CN116611369B - Interpolation method and device based on smoothness magnitude and candidate template point number - Google Patents

Interpolation method and device based on smoothness magnitude and candidate template point number Download PDF

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CN116611369B
CN116611369B CN202310872199.6A CN202310872199A CN116611369B CN 116611369 B CN116611369 B CN 116611369B CN 202310872199 A CN202310872199 A CN 202310872199A CN 116611369 B CN116611369 B CN 116611369B
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interpolation
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node
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CN116611369A (en
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马燕凯
白进维
燕振国
毛枚良
孙堃
闵耀兵
朱华君
江定武
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Computational Aerodynamics Institute of China Aerodynamics Research and Development Center
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Abstract

The application discloses an interpolation method and device based on smoothness magnitude and candidate template point number, which are used for obtaining a flow field area to be simulated and splitting the flow field area to be simulated to obtain a plurality of structural grids; determining a target interpolation template, wherein the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring the smoothness degree of the candidate templates; the method solves the Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through a target interpolation template, a new candidate template set and a new method for calculating a smoothness metric factor are adopted, a new nonlinear weighting mechanism is realized, and a weighted combination format is promoted to approach a linear format under the combined action of two mechanisms in a smooth area of a flow field.

Description

Interpolation method and device based on smoothness magnitude and candidate template point number
Technical Field
The application belongs to the technical field of data processing, and particularly relates to an interpolation method and device based on smoothness magnitude and candidate template points.
Background
Computational Fluid Dynamics (CFD) is one of the important means for developing flow physics problem research, and plays an increasingly important role in the fields of advanced aircraft aerodynamic performance assessment, aerodynamic thermal environment prediction, flow mechanism and influence law research and the like. Because of the characteristics of larger value dissipation, low precision, lower resolution and the like of the low-order precision method, the low-order precision method has obvious defects when solving the pneumatic problem with space-time multi-scale or strong nonlinearity, and is easy to generate strong grid dependence. With the increasing demands of engineering applications on numerical simulation accuracy, people prefer to use a high-order accuracy numerical simulation method with low dissipation, low dispersion and high resolution.
The core of the high-order precision numerical simulation method is a high-order precision format, and after decades of continuous development and improvement, numerous high-order precision formats including high-order precision finite difference, high-order precision finite volume, high-order precision finite element and the like are formed, wherein the high-order precision finite difference format based on the structural grid is most widely applied due to relatively small calculation cost. The high-order precision finite difference format can be divided into a high-order precision linear format and a high-order precision nonlinear format, and when the simulated flow problem contains shock waves and other discontinuous structures, the high-order precision nonlinear format needs to be adopted.
The most commonly used high-order precision nonlinear format design method adopts a weighted combination design mode: dividing the overall template with high-order precision into a plurality of candidate templates with lower-order precision, and combining the candidate templates with low-order precision to obtain the template with high-order precision; giving dynamic weight to each candidate template according to the smoothness degree of each candidate template, wherein the smoother a certain candidate template is, the greater the weight given to the candidate template is, and otherwise, the smaller the weight given to the candidate template is; if the integral template does not contain shock wave and other discontinuous structures, each candidate template is smooth, the given weight is close to ideal weight, and the result of the weighted combination is close to a high-order precision linear format; if the integral template contains shock wave and other discontinuous structures, the candidate template containing the discontinuity is given smaller weight, and other candidate templates not containing the discontinuity are given larger weight, so that cross-discontinuity interpolation is avoided, and therefore, the high-order precision format of the weighted combination can basically capture shock wave and other discontinuous structures without oscillation.
The nonlinear weighted high-order precision format design method has a disadvantage that even if no discontinuous structures such as shock waves exist in the local flow field, the smoothness of each candidate template is different, and the given dynamic weight is different from the ideal weight to some extent. As a result, this nonlinear weighted higher order precision format design approach introduces excessive levels of dissipation in the smooth regions, resulting in reduced numerical simulation precision, which is detrimental to long-term fine simulation of multi-scale flow structures.
Disclosure of Invention
The application aims to provide an interpolation method, an interpolation device, terminal equipment and a storage medium based on smoothness magnitude and candidate template point numbers, so as to solve the defects in the prior art.
