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This is the Helmholtz equation, and it is also one of the tools commonly used in mathematics to solve electromagnetic field scattering problems.
In layman's terms, if a problem involves the inverse problem of a partial differential equation (PDE).
Then this type of problem generally has the following form: given a PDE and some information about the equation solution u (based on practical application considerations, this information should be easier to obtain through measurement, such as boundary values or asymptotic behavior at infinity, etc.
This is then used to invert some unknown information in the PDE, such as coefficients, domain, and even the model itself.
As for the backscattering problem, it is generally assumed that the wave cannot penetrate the scatterer, that is, the scattered wave field only exists outside the scatterer.
But obviously, this kind of "limited" calculation method is not what Xu Chuan needs.
For electromagnetic railguns, various issues such as internal magnetic field reflection and derivation are much more complicated than this.
In the study, the soft light illuminated the manuscript paper. While thinking, Xu Chuan wrote on the paper and said to himself:
"...Give appropriate boundary conditions on the boundary ?D of the scatterer. If the scatterer is acoustically soft, u|?D = 0 can be considered; and when the scatterer is sound-hard
),We have
?u?ν|?D =0.”
"But on top of this, the so-called impedance boundary condition needs to be considered, that is, (?u/?v λu)·|?D = 0,λ∈ C, Imλ> 0...."
"Then the scattering field at infinite?ν distance has the following asymptotic expression: u^s(x)= e^ik|x|/|x^(n?1)/2{u∞(?x) O(
1/|x|)......"
Looking at the manuscript paper under his pen, Xu Chuan's eyes showed a hint of joy.
According to his experience, before solving a complex problem, finding the entrance to the complex problem is the most effective and fastest way.
And as long as he found this opening, at least he would be able to see how to take the next step.
Regarding the magnetic field data problem of the electromagnetic railgun, he has successfully found the thread.
...
For Xu Chuan, it was only a year ago that he devoted himself wholeheartedly to theoretical research in mathematics.
After the Weak Riemann Hypothesis was proved, more of his work was in the field of aerospace and physics.
However, for him, the familiar feeling of immersing himself in mathematical research work is not unfamiliar.
Especially in the fields he is interested in, every piece of additional knowledge acquired is like a dose of dopamine, bringing him satisfaction and happiness.
Especially when all his attention was focused on the black mathematical symbols on the white manuscript paper, it seemed as if the whole world had disappeared, leaving only the Arabic numerals and ancient Greek symbols in front of him.
The pen slides smoothly across the paper, leaving wonderful characters one after another, as if every stroke is a poem, and every word is a bright star, lighting up the whole world.
Late at night, a gentle lamp lights up in the quiet study, and the Purple Mountain outside the window seems to be sleeping. Occasionally, there are some rustling sounds, just like love words in a dream.
Staring at the manuscript paper on the desk, Xu Chuan's eyes were bright and he murmured softly:
"...The first unique result can be obtained with the help of the spectral theory of the Laplace operator: the far field generated by infinitely many plane waves can uniquely determine an acoustically soft scatterer."
"Using the singular source field method, we can solve the problem that the far field generated by infinite plane waves can uniquely determine an acoustic hard scatterer."
"Then if the incident wave is a time-harmonic electromagnetic wave, the corresponding PDE model is a time-harmonic Maxwell equation. That is..."
"..."
For Xu Chuan, he has to consider not only the analysis and calculation problems of mathematically solving the three-dimensional elliptical electromagnetic field and high-dimensional large-scale backscattering problems, but also the mathematical model part.
In the field of electromagnetic field application research, based on electromagnetic field theory and the integration of new achievements in numerical calculation methods and computer software technology, a new branch of disciplines called electromagnetic field numerical calculation has been derived.
Numerical integration method is one of the basic contents in the application of numerical calculation methods. It not only lays the foundation for numerical quadrature of various types of integral expressions.
And with the increasing development of numerical calculation methods, it has become an indispensable component in the construction of various numerical methods.
However, for the analysis and calculation problems of three-dimensional elliptical electromagnetic fields and high-dimensional large-scale backscattering problems, due to the lack of compactness conditions caused by critical nonlinear terms, it is difficult to solve the problems through trapezoidal quadrature, Simpson quadrature, and Gaussian quadrature.
Product method and elliptic integral method are used to solve this problem.
But for Xu Chuan, it is not too difficult to creatively complete a new method for the lack of compactness conditions caused by critical nonlinear terms.
The only thing that needs to be considered is to combine computer software technology with numerical calculations of anisotropic electromagnetic fields, and use the fast calculation speed and high accuracy of computers to improve the speed and accuracy of numerical calculations of anisotropic electromagnetic fields.
The numerical calculation of rectangular long direct current-carrying conductors is realized through the direct integration method of field source discretization, and these numerical calculation methods are compared.
According to the running results of the analytical solution and numerical solution calculation program for the magnetic field of long direct current-carrying wires, it can be seen that the numerical solution and the analytical solution are very close and can meet the general requirements of electromagnetic field engineering.
Although this may not completely solve the problem, at least it can solve part of the problem!
Thinking of this, Xu Chuan pushed aside the full draft paper and wrote a new line of calculations on a piece of white paper.
【?x E^s? ikH^s = 0,?x H^s ikE^s = 0...】
Looking at the mathematical formulas on the manuscript paper, a smile gradually appeared on Xu Chuan's lips.
It seems that he will be studying this issue in seclusion for a few days.
Stretching and yawning, Xu Chuan picked up his phone from the table, sent a text message to Zheng Hai, telling him that he needed to study some math problems, then stood up and walked towards the bathroom.
Although at the beginning, he did not expect to find the answer to this question in just a few days.
But unexpectedly, the analysis and calculation problems of three-dimensional elliptical electromagnetic fields and high-dimensional large-scale backscattering problems have been quite smooth.
In just two days, he had successfully found the solution to this problem, and even completed preliminary simulation calculations on various issues such as magnetic field reflection and derivation in stages.
The process went so smoothly that Xu Chuan himself couldn't believe it.
As for now.
He was going to take a shower and then sleep!
Wait until you get up tomorrow to start formally solving this problem.