sensitivity_analysis
TRANSCRIPT
1. Sensitivity analysis of electromagnetic fields In last years, the methods for shape and material structure recognition using the modern CAD technology has made remarkable progress in achieving automatically optimum designs. The shape recognition, so as recognition and optimization of structure, are done in order to fine tune the optimum layout. The sensitivity analysis is the tool used to accelerate optimization process. The optimization takes place in iterative manner, basing on gradient information derived from the sensitivity analysis [6], [10]. Sensitivity analysis determines the dependence of global or local electromagnetic quantities on geometrical or physical parameters expressed in the form of an objective function. This chapter deals with sensitivity evaluation in conjunction with finite element algorithms. It will be shown, that the sensitivity analysis necessary to solve an inverse problem, requires only a low additional calculation cost.
1.1. Definitions
The absolute sensitivity value of electromagnetic quantity U versus parameter p we define as partial derivative [5]
Up
US
p
∂=∂
. (1.1)
The sensitivity defined in such manner is appointed by both quantities: U and p. To avoid this dependence the relative sensitivity may be used:
lnln
ln
lnU Up p
UU pUS S
pp Up
∂∂= = =∂∂
, (1.2)
which can be easily find knowing (1.1). These are the sensitivities of first order, also called linear sensitivities. In this work we will use only absolute sensitivity (1.1). For evaluation of sensitivity the finite element method will be used. As the parameter p we will understand either material parameter, as electric conductivity γe inside the finite element e, or the geometric parameter, as the coordinate of the node xn , which can be part of the border line. The electromagnetic quantities, for which the sensitivity is calculated, may be of two types: quantity defined in the point m of the area, for example the modified magnetic vector
potential in the node m of finite element,
lubm mm mn
u uS S
xγ∂ ∂= =∂ ∂e e n
, (1.3)
quantity defined for some area, for example the voltage induced in measurement coil c
evaluated as the integral on the coil area Sc,
( ) ( )1 1
d or dnS u S uxγ
∂ ∂= =
∂ ∂∫∫ ∫∫c c
c ce e n
c cS S
S SS S
. (1.4)
Sensitivity may be evaluated directly, when we disturb the chosen parameter of the model, and then evaluate effect of this disturbance. In the Fig.1 the analyzed model, containing conductive region (SE), non-conductive region and the area of measurement coil (Sc) is shown.
Fig. 1.1. Analyzed model and conductivity disturbance in element e
Conductivity disturbance δγ inside element e causes the change of the voltage on the measurement coil with cross-section Sc. Value of this voltage may be evaluated using any commercial finite elements package. When the number of changeable parameters is large, such treatment may be not effective, and the obtained sensitivities ballasted with additional error. All these reasons together preclude the use of direct method to solve real problems. However, it is useful for testing the correctness of other algorithms for determining the sensitivity, described in this work.
2. The sensitivity analysis in the frequency domain Basics of sensitivity analysis, as well as its implementations in numerical algorithms have been extensively described in textbooks on the circuits theory . Analogous methods for the analysis of electromagnetic fields in the frequency domain appeared in the works. Similarly, as in circuits theory, the evaluation methods may be divided on two types, described below.
2.1. Adjoint model method Adjoint models used in circuit theory are basing on the Tellegen’s theorem [16]. The sensitivity equation for electromagnetic fields may be also derived using Lorenz lemma [14]. Exploiting divergence theorem this can written as follows:
( ) ( )
( )
d div d
rot rot rot rot d .
E H E H n E H E H
H E E H H E E H
S V
V
S V
V
+ + + +
+ + + +
× − × = × − × =
= − − +
∫ ∫∫
(2.1)
Taking into account the Maxwell’s equations containing magnetic current L (it should be equal to zero in every physical system, but the adjoint system may be non-physical), and writing them in the complex form
( )S
S
rot j
rot j .
γ ωεωµ
= + += − −
H J E
E L H (2.2)
yields
( )
( ) ( ) ( )( )S S S
d
j ( ) j ( j ) d .jωµ γ ωε ωµ γ ωε
E H E H n
HH E J E H L H E J E
S
V
S
V
+ +
+ + + + + + + + +
× − × =
= − − + + + + + + +
∫∫
(2.3)
Equation (2.3) is valid for small perturbations of material parameters δγ, δµ, δε , too:
( ) ( )( )
( )( ) ( )( )(( )( ) ( )( ))
S
S S
δ δ d
j δ δ δ ( )
δ j ( δ ) j ( δ ) δ d .
jω µ µ γ ωε
ωµ γ γ ω ε ε
E E H E H H n
H H H E E J E
H H L H E J E E
S
V
S
V
+ +
+ + + + +
+ + + +
+ × − × + =
= − + + − + + + +
+ + + + + + + + +
∫∫ (2.4)
When the disturbance is small enough, the small terms of high orders may be neglected. Subtracting the equation (2.3) form (2.4) yields:
( )
( )(
( ) ( ) )S
S
δ δ d
j δ j δ δ ( )
δ j δ j δ δ d .
j
j
ω µ ω µ γ ωε
ω µ γ ω ε γ ωε
E H E H n
H H H H E J E
H L H E E E E E E
S
V
S
V
+ +
+ + + + + +
+ + + + + +
× − × =
= − − − + + +
+ + + + + +
∫∫ (2.5)
If the adjoint model has the same material parameter, as original, we obtain the sensitivity equation in the form similar to [6]:
( )
( )
S Sδ δ d
δ j δ j δ d .γ ω µ ωγ ε
J E L H
E E H H E E
V
V
V
V
+ +
+ + +
− =
= − +
∫∫
(2.6)
This equation expresses the changes in the field of original model δE and δH caused by variation of material parameters γ, µ, ε . The numerical examples shown below utilize only the sensitivity versus electric conductivity γ . In this case the sensitivity equation takes simple form of
( )S Sδ δ d δ d .γJ E L H E EV V
V V+ + +− =∫ ∫ (2.7)
The above equation determines how to construct the adjoint model. The excitation current JS
+ should be driven into this node, where the sensitivity value of E has to be obtained. Similarly, the magnetic current LS
+ makes possible the sensitivity calculation of H. The proper boundary conditions of both systems cause vanishing of the surface integral in Eq.(2.5). It means that the original and adjoint systems differ only in excitations and eventually in boundary conditions. The geometric properties and material parameters are the same. Further, the stiffness matrix of both systems is the same and requires only one factorization. The relationship of vector magnetic potential to the intensity of the electric field
jω= −E A , (2.8) makes possible to write the sensitivity equation for magnetic vector potential
S δ d j δ d .ω γJ A A AV V
V V+ +=−∫ ∫ (2.9)
So, the adjoint model(+) owns the same topology, as original. It differs only in excitations. Measurement of field distribution by means of voltage induced in a circular wire The voltage induced in circular wire of radius rck is
11 c c c2π 2πj , and its differential: 2πj .kk k k k k
k
VV r E r A r
Aω ωϕ ϕ
ϕ
∂= = − = −
∂ (2.10)
The objective function used in the optimization process may take the form
( )2
1
=1
1 = ,
2
zw
k k k
k
W A Aϕ ϕℜ −∑
(2.11)
with A
- the result obtained by means of measurement, and zw - the number of wires used to measure the field. Weighting functions Wk can be selected in accordance with the requirements of the user. In order to comply the left side of equation (1.9) with equation (1.10) there should be applied the excitation of adjoint model JS
+ as a Dirac impulse located at the point (rck, zck) of intersection of circular wire with plane r−z. It is convenient to adopt current ISk
+ introduced at this point of value 1. In this way, the sensitivity of the potential A at the point k, evaluated in terms of conductivity γ e inside finite element e, with the area Se, was derived
δd
δ
k
ck
A jA A r
rϕ
ϕ ϕω
γ+=− ∫
e
e
e
S
S , (2.12)
and then, the sensitivity of voltage induced in a circular wire versus conductivity variation:
1 1 2δ δ δ2π d
δ δ δ
k k k
k
V V AA A r
Aω
γ γϕ
ϕ ϕϕ
+= =− ∫e
e
e e
S
S . (2.13)
The structure of the adjoint model for field measuring with the circular wire is shown in Figure 2.1. Excitation current is now introduced to the node corresponding to the position of the wire.