In a first aspect, an embodiment of the present application provides an interpolation method based on a smoothness magnitude and a candidate template point number, where the method includes:
acquiring a flow field region to be simulated, and splitting the flow field region to be simulated to obtain a plurality of structural grids;
determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates;
and solving a Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through the target interpolation template.
Optionally, the obtaining the flow field area to be simulated, and splitting the flow field area to be simulated to obtain a plurality of structural grids includes:
under a calculation coordinate system, solving each dimension of each structural grid based on a finite difference method of the structural grids to obtain a plurality of structural grids, wherein the structural grids comprise real solving points and virtual solving points, the values of the real solving points are determined according to the current initial value conditions, and the values of the virtual solving points are determined according to the edge value conditions.
Optionally, the solving the Navier-Stokes equation by using the target interpolation template and adopting a nonlinear weighted high-order precision format includes:
determining the position of a solution point and the position of a flux point corresponding to each structural grid according to the structural grids;
obtaining a physical quantity value at a half node by adopting a nonlinear weighted high-order precision interpolation method according to the physical quantity value at the node;
determining the physical quantity value of a half node according to the physical quantity number of the node by adopting a nonlinear weighted high-order precision interpolation method;
and determining the numerical flux at the node according to the physical quantity value of the node and the physical quantity value of the half node.
Optionally, the interpolation precision corresponding to the first number of candidate templates is a first order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second order smoothness metric factor, including:
determining the first number as M and the second number as N;
the smoothness metric factor corresponding to the M-point candidate template is determined according to the sum of squares of the difference operators of the M-2 order derivative and the M-1 order derivative.
Optionally, the calculation formula of the candidate template is as follows:
the method comprises the steps of carrying out a first treatment on the surface of the Wherein: j represents a node index, h represents a grid spacing, U j Representing the physical quantity at node j +.>Representing the actual physical quantity at the half node S k Representing the kth candidate template,/->Representing the approximate physical quantity at the half-node obtained using the kth candidate template.
Optionally, the method for calculating the smoothness metric factor includes:
wherein: IS k Representing the smoothness factor on the kth candidate template,/->And respectively representing 1-5 derivatives of the physical quantity at the j node.
In a second aspect, an embodiment of the present application provides an interpolation apparatus based on a smoothness magnitude and a candidate template point number, where the apparatus includes:
the acquisition module is used for acquiring a flow field region to be simulated, and splitting the flow field region to be simulated to obtain a plurality of structural grids;
the determining module is used for determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates;
and the calculation module is used for solving the Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through the target interpolation template.
Optionally, the acquiring module is configured to:
under a calculation coordinate system, solving each dimension of each structural grid based on a finite difference method of the structural grids to obtain a plurality of structural grids, wherein the structural grids comprise real solving points and virtual solving points, the values of the real solving points are determined according to the current initial value conditions, and the values of the virtual solving points are determined according to the edge value conditions.
Optionally, the computing module is configured to:
determining the position of a solution point and the position of a flux point corresponding to each structural grid according to the structural grids;
obtaining a physical quantity value at a half node by adopting a nonlinear weighted high-order precision interpolation method according to the physical quantity value at the node;
determining the physical quantity value of a half node according to the physical quantity number of the node by adopting a nonlinear weighted high-order precision interpolation method;
and determining the numerical flux at the node according to the physical quantity value of the node and the physical quantity value of the half node.
Optionally, the interpolation precision corresponding to the first number of candidate templates is a first order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second order smoothness metric factor, including:
determining the first number as M and the second number as N;
the smoothness metric factor corresponding to the M-point candidate template is determined according to the sum of squares of the difference operators of the M-2 order derivative and the M-1 order derivative.
Optionally, the calculation formula of the candidate template is as follows:
the method comprises the steps of carrying out a first treatment on the surface of the Wherein: j represents a node index, h represents a grid spacing, U j Representing the physical quantity at node j +.>Representing the actual physical quantity at the half node S k Representing the kth candidate template,/->Representing the approximate physical quantity at the half-node obtained using the kth candidate template.
