4 Variational Calculus
4.5 Calculus of Variations in Adjustment Theory
= ˆb
a
∂Z
∂ Φ00+εδΦ00δΦ00dx ε=0
= ˆb
a
∂Z
∂Φ00δΦ00dx= 0! . (4.71)
By applying the fundamental lemma of calculation of variations to the above equation, the Euler-Lagrange equation for this case is obtained
∂Z
∂Φ00 = 0. (4.72)
Equating both strong formulations, the above Euler-Lagrange equation and the Poisson equation in Eq. (4.53), it reads
∂Z
∂Φ00 =Φ00−L . (4.73)
Hence, the Lagrangian density functional is Z = 1
2Φ00Φ00−LΦ00+C , (4.74)
whereC = C x
is an arbitrary function. In a special case whereC = 12LL, the Lagrangian density functional becomes
Z = 1
2Φ00Φ00−LΦ00+ 1
2LL= 1
2 Φ00−L2
. (4.75)
The extremal formulation of the Poisson equation for the “least squares” finite element method is
J = ˆb
a
1
2 Φ00−L2
dx → minimal. (4.76)
The same objective function as above can be determined if the residual functionvis introduced to Poisson equa-tion in Eq. (4.53) as
d2Φ
dx2 =L+v . (4.77)
The same result as in Eq. (4.76) is obtained for minimizing the integrated squared residual function
J = ˆb
a
1
2v2dx= ˆb
a
1 2
Åd2Φ dx2 −L
ã2
dx= ˆb
a
1
2 Φ00−L2
dx → minimal. (4.78)
Due to this similarity with the method of least squares, this approach to solve the Poisson’s differential equation is called theleast squaresfinite element method. One notices that both objective functions, in Eq. (4.67) and in Eq. (4.76), are fundamentally different. Although both extremal formulations are used to approximate solution for same strong formulation, the Poisson equation.
In conclusion, there is a unique strong formulation for a specific problem. But, different extremal formulations of the same strong formulation can be found. Consequently, there exists many variational formulations. This in turn leads to different approximate solutions with various qualities for a specific problem. For a unique numerical solution, however, an unambiguous extremal formulation (apart from scaling and shifting) must be established.
This is accomplished by specifying the trial and test function,ΦandδΦ.
the scheme of variational calculation. In this dissertation, the least squares and finite element method are analysed in the context of the variational calculus.
The finite element method is used to approximate a solution of differential equations. While the least squares adjustment is applied for solving overdetermined system of algebraic equations. Both methods share the same chal-lenge of not knowing the “true” solution. Even though a system of differential equations for a specific engineering problems can be formulated by means of, for example, continuum physics. But afterwards, there are generally two obstacles: Does a solution exist at all for the given differential equations? And, if it is the case, how do we get it?
In fact, one may not even know if there is a solution at all. For example, the existence of an exact solution of the Navier-Stokes equation in fluid mechanics is still unknown, see (Fefferman 2000). Only for a limited case such as the one-dimensional Poisson equation respectively Euler-Bernoulli beam equation, it is possible to obtain close-form solution. But, these types of differential equations result from simplification of an engineering problem with debatable assumptions. In order to obtain quantitative results without any questionable simplifi-cations, a numerical approximation technique such as the finite element method must be used. The differential equations are multiplied by the test function and integrated over the region of interest. This transforms the dif-ferential equations into their corresponding variational form. It is then possible to obtain numerical results. In practice, it usually remains unnoticed that the procedure of variational calculus is applied in reverse. In the con-text of variational calculus, the differential equations represent the strong formulation of the problem. Since their variational formulation ultimately leads to the approximate solution, it is understandable that their extremal formu-lation becomes irrelevant in the finite element method. There are other numerical methods for solving differential equations such as the least squares finite element method. By using different trial and test functions,ΦandδΦ, various numerical methods result. This approach is the so-calledmethod of weighted residuals, see (B. Finlayson and Scriven 1966). Theresidualsrefer to the discrepancy arising from the fact that any trial functions fails to satisfy the strong formulation. Theweightreferred to the possibility of using different test functions. And, the integratedweighted residualslead to various variational formulations. These in turn lead to different approximate methods. Also, in a similar situation concerning the solutions of differential equations, overdetermined system of algebraic equations has no solution at all due to their discrepancies. In adjustment calculation, this deficiency can be eased by introducing residuals into the algebraic equations as an additive quantity to the observations. At the same time, the sum of weighted squared residuals must be minimal in order to obtain an adjusted solution. In the adjustment