Fig. 2.1. Adjoint model structure for measurement with circular wire
Measurement of field distribution using coil with finite cross section Assuming that the coil windings were wound with sufficiently thin wire, the voltage induced in the k-th circular coil of cross-section Sck and n turns can be expressed as
c
2 c cc
2πjd
k
k k k kk
nV r A
ωϕ=− ∫
S
SS
. (2.14)
The objective function may be defined similarly as in (2.11)
( )22
=1
1 = ,
2
m
k k k
k
W V V
ℜ −∑ (2.15)
with: m number of measurement coils, and V
measured coil voltage. In order to comply the
left side of equation (2.9) with equation(2.14) there should be applied excitation of adjoint model JS
+ existing in the area of coil Sck . If in addition, the current density in this area will be chosen in such a way, that it will give rise to a total current equal to 1,
S S 1k k kI J+ += ⋅ =cS , (2.16)
then the dependence, similar as (2.13), for the sensitivity of voltage induced in the coil of finite cross section is:
2 2 2δ δ δ2π d
δ δ δ
k k k
k
V V An A A r
Aω
γ γϕ
ϕ ϕϕ
+= =− ∫e
e
e e
S
S . (2.17)
The structure of adjoint model is shown in the Fig. 2.2. Through the excitation coil does not flow the current, while the excitation is in the area of the measuring coil.
Fig. 2.2. Adjoint model structure for measurement with the coil of finite cross-section
Measuring the impedance change of coils exciting the field The role of the measuring coils can take the exciting coil. If it will be supplied by a fixed current value, the voltage induced in it will depend, inter alia, from the distribution of conductivity of the sample material. In this way we can determine the impedance of the coil with the cross-section of SS excited by the current I,
S
S SS
2πjd
nZ rA
I
ωϕ=−
⋅ ∫S SS
. (2.18)
With this method of measurement there is not needed to apply an additional measuring coil, so the excitation of adjoint system is attached the same, as this of the original system. The only difference is the value of exciting current density JS
+, which as before must be chosen so, as to give rise the total current IS
+ to be equal to 1. Then, the sensitivity of the coil impedance versus conductivity in the element e is given by
2S Sδ δ δ 2π
dδ δ δ
Z Z A nA A r
A I
ωγ γ
ϕϕ ϕ
ϕ
+= =− ∫e
e
e e
S
S . (2.19)
The structure of the model is in this case the same as the original model, only the value of excitation is different. It was presented in Figure 2.3.
Fig. 2.3. Adjoint model structure for impedance measurement of the coil
Presented here three methods for measuring the distribution of the magnetic field can be reduced utilizing adjoint model to a common algorithm, differing only with excitation of adjoint model. In order to present, how adjoint model works, in the subsequent figures there are shown the equipotential lines of magnetic vector potential A in the analysis of a simple model consisting of a circular coil placed over a conducting plate. Fig. 2.4 shows the adjoint model field lines for the circular wire. For the point, where the current was introduced, the vector potential gives rise to an infinitely large value. While using the finite element method, it takes a finite value, depending on the discretization of the area. However, this does not prove the
inadequacy of this model, because the area of conductivity, where the sensitivity is determined, is usually far from that point. Diagram shown in Figure 2.5 corresponds to the field of adjoint model with a rectangular coil, while the field distribution obtained by the coil impedance measurements shown in Fig. 2.6 is in fact re-scaled field distribution of the original model.
Rys. 2.4. Magnetic field of adjoint system for circular wire
Rys. 2.5. Magnetic field of adjoint system for circular coil
Rys. 2.6. Magnetic field of adjoint system for coil impedance measurement
Due to the fact, that the integrand in formulas (2.13), (2.17), (2.19) includes, using the first-order finite element, product of two linear functions, determining the integrals must be done by using second-order Gaussian quadrature. Application of the adjoint model allows for the evaluation in a single pass the sensitivity of induced voltage versus conductivity of all elements. Determination of the sensitivity of the next measurement coil requires to build a new adjoint system, differing only by excitation. For this reason, an extremely effective method of solving the equations of adjoint system is the L/U decomposition. Determination of gradient of objective function If the objective function may be approximated nearby stationary point by the quadratic function (2.11), (2.15), the direct calculation of its gradient components using adjoint model is possible. Gradient of objective function with respect to the material parameter γ is a vector of the form
1
,...,mγ γ
∂ℜ ∂ℜ∇ℜ = ∂ ∂ . (2.20)
Variations of first order can be expressed using the gradient components
11
... mm
γ γγ γ
∂ℜ ∂ℜ∆ℜ = ∆ + + ∆∂ ∂
. (2.21)
Re-writing the objective function (2.11) into matrix form
( ) ( )T
1
1 =
2ℜ − −A A W A A
, (2.22)
with the vector A
meaning the values obtained from the measurement, we can apply the following first order approximation
( )T
1 =∆ℜ − ∆A A W A
. (2.23)
The component of the gradient related to element e may be evaluated as
( )T1 1
γ γ γ∂ℜ ∂ℜ ∂ ∂= = −∂ ∂ ∂ ∂
A AA A W
Ae e e
. (2.24)
If the gradient components would be determined using sensitivity equation (2.9), a comparison of the left hand side of this equation with equation (2.24) leads to the conclusion, that the exciting current value of the k-th circular coil in adjoint model, ISk
+, should be
( )Sk k k kI A A W+ = −
. (2.25)
Similar reasoning can be applied to the objective function (2.15), for coils of finite cross-section, and concluded, that the total current of the adjoint model spread on the area of the coil k − Sck should be
( )Sk k k kI V V W+ = −
. (2.26)
2.2. Method of incremental systems Solution to a problem described by Helmholtz differential equation using FEM may be reduced to solution the system of equations formed by equations items
( )+ =C M u f . (2.27)
C and M are respectively stiffness and mass matrices, and in writing mass matrix used the symbolic (complex) notation. Differentiating this equation versus the material parameter x gives
( )x x
∂ ∂+ + =∂ ∂M u
u C M 0 , (2.28)
because the material parameters are contained only in the matrix M. In this way we can give a definition of the sensitivity of potential u versus the parameter x
( ) 1
x x x
−∂ ∂= = +∂ ∂
u u MS C M u . (2.29)
Determination of sensitivity on the basis of the definition (2.29) requires a inversion of the stiffness matrix C and of the mass M. This can be avoided, evaluating the sensitivity ∂u/∂x directly with the system of equations (2.28). This system differs from the FEM system of
equations (2.27) only by excitation, which contains a derivative of mass matrix. Mass matrix is a linear function of material parameters, and hence the calculation of the derivative can be done using the same procedure as used to create the matrix M. In addition, if the FEM system of equations (2.27) is solved using the L/U decomposition, with the same decomposition can be solved (2.28). The structure of incremental system is shown in Fig 2.7. Its excitation is located in an area, where there is disturbance of x, in this case, the disturbance of conductivity inside element e, ie γe. This analysis delivers the sensitivity of the nodal values versus the parameter x. Obtaining sensitivity for an area, such as the sensitivity of induced voltage in the measuring coil, requires when using this method, an additional integration. The method of incremental system delivers in a single pass the sensitivity of all nodal potentials versus one parameter x. The solution of the sensitivity for the next parameter requires the derivation of the matrix M and a solution of (2.28) again.
Fig. 2.7. The structure of incremental system
In the majority of inverse tasks there is an excess of the measurement data, ie the need for sensitivity evaluation for multiple coils or wires. This suggests advantage of an incremental method over the adjoint system method. But that is not, because the evaluation of sensitivity in an area, such as voltage induced in the measurement coil of finite cross-section, with this method requires an additional integration.
2.3. Method of incremental systems for geometrical parameter For the inverse tasks of shape optimization the sensitivity terms versus geometrical parameters, here the coordinates of FEM mesh nodes, are necessary. Method of incremental system for geometric parameter is also based on the differentiation of equations (2.27). Since the geometric coordinates of nodes take part in the formation of both matrices C and M , the sensitivity versus this geometric parameter must be evaluated from the equations system,
( )x x x
∂ ∂ ∂ + = − + ∂ ∂ ∂
u C MC M u . (2.30)
The matrices C and M for Cartesian coordinate system are given by [>>>]. Because the derivative of element area ∆e takes the form of
1 1and , where: , ,
2 2m mm m
b c m i j kx y
∂∆ ∂∆= = =∂ ∂
e e
, (2.31)
we obtain the analytical form of stiffness and mass matrix derivatives for element e versus coordinates of node i :
( )
2 2
2 22
2 2
0 2 1 11
2 1 2 1 ,2 124 2 1 1 2
i i
i i i j i j i k i k i i
i ii j i j j j j k j k i j j k
i k i k j k j k k k i j k k
x x
b c b b c c bb c c c cb j b
bb c c b c b b c c c c c c
bb c c b b c c b c c c c c
ωµ γ
∂ ∂+ =∂ ∂
+ + + − = + + + + − − − + ∆∆ + + + −
C Me e
e e
ee
(2.32)
( )
2 2
2 22
2 2
0 2 1 11
2 1 2 1 .2 124 2 1 1 2
i i
i i i j i j i k i k i i
i ii j i j j j j k j k i j j k
i k i k j k j k k k i j k k
y y
b c b b c c bb c c b bc j c
bb c c b c b b c c b b b b
bb c c b b c c b c b b b b
ωµ γ
∂ ∂+ =∂ ∂
+ + + − − = + + + + − − − − ∆∆ + + + −
C Me e
e e
ee
(2.33)
Derivatives relating to the other element nodes can be obtained by cyclic permutation of subscripts and elements of the matrix.