Optionally, the fourth method for calculating a smoothness metric factor includes:
wherein: IS k Representing the smoothness factor on the kth candidate template,/->And respectively representing 1-5 derivatives of the physical quantity at the j node.
In a third aspect, an embodiment of the present application provides a terminal device, including: at least one processor and memory;
the memory stores a computer program; the at least one processor executes the computer program stored by the memory to implement the interpolation method provided in the first aspect based on the magnitude of the smoothness and the number of candidate template points.
In a fourth aspect, an embodiment of the present application provides a computer readable storage medium having stored therein a computer program that, when executed, implements the interpolation method based on the magnitude of smoothness and the number of candidate template points provided in the first aspect.
The embodiment of the application has the following advantages:
the interpolation method, the interpolation device, the interpolation terminal equipment and the interpolation storage medium based on the smoothness magnitude and the candidate template point number, which are provided by the embodiment of the application, are used for obtaining a plurality of structural grids by obtaining a flow field region to be simulated and splitting the flow field region to be simulated; determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates; the method solves the Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through a target interpolation template, a new candidate template set and a new method for calculating a smoothness metric factor are adopted, a new nonlinear weighting mechanism is realized, and a weighted combination format is promoted to approach a linear format under the combined action of two mechanisms in a smooth area of a flow field.
Drawings
In order to more clearly illustrate the embodiments of the application or the prior art solutions, the drawings which are used in the description of the embodiments or the prior art will be briefly described below, it being obvious that the drawings in the description below are only some of the embodiments described in the present application, and that other drawings can be obtained according to these drawings without inventive faculty for a person skilled in the art.
FIG. 1 is a flow chart of an interpolation method based on smoothness magnitude and candidate template points in an embodiment of the application;
FIG. 2 is a schematic diagram of an equal portion of a calculation region according to an embodiment of the present application;
FIG. 3 is a diagram illustrating a method for determining half-node values according to an embodiment of the application;
FIG. 4 is a schematic diagram of candidate template distribution according to an embodiment of the application;
FIG. 5 is a block diagram illustrating an embodiment of an interpolation apparatus according to the present application based on smoothness magnitude and number of candidate templates;
fig. 6 is a schematic structural diagram of a terminal device of the present application.
Detailed Description
In order to make the objects, technical solutions and advantages of the present application more apparent, the technical solutions of the present application will be clearly and completely described below with reference to specific embodiments and corresponding drawings. It will be apparent that the described embodiments are only some, but not all, embodiments of the application. All other embodiments, which can be made by those skilled in the art based on the embodiments of the application without making any inventive effort, are intended to be within the scope of the application.
Noun interpretation:
Navier-Stokes equation: a system of hydrodynamic control equations consisting of conservation of mass, conservation of momentum, and conservation of energy.
Physical quantity of flow field: the physical variables such as density, velocity, pressure, temperature, etc. in the flow are collectively referred to.
CFD: computational Fluid Dynamics, computational fluid dynamics, a discipline for approximating a solution to a physical flow problem using a numerical discrete flow control equation.
High order precision format: the precision of the space discrete format is a numerical discrete method with three or more orders.
Nonlinear weighted format: the weights of the individual candidate templates are not fixed in a format in which a plurality of candidate templates are dynamically combined.
Ideal weights: each candidate template in the nonlinear weighting format is strictly equivalent to the linear format under the action of ideal weight, and the value of the dynamic weight is equal to the ideal weight in an absolute smooth flow field or a uniform flow field.
IS: indicator of Smoothness, a smoothness measurement factor, a parameter that measures the smoothness of the candidate templates.
An embodiment of the application provides an interpolation method based on smoothness magnitude and candidate template point numbers, which is used for interpolation processing. The execution subject of the embodiment is an interpolation device based on the magnitude of smoothness and the number of candidate template points, and is provided on a terminal device, for example, the terminal device at least includes a computer terminal and the like.
Referring to fig. 1, a step flow diagram of an embodiment of an interpolation method based on smoothness magnitude and candidate template points according to the present application is shown, and the method specifically may include the following steps:
s101, acquiring a flow field region to be simulated, and splitting the flow field region to be simulated to obtain a plurality of structural grids;
specifically, the terminal equipment acquires a flow field area to be simulated, generates a body-attached structural grid, namely a structural grid unit, around the outline flow field area to be simulated, and covers all the flow field areas with complex outline by a plurality of butt joint structural grids.