calculation, the variational calculus procedure is followed in a special way. The strong formulation is the overdetermined system of algebraic equations. Instead of following the variational calculus in reverse as in the finite element method, i. e. by multiplying the test functions with the strong formulation in order to obtain the variation formulation respectively the normal equations. A “detour” is made by introduction of residuals to the algebraic equations. And a “special” extremal formulation that the sum of weighted squared residuals has to be minimal is then postulated. The directional derivative respectively the derivation with respect to the unknowns of this objective function also leads to the normal equations. This target function is in so far “special” as other objective functions are also possible that result in exactly the same numerical solution. Other kinds of objective functions can be formulated by scaling and shifting. Ultimately, this special objective function is required for statis-tical evaluations. By using different trial and test functions,ΦandδΦ, various analysis methods result. In particular, the Fourier series belongs to the problem domain, which can be expressed by an algebraic equation as strong for-mulation. The trial and test functions,ΦandδΦ, are represented as the sum of a set of complex exponentials. In summary, it can be concluded that both the finite element method and the least squares adjustment follow the same variational calculus procedure. The normal equations of the least squares adjustment and the system of linear equa-tions of the finite element method are similar insofar as both methods use their corresponding variational forms to calculate their respective numerical solutions. The differences are that both solve different types of equations in conjunction with different representations for the trial and test function. Hence, the relationship between finite element method and least squares adjustment can be associated by variational calculus.
A few final remarks are in order: First, variational calculus is to be considered as a universal procedure of the adjust-ment theory. Many methods can be explained by answering the following questions:
• What is the formulation of the problem and, if necessary, the conditions?
• Is it a continuous or discrete problem?
• Which trial function is used for the solution?
Therefore, variational calculus should be considered as a unifying method in the adjustment theory. An overview of the discussed methods is shown in Tab. 4.1 . Second, methods have been developed from different scientific fields to solve problems of their discipline. Accordingly, the methods are strongly motivated by the specific perspectives of the respective discipline. From an abstract perspective, i. e. the methods are presented without mechanical or geodetic accents, it becomes clear that the methods are actually identical. Therefore, the finite element method should be regarded as part of the adjustment theory rather than part of physics, because this method requires no physical justification. In addition, the adjustment theory should be understood as a method theory. Third, B.
Finlayson and Scriven (1966) as well as B. A. Finlayson (1972) describe in their paper the method of weighted residuals. Their focus is on the different methods of solving differential equations numerically. The fact that the methods are also suitable for data analysis, i. e. algebraic equations can also be solved numerically with them, is ignored. As far as Boljen (1993) is concerned, in his work he presents a short treatment of the variational calculus.
A comparison between the continuous and discrete version of least squares method are also shown. He realizes that what he considered as “finite element method” is a continuous problem. But if one takes a closer look at his extremal formulation, it turns out that this is least squares finite element method rather than Ritz-Galerkin finite element method. Also, Milev (2001) recognizes the analogies between the principle of virtual work from mechanics and the method of least squares in geodesy. But in his work he introduces the variational calculus by means of Lagrangian mechanics for non-expert in mechanics. Through this representation, the methodological similarities between the two methods are difficult to comprehend. A more detailed presentation of the variational calculus and the abstraction of important elements of it were absent in the work of the above-mentioned authors.
82 CALCULUS OF VARIATIONS IN ADJUSTMENT THEORY|VARIATIONAL CALCULUS
Table4.1:Anoverviewofthediscussedmethods.Gauss-Markovmodel(GMM),con- tinuousGauss-Markovmodel(cGMM),finiteelementmethod(FEM),least squaresfiniteelementmethod(LSFEM) DGMMCGMMFEMLSFEM StrongFormΦ=LΦ=LΦ00 =LΦ00 =L VariationalFormΦT P−LT P δΦ=0´ Φ−L δΦdx=0´ Φ0 δΦ0 +LδΦ dx=0´ Φ00 −L δΦ00 dx=0 possibleExtremalFormΩ=Φ−LT PΦ−L Ω=´ Φ−L2 dxJ=´ −1 2Φ0Φ0−LΦ+C dxJ=´ 1 2Φ00−L2 dx TrialFunctionΦ=AXΦ=P iciXiΦ=P iciXiΦ=P iciXi TestFunctionδΦ=AYδΦ=P jbjXjδΦ=P jbjXjδΦ00 =P jbjX00 j
x
X1 X2 X3 ...
Xi
+ Φ
1 1 1 1
c1 c2 c3 ci
Figure 4.1: The representation of the trial functionΦ=P
i
ciXi as a neuron network.