2.4. Sensitivity of energetic functional versus coordinates of nodes In addition to the sensitivity evaluation of the of punctual and area quantities, it is also possible to determine the sensitivity of the global quantity, such as a energetic functional of finite element method [7]. The purpose of such proceedings is to optimize additionally the value of this functional for the problem solved. The idea of the finite element method is to minimize the functional value in terms of nodes potentials, and established nodes positions. Additional reduction in the value of the functional can be obtained by optimizing the location of nodes. This task can be performed iteratively on the basis of information obtained from the sensitivity analysis. Suppose, that three-dimensional magnetostatic field is analyzed using the magnetic scalar potential Ω . Let us write the global functional as the sum of functionals for tetrahedral elements as follows:
( ) ( )
( )2 2 2
2
S
1 + + 2div .
x y z
Ω Ω
Ω Ω Ω ΩT
e e
e
e e ee e e e e
e
e
V V V VV
= =
∂ ∂ ∂ = − ∂ ∂ ∂
∑
∑
F F
(2.34)
For the elements of first order the derivatives included in the functional give values independent from x, y, z .
, , , , , , , , , = = =
6 6 6
i i i
j j ji j k l i j k l i j k l
k k k
l l l
b b b b c c c c d d d d , ,
x y z
Ω Ω ΩΩ Ω ΩΩ Ω ΩΩ Ω ΩΩ Ω Ω
∂ ∂ ∂ ∂ ∂ ∂
e e e
e e eV V V. (2.35)
Derivative of the functional in relation to x coordinate of the i-node position takes the form
( )
( ) ( )2
S
=
2 +2 +2
2 1div ( ) .
i
i i i
i i
x
x x x y x y z x z
x x
Ω
Ω Ω Ω Ω Ω Ω
ΩΩ Ω
∂∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂− −∂ ∂
T
e e
e e e e e ee e e
ee e e e
e e
V V V
VV V
F
F
(2.36)
Components of the derivative are
= =6 6i i i
i i
b b ,
x x x x
Ω Ω Ω ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
e eeV (2.37)
[ , , , ] =
6
i
i
k l l j j kij
ik
l
xc z z z z z z
x y
Ω
Ω ΩΩΩ
∂ ∂ − − −∂ ∂ ∂ ∂
ee
eV
V, (2.38)
[ , , , ] =
6
i
ii k l l j j k
j
ik
l
xy y y y y yd
x z
Ω
Ω ΩΩΩ
∂ ∂ − − − ∂ ∂ ∂ ∂
ee
eV
V. (2.39)
The remaining components are obtained by cyclic permutations, while the derivatives of tetrahedral element volume are
= , = , = .6 6 6i i i
i i i
b c d
x y z
∂ ∂ ∂− − −∂ ∂ ∂
e e eV V V (2.40)
Presented formulas are easily to be programmed. Knowledge of the functional sensitivity versus the coordinates of nodes allows us to optimize the mesh nodes of finite elements. In
order to carry out the optimization, we can use one of the known algorithms, eg gradient method. After each iteration, it is necessary to re-solve of the problem using FEM. Optimization of the functional test was carried out using the three-dimensional model of the shape of "L" shown in Figure 2.8. Inside the area the Laplace equation is satisfied, and the model contains only three nodes with coordinates (0.75, 0.75, *), whose position can be optimized. An asterisk (*) denotes the z-coordinates equal to 0, 0.5 or 1
Fig. 2.8. Test area in the shape of „L”.
Optimal location of a node was found after three iterations. The result was also reducing calculation error. Optimizing the more dense mesh is shown in Figure 2.9. We can see, as the nodes tend to concentrate near the inner edge of the model. The original location of nodes and elements is indicated by a dotted line. The proposed method of finite elements mesh optimization is, due to the significant effort of numerical calculations, profitable only if the models are used for multiple calculations.
Tab. 4.1. Optimization results of the mesh of shape „L”
Iteration number
Position
of nodes x, y, z
Energy
value [J]
Estimated
calculations error % 1
0,75; 0,75; *
3,18·10
-7
10,2
2
0,52; 0,52; *
3,14·10
-7
9,0
3
0,58; 0.58; *
3,11·10
-7
8,1
Fig. 2.9. Optimization of tetrahedral meshes
3. Sensitivity analysis in time domain Some elements of the sensitivity analysis in the time domain were given for mechanical systems in [12], and development of this method is in the works [9] ,[13]. Determination of the sensitivity of time domain can be done with the direct method, similar to this one described in chapter 2.1. The disadvantage of this direct solution is its low efficiency, and obtained only approximate numerical values of sensitivity. The direct method is, however, well suited to validate the results obtained by other methods. As in the frequency domain, here are applicable two basic methods.
3.1. Adjoint model method
Presented derivation of the sensitivity equation is based on the Lorentz lemma [14]. Using the divergence theorem this relationship can be represented as follows
( )
( )
(
)
0
0
0
( , ) ( , ) ( , ) ( , ) d
div ( , ) ( , ) ( , ) ( , ) d d
( , )rot ( , ) ( , )rot ( , )
( , )rot ( , ) ( , )rot ( , ) d d ,
while the time is opposite to ; .
E x H x E x H x n
E x H x E x H x
H x E x E x H x
H x E x E x H x
+ +
+ +
+ +
+ +
× − ×
= × − ×
= −
− +
= −
∫ ∫∫ ∫∫ ∫
T
S
T
V
T
V
t τ τ t S
t τ τ t V t
τ t t τ
t τ τ t V t
τ t τ T t
(3.1)
According to (3.1) there exist fields of two systems: the original (physical) and fictitious, called the adjoint system, indicated by (+). Adjoint system has not to be a physical system. It can be excited using the quantities does not occur in physical systems, such as a magnetic current density L+. The introduction of such excitations allows for easy determination of the sensitivity of the desired field quantity. Taking into account the Maxwell equations we obtain the following:
( )0
S0
S S
( , ) ( , ) ( , ) ( , ) d
( , ) ( , )( , ) ( , ) ( , ) ( , )
( , ) ( , )( , ) ( , ) ( , ) ( , )
µ γ ε
µ γ ε
E x H x E x H x n
H x E xH x E x J x E x
H x E xH x L x E J x E x
T
S
T
V
t τ τ t S
t ττ t τ τ
t τ
τ tt τ t t
τ t
+ +
++ + + + +
++ + +
× − × =
∂ ∂ = − − + + + ∂ ∂ ∂ ∂ + + + + + ∂ ∂
∫ ∫
∫ ∫d dV t.
(3.2)
If in the original model there are changing the material parameters δγ, δµ, δε, it causes the changes in the intensities of fields δE and δH, so that excitation current values JS remain unchanged,
( )0
S0
S S
( δ ) ( δ ) d
( δ )( δ ) ( δ )
( δ )( δ ) ( δ )( δ ) ( δ ) d d
µ µ γ ε
µ γ γ ε ε
E E H E H H n
H H EH E E J E
H E EH H L E J E E
T
S
T
V
S
t τ
V t.τ t
+ +
++ + + + +
++ + +
+ × − × + =
∂ + ∂ = − + − + + + + ∂ ∂ ∂ ∂ + + + + + + + + + + ∂ ∂
∫ ∫
∫ ∫ (3.3)
With proper boundary condition Eq.(3.3) equals zero and subtraction (3.3) and (3.2) by neglecting of small values of second order leads to
( )S S0
0
δ δ d d
δ δ δ
(δ ) (δ )δ δ δ δ d d
γ µ ε
γ γ µ µ ε ε
J E L H
H EEE H E
H H E EEE EE H H E E
T
V
T
V
V t
t t
V t.τ t t τ
+ +
+ + +
+ ++ + + + + + +
− =
∂ ∂= − + + ∂ ∂∂ ∂ ∂ ∂ + − + − + − ∂ ∂ ∂ ∂
∫ ∫∫ ∫ (3.4)
We assume that the adjoint model not only has the same topology as the original, but also the same material parameters, ie γ = γ+, µ = µ+, ε = ε+. Then, the components containing γ are vanishing, and the last four terms, including µ and ε can be integrated by parts,
0
00
00
(δ ) (δ )δ δ d d
(δ ) (δ )δ d
δ δ δ d d
µ ε
µ µ
ε ε
H H E EH H E E
H HHH H H
E EEE E E
T
V
TT
V
TT
V t=τ t t τ
tτ t
t V.t τ
+ ++ +
+ + +
+ ++
∂ ∂ ∂ ∂ − + − ∂ ∂ ∂ ∂ ∂ ∂ = − + + ∂ ∂
∂ ∂ + − + ∂ ∂
∫ ∫
∫ ∫
∫
(3.5)
Taking into account, that ∂t = −∂τ and initial conditions for the diffusion problem in both, the original as adjoint model, are zero, the components of the integral (3.5) are vanishing. The remaining part of the equation (3.4) takes the form of the sensitivity equation
( )S S0
0
( , )δ ( , ) ( , )δ ( , ) d d
( , ) ( , )( , ) ( , )δ ( , )δ ( , )δ d d .γ µ ε
J x E x L x H x
H x E xE x E x H x E x
T
V
T
V
τ t τ t V t
t tt τ τ τ V t
t t
+ +
+ + +
− =
∂ ∂ = − + ∂ ∂
∫ ∫∫ ∫
(3.6)
Equation (3.6) describes the changes in the electric field and magnetic field caused by changes in the original model, of material parameters γ, µ, ε, ie the sensitivity of the electromagnetic field intensities. In order to obtain the sensitivity for area of interest to us, or for the point, we use the appropriate choice of spatial excitation in adjoint model. The
structure of adjoint model results from the above described sensitivity equation. Boundary conditions must be chosen so, that the Lorenz lemma be satisfied. Initial conditions should be zero,
0 0
0 0
( , ) 0, ( , ) 0,
( , ) 0, ( , ) 0.
