CN118377991B - High-order constitutive equation source term reconstruction method based on deep learning technology - Google Patents
High-order constitutive equation source term reconstruction method based on deep learning technology Download PDFInfo
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Abstract
The invention discloses a high-order constitutive equation source term reconstruction method based on a deep learning technology, which relates to the field of fluid dynamics and comprises the following steps: s1, reconstructing an original control equation to enable the reconstructed control equation to have an unknown term number X 1 and a coefficient X 2; s2, correcting free parameters of a high-order constitutive equation source term based on a deep learning mode; s3, embedding the free parameters obtained in the step S2 back into a control equation source term, and solving the control equation by using a corrected high-order constitutive equation. The invention provides a high-order constitutive equation source term reconstruction method based on a deep learning technology, which is characterized in that a nonlinear term in a high-order constitutive equation is fused into an NS equation source term by introducing the deep learning technology, so that the high-order constitutive equation which is originally solved by an iterative method is replaced by a model training mode, the solving difficulty of the iterative method is reduced, the calculation efficiency is improved, and the source term precision is improved.
Description
Technical Field
The present invention relates to the field of fluid dynamics. More particularly, the invention relates to a high-order constitutive equation source term reconstruction method based on a deep learning technology.
Background
In the field of current hydrodynamic research, higher-order constitutive equations play a critical role in describing complex fluid behavior, particularly under extreme conditions, such as high-velocity flow or complex flow phenomena. Although the higher-order constitutive equation can improve the accuracy of the traditional NS equation, it has low computational efficiency and limited accuracy. The high-order constitutive equation relates to a nonlinear term, and an iteration method, such as a fixed point iteration method, a Newton iteration method, a conjugate gradient method and the like, is usually adopted in the traditional solution, however, the problems of high time complexity and low calculation efficiency exist in the algorithms.
In view of the foregoing, the prior art has limitations in determining and correcting the source term of the higher-order constitutive equation, and a more accurate and reliable method is needed to improve the accuracy and reliability of model description, especially in the study of complex hydrodynamic behavior.
Disclosure of Invention
It is an object of the present invention to address at least the above problems and/or disadvantages and to provide at least the advantages described below.
To achieve these objects and other advantages and in accordance with the purpose of the invention, a high-order constitutive equation source term reconstruction method based on a deep learning technique is provided, including:
s1, reconstructing an original control equation to enable the reconstructed control equation to have an unknown term number X 1 and a coefficient X 2;
S2, correcting free parameters of a high-order constitutive equation source term based on a deep learning mode;
s3, embedding the free parameters obtained in the step S2 back into a control equation source term, and solving the control equation by using a corrected high-order constitutive equation.
Preferably, in S1, the method of reconstructing the original control equation is as follows:
s10, developing a general form of an NS equation according to the law of conservation of mass, momentum and energy into:
In the above formula, ρ is density, E is energy, p is pressure, n is viscous stress, Δ is additional volumetric stress, Q is heat flow term, U is velocity term, The method is a Laplacian, I is an identity matrix, and t is a time item;
s11, based on a nonlinear coupling constitutive relation model NCCR equation, the constitutive model list of the NS equation is adjusted to be:
in the above formula, S 1、S2、S3 represents a nonlinear part in the constitutive model, where the number of terms and coefficients of the unknown source term are contained, and:
in the above formula, X 1 and X 2 represent unknown terms and coefficients, λ is the thermal conductivity coefficient, T is the temperature, η is the viscosity coefficient, η b is the volumetric viscosity coefficient, Is a second order traceless symmetric tensor;
s12, substituting the equation obtained in the S11 into the S10 to obtain a reconstructed control equation:
Wherein, Representing the reconstructed source term;
Wherein S represents a source item group, and a specific source item S 1、S2、S3 is contained.
Preferably, in S2, the way to correct the free parameters of the source term of the higher-order constitutive equation based on the deep learning method is as follows: s20, collecting and cleaning experimental data to obtain a data set;
S21, based on the research of the data relevance in the data set, mutually verifying the data layer and the source item form;
s22, analyzing the influence of different input forms on the model performance so as to design a model structure according to the data characteristics in the data set;
S23, based on a theoretical model of computational fluid dynamics and experimental data errors L, constructing a high-order constitutive equation source term free parameter corrected by the neural network on a learning machine in a deep learning mode.
