Inverse Problem for Coefficient Identification in

Inverse Problem for Coefficient Identification in Euler-Bernoulli Equation by Linear Spline Approximation Tchavdar T. Marinov1 and Rossitza Marinova2 1

2

Southern University at New Orleans, Department of Natural Sciences, 6801 Press Drive, New Orleans, LA 70126 [email protected] Department of Mathematical and Computing Sciences, Concordia University College of Alberta, Edmonton, AB, Canada T5B 4E4, Adjunct professor at the Department of Computer Science, University of Saskatchewan, SK, Canada [email protected]

Abstract. We display the performance of the technique called Method of Variational Imbedding for solving the inverse problem of coefficient identification in Euler-Bernoulli equation from over-posed data. The original inverse problem is replaced by a minimization problem. The EulerLagrange equations comprise an eight-order equation for the solution of the original equation and an explicit system of equations for the coefficients of the spline approximation of the unknown coefficient. Featuring examples are elaborated numerically. The numerical results confirm that the solution of the imbedded problem coincides with the exact solution of the original problem within the order of approximation error.

1

Introduction

Consider the simplest form of Euler-Bernoulli equation d2 d2 u σ(x) = f (x), 0 ≤ x ≤ 1. dx2 dx2

(1)

The function f (x) represents the transversely distributed load. The coefficient σ(x), called flexural rigidity, is the product of the modulus of elasticity E and the moment of inertia I of the cross-section of the beam about an axis through its centroid at right angles to the cross-section. If the coefficient σ(x) > 0 and the right-hand side function f (x) ≥ 0 are given, under proper initial and/or boundary conditions, the problem possesses a unique solution, usually referred as a direct solution. In practice, there exist lots of interesting problems, in which the coefficient σ(x) is not exactly known. In reality, Euler-Bernoulli equation models a tensioned “beam”. Under environmental loads, caused by environmental phenomena such as wind, waves, current, tides, earthquakes, temperature, ice, the structure of the I. Lirkov, S. Margenov, and J. Wa´ sniewski (Eds.): LSSC 2009, LNCS 5910, pp. 588–595, 2010. c Springer-Verlag Berlin Heidelberg 2010

Inverse Problem for Coefficient Identification in Euler-Bernoulli Equation

589

ingredients of the “beam” is changing. Usually it is expensive, even not possible, to measure the changes of the properties of the materials directly. On the other hand, the changes in the physical properties of the materials cause changes in the coefficient σ in equation (1) and, respectively, changes in the solution. Thus, a new, so called inverse, problem appears: to find simultaneously the solution u and the coefficient σ of the Euler-Bernoulli equations. A method for transforming an inverse problem into a correct direct problem, but for a higher-order equation, was proposed in [1] and called Method of Variational Imbedding (MVI). The idea of MVI is to replace the incorrect problem with a well-posed problem for minimization of a quadratic functional of the original equations, i.e. we “embed” the original incorrect problem in a higher-order boundary value problem which is well-posed (see [1,7,8]). For the latter, a difference scheme and a numerical algorithm for its implementation can easily be constructed. The advantage of MVI comparing to a regularization method (see, for example, [10,4]) is that there are no “boundary layers” at the two ends of the interval as it was observed for a similar problem in [5]. Recently, in [9], the MVI was applied to the problem for identifying the coefficient σ in (1) in the case when the coefficient is a piecewise constant function. In the present work we are considering the case when the coefficient is a piecewise linear function. Although this paper is focused on a fourth order ordinary differential equation, the proposed method can be generalized for identification of coefficient in partial differential equations. Similar to the procedure proposed here, the approach for the identification of a coefficient in parabolic partial differential equation is given in [2]. In [8] MVI was successfully applied to the problem for identification of a coefficient in elliptic partial differential equation. The paper is organized as follows. In Section 2 the inverse problem for identification of the unknown coefficient is formulated. The application of the MVI to the inverse problem is described in Section 3. The numerical scheme is given in Section 4. Illustration of the constructed numerical scheme is given in Section 5.