+= =
+= =
= =
= =
E x E x
H x H x
t τ
t τ
t τ
t τ
(3.7)
The shape of the excitation of adjoint system in time S S( , ) and ( , )+ +J x L xτ τ may be chosen in
an arbitrary way, convenient for further calculations. Since the adjoint system is a fictitious system, in particular, we can choose physically not realizable excitation, such as the Dirac impulse, or a unit step impulse . In further considerations, let's take simplified form of sensitivity equation (3.6), limited to the sensitivity of the electric field E versus material conductivity γ,
S0
0 0
( , )δ ( , )d d
( , ) ( , )δ d d δ ( , ) ( , )d d ,γ γ
+
+ +
=
= =
∫ ∫∫ ∫ ∫ ∫
J x E x
E x E x E x E xe
T
V
T Te
V V
τ t V t
t τ V t t τ V t
(3.8)
while V e denotes the area, where a change in conductivity occurs. In further considerations it will be usually area of one finite element e. Conductivity γe remains constant for the whole of V e and the entire analysis time T. Since the analysis of electromagnetic fields is usually carried out using the auxiliary potential A, the implementation of this method for the magnetic vector potential A is presented below. For two-dimensional analysis the scalar magnetic potential φ = 0, so the relationship for the original model and adjoint model simplifies to
( , ) ( , ) ( , )( , ) and ( , ) .
+ ++∂ ∂ ∂= − = − =
∂ ∂ ∂A x A x A x
E x E xt τ τ
t τt τ t
(3.9)
In this way, the version of the sensitivity equation for the magnetic vector potential A by the two-dimensional analysis is obtained
S0 0
( , ) δ ( , )d d δ ( , ) ( , )d dγ+ +∂ ∂ ∂=
∂ ∂ ∂∫ ∫ ∫ ∫J x A x A x A xe
T Te
V V
τ t V t t τ V tt t t
. (3.10)
Excitation of adjoint models in the time domain At last it remains to consider the excitation shape of the adjoint model in the time domain. We utilize two shapes of pulses, which lead to a simplification of the left hand side of equation (3.10). Let us introduce the unit step current
S S
0 for 0( , ) ( )1( ), where: 1( )
1 for 0.+ + ≤= = >
J x J xτ
τ τ ττ
(3.11)
Using unit step allows us to integrate with respect to the time,
( )S0
( ) δ ( , ) δ ( , ) d δ ( , ) ( , )d dγ+ +∂ ∂− =
∂ ∂∫ ∫ ∫J x A x A x A x A xe
Te
V V
T V t τ V tt t
0 . (3.12)
With zero initial condition for the potential A also the sensitivity δ ( , ) 0=A x 0 . The obtained sensitivity equation determines the sensitivity for the analysis time T and its left side is analogous to the left side of the sensitivity equation (2.9). Here we can also introduce other types of spatial excitations, receiving directly the sensitivity for voltage measured in the coils. Another way is to use a current in the shape of the Dirac impulse:
+S S
for 0( , ) ( ) ( ), where: ( ) and ( )d 1.
0 for 0
∞+
−∞
∞ == = = > ∫J x J xτ
τ δ τ δ τ δ τ ττ
(3.13)
In this case the sensitivity equation takes the form
S0
( ) δ ( , )d δ ( , ) ( , )d dγ+ +∂ ∂ ∂=
∂ ∂ ∂∫ ∫ ∫J x A x A x A xe
Te
V V
T V t τ V tt t t
. (3.14)
Since the adjoint system is analyzed for backward time: τ = T – t, the Dirac pulse τ = 0 corresponds to the time of analysis of the original system t = T. The right side of this equation is the same as (3.12), while the sensitivity evaluation using this method takes place after the time discretization recursively, for subsequent analysis up to the time T. Time discretization Discretization of equations (3.12) and (3.14) has been performed in such a way, that the time T, for which sensitivity is determined, was divided into I equal segments: T = I.
∆t, where I = 1,2,...,N. The analysis of the original system took place for the current time and the nodal potentials were calculated at the time points n.
∆t, while the adjoint system in the backward time at the time points n.
∆τ, with n = 1,2,...,N. While using finite elements linear in the time domain, the derivative of the potential in the i-th time element is constant, and equation (3.12) for unit step excitation can be shown as
11S
1
( ) ( )( ) ( )( )δ ( , )d δ dγ
∆ ∆
+ ++ − + −
=
−−= ∑∫ ∫ A x A xA x A x
J x A xe
I
e I i I ii i-
V Vi
T V Vt t
. (3.15)
An analogous equation, we obtain for forcing the adjoint model by the Dirac impulse (3.14)
( )( )
S S
1 1
1
( )δ ( , )d ( )δ ( , )d
1δ ( ) ( ) ( ) ( ) d .γ
∆
∆
J x A x J x A x
A x A x A x A x
+ +
+ +− + −
=
= +
+ − −
∫ ∫∑∫ e
V V
I
ei i- I i I i
Vi
T V T- t V
Vt
(3.16)
While exciting by the Dirac impulse, calculations take place in a recursive manner, based on knowledge of the sensitivity of the previous time step, with zero initial condition. Given equations (3.15) and (3.16) do not depend on the differential scheme Θ used for time stepping. There remains still time discretization of exciting pulses. Since the solution of differential scheme Θ corresponds to the differential equation solving in collocation point t = (n − 1+Θ)⋅∆t, the value of excitation for the collocation point should correspond to the value of the pulse. Figure 3.1 shows the collocation points for different values of Θ and unit step excitation 1(τ). Value of the function in collocation points is 1. Linear shape functions in the time elements are forcing in a linear excitation change in each element and the continuity on elements border, which consequently leads to the waveforms shown in Figure 3.1 [13].
Fig. 3.1. Excitation discretization for unit step pulse for the various schemes Θ
Potential values obtained using the so-defined excitation, as shown in Figure 3.1a, exhibit similar oscillations within the time elements, but providing the correct solutions in collocation points. If we use the excitation shape of the Dirac impulse δ(τ) (3.13), we assume, that the pulse occurs for i = 1, and the following condition will be met:
( )d 1.δ τ τ
∞
−∞
=∫ (3.17)
The shape of excitation and its values in the collocation points for the three differential schemes Θ are shown in Figure 3.2.
Fig. 3.2. Excitation discretization for Dirac pulse to the various schemes Θ
The condition (3.17) for the waveform Fig. 3.2a takes the form
2
1 2 2 2( )d 1
2∆
∆ ∆ ∆i
δ τ τ tt t t
∞∞
−∞ =
= + − = ∑∫ , (3.18)
and for the waveform Fig. 3.2b
1 3 3 3 3 3 3 3 3( )d ...
2 2 2 4 4 8 8 16 16
3 1 1 1 3 21 ... 1,
2 2 4 8 2 3
∆∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
δ τ τ tt t t t t t t t
∞
−∞
= + − − + + − − + = = − + − + = ⋅ =
∫ (3.19)
and for the waveform Fig. 3.2c
1 1 1( )d 1.
2∆
∆ ∆δ τ τ t
t t
∞
−∞
= + = ∫ (3.20)
Since the condition (3.17) is fulfilled, all three waveforms can be considered as discretized Dirac impulse. There are no clear grounds for the choice of excitation of adjoint model as one of the two pulses described above, the Dirac pulse or unit step pulse. Contribution of numerical calculations is the same for both cases, and the same values of sensitivity are obtained.