The invention at least comprises the following beneficial effects: the patent provides a high-order constitutive equation source term reconstruction method based on deep learning, which aims to redefine a solving method of a high-order constitutive equation, and the source term is corrected through the deep learning, so that the calculation efficiency and the calculation precision are improved. The method is not only helpful to overcome the defects of the traditional method, but also has wide significance in practical application.
The accuracy of the traditional NS equation can be improved by the high-order constitutive equation, and the high-order constitutive equation correction part is incorporated into the source term of the NS equation, so that the high-order constitutive equation which is needed to be solved by an iteration method originally is replaced by a deep learning method. Based on CFD and experimental data errors, a neural network is constructed to correct free parameters of a source term of a high-order constitutive equation, so that the precision of the source term is improved.
According to the method, an iteration method which is commonly adopted in traditional solving is replaced, in order to improve efficiency, a deep learning thought is introduced, nonlinear items in a high-order constitutive equation are integrated into source items, and training is carried out by adopting a deep learning model, so that the solving difficulty of the iteration method is reduced, and therefore the computing efficiency is improved.
Meanwhile, compared with the traditional method, the method based on the deep learning also reduces the dependence on professional knowledge, and enables the reconstruction process of the source item to be more automatic.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention.
Drawings
FIG. 1 is a schematic diagram of a source item correction procedure according to the present invention;
FIG. 2 is a flow chart of the network training method of the present invention;
FIG. 3 is a schematic flow chart of the technical scheme of the model verification of the invention;
fig. 4 is a schematic diagram of the internal structure of a normal shock wave according to an embodiment of the present invention.
Detailed Description
The present invention is described in further detail below with reference to the drawings to enable those skilled in the art to practice the invention by referring to the description.
1. Construction of a control equation. Based on the deep physical principle and the specific problem situation, error sources such as cut-off errors are considered, factors which possibly influence the accuracy of the source item are identified and solved in the correction process, and the mathematical model of the source item is considered how to adjust, so that the physical rationality of the corrected model is ensured.
NS (Navier-Stokes) equation general form:
wherein U is a velocity term; t is a time term; representing the gradient of each vector for the Laplace operator; f c is the convection term flux; f v is the viscous term flux;
the method is developed according to the law of conservation of mass, momentum and energy:
In the above formula, ρ is density, E is energy, p is pressure, n is viscous stress, Δ is additional volumetric stress, Q is heat flow term, U is velocity term, The method is a Laplacian, I is an identity matrix, and t is a time item;
The constitutive model of the NS equation is listed below:
Wherein λ is the thermal conductivity, T is the temperature, η is the viscosity coefficient, η b is the volumetric viscosity coefficient, mathematical notation Defined as a second order unscented symmetric tensor, whose expression is as follows:
in the above formula, I is an identity matrix, tr [ A ] is the trace of matrix A, and A t is the transpose of matrix A;
The constitutive equation is adjusted according to the nonlinear coupling constitutive relation model (NCCR equation) by considering error sources such as truncation error, and S 1、S2、S3 contains unknown source term numbers and coefficients.
Wherein S 1、S2、S3 represents a nonlinear portion in the constitutive model, which contains the number of terms and coefficients of the unknown source term, and:
X 1 and X 2 represent unknown terms and coefficients, λ is the thermal conductivity, T is the temperature, η is the viscosity coefficient, η b is the volumetric viscosity coefficient, Is a second order unscented symmetric tensor.
The substitution of the above back into the control equation is generalized to the following form,
Wherein, Representing the reconstructed source term;
where S represents a source term group, including a specific source term S 1、S2、S3, and a specific source term S 1、S2、S3 includes an unknown term number and a coefficient X 1、X2.
2. Application of artificial intelligence. And combining the source item form of physical derivation, determining the number and the coefficient of the source item by means of big data and artificial intelligence technology, and improving the description precision of the model.
The high-order constitutive equation reconstruction method can be divided into a learning machine and a proxy machine, as shown in fig. 1, wherein q n in fig. 1 represents an input feature set, the feature set further outputs the source term number and the coefficient X i through a network model, i=1 and 2, and the network model is shown in fig. 2 in detail.
