bands are present in most collapse situations (see for example Borges et al.18 and papers therein). Accuracy in the numerical solution of limit analysis is seriously affected by local singularities arising from these localized plastic deformations. One possible approach in order to overcome this problem is to add more grid-points where the solution presents those singularities. So, it becomes necessary not only to identify these regions, but also to obtain a good equilibrium between the refined and unrefined regions for an optimal overall accuracy5 . In limit analysis, an a priori error estimate, as provided by the standard error analysis in the finite element method, is often insufficient to assure reliable estimates of the computed solution accuracy. This is due to the fact that it only yields information on the asymptotic error behaviour and requires regularity conditions of the solution, which are not satisfied in the presence of singularities such as the abovementioned ones. Those facts disclose the need of an estimator which can a posteriori be extracted from the computed numerical solution. We discuss in this paper, firstly, the abstract framework of the adopted error estimation techniques, by considering the derivatives recovery and the proposal of a directional error estimator. Together with limit analysis applications, we also select some adaptive mesh-refinement solutions for interpolation problems in order to show that the pro-
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
posed adaptive strategy using our anisotropic error estimator recovers optimal and/or superconvergence rates. 2 ESTIMATORS BASED ON DERIVATIVES RECOVERY In recovery based error estimation methods the gradients and/or Hessians of solutions, obtained on a given mesh, are smoothed and after that the smoothed solution is used in error estimation. It is well known that the derivatives of the uh function is superconvergent in some interior points of the mesh elements, that is, in these points the derivatives of the finite element solutions exhibit higher accuracy than normally expected. Taking advantage of superconvergence, the main idea of the proposed estimator is focused on the recovering of the Hessian with a higher order of accuracy than that naturally obtained from the finite element approximation. So, in order to obtain the proposed error estimator, it is necessary to recover the Hessian matrix from the information given by the finite element solution itself uh . Almost every algorithm used to recover the Hessian matrix use first derivatives information. Recovering first and second derivatives are main issues of the following sections. 2.1 The interpolation error as an error indicator of the approximate solution We present an anisotropic a posteriori error estimator for the diference between a given function u and a discrete function U ∈ Vh which is a good approximation of u in Ω in the sense that ku − U kLp (Ω) ≤ Cku − ΠukLp(Ω)
(1)
with Π : W 2,p (Ω) → Vh denoting an operator whose approximation properties are similar to the Cl´ement interpolation operator6 , i.e.
ku − U kLp (≤) ≤ C
≤C
X T ∈T
1/p
kH(u)k
p Lp (T˜ )
X T ∈T
kH(u) − HR (U )kLp (T˜) + kHR (U )kLp(T˜)
p 1/p
(2)
where H(u)(x) = D2 u(x)(x − xK ) · (x − xK ), HR (U ) = D2R U (x)(x − xK ) · (x − xK ), D2 U is the hessian operator and D2R U is a recovery hessian operator. We denote by T˜ the patch formed by T and its neighbour elements and xK is the centroid of the element K. Based on (2) we assume that there exists a constant C such that6 ku − uh kLp (Ω) ' CkHR (uh (x))(x − x0 ) · (x − x0 )kLp (Ω)
3
(3)
Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
where HR (uh (x)) denotes the recovered Hessian matrix, obtained from the information given by the finite element solution uh . This shows that the interpolation error in one point x, where |x −x0 | is small enough, is governed by the behaviour of the second order derivative in such point. Thus, the interpolation error is not distributed in an isotropic way around the point x0 ; i.e. the error depends on direction x − x0 and the recovered Hessian matrix value in this point, HR (u(x)). The above result suggests the use of expression (3) as a directional error estimator in the terminology used by Peir´ o1 . Since the recovered Hessian matrix is not positive definite it can not be taken as a metric tensor. As an alternative, Peir´o introduced the metric tensor G = QΛQT
(4)
where Q is the matrix of eigenvectors of the recovered Hessian matrix, the matrix Λ = diag{|λ1|, |λ2 |} and |λi |, i = 1, 2, are the absolute value of the associated eigenvalues (|λ1 | ≤ |λ2 |). According to this definition, the metric tensor field G is at least positive semi-definite. In particular, a zero eigenvalue poses no difficulty since it leads to infinite mesh sizes that are inhibited by the limitation in the element size or by the finiteness of the computational domain. Following the above ideas, Dompierre et al. 2 , introduce an error estimator associated to the size of element edges and Buscaglia et al.3 use (4) in a quality mesh indicator. The error estimator introduced herein is a variation of the first one. Instead of considering an error estimator associated to the element edge length, it is proposed another one that provides a measure of the second derivative contribution in each element4 . Considering a given finite element mesh T of the domain Ω, the error estimator at element level T ∈ T , is defined by the expression4,5,7,8 : ηT = {
Z ΩT
1
(G(uh (x0 ))(x − x0 ) · (x − x0 ))p dΩ} p
(5)
and the global error estimator η is given by: η=(
X T ∈T
p
1
ηT ) p
where ΩT is the area of the element T and x0 is the center of this element.
