CN114036815B - True and false dual particle model modeling method for coupling physical field quick solving - Google Patents
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Abstract
The modeling method of the true and false dual particle model for the rapid solving of the coupled physical field comprises the steps of firstly dividing different patches according to each boundary coordinate and shape of CAD configuration, establishing spline curve node vectors and geometric control points, calculating interpolation basis functions, and obtaining an accurate analysis model of the physical problem to be solved after refinement; then, according to the calculation precision requirement and the physical field continuity requirement, giving the number and arrangement mode of the physical field particles, and performing particulation and dispersion on the physical field and the analysis domain; then constructing an interpolation mapping relation between the control particle information and the physical particle information; marking the boundary layer of the analysis domain based on the linear combination of the shape function in the numerical analysis model and the physical field information of the control particles corresponding to the boundary, applying boundary conditions, and applying physical properties and interaction of the physical particles to form an authenticity dual particle model for quickly solving the coupled physical field; the invention completes partial differential equation calculation with a small amount of degrees of freedom to obtain the numerical value of the high-continuity physical field distribution.
Description
Technical Field
The invention belongs to the technical field of numerical analysis of complex physical systems, and particularly relates to a true and false dual particle model modeling method for rapidly solving a coupled physical field.
Background
Since the 40 s of the 20 th century, efforts have been made to understand various physical phenomena using computers. When the traditional finite element analysis based on grid calculation encounters large deformation such as a metal forming stamping process and problems such as dynamic crack expansion, explosion impact and the like, the problems of grid rapid implementation repartition, grid distortion and the like are faced, the calculation efficiency and the calculation accuracy are seriously affected, a grid-free method can effectively cope with the problems, but in order to obtain higher calculation accuracy, the grid-free method usually needs hundreds of thousands of degrees of freedom to meet the calculation requirement, and the calculation reliability and the calculation efficiency cannot support multi-physical field real-time simulation calculation and structural design.
In the traditional numerical analysis method, units or particles participating in the calculation of the partial differential equation are consistent with units or particles carrying discrete physical field information, so that calculation and characterization are mutually coupled, the discrete degree is required to be increased, a huge amount of grids are divided or the huge amount of particles are filled in order to obtain a physical field calculation result with enough continuity, however, the convergence of the partial differential equation does not need such a huge amount of freedom, so that the calculation cost is increased excessively, and the calculation efficiency is severely limited. Therefore, if calculation can be stripped and characterized, the algorithm trap of mutual coupling constraint can be broken, the calculation cost can be reduced while the calculation accuracy is ensured, and the calculation efficiency is improved. Therefore, in order to support the calculation and structural design work of complex physical systems, a numerical analysis method capable of completing partial differential equation calculation with a small amount of degrees of freedom and obtaining high-continuity physical field distribution is urgently needed.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention aims to provide an authenticity dual particle model modeling method for rapidly solving a coupling physical field, which can complete partial differential equation solving with a small amount of degrees of freedom and obtain a numerical value of high-continuity physical field distribution.
In order to achieve the purpose, the invention adopts the following technical scheme:
a modeling method of an true and false dual particle model for rapidly solving a coupled physical field comprises the following steps:
1) Determining an analysis model of the physical problem to be solved:
1.1 The method comprises the steps of) importing the real geometric configuration of a physical problem model to be solved, dividing the shape of each boundary into different patches according to the CAD configuration, defining geometric control points including different patches and boundary contour lines to obtain a CAE model, ensuring the consistency of the CAD configuration and the CAE model, and establishing a complete analysis node vector and spline curve interpolation basis function under a parameter coordinate system according to modeling precision requirements, wherein the node vector is a non-decreasing sequence xi = { xi between 0 and 1 1 ，ξ 2 ，…，ξ m+p+1 Adopting a B-spline basis function as an interpolation function, wherein a specific recurrence formula is as follows:
wherein N is a B spline curve basis function, p is the order of the basis function, and ζ is a node under a parameter coordinate system;
b spline curve interpolation division is carried out on boundary contour lines and internal key configurations on different patches in CAD configuration to obtain node vectors and a shape control point distribution area, each B spline curve is composed of at least 4 nodes, the order of the fitted B spline curve is not less than 2, the internal space patches of the area are composed of two groups of similar B spline basis functions and weight coefficients, a two-dimensional spline surface is obtained through bilinear interpolation to form an original numerical analysis model for calculation and analysis, and the bilinear rational B spline basis functions are expressed as follows:
wherein R is a bilinear rational B spline basis function, N is a B spline curve basis function, p is the order of the basis function, and ω is a projection weight;
1.2 Building an accurate analysis model of the physical problem to be solved:
