CN114036815A - True and false double-particle model modeling method for coupled physical field rapid solving - Google Patents
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Abstract
Firstly, dividing different patches according to boundary coordinates and shapes of a CAD (computer-aided design) configuration, establishing spline curve node vectors and geometric control points, calculating interpolation basis functions, and refining to obtain an accurate analysis model of a physical problem to be solved; according to the calculation precision requirement and the physical field continuity requirement, the number and the arrangement mode of the particles in the physical field are given, and the physical field and the analysis domain are subjected to particle dispersion; then, an interpolation mapping relation between the control particle information and the physical particle information is established; marking a boundary layer of an analysis domain based on a shape function in the numerical analysis model and a linear combination of physical field information of control particles corresponding to the boundary, applying boundary conditions, and applying physical properties and interaction of the physical particles to form a coupled physical field fast solving-oriented true and false double particle model; the method completes partial differential equation solution with a small amount of freedom degrees, and obtains the numerical value of 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 method for modeling an authenticity double-particle model for rapid solution of 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. In the traditional finite element analysis based on grid calculation, when large deformation such as a metal forming stamping process, dynamic crack propagation, explosion impact and other problems are encountered, the problems of quick grid repartitioning, grid distortion deformation and the like are encountered, the calculation efficiency and the solution precision are seriously influenced, and a grid-free method can effectively solve the problems.
In the traditional numerical analysis method, the units or particles participating in the partial differential equation calculation are consistent with the units or particles bearing the discrete physical field information, so that the calculation and the characterization are mutually coupled, in order to obtain a physical field calculation result with high enough continuity, the discrete degree needs to be increased, a huge amount of grids need to be divided or huge amount of particles need to be filled, however, the convergence of the partial differential equation does not need the huge amount of freedom, so that the calculation cost is increased unnecessarily, and the calculation efficiency is limited seriously. Therefore, if the calculation can be stripped for representation, the algorithm trap of mutual coupling restriction can be broken, the calculation cost can be reduced while the calculation precision is ensured, and the calculation efficiency is improved. Therefore, in order to support the calculation and structural design work of a complex physical system, a numerical analysis method which can complete partial differential equation calculation with a small number of degrees of freedom and can obtain 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 a method for modeling a true and false double particle model for quickly solving a coupled physical field, which can complete partial differential equation solution with a small amount of freedom and obtain a value of high-continuity physical field distribution.
In order to achieve the above purposes, the technical scheme adopted by the invention is as follows:
a coupled physical field fast solving-oriented true and false dual particle model modeling method comprises the following steps:
1) determining an analysis model of a physical problem to be solved:
1.1) importing a real geometric configuration of a physical problem model to be solved, dividing the shape of each boundary into different patches according to a CAD configuration, defining geometric control points including different patches and boundary contour lines to obtain a CAE model, ensuring the CAD configuration to be consistent with the CAE model, establishing a complete analysis node vector and a spline curve interpolation basis function under a parameter coordinate system according to the requirement of modeling precision, wherein the node vector is a non-decreasing sequence xi between 0 and 1 and xi { xi ═ xi-1，ξ2，…，ξm+p+1And B-spline basis functions are adopted as interpolation functions, and a specific recursion formula is as follows:
wherein N is a B spline curve base function, p is a base function order, and xi is a node in a parameter coordinate system;
b spline curve interpolation division is carried out on boundary contour lines and internal key configurations on different surface patches in the 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, a surface patch in the internal space of the area is 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 expression form of the bilinear rational B spline basis functions is as follows:
wherein R is a bilinear rational B spline basis function, N is a B spline curve basis function, p is a basis function order, and omega is a projection weight factor;
1.2) constructing 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 method of inserting more control nodes and improving NURBS (non-uniform rational B-spline) basic function orders at different nodes is adopted to realize the improvement of curve and surface fitting orders, refine boundary structures and obtain an accurate analysis model of a physical problem to be solved;
2) establishing a two-layer particlized non-grid analysis model of the problem to be solved:
2.1) setting virtual particles with fixed spatial positions at each geometric control node on the accurate analysis model obtained in the step 1) to form a control layer particle network, wherein physical information of the control layer particles is determined by an initial moment physical field, so that the control layer particles have a 'pseudo particle' characteristic; the method comprises the following steps of constructing a physical layer particle model which truly participates in particle method calculation on the basis of control layer particles, performing particle discretization on a current physical field and an analysis region, giving the number and arrangement mode of physical field particles according to calculation precision requirements and physical field continuity requirements, keeping consistent interpolation basis functions used in physical field discretization and shape functions adopted in analysis region discretization on the basis of an isoparametric transformation idea, and obtaining a functional relation between the physical field and the analysis region through discretization as follows:
wherein M isiAs a function of shape, ciIs a discrete particle coordinate vector, u, v are coordinates in a parametric coordinate system, D0For parametric coordinate systems, D is the physical analysis domain and F is the coordinate based on NURBS basis functionStandard transformation, p is unknown physical field distribution to be solved, diIs 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 the transmission of an interpolation relation when needed;
2.2) after corresponding physical information is given to the control layer particle virtualized in the step 2.1), constructing a mapping relation between the control layer particle information and the physical layer particle information; setting the order of the mapping relation according to the calculation precision requirement and the restriction of the calculation capacity, wherein the specific two-dimensional mapping form is as follows:
wherein QkjTo control the interpolation mapping function of particles to physical particles, gkjFor controlling the physical information vector carried by the particle, u, v are coordinates in a parametric coordinate system, diIs a discrete physical particle information vector;
if the order of a certain direction of the interpolation mapping function is q, the number m of the control particles corresponding to a certain physical particle in the 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 a large amount of physical particle information is obtained by mapping a small amount of control particle information through an interpolation relation related to a spatial position, and physical field distribution with high-order continuity is further obtained;
3) analysis zone boundary conditions apply:
obtaining a two-layer particle non-grid 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 particle non-grid analysis model and physical field information of the control particles corresponding to the boundary, dividing a structural grid for a non-physical field calculation area in the analysis area, and applying a wall surface collision boundary condition or a thermodynamic condition based on a boundary force method idea;
4) physical particle physical properties and interactions apply:
according to the real physical property condition of an analysis area, screening physical particles, applying corresponding physical property parameters, setting the interference radius of the particles, combining a phase shift field thought, and calculating and applying the interaction among the physical particles;
5) constructing a calculation model of the physical field distribution to be solved:
according to the mapping interpolation function obtained in the step 2), a Jacobian matrix is combined to disperse and calculate a high-order partial differential term in the unsteady partial differential control equation, and the specific form is as follows:
wherein QkjTo control the interpolation mapping function of particles to physical particles, gkjFor controlling the physical information vector carried by the particle, u, v are coordinates in a parametric coordinate system, diThe DF is a space transformation Jacobian matrix for a discrete physical particle information vector;
and (3) gradually solving the development condition of the unknown physical field control particle information by combining a given time step length, then calculating the physical particle information according to the interpolation mapping relation in the step 2), realizing that a large amount of physical particle information is obtained by solving a small amount of control particle physical information and mapping, and obtaining the smooth physical field distribution and development condition in each time step.
The invention has the following beneficial technical results:
the invention provides a method for modeling an authenticity double particle model for quickly solving a coupled physical field for the first time; because the control particles used for calculation do not bear the physical information of the particles, the decoupling of partial differential control equation calculation and physical field information representation is realized, the calculation cost is reduced and the calculation efficiency is improved while the obtained physical field distribution result is ensured to have high continuity and accuracy; because the invention uses the spline curve-based geometric configuration description method, the actual geometric analysis area and the numerical analysis calculation model are accurately unified, the calculation precision is improved, and the calculation error is reduced, so the solution result is more accurate.
The invention breaks through an algorithm trap of mutual coupling of calculation and representation in the traditional numerical analysis method, provides a double-particle network model based on authenticity for the first time, provides a good theoretical basis and a research direction for the development of a high-efficiency numerical analysis tool, provides a calculation solving mode corresponding to the model, and completes the calculation task of the physical quantity of the structural design in the subsequent complex physical system.
Drawings
FIG. 1 is a partial schematic view of a fluid flow meter model according to an embodiment of the invention.
FIG. 2 is a flow chart of the present invention.
FIG. 3 is a diagram illustrating a pseudo particle distribution of a control layer according to an embodiment of the invention.
Fig. 4 is a physical layer true particle distribution diagram obtained based on interpolation mapping according to an embodiment of the present invention.
Fig. 5 is a schematic diagram of a true-false dual particle model of a flow meter according to an embodiment of the present invention.
Detailed Description
The invention is described in detail below with reference to the accompanying drawings and embodiments, and the method of the invention can be used for solving coupling problems of various complex physical fields or multiple physical fields, in the embodiment, the simulation of the working state of a liquid flowmeter of a certain type is taken as an example, and the transient numerical simulation of the complex fluid-solid coupling problem is completed, as shown in fig. 1, the total length of a pipeline where the flowmeter is located is 3000mm, the inner diameter of the pipeline is 300mm, the flowmeter is made of stainless steel, as shown in fig. 1, the flowmeter is composed of a front flow guide frame 1, a turbine 2 and a rear flow guide frame 4, the turbine 2 is connected with 20 blades 3, a flowing medium is water, the temperature is 25 ℃, the flowing-in is performed at a speed of 1m/s from a left cross section, the flowing-out is performed freely from a right outlet, the flowmeter is static at an initial moment, and the change process of the rotation speed of the flowmeter is analyzed, and the stable rotation speed of the flowmeter under the current flow speed is obtained.
Referring to fig. 2, a method for modeling a true-false dual particle model for coupled physical field fast solution includes the following steps:
1) determining an analysis model of a physical problem to be solved:
1.1) establishing a pipeline model and importing a flowmeter entity model, dividing surface patches of the flowmeter according to coordinates and shapes of each boundary of a CAD (computer-aided design) configuration, defining different patches and geometric control points in the boundaries to obtain a CAE (computer-aided engineering) model, ensuring that the CAD configuration is consistent with the CAE model, establishing a complete analysis node vector and a spline curve interpolation basis function under a parameter coordinate system according to the requirement of modeling precision, wherein the node vector is a non-decreasing sequence xi (xi) between 0 and 11，ξ2，…，ξm+p+1And B-spline basis functions are adopted as interpolation functions, and a specific recursion formula is as follows:
wherein N is a B spline curve base function, p is a base function order, and xi is a node in a parameter coordinate system;
b spline curve interpolation division is carried out on a surface patch on the outer surface of the flow meter and the inner side wall surface of the pipeline which are divided in the CAD configuration, so that a distribution area of each node vector and a shape control point is 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 two-dimensional spline surface patch can be obtained by two groups of similar B spline basis functions and weight coefficients through bilinear interpolation, an original numerical analysis model for calculation and analysis is formed, and the expression form of the bilinear rational B spline basis functions is as follows:
wherein R is a bilinear rational B spline basis function, N is a B spline curve basis function, p is a basis function order, and omega is a projection weight factor;
1.2) constructing 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 method of inserting more control nodes and improving NURBS (non-uniform rational B-spline) basic function orders at different nodes is adopted, so that the improvement of curve and surface fitting orders can be realized, the curved surface and corresponding boundary contour lines of the turbine blade can be refined, and an accurate analysis model of the physical problem to be solved can be obtained;
2) establishing a two-layer particlized non-grid analysis model of the problem to be solved:
2.1) setting virtual particles with fixed spatial positions at each geometric 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 FIG. 3; the control layer particles do not really exist or directly participate in physical field calculation, and physical information on the control layer particles is determined by a physical field at an initial moment, so that the control layer particles have the characteristic of pseudo particles; the method comprises the following steps of constructing a physical layer particle model actually participating in particle method calculation on the basis of control layer particles, performing particle discretization on a current physical field and an analysis region, giving the number and arrangement mode of physical field particles according to calculation precision requirements and physical field continuity requirements, keeping consistent interpolation basis functions used in physical field discretization and shape functions adopted in analysis region discretization on the basis of an isoparametric transformation idea, and discretely obtaining the physical field and the analysis region as follows:
wherein M isiAs a function of shape, ciIs a discrete particle coordinate vector, u, v are coordinates in a parametric coordinate system, D0Is a parameter coordinate system, D is a physical analysis domain, F is coordinate transformation based on NURBS basis function, p is unknown physical field distribution to be solved, DiIs a discrete physical particle information vector;
the distribution of the particles in the physical layer is shown in fig. 4, wherein the particles, although participating in the calculation by the particle method, do not store physical information on the particles, and are only obtained by the upper layer particle control network through a certain interpolation relation when needed;
2.2) after corresponding physical information is given to the control layer particle virtualized in the step 2.1), constructing a mapping relation between the control layer particle information and the physical layer particle information; setting the order of the mapping relation according to the calculation precision requirement and the restriction of the calculation capacity, wherein the specific two-dimensional mapping form is as follows:
wherein QkjTo control the interpolation mapping function of particles to physical particles, gkjFor controlling the physical information vector carried by the particle, u, v are coordinates in a parametric coordinate system, diIs a discrete physical particle information vector;
if the order of a certain direction of the interpolation mapping function is q, the number m of the control particles corresponding to a certain physical particle in the certain direction is specifically:
m＝q+1 (7)
the number of particles of the control layer is far smaller than that of particles of the physical layer, so that a large amount of physical particle information is obtained by mapping a small amount of control particle information through an interpolation relation related to a spatial position, and further physical field distribution with high-order continuity is obtained, a discrete physical field is ensured to have certain smoothness, and finally, a true and false double particle model is constructed as shown in fig. 5;
3) analysis zone boundary conditions apply:
obtaining a two-layer particle non-grid 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 particle non-grid analysis model and physical field information of the control particles corresponding to the boundary, dividing a structural grid for a non-physical field calculation area in the analysis area, and applying a wall surface collision boundary condition and a wall surface friction condition based on a boundary force method idea;
4) physical particle physical properties and interactions apply:
according to the real physical property condition of an analysis area, screening physical particles, applying corresponding physical property parameters, setting the interference radius of the particles, combining a phase shift field thought, and calculating and applying the interaction among the physical particles;
5) constructing a calculation model of the physical field distribution to be solved:
according to the mapping interpolation function obtained in the step 2), a Jacobian matrix is combined to disperse and calculate a second-order partial differential term in the incompressible N-S unsteady partial differential control equation, and the specific form is as follows:
wherein QkjTo control the interpolation mapping function of particles to physical particles, gkjFor controlling the physical information vector carried by the particle, u, v are coordinates in a parametric coordinate system, diThe DF is a space transformation Jacobian matrix for a discrete physical particle information vector;
by combining a given time step length, the development condition of unknown physical field control particle information is gradually solved, then the physical particle information is calculated according to the interpolation mapping relation in the step 2), the purpose that the physical information of a small amount of control particles is solved, a large amount of physical particle information is obtained through mapping, smooth physical field distribution and development conditions are obtained in each time step, and the stable rotating speed of the flowmeter is 2.73r/s under the background of the flow speed of 1m/s is obtained in the embodiment.
Claims (1)
1. A coupled physical field fast solving-oriented true and false dual particle model modeling method is characterized by comprising the following steps:
1) determining an analysis model of a physical problem to be solved:
1.1) importing a real geometric configuration of a physical problem model to be solved, dividing the shape of each boundary into different patches according to a CAD configuration, defining geometric control points including different patches and boundary contour lines to obtain a CAE model, ensuring the CAD configuration to be consistent with the CAE model, establishing a complete analysis node vector and a spline curve interpolation basis function under a parameter coordinate system according to the requirement of modeling precision, wherein the node vector is a non-decreasing sequence xi between 0 and 1 and xi { xi ═ xi-1，ξ2，···，ξm+p+1And B-spline basis functions are adopted as interpolation functions, and a specific recursion formula is as follows:
wherein N is a B spline curve base function, p is a base function order, and xi is a node in a parameter coordinate system;
b spline curve interpolation division is carried out on boundary contour lines and internal key configurations on different surface patches in the 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, a surface patch in the internal space of the area is 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 expression form of the bilinear rational B spline basis functions is as follows:
wherein R is a bilinear rational B spline basis function, N is a B spline curve basis function, p is a basis function order, and omega is a projection weight factor;
1.2) constructing 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 method of inserting more control nodes and improving NURBS (non-uniform rational B-spline) basic function orders at different nodes is adopted to realize the improvement of curve and surface fitting orders, refine boundary structures and obtain an accurate analysis model of a physical problem to be solved;
2) establishing a two-layer particlized non-grid analysis model of the problem to be solved:
2.1) setting virtual particles with fixed spatial positions at each geometric control node on the accurate analysis model obtained in the step 1) to form a control layer particle network, wherein physical information of the control layer particles is determined by an initial moment physical field, so that the control layer particles have a 'pseudo particle' characteristic; the method comprises the following steps of constructing a physical layer particle model which truly participates in particle method calculation on the basis of control layer particles, performing particle discretization on a current physical field and an analysis region, giving the number and arrangement mode of physical field particles according to calculation precision requirements and physical field continuity requirements, keeping consistent interpolation basis functions used in physical field discretization and shape functions adopted in analysis region discretization on the basis of an isoparametric transformation idea, and obtaining a functional relation between the physical field and the analysis region through discretization as follows:
wherein M isiAs a function of shape, ciIs a discrete particle coordinate vector, u, v are coordinates in a parametric coordinate system, D0As a parametric coordinate system, D as a physical analysisDomain, F is coordinate transformation based on NURBS basis function, p is unknown physical field distribution to be solved, diIs 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 the transmission of an interpolation relation when needed;
2.2) after corresponding physical information is given to the control layer particle virtualized in the step 2.1), constructing a mapping relation between the control layer particle information and the physical layer particle information; setting the order of the mapping relation according to the calculation precision requirement and the restriction of the calculation capacity, wherein the specific two-dimensional mapping form is as follows:
wherein QkjTo control the interpolation mapping function of particles to physical particles, gkjFor controlling the physical information vector carried by the particle, u, v are coordinates in a parametric coordinate system, diIs a discrete physical particle information vector;
if the order of a certain direction of the interpolation mapping function is q, the number m of the control particles corresponding to a certain physical particle in the 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 a large amount of physical particle information is obtained by mapping a small amount of control particle information through an interpolation relation related to a spatial position, and physical field distribution with high-order continuity is further obtained;
3) analysis zone boundary conditions apply:
obtaining a two-layer particle non-grid 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 particle non-grid analysis model and physical field information of the control particles corresponding to the boundary, dividing a structural grid for a non-physical field calculation area in the analysis area, and applying a wall surface collision boundary condition or a thermodynamic condition based on a boundary force method idea;
4) physical particle physical properties and interactions apply:
according to the real physical property condition of an analysis area, screening physical particles, applying corresponding physical property parameters, setting the interference radius of the particles, combining a phase shift field thought, and calculating and applying the interaction among the physical particles;
5) constructing a calculation model of the physical field distribution to be solved:
according to the mapping interpolation function obtained in the step 2), a Jacobian matrix is combined to disperse and calculate a high-order partial differential term in the unsteady partial differential control equation, and the specific form is as follows:
wherein QkjTo control the interpolation mapping function of particles to physical particles, gkjFor controlling the physical information vector carried by the particle, u, v are coordinates in a parametric coordinate system, diThe DF is a space transformation Jacobian matrix for a discrete physical particle information vector;
and (3) gradually solving the development condition of the unknown physical field control particle information by combining a given time step length, then calculating the physical particle information according to the interpolation mapping relation in the step 2), realizing that a large amount of physical particle information is obtained by solving a small amount of control particle physical information and mapping, and obtaining the smooth physical field distribution and development condition in each time step.
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