3.2. Method of incremental model Let us start with following scheme Θ
( ) ( )
1
11 1 ,
nΘ
Θ Θ Θ
+ = ∆
= − − + − + ∆
C M u
M C u f fn-1 n-1 n
t
t
(3.21)
with: C and M – stiffness and mass matrices of finite elements, respectively. u denotes the modified magnetic vector potential. Differentiating it versus material parameter γe of the element e we become
( ) ( )
1
11 1 .
nΘγ
Θ Θ Θγ
∂ + = ∂ ∆
∂ = − − + − + ∂ ∆
C M u
M C u f f
e
n-1 n-1 ne
t
t
(3.22)
In a similar way, we can set the derivatives of the differential scheme in relation to other material and geometrical parameters. In the case of derivative terms γ, stiffness matrix C does not contain this parameter, besides that we assume, that due to changes in the material parameter the excitation f does not change. Then equation (3.22) simplifies to the form
( ) 1
1 1
1 11 .
nn
nn
Θγ γ
Θγ γ
−
∂ ∂ + + = ∆ ∂ ∆ ∂
∂ ∂ = + − − ∆ ∂ ∆ ∂
M uu C M
M uu M C
e e
-1e e
t t
t t
(3.23)
By introducing the definition of the sensitivity for the point referred to in Eq. (1.3) we obtain the system of equations
( ) ( ) 1
1
1 11 ,
n
nn n
S
S
Θ
Θγ
−
+ = ∆
∂ = − + − − ∆ ∂ ∆
C M
Mu u M C
e
-1 ee
t
t t
(3.24)
from which with the initial condition S0
e = 0 can be recursively evaluated the value of the sensitivity for the subsequent time steps tn. Keeping constant time step, preferably the same as in the analysis of the original model, one can calculate the sensitivity with the same decomposition of the matrix C and M, as determined by analysis of the original model. Calculation of the sensitivity for the area, for example, the sensitivity of voltage induced in the measurement coil of finite cross-section, by this method requires the integration over the coil area.
3.3. Comparison of described methods To determine the sensitivity the methods described above make use either of adjoint or incremental model. Both models are fictitious, and may take the form unacceptable for a physical system. Also excitations of these systems need not be physically exercisable. However, this does not disturb the analysis with finite elements. Compared with the direct evaluation of sensitivity (Chapter 1.1), both methods are very effective. In both cases, the first is an analysis of the original (physical) model. Vector potentials obtained by this analysis are used in the components of the sensitivity equation in
the method of adjoint model or in the excitation of incremental model. The adjoint model method provides in a single pass sensitivity of voltage induced in one coil in terms of all parameters of the system. Method of incremental model, on the contrary, provides the sensitivity of all nodes of a model versus one parameter. Determination of the sensitivity against another one parameter requires the re-calculation of the right side vector of equation (3.24), including derivative of matrix M, but this act does not requires complicated calculations, and even can be achieved using a standard package for the fields analysis. Due to the excess of the measurement data, desired by optimization, and a small number of variable parameters, it would indicate a predominance of incremental systems. However, when analyzed in the time domain, the effectiveness of the method is determined by the calculations repeated for each time step. If there is desired the area sensitivity, then the advantage of the adjoint model method, which does not require integration in the coils, may be demonstrated. On the other hand, any further decomposition of the matrix caused by, for example, changing the position of the probe, there is not of great importance for the global execution time. An additional advantage of the adjoint system method is the fact, that in determining the sensitivity for the application to the optimization task, this method allows the direct determination of a gradient of the square objective function.
3.4. Example for sensitivity analysis The presented example concerns the model of ring coils inside an infinitely long pipe. System consists of two excitation coils and located in between the measuring coil (Fig. 3.3). Currents of opposite direction are forming differential system, what results in measuring of zero coil voltage for a homogeneous tube structure. These coils systems are used in eddy current flaw detection to detect welding defects, nipples, etc. The nearing of differential coils to the flaw causes inducing of eddy currents and the voltage on the measurement coil. Presented probe does not allow to detect a defect location on the circumference of the tube, but only its location along the tube. In the absence of a flaw the system has the circular-cylindrical symmetry, and therefore it can be analyzed using 2D algorithms.
Fig. 3.3. Differential coils inside pipe
Fig. 3.4. Excitation current shape For the excitation the current pulse in the shape of half-sine was chosen. The analyzed model is shown in Figure 3.5. Material parameters correspond to the copper pipe wall. It was
assumed, that the coils are very thin wire wound, so that the skin effect is negligible and the current density in the coil is uniform 1A/mm2, and this area can be treated as non-conductive (γ = 0). Since the calculations will be done in time domain, the complexity of the model and size of the area is more important, than by the calculation models with the harmonic impulse. Therefore, the boundary is placed relatively close, causing the error associated with this. On the axis of symmetry the zero Dirichlet condition was forced, while other boundaries were selected for the zero Neumann condition, constant throughout the whole analysis in time domain.
Fig. 3.5. Contours and parameters of analyzed model
The area under consideration was divided into 2304 triangular elements of the "envelope". Application of these elements provides a symmetrical digitization, causing the calculated magnetic field of differential coils is symmetrical, and the voltage in the measuring coil equal to zero in the absence of flaw. In the air area the Laplace equation is taken, Poisson’ in excitation areas, and in the pipe wall the homogeneous diffusion equation is assumed.
Fig. 3.6. Discretization with „envelope” elements
Regardless of the method, which is used to determine the sensitivity, analysis begins with the original system. The result are transient values of the modified vector potential u. A fixed value of time step ∆t = 20 µs is adopted, and the analysis was continued in 250 time steps. As differential scheme Crank-Nicolson’ (Θ = ½) is used.
Fig. 3.7. Modified magnetic vector potential u for time moments of: a) t = 200 µs, b) t = 400 µs, c) t = 700 µs,
d) t = 1000 µs, original model Figure 3.7 shows a gradual field diffusion to the pipe wall, confirming the appropriate selection of time step ∆t. After the disappearance of excitation currents, for the time t = 1000 µs, is seen only eddy current field excited in the pipe wall (Fig. 3.7d). The analysis was performed with the adjoint system with the excitation current shape of unit step pulse (fig. 3.1a) for the scheme Θ = ½. The obtained results are shown in fig. 3.8.
Fig. 3.8. Modified magnetic vector potential u for time moments of: a) τ = 20 µs, b) τ = 40 µs, c) τ = 60 µs,
d) τ = 80 µs, adjoint model Excitation has been placed in the area of measuring coil. It should be noted, that these results do not correspond to the collocation points at Θ = ½. There were selected other moments of time, in order to show the discretization effect of an impulse excitation corresponding to the unit step at Θ = ½. As is apparent from Figure 3.1a, for even times ∆t excitation reaches zero, as can be seen clearly in Figure 3.8b at τ = 2·20µs, and in Figure 3.8d at τ = 4·20µs. In turn, for the odd times the excitation reaches a double value. Of course, the potential values of the collocation points do not show these oscillations. Implementation and analysis of the original and adjoint system allows us to evaluate in an area of the measurement coil the sensitivity versus the conductivity of all finite elements (3.15). It should be noted, that these are not the sensitivity values of voltage induced in the coil, but only some integral values of the vector potential on the area of the coil. For the initial analysis time (Figure 3.9a, b) the terms of conductivity is dominated by the sensitivity of surface area of wall. Since the field is not yet diffused into the interior walls, the sensitivity of the elements of this area is negligible. In the final times of analysis (Fig. 3.9d, e, f) field have diffused the whole wall thickness. Although the sensitivity of the elements lying on the surface are still the greatest, but we can now observe the impact of items that fall inside the wall, or even at the opposite side.
Fig. 3.9. Sensitivity for measurement coil area versus conductivity in elements for times: a) t = 200 µs,
b) t = 400 µs, c) t = 700 µs, d) t = 1000 µs, e) t = 2000 µs, f) t = 3000 µs The dependence of the sensitivity on of the position of the element may by better presented as a function of time for specific items. For comparison, four elements on the inner surface of the tube wall, the four inside wall and two elements on the opposite side of the wall were chosen. Figure 3.10a shows, that the sensitivity for the elements lying inside the wall, closer to the coil, occurs at first, while for the elements lying on the opposite side at the end. By contrast, the position of the element against coil is associated with the sign of sensitivity, which is the result of differential excitation. Figure 3.10b shows the sensitivity against conductivity for elements lying directly under the coil. They appear almost simultaneously with the occurrence of excitation pulse. The presented example using the pulsed excitation and time domain analysis shows, how the position of the conductive elements is associated with the sensitivity output value. This dependence can be used to identify the structure of conductive material, including discontinuities located deep below the surface.