The finite difference method based on the structural grid is solved dimension by dimension, and the discrete process and algorithm of each dimension direction are the same, so that only one-dimensional calculation is taken as an example. The discrete area under the calculation coordinate system is divided into N-1 structural grids.
S102, determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates;
specifically, a nonlinear weighted high-order precision interpolation method is adopted according to the physical quantity value at the node to obtain the physical quantity value at the half node. For example, interpolation of 5 node values is used to obtain 1 half node value.
When a nonlinear weighted high-order precision format is adopted to solve a Navier-Stokes equation, a new candidate template set, three 3-point templates and a synthesized 5-point template are adopted, wherein the 3-point candidate templates are 2-order smoothness measurement factors, and the 5-point templates are 6-order smoothness measurement factors. By adopting the design method, the dissipation level introduced by the original nonlinear weighted 5-order precision interpolation method in a smooth area can be reduced, and the long-time fine simulation effect of the multi-scale flow structure is improved.
S103, solving a Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through a target interpolation template.
According to the interpolation method based on the smoothness magnitude and the number of candidate template points, which is provided by the embodiment of the application, a flow field area to be simulated is obtained, and split is carried out on the flow field area to be simulated, so that a plurality of structural grids are obtained; determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates; the method solves the Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through a target interpolation template, a new candidate template set and a new method for calculating a smoothness metric factor are adopted, a new nonlinear weighting mechanism is realized, and a weighted combination format is promoted to approach a linear format under the combined action of two mechanisms in a smooth area of a flow field.
The interpolation method based on the smoothness magnitude and the candidate template points provided by the embodiment of the application is further described in a further embodiment of the application.
Optionally, acquiring a flow field area to be simulated, splitting the flow field area to be simulated to obtain a plurality of structural grids, including:
under a calculation coordinate system, solving each dimension of each structural grid based on a finite difference method of the structural grids to obtain a plurality of structural grids, wherein the structural grids comprise real solving points and virtual solving points, the values of the real solving points are determined according to the current initial value conditions, and the values of the virtual solving points are determined according to the edge value conditions.
Optionally, solving the Navier-Stokes equation by using a nonlinear weighted higher order precision format through a target interpolation template includes:
determining the position of a solution point and the position of a flux point corresponding to each structural grid according to the structural grids;
obtaining a physical quantity value at a half node by adopting a nonlinear weighted high-order precision interpolation method according to the physical quantity value at the node;
determining the physical quantity value of a half node according to the physical quantity number of the node by adopting a nonlinear weighted high-order precision interpolation method;
and determining the numerical flux at the node according to the physical quantity value of the node and the physical quantity value of the half node.
Optionally, the interpolation precision corresponding to the first number of candidate templates is a first order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second order smoothness metric factor, including:
determining the first number as M and the second number as N;
the smoothness metric factor corresponding to the M-point candidate template is determined according to the sum of squares of the difference operators of the M-2 order derivative and the M-1 order derivative.
Alternatively, the calculation formula of the candidate template is as follows:
the method comprises the steps of carrying out a first treatment on the surface of the Wherein: j represents a node index, h represents a grid spacing, U j Representing the physical quantity at node j +.>Representing the actual physical quantity at the half node S k Representing the kth candidate template,/->Representing the approximate physical quantity at the half-node obtained using the kth candidate template.
Optionally, the method for calculating the smoothness metric factor comprises:
wherein: IS k Representing the smoothness factor on the kth candidate template,/->And respectively representing 1-5 derivatives of the physical quantity at the j node.
The embodiment of the application provides a nonlinear weighted 5-order precision interpolation method based on a structural grid high-order precision finite difference method, which comprises the following specific contents:
step one, dividing a calculation area.
The finite difference method based on the structural grid is solved dimension by dimension, and the discrete process and algorithm of each dimension direction are the same, so that only one-dimensional calculation is taken as an example. The discrete area under the calculated coordinate system is divided into N-1 units as shown in fig. 2.
