Explicit solutions in optimal design problems for ...
Transcript of Explicit solutions in optimal design problems for ...
University of Osijek - Department of Mathematics
Explicit solutions in optimal designproblems for stationary diffusion equation
Kresimir Burazin
University J. J. Strossmayer of OsijekDepartment of MathematicsTrg Ljudevita Gaja 631000 Osijek, Croatia
http://www.mathos.unios.hr/˜[email protected]
Joint work with Marko Vrdoljak
Kresimir Burazin Benasque, September 2015 1/18
University of Osijek - Department of Mathematics
Outline
Compliance optimization, composite materials and relaxation
Multiple states - spherically symmetric case
Examples
Kresimir Burazin Benasque, September 2015 2/18
University of Osijek - Department of Mathematics
Optimal design problem (single state)
� � Rd open and bounded, f 2 L2(�) given;
Kresimir Burazin Benasque, September 2015 3/18
University of Osijek - Department of Mathematics
Optimal design problem (single state)
� � Rd open and bounded, f 2 L2(�) given; stationary diffusionequation with homogenous Dirichlet b. c.:
�� div (Ar u) = fu 2 H1
0(�) ; (1)
Kresimir Burazin Benasque, September 2015 3/18
University of Osijek - Department of Mathematics
Optimal design problem (single state)
� � Rd open and bounded, f 2 L2(�) given; stationary diffusionequation with homogenous Dirichlet b. c.:
�� div (Ar u) = fu 2 H1
0(�) ; (1)
where A is a mixture of two isotropic materials with conductivities0 < � < �:
Kresimir Burazin Benasque, September 2015 3/18
University of Osijek - Department of Mathematics
Optimal design problem (single state)
� � Rd open and bounded, f 2 L2(�) given; stationary diffusionequation with homogenous Dirichlet b. c.:
�� div (Ar u) = fu 2 H1
0(�) ; (1)
where A is a mixture of two isotropic materials with conductivities0 < � < �: A = ��I + (1 � �)�I, where � 2 L1 (�; f 0; 1g),R
� � dx = q� , for given 0 < q� < j�j.
Kresimir Burazin Benasque, September 2015 3/18
University of Osijek - Department of Mathematics
Optimal design problem (single state)
� � Rd open and bounded, f 2 L2(�) given; stationary diffusionequation with homogenous Dirichlet b. c.:
�� div (Ar u) = fu 2 H1
0(�) ; (1)
where A is a mixture of two isotropic materials with conductivities0 < � < �: A = ��I + (1 � �)�I, where � 2 L1 (�; f 0; 1g),R
� � dx = q� , for given 0 < q� < j�j.For given � , �, �, q� and f we want to find such material A whichminimizes the compliance functional (total amount of heat/electrical energydissipated in � ):
J(�) =Z
�f (x)u(x)dx =
Z
�A(x)r u(x) �r u(x) dx �! min ;
where u is the solution of the state equation (1).Kresimir Burazin Benasque, September 2015 3/18
University of Osijek - Department of Mathematics
Relaxation by homogenisation
� 2 L1 (�; f 0; 1g) ��� � 2 L1 (�; [0; 1])A = ��I + (1 � �)�I A 2 K(�) a.e. on �
classical material composite material - relaxation
Kresimir Burazin Benasque, September 2015 4/18
University of Osijek - Department of Mathematics
Relaxation by homogenisation
� 2 L1 (�; f 0; 1g) ��� � 2 L1 (�; [0; 1])A = ��I + (1 � �)�I A 2 K(�) a.e. on �
classical material composite material - relaxation
DefinitionA sequence of matrix functions A" is said to H-converge to A� if for every fthe sequence of solutions of
�� div (A" r u" ) = fu" 2 H1
0(�)
satisfies u" * u in H10(�), A" r u" * A�r u in L2(�;Rd ), where u is the
solution of the homogenised equation�
� div (A�r u) = fu 2 H1
0(�) :Kresimir Burazin Benasque, September 2015 4/18
University of Osijek - Department of Mathematics
Composite material
DefinitionIf a sequence of characteristic functions �" 2 L1 (�; f 0; 1g) andconductivities
A" (x) = �" (x)�I + (1 � �" (x))�I
satisfy �" * �weakly �and A" H-converges to A�, then it is said that A�
is homogenised tensor of two-phase composite material with proportions �of first material and microstructure defined by the sequence (�" ).
