The Use of Fast Automatic Differentiation Technique for Solving Coefficient Inverse Problems VLADIMIR ZUBOV, ALLA ALBU Dorodnicyn Computing Centre of Russian Academy of Sciences Moscow, Russia
In studying and modeling heat propagation in complex porous composite materials both the convective and radiative heat transfer must be taken into account. The thermal conductivity coefficients in this case typically depend on the temperature. To estimate these coefficients, various models of the medium are used. As a result, one has to deal with a complex nonlinear model that describes the heat propagation in the composite material.
However, another approach is possible: a simplified model is constructed in which the radiative heat transfer is not taken into account, but its effect is modeled by an effective thermal conductivity coefficient that is determined based on experimental data. We consider one possible statement of the inverse coefficient problem. It is considered based on the Dirichlet problem for the one-dimensional unsteady-state heat equation. The inverse coefficient problem is reduced to a variational problem.
The identification problem of the model parameters Direct problem: ( ) T x t ( , ) T x t ( , ) ∂ ∂ ∂ C K T ( ) 0, ( , ) x t Q , ρ − = ∈ t x x ∂ ∂ ∂ ( 1 ) T x ( , 0) w x ( ), 0 x L , = ≤ ≤ 0 T (0, ) t w t ( ), T L t ( , ) w t ( ), 0 t . = = ≤ ≤ Θ 1 2 L A layer of material of width is considered Q ( 0 x L ) ( 0 t ) x - the Cartesian coordinate of the point in the layer x T x t ( , ) - the temperature of the material at the point with the coordinate t at the time ρ C and are the density and the heat capacity of the material, respectively K T ( ) - the coefficient of the convective thermal conductivity w 2 t ( ) w 0 x ( ), w 1 t ( ), - are given
The cost functional : L 2 K ( T ) T ( x , t ) ( x , t ) ( x , t ) dxdt 0 0 2 T K T ( 0 , t ) ( 0 , t ) ( t ) d t x 0 ( , ) x t 0 µ ≥ - a given weighting function 0 - a given number; β ≥ ( ) t Ρ - the known heat flux on the left boundary of the domain The optimal control problem is to find the optimal control T x t ( , ) K T ( ) and the corresponding optimal solution K ( T ) of problem (1) that minimizes functional
THE GRADIENT OF FUNCTIONAL IN THE CONTINUOUS CASE { } Let: a min min w x ( ), min w t ( ), min w t ( ) = 0 1 2 x [0, ] L t [0, ] t [0, ] ∈ ∈ Θ ∈ Θ { } b max max w x ( ), max w t ( ), max w t ( ) = 0 1 2 x [0, ] L t [0, ] t [0, ] ∈ ∈ Θ ∈ Θ 2 1 T ( x , t ) С ( Q ) C ( Q ) { } 1 - the class of the G K z ( ) : K z ( ) C ([ , ]), a b K z ( ) 0, z [ , ] a b = ∈ > ∈ feasible control functions z 1 - the specific internal energy E ( z ) ( ) C ( ) d C [ a , b ] 0 K ( z ) - the variation of the control function z T ( x , t ) - the variation of the phase variable xt
The Lagrange functional: L E ( T ) T ( x , t ) I K ( T ) p x , t K ( T ) d x d t t x x 0 0 2 1 p x t , C ( ) Q C Q ( ) ( ) - an arbitrary function I : The first variation of L E ( T ) T ( x , t ) I K ( T ) p x , t K ( T ) d x d t xt t x x 0 0 T ( x , t ) Denote: ( x , t ) K T ( x , t ) x
The adjoint problem 2 p ( x , t ) p ( x , t ) , E T ( x , t ) K ( T ) 2 ( x , t ) T ( x , t ) ( x , t ) 2 t x ( x , t ) Q , p ( x , ) 0 , ( 0 x L ), , p ( L , t ) 0 , t p ( 0 , t ) 2 ( 0 , t ) ( t ) ( 0 ) The first variation of the Lagrange functional p ( x , t ) T ( x , t ) I K ( T ) dx dt z x x Q
THE GRADIENT OF FUNCTIONAL I K T ( ) ( ), T T [ , ] a b ( ) ∇ = Μ ∈ ( i ) ( ) Fin p x ( , ), t ( , ) t ( , ) i i i ( ) sign J ( , ) d , i x i i ( i ) ( ) Ini i j Q Q , Q Q , i i j x ( , ) t ( , ) i ∂ ξ η ∂ ξ η i i x x ( , ) t t ( , ) = ξ η = ξ η ∂ξ ∂ξ i i J ( , ) ξ η = ∂ i x ( , ) t ( , ) ξ η ∂ ξ η i i x - the isolines of the field const η = η ∂η ∂η of temperature const - the lines orthogonal to ξ = const η = ( ) ( ) i ( ) i Q , η η - the interval o variation of the isothrems in Ini Fin i , ( i ) i ( ) - the coordinates of the endpoints of the isotherm ( ), ( ), Fin Ini ( , Q i ) ( i ) contained in const i Ini Fin
THE DISCRETE OPTIMAL CONTROL PROBLEM ~ ~ ~ ~ [ a , b ] T …, T a , T , , is partitioned by the points The interval T N b 0 2 1 into N parts. K ( T ) The function to be found is approximated by a continuous ~ N piecewise linear functions with the nodes at the points ( T n k , ) n n 0 so that k k ~ ~ ~ n 1 n T T T , ( n 0 ,..., N 1 ) K ( T ) k ( T T ), ~ ~ n n 1 n n T T n 1 n % k K T ( ) = n n
THE DISCRETE OPTIMAL CONTROL PROBLEM T was approximated by a continuous K T ( ) ( [ a , b ]) The function piecewise linear function ~ ~ ~ I j ~ j Nonuniform grid : x J T T ( x , t ) j t i i i i 0 j 0 Finite difference scheme that approximates the direct problem: j j T T j j j j i i 1 C ( h h ) T K ( T ) K ( T ) i i i i 1 i i i 1 h i j j T T j j j j i i 1 j 1 J , , K ( T ) K ( T ) D , I i 1 , 1 , i i i 1 h i 1 j 1 j 1 T T j j 1 j j 1 j 1 i i 1 D C ( h h ) T ( 1 ) K ( T ) K ( T ) i i i i i 1 i i i 1 h i j 1 j 1 T T j j 1 j 1 ( 1 ) K ( T ) K ( T ) i i 1 i i 1 h 0 i T ( w ) , i 1 ( 0 , I ), i 0 i j j T w , j j T w , j ( 0 , J ) 0 1 I 2 j % j % j 1 − % % h x x , i 0, I 1, t t , j 1, , J = − = − τ = − = i i 1 i +
Cost functional is approximated by the function F F ( k , k ,..., k ) 0 1 N with the aid of the trapezoids method: J I 1 j j 2 j j K ( T ) F ( T ) h i i i i j 1 i 1 J I 1 j j 2 j j K ( T ) F ( T ) h i i i i j 1 i 1 2 C h 1 j 1 j 1 j 1 j 1 j j 1 j j K ( T ) K ( T ) T T 0 0 0 T T 0 1 1 0 0 0 j 2 h 2 0
Fast Automatic Differentiation technique j T i , j , Z , U i i , j i , j F v F q , v , Z , U p k ( q , v ) ( q , v ) q k k n n n ( q , v ) K ( q , v ) j v p q , v , Z , , U p F i q v q , v q j j T T i i q , v Q q , v j Q q , v : T Z K q , v : k U i , j q , v i i , j n q , v j p - the values of conjugate variables (impulses) i
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