2

Inverse Problem Formulation

Consider the Euler-Bernoulli equation (1) where the function f (x) is given. We expect that the functions under consideration are as many time differentiable as necessary. If the coefficient σ is not given, in order to identify it one needs additional information. Suppose the solution satisfies the conditions u(0) = α0,0 ,

u(1) = α1,0 ,

(2)

u (0) = α0,1 ,

u (1) = α1,1 ,

(3)

u (0) = α0,2 , u(ξi ) = γi ,

u (1) = α1,2 , i = 1, 2, . . . , n − 1,

(4) (5)

where the points 0 < ξi < 1 are given. We suppose that the coefficient σ is a piecewise linear function σ(x) = σi (x) = ai + bi (x − ξi−1 ) for ξi−1 < x < ξi , i = 1, 2, . . . , n,

(6)

590

T.T. Marinov and R. Marinova

where the constants ai , bi , are unknown (ξ0 = 0, ξn = 1). The number of unknown constants ai , bi is 2n. On the other hand, the additional number of conditions in (2)-(5) for the fourth order ordinary differential equation is n + 1 = (n + 5) − 4. Therefore, if we add the condition for continuity of the function σ, i.e., σi (ξi ) = σi+1 (ξi ),

(7)

i = 1, 2, . . . , n − 1 the number of the conditions is exactly equal to the number of unknown constants. There may be no solution (u, σ), satisfying all of the conditions (2)–(5), for arbitrary αk,l , (for k = 0, 1 and l = 0, 1, 2), and γi , (for i = 1, 2, . . . , n − 1). For this reason, we suppose that the problem is posed correctly after Tikhonov, [10] i.e., it is known a-priori that the solution of problem exists. In other words, we assume that the data in the boundary conditions (2)–(5) have “physical meaning” and, therefore, the solution exists. The problem is how to convert this additional information to the missing information on the coefficients. The solution approach proposed here is a generalization of the implementation of MVI to a similar problem given in [6] for identification of a piecewise coefficient of two parts (see also [7] and [8]), and continuation of the idea proposed in [9].

3

Variational Imbedding

Following the idea of MVI, we replace the original problem with the problem of minimization of the functional 1

1 A (u, σ)dx =

I(u, σ) =

2

0

d2 dx2

2 d2 u σ(x) 2 − f (x) −→ min , dx

(8)

0

where u satisfies the conditions (2)–(5), σ is an unknown piecewise function, defined with equation (6). The functional I(u, σ) is a quadratic and homogeneous function of A(u, σ) and, hence, it attains its absolute minimum if and only if A(u, σ) ≡ 0. In this sense there is an one-to-one correspondence between the original equation (1) and the minimization problem (8). Since σ(x) is a piecewise linear function, we can rewrite the functional I as ξ 2 n i 2 d d2 u σi − f (x) −→ min . I(u, σ) = dx2 dx2 i=1

(9)

ξi−1

The necessary condition for minimization of the functional I is expressed by the Euler-Lagrange equations for the functions u(x) and σ(x). 3.1

Equation for u

The Euler-Lagrange equation with respect to the function u reads 2 d d2 d2 d2 d2 d2 u σ A= σ σ − f (x) = 0. dx2 dx2 dx2 dx2 dx2 dx2

(10)


591

Therefore, in each interval ξi−1 < x < ξi , the function u satisfies the equation d4 d2 u d2 d2 d2 σi 4 σi 2 = σi 2 f (x), 2 2 dx dx dx dx dx