Kresimir Burazin Benasque, September 2015 5/18
University of Osijek - Department of Mathematics
Composite material
DefinitionIf a sequence of characteristic functions �" 2 L1 (�; f 0; 1g) andconductivities
A" (x) = �" (x)�I + (1 � �" (x))�I
satisfy �" * �weakly �and A" H-converges to A�, then it is said that A�
is homogenised tensor of two-phase composite material with proportions �of first material and microstructure defined by the sequence (�" ).Example – simple laminates: if �" depend only on x1, then
A� = diag(��� ; �+
� ; �+� ; : : : ; �+
� ) ;where
�+� = �� + (1 � �)� ; 1
���
= �� + 1 � �
� :
Kresimir Burazin Benasque, September 2015 5/18
University of Osijek - Department of Mathematics
Composite material
DefinitionIf a sequence of characteristic functions �" 2 L1 (�; f 0; 1g) andconductivities
A" (x) = �" (x)�I + (1 � �" (x))�I
satisfy �" * �weakly �and A" H-converges to A�, then it is said that A�
is homogenised tensor of two-phase composite material with proportions �of first material and microstructure defined by the sequence (�" ).Example – simple laminates: if �" depend only on x1, then
A� = diag(��� ; �+
� ; �+� ; : : : ; �+
� ) ;where
�+� = �� + (1 � �)� ; 1
���
= �� + 1 � �
� :
Set of all composites:
A := f (�;A) 2 L1 (�; [0; 1]� Md(R)) :Z
��dx = q� ; A 2 K(�) a.e.g
Kresimir Burazin Benasque, September 2015 5/18
University of Osijek - Department of Mathematics
Effective conductivities – set K(�)
G–closure problem: for given �find allpossible homogenised (effective)tensors A�
Kresimir Burazin Benasque, September 2015 6/18
University of Osijek - Department of Mathematics
Effective conductivities – set K(�)
G–closure problem: for given �find allpossible homogenised (effective)tensors A�
K(�) is given in terms of eigenvalues(Murat & Tartar; Lurie & Cherkaev):
��� � �j � �+
� j = 1; : : : ; ddX
j= 1
1
�j � � � 1
��� � �
+ d � 1
�+� � �
dX
j= 1
1
� � �j� 1
� � ���
+ d � 1
� � �+�
;
2D:
O �1
�2
�� = 1
�
�� = 0
�
�+�
�+�
���
���
K(�)
Kresimir Burazin Benasque, September 2015 6/18
University of Osijek - Department of Mathematics
Effective conductivities – set K(�)
G–closure problem: for given �find allpossible homogenised (effective)tensors A�
K(�) is given in terms of eigenvalues(Murat & Tartar; Lurie & Cherkaev):
��� � �j � �+
� j = 1; : : : ; ddX
j= 1
1
�j � � � 1
��� � �
+ d � 1
�+� � �
dX
j= 1
1
� � �j� 1
� � ���
+ d � 1
� � �+�
;
2D:
�1
�2
���
�+�
��� �+
�
Kresimir Burazin Benasque, September 2015 6/18
University of Osijek - Department of Mathematics
Effective conductivities – set K(�)
G–closure problem: for given �find allpossible homogenised (effective)tensors A�
K(�) is given in terms of eigenvalues(Murat & Tartar; Lurie & Cherkaev):
��� � �j � �+
� j = 1; : : : ; ddX
j= 1
1
�j � � � 1
��� � �
+ d � 1
�+� � �
dX
j= 1
1
� � �j� 1
� � ���
+ d � 1
� � �+�
;
2D:
�1
�2
���
�+�
��� �+
�
3D:
�1
�2
�3
Kresimir Burazin Benasque, September 2015 6/18
University of Osijek - Department of Mathematics
Effective conductivities – set K(�)
G–closure problem: for given �find allpossible homogenised (effective)tensors A�
K(�) is given in terms of eigenvalues(Murat & Tartar; Lurie & Cherkaev):
