Brief notes
Struct Multidisc Optim 24, 72–77 Springer-Verlag 2002 Digital Object Identifier (DOI) 10.1007/s00158-002-0215-1
An optimization technique for the solution of the Signorini problem using the boundary element method A. Leontiev, W. Huacasi and J. Herskovits
Abstract We present an optimization technique for the numerical solution of the Signorini problem for the Laplacian via boundary element discretization. The discretized problem is a mixed linear complementarity problem with potential and flux at the contact region as complementary variables. This complementarity problem is reformulated as a bilinear optimization program, which we solve with an interior point algorithm. Numerical results for a test problem and a comparison with another solution technique are given. Key words Signorini problem, boundary elements method, linear complementarity problem, bilinear optimization program, interior point algorithm
1 Introduction In recent years, much interest has been devoted to the mathematical formulation of problems involving unilateral constraints and their numerical solution. An important class is the Signorini problem, where unilateral constraints are imposed at the boundary. This is the case of contact problems for solids. Since the constraints only concern values on the boundary, it is quite natural to look for a numerical solution of this kind of problem by means of the direct Boundary Elements Method (BEM). In the case of the Signorini problem for the Laplacian, the unknown boundary values Received February 9, 2001 A. Leontiev1 , W. Huacasi2 and J. Herskovits3 1
Institute of Mathematics, Federal University of Rio de Janeiro, 21945 970, Rio de Janeiro, RJ, Brazil e-mail:
[email protected] 2 Mathematical Department, State University of Norte Fluminense, 28015 620, Campos dos Goytacazes, RJ, Brazil e-mail:
[email protected] 3 Mechanical Engineering Program, COPPE/Federal University of Rio de Janeiro, 21945 970, Rio de Janeiro, RJ, Brazil e-mail:
[email protected]
are potential and flux, which are considered primary variables in BEM. Then they are obtained directly. This is not the case for domain methods. As a consequence, BEM yields higher accuracy. Moreover, the size of the discrete model obtained with boundary elements is smaller. Boundary variational inequality formulations for the Laplace equation with Signorini boundary conditions were studied by several authors (e.g. Gakwaya and Lambert 1989; Han 1990; Simunovic and Saigal 1992; Spann 1993). Some variational principles for unilateral BEM contact problems in elasticity were obtained by Polizzotto (1993), for the direct determination of the unknown boundary quantities, taking the relative displacements in the contact region as independent variables. Other authors developed variational treatments of BEM contact problems based on the use of Green’s functions (for example Alliney et al. 1990). In general, all these approaches lead to the solution of a linear complementarity problems (LCP). Depending on the method, LCP has different properties [nonsymmetric matrix (Alliney et al. 1990), symmetric sign-definite matrix (Polizzotto 1993), etc.]. To solve the linear complementarity problem, different techniques can be employed. We mention Lemke’s method (Bazaraa and Shetty 1979), gradient projection, quasi-Newton and conjugate gradient projection algorithms (Xiao et al. 1999), the decomposition-coordination method (Spann 1993) and some heuristic iterative procedures with trial and error. Another approach to treat LCP is the reformulation as an optimization problem (Friedlander et al. 1995; Solodov 1999). The most natural way is to state a bilinear mathematical program (Cottle et al. 1992). In this paper we deal with the Signorini problem for the Laplacian. After boundary element discretization we obtain a linear complementarity problem. Since not all variables are complementary, we have to deal with a mixed linear complementarity problem (MLCP). It is possible to modify existing LCP algorithms to solve MLCP. Our approach is quite different. We reformulate MLCP as a bilinear mathematical program. This program is nonconvex, but it has a unique solution. Moreover, it satisfies linear independence regularity conditions
73 (Herskovits 1995). Herskovits’ interior point algorithm is employed to solve this problem (Herskovits 1998). We present numerical results for an example that has an explicitly know solution, constructed by Spann (1993).