(1) Data preparation and cleansing
Experimental data related to the system under study are collected, and the quality and consistency of the data are ensured by preprocessing operations such as classification, quality evaluation, calibration, completion and the like. Through the series of data cleaning steps, high-quality and reliable data are obtained, and a solid foundation is laid for subsequent research and analysis.
The main contents of data preparation and cleansing include: firstly, defining the target and the requirement of data collection, including the scale and the detail level of the required data; designing a data collection scheme based on the target, including determining a data source (e.g., experiment, existing database, etc.), a data category (feature requirement), etc.; according to the scheme, the design comprises the steps of selecting control variables and independent variables, designing an experiment flow, presetting the repetition times of experiments, random seeds and the like, so as to ensure the repeatability and the reliability of data; secondly, the collected raw data needs to be subjected to a cleaning process to remove inaccurate, missing or irrelevant data, which can be accomplished by manual inspection or an automated script; and finally, the cleaned data are arranged into a proper format and stored.
(2) Data correlation study
And developing characteristic research on the obtained high-quality data set, researching the relevance among different data through the extracted data characteristic, mutually verifying the source item form from the data layer, and ensuring the rationality of the source item form.
The content of the data correlation study includes: first, the assumptions validated or explored from the data are clarified, based on previous studies, theoretical predictions, or insight and intuition into the problem domain; analyzing the dataset using statistical methods and machine learning techniques to identify possible associations and patterns, including using correlation analysis, shap analysis, etc. techniques to discover potential links in the data; establishing a mathematical model or a theoretical model according to the research hypothesis to describe the relationship between variables in the hypothesis; finally, the built model is verified by utilizing the collected data, which involves comparing the predicted result of the model with the actual observed data, and checking whether the model can accurately reflect the data relevance or not, wherein the repeated verification may be needed in the process.
(3) Data characterization study
And selecting different characteristics or fusing, exploring the influence of different input forms on the model performance, designing a more reasonable model structure according to the data characteristics in the data set, and increasing the interpretability and generalization capability of the model. Specific means include correlation analysis, scatter diagram, feature fusion, and the like.
The content of the data characterization study included: first, the characteristics of the data are deeply understood, including the type, structure and distribution characteristics of the data, and potential problems existing in the data; selecting a proper data preprocessing method according to the understanding of the data characteristics; then selecting the model type most suitable for the data characteristics, further designing the structure of the model, determining the depth, width, connection mode and the like of the model according to the complexity of tasks and the scale of data, and simultaneously considering the introduction of regularization, dropout and other technologies to prevent overfitting; and continuously adjusting the model structure and parameters according to the performance of the model on the verification set until the satisfactory performance is achieved.
(4) Determining the number and coefficients of source items
Based on Computational Fluid Dynamics (CFD) and experimental data errors L, a theoretical model and actual data are fused, free parameters of a neural network modified high-order constitutive equation source term are constructed in a big data and artificial intelligence mode, the number and coefficients of the source term are determined, fine optimization is carried out on the source term, and a learning machine network training mode is shown in figure 2. And integrating the corrected source item to obtain a complete model.
Step1 (learning machine): the Mach number Ma, the Reynolds number Re, the Knudsen number kn, the error L and the like are taken as network inputs, and the proper network structure and super parameters are adjusted to output X 1 and X 2.
Step2 (agent): the network inputs give incoming flow conditions, the network outputs X 1 and X 2 are embedded back into the control equation source term, and CFD solution is performed with the modified higher-order constitutive equation.
3. Model verification and evaluation
As shown in fig. 3, the technical solution of model verification and evaluation mainly includes:
(1) And (5) model verification. Embedding the corrected high-order constitutive equation into the NS equation, selecting a proper numerical method, performing experimental simulation, comparing the matching degree between the simulation result and the actual data, and verifying and evaluating the effectiveness and accuracy of the correction method.
(2) Generalization evaluation. The size and quality of the training data set are considered, the application range of the model is explored, the size of a proper training data set is determined through methods such as experiments and cross verification, the optimal prediction capacity of the model on unknown data is ensured, and the model with high generalization capacity is obtained.