��� � �j � �+
� j = 1; : : : ; ddX
j= 1
1
�j � � � 1
��� � �
+ d � 1
�+� � �
dX
j= 1
1
� � �j� 1
� � ���
+ d � 1
� � �+�
;
minA J is a proper relaxation ofminL1 (� ;f 0;1g) I
2D:
�1
�2
���
�+�
��� �+
�
3D:
�1
�2
�3
Kresimir Burazin Benasque, September 2015 6/18
University of Osijek - Department of Mathematics
Multiple state optimal design problem
State equations�
� div (Ar ui ) = fiui 2 H1
0(�) i = 1; : : : ; m
State function u = (u1; : : : ; um)
Kresimir Burazin Benasque, September 2015 7/18
University of Osijek - Department of Mathematics
Multiple state optimal design problem
State equations�
� div (Ar ui ) = fiui 2 H1
0(�) i = 1; : : : ; m
State function u = (u1; : : : ; um)8>>>><
>>>>:
I(�) =P m
i= 1 �iR
� fiui dx ! min
u = (u1; : : : ; um) state function for A = ��I + (1 � �)�I
� 2 L1 (�; f 0; 1g) ;Z
�� dx = q� ;
for some given weights �i > 0.
Kresimir Burazin Benasque, September 2015 7/18
University of Osijek - Department of Mathematics
Multiple state optimal design problem
State equations�
� div (Ar ui ) = fiui 2 H1
0(�) i = 1; : : : ; m
State function u = (u1; : : : ; um)8>>>><
>>>>:
I(�) =P m
i= 1 �iR
� fiui dx ! min
u = (u1; : : : ; um) state function for A = ��I + (1 � �)�I
� 2 L1 (�; f 0; 1g) ;Z
�� dx = q� ;
for some given weights �i > 0. Proper relaxation:
J(�;A) =mX
i= 1
�i
Z
�fiui dx ! min on
A := f (�;A) 2 L1 (�; [0; 1] � Md(R)) :Z
��dx = q� ; A 2 K(�) a.e.g
Kresimir Burazin Benasque, September 2015 7/18
University of Osijek - Department of Mathematics
How do we find a solution?
A. Single state equation: [Murat & Tartar,1985] This problem can be rewritten as asimpler convex minimization problem.
Kresimir Burazin Benasque, September 2015 8/18
University of Osijek - Department of Mathematics
How do we find a solution?
A. Single state equation: [Murat & Tartar,1985] This problem can be rewritten as asimpler convex minimization problem.
I(�) =Z
�fu dx �! min
T =�
� 2 L1 (�; [0; 1]) :R
� � = q��
� 2 T ; and u determined uniquely by8<
:� div (�+
� r u) = f
u 2 H10(�)
Kresimir Burazin Benasque, September 2015 8/18
University of Osijek - Department of Mathematics
How do we find a solution?
A. Single state equation: [Murat & Tartar,1985] This problem can be rewritten as asimpler convex minimization problem.
I(�) =Z
�fu dx �! min
T =�
� 2 L1 (�; [0; 1]) :R
� � = q��
� 2 T ; and u determined uniquely by8<
:� div (�+
� r u) = f
u 2 H10(�)
B. Multiple state equations: Simplerrelaxation fails:
Kresimir Burazin Benasque, September 2015 8/18
University of Osijek - Department of Mathematics
How do we find a solution?
A. Single state equation: [Murat & Tartar,1985] This problem can be rewritten as asimpler convex minimization problem.
I(�) =Z
�fu dx �! min
T =�
� 2 L1 (�; [0; 1]) :R
� � = q��
� 2 T ; and u determined uniquely by8<
:� div (�+
� r u) = f
u 2 H10(�)
B. Multiple state equations: Simplerrelaxation fails:
I(�) =mX
i= 1
�i
Z
�fiui dx �! min
T =�
� 2 L1 (�; [0; 1]) :R
� � = q��
� 2 T ; and ui determined uniquely by8<
:� div (�+
� r ui ) = fi
ui 2 H10(�)
i = 1; : : : ; m ;
Kresimir Burazin Benasque, September 2015 8/18
University of Osijek - Department of Mathematics
How do we find a solution?