(11)

under the boundary conditions (2)–(5). Since each equation (11) is of eight order we need some additional boundary conditions. From the original problem we have d2 d2 u σi − = f (ξi ), dx2 dx2 ξi

d2 u d2 σ + = f (ξi ), i+1 dx2 dx2 ξi

(12)

where ξi− and ξi+ stand for the left-hand and right-hand derivatives and d2 d4 d2 u σ f (ξi ), − = i dx4 dx2 ξi dx2 2 2 d u d u σi 2 ξ− = σi+1 2 ξ+ , dx i dx i

d4 d2 d2 u σ f (ξi ), + = i+1 dx4 dx2 ξi dx2 2 2 d d u d d u σi 2 ξ− = σi+1 2 ξ+ , dx dx i dx dx i

(13) (14)

where i = 1, 2, . . . , n − 1. 3.2

Equation for σ

The problem is coupled by the equation for σ. Since σ is a piecewise function, for the functional I one arrives at the problem for minimization of the function q(a1 , . . . , an , b1 , . . . , bn ) =

n

(Ai11 a2i +2Ai12 ai bi +Ai22 b2i +Ai1 ai +Ai2 bi +Ai0 ), (15)

i=1

with respect to a1 , . . . , an , b1 , . . . , bn under the continuity conditions (7) which we rewrite in the form ai + bi (ξi − ξi−1 ) − ai+1 = 0.

(16)

Minimizing the function q under the constraints (16) we introduce Lagrange multipliers μi and introduce the function: Q=

n i 2

A11 ai + 2Ai12 ai bi + Ai22 b2i + Ai1 ai + Ai2 bi + Ai0

(17)

i=1

+

n−1

μi (ai + bi (ξi − ξi−1 ) − ai+1 ) .

i=1

We obtain the following five-diagonal system of linear equations for ai , bi , μi : ∂Q = 2A111 a1 + 2A112 b1 + A11 + μ1 = 0, ∂a1 ∂Q = 2A112 a1 + 2A122 b1 + A12 + μ1 ξ1 = 0, ∂b1 ∂Q = a1 + b1 ξ1 − a2 = 0, ∂μ1

(18) (19) (20)

592


and ∂Q = 2Ai11 ai + 2Ai12 bi + Ai1 + μi − μi−1 = 0, ∂ai ∂Q = 2Ai12 ai + 2Ai22 bi + Ai2 + μi (ξi − ξi−1 ) = 0, ∂bi ∂Q = ai + bi (ξi − ξi−1 ) − ai+1 = 0, ∂μi

(21) (22) (23)

for i = 2, . . . , n − 1, and ∂Q = 2An11 an + 2An12 bn + An1 − μn−1 = 0, ∂an ∂Q = 2An12 an + 2An22 bn + An2 = 0. ∂bn

4

(24) (25)

Difference Scheme

We solve the formulated eight-order boundary value problem using finite differences. It is convenient for the numerical treatment to rewrite the eight order equation (11) as a system of four second order equations. In each of the subintervals [ξi−1 , ξi ], i = 1, 2, . . . , n we solve the following system of four equations u = v, 4.1

(σv) = w,

w = z,

(σz) = (σf ) .

(26)

Grid Pattern and Approximations

We introduce a regular mesh with step hi (see Fig. 1) in each of the subintervals [ξi−1 , ξi ], i = 1, 2, . . . , n, allowing to approximate all operators with standard central differences with second order of approximation. i−1 , where ni For the grid spacing in the interval [ξi−1 , ξi ] we have hi ≡ ξin−ξ i −2 is the total number of grid points for the i-th interval. Then, the grid points are defined as follows: xij = ξi−1 + (j − 1.5)hi for j = 1, 2, . . . , ni . Let us introduce the notation uij = u(xij ) for i = 1, 2, . . . , n, and j = 1, . . . , ni . We employ symmetric central differences for approximating the differential operators. The differential operators in the boundary conditions are approximated with second order formulae using central differences and half sums.

•

ξi−1 | •

•

•

xi1

xi2

xi3

xi4

...