4
(6)
Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
2.2 First order derivative recuperation Several approaches have been proposed, in the framework of the Finite Element Method, in order to recover first derivatives 4,5,7,8 . Among them, there are the Weighted Average and Patch Recovering techniques, which we briefly summarize in what follows. • Weighted Average. The approach quoted Weighted Average, consists on turning the inter-elements discontinuous field ∇uh into the continuous field ∇R uh . This is made by employing the same element basis functions, used to construct uh , to compute ∇uh , and then adopting a weighted average of the ∇uh , computed on the elements surrounding a node N , as the value ∇R uh (XN ) of the recovered gradient in this node (XN is the coordinate of node N ). The weighted average is computed using weights given by the inverse of the distance between the node N and the point of superconvergence of the gradient (the center of the element in the case of linear triangles) or, instead, the area of the surrounding elements. • Patch Recovering This type of recovery estimator was introduced by Zienkiewicz and Zhu9,10,11 . In this method, it must be identified the node patch Ωp , which is constructed with the elements having a vertex at node N (Figure 1a). Over this local patch, it is possible to obtain improved solutions, ∇R uh , in the form of a polynomial expansion, i.e., [∇R uh (x)]i = P(x)ai
(7)
where [∇R uh (x)]i is the i–th component of the gradient defined over the patch and ai is the column–vector of unknown polynomial coefficients, to be determined. The monomial terms in the row–matrix P are those present in the basis functions of the applied elements. For instance, in linear triangles P = [1, x, y] and for quadratic triangles P = [1, x, y, x2, xy, y 2].
N
N
Figure 1. Local patch for typical nodes: (a) Interior Node (b)Boundary Node
The unknown coefficients ai can be determined by a least square fit of the derivatives at superconvergent points inside the patch, where locations are known in advance4,9,10,11 . Herein it is adopted an alternative approach, computing the vector of unknown coefficients ai , by the minimization of the functional:
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
Z
Ψ(ai ) =
Ωp
([∇R uh (x)]i − [∇uh (x)]i )2 dΩ
(8)
over the local patch Ωp , which implies in the solution of a linear system of equations given by: Pp ai = bp
Pp =
elp Z X
(
n=1
bp =
elp Z X
(
n=1
Ωn
(9)
PT (x)P(x)dΩ)
(10)
PT (x)[∇uh (x)]i dΩ)
(11)
Ωn
where elp is the number of elements in the patch. The index p is used to place emphasis on the dependence of the matrix and the independent term with respect to the patch. It is worth noting that the recovered gradient depends only on values of ∇uh sampled on the patch. Remark 2.1 - The matrix Pp is symmetric, positive definite and it is independent of any particular i–th component to be recovered. Remark 2.2 - The number of equations to be simultaneously solved is small ( three for linear triangles) and the cost of recovery evaluation is proportional to the number of nodes in the grid. Remark 2.3 - Once the vectors of parameters a have been computed, the gradient in the node is easily determined, since it corresponds to the component of the vectors ai representing the independent term of the polynomial expansion. Remark 2.4 - The computed recovered gradient performs better in the interior of the domain rather than on patches near the boundaries. The main reason to this is the existence of vertex boundary nodes that lead to poor results for the polynomial expansion whenever there are too few elements in the patch. Trying to overcome this effect, we adopt, for the value of the gradient at any boundary node N , a weighted average of the values (nvi), N , xN (figura 2b). For example , for linear triangles, it is computed the recovered gradient in the patch of the interior node j, that is: [∇R uh (xN , yN )]i = (ai )(j) (1) + (ai )(j) (2)xN + (ai )(j) (3)yN
(12)
Thereafter, a weighted average of these gradients is computed by using, for instance, the inverse of the distance of the neighbor node to the node N . Whichever alternative is used, the efficiency of the algorithms adopted to identify new nodal patches is extremely important.