Fig. 3.10. Sensitivity for measurement coil area versus conductivity in elements: a) inside tube wall and on the
opposite side, b) on the inner side of tube wall, c) placement of chosen elements
4. Solution of inverse tasks using sensitivity analysis As the inverse task will be seen the identification task of material parameters of the environment, of the material contours, or sources distribution that cause the existence of electromagnetic fields. From the viewpoint of applications there are of particular importance, the first two types of inverse tasks. A way of solving this task proposed in this paper is to match the distribution of fields on a simulation model to the measured distribution of the real object. But we can not prove in general case, that the same distribution of the electromagnetic field corresponds to the same structures of the simulation and the original model. This means, that the solution of the inverse may be ambiguous, and evaluation of obtained solutions must often leave the user experience.
4.1. Iterative algorithm The inverse task is therefore a task to seek the vector of parameters p belonging to the space of available solutions [ ]Π , and minimizing the norm
[ ] 2min , where: is the measured field distribution,
the field solution of simulated model.
Π∈− −
−p
u u u
u
(4.1)
Solution of the inverse task is usually one of the following methods:
− stochastic methods (genetic algorithms, simulated annealing, Monte-Carlo method), − artificial neural network, − deterministic method (gradient-less and gradient optimization methods).
Sensitivity analysis is used in the last group of methods, which are deterministic gradient methods. Stochastic methods demonstrate their usefulness in multimodal cases, while the technical task of identifying and optimizing the shape, as well as identifying the structure of the material, should be so defined, as to be unimodal case, which of course does not always succeed. By unimodal tasks the efficiency of stochastic methods, as well as efficiency of gradient-less optimization methods, is considerably worse. Artificial neural networks are unrivaled in speed, but the answer provided by the network to an unknown input signal is uncertain and depends on many factors. Among the well-known gradient optimization methods we have selected Gauss-Newton method. Way to seek solution of the problem using a iterative gradient algorithm is shown in simplified form in Figure 4.1.
Fig. 4.1. Iterative algorithm for solution of inverse task
In order to clarify the iterative algorithm, suppose the inverse task is to identify the structure of conductive material, and to that end with each of the C eddy current transducers were made N measurements. Number N will mean either the number of frequencies used in the measurement, when the excitation current is harmonic, or N is number of discrete time moments for which there was measured the transducer voltage with pulse excitation of a different shape. For the objective function we can then use the following definition of quadratic error:
( ) ( )2 2( ) ( )
1 1 1
1 1,
2 2u u u u
C N CN
c cn n v v
c n v
⋅
= = =
ℜ= − = −∑∑ ∑ (4.2)
where index v replaces indices c and n: v = (c − 1)·N + n . Suppose, that in the inverse task occur M parameters pm. It may be material conductivity in particular finite elements, or the coordinates of the nodes of finite elements. The first derivatives of the parameters pm can be determined from the formula
( )1
, 1,2,..., ,u
u up p
CN
vv v
m mv
m M⋅
=
∂ℜ ∂= − =
∂ ∂∑ (4.3)
while the second derivatives of the formula
( )2 2
1
, 1,2,..., , 1,2,..., .u u u
u up p p p p p
CN
v v vv v
m w m w m wv
m M w M
⋅
=
∂ ℜ ∂ ∂ ∂ = − + = = ∂ ∂ ∂ ∂ ∂ ∂ ∑ (4.4)
The essence of the Gauss-Newton method is to approximate the function near the stationary point p0 with the square form [2]
( ) ( ) ( )T 20 0 0 0 0 0
1( ) ( ) ( ) ( )
2ℜ ≅ ℜ +∇ℜ − + − ∇ ℜ −p p p p p p p p p p . (4.5)
The necessary condition for the existence of a minimum pmin is zeroing the gradient of the objective function,
( )2min 0 0 min 0( ) ( ) ( ) 0∇ℜ = ∇ℜ +∇ ℜ − =p p p p p . (4.6)
The Gauss-Newton method assumes, that the vector p lies near a minimum of p0, and then in formula (4.4) can be omitted the small of second order, approximating the second derivatives in following manner
( )2
1
, 1,2,..., and 1,2,..., ,u u
u up p p p
CN
v vv v
m w m wv
m M w M
⋅
=
∂ ℜ ∂ ∂= − = =
∂ ∂ ∂ ∂∑ (4.7)
or in vector notation
( )22 ( ) ( )∇ ℜ = ∇ℜp p . (4.8)
Knowledge of the gradient and Hessian of the objective function allows us to evaluate a corrections vector δp0 added in the current position p0,
( )2 20 min 0 0 0 0( ) ( )δ ( )∇ ℜ − = ∇ ℜ = −∇ℜp p p p p p . (4.9)
In real applications, the vector of parameters p0 is not close to a minimum. The approximation (4.9) results, that after addition of corrections we did not find the minimum, and the only new point p1
1 0 0δ= +p p p . (4.10)
Considerable distance from the vector of parameters p0 to minimum pmin can cause even the iterative process is not convergent. To determine the corrections vector the sensitivity analysis is used. We define the following matrix called the sensitivity matrix S:
1 1
1
1
...
... ... ...
...
u u
p p
u u
p p⋅ ⋅
∂ ∂ ∂ ∂
= ∂ ∂
∂ ∂
SM
C N C N
M
. (4.11)
In the simplest case, the matrix S may be a square matrix with number of columns M = C·N. This condition is practically easy to meet, because we can always choose the appropriate number of frequencies or the number of discretization time points. Usually, taking into account the frequencies or time discretization points, is left to the methods of solving of over-determined equation systems, providing an excess of measurement data and resulting in the matrix S in a rectangular form M < C·N. Utilizing the sensitivity matrix, the gradient of the objective function (4.3) can be summarized as follows:
( )T∇ℜ = −S u u
, (4.12)
but on the basis of (4.8) Hessian can be approximated by
2 T( )∇ ℜ =p S S . (4.13) The corrections vector can therefore be determined from the system of equations
( )T Tδ ,= −S S p S u u
(4.14)
or from:
δ .δ= − =S p u u u
(4.15) The first of these formulas, (4.14), is a system of M linear equations with M unknowns, while the second, (4.15), system of C·N equations with M unknowns, ie over-determined system.
4.2. Methods for solution of over-determined equation systems Over-determined system of linear equations with the right hand side containing the measured data (4.15) is mathematically inconsistent. Its exact solution does not exist, and we can only be seeking for a solution minimizing error in the selected norm. Generally it is l2 norm, although in some applications there are useful other norms, such as l∞, which minimize the maximum solution error. One method of solving over-determined systems in l2 is the method of normal equations, which leads to a system of equations (4.14). Normal equations method causes significant numerical errors, additionally in the event of poor condition the created system of equations is difficult to regularization. In the literature we can find examples of over-determined equations whose solution by normal equations leads to a singular matrix, and the use of other methods, such as singular value decomposition of, leads to a solution. For this reason, discussion of methods for solving of over-determined systems focuses on method of decomposition of the matrix on singular values.
Any rectangular matrix S of dimensions C·N × M (C·N ≥ M) can be orthogonally decomposed into three matrices [2]
T=S U Σ V , (4.16) where U is orthogonal matrix of size C·N × C·N , V is the orthogonal matrix of dimensions M × M, which means that U U T = I and V V T = I, with: I - identity matrix. The matrix Σ has the same dimension as S, but it is a diagonal matrix with the following structure:
( )1 2, diag , ,..., .σ σ σ = =
DΣ D
0
0 0 K (4.17)
Diagonal matrix D of dimensions K × K, K ≤ M contains the singular values descending ordered
1 2 ... 0.σ σ σ≥ ≥ ≥ ≥K (4.18)
Substituting the orthogonal decomposition (4.16) to the system of equations (4.15) and multiplying both sides by UT continues to get still the system of C·N equations,
T Tδ δ=Σ V p U u , (4.19) the vector of unknowns, marked x, and excitation vector g can be represented as:
1 1T T
2 2
, where: δ , δ
= = = = = − ⋅ −
x gΣ x g x V p g U u
x g
K K
M K C N K. (4.20)
Utilizing the structure of the matrix Σ we can split system of equations (4.20) into two systems:
1 1
2 2.
=⋅ =
D x g
x g0 (4.21)
If M = K = C·N , the vectors labeled with index (2) do not exist. The first of equations (4.21) forms a square system of equations. Solution of the second is indefinite, therefore, we adopt a zero vector as a solution with the smallest norm. As we can see, the norm of the solution rest of over-determined equations system is then |g2|. Remembering, that the solution of over-determined system of equations delivers a corrections vector of the iterative algorithm, it can be stated, that such approach is fully reasonable. It may be a problem, to find a relevant number of K, it is the rank of the matrix D, which may be maximum M. However, to achieve the maximum rank of M is not beneficial in poorly conditioned systems. There are two possibilities here:
analysis of singular value σ (4.18) allows to find K corresponding to a so small singular value, that with the numerical accuracy it is equal to zero,
assumption for an iterative algorithm a priori value of the rank K for a further iteration.