the original numerical analysis model established in the step 1.1) does not consider the order requirement of a physical partial differential control equation, and adopts a method of inserting more control nodes and improving NURBS basis function orders at different nodes to realize the improvement of curve and curve fitting orders and refine a boundary structure so as to obtain an accurate analysis model of a physical problem to be solved;
2) Establishing a two-layer particulated gridless analysis model of a problem to be solved:
2.1 Setting virtual particles with fixed space positions at each geometrical control node on the accurate analysis model obtained in the step 1) to form a control layer particle network, wherein the physical information of the control layer particles is determined by an initial moment physical field, so that the control layer particles have the characteristic of pseudo particles; constructing a physical layer particle model actually participating in particle method calculation on the basis of control layer particles, carrying out particulation and dispersion on a current physical field and an analysis area, giving the number and arrangement mode of the physical field particles according to the calculation precision requirement and the physical field continuity requirement, and keeping consistent interpolation basis functions used by the physical field dispersion and shape functions adopted by the analysis area dispersion based on the principle of equal parameter transformation, wherein the function relation between the discrete physical field and the analysis area is as follows:
wherein M is i C is a shape function i Is a discrete particle coordinate vector, u, v are coordinates in a parameter coordinate system, D 0 For the parameter coordinate system, D is the physical analysis domain, F is the coordinate transformation based on NURBS basis function, p is the unknown physical field distribution to be solved, D i Is a discrete physical particle information vector;
the constructed physical layer particles do not store physical information, and are only obtained by the upper layer control particle network through interpolation relation transmission when needed;
2.2 After the virtual control layer particles in the step 2.1) are endowed with the corresponding physical information, the mapping relation between the control layer particle information and the physical layer particle information is constructed; according to the calculation precision requirement and the calculation capacity constraint, setting a mapping relation order, wherein a specific two-dimensional mapping form is as follows:
wherein Q is kj To control the interpolation mapping function of particles to physical particles g kj For controlling the physical information vector carried by the particles, u, v are the coordinates in the parameter coordinate system, d i Is a discrete physical particle information vector;
if the order of a certain direction of the interpolation mapping function is q, the number m of control particles corresponding to a certain physical particle in a certain direction is specifically:
m＝q+1 (7)
the number of the control layer particles is far smaller than that of the physical layer particles, so that huge physical particle information is obtained after a small amount of control particle information is mapped through interpolation relation related to space positions, and further physical field distribution with high-order continuity is obtained;
3) Analysis region boundary condition application:
obtaining a two-layer particulated gridless analysis model according to the step 2), searching physical particles at the boundary of an analysis area, extracting corresponding control particles, marking the boundary layer of the analysis area based on the linear combination of a shape function in the two-layer particulated gridless analysis model and physical field information of the control particles corresponding to the boundary, dividing a structured grid in a non-physical field calculation area in the analysis area, and applying wall collision boundary conditions or thermodynamic conditions based on the idea of a boundary force method;
4) Physical particle physical properties and interactions apply:
screening physical particles according to the real physical property condition of an analysis area, applying corresponding physical property parameters, setting particle interference radius, combining with a phase shift field idea, and calculating and applying interaction among the physical particles;
5) Constructing a calculation model of physical field distribution to be solved:
according to the mapping interpolation function obtained in the step 2), the higher-order partial differential term in the unsteady partial differential control equation is discretized and calculated by combining with the jacobian matrix, and the specific form is as follows:
wherein Q is kj To control the interpolation mapping function of particles to physical particles g kj For controlling the physical information vector carried by the particles, u, v are the coordinates in the parameter coordinate system, d i The DF is a space conversion jacobian matrix;
combining a given time step length, gradually solving the development condition of the unknown physical field control particle information, and calculating the physical particle information according to the interpolation mapping relation in the step 2), so that the huge amount of physical particle information is obtained by solving a small amount of control particle physical information and mapping, and the smooth physical field distribution and development condition are obtained in each time sub-step.
The invention has the following beneficial technical results:
the invention provides a modeling method of an true and false dual particle model for rapidly solving a coupling physical field for the first time; the control particles used for calculation do not bear particle physical information, so that partial differential control equation calculation and decoupling of physical field information characterization are realized, the obtained physical field distribution result is ensured to have high continuity and accuracy, the calculation cost is reduced, and the calculation efficiency is improved; the invention uses the geometric configuration description method based on spline curve to accurately unify the actual geometric analysis area and the numerical analysis calculation model, thereby improving the calculation precision, reducing the calculation error and ensuring more accurate solving result.
The invention breaks through the algorithm trap of mutual coupling of calculation and characterization in the traditional numerical analysis method, provides a true-false dual particle network model for the first time, provides a good theoretical basis and research direction for the development of a high-efficiency numerical analysis tool, provides a corresponding calculation solving mode of the model, and completes the physical quantity calculation task of the structural design in the subsequent complex physical system.
Drawings
FIG. 1 is a schematic view of a portion of a model of a fluid flow meter according to an embodiment of the invention.
Fig. 2 is a flow chart of the present invention.
FIG. 3 is a graph showing a control layer pseudo-particle distribution diagram according to an embodiment of the present invention.
FIG. 4 is a graph of a physical layer true particle distribution based on interpolation mapping according to an embodiment of the present invention.
FIG. 5 is a schematic diagram of an exemplary embodiment of a dual particle model for flow meter authenticity.
Detailed Description
The method can be used for solving various complex physical fields or multi-physical field coupling problems, the working state simulation of a liquid flowmeter of a certain model is taken as an example to complete transient numerical simulation of the complex fluid-solid coupling problem, the total length of a pipeline where the flowmeter is positioned is 3000mm as shown in fig. 1, the inner diameter of the pipeline is 300mm, the flowmeter is made of stainless steel, the flowmeter is composed of a front diversion frame 1, a turbine 2 and a rear diversion frame 4 as shown in fig. 1, 20 blades 3 are connected to the turbine 2, the flowing medium is water, the temperature is 25 ℃, the flowing medium flows in at a speed of 1m/s from a left section and flows out freely from a right outlet, the flowmeter is stationary at the initial moment, and the change process of the rotating speed of the flowmeter is analyzed to obtain the stable rotating speed of the flowmeter under the current flow speed.
Referring to fig. 2, a method for modeling an authenticity dual particle model for fast solving a coupled physical field includes the following steps:
1) Determining an analysis model of the physical problem to be solved:
1.1 A pipeline model is established and is imported into a flowmeter entity model, the surface patches of the outer surface of the flowmeter are divided according to the coordinates and the shapes of all boundaries of a CAD (computer aided design) configuration, geometric control points in different surface patches and the boundaries are defined, a CAE model is obtained, the CAD configuration and the CAE model are guaranteed to be consistent, a complete analysis node vector and spline curve interpolation basis function under a parameter coordinate system is established according to modeling precision requirements, and the node vector is a non-decreasing sequence xi= { xi between 0 and 1 1 ，ξ 2 ，…，ξ m+p+1 Using B-spline basis functions as interpolationThe function, the specific recurrence formula, is as follows:
wherein N is a B spline curve basis function, p is the order of the basis function, and ζ is a node under a parameter coordinate system;
b spline curve interpolation division is carried out on the surface piece of the outer surface of the flowmeter and the inner side wall surface of the pipeline which are divided in the CAD configuration, so that node vectors and shape control point distribution areas are obtained, each B spline curve is composed of at least 4 nodes, the order of the fitted B spline curve is not less than 2, a space surface piece in the area can be obtained by two groups of similar B spline basis functions and weight coefficients through bilinear interpolation, a two-dimensional spline surface is formed, an original numerical analysis model which can be used for calculation and analysis is formed, and the bilinear rational B spline basis functions are expressed as follows:
wherein R is a bilinear rational B spline basis function, N is a B spline curve basis function, p is the order of the basis function, and ω is a projection weight;
1.2 Building an accurate analysis model of the physical problem to be solved:
the original numerical analysis model established in the step 1.1) does not consider the order requirement of a physical partial differential control equation, and the improvement of curves and curve fitting orders can be realized by adopting a method of inserting more control nodes and improving NURBS basis function orders at different nodes, so that the curve surfaces of turbine blades and corresponding boundary contour lines are refined, and an accurate analysis model for solving the physical problem is obtained;
2) Establishing a two-layer particulated gridless analysis model of a problem to be solved:
2.1 Setting virtual particles with fixed space positions at each geometrical control node on the accurate analysis model obtained in the step 1) to form a control layer particle network, wherein the distribution situation of the control layer particles in the embodiment is shown in figure 3; the control layer particles do not actually exist or directly participate in physical field calculation, and physical information on the control layer particles is determined by the physical field at the initial moment, so that the control layer particles have the characteristic of pseudo particles; constructing a physical layer particle model actually participating in particle method calculation on the basis of control layer particles, carrying out particulation and dispersion on a current physical field and an analysis area, giving the number and arrangement mode of the physical field particles according to the calculation precision requirement and the physical field continuity requirement, and keeping consistent an interpolation basis function used by the physical field dispersion and a shape function adopted by the analysis area dispersion based on a constant parameter transformation idea, wherein the physical field and the analysis area obtained by the dispersion are shown as follows:
wherein M is i C is a shape function i Is a discrete particle coordinate vector, u, v are coordinates in a parameter coordinate system, D 0 For the parameter coordinate system, D is the physical analysis domain, F is the coordinate transformation based on NURBS basis function, p is the unknown physical field distribution to be solved, D i Is a discrete physical particle information vector;
the distribution of the constructed physical layer particles is shown in fig. 4, wherein the particles participate in the particle method calculation, but the physical information is not stored on the particles, and the particles are only transmitted by an upper layer control particle network through a certain interpolation relation when needed;
2.2 After the virtual control layer particles in the step 2.1) are endowed with the corresponding physical information, the mapping relation between the control layer particle information and the physical layer particle information is constructed; according to the calculation precision requirement and the calculation capacity constraint, setting a mapping relation order, wherein a specific two-dimensional mapping form is as follows:
wherein Q is kj To control the interpolation mapping function of particles to physical particles g kj For controlling the physical information vector carried by the particles, u, v are the coordinates in the parameter coordinate system, d i Is a discrete physical particle information vector;
if the order of a certain direction of the interpolation mapping function is q, the number m of control particles corresponding to a certain physical particle in a certain direction is specifically:
m＝q+1 (7)
the number of the control layer particles is far smaller than that of the physical layer particles, so that huge amount of physical particle information is obtained after a small amount of control particle information is mapped through interpolation relation related to space positions, and further physical field distribution with high-order continuity is obtained, a certain smoothness of a discrete physical field is ensured, and a finally constructed true and false dual particle model is shown in fig. 5;
3) Analysis region boundary condition application:
obtaining a two-layer particulated gridless analysis model according to the step 2), searching physical particles at the boundary of an analysis area, extracting corresponding control particles, marking the boundary layer of the analysis area based on the linear combination of a shape function in the two-layer particulated gridless analysis model and physical field information of the control particles corresponding to the boundary, dividing a structured grid in a non-physical field calculation area in the analysis area, and applying wall collision boundary conditions and wall friction conditions based on a boundary force method idea;
4) Physical particle physical properties and interactions apply:
screening physical particles according to the real physical property condition of an analysis area, applying corresponding physical property parameters, setting particle interference radius, combining with a phase shift field idea, and calculating and applying interaction among the physical particles;
5) Constructing a calculation model of physical field distribution to be solved:
according to the mapping interpolation function obtained in the step 2), the second order partial differential term in the incompressible N-S unsteady partial differential control equation is discretized and calculated by combining with a jacobian matrix, and the specific form is as follows:
wherein Q is kj To control the interpolation mapping function of particles to physical particles g kj For controlling the physical information vector carried by the particles, u, v are the coordinates in the parameter coordinate system, d i The DF is a space conversion jacobian matrix;
combining a given time step length, gradually solving the development condition of unknown physical field control particle information, calculating physical particle information according to the interpolation mapping relation in the step 2), obtaining huge amount of physical particle information by solving a small amount of control particle physical information and mapping, and obtaining smooth physical field distribution and development condition in each time sub-step, wherein the stable rotating speed of the flowmeter is 2.73r/s under the flow speed background of 1 m/s.
Claims (1)
1. The modeling method of the true and false dual particle model for rapidly solving the coupled physical field is characterized by comprising the following steps of:
1) Determining an analysis model of the physical problem to be solved:
1.1 The method comprises the steps of) importing the real geometric configuration of a physical problem model to be solved, dividing the shape of each boundary into different patches according to the CAD configuration, defining geometric control points including different patches and boundary contour lines to obtain a CAE model, ensuring the consistency of the CAD configuration and the CAE model, and establishing a complete analysis node vector and spline curve interpolation basis function under a parameter coordinate system according to modeling precision requirements, wherein the node vector is a non-decreasing sequence xi = { xi between 0 and 1 1 ，ξ 2 ，···，ξ m+p+1 Adopting a B-spline basis function as an interpolation function, wherein a specific recurrence formula is as follows:
wherein N is a B spline curve basis function, p is the order of the basis function, and ζ is a node under a parameter coordinate system;
b spline curve interpolation division is carried out on boundary contour lines and internal key configurations on different patches in CAD configuration to obtain node vectors and a shape control point distribution area, each B spline curve is composed of at least 4 nodes, the order of the fitted B spline curve is not less than 2, the internal space patches of the area are composed of two groups of similar B spline basis functions and weight coefficients, a two-dimensional spline surface is obtained through bilinear interpolation to form an original numerical analysis model for calculation and analysis, and the bilinear rational B spline basis functions are expressed as follows:
wherein R is a bilinear rational B spline basis function, N is a B spline curve basis function, p is the order of the basis function, and ω is a projection weight;
1.2 Building an accurate analysis model of the physical problem to be solved:
the original numerical analysis model established in the step 1.1) does not consider the order requirement of a physical partial differential control equation, and adopts a method of inserting more control nodes and improving NURBS basis function orders at different nodes to realize the improvement of curve and curve fitting orders and refine a boundary structure so as to obtain an accurate analysis model of a physical problem to be solved;
2) Establishing a two-layer particulated gridless analysis model of a problem to be solved:
2.1 Setting virtual particles with fixed space positions at each geometrical control node on the accurate analysis model obtained in the step 1) to form a control layer particle network, wherein the physical information of the control layer particles is determined by an initial moment physical field, so that the control layer particles have the characteristic of pseudo particles; constructing a physical layer particle model actually participating in particle method calculation on the basis of control layer particles, carrying out particulation and dispersion on a current physical field and an analysis area, giving the number and arrangement mode of the physical field particles according to the calculation precision requirement and the physical field continuity requirement, and keeping consistent interpolation basis functions used by the physical field dispersion and shape functions adopted by the analysis area dispersion based on the principle of equal parameter transformation, wherein the function relation between the discrete physical field and the analysis area is as follows:
wherein M is i C is a shape function i Is a discrete particle coordinate vector, u, v are coordinates in a parameter coordinate system, D 0 For the parameter coordinate system, D is the physical analysis domain, F is the coordinate transformation based on NURBS basis function, p is the unknown physical field distribution to be solved, D i Is a discrete physical particle information vector;
the constructed physical layer particles do not store physical information, and are only obtained by the upper layer control particle network through interpolation relation transmission when needed;
2.2 After the virtual control layer particles in the step 2.1) are endowed with the corresponding physical information, the mapping relation between the control layer particle information and the physical layer particle information is constructed; according to the calculation precision requirement and the calculation capacity constraint, setting a mapping relation order, wherein a specific two-dimensional mapping form is as follows:
wherein Q is kj To control the interpolation mapping function of particles to physical particles g kj For controlling the physical information vector carried by the particles, u, v are the coordinates in the parameter coordinate system, d i Is a discrete physical particle information vector;
if the order of a certain direction of the interpolation mapping function is q, the number m of control particles corresponding to a certain physical particle in a certain direction is specifically:
m＝q+1 (7)
the number of the control layer particles is far smaller than that of the physical layer particles, so that huge physical particle information is obtained after a small amount of control particle information is mapped through interpolation relation related to space positions, and further physical field distribution with high-order continuity is obtained;
3) Analysis region boundary condition application:
obtaining a two-layer particulated gridless analysis model according to the step 2), searching physical particles at the boundary of an analysis area, extracting corresponding control particles, marking the boundary layer of the analysis area based on the linear combination of a shape function in the two-layer particulated gridless analysis model and physical field information of the control particles corresponding to the boundary, dividing a structured grid in a non-physical field calculation area in the analysis area, and applying wall collision boundary conditions or thermodynamic conditions based on the idea of a boundary force method;
4) Physical particle physical properties and interactions apply:
screening physical particles according to the real physical property condition of an analysis area, applying corresponding physical property parameters, setting particle interference radius, combining with a phase shift field idea, and calculating and applying interaction among the physical particles;
5) Constructing a calculation model of physical field distribution to be solved:
according to the mapping interpolation function obtained in the step 2), the higher-order partial differential term in the unsteady partial differential control equation is discretized and calculated by combining with the jacobian matrix, and the specific form is as follows:
wherein Q is kj To control the interpolation mapping function of particles to physical particles g kj For controlling the physical information vector carried by the particles, u, v are the coordinates in the parameter coordinate system, d i The DF is a space conversion jacobian matrix;
combining a given time step length, gradually solving the development condition of the unknown physical field control particle information, and calculating the physical particle information according to the interpolation mapping relation in the step 2), so that the huge amount of physical particle information is obtained by solving a small amount of control particle physical information and mapping, and the smooth physical field distribution and development condition are obtained in each time sub-step.
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