We consider the duality method for solving a model f elastic problem with a crack based on the use of duality methods. An article presents the theorems, allowing to use Uzawa method for search a saddle point of the modified Lagrangian functional. The results of numerical experiments are given.
The methodological basis of study is duality scheme, based on replacing the problem of constrained optimization to finding a saddle point of the modified Lagrangian functional. This method was considered to the finite-dimensional problems of linear and convex programming in books by authors E.G. Gol’stein and N.V. Tret’yakov [Gol’stein & Tret’yakov, 1989], A.A. Kaplan and Ch. Grossman [Grossman & Kaplan, 1981]. In works of R.V. Namm, G. Woo and E.M. Vikhtenko [Namm & Vikhtenko, 2011, Vikhtenko et al., 2014] these studies were extended to the infinite-dimensional variational inequalities of mechanics. The regularity of the solution may be arbitrary bad in the neighborhood of the crack edges, and the dual problem may be unsolvable. Therefore the applying a similar scheme for solving the crack problem may be complicated. Despite this problem, it is possible to justify the duality scheme to solve the crack problem, as well as equality of duality for the original and dual problems. 2
Formulation of the Problem Let Ω ⊂ R2 be a bounded domain with Lipschitz boundary Γ and γ ⊂ Ω be a cut (crack) inside of Ω. For simplicity we assume
γ = { (x1, x2) ∈ Ω : a < x1 < b, x2 = const} . and suppose that both end points (a, 0) and (b, 0) do not belong to the boundary Γ. Denote Ω Consider the minimizing problem = Ω\γ¯. J (v) = 1 ∫ |∇v|2 dΩ −
∫ 2 Ω Ω v ∈ K = {v ∈ H1 (Ω ) : [v] ≥ 0 on γ, v = 0 on Γ}.
f v dΩ − min, Here [v] = v+ − v is the jump of v across γ (v+ is a function value v on upper crack face, v is a function value v on lower crack face, marks ± correspond to positive and negative directs of normal vector on cut γ); f ∈ L2 (Ω) is a given function.
Problem (1) has a unique solution u ∈ K, which is, simultaneously, a solution of variational inequality [Khludnev, 2010] ∫ Ω ∫
Ω ∇u∇ (v − u) − f (v − u) dΩ ≥ 0 ∀v ∈ K.
Assuming H2-regularity of function u it can be shown that the problem (2) is equivalent to the boundary value problem [Khludnev, 2010] −∆u = f
u = 0 [u] ≥ 0, [ux2 ] = 0, ux2 ≤ 0, ux2 [u] = 0 in Ω , on Γ, on γ.
(1) (2) (3)
The question of solvability of problem (1) is investigated in detail in [Khludnev, 2010]. There is a theorem that establishes the existence of the solution of problem (1).
Theorem 1. Let K is a convex and closed set in Hilbert space H1 (Ω ), functional J (v) is weakly lower semicontinuous and coercive. Then problem (1) is solvable.
Applying Friedrich’s inequality ∫ Ω ∫
Ω v2dΩ ≤ C |∇v|2 dΩ ∀v ∈ H1 (Ω ) , v = 0 on Γ, it can be proved that the problem (1) has unique solution. 3
Duality Scheme for Solving Model Crack Problem For arbitrary m ∈ L2 (γ) construct the set
Km = {v ∈ H1 (Ω ) : v = 0 on Γ, − [v] ≤ m a.e. on γ}.
It is easy to show that Km is a convex and closed set in H1 (Ω ). If the function [m] ∈ L2 (γ) \H1=2 (γ) the corresponding set Km may be empty. On space L2 (γ) define the sensitivity functional χ (m) = {
inf J (v) , if Km ̸= ∅, v2Km +∞, otherwise.
Functional χ (m) is a proper convex and coercive functional on L2 (γ), but it’s effective domain domχ = {m ∈ L2 (γ) : χ (m) < +∞} does not coincide with L2 (γ). Notice that domχ is a convex but not closed set. In this case, domχ = L2 (γ).
Theorem 2. The functional χ (m) is weakly lower semicontinuous on L2 (γ).
The proof of this statement is presented in [Vikhtenko & Namm, 2016]. This property of the sensitivity functional is the basis of some theorems allows constructing and justifying the search algorithms of the saddle points of the modified Lagrangian functional.
We define the following functional on the space H1 (Ω ) × L2 (γ) × L2 (γ) [Namm & Vikhtenko, 2011, Vikhtenko et al., 2014]
K (v, l, m) = 1 ∫ ( 2r (l + r m)2 − l2) dΓ, if − [v] ≤ m a.e. on γ,
otherwise, 1 ∫ (( 2r (l − r [v])+)2 − l2) dΓ. J (v) + +∞,
inf m2L2( ) and modified Lagrangian functional Here r > 0 is a constant, (l − r [v])+ = max {0, l − r [v] , }.
Let us introduce the modified dual functional
M (v, l) =
K (v, l, m) = J (v) +
v2H1(Ω )
M (v, l) =
inf inf v2H1(Ω ) m2L2( )
K (v, l, m) = =
inf v2H1(Ω ) {
J (v) + 1 ∫ (( 2r (l − r [v])+)2 − l2)
} dΓ .
inf inf v2H1(Ω ) m2L2( )
K (v, l, m) =
inf inf m2L2( ) v2H1(Ω )
K (v, l, m) , M (l) =
inf m2L2( ) { χ (m) + 1 ∫ ( 2r (l + r m)2 − l2) dΓ} .
For an arbitrary l ∈ L2 (γ), we consider the functional Then the dual functional M (l) has the form
Fl (m) = χ (m) + 1 ∫ ( 2r
(l + r m)2 − l2) dΓ.
M (l) = m2iLn2f( ) Fl (m) . (4) (5) (6)
Fl (m) is a weakly lower semicontinuous and coercive functional on L2 (γ). Therefore the problem (6) has a solution m (l) for every l ∈ L2 (γ). It is easy to see, that element m (l) is unique. Further, we mention one important property of dual functional [Vikhtenko & Namm, 2016].
Theorem 3. The dual functional M (l) is Gateaux differentiable in L2 (γ) and it’s derivative ∇M (l) satisfies the Lipschitz condition with the constant r 1; that is, for all l1, l2 ∈ L2 (γ), it holds that
∥∇M (l1) − ∇M (l2)∥L2( ) ≤ r 1 ∥l1 − l2∥L2( ) .
In the proof of this theorem it shows that the functional subdifferential consists of a single element ∂ (M (l)) = { m (l) }. Therefore ∇M (l) = m (l).
Using equations (4) and (5) for the dual functional, we get ∇M (l) = m (l) = max { l }
− r , − [v] . Let us consider the dual problem { M (l) − sup,
l ∈ L2 (γ) .
Since the gradient of the functional M (l) satisfies the Lipschitz condition, the dual problem (7) can be solved by using the gradient method for maximizing a functional lk+1 = lk + r ∇M (lk) = lk + r m (lk) , k = 0, 1, 2, . . .
Here l0 ∈ L2 (γ) is an arbitrary initial value.
Theorem 4. The sequence {lk} constructed by the gradient method (8) satisfies the limit equality (7) (8)
Using gradient method (8), we can construct the following algorithm Uzawa method for solving the problem (1) lim ∥m (lk)∥L2( ) = 0.
k!1 (i)
uk+1 = arg (ii) lk+1 = lk + r max
min M (v, lk) ; v2H1(Ω ) {− lrk , − [uk+1]} .
Justifying the convergence presented method is complicated by the fact that the problem (7) can be unsolvable. The question of solvability of problem (7) is closely connected with the regularity of a solution u of problem (1). It can be proved if u ∈ H2 (Ω ), then −[ux2 ] ∈ H1=2 (γ) is a solution of problem (7) [Khludnev, 2010]. This assumption looks unnatural in crack problem. Despite this problem, the duality ratio can be proved for initial and dual problem [Vikhtenko & Namm, 2016].
Theorem 5. There is a duality ratio
sup l2L2( )
M (l) = inf J (v) .
v2K lim J (uk) = inf J (u) .
k!1 v2K
Note that if the dual problem (7) has a solution, then {lk} is a bounded sequence in L2 (γ) [4]. Together with theorem 4 it means that method (i), (ii) converges according the initial functional J (u), that is 4
Numerical Implementation of Method Ω = {(x, y) ∈ R2: 0 ≤ x ≤ 1; 0 ≤ y ≤ 1} and γ = {(x, y) ∈ Ω: a < x < b, y = c}. Approximation of the problem made by means of finite element method. To perform a triangulation of the region by step h, as shown in figure 1.
I is the set of indices of internal cracks nodes; IΩ is the set of indices of all other nodes of the triangulation.
For each node of triangulation we define a piecewise affine basis function φk (x, y). Also define functions φk+ (x, y) and φk (x, y) for each top and bottom node on the crack faces.
Construct an approximate solution as linear combination of basis functions φk (x, y).
vh (x, y) = ∑ vk φk (x, y) +
∑ vk+ φk+ (x, y) + k2IΩ k2I ∑ vk φk (x, y) . k2I Let us substitute function vh (x, y) in a modified Lagrangian functional M (v, l) taking change of variables vi+ = vi + ti, ti = [vi] ≥ 0, i ∈ I .
Also we perform trapezoid approximation of the integral ∫ (( (l − r[v])+)2 − l 2) dΓ ≈ h2 ∑ {((li − r ti)+)2 − li2} .
i2I As result we obtain finite-dimensional functional of the following form
M (vh, vh , th, lh)=
1 ∑
∑ aijvivj + ∑ 2 i2IΩ j2IΩ i2I j2I ∑ ai+j (vi + ti) (vj + tj)+ ∑
∑ aijvi vj + i2I j2I +2 ∑
∑ bi+jvi (vj + tj)+ 2 ∑ i2IΩ j2I
∑ bijvivj i2IΩ j2I − − ∑ fivi −
∑ fi+ (vi + ti)− i2IΩ ∑ fi vi + i2I 2r i2I h ∑ {((li − r ti)+)2 − li2} , where i2I ∫
Ω ai+j = ∫
Ω bi+j = ∫ Ω ∫ Ω ∫
Ω aij = ∇φi∇φjdΩ, i, j ∈ IΩ, ∫ Ω ∫
Ω ∇φi+∇φj+dΩ, aij =
∇φi ∇φj dΩ, i, j ∈ I , ∇φi∇φj+dΩ, bij =
∇φi∇φj dΩ, i ∈ IΩ, j ∈ I , fi = f ∇φidΩ, fi+ = f ∇φi+dΩ, fi =
f ∇φi dΩ, i ∈ I . ∫ Ω According to the Uzawa method on the iteration number k it is necessary to solve auxiliary problem { M (vh, vh , th, lhk)− min,
−ti ≤ 0, i ∈ I .
To solve this problem we apply the pointwise relaxation method.
Computational experiments were conducted with the following fixed parameters, which are presented in the table 1.
Consider three examples differing in way of defining function f in the computational domain Ω. The values of the function and the corresponding solution of the problem (1) is shown in figures 2–4.
In example 1 we attempt to raise the portion positioned below the crack. For this purpose, we define a positive value of the function f on the lower face of the crack, and negative on the top face. As seen from the corresponding graph solution in figure 2, the condition [v] = v+ − v ≥ 0 is satisfied on γ. Disconnection of crack faces does not occur; we obtain the solution of the Dirichlet problem.
In example 2, the value of the function f is reversed. In this case, we observe opening faces along the entire length of the crack. Corresponding graph solution is presented in figure 3. 1. The external iterations is the number of step (ii) of the Uzawa method to achieve the required accuracy. 2. The internal iterations is the number of step (i) of the Uzawa method to achieve the required accuracy.
The table 2 shows the calculation results for example 3 for different values of parameter r. Here null vectors chosen as an initial approximation.
Conducted computing experiments have shown that the convergence of the method to the saddle point of the Lagrangian functional is good enough. At that, increasing the value of the parameter r leads to growth convergence rate. For considered example on the grid in steps of h = 2 6, with an accuracy of ε = 10 8, it is necessary less than sixty external iterations. 5
These results allow us to conclude about the effectiveness of the chosen approach for solving model crack problem, one of the formulations of this is a minimization problem (1). The main advantage of using a modified Lagrangian functional is the shift step value by the dual variable is large positive value in distinct from the classical scheme of duality. Therefore duality method with a modified Lagrangian functional superior to the standard scheme uses classical functional for computing speed.