For the aim of regularization we propose in the paper to assume in each iteration the condition factor κ for the sensitivity matrix S,
1,
q
a
bκ −= (4.22)
while q is the number of iteration. The parameters a and b are adopted a priori for the whole process of iteration. Basing on the condition factor κ there is then determined a threshold for singular values, below which the singular values are reset,
1ε κσ= . (4.23)
Described method was used in the example shown below.
5. Examples of inverse tasks
5.1. Recognition of material parameter using multi-frequency method The coil over the conducting plate The aim of the first example is to show of a well-conditioned iterative algorithm, with the same number of variables, as in the system of equations (4.15). As variables the values of conductivity in particular finite elements discretizing a part of conductive plate are taken. This part of the whole area will be hereinafter referred as search area. We have selected a simple model of eddy current flaw detection system [10], which consists of a coil placed above a conducting plate, as shown in Figure 5.1.
Fig. 5.1. Coils over conducting plate
Below the exciting coil the field measuring coil was placed, to determine the value of the magnetic flux over the defect. Arrangement of coils and plates have circular-cylindrical symmetry and can be analyzed in two dimensions r, z (2D). This type of symmetry means, that reference defect has the shape of circular groove of the axis of symmetry coinciding with the axis of the coil. As we can see, the limitations of 2D analysis does not allow for modeling a wide class of real defects. For discretization of the model the 10000 triangular finite elements with 5101 nodes were used. Bandwidth of the FEM matrix was equal 102. The model consists of an air area with the parameters µ = µ0, γ = 0, area of the excitation coil, wound with a very thin copper wire, which allows for the assumption of the same material parameters, such as air, and from the conductive plate with the parameters: µ = µ0, γ = 2·107S/m, which correspond to the non-magnetic steel. Exciting current density in the coil cross-section is constant JS = 106A/m2. Using the modified potential u in these areas, Laplace, Poisson and Helmholtz equations were constituted, respectively. Symmetry conditions require the assumption on the symmetry axis of the zero Dirichlet boundary condition. The same condition was adopted at the plate edges, while because of relatively close area cut in the air, the infinite elements were used there.
Fig. 5.2. Model discretization, material parameters and boundary conditions
The described model was then used to simulate field measurements over the plate with the defect. There were introduced two types of test defects and a set of C = 12 induced voltages from the measurement coils, placed directly under the excitation coil, was read off. The simulations were carried out for the following six frequencies (N = 6): 10, 20, 100, 200, 1000, 2000 [kHz]. The defect was then removed from the test plate and launched an iterative algorithm, which is illustrated in Figure 4.1. Initial value of conductivity γe = 2·107S/m corresponded to the full board. The search area was selected, consisting of M = 72 elements. It reveals in Figure 5.2 darker area in the plate. The sensitivity of voltage induced in the measuring coils in terms of conductivity in all finite elements of in the search area, for the six frequencies used, has been evaluated. In this way, a square system of equations (M = C·N) was formed, from which we could evaluate the corrections of conductivity for each element.
1 111 12 13 1
2 221 22 23 2
3 331 32 33 3
31 2
δ δ....
δ δ....
δ δ.... . .
.... ............ .... .... ....
δ δ....
uS S S S
uS S S S
uS S S S
uSS S S
γγγ
γ
=
M
M
M
M MMM M MM
(5.1)
Since a voltage variations δuM were calculated using the same numeric model, as original, the system of equations (5.1) is relatively well conditioned and can be easily solved. The corrections δγ obtained by using the symbolic method also have a complex value, and
therefore the corrections made to the model conductivity are real part: δγ = Re(δγ). The iterative algorithm converges fast to the shape of a test defect.
Fig. 5.3. Conductivity distribution after five iterations (two defects)
Fig. 5.4. Conductivity distribution after five iterations (skew defect)
Conductivity distributions shown in Figure 5.3 and Figure 5.4 were obtained after five iterations. These shapes correspond well to the assumed defects. Differential coils inside conducting pipe Another model is showing the operation of algorithms on the model of differential coils placed concentric in conductive pipe, similar to that described previously. Its contours are shown in Figure 5.5. This system owns the circular-cylindrical symmetry, coinciding with the
axis of the tube. Taking into account this kind of symmetry for a defect means, that it has the form of concentric groove running around the pipe. Such model may be responsible for a round rupture or weld.
Fig. 5.5. The model of differential probe inside conducting pipe
The model was discretized using 3864 finite elements with 1998 nodes. Bandwidth of the MES matrix was equal 49. Pipe material parameters were: µ = µ0 and γ = 2·107S/m. Excitation current density in the differential coils the was JS = 106A/m2. As in the previous example, the value of voltage induced in the measurement coil was obtained by the numerical analysis. There were used four frequencies: 10, 20, 100 and 200[Hz]. Using the so-called envelope finite element was found necessary to maintain the symmetry of the model so, as to obtain the voltage induced in the measuring coil by absence of defect equal to zero. In Figure 5.6 shows an example of field distribution around a differential coils system at a frequency f = 10Hz in the absence of a defect in the wall of the tube.
Fig. 5.6. Equipotential lines of vector potential
Search area is defined so, that it consisted of 36 finite elements. To obtain a sufficient number of measurements required for the formulation of the inverse task, the defect has been shifted against coils in nine steps, as 2mm. In this way the 36 equations (5.1) with 36 unknown corrections δγ were obtained. For a defect located on the same side of the wall, as the sensor, an iterative algorithm converges very quickly and after three iterations the shape of the test defect was recognized.
Fig. 5.7. Recognized test defect inside the wall
System of equations (5.1) in solving the inverse task for a defect located on the opposite side of the pipe wall has proved to be ill-conditioned and the algorithm does not converge. Therefore the simple Tikhonov regularization algorithm was applied, modifying elements of the sensitivity matrix and right hand vector as follows:
* * *
=1 =1
= + , = and δ = δ .M M
kk kk kl mk ml k mk k
m m
S S S S S u S uα ∑ ∑ (5.2)
There was experimentally selected the lowest value of coefficient α, at which the iterative algorithm converges. After completing five iterations the conductivity distribution corresponding to the test defect outside the pipe was obtained and shown in Figure 5.8.
Fig. 5.8. Recognized defect shape outside the pipe wall
The two presented examples show the operation of an iterative algorithm for the inverse task, when input data are affected by at most a small error of numerical analysis of the model. The real measurement data, applied without improving the condition factor of the equations system, lead to divergence of the algorithm.
5.2. Shape recognition with multi-frequency method By optimizing of the shape in the example below sensitivity dependences derived above are used. Iterative algorithm for inverse tasks is the same, as this for the material parameter recognition using the incremental system. The main difference is, that in the inverse task of the shape recognition usually occurs only a little number of variable parameters, which are in this case the geometric coordinates of the finite element nodes. This often allows us to build a over-determined system of equations and, as a result, to find the optimal solution. The values of relative sensitivity in the shape identification of the conductive, or additionally ferromagnetic bodies, are clearly greater than the sensitivity calculated in terms of conductivity of the finite elements. This is because the nodes, which are optimized, lie on the boundary, and their displacement results in large changes of the field in the measurement coil. Described reasons mean, that an inverse algorithm used for shape optimization is usually a well-convergent. An important problem is changing the topology due to shifting the boundary line. Generating a new digitization in each iteration is inefficient, besides that it would result in abrupt changes
of FEM discretization error, making impossible the convergence of algorithm. Therefore, if a displacements are small, a local change of discretization is undertaken, so as not to destroy the continuity of the mesh. How local changes in discretizing mesh works is shown in Figure 5.9. Suppose, that there has been designated the new location of node P0. Shifting the node along the y-axis is accompanied by a forced transfer of nodes P1, P2 and nodes located below, P-1, P-
2. Forced shifting depends on the distance of nodes from P0, while outside a certain circle nodes are not moved.
Fig. 5.9. Local discretization changes of finite element mesh
This method is suitable for use also in tetrahedral networks, it avoids the generation of discontinuity in meshes, but causes adverse changes in refinement of elements. As shown in Figure 5.9, there was the density increase over node P0 and density decrease underneath. The model used to the shape recognition of the surface corresponds to the arrangement of measuring used in the work [15]. It has been shown in Figure 5.10. It consists of two differential coils wound at very highly ferromagnetic core, between which there is placed a measuring coil. Value of the exciting current I = 60mA, frequency f = 1kHz. Defect was in the form of coaxial groove of dimensions 1×3mm. Measurements of voltage induced in the coil were performed by moving the probe against defects with 2mm steps at distance 40mm (C = 20 measurements). Measured was real and imaginary part of the voltage induced in the coil.
