At present, two-dimensional electrical forward modeling is the most important problem. It corresponds to Poisson equation or Laplace equation under the first, second or third boundary conditions. In mathematical language, these are all elliptic partial differential equations, so here we discuss the variational problems related to two-dimensional elliptic equations. Firstly, the equivalence between the boundary value problem of elliptic equation and the corresponding variational problem is discussed.
An elliptic differential equation satisfied by an objective function (such as potential). ) in the exploration area d electrical exploration is
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On the boundary γ of D domain (γ is a piecewise smooth closed curve), one of the following conditions is satisfied:
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In the above formula, u(x, y) is the expected objective function, and α, β, γ and F are all known functions of X and Y, and α > 0, β≥0 and γ≥0 are required. Equation (9.3. 1) is also called Helmholtz equation.
Mathematically, it can be proved that if the function (x, y) is the solution of the equation Lu=f (i.e. (9.3.4) or (9.3.2) or (9.3.3) with boundary conditions, then the function makes the corresponding functional.
J[u]=(Lu，u)-2(f，u) (9.3.5)
Reach the minimum, where parentheses represent the inner product, defined as
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Or conversely, if the functional j [u] is minimal, then Lu=f is the solution of the equation Lu=f under the corresponding boundary conditions, that is, the boundary value problem of Helmholtz equation and the variational problem of quadratic functional j [u] are equivalent.
We don't do strict proof here, but only deduce its equivalence from the formula. First of all,
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because
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therefore
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Figure 9.3 Geometric relationship of regional boundaries
(dl is ds in the text)
Using plane Green's formula
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The middle two terms of the above integral can be written as
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As can be seen from Figure 9.3.
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therefore
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Substitute the third boundary condition, and then
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In the above formula, ds is the micro line segment on γ, and n is the outer normal direction of the line segment, so
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In fact, j [u] is a quadratic functional, and its first-order variation and second-order variation can be obtained. Let δu(x, y) be the increment of u(x, y), then
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Among them:
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The necessary condition (9.3.6) of functional extremum is
δJ=0
Because α > 0, β≥0, γ≥0 and δu? 0, so δ 2j > 0, so the necessary and sufficient conditions for the functional (9.3.6) to reach the minimum value are δJ=0 and δ 2j > 0.
Now rewrite the first-order variation Δ j because
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therefore
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And δu is arbitrary and α > 0, so it is deduced from δJ=0.
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This is the solution of the third boundary value problem of equation (9.3. 1).
When γ=0, there is no linear integral term in equation (9.3.6), so it is easy to see that δJ=0 is equivalent to solving the differential equation under the second boundary condition, namely:
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For the first kind of boundary value problems, the corresponding functional (9.3.6) also has no line integral term, so considering δ u | γ = 0, there is.
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The minimal function u(x, y) corresponding to Δ δj = 0 is the solution of the equation under the first boundary condition.
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In the above discussion, there is a case where the boundary conditions are automatically satisfied by the extreme value function when looking for a function to minimize the functional (9.3.6), and there is no need to list the definite solution conditions. This boundary condition is called natural boundary condition.
Figure 9.4 Interface of Two Media
The first boundary condition, like the boundary value problem of differential equations, must be listed as a definite solution condition, that is, the extreme solution must be found in the function class that meets this boundary condition, which is called imposed boundary condition.
The foregoing discussion applies to media with continuously changing physical properties. Now consider the case where the partitions are unified and have physical interfaces. At this time, the influence of the interface must be considered, and the physical parameters of the medium will appear in the formula. Now, taking the steady current field as an example, we discuss two universal boundary conditions that the current field must meet at the interface between two conductive media, namely (8.3.9) and (8.3. 10). At this time, the dielectric parameters α 1=σ 1 and α2=σ2 are the conductivities of the two media respectively, which is the normal direction of the interface, and it is specified that the medium 1 points to the medium 2. As shown in Figure 9.4. For the sake of simplicity, the first boundary value problem of two-dimensional Laplace equation is discussed, which is equivalent to the line source problem without field source space.
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At this point, β=0, f = 0 ,( 9. 3. 6) is simplified as follows.
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In the formula, the d domain consists of regions D 1 and D2, where α = α 1 on d1and α=α2 on D2. A variant of the above formula can be written as
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Since the normal direction of the specified interface points from medium 1 to medium 2, the last term of the above formula is negative. On the interface γ 12, the potential change in medium 1 near the boundary must be equal to the potential change in medium 2, that is, Δ u = Δ u. The last two integrals in the above formula cancel each other because of the continuous normal component of the current, so the variation of the corresponding functional can be obtained.