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
2.2.1 The algorithm In order to approximate functions presenting strong variations in the derivatives the adapted mesh turns to be oriented by means of the stretching of its elements in the direction of maximum curvature of the function. Whenever this stretching is very large, the matrix Pp may become ill–conditioned. In addition, large stretching may also cause poor precision when computing weighted averages or least square fitting. To overcome this situation, the original domain Ωp is locally transformed into a standard unstretched domain Ω∗p where the polynomial expansion is initially adopted. For instance, when using mesh generators based on the advancing front technique 1,13,14,15,16,17 this domain transformation is naturally choosed compatible with the domain mapping which is part of the mesh generation algorithm. This is the procedure adopted in the present work and is described in the following. Considering the neighborhood of an arbitrary point N , the mesh generation algorithm tends to create triangular elements which are equilateral when viewed in the transformed domain given by the following operator. S(N ) =
1 1 e1 ⊗ e1 + e ⊗ e2 s(N ) ∗ h(N ) h(N ) 2
(13)
where ei (N ), (i = 1, 2) are the eigenvectors of the Hessian matrix H(uh (N )),h(N ) is the size in the e2 direction and s ≥ 1 is the stretching in the e1 direction of an element generated at node N . Those parameters are all dynamically defined along the mesh adaptation process. The selection of these parameters is discussed in the next section.
S(N) N^ ^ M
N M
Figure 2. Stretched mesh and the transformed domain defined by the advancing front technique
Consequently, each point M belonging to some element adjacent to a node N of the mesh (figure 2), is mapped in a point xM given by: xM = S(N )(XM − XN )
(14)
where XM , XN and xM denote the coordinates of the point N e M before and after the mapping, , respectively. The algorithm to recover first derivatives is then summarized in the BOX 1.
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
123456-
BOX 1 - First Derivative Recovery Algorithm Define the patch associated to node N (figure 1). Build the metric tensor S(N ), from information about the mesh shape around of the node N , defined by the known parameters s(N ) , h(N ), e1 e e2 . Transform all of the elements of the patch . Compute the gradients graduh in each one of the transformed elements. Using anyone of the recovering algorithms, compute gradRuh (N ) . By ∇R uh (N ) = ST (N )gradRuh (N ) transform the gradient gradRuh (N ) to the original domain.
Second derivatives can also be recovered by using the same approaches used for the first derivative recuperation. In fact, taking ∇R uh as a new field, we can reapply the algorithm in order to find ∇R (∇R uh ). The symmetric part of the approximation is retained, in order to ensure the symmetry of recovered Hessian matrix12. 3 ADAPTIVE PROCEDURE As mentioned before, it is taken as a local error indicator at element level, ηT , the value expressed by equation (5). Denoting η the global error indicator in the triangulation Tk , given by equation (6), the main idea to adapt the mesh, according to this error indicator, is to find a new mesh Tk+1 , with a given number of elements Nel. The new finite element mesh is generated trying to produce a uniform distribution of the local error estimator over all elements8 . As mentioned before, our remeshing algorithm is based on the advancing front technique. In this technique, the mesh generator tries to build equilateral triangles in the metric defined by the variable metric tensor S which, at node N of the actual mesh, takes the value defined by (13). To evaluate the parameters we proceed as follows( see Feij´ oo et al.4 , for more details). 1 - For each element compute the local error ηT and the global error η. 2 - Given a number of elements Nel in the new adapted mesh, the expected local error indicator, equally distributed on all elements in the triangulation Tk+1 , is given by η∗ = √
η Nel
(15)
3 - The decreasing or increasing rate of the element size is estimated by βT = (η ∗ /ηT )) 3 1
.