•

•

•

xini −3

xini −2

xini −1

Fig. 1. The mesh used in our numerical experiments

ξi |

• xini


4.2

593

General Construction of the Algorithm

(I) With the obtained “experimentally observed” values of αk,l , (for k = 0, 1 and l = 0, 1, 2, 3), and γi , (for i = 1, 2, . . . , n − 1) the eight-order boundary value problem (10), (2)–(5), (12)–(14) is solved for the function u with an initial guess for the function σ. (II) The approximation of the function σ for the current iteration is calculated from the system (18)-(25). If the l2 -norm of the difference between the new and the old field for σ is less than ε0 then the calculations are terminated. Otherwise, the algorithm returns to (I) with the new calculated σ.

5

Numerical Experiments

The accuracy of the difference scheme is validated with tests involving various grid spacing h. We have run a number of calculations with different values of the mesh parameters and verified the practical convergence and the O(h2 ) approximation of the difference scheme. For all calculations presented here, ε0 = 10−12 and the initail values for the coefficients in σ are ai = 0.5 and bi = 1.5. Here we illustrate the developed difference scheme using two coefficient identification problems. 5.1

Linear Coefficient

Consider the case when σ(x) = 1 + x and f (x) = (3 + x) exp(x) for which under proper boundary conditions the exact solution is u(x) = exp(x).

(27)

For this test we let the number of intervals n in the definition (6) of σ equal to 1, i.e., n = 1. In other words, we know a-priori that the coefficient is a linear function. The goal of this test is to confirm second order of approximation of the proposed scheme. The values of the identified coefficient σ = a + bx with four different steps h are given in Table 1. The rates of convergence, calculated as a2h − aexact b2h − bexact , , rate = log (28) rate = log2 2 ah − aexact bh − bexact Table 1. Obtained values of the coefficients a and b, and the rate of convergence for four different values of the mesh spacing h exact 0.1 0.05 0.025 0.0125

a 1.0 0.996671245 0.999166953 0.999791684 0.999947911

|a − aexact | rate b |b − bexact | rate — — 1.0 — — 3.328754087E-03 — 0.998334860 1.665139917E-03 — 8.330468464E-04 1.99851 0.999583428 4.165712337E-04 1.99901 2.083158588E-04 1.99962 0.999895839 1.041605270E-04 1.99975 5.208818084E-05 1.99974 0.999973961 2.603891276E-05 2.00007

594


Table 2. l2 norm of the difference u − uexact and the rate of convergence for four different values of the mesh spacing h 0.1 0.05 0.025 0.0125

||u − uexact ||l2 rate 1.018490804573E-04 — 2.367271717866E-05 2.10514 5.686319062981E-06 2.05766 1.392183281145E-06 2.03015

Table 3. l2 norm of the differences u−uexact and σ −σexact and the rate of convergence for four different values of the mesh spacing h 0.01 0.005 0.0025 0.00125

||σ − σexact ||l2 rate ||u − uexact ||l2 rate 1.069187970901E-08 — 4.552581248664E-05 — 2.584956511555E-09 2.0483 1.138599436962E-05 1.99942 6.379806228978E-10 2.01856 9.614682942106E-07 3.56588 1.586000035587E-10 2.00812 1.624871713448E-07 2.56491

are also shown in Table 1. Similar results for the l2 norm of the difference between the exact and the numerical values of the function u are presented in Table 2. This test clearly confirms the second order of convergence of the numerical solution to the exact one. 5.2

Linear Coefficient as a Piecewise Linear Function

Consider again the solution (27) but now we do not assume a-priori that the coefficient is the same function in the whole interval. We identify the coefficient as a piecewise linear function, as defined in (6), for n = 10. In each subinterval, the expected values of the coefficient σ are ai = 1 and bi = 1. For this test we performed a number of calculations with different spacings h. The l2 norm of the difference between the exact and the numerical values of the functions u and σ, and the rate of convergence, calculated using the norm of the difference, for four different steps h, are given in Table 3. The fact that the numerical solution approximates the analytical one with O(h2 ) is clearly seen from the Table 3.

Acknowledgment This work was partially supported by MITACS and NSERC.

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