Chapter 306 Dimensionality reduction problem of nonlinear partial differential equations

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

Within a few days, the external network connected to the school could already access several of the more mainstream academic databases.

About half a year ago, when Chang Haonan began to seriously consider writing a new simulation modeling software, he realized very clearly that the study of multiphysics, especially strongly coupled multiphysics problems, is essentially a study of nonlinear partial differentials. Solution of systems of equations.

But when this kind of thing is implemented in the engineering field, it is often not as simple as a sentence.

Especially considering that the computing speed of today's supercomputers is not optimistic.

There is no way to obtain analytical solutions for most partial differential equations. At least in a short period of time, we can only work on numerical solutions.

Many solutions that are very aesthetic in mathematics may not be practical.

Traditionally, the dimensionality reduction of dynamic systems of nonlinear partial differential equations mainly uses methods based on variable discretization, typically such as the finite element method, the finite volume method and the finite difference method, which can be called the three masters in this field.

But it’s not like there aren’t other ideas.

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

Although the article is 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 would become famous more than ten years later, but at this time was just a humble magazine.

The reason why he is attracted to click in and download it at a speed of several Kb per second is mainly because the summary is so attractive.

“The current commonly used finite difference method and finite element method can only obtain high-dimensional ordinary differential equation systems for dimensionality reduction of nonlinear partial differential equation dynamic systems. In the past forty years, system reduction based on variable separation has become more and more difficult. The dimensional method has been developed rapidly. Under certain conditions, it can avoid some essential problems caused by the spatial discretization method and reduce a type of nonlinear partial differential equation dynamic system to a lower dimension, which facilitates rapid analysis, calculation and optimization. And the implementation of active controllers can be applied to numerical analysis of common mechanical and thermal coupling problems in the field of chemical engineering..."

Although the specific issues involved have nothing to do with aircraft design, the force-thermal coupling mentioned in it is originally the most basic and most urgent issue that Chang Haonan currently faces.

This excerpt simply spoke to his heart.

He believed that a few months ago, when Lu Yuying saw his paper for the first time in Chengdu, the sense of transparency in her heart was nothing more than this.

Minutes to download have never been so long.

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

"It is well known that any continuous function can be approximately represented by the expansion of the Fourier series sequence. Based on the above principles, the space-time affinity variables in nonlinear partial differential equations can be expanded into an infinite-dimensional space basis function set and its corresponding The form of the series sum of time coefficients is:

X(z,t)=(i=1,∞)∑φi(z)xi(t)

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

Very basic indeed.

Space-time variable separation technology is not a new thing. It can be found in any mathematical physics method or similar textbook. However, it is generally believed that partial differential equations suitable for using the separation variable method should have certain forms and characteristics, such as linear and homogeneous. , separable, coefficients only depend on one variable, etc., which greatly limits the application of such methods.

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

In order to explain the dimensionality reduction method of dynamic systems of nonlinear partial differential equations in detail and clearly, this section uses the parabolic nonlinear partial differential equation system as the object for explanation...

"coming!"

When he saw the content he was interested in, he became energetic, and even the slight sleepiness he felt just now disappeared instantly.

The boundary conditions and initial conditions are:

Among them, x(z, t) represents the space-time state variable and is a continuous function on the infinite-dimensional Hilbert space defined in the space region [a, b]. represents the spatial coordinate, z∈[a, b] represents the spatial coordinate, which is the subspace on the real number domain defined by the process, t∈[0,∞) represents the time variable...

……

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

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 field of a non-isothermal tubular reactor...

"Well...there's something..."

Chang Haonan saw the back and nodded with understanding.

"In general."

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

“First, select an appropriate space orthogonal basis function and use space-time separation technology to separate the space-time variables of the dynamic system of nonlinear partial differential equations, that is, expand the space-time affinity variables of the system on the selected or obtained orthogonal space basis function , substitute the expansion into the original system and combine it with the nonlinear Galerkin method..."

An hour passed quickly while he was writing and drawing.

Although the object used to illustrate the theory in the article is only a very simple parabolic system, the two application examples cited later are indeed quite good and worthy of the author's boast in the abstract.

This article even deserves to be submitted to a higher-impact journal. The reason why it appears here is most likely because the author and the editor-in-chief are from the same school and received an invitation to submit a manuscript.

In fact, at the end of the summary, Chang Haonan also discovered parts that the author himself had not written.

The method in the article can not only be applied to heat transfer and flow field calculations, but with slight modifications, it can even be used to deal with mass transfer problems and the chemical reaction process itself.

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

Of course, the fact that it was not written down may not be because the author did not discover it. It is possible that he was saving it to prepare another article...

"But... there are still problems."

Chang Haonan looked at the three pieces of draft paper in front of him that were already filled with writing.

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 can be directly transferred to the scene Chang Haonan needs.

"Using eigenfunctions as spatial basis functions combined with the weighted residual method to reduce the dimensionality of the nonlinear partial differential equation dynamic system can obtain a finite-dimensional ordinary differential equation dynamic system to approximate the infinite-dimensional dynamics of the original system, but essentially linear methods are still used Approximation is still not enough for real strongly nonlinear problems. However, if other spatial basis functions are used in the dimensionality reduction process, such as Fourier sequence functions and orthogonal basis functions, it may be dynamically related to nonlinear partial differential equations. The characteristics of the system itself are irrelevant..."

Thinking of this, he turned his head and glanced at another room next to him where a supercomputer was placed.

Of course there is no problem in theory, but if you really start to calculate...

Since the computing center that I am responsible for has just been launched, there are not many projects currently using it. However, the load on this supercomputer has been reduced to a considerable level.

If it were left ten years later, hard calculation according to the ideas in the article may not be impossible, but with the current domestic supercomputing level, I am afraid it would take several years to calculate the force-thermal coupling level of the phased array radar array...

When I have this time, I will create a test version.

It's definitely not going to work.

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

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

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

There was the sound of creaking equipment running in the computer room.

The moon outside the window climbed from the horizon to the mid-air, then gradually set, and finally ushered in a rising sun.

"I see."