The solving points with the reference number of 1-N-1 are real solving points, and the other solving points are virtual solving points. The value of the true solution point at the current moment tis given by an initial value condition, and the value of the virtual solution point is given by an edge value condition.
And step two, obtaining the physical quantity value at the half node by adopting a nonlinear weighted high-order precision interpolation method according to the physical quantity value at the node. As shown in fig. 3, 1 half-node value is interpolated using 5 node values.
(1) The design method of interpolation templates. As shown in fig. 4, three 3-point candidate templates and one global 5-point candidate template. Interpolation formulas on each candidate template are
The method comprises the steps of carrying out a first treatment on the surface of the Wherein: j represents a node index, h represents a grid spacing, U j Representing the physical quantity at node j +.>Representing the actual physical quantity at the half node S k Representing the kth candidate template,/->Representing the approximate physical quantity at the half-node obtained using the kth candidate template.
Wherein the interpolation on the 3-point candidate templates is 3-order precision and the interpolation on the 5-point candidate templates is 5-order precision.
(2) Calculation of a smoothness metric factor (IS) for the candidate template. The smoothness metric factor is a parameter that measures the smoothness of the candidate templates, the M-point candidate template smoothness metric factor is constructed from the sum of squares of the difference operators of the M-2 and M-1 derivatives, i.e., the 3-point candidate template involves first and second derivatives, the 5-point candidate template involves 3 and 4 derivatives, and not all the derivatives. The specific calculation formula is that
The method comprises the steps of carrying out a first treatment on the surface of the Taylor expansions are respectively
Wherein: IS k Representing the smoothness factor on the kth candidate template,/->And respectively representing 1-5 derivatives of the physical quantity at the j node.
In the method provided by the embodiment of the application,the candidate template sets are different and mainly adopt 33 point candidate templates, which are composed of 3 points and 1 5 point candidate templates; />The smoothness metric factors are different. For the case of containing an M-point candidate template, the literature computes derivative difference operators that will involve all possible orders of 1 to M-1, the magnitude of the smoothness factor being h 2 The magnitude of the magnitude is irrelevant to the number of candidate template points; in this patent, only the differential operators of the derivatives of the two orders M-2 and M-1 are involved, the magnitude of the smoothness factor is h 2（M-2） The magnitude of which is related to the number of points contained in the candidate template. />The weighting mechanism is different. In smooth areas, IS 1 、IS 2 And IS 3 Are all h 2 Magnitude, its value is equivalent, so that three 3-point candidate templates can be combined to be close to 5 th A result of the linear interpolation; IS on the other hand 4 Is h 6 In order, the magnitude of which is much smaller than the other three smoothness factors, a 5-point candidate template would be given a weight close to 1. Under the combined action of the two mechanisms, the interpolation method in the patent is closer to 5 in a smooth area th Linear interpolation and thus the level of dissipation introduced would be significantly lower than in the prior art. And the method solves the problem that the interpolation format order is not reduced near the extreme point (the first derivative is zero and the second derivative is not zero).
(3) And determining the dynamic weight of each candidate template according to the smoothness metric factor. The dynamic weights are expressed and determined as follows:
；
wherein, the liquid crystal display device comprises a liquid crystal display device,the method comprises the steps of carrying out a first treatment on the surface of the Epsilon is a small amount preventing the denominator from zero, C k Ideal weights representing the kth candidate template, IS k A smoothness metric factor, alpha, representing the kth candidate template k Non-normalized dynamic weights, ω, representing the kth candidate template k Representing normalized dynamic weights of the kth candidate template.
(4) And calculating to obtain the physical quantity value at the half node.
Wherein j represents a node number, ω k Normalized dynamic weight representing kth candidate template, +.>Representing the approximate physical quantity at half node obtained using the kth candidate template, ++>Representing the approximate physical quantity at the resulting half-node.
Step three, repeating step two for j=0, 1, …, N-1.
And step four, calculating the numerical flux of each node and each half node. The numerical flux at the node is directly calculated by the flow field variable and the grid derivative at the node, and the numerical flux at the half node can be calculated by adopting various common numerical flux calculation methods, such as Roe, steger-Warming and the like.
