Although Chang Haonan's award has given the school a lot of publicity work to do, Tang Lintian naturally will not forget what he promised a few days ago.

It didn't take a few days to access the more mainstream academic databases on the extranet connected to the school.

Chang Haonan realized very clearly from about half a year ago, when he first seriously considered writing a new simulation modeling software, that the study of multiphysics, especially strongly coupled multiphysics problems, is essentially the solution of a system of nonlinear partial differential equations.

But this kind of thing, when it comes to the operation of the engineering field, is often not as simple as a sentence.

Especially considering that today's supercomputers are not very fast.

The vast majority of partial differential equations have no way to find an analytical solution, at least for a short time, and can only work in the direction of numerical solution.

Many mathematically aesthetically pleasing solutions may not be practical.

Traditionally, the method of dimensionality reduction based on variable discretization is mainly used for the dimensionality reduction of dynamic systems of nonlinear partial differential equations, such as the finite element method, the finite volume method and the finite difference method.

But it's not like there's no other way of thinking.

For example, Chang Haonan accidentally saw this paper when he was resting in the computer room one night.

Although it was an article in the field of applied mathematics, it was published in a seemingly unrelated journal in the field of chemical engineering.

Chemical Engineering Journal

A magazine that is considered famous after more than a decade, but this is only a new and inconspicuous magazine.

The main reason why he clicked in and downloaded it at a speed of a few kilobytes per second was that the abstract was so attractively written.

In the past 40 years, the system dimensionality reduction method based on variable separation has developed rapidly, and under certain conditions, it can avoid some essential problems caused by the spatial discretization method, and reduce a class of nonlinear partial differential equation dynamic systems to a lower dimension, which is convenient for rapid analysis and calculation, optimization and the realization of active controllers. It can be applied to the numerical analysis of common mechanical-thermal coupling problems in the field of chemical engineering......"

Although the specific problems involved are not related to the design of the aircraft, the force-thermal coupling mentioned in it is also the most basic and urgent problem that Chang Haonan is currently facing.

This summary simply speaks to his heart.

He believed that a few months ago, when Lu Yuying first saw his paper in Rongcheng, his inner sense of transparency was nothing more than that.

A few minutes of download time has never been as long as it is now.

Chang Haonan stared at the progress bar on the screen, and clicked on the document almost the moment the download was completed.

"It is well known that any continuous function can be approximated by the expansion of a Fourier series sequence, and based on the above principle, the space-time affinity variables in a nonlinear partial differential equation can be expanded into the form of an infinite-dimensional set of basis functions and the series sum of their corresponding time coefficients:

X(z，t)=(i=1，∞)∑φi(z)xi(t)

where xi(t) denotes the time coefficient corresponding to each basis function φi(z)......"

It's really basic.

The technique of separating spatiotemporal variables is not a new thing, and can be found in any mathematical and physical methods or similar textbooks, but it is generally believed that partial differential equations suitable for using the method of separating variables should have certain forms and characteristics, such as linearity, homogeneity, separability, coefficients depend on only one variable, etc., which greatly limits the application of such methods.

Therefore, Chang Haonan quickly skipped this part of the content and looked directly at the third section, which is often the first section of the text:

In order to elaborate and clearly describe the method of dimensionality reduction of the dynamic system of nonlinear partial differential equations, this subsection uses the parabolic nonlinear partial differential equation system as the object......

"It's coming!"

When he saw the content he was interested in, his spirits lifted, and even the little sleepiness he had just had disappeared in an instant.

The boundary conditions and initial conditions are:

where x(z,t) represents the space-time state variables and is a continuous function on the infinite Vielbert space defined on the spatial region [a,b]. denotes the spatial coordinates, z∈[a,b] denotes the spatial coordinates, is the subspace on the real domain defined by the process, and t∈[0,∞) denotes the temporal variable......

……

Finally, we can get the expression of the above nonlinear partial differential equation system in Hilbert space H([a,b]):

x(z，t)/t=Ax(z，t)+Bu(z，t)+(x，z，t)

x(z，0)=x0(z)

Two simulation examples are given below, namely the dimensionless Kuramoto-Sivashinsky equation in one-dimensional space, and the temperature and pressure fields of a non-isothermal tubular reactor......

"Hmm...... Something......"

Chang Haonan saw the back and nodded inwardly.

"In general."

He pulled out a piece of paper from the printer next to him and began to summarize it to himself,

"Firstly, the appropriate spatial orthogonal basis function is selected and the spatiotemporal separation technique is used to separate the spatiotemporal variables of the dynamic system of the nonlinear partial differential equation, that is, the spatiotemporal affinity variables of the system are expanded on the selected or obtained orthogonal space basis functions, and the expanded formula is substituted into the original system and combined with the nonlinear Galerkin method......"

An hour passed quickly in his writing, writing, drawing, and painting.

Although the object used to illustrate the theory in the article is only a very simple parabolic system, the two application examples given later are indeed quite good, and they are worthy of the awesomeness that the author bragged in the abstract.

This article is even worthy of being submitted to a higher-impact journal, most likely because the author and the editor-in-chief are from the same school and have received an invitation to submit a manuscript.

In fact, Chang Haonan summed it up to the end and found the parts that the author himself did not write.

The method described in the article can be applied not only to heat transfer and flow field calculations, but also to mass transfer problems and the chemical reaction process itself, with a slight modification.

In other words, all the characteristics involved in the chemical production process can be included.

Of course, if you don't write it, it's not necessarily that the author didn't find it, it's probably because you have something left for another article......

"But...... There are still problems. ”

Chang Haonan looked at the three pieces of scratch paper that had been written in front of him.

Although it can be applied in a wide range of fields, it does not mean that the method mentioned in this article is a master key, and it can be directly moved to the scene that Chang Haonan needs.

"By using the eigenfunction as the space basis function combined with the weight residual method to reduce the dimension of the dynamic system of nonlinear partial differential equations, the finite-dimensional ordinary differential equation dynamic system can be obtained to approximate the infinite dimensional dynamics of the original system, but in essence, it is still approximated by linear means, which is still not enough for the real strong nonlinear problem, but if other spatial basis functions are used in the dimension reduction process, such as Fourier sequence function and orthogonal basis function, it may have nothing to do with the characteristics of the dynamic system of nonlinear partial differential equations itself......"

Thinking of this, he glanced sideways at another room next to him with a supercomputer.

Theoretically, of course, there is nothing wrong with that, but if you really start calculating......

Since this computing center in charge of itself has only just been opened, there are not many projects that use it at present, but despite this, the load of this supercomputer has been pulled to a high level.

If it is left in ten years, it may not be feasible to calculate hard according to the ideas in the article, but with the current level of supercomputing in China, I am afraid that it will take several years to calculate the level of force-thermoelectric coupling of a phased array radar array......

I had the time to make a test version and come out.

It's definitely not going to work.

"What if we use the balanced truncation method or the optimization method......

The tip of the pen in Chang Haonan's hand began to slide back on the paper.

Soon, the fourth and fifth pages of scratch paper were also filled.

There is a creaking sound of equipment running in the machine room.

Outside the window, the moon climbs from the horizon to the sky, and then gradually sets, finally ushering in a round of rising sun.

"I see."

(End of chapter)