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Where γ = γ 1+γ 2, including only the outer boundary and excluding the dielectric interface.
Therefore, as long as the functional expression contains physical parameters, the functional variation has nothing to do with the interface of the medium, but only with the outer boundary. In the process of functional extremum, the boundary conditions on the interface will be automatically satisfied, which also belongs to natural boundary conditions.
Ritz-Galerkin method
Ritz method is a method to find the approximate minimum value of functional. Galerkin method comes from "virtual displacement principle" in mechanics and has nothing to do with variational problems, so it is not necessary to solve differential equation problems as functional variational problems. But when the differential equation problem is equivalent to the variational problem, it is the same as the Ritz method, so they are described side by side here.
Poisson equation under the first boundary condition is considered.
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The corresponding variational problem is to find the functional.
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The minimum value of.
Let the approximation of the minimum function be
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Generally, for Wi(x, y), trigonometric functions or polynomials satisfying boundary conditions can be selected, and αi is the coefficient to be determined. If (9.3. 14) is substituted into (9.3. 13), then
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In ...
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It can be seen that J(un) is a multivariate quadratic function of α 1, α2, ..., αn, and it must meet the necessary conditions for finding the functional minimum.
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From this, you can get
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or
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This is a system of linear algebraic equations. If we solve this system of equations, we can get the coefficients α 1, α2, …, αn, and substitute them into the formula (9.3. 14), which is our required solution.
In fact, substituting the expression of λk, s and μk into (9.3. 15) can be written as follows.
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or
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According to Green's first formula
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Let u=un, v=Ws, and consider the boundary condition, and Wi should also satisfy the boundary condition ws | γ = 0, so there is
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Substitute (9.3. 16) to get.
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This is another form of the system of equations (9.3. 15), the so-called Galerkin system of equations.
For the first boundary value problem of general elliptic equations (9.3. 1)
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The corresponding variational problem is to find the functional.
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The minimum value of.
Similarly, assuming that its minimum function is approximately (9.3. 14), it can be proved that the equation for determining αk at this time is
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In ...
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The Galerkin form of the system of equations (9.3.20) is
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Minimum energy principle
There is a basic principle in electromagnetism, that is, when the electromagnetic field reaches the equilibrium state, it is required to meet the condition of minimum energy, or the condition of minimum electromagnetic energy is equivalent to Maxwell equations, which is the basic law of expressing the state and distribution of electromagnetic field in different forms.
It is known that the power density flux of electromagnetic field is given by Poynting vector.
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The power per unit volume ψ is the negative divergence of density flux, that is
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Due to, and
Equation (9.3.22) can be written as follows
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The energy density can be obtained by integrating the above formula with time t.
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The three terms on the right side of the above formula represent the energy density of magnetic field, electric field and conduction current respectively.
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Where γ2 =-μω(ωω+Iσ), which is the expression of total energy expressed by magnetic field strength. The expression of total energy expressed by electric field intensity can also be obtained:
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When considering the existence of field sources, there are
Magnetic source energy
Power supply energy
Current source energy
Current source power
Where: is the source magnetization; Is the polarization intensity; Is the source current density.
The total electromagnetic energy DT consists of the energy of the field and the energy of the source. For example, the total energy of the electric field is the sum of the energy of the electric field and the energy of the polarization source.
DT=DF+DE
As mentioned earlier, the state of electromagnetic field is determined by the principle of minimum energy. Therefore, according to the necessary condition of extreme value, the variation δ DT of energy dt must be zero. For electric field, its variational equation is
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The magnetic field of the magnetic source is
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Given the field source distribution and appropriate boundary conditions, the field values of each part of the medium can be obtained by the above variational equation.
In fact, it can be proved that the field value satisfying variational equation must satisfy Helmholtz equation with field source term. For example, in equation (9.3.25), because of the variation and integration of different variables, they can be obtained in an interchangeable order.
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Using Gaussian formula, the second term of the above formula is
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Since the variational δE is zero everywhere on the boundary S plane of the region, the above formula is zero. So you can write
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Since Δ e is not zero and Δ dt is zero in this region, it is required that
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In the region of uniform permeability, it is simplified as
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or
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This is the Helmholtz equation of the field source. It can be seen that the variational equation derived from the principle of energy minimization and the differential equation derived from Maxwell equations are equivalent to determining the state of the field.