8
(16)
Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
From the rate βT , computed at elementwise, different approaches can be selected to find the distribution at nodal level. Those approaches can be the same used to perform the derivatives. This rate at nodal level is noted β(N ). 4 - The size of the new element, to be generated at node N , is defined by hk+1 (N ) = β(N )hk (N )
(17)
In the computational implementation of the present method, two threshold values for the computed new element size are used α ∗ hk ≤ hk+1 ≤ α ∗ hk ≤ L
(18)
This two parameters are used in order to ensure a progressively mesh adaptation to the solution. Moreover, L represents the characteristic length of the domain Ω. 5 - The stretching factor s at node N is defined by q
|λ2 |/|λ1 |
s(Nk ) =
(19)
where |λ1 | ≤ |λ2 | are the absolute eigenvalue of the Hessian matrix H(uh (N )). This stretching factor must bounded in order to ensure that the length of the element in the direction e1 , shk+1 , must be no greater than the characteristic length L of the domain Ω, i.e., s ≤ L/hk+1 . 6 - hk+1 -scaling. Due to the limitation on the values of h and s, the number of elements in the new adapted mesh may be different from the expected Nel. To force the equality between these two numbers, the elements size h at nodal level, must be scaled. In particular, the number of elements in the new finite element mesh is given by Z
2 4 N elnew = √ dΩ 3 Ω sh2
(20)
Then, the scaled value for the element size is given by s
hk+1 ←
N elnew × hk+1 Nel
(21)
The adaptive strategy described is repeated until the error estimator in the mesh Tk , relative to the solution of the problem analyzed, becomes lower than a given admissible relative error γ, that is, ηk 1
{kuh kL2 (Ω) + ηk } 2
9
≤
γ
(22)
Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
4 NUMERICAL APPLICATIONS This section is devoted to numerical applications using the proposed mesh-adaptive strategy. Before we apply the adaptive procedure for Limit Analysis problems, we select test cases in order to analyze the performance of the recovering techniques for first and second order derivatives. This comparison will allow us to select the most appropriate for our particular application. 4.1 Test Cases In order to compare the behaviour of the recovering techniques for first and second order derivatives presented in this paper, we consider de following functions:
Case 1: u(x, y) = (1 − x40 )
(x, y) ∈ [0, 1] × [0, 1]
Case 2: u(x, y) = (1 − x40 )(1 − y 40)
(x, y) ∈ [0, 1] × [0, 1] 2
2
Case 3: u(x, y) = (1 − exp [−(1 − 500(x − 0.5) + (y − 0.5)] )
(x, y) ∈ [0, 1] × [0, 1]
In the following, these function will be referred just by their correspondent label. As an approximation for each function above, we adopt the finite element linear interpolant uh ∈ Vh = {v ∈ H0 1 (Ω)(: v|T ∈ P1 (T )} where P1 (T ) denotes the space of polynomials, of degree not greater than 1, defined in the triangle T ∈ T , i.e. uh (xN , yN ) = u(xN , yN ), where (xN , yN ) are the coordinates of the nodal point N in the finite element mesh T . For the quoted functions, by using graphics in log × log scale, we present the global error η and the following estimators, versus the total number of nodes: k(u − uh )kL2 = (
Z Ω
1
(u − uh ) · (u − uh )dΩ) 2
kG(u) − Gr (uh )kL2 = ( kG(u) − G(uh )kL2 = (
Z
1
Z Ω
kH(u) − Hr (uh )kL2 = (
Ω Z
(∇u − ∇R uh ) · (∇u − ∇R uh )dΩ) 2 1
(∇u − ∇uh ) · (∇u − ∇uh )dΩ) 2
Ω
1
(H(u) − HR (uh )) · (H(u) − HR (uh ))dΩ) 2
where (.)r means some of the recovering techniques described in this work. We adopt an inicial coarse mesh with 25 nodes and 32 elements, as shown in figures 3, 4 and 5. All the adaptive process was perfomed considering an expected number of elements in the new adapted mesh (Nel) twice the number of elements in the preceding one. The threshold values adopted for the new element size, as defined in equation (18), are L = 1 and α = 0.05.