Fig. 5.10. Model of the probe inside the pipe with coaxial defect
Then, using the model shown in Figure 5.10, there was simulated measurement with respect to the probe shift versus defect. In order to eliminate the discretization error and the impact of FEM mesh cutting, the results are codified in terms of the maximum value of the measurement signal. The result are the following characteristics, separately for real and imaginary parts of voltage [11]:
Fig. 5.11. The voltage induced in measurement coil for different positions of probe
Basing on simulation model, the minimization of following error function was done
*1
1
1( )( )
2u u u u
C
v v v v
v=
ℜ = − −∑ , (5.3)
where uv denotes the voltage value in measurement coil for position number v.
The width of search area was 6mm, which with a elements width equal to 0.5mm gave the number of moving nodes M = 13. From the 20 measurements there were rejected those corresponding to the maximum probe distance from the defect, leaving C = 13. Although in this way the square system of equations was received, but it required regularization. To solve this problem, the TSVD with variable rank of the matrix S was used. After 15 iterations the trajectory of the signal received from a defect became similar to the measured trajectory.
Fig. 5.12. Trajectory of the signal induced in measurement coil
In order to illustrate and evaluate an iterative process, the two error rates were introduced. The first is based on the difference of voltage induced in the measuring coil in the present iteration and measured voltage,
*
2
*
( )( )
100%( )( )
śr śr
U
u u u u
u u u u
Ż
v v v v
v
v v v v
v
− −
= ⋅− −
∑∑
. (5.4)
The second one was defined knowing the actual shape of the surface defect,
*
2
*
( )( )
100%,( )( )
śr śr
P
x x x x
x x x x
Ż
v v v v
v
v v v v
v
− −
= ⋅− −
∑∑
(5.5)
where: vx
– denotes the known coordinates of the surface.
Fig. 5.13. Error rates (5.4) and (5.5) for consecutive iterations
Both, the field error rate (5.4) and shape error rate (5.5) reach a minimum in the 15-th iteration, when the process should be terminated. The surface shape reached at 15-th iteration is shown in Figure 5.14.
Fig. 5.14. Surface shape after 15 iterations. Test defect marked with thick line
Then, an iterative process was repeated utilizing only module of the voltage measured in the measuring coil, minimizing the following objective function of the form:
22
1
1(| | | |)
2u u
C
v v
v=
ℜ = −∑ . (5.6)
To show an iterative process, there were used two indices similar to previously defined, except that the error rate of the voltage in the measuring coil was in this case of the form
2
2| | 2
(| | | |)
100%(| | | |)
śr
U
u u
u u
Ż
v v
v
v v
v
−
= ⋅−
∑∑
. (5.7)
Iterative process was similar as before, the minimum value of both error rates was obtained at 15-th iteration (Figure 5.15).
Fig. 5.15. Error rates (5.7) and (5.5) for consecutive iterations
After stopping the iterative process on the 15-th iteration, we compared the voltage module measured in the coil and the value determined using FEM, obtaining a good agreement of the two graphs (Figure 5.16). Despite this, the shape of surface defect, shown in Figure 5.17, differs from the test defect more than the shape shown in Figure 5.14. The reason for this is the amount of information carried by measuring the voltage module less than this for measuring both, the module and phase shift (or the real and imaginary) of voltage. Algorithm used to move nodes does not allow for modeling defect of rectangular sides, but their shape approximates a diagonal line. Unfortunately, here is a significant disturbance of finite elements discretization and mesh elements would require further adaptation of.
Fig. 5.16. Voltage module measured in the coil and the value determined using FEM
Fig. 5.17. Surface shape recognized basing only on module of voltage. Test defect marked with thick line
5.3. Recognition of material parameters in the time domain The identified parameter of material will be electrical conductivity γ. We will use two-dimensional model of the system, similar to that shown in (Fig. 3.3) and in the work [13], modified slightly so, that it contains additionally a core concentrating magnetic field. It is a differential measurement system, which includes two opposite connected measurement coils. In the absence of defect, the system is symmetrical and its output voltage is zero. The measurement with described sensor was simulated using 189696 finite element and 95409 nodes. The assumed finite element size is suitable for moving the search area including a defect in steps of 0.125mm. Measurement of the output sensor voltage was made in this way for Cp = 97 sensor locations.
Fig. 5.18. Model dimensions and parameters
Fig. 5.19. Positions of the search area
The measuring coil had 500 turns. Exciting current density was JSmax = 0,14 A/mm2. Number of turns of exciting coil was n = 230, what correspond to a driving current of 10mA. Pulse duration has been chosen so, that the field has time during its lifetime to diffuse the entire pipe wall. The parameters of the wall correspond to Inconel 600: µr = 1 and γ = 1MS/m, which after taking into account the wall thickness δ = 1,27mm gave the pulse duration of T:
2
1,5µs4
πδ µγ= ≈T . (5.8)
Analysis time T was divided into n = 150 equal time elements. Pulse shape is shown in Figure 5.20.
Fig. 5.20. Excitation current shape
To show, that the identification of the conductivity distribution is possible with other discretization elements, the process used to identify a network containing only 128700 finite elements and 64775 nodes. Grid step was 0.2mm (Fig. 5.19), which gave the distance between the start and stop of C = 61 positions. Only some of them coincide to the sensor positions during simulation of measurement, the remaining measurement data were interpolated using the method of sections. In practice it is hard to expect, that the finite elements discretization allows for accurate representation of the real defect, thus mesh used in the identification process, marked in Figure 5.21 color white was chosen so, as not to be able to imitate its shapes shown in the test defects fig. 5.21a, b, c, d. The grid used in the simulation of the measurement is marked in black.
Fig. 5.21. The test defects in the mesh elements used in the simulation of measurement (black) and mesh
used to identify defects (white) We assumed, that the iterative process can be maximally continued up to 35-th iteration. The parameters of regularization were assumed experimentally (4.22) a = 0,99 and b = 1,5. Choice of parameters was carried out in such a way that, at first, not to carry out too many iterations
with a constant rank of the matrix, and at last, not to cause a divergence process. The values of the condition factor imposed under this circumstances are shown in Figure 5.26.
Fig 5.22. Identification process of defect 5.21a: a) the assumed distribution of conductivity, b) the distribution of conductivity after 5 iterations, c) after 15 iterations, d) after 23 iterations
Fig. 5.23. Identification process of defect 5.21b: a) the assumed distribution of conductivity, b) the distribution of conductivity after 5 iterations, c) after 15 iterations, d) after 25 iterations
Fig. 5.24. Identification process of defect 5.21c: a) the assumed distribution of conductivity, b) the distribution of conductivity after 5 iterations, c) after 15 iterations, d) after 21 iterations
Fig. 5.25. Identification process of defect 5.21d: a) the assumed distribution of conductivity, b) the distribution of conductivity after 5 iterations, c) after 15 iterations, d) after 23 iterations
In Figure 5.22 is shown the process of identifying a defect 5.21a. Only 23 iterations were performed, because by the 24-th in the parameters appeared negative conductivity. Achieved rank of the matrix for various types of defects is shown in Figure 5.27. We can conclude, that the position and shape of the defect has been reproduced exactly. The value of conductivity was illustrated in Figures 5.22 - 5.25 with the degrees of gray. A white color corresponds to conductivity γ = 0, while the black color was attributed to conductivity γ > 1MS/m. Then we tried to identify a defect lying on the opposite side of the pipe wall (Fig. 5.21b). In this case up to 31 iterations succeeded to perform, but in the final iterations, the increasing of the objective function was observed, which can be clearly seen in Figure 5.28b. This is due to the fact, that the process used in the iterative Gauss-Newton algorithm does not guarantee the minimization of the objective function from iteration to iteration. Basing on the analysis of the chart in Figure 5.28b, as the final result of iterations is considered the distribution of conductivity after the 25th iteration (Fig. 5.23d). There was correctly reproduced the shape of a skew crack (Fig. 5.21c). Distribution of conductivity after 21 iterations is shown in Figure 5.24d. But there was no success identifying the group of defects 5.21d. The resulting distribution (Fig. 5.25d) shows a large defect within the pipe wall. For the lack of success may be to blame in this case not sufficiently accurate discretization of the area. The graphs 5.26 - 5.28 illustrate the process of identifying of exemplary defects. In any case, the process did not reach a ceiling of 35 iterations, at which the matrix of the sensitivity reaches a full rank. Sensitivity matrix consisted of 9150 rows and 360 unknowns. Interruption of iterations was usually caused by a occurrence of negative conductivity, which has prevented a solution of the direct problem.
Fig. 5.26. The condition factor assumed for consecutive iterations
Fig. 5.27. Matrix rank reached during iterations for different defects
Fig. 5.28. Objective function minimization as function of iteration and for different defects
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