And fifthly, calculating the numerical flux derivative at the node. The following method is adopted
Wherein j represents a node number, h represents a mesh spacing, +.>Representing the numerical flux at the node, +.>Representing the numerical flux at half-node, +.>Representing the first derivative of the numerical flux at the node.
And step six, updating the physical quantity value at the node at the next moment.
Wherein j represents a node index, t represents a current time,/->Representing the physical quantity at the node at the current moment, t+1 representing the next moment,/i->Representing the physical quantity at the node at the next moment, fatt representing the time step, ++>Representing the first derivative of the numerical flux at the node.
The embodiment of the application adopts a new candidate template set and a new method for calculating the smoothness metric factor, thereby realizing a new nonlinear weighting mechanism. The method has the effect that the two mechanisms are combined in the smooth area of the flow field to promote the weighted combination format to approach the linear format, and compared with the original method, the method has higher resolution and lower dissipation level in the smooth area. Near the extreme point (the first derivative is zero, the second derivative is not zero), the phenomenon that the interpolation format order is reduced in the original design method can not occur.
It should be noted that, for simplicity of description, the method embodiments are shown as a series of acts, but it should be understood by those skilled in the art that the embodiments are not limited by the order of acts, as some steps may occur in other orders or concurrently in accordance with the embodiments. Further, those skilled in the art will appreciate that the embodiments described in the specification are presently preferred embodiments, and that the acts are not necessarily required by the embodiments of the application.
According to the interpolation method based on the smoothness magnitude and the number of candidate template points, which is provided by the embodiment of the application, a flow field area to be simulated is obtained, and split is carried out on the flow field area to be simulated, so that a plurality of structural grids are obtained; determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates; the method solves the Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through a target interpolation template, a new candidate template set and a new method for calculating a smoothness metric factor are adopted, a new nonlinear weighting mechanism is realized, and a weighted combination format is promoted to approach a linear format under the combined action of two mechanisms in a smooth area of a flow field.
Another embodiment of the present application provides an interpolation device based on a smoothness magnitude and a candidate template point number, which is configured to execute the interpolation method based on the smoothness magnitude and the candidate template point number provided in the foregoing embodiment.
Referring to fig. 5, a block diagram illustrating an embodiment of an interpolation apparatus according to the present application based on smoothness magnitude and candidate template points may specifically include the following modules: an acquisition module 501, a determination module 502, and a calculation module 503, wherein:
the acquisition module 501 is configured to acquire a flow field area to be simulated, and split the flow field area to be simulated to obtain a plurality of structural grids;
the determining module 502 is configured to determine a target interpolation template, where the target interpolation template includes at least a first number of candidate templates and b global second number of candidate templates, where a, b are natural numbers greater than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates;
the calculation module 503 is configured to solve the Navier-Stokes equation by using a nonlinear weighted high-order precision format through the target interpolation template.
According to the interpolation device based on the smoothness magnitude and the number of candidate template points, which is provided by the embodiment of the application, a flow field area to be simulated is obtained, and split is carried out on the flow field area to be simulated, so that a plurality of structural grids are obtained; determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates; the method solves the Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through a target interpolation template, a new candidate template set and a new method for calculating a smoothness metric factor are adopted, a new nonlinear weighting mechanism is realized, and a weighted combination format is promoted to approach a linear format under the combined action of two mechanisms in a smooth area of a flow field.
The interpolation device based on the smoothness magnitude and the candidate template point number provided by the embodiment of the application is further described in a further embodiment of the application.
Optionally, the acquiring module is configured to:
under a calculation coordinate system, solving each dimension of each structural grid based on a finite difference method of the structural grids to obtain a plurality of structural grids, wherein the structural grids comprise real solving points and virtual solving points, the values of the real solving points are determined according to the current initial value conditions, and the values of the virtual solving points are determined according to the edge value conditions.
Optionally, the computing module is configured to:
determining the position of a solution point and the position of a flux point corresponding to each structural grid according to the structural grids;
obtaining a physical quantity value at a half node by adopting a nonlinear weighted high-order precision interpolation method according to the physical quantity value at the node;
determining the physical quantity value of a half node according to the physical quantity number of the node by adopting a nonlinear weighted high-order precision interpolation method;
and determining the numerical flux at the node according to the physical quantity value of the node and the physical quantity value of the half node.
