Extension of the adjoint method Stanislas Larnier Institut de - - PowerPoint PPT Presentation
Extension of the adjoint method Extension of the adjoint method Stanislas Larnier Institut de Mathmatiques de Toulouse Universit Paul Sabatier, France PICOF12 April 4, 2012 1 / 27 Extension of the adjoint method Introduction 1
Extension of the adjoint method
Extension of the adjoint method
Stanislas Larnier
Institut de Mathématiques de Toulouse Université Paul Sabatier, France
PICOF’12 April 4, 2012
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Extension of the adjoint method
1
Introduction
2
Adjoint method
3
Theoretical part
4
Numerical results
5
Conclusions and perspectives
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Extension of the adjoint method Introduction
1
Introduction
2
Adjoint method
3
Theoretical part
4
Numerical results
5
Conclusions and perspectives
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Extension of the adjoint method Introduction
Topology optimization formulates a design problem as an optimal material distribution problem. The search of an optimal domain is equivalent to finding its characteristic function, it is a 0-1 optimization problem. Different approaches make this problem differentiable: Relaxation, homogenization, Level set, Topological derivatives.
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Extension of the adjoint method Introduction
To present the basic idea, let Ω be a domain of Rd, d ∈ N\{0} and j(Ω) = J(uΩ) a cost function to be minimized, where uΩ is a solution to a given partial differential equation defined in Ω. Let x be a point in Ω and ω1 a smooth open bounded subset in Rd containing the origin. For a small parameter ρ > 0, let Ω\ωρ be the perturbed domain obtained by making a perforation ωρ = ρω1 around the point x. The topological asymptotic expansion of j (Ω\ωρ) when ρ tends to zero is the following: j (Ω\ωρ) = j(Ω) + f(ρ)g(x) + o(f(ρ)). where f(ρ) denotes an explicit positive function going to zero with ρ and g(x) is called the topological gradient or topological derivative. It is usually simple to compute and is obtained using the solution of direct and adjoint problems defined on the initial domain.
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Extension of the adjoint method Introduction
In topology optimization, there are some drawbacks of topological derivatives approaches: The asymptotic topological expansion is not easy to obtain for complex problems. It needs to be adapted for many particular cases such as the creation of a hole on the boundary of an existing one or on the
It is difficult to determine the variation of a cost function when a hole is to be filled. In real applications of topology optimization, a finite perturbation is performed and not an infinitesimal one.
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Extension of the adjoint method Adjoint method
1
Introduction
2
Adjoint method
3
Theoretical part
4
Numerical results
5
Conclusions and perspectives
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Extension of the adjoint method Adjoint method
Consider the following steady state equation F(c, u) = 0 in Ω, where c is a distributed parameter in a domain Ω. The aim is to minimize a cost function j(c) := J(uc) where uc is the solution of the direct equation for a given c. Let us suppose that every term is differentiable. We are considering a perturbation δc of the parameter c. The direct equation can be seen as a constraint, and as a consequence, the Lagrangian is considered: L(c, u, p) = J(u) + (F(c, u), p), where p is a Lagrange multiplier and (·, ·) denotes the scalar product in a well-chosen Hilbert space.
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Extension of the adjoint method Adjoint method
To compute the derivative of j, one can remark that j(c) = L(c, uc, p) for all c, if uc is the solution of the direct equation. The derivative of j is then equal to the derivative of L with respect to c: dcj(c)δc = ∂cL(c, uc, p)δc + ∂uL(c, uc, p)∂cu δc. All these terms can be calculated easily, except ∂cu δc, the solution of the linearized problem: ∂uF(c, uc)(∂cu δc) = −∂cF(c, uc) δc. To avoid the resolution of this equation for each δc, the term ∂uL(c, uc, p) is cancelled by solving the following adjoint equation. Let pc be the solution of the adjoint equation: ∂uF(c, uc)Tpc = −∂uJT. So the derivative of j is explicitly given by dcj(c)δc = ∂cL(c, uc, pc)δc.
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Extension of the adjoint method Adjoint method
Note that if the Lagrangians L(c + δc, . . . , . . . ) and L(c, . . . , . . . ) are defined on the same space, we have
j(c + δc) − j(c) =L(c + δc, uc+δc, pc) − L(c, uc, pc) =(L(c + δc, uc+δc, pc) − L(c + δc, uc, pc)) + (L(c + δc, uc, pc) − L(c, uc, pc)).
In the case of a regular perturbation δc, the second term gives the main variation and the first term is of higher order. In the case of a singular perturbation, the first term is of the same
uc has to be estimated. The basic idea of the numerical vault is to update the solution uc by solving a local problem defined in a small domain around x0.
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Extension of the adjoint method Theoretical part
1
Introduction
2
Adjoint method
3
Theoretical part
4
Numerical results
5
Conclusions and perspectives
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Extension of the adjoint method Theoretical part
In the linear case is studied, consider the variational problem depending on a parameter ε: aε(uε, v) = ℓε(v) ∀v ∈ Vε, where Vε is a Hilbert space, aε is a bilinear, continuous and coercive form and ℓε is a linear and continuous form. Typically, Vε est tel que H1
0 ⊂ Vε ⊂ H1.
The aim is to minimize a cost function which depends of ε. j(ε) := Jε(uε). The cost function Jε is of class C1, the adjoint problem associated to the problem is aε(w, pε) = −∂uJε(uε)w ∀w ∈ Vε, where pε is the solution of this problem.
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Extension of the adjoint method Theoretical part
Suppose that aε, ℓε and Jε are integrals over a domain Ω. The domain Ω is split into two parts, a part D containing the perturbation, and its complementary Ω0 = Ω\D. The forms aε, ℓε, and the cost function Jε are decomposed in the following way: aε = aΩ0 + aε
D,
ℓε = ℓΩ0 + ℓε
D,
Jε = JΩ0 + Jε
D,
where aΩ0, lΩ0 et JΩ0 are independants of ε.
VΩ0 , the space consisting of functions of Vε and V0 restricted to Ω0, Vε
D , the space consisting of functions of Vε restricted to D,
V0
D , the space consisting of functions of V0 restricted to D, 13 / 27
Extension of the adjoint method Theoretical part
We assume that V0 ⊂ Vε. Let us consider uε
D, the local update of u0:
Find uε
D ∈ Vε D solution of
aε
D(uε D, v)
= ℓε
D(v),
∀v ∈ Vε
D,0,
uε
D
= u0
The update of u0, named ˜ uε, is given by: ˜ uε = uε
D
in D, u0 in Ω0.
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Extension of the adjoint method Theoretical part
Hypotheses: There exist three positive constants η, C and Cu independent of ε and a positive real valued function f defined on R+ such that lim
ε→0 f(ε) = 0,
Jε(v) − Jε(u) − ∂uJε(u)(v − u)Vε ≤ Cv − u2
Vε, ∀v, u ∈ B(u0, η),
uε − u0VΩ0 ≤ Cuf(ε), lim
ε→0 pε − p0Vε = 0.
Proposition Under these hypotheses, we have uε − ˜ uεVε = O(f(ε)). Theorem (Update of the direct solution) Under these hypotheses, we have j(ε) − j(0) = Lε(˜ uε, p0) − L0(u0, p0) + o (f(ε)) .
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Extension of the adjoint method Theoretical part
V0 is not necessary a sub-space of Vε. The definition of ˜ uε stays the same. Let us consider pε
D, the local update of p0
Find pε
D ∈ Vε D solution of
aε
D(w, pε D)
= −∂uJε
D(uε D)w,
∀w ∈ Vε
D,0,
pε
D
= p0
The update of p0, named ˜ pε, is given by: ˜ pε = pε
D
in D, p0 in Ω0.
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Extension of the adjoint method Theoretical part
Hypotheses: There exist four positive constants η, C, Cu and Cp independent of ε and a positive real valued function f defined on R+ such that lim
ε→0 f(ε) = 0,
Jε(v) − Jε(u) − ∂uJε(u)(v − u)Vε ≤ Cv − u2
Vε, ∀v, u ∈ B(u0, η),
uε − u0VΩ0 ≤ Cuf(ε), pε − p0VΩ0 ≤ Cpf(ε). Proposition Under these hypotheses, we have pε − ˜ pεVε = O(f(ε)). Theorem (Update of the direct and adjoint solutions) Under these hypotheses, we have j(ε) − j(0) = Lε(˜ uε, ˜ pε) − L0(u0, p0) + O(f(ε)2).
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Extension of the adjoint method Numerical results
1
Introduction
2
Adjoint method
3
Theoretical part
4
Numerical results
5
Conclusions and perspectives
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Extension of the adjoint method Numerical results
Let Ω be a rectangular bounded domain of R2 and Γ be its boundary, composed of two parts Γ1 and Γ2. The points of the rectangle are submitted to a vertical displacement u solution of the following equation: −∇.σ(u) = 0 in Ω, u = 0
σ(u)n = µ
where φ(u) = 1
2 (Du + DuT ), σ(u) = hH0φ(u), σ(u) is the stress distribution, H0 is the
Hooke tensor and h(x) represents the material stiffness. The optimization problem is to minimize the following cost function: j(h) =
g.u dx,
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Extension of the adjoint method Numerical results
−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5 1 1.5 2 2.5 −1 −0.5 0.5 1 1.5 x 10
−70.5 1 1.5 2 2.5 −6 −4 −2 2 4 6 8 10 x 10
−120.5 1 1.5 2 2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 x 10
−7From up to down, the perturbations and the associated curves
The abscissa is the inhomogeneity stiffness. The stiffness varies from 0 to 2.5 with the material stiffness equal to 1. The ordinate is the variation of the cost function.
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Extension of the adjoint method Numerical results
−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5 1 1.5 2 2.5 −1 −0.5 0.5 1 1.5 2 x 10
−60.5 1 1.5 2 2.5 −5 5 10 x 10
−70.5 1 1.5 2 2.5 −3 −2 −1 1 2 3 4 5 x 10
−7From up to down, the perturbations and the associated curves
The abscissa is the inhomogeneity stiffness. The stiffness varies from 0 to 2.5 with the material stiffness equal to 1. The ordinate is the variation of the cost function.
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Extension of the adjoint method Numerical results
−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.2 0.4 0.6 0.8 1 −6 −5 −4 −3 −2 −1 0 x 10
−70.2 0.4 0.6 0.8 1 −2.5 −2 −1.5 −1 −0.5 0.5 x 10
−7From up to down, the perturbations and the associated curves
The abscissa is the inhomogeneity stiffness. The stiffness varies from 0 to 1 with the material stiffness equal to 1. The ordinate is the variation of the cost function.
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Extension of the adjoint method Numerical results
Let Ω be a rectangular bounded domain of R2 and Γ be its boundary, composed of two parts Γ1 and Γ2. The points of the rectangle are submitted to a horizontal displacement u solution of the following equation: −∇.σ(u) = 0 in Ω, u = 0
σ(u)n = µ
where φ(u) = 1
2 (Du + DuT ), σ(u) = hH0φ(u), σ(u) is a stress distribution, H0 is the
Hooke tensor and h(x) = 1 or ρ represents the material stiffness. The optimization problem is to minimize the following cost function: J(h) =
|uh − uobs|2 dx.
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Extension of the adjoint method Numerical results
−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −4 −3 −2 −1 x 10
−4−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 x 10
−4−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −5 −4 −3 −2 −1 1 x 10
−11From left to right, in the first row, the horizontal displacements for the uniform material and the perturbed material. In the second row, the inclusion and the detection obtained with the numerical vault.
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Extension of the adjoint method Conclusions and perspectives
1
Introduction
2
Adjoint method
3
Theoretical part
4
Numerical results
5
Conclusions and perspectives
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Extension of the adjoint method Conclusions and perspectives
Creation of a new method to solve some difficulties of topological derivatives. Major advantages of the method are that the numerical vault is non invasive and can be used in different kinds of problems with a parallel computing implementation. Our method could be a tool in theoretical investigations.
Perspectives coupling with multiscale algorithms, semilinear equations.
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Extension of the adjoint method Conclusions and perspectives
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