Examples: one-dimensional forward shock is a physical flow phenomenon with strong compression characteristics, as shown in fig. 4. When the airflow passes through the interior of the real shock wave at the supersonic speed, the flow variable is smoothly transited from the wave front to the wave back, the wave front and the wave back are in a thermal equilibrium state, and the macroscopic physical quantity Ma 2 of the wave back is determined by the relationship of Rangkine-Hugoniot through the wave front Mach number Ma 1. The shock wave is in an unbalanced state, and the shock wave thickness delta is only a few mean free paths.
1. Control equation
In one dimension, the control equation is as follows:
The source item S is as follows:
The nonlinear part is set as X 1 and X 2, and the nonlinear part is solved by using a neural network training output nonlinear part instead of the traditional iteration method.
2. Network training (learning machine)
As shown in fig. 2, a control equation with a certain physical form is used, and mach number, reynolds number, nusen number and the like are taken as network inputs, and appropriate network structures and super parameters are adjusted, and outputs X 1 and X 2 are output.
3. CFD solver (agency machine)
Given the incoming flow condition, the network inputs the incoming flow condition, and embeds the X 1 and X 2 output by the network back into the control equation source term to obtain a correction equation with the neural network replacing the unknown variable. Defining proper dimensionless parameters, selecting proper boundary conditions, and carrying out CFD solving.
In order to calculate and solve the internal flow of the shock wave in fig. 4, the method replaces the iteration method which is commonly adopted in the traditional solving, so that the high-order constitutive equation which is originally solved by the iteration method is replaced by the deep learning method, and the dependence on professional knowledge is reduced by the method based on the deep learning, so that the reconstruction process of the source term is more automatic.
The above is merely illustrative of a preferred embodiment, but is not limited thereto. In practicing the present invention, appropriate substitutions and/or modifications may be made according to the needs of the user.
Although embodiments of the invention have been disclosed above, they are not limited to the use listed in the specification and embodiments. It can be applied to various fields suitable for the present invention. Additional modifications will readily occur to those skilled in the art. Therefore, the invention is not to be limited to the specific details and illustrations shown and described herein, without departing from the general concepts defined in the claims and their equivalents.
Claims (2)
1. The high-order constitutive equation source term reconstruction method based on the deep learning technology is characterized by comprising the following steps of:
s1, reconstructing an original control equation, namely adjusting an NS equation according to a NCCR equation so that the reconstructed control equation has an unknown term number X 1 and a coefficient X 2;
S2, correcting free parameters of a high-order constitutive equation source term based on a deep learning mode;
S3, embedding the free parameters obtained in the step S2 back into a control equation source term, and solving the control equation by using a corrected high-order constitutive equation;
in S2, the way to correct the free parameters of the source term of the higher-order constitutive equation based on the deep learning way is:
S20, collecting and cleaning experimental data to obtain a data set;
S21, based on the research of the data relevance in the data set, mutually verifying the data layer and the source item form;
s22, analyzing the influence of different input forms on the model performance so as to design a model structure according to the data characteristics in the data set;
S23, based on a theoretical model of computational fluid dynamics and experimental data errors L, constructing neural network modified higher-order constitutive equation source term free parameters X 1 and X 2 on a learning machine in a deep learning mode.
2. The method for reconstructing a source term of a higher-order constitutive equation based on a deep learning technology as set forth in claim 1, wherein in S1, a method for reconstructing an original control equation is as follows:
s10, developing a general form of an NS equation according to the law of conservation of mass, momentum and energy into:
In the above formula, ρ is density, E is energy, p is pressure, n is viscous stress, Δ is additional volumetric stress, Q is heat flow term, U is velocity term, The method is a Laplacian, I is an identity matrix, and t is a time item;
s11, based on a nonlinear coupling constitutive relation model NCCR equation, the constitutive model list of the NS equation is adjusted to be:
in the above formula, S 1、S2、S3 represents a nonlinear part in the constitutive model, where the number of terms and coefficients of the unknown source term are contained, and:
in the above formula, X 1 and X 2 represent unknown terms and coefficients, λ is the thermal conductivity coefficient, T is the temperature, η is the viscosity coefficient, η b is the volumetric viscosity coefficient, Is a second order traceless symmetric tensor;
s12, substituting the equation obtained in the S11 into the S10 to obtain a reconstructed control equation:
Wherein, Representing the reconstructed source term;
Wherein S represents a source item group, and a specific source item S 1、S2、S3 is contained.
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