Using the notations Hij = Γ qi∗ dΓj for i = j, Hii = j 0.5 and Gij = Γ u∗i dΓj , we can write this equation in j matrix form as Hu = Gq ,
2 Problem formulation and the BEM discretization We consider the Signorini problem for the Laplace operator in the bounded domain Ω with a smooth boundary Γ . Let ΓD , ΓN and ΓC be parts of Γ such that ΓD ∪ ΓN ∪ ΓC ≡ Γ and ΓC = ∅. Given q◦ ∈ H −1/2 (ΓN ∪ ΓC ) and u◦ ∈ H 1/2 (ΓD ∪ ΓC ), we look for u ∈ H 1 (Ω) that verifies ∆u = 0 in Ω , u = u on ΓD , ◦ (1) q = q on ΓN , ◦ u ≥ u◦ , q ≥ q◦ , (u − u◦) (q − q◦ ) = 0 on ΓC , where q ≡ ∂u/∂n. The conditions at ΓC represent the contact with an unilateral boundary obstacle given by u◦ . It was proved that this problem has an unique solution if and only if ΓD = ∅ or Γ ∪Γ q◦ ds < 0 (Spann 1993). N C In the two-dimensional case for the problem governed by the Laplace equation, flux and potential verify on the boundary Γ the integral equation ∗ 0.5u(ξ) + q (ξ, χ)u(χ) dΓ = u∗ (ξ, χ)q(χ) dΓ , (2) Γ
Γ
where χ ≡ (x, y) ∈ Γ , u∗ (ξ, χ) is the fundamental solution of Laplace equation, q ∗ (ξ, χ) its normal derivative, and ξ ∈ Γ is the collocation point (Brebbia et al. 1984). In the present approach, we first make a boundary element discretization based on the previous equation and then introduce the contact conditions on the discrete model. Let be E the number of (geometrical) nodes and elemE ents Γi , such that Γ = Γj . Considering that the flux j=1
and the potential are approximated by constant functions for each Γj , j = 1, . . . , E, we perform the following discretization of the integral equation: E ∗ 0.5ui + qi dΓj uj = j=1
Γj
j=1
where H, G ∈ RE×E and u, q ∈ RE . Let (xi , yi ) be the coordinates of the geometrical nodes i = 1, . . . , E and xE+1 = x1 , yE+1 = y1 . Then, we can obtain explicit formulae for the elements of G and H. When i = j: Gij = −
4
1/2 0.5ωk a2x + a2y ln (xc − axγk − bx)2 +
k=1
(yc − ay γk − by )2 ,
(5)
Hij =
(6)
4 ωk ay (ax γk + bx − xc ) − ax (ay γk + by − yc ) − , 2 2 k=1 xc − ax γk − bx + yc − ay γk − by
and when i = j: 1/2 2 2 2 2 Gii = 2 ax + ay 1 − ln ax + ay , Hii = π , where ax = 0.5 xj+1 − xj , bx = 0.5 xj+1 + xj , ay = 0.5 yj+1 − yj , by = 0.5 yj+1 + yj , xc = 0.5 xi + xi+1 , yc = 0.5 yi + yi+1 , and γk , ωk are the abscissa and weight of the Gauss quadrature. Let n, m and be the numbers of the boundary elements located at the segments ΓD , ΓN and ΓC , respectively and N := n, M := N + m, L := M + , K := L + . We consider as independent variables the flux at the boundary elements of ΓD : X1 . . . XN , the potential at the boundary elements of ΓN : XN +1 . . . XM , and potential and flux at the boundary elements of ΓC : XM+1 . . . XL and XL+1 . . . XK , respectively. Let X = (X1 . . . XK ), U = (u1 . . . uN , XN +1 . . . XM , XM+1 . . . XL ) ,
(7)
Q = (X1 . . . XN , qN +1 . . . qM , XL+1 . . . XK ) ,
E u∗i dΓj qj ,
(4)
i = 1, . . . , E ,
(3)
Γj
where ui ≡ u(ξi ), u∗i ≡ u∗ (ξi , χ), qi∗ ≡ q ∗ (ξi , χ), ξi ∈ Γi and uj ≡ u(χ), qj ≡ q(χ), χ ∈ Γj , j = 1, . . . , E.
where the values of potential u1 . . . uN at the segment ΓD and of flux qN +1 . . . qM at the segment ΓN are defined corresponding to the boundary conditions of the problem. For the unknown values of potential and flux at the contact segment ΓC we have, in addition to the boundary equation, the contact conditions
74
Fig. 1 Analytical solution of the problem
◦ XM+i ≥ UM+i , XL+i ≥ Q◦L+i ,
◦ XM+i − UM+i XL+i − Q◦L+i = 0 ,
i = 1... .
(8)
The values U ◦ and Q◦ correspond to discretization of the functions u◦ and q◦ at ΓC . This way, we can formulate our problem in the following form: HU = GQ , X ◦ i = 1... , M+i − UM+i ≥ 0, ◦ X − Q ≥ 0, i = 1... , L+i L+i ◦ XM+i − UM+i XL+i − Q◦L+i = 0, i = 1 . . . , (9)
variables of the mathematical program. Due to the objective function F (X), this problem is nonconvex. To solve this problem we use Herskovits’ interior point algorithm for nonlinear mathematical programming (Herskovits 1998).
3 Numerical tests As a numerical test we take the example from Spann (1993), whose analytical solution is known. This is a Signorini problem for the Laplacian ∆u = 0 in the semiannular domain Ω = {x ∈ R2 : a < |x| < b, x2 ≤ 0} with Dirichlet conditions at the noncontact part of the boundary ΓD = {x ∈ R2 : |x| = b, x2 < 0, a < |x1 | < b, x2 = 0} and contact condition u ≥ 0, q ≥ 0, uq = 0 at the contact boundary ΓC = {x ∈ R2 : |x| = a, x2 < 0}. We compute the Dirichlet data accordingly to the analytical solution (Fig. 1)
that is a mixed linear complementarity problem (MLCP), due to the fact that X1 . . . XM are not complementary variables. To solve this problem we state the equivalent bilinear mathematical program min F (X) X HU = GQ , (10) ◦ XM+i − UM+i ≥ 0, i = 1 . . . , i = 1... , XL+i − Q◦L+i ≥ 0,
where
with the objective function
w(x1 + ix2 ) =
F (X) =
◦ XM+i − UM+i XL+i − Q◦L+i . i=1
This problem has K variables, E linear equality constraints, and 2 bound constraints. We note that the potential XM+1 . . . XL and flux XL+1 . . . XK at the boundary elements at ΓC are considered as independent
u(x1 , x2 ) := Im w(x1 + ix2 )3 ,
(11)
√ √ A sgn x1 + i B sgn x2 ,
with
2 2 2 x21 − x22 r a2 + 0.25 − + r2 a2 r2 x21 − x22 r2 a2 0.25 + , r2 a2 r2 A = 0.5
(12)
75
Fig. 2 Analytical solution without contact condition
B = 0.5
x21 − x22 r2
2
+ 0.25
r2 a2 − a2 r2
2 −
x21 − x22 r2 a2 + , (13) 0.25 r2 a2 r2 and r := x21 + x22 , |x| ≥ a. We will now compare our numerical results with the analytical solution at the contact boundary ΓC , that is x22 − x21 u(x1 , x2 ) = − max 0, sign x2 , a2 6 x2 − x2 q(x1 , x2 ) = − 3 max 0, 1 2 2 |x1 |x2 . (14) a a Let us note, that in the absence of the contact conditions at ΓC (the case of the natural conditions q = 0), the solution is given in Fig. 2, and the “penetration” (u < 0) at the boundary ΓC is evident. We take a = 0.1 and b = 0.25 to compare our numerical results with the ones carried out by Spann (1993). Our discretization includes 39 boundary elements (Fig. 3). Thus, the mathematical program has 54 variables, 39 equality constraints and 30 “box” constraints. We stop the algorithm when the norm of the gradient of the Lagrangian and the absolute values of the equality constraints are all less that 10−6 (see for details Herskovits 1998). With different initial data, the convergence of the algorithm was obtained in no more than 42 iterations. Table 1 shows the history of iterations: the first column gives the number of iteration, the second one shows the objective function value. The third column is the
value of the corresponding Lagrangian and the fourth one presents the maximal error in the equality constrains that corresponds to the residual error of the discrete boundary integral equation. Figure 4 shows the results of the analytical solution for the potential (curve 1) and the flux (curve 2) at the contact boundary, the numerical results (circles for potential and squares for flux) and the location of the boundary without contact conditions (curve 3). In this figure we use the parameterization of the contact boundary t → {0.1 cos[(2πt)/30], 0.1 sin[(2πt)/30)]} and scale the graphic of the flux by 0.1.
Table 1 History of iterations Iteration 01 02 03 04 05 10 15 20 25 30 35 36 37 38 39 40 41 42
Object
Lagrangian
Equalities
2.50303 × 10+04 1.14333 × 10+04 3.01613 × 10+03 4.86057 × 10+02 2.02769 × 10+02 8.52773 × 10+01 6.78598 × 10+01 2.05065 × 10+01 5.97883 × 10+00 1.29372 × 10+00 1.34022 × 10−01 3.60880 × 10−02 1.06892 × 10−02 3.20328 × 10−03 9.60910 × 10−04 9.96230 × 10−05 1.06966 × 10−06 1.23431 × 10−10
2.30290 × 10+02 3.50352 × 10+02 3.90575 × 10+02 2.03048 × 10+02 7.64234 × 10+01 2.49596 × 10+01 1.64723 × 10+01 1.67482 × 10+01 9.19378 × 10+00 2.31347 × 10+00 1.64539 × 10+00 3.95411 × 10−01 8.80828 × 10−02 9.02224 × 10−03 3.19281 × 10−03 9.39293 × 10−04 9.67516 × 10−05 1.03952 × 10−06
2.37636 × 10−11 7.53705 × 10+01 2.15748 × 10−13 1.44919 × 10−13 3.95233 × 10−14 2.78385 × 10−14 2.05715 × 10−14 1.64680 × 10−14 1.31497 × 10−14 2.96425 × 10−14 2.65557 × 10−14 1.97616 × 10−14 2.63488 × 10−14 4.64725 × 10−14 1.98592 × 10−14 1.68820 × 10−14 3.31947 × 10−14 3.30987 × 10−14
76 Y
0.25 4
3
2
0.1
1
25 O
39 5
X
26 21
27
38
28
37
6
24 23 22
36
20
29 35
7
34 33 32 31
30
19 18
8
Ω
9
17 16
10 11
12
13
15
14
Fig. 3 BEM discretization 2.0
2 1.5
1.0
0.5
1
0.0
3
– 0.5
– 1.0
t 0
5
10
15
20
25
30
Fig. 4 Analytical and numerical solution at the contact boundary
In our experiments the error of the computed numerical potential is less than 0.008. Spann (1993) obtained a similar result with a discretization having 128 nodes (for one-half of the annular domain) nodes. He employed a decomposition-coordination algorithm for solving LCP. The number of iterations is of the same order [48 for Spann (1993)]. We note that our iterates verify the equality constraints from the fourth iteration (see Table 1). The error for the flux at the corner elements in our method is related to the chosen discretization by constant elements, and can be essentially reduced by using the discretization with elements of the highest order, like linear or quadratic ones.
4 Conclusions The approach proposed in this paper for the numerical solution of the unilateral problem via the boundary element method is based on the reformulation of the linear complementarity problem, which appear after discretization, as a bilinear mathematical program. The interior point algorithm is used to find the solution. The method is simple to apply and can be used for 2D and 3D problems. The example tested shows computational efficient, and good numerical accuracy among other advantages. Moreover, the proposed method can be adapted to linear or quadratic boundary elements.
77 Acknowledgements The authors gratefully acknowledge the financial support provided by CNPq (Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico, Brazil), FAPERJ (Funda¸ca ˜o da Amparo a ` Pesquisa do Estado do Rio de Janeiro) and FENORTE (Funda¸ca ˜o Estadual do Norte Fluminense, Rio de Janeiro).
References Alliney, S.; Tralli, A.; Alessandri, C. 1990: Boundary variational formulation and numerical solution techniques for unilateral contact problems. Comp. Mech 6, 247–257 Bazaraa, M.S.; Shetty, C.M. 1979: Nonlinear programming. Theory and algorithms. New York: John Willey & Sons Brebbia, C.A.; Telles, J.C.F.; Wrobel, L.C. 1984: Boundary elements techniques – theory and applications in engineering. Berlin, Heidelberg, New York: Springer Cottle, R.W.; Pang, J.-S.; Stone, R.E. 1992: The linear complementarity problem. New York: Academic Press Friedlander, A.; Mart´ınez, J.M.; Santos, S.A. 1995: Solution of linear complementarity problems using minimization with simple bounds. J. Global Opt. 6, 253–267 Gakwaya, A.; Lambert, D. 1989: A boundary element and mathematical programming approach for frictional contact problems. In: Advances in boundary elements, Vol. 3, pp. 163–179 Han, H. 1990: A direct boundary method for Signorini problems. Math. Comp. 55, 115–128 Herskovits, J. 1995: A view on nonlinear optimization. In: Herskovits, J. (ed.) Advances in structural optimization, pp. 71–117. Dordrecht: Kluwer Herskovits, J. 1998: Feasible direction interior-point technique for nonlinear optimization. JOTA 99, 121–146 Polizzotto, C. 1993: Variational boundary-integral-equation approach to unilateral contact problems in elasticity. Comp. Mech. 13, 100–115 Simunovic, S.; Saigal, S. 1992: Frictionless contact with BEM using quadratic programming. J. Eng. Mech. Div. ASCE 118, 1876–1891 Solodov, M.V. 1999: Some optimization reformulations of the extended linear complementarity problem. Comp. Optimiz. Applic. 13, 187–200 Spann, W. 1993: On the boundary element method for the Signorini problem of the Laplacian. Numerische Mathematik 65, 337–356 Xiao, J.-R.; Lok, T.-S.; Xu, S. 1999: Boundary elements-linear complementary equation solution to polygonal unilaterally simply supported plates. Engrg. Anal. Boundary Elem. 23, 297–305
5 Appendix: Herskovits’ feasible direction interior point algorithm (Herskovits 1998) Let the constrained optimization problem be min f (X) X s.t. g(X) ≤ 0, h(X) = 0 .
(15)
Here h : Rq → Rp , g : Rq → R . We introduce the function φc (X) = f (X) + pi=1 ci |hi (X)|, where ci are positive constants and use the following notations: G(X) is a diagonal matrix with Gii (X) = gi (X), Λ is a diagonal matrix Λii = λi . Parameters. γ ∈ (0, 1), η ∈ (0, 1), ϕ > 0 and ν ∈ (0, 1). Data. X ∈ ∆◦ , 0 < λ ∈ R , B ∈ Rq×q symmetric and positive definite, 0 < ω i ∈ R , 0 < ω e ∈ Rp and 0 < c ∈ Rp . Step 1. Computation of a search direction 1. Compute (d◦ , λ◦ , µ◦ ) by solving the linear system Bd◦ + ∇g(X)λ◦ + ∇h(X)µ◦ Λ∇g t (X)d◦ + G(X)λ◦ ∇ht (X)d◦
= −∇f (X) , = 0, = −h(X) .
(16)
If d◦ = 0, stop. 2. Compute (d1 , λ1 , µ1 ) by solving the linear system Bd1 + ∇g(X)λ1 + ∇h(X)µ1 Λ∇g t (X)d1 + G(X)λ1 ∇ht (X)d1
= 0, = −Λω i , = −ω e .
(17)
3. If ci < −1.2µ◦i, then set ci = −2µ◦i ; i = 1, . . . , p. 4. If dt1 ∇φc (X) > 0, set 2 ρ = min ϕ d◦ 2 ; (γ − 1)dt◦ ∇φc (X)/dt1 ∇φc (X) . (18) 2
5. Otherwise, set ρ = ϕ d◦ 2 . 6. Compute the search direction d = d◦ + ρd1 and also λ = λ◦ + ρλ1 . Step 2. Line search Compute t, the first number of the sequence {1, ν, ν 2 , ν 3 , . . . } satisfying φc (X + td) h(X + td) gi (X + td) gi (X + td)
≤ ≤ < ≤
φc (X) + tηdt ∇φc (X) , 0, ¯i ≥ 0 , 0 if λ gi (X) otherwise .
(19)
Step 3. Updates 1. Set X := X + td and define new values for ω i > 0, ω e > 0, λ > 0 and B symmetric and positive definite. 2. Go to Step 1.