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
1 - x40
Inicial Mesh
After Five Steps:
Elements 762
Maximal local error 3.88e-7
Maximal stretch 2767.90
Function
1E-01 Global Estimator 1E-02 1E-03 (-3)
1E-04 Patch recovering Inverse distance Area
1E-05 1E-06 10
100
1000
NODES 1E+03
1E+03 (-3/4)
1E+02
(-3/4)
1E+02
1E+01
1E+01
1E+00
1E+00
(-3/2)
1E-01
(-3/2)
1E-01
1E-02
|(U_ex)-(U_h)|_L2 Global Error |G(U_ex)-G(U_h)|_L2 |G(U_ex)-G(U)_r|_L2 |H(U_ex)-H(U)_r|_L2
1E-03 1E-04 1E-05 1E-06 10
1E-02 |U_ex-U_h|_L2 Global Error |GrU_ex - GrU_h|_L2 |GrU_ex - GrU_r|_L2 |H_ex - H_r|_L2
1E-03 (-3) 1E-04
Patch Recovering 100
1E-05 1E-06
1000
10
(-3)
Area 100
NODES
NODES
1E+03 (-3/4)
1E+02 1E+01 1E+00
(-3/2) 1E-01 1E-02 |U_ex-U_h|_L2 Global Error |GrU_ex - GrU_h|_L2 |GrU_ex - GrU_r|_L2 |H_ex - H_r|_L2
1E-03 1E-04 1E-05 1E-06 10
(-3)
Inverse distance 100
1000
NODES
Figura 3. Case 1: u(x, y) = (1 − x40 )
11
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
(1 - x40 )(1 - y40 ) Inicial Mesh
After five Steps
Maximal stretch 3325
Elements 3330
Maximal local error 1.35e-4
Function
1E-01 Global Estimator 1E-02 1E-03 Area Inversa da Distancia
1E-04
Patch Recovering Patch Recovering without stretching
1E-05 10
100
1000
10000
Nodes 1E+03 1E+02 1E+01 1E+00 1E-01 1E-02 1E-03 1E-04 1E-05
(-1/2)
(-1)
|U_ex-U_h|_L2 Erro Global |GrU_ex - GrU_h|_L2 |GrU_ex - GrU_r|_L2 |H_ex - H_r|_L2
10
(-3/2)
Inverse distance 100
1E+03 1E+02 1E+01 1E+00 1E-01 1E-02 1E-03 1E-04 1E-05
(-1/2)
(-1)
|U_ex-U_h|_L2 Erro Global |GrU_ex - GrU_h|_L2 |GrU_ex - GrU_r|_L2 |H_ex - H_r|_L2
10
1000
(-3/2)
Area
100
1000 Nodes
Nodes
1E+03 (-1/2)
1E+02 1E+01
(-3/2)
1E+00 1E-01 |U_ex-U_h|_L2 Erro Global |GrU_ex - GrU_h|_L2 |GrU_ex - GrU_r|_L2 |H_ex - H_r|_L2
1E-02 1E-03 1E-04 10
(-2)
Patch Recovering 100
1000
Nodes
Figura 4. Case 2: u(x, y) = (1 − x40 )(1 − y 40)
12
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
For all these functions, we present in figures 3 , 4 and 5, the mesh and the isovalues of the functions , as well as the distribution of the local error after five steps of adaptation, considering the path recovering technique. For function 2, we can observe in figure 4, a fine homogeneity obtained in the error distribution. For the others two examples this fine homogeneity is only observed far from the region of strong gradients. We observe a good convergence rate for all the estimatores plotted in the above mentioned figures. For function 1, all of the proposed recovering techniques presented the same convergence rate, nevertheless, for functions 2 and 3, a convergenge rate a bit higher was obtained by using the patch recovering procedure compared to that obtained by weighted average approaches. We demonstrate in figure 4 the effectiveness of using stretched meshes, a crucial tool in our strategy, by means of a comparison among the performance of the global error, with the patch recovering procedure, when using nonstretched or stretched finite element mesh refinements. The improvement in the convergence rate, by the adoption of stretched finite elements, becomes clear from the analysis of the plotted results. To conclude this section, it is worth to observe that in these tests, the ratio between the global error and the norm k(u−uh )kL2 becomes very close to one after five remeshing steps. This is a numerical evidence that confirms the validity of expression (3) and the adequacy of the proposed directional error estimator. f ( x, y) = 1 −
1 e p( x , y )
p( x , y ) = 1 − 500( x − 0.5) 2 + ( y − 0.5) 2 1E+02
1E-01
Estimador Global
(-1)
1E+01 1E+00
1E-02 (-3/2)
1E-01 1E-02
|U_ex-U_h|_L2 Global Error |GrU_ex - GrU_h|_L2 |GrU_ex - GrU_r|_L2 |H_ex - H_r|_L2
1E-03 1E-04 10
1E-03 (-2)
Area Inverse distance Patch Recovering
Patch Recovering 100
1E-04 1000
10
Nodes
1000
Nodes
Initial
function
100
After Five Steps
function
error
3.59 E-3 (max)
error
8.2 E-5 (max)
Figura 5. Case 3: u(x, y) = (1 − e−p(x) )
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
4.2 Limit Analysis We begin this section by briefly introducing the theoretical framework of limit analysis. Afterwards, two examples of limit analysis are presented, where the finite element adaptive strategy previously outlined is applied. Under the assumption of proportional loading, the limit analysis problem consists in finding a load factor α such that the body undergoes plastic collapse when subject to the reference loads F uniformly amplified by α. In turn, a system of loads produces plastic collapse if there exists a stress field in equilibrium with these loads, which is plastically admissible and related, by the constitutive equations, to a plastic strain rate field being kinematically admissible. Thus, the limit analysis problem consists in finding α ∈ IR, a stress field T ∈ W 0 , a plastic strain rate field Dp ∈ W and a velocity field v ∈ V such that v∈V (23) Dp = Dv, Z B
T · Dv dB = α ( T
∈
Z B
b · v dB +
Z Γτ
τ · v dΓ ) ∀ v ∈ V
P = {T ∈ W 0 | f (T ) ≤ 0 in B} Dp = ∇f (T ) λ˙ f · λ˙ = 0
λ˙ ≥ 0
(24) (25) (26)
f ≤0
(27)
We explain, in what follows, the above notation and the meaning of these relations. Equation (23) imposes that the collapse plastic strain rate is related to a kinematically admissible velocity field v by means of the tangent deformation operator D. The Principle of Virtual Power (24), imposes the equilibrium of the stress fields with a given body loads αb and surface loads ατ prescribed in region Γτ of the body. For elastic ideally-plastic materials the plastic admissibility of stress fields is defined by the set P , as in (25). The admissibility function f is a m−vector field, comprising in each component a plastic mode. The inequality in (25) is understood as imposing that each one of those components fk is non-positive. The constitutive relation is expressed by the normality rule (26) and complementarity condition (26). In (26), ∇f (T ) denotes the gradient of f , and λ˙ is the m−vector field of plastic multipliers. Also, in (27) the inequalities hold componentwise. The kinematical principle for limit analysis was discretized by using a three-node interpolation finite element basis for the velocity field. In this model the approximate collapse factor α is greater than or equal to the true α, ˆ defined by the continuum model18 . In both aplications, it was adopted the Von Mises criterion for plastic admissibility. The discretized version of the limit analysis formulation leads to a finite dimensional problem that can be seen as a discret version of the system (13-17). A Newton-like strategy for solving this discrete problem is describe by Borges et al.18 .
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
The choice of a variable to be used to control the adaptive process is not obvious. To capture the discontinuities, the natural choice is one of the components of the velocity vector field, provided that we can estimate in advance which component is suitable to this aim. However, this is not an easy task for a general problem. Unlike the velocity field, the scalar field of plastic multipliers may be used as control variable without the previous disadvantage. Because we observe in (26), that λ˙ k is positive only when the function fk is zero, that is, when the stress is on the yield surface. So, the plastic multipliers clearly indicate the region where localized plastic deformations or slip bands are present. As a consequence, the local singularities arising from these localized plastic deformations are also detected in this way. Based on this argumet, the scalar field of plastic multipliers appears to be a good choice as control variable and is adopted for the most of the applications. Finally, based on the adaptive procedure explained in the previous section, the sketch of our adaptive process for limit analysis is summarized in BOX 2. BOX 2 - Adaptive Strategy for Limit Analysis Repeat 1- Apply the Newton-like strategy for solving the discrete limit analysis. 2- Choose the control variable and compute the local estimator error ηT and the global error η 3- For a given Nel, for each node of the mesh Tk estimate η ∗ , hk+1 and sk+1 . 4- Scale hk+1 5- Generate a new mesh Tk+1 . Until { ηk
≤
1
γ{kuh kL2(Ω) + ηk } 2 }
4.2.1 - Square slab with symmetrical internal slit A square slab with a symmetrical internal slit subjected to traction, presents localized deformation in the form of slip bands emanating from the roots of the crack. This collapse mechanism, as obtained in the numerical solution, is shown in figure 6. We analyze the behaviour of the adaptive strategy, when choosing the x-component of the velocity field or the plastic multipler field as the control variable. As shown in figure 7, both are effective as a controller for the directional error estimator. The isovalues for the plastic deformation and local error show that the proposed procedure is also effective in capturing the localized plastic deformation. Indeed, it increases the nodal points of the mesh only near the region where the discontinuities take place, as well as it constructs elements which are aligned and stretched in the direction of slip bands.
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
B C
2a
a p
B αp/aσy Field analyzed/Recovering Method Velocity/patch recovering Plastic multiplier/patch recovering Plastic multiplier/Inverse distance
0,65
0,6
C 0,55
0,5 0
300
600
900
1200
1500
1800
DOF
velocity field
σy - material limit yield in pure traction
Figura 6. Slab with an internal slit – Collapse load evolution and velocity field.
We obtain a 27% reduction in the collapse load approximations (figure 6) by means of five adaptation steps. It is worth noting that the kinematical principle adopted yields an upper bound to the exact collapse load, so, this reduction during the adaptive process is expected. 4.2.2 A thin square slab, with a central circular hole and subject to traction. This is a plane stress problem, that were analyzed considering three values of the ratio between the diameter d of the hole and the length L of the slab side. The numerical results for the collapse factor are plotted in figure 8 and compared to analytical lower and upper bounds19 . For d/L = 0.2 the velocity field presents a slip band which runs from the unstressed edge of the slab and meets the hole in the transversal axis of symmetry. The behaviour of this numerical velocity field complies with that foreseen in the analytical solution that identifies slip bands for ratios d/L lower than 0.4 . The analitycal collapse mechanism for d/L > 0.6 imposes that each quarter of the slab rotates as a rigid body, producing plastic hinges at its narrowest parts, that is, along the axes of symmetry. All of those kind of mechanisms are detected in the numerical solution, as can be observed in figure 3, where plastic region is the darkest one.
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
0,035
0,005 Field analyzed/Recovering Method Velocity/patch recovering Plastic multiplier/patch recovering
0,03
0,004
Original Mesh : 42 DOF
0,003
0,02 0,015
0,002
Global Error
Global Error
0,025
0,01 0,001
Plastic deformation
0,005 0 0
300
600
Field analyzed by the estimator:
900
1200
1500
Local error
0 1800
DOF
After Five steps Plastic deformation
Local error
Velocity X: 1725 DOF error max:1.6 e-4 Max. stretching 370,2
Plastic Multiplier: 1579 DOF error max:1.66 e-4 Max. stretching 125,9
Figura 7. Slab with an internal slit – Erro, mesh and plastic deformation. INITIAL
AFTER 4 STEPS plastic multiplier field
d / L=0.2
p
d / L=0.4
αp aσy
Present-Kin. 1258 dof U/L Bounds-[G & M C, 1954]
d / L=0.8
Figura 8. Slab with circular cutout – Collapse load, final meshes and plastic region.
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
5 CONCLUSIONS The adaptive technique proposed shows to be appropriated to capture discontinuities arising from the localized plastic deformations during plastic collapse, and as a consequence, i deeply improves the numerical evaluation of the collapse load. The numerical applications confirmed the feasibility of process, that is, the computation of the error estimate was less expensive than the calculation of the numerical solution. REFERENCES [1 ] Peir´ o, J.; A Finite Element Procedure for the Solution of the Euler Equations on the Unstructured Meshes. Ph.D. these, Department of Civil Engineering, University of Swansea, UK, (1989). [2 ] Dompierre, J.,Vallet, M. G., Fortin, M., Habashi W.G., A¨It-Ali-Yahia, D., Boivin, S., Bourgault, Y. & Tam A.; Edge -Based Mesh Adaptation for CFD, Conference on Numerical Methods for the Euler and Navier-Stokes Equations, Montreal, 265299, (1995). [3 ] Buscaglia, G.C., Dari, A; Anisotropic Mesh Optimization and its Application in Adaptivity, International Journal for Numerical Methods in Engineering, 40, 41194136, (1997). [4 ] Feij´oo, R.A., Borges, L., Zouain, N.; Estimadores a posteriori y sus Aplicaciones en el An´ alisis Adaptativo. Internal Report N0 1/97 - COPPE/UFRJ, (1997). [5 ] Verf¨ urth, R.; A Review of a Posterriori Error Estimation and Adaptive Mesh-Refinement Techniques,Publication of Institut f¨ ur Angewandte Mathematik der Universit¨ at Z¨ urich, (1995). [6 ] Regina C. de Almeida, Ra´ ul A. Feij´oo, Augusto C. N. Gale˜ao, Claudio Padra and Renato Sim˜ oes; Adaptive Finite element Computacional Fluid Dynamics using an Anisotropic Error Estimator, Computacional Mechanics - New Trends and Applications - Fourth World Congress on Computacional Mechanics, (1998) [7 ] Ainsworth, M, Oden , J.T.; A posteriori eeror estimation in finite element analysis, Comput. Methods Appl. Mech. Engrg.,142,1-88, (1997). [8 ] Zienkiewicz, O. C. & Zhu, J. Z.; A Simple Error Estimator and Adaptive Procedure for Practical Engineering Analyses,Int. J. Numer. Methods Eng.,24, 333-357, (1987). [9 ] Zienkiewicz, O. C. & Zhu, J. Z.; The Superconvergent Patch Recovery and a posteriori error estimates. Part 1: The recovery technique ,Int. J. Numer. Methods Eng.,33, 1331-1364, (1992).
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Lavinia Borges, Ra´ ul Feij´ oo, Claudio Padra and Nestor Zouain
[10 ] Zienkiewicz, O. C. & Zhu, J. Z.; The Superconvergent Patch Recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity ,Int. J. Numer. Methods Eng.,33, 1365 -1382, (1992). [11 ] Zienkiewicz, O. C. & Zhu, J. Z.; Superconvergent Patch Recovery Techniques – Some Further Tests,Commun. Appl. Numer. Methods, 9, 251-258, (1992). [12 ] Zienkiewicz, O. C. & Taylor, R. L. ; The Finite Element Method, Fourth Edition, McGraw Hill Int. Ed., (1991). [13 ] Oliveira, M., Guimares, A.C.S., Feij´ oo, R.A, V´enere, M.J., Dari, E.; An Object Oriented Tool for Automatic Surface Mesh Generation Using the Advancing Front Technique,Report of P&D, LNCC/CNPq N0 18/95, to appear in Int. Journal for Latin America Applied Research, (1995). [14 ] Fancello, E.; Guimar˜ aes, A.C.S.; Feij´oo, R.A.; ARANHA - Gerador de Malhas 2D para Elementos Finitos Triangulares de 3 e 6 n´ os, ENIEF 90 - Encuentro Nacional de Investigadores y Usuarios del M´etodo de los Elementos Finitos, Mar del Plata, Argentina, 05-09 de novembro de 1990. Anais do XI Congresso Ibero Latino Americano sobre M´etodos Computacionais para Engenharia, p´ ag. 983 996, 29-31, Rio de Janeiro.Relat´ orio de P&D no.021/90,LNCC/CNPq, Rio de Janeiro,Brasil, (1990). [15 ] Fancello, E.; Guimar˜ aes, A.C.S.; Feij´oo, R.A.; Venere, M.J.; Gera¸c˜ao de Malhas 2-D em Programa¸c˜ao Orientada a Objetos. Anais do XI Congresso Brasileiro de Engenharia Mecˆ anica, San Paulo, Brasil,pp. 635-638, (1991). [16 ] Dari, E.; V´enere, M.J.; Finite Element 3-D Mesh Generation Using the Advancing Front Technique, MECOM’94,Mar del Plata, Argentina (1994). [17 ] V´enere, M.J.; T´ecnicas Adaptativas para el M´etodo de Elementos Finitos en dos y tres Dimensiones, Tesis de Doctorado, Instituto Balseiro, Centro At´omico Bariloche, Argentina,(1996). [18 ] Borges, L.A., Zouain,N. e Huespe, A.E.; A nonlinear optimization procedure for limit analysis , European Journal of Mechanics A/Solids,15, no 3, 487-512,(1996). [19 ] Gaydon, F. A., McCrum, A. W.; A Theoretical Investigation of the Yield Point Loading of a Square Plate with a Central Circular Hole, J. Mech. and Phys. Solids, 2, 156-169,(1954). Acknowledgement: The authors would like to acknowledge the support of this research by their own institutions and by CNPq and FAPERJ. Also they are grateful to TACSOM group (www.lncc.br/˜tacsom) by the softwares facilities offered.
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