Optionally, the interpolation precision corresponding to the first number of candidate templates is a first order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second order smoothness metric factor, including:
determining the first number as M and the second number as N;
the smoothness metric factor corresponding to the M-point candidate template is determined according to the sum of squares of the difference operators of the M-2 order derivative and the M-1 order derivative.
Alternatively, the calculation formula of the candidate template is as follows:
the method comprises the steps of carrying out a first treatment on the surface of the Wherein: j represents a node number, h represents a mesh pitch, uj represents a physical quantity at a node j, +.>Representing the actual physical quantity at the half node S k Representing the kth candidate template,/->Representing the approximate physical quantity at the half-node obtained using the kth candidate template.
Optionally, a fourth method of calculating a smoothness metric factor includes:
wherein: representing IS k Representing the smoothness factor on the kth candidate template,/->And respectively representing 1-5 derivatives of the physical quantity at the j node.
For the device embodiments, since they are substantially similar to the method embodiments, the description is relatively simple, and reference is made to the description of the method embodiments for relevant points.
According to the interpolation device based on the smoothness magnitude and the number of candidate template points, which is provided by the embodiment of the application, a flow field area to be simulated is obtained, and split is carried out on the flow field area to be simulated, so that a plurality of structural grids are obtained; determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates; the method solves the Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through a target interpolation template, a new candidate template set and a new method for calculating a smoothness metric factor are adopted, a new nonlinear weighting mechanism is realized, and a weighted combination format is promoted to approach a linear format under the combined action of two mechanisms in a smooth area of a flow field.
An embodiment of the present application provides a terminal device, configured to perform the interpolation method provided in the above embodiment, based on the smoothness magnitude and the number of candidate template points.
Fig. 6 is a schematic structural view of a terminal device of the present application, as shown in fig. 6, the terminal device includes: at least one processor 601 and memory 602;
the memory stores a computer program; at least one processor executes a computer program stored in a memory to implement the interpolation method based on smoothness magnitude and number of candidate templates provided in the above embodiment.
The terminal equipment provided by the embodiment obtains a flow field area to be simulated, and splits the flow field area to be simulated to obtain a plurality of structural grids; determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates; the method solves the Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through a target interpolation template, a new candidate template set and a new method for calculating a smoothness metric factor are adopted, a new nonlinear weighting mechanism is realized, and a weighted combination format is promoted to approach a linear format under the combined action of two mechanisms in a smooth area of a flow field.
Still another embodiment of the present application provides a computer readable storage medium having a computer program stored therein, where the computer program when executed implements the interpolation method based on the smoothness metric magnitude and the candidate template point number provided in any one of the above embodiments.
According to the computer readable storage medium of the embodiment, a plurality of structural grids are obtained by acquiring a flow field area to be simulated and splitting the flow field area to be simulated; determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates; the method solves the Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through a target interpolation template, a new candidate template set and a new method for calculating a smoothness metric factor are adopted, a new nonlinear weighting mechanism is realized, and a weighted combination format is promoted to approach a linear format under the combined action of two mechanisms in a smooth area of a flow field.
It should be noted that the foregoing detailed description is exemplary and is intended to provide further explanation of the application. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of exemplary embodiments according to the present application. As used herein, the singular is intended to include the plural unless the context clearly indicates otherwise. Furthermore, it will be further understood that the terms "comprises" and/or "comprising," when used in this specification, specify the presence of stated features, steps, operations, devices, components, and/or groups thereof.
It should be noted that the terms "first," "second," and the like in the description and the claims of the present application and the above figures are used for distinguishing between similar objects and not necessarily for describing a particular sequential or chronological order. It is to be understood that the terms so used are interchangeable under appropriate circumstances such that the embodiments of the application described herein are capable of operation in sequences other than those illustrated or otherwise described herein.
Furthermore, the terms "comprise" and "have," as well as any variations thereof, are intended to cover a non-exclusive inclusion. For example, a process, method