A. Single state equation: [Murat & Tartar,1985] This problem can be rewritten as asimpler convex minimization problem.
I(�) =Z
�fu dx �! min
T =�
� 2 L1 (�; [0; 1]) :R
� � = q��
� 2 T ; and u determined uniquely by8<
:� div (�+
� r u) = f
u 2 H10(�)
B. Multiple state equations: Simplerrelaxation fails:
I(�) =mX
i= 1
�i
Z
�fiui dx �! min
T =�
� 2 L1 (�; [0; 1]) :R
� � = q��
� 2 T ; and ui determined uniquely by8<
:� div (�+
� r ui ) = fi
ui 2 H10(�)
i = 1; : : : ; m ;
In spherically symmetric case the simplerrelaxation can be done!
Kresimir Burazin Benasque, September 2015 8/18
University of Osijek - Department of Mathematics
Relaxed designs
A := f (�;A) 2 L1 (�; [0; 1]� Md(R)) :Z
��dx = q� ; A 2 K(�) a.e.g
O �1
�2
�� = 1
�
�� = 0
�
�+�
�+�
���
���
K(�)
Kresimir Burazin Benasque, September 2015 9/18
University of Osijek - Department of Mathematics
Relaxed designs
A := f (�;A) 2 L1 (�; [0; 1]� Md(R)) :Z
��dx = q� ; A 2 K(�) a.e.g
Further relaxation:
B : : :R
� �dx = q�
��� � �min(A) ; �max(A) � �+
�
O �1
�2
�� = 1
�
�� = 0
�
�+�
�+�
���
���
K(�)
Kresimir Burazin Benasque, September 2015 9/18
University of Osijek - Department of Mathematics
Relaxed designs
A := f (�;A) 2 L1 (�; [0; 1]� Md(R)) :Z
��dx = q� ; A 2 K(�) a.e.g
Further relaxation:
B : : :R
� �dx = q�
��� � �min(A) ; �max(A) � �+
�
B is convex and compact and J iscontinuous on B, so there is asolution of minB J.
O �1
�2
�� = 1
�
�� = 0
�
�+�
�+�
���
���
K(�)
Kresimir Burazin Benasque, September 2015 9/18
University of Osijek - Department of Mathematics
Equivalence of minB J and minT I
Theorem
I There is unique u� 2 H10(�;Rm) which is the state for every solution
of minB J and minT I.
Kresimir Burazin Benasque, September 2015 10/18
University of Osijek - Department of Mathematics
Equivalence of minB J and minT I
Theorem
I There is unique u� 2 H10(�;Rm) which is the state for every solution
of minB J and minT I.I If (��;A�) is an optimal design for the problem minB J, then �� is
optimal design for minT I.
Kresimir Burazin Benasque, September 2015 10/18
University of Osijek - Department of Mathematics
Equivalence of minB J and minT I
Theorem
I There is unique u� 2 H10(�;Rm) which is the state for every solution
of minB J and minT I.I If (��;A�) is an optimal design for the problem minB J, then �� is
optimal design for minT I.I Conversely, if �� is a solution of optimal design problem minT I, then
any (��;A�) 2 B satisfying A�r u�i = �+ (��)r u�
i almosteverywhere on � (e.g. A� = �+ (��)I) is an optimal design for theproblem minB J.
Kresimir Burazin Benasque, September 2015 10/18
University of Osijek - Department of Mathematics
Equivalence of minB J and minT I
Theorem
I There is unique u� 2 H10(�;Rm) which is the state for every solution
of minB J and minT I.I If (��;A�) is an optimal design for the problem minB J, then �� is
optimal design for minT I.I Conversely, if �� is a solution of optimal design problem minT I, then
any (��;A�) 2 B satisfying A�r u�i = �+ (��)r u�
i almosteverywhere on � (e.g. A� = �+ (��)I) is an optimal design for theproblem minB J.
I If m < d, then there exists minimizer (��;A�) for J on B, such that(��;A�) 2 A , and thus it is also minimizer for J on A .
Kresimir Burazin Benasque, September 2015 10/18
University of Osijek - Department of Mathematics
Simpler relaxation in case of spherical symmetry
TheoremLet � � Rd be spherically symmetric, and let the right-hand sidesfi = fi (r ), r 2 ! , i = 1; : : : ; m be a radial function. Then there exists aminimizer (��;A�) of the optimal design problem minA J which is a radialfunction.
Kresimir Burazin Benasque, September 2015 11/18
University of Osijek - Department of Mathematics
Simpler relaxation in case of spherical symmetry
TheoremLet � � Rd be spherically symmetric, and let the right-hand sidesfi = fi (r ), r 2 ! , i = 1; : : : ; m be a radial function. Then there exists aminimizer (��;A�) of the optimal design problem minA J which is a radialfunction. More preciselya) For any minimizer �of functional I over T , let us define a radial
function �� : � �! R as the average value over spheres of �: forr 2 ! we take
��(r) := �Z
@B(0;r )�dS ;
where S denotes the surface measure on a sphere. Then �� is alsominimizer for I over T .
Kresimir Burazin Benasque, September 2015 11/18
University of Osijek - Department of Mathematics
Simpler relaxation in case of spherical symmetry. . . cont.
Theorem
b) For any radial minimizer �� of I over T , let us define A� as a simplelaminate with layers orthogonal to a radial direction er and localproportion of the first material ��. To be specific, we can defineA� : � �! Md(R) in the following way:
Kresimir Burazin Benasque, September 2015 12/18
University of Osijek - Department of Mathematics
Simpler relaxation in case of spherical symmetry. . . cont.
Theorem
b) For any radial minimizer �� of I over T , let us define A� as a simplelaminate with layers orthogonal to a radial direction er and localproportion of the first material ��. To be specific, we can defineA� : � �! Md(R) in the following way:
I If x = re1 = (r; 0; 0; : : : ; 0), then
A�(x) := diag(�+ (��(r)); �� (��(r)); �+ (��(r)); : : : ; �+ (��(r))) :
Kresimir Burazin Benasque, September 2015 12/18
University of Osijek - Department of Mathematics
Simpler relaxation in case of spherical symmetry. . . cont.
Theorem
b) For any radial minimizer �� of I over T , let us define A� as a simplelaminate with layers orthogonal to a radial direction er and localproportion of the first material ��. To be specific, we can defineA� : � �! Md(R) in the following way:
I If x = re1 = (r; 0; 0; : : : ; 0), then
A�(x) := diag(�+ (��(r)); �� (��(r)); �+ (��(r)); : : : ; �+ (��(r))) :
I For all other x 2 �, we take the unique rotation R(x) 2 SO(d) suchthat x = jxjR(x)e1, and define
A�(x) := R(x)A�(R�(x)x)R�(x) :
Kresimir Burazin Benasque, September 2015 12/18
University of Osijek - Department of Mathematics
Simpler relaxation in case of spherical symmetry. . . cont.
Theorem
b) For any radial minimizer �� of I over T , let us define A� as a simplelaminate with layers orthogonal to a radial direction er and localproportion of the first material ��. To be specific, we can defineA� : � �! Md(R) in the following way:
I If x = re1 = (r; 0; 0; : : : ; 0), then
A�(x) := diag(�+ (��(r)); �� (��(r)); �+ (��(r)); : : : ; �+ (��(r))) :
I For all other x 2 �, we take the unique rotation R(x) 2 SO(d) suchthat x = jxjR(x)e1, and define
A�(x) := R(x)A�(R�(x)x)R�(x) :
Then (��;A�) is a radial optimal design for minB J.
Kresimir Burazin Benasque, September 2015 12/18
University of Osijek - Department of Mathematics
Simpler relaxation in case of spherical symmetry. . . cont.
Theorem
b) For any radial minimizer �� of I over T , let us define A� as a simplelaminate with layers orthogonal to a radial direction er and localproportion of the first material ��. To be specific, we can defineA� : � �! Md(R) in the following way:
I If x = re1 = (r; 0; 0; : : : ; 0), then
A�(x) := diag(�+ (��(r)); �� (��(r)); �+ (��(r)); : : : ; �+ (��(r))) :
I For all other x 2 �, we take the unique rotation R(x) 2 SO(d) suchthat x = jxjR(x)e1, and define
A�(x) := R(x)A�(R�(x)x)R�(x) :
Then (��;A�) is a radial optimal design for minB J.Moreover, (��;A�) 2 A , and thus it is also a solution for minA J.
Kresimir Burazin Benasque, September 2015 12/18
University of Osijek - Department of Mathematics
Optimality conditions for minT I
Lemma�� 2 T is a solution minT I if and only if there exists a Lagrange multiplierc � 0 such that
�� 2 h0; 1i )mX
i= 1
�i jr u�i j2 = c ;
�� = 0 )mX
i= 1
�i jr u�i j2 � c ;
�� = 1 )mX
i= 1
�i jr u�i j2 � c ;
or equivalentlymX
i= 1
�i jr u�i j2 > c ) �� = 0 ;
mX
i= 1
�i jr u�i j2 < c ) �� = 1 :
Kresimir Burazin Benasque, September 2015 13/18
University of Osijek - Department of Mathematics
Ball with nonconstant right-hand side
In all examples � = 1, � = 2.
� = B(0; 2) � R2, one state equation, f (r ) = 1 � r
Kresimir Burazin Benasque, September 2015 14/18
University of Osijek - Department of Mathematics
Ball with nonconstant right-hand side
In all examples � = 1, � = 2.
� = B(0; 2) � R2, one state equation, f (r ) = 1 � r
State equation in polar coordinates � 1
r
�r�+
�(r)u0�0
= 1 � r :
Integration gives ju0(r)j = (r)��(r)+ �(1� �(r)) ; where (r ) = j2r2� 3r j
6 :
Kresimir Burazin Benasque, September 2015 14/18
University of Osijek - Department of Mathematics
Ball with nonconstant right-hand side
In all examples � = 1, � = 2.
� = B(0; 2) � R2, one state equation, f (r ) = 1 � r
State equation in polar coordinates � 1
r
�r�+
�(r)u0�0
= 1 � r :
Integration gives ju0(r)j = (r)��(r)+ �(1� �(r)) ; where (r ) = j2r2� 3r j
6 :
Conditions of optimality: there exists a constant � :=p
c > 0 such thatfor optimal �� we have:
ju0(r)j > � ) ��(r) = 0
) g� := � > �
ju0(r)j < � ) ��(r) = 1
) g� := � < �
�� 2 h0; 1i ) ju0(r)j = �) ��(r) = ��� (r)
�(�� �)Kresimir Burazin Benasque, September 2015 14/18
University of Osijek - Department of Mathematics
Ball with nonconstant right-hand side
In all examples � = 1, � = 2.
� = B(0; 2) � R2, one state equation, f (r ) = 1 � r
State equation in polar coordinates � 1
r
�r�+
�(r)u0�0
= 1 � r :
Integration gives ju0(r)j = (r)��(r)+ �(1� �(r)) ; where (r ) = j2r2� 3r j
6 :
Conditions of optimality: there exists a constant � :=p
c > 0 such thatfor optimal �� we have:
ju0(r)j > � ) ��(r) = 0
) g� := � > �
ju0(r)j < � ) ��(r) = 1
) g� := � < �
�� 2 h0; 1i ) ju0(r)j = �) ��(r) = ��� (r)
�(�� �)34
20
r
ju0j
�1
�2
g��3
�4g�
Kresimir Burazin Benasque, September 2015 14/18
University of Osijek - Department of Mathematics
Ball with nonconstant right-hand side
Lagrange multiplier � is uniquely determined by the constraint�R
� �� dx = � := q�j� j 2 [0; 1], which is algebraic equation for �.
Kresimir Burazin Benasque, September 2015 15/18
University of Osijek - Department of Mathematics
Ball with nonconstant right-hand side
Lagrange multiplier � is uniquely determined by the constraint�R
� �� dx = � := q�j� j 2 [0; 1], which is algebraic equation for �.
�1 �2 �3 10 �
�
�1
�2
�3
�4
Kresimir Burazin Benasque, September 2015 15/18
University of Osijek - Department of Mathematics
Ball with nonconstant right-hand side
Lagrange multiplier � is uniquely determined by the constraint�R
� �� dx = � := q�j� j 2 [0; 1], which is algebraic equation for �.
�1 �2 �3 10 �
�
�1
�2
�3
�4
20r
ju0j
�
g�g�
p�1 q�
1 p�2q�
2 p�3 q�
3
��
��
Kresimir Burazin Benasque, September 2015 15/18
University of Osijek - Department of Mathematics
Multiple states
Two state equations on a ball � = B(0; 2)I f1 = �B(0;1) ; f2 � 1 ;
Kresimir Burazin Benasque, September 2015 16/18
University of Osijek - Department of Mathematics
Multiple states
Two state equations on a ball � = B(0; 2)I f1 = �B(0;1) ; f2 � 1 ;
I
�� div (�+
� r ui ) = fiui 2 H1
0(�) i = 1; 2
Kresimir Burazin Benasque, September 2015 16/18
University of Osijek - Department of Mathematics
Multiple states
Two state equations on a ball � = B(0; 2)I f1 = �B(0;1) ; f2 � 1 ;
I
�� div (�+
� r ui ) = fiui 2 H1
0(�) i = 1; 2
I �Z
�f1u1 dx +
Z
�f2u2 dx ! min
Kresimir Burazin Benasque, September 2015 16/18
University of Osijek - Department of Mathematics
Multiple states
Two state equations on a ball � = B(0; 2)I f1 = �B(0;1) ; f2 � 1 ;
I
�� div (�+
� r ui ) = fiui 2 H1
0(�) i = 1; 2
I �Z
�f1u1 dx +
Z
�f2u2 dx ! min
Solving state equation
u0i (r ) = i (r )
�(r)� + (1 � �(r))� ; i = 1; 2 ;
with
1(r) =
8>><
>>:
� r2
; 0 � r < 1 ;
� 1
2r; 1 � r � 2 ;
and 2(r) = � r2
:
Similarly as in the first example: := � 21 + 2
2, g� := �2 , g� :=
�2 .Kresimir Burazin Benasque, September 2015 16/18
University of Osijek - Department of Mathematics
Geometric interpretation of optimality conditions
20 r
c
g�
g�
A: 0 < � � 120 r
c
4p �
g�
g�
B: 1 < � � 4
20 r
c
4p �
g�
g�
C: 4 < � � 16 20 r
c
g�
g�
D: 16 < �
Kresimir Burazin Benasque, September 2015 17/18
University of Osijek - Department of Mathematics
Geometric interpretation of optimality conditions
20 r
c
g�
g�
A: 0 < � � 120 r
c
4p �
g�
g�
B: 1 < � � 4
20 r
c
4p �
g�
g�
C: 4 < � � 16 20 r
c
g�
g�
D: 16 < �
As before, Lagrange multiplier can be numerically calculated fromcorresponding algebraic equation �
R� �� dx = �.
Kresimir Burazin Benasque, September 2015 17/18
University of Osijek - Department of Mathematics
Optimal �� for case B
20 r
c
cg�
g�
pc1 qc
1 qc2 qc
3
�� �
g�
Kresimir Burazin Benasque, September 2015 18/18
University of Osijek - Department of Mathematics
Optimal �� for case B
20 r
c
cg�
g�
pc1 qc
1 qc2 qc
3
�� �
g�
In orange region:
��(r) = 1
� � �
� �
r (r)
c
!
20 r
��
1
pc1 qc
1 qc2 qc
3
Kresimir Burazin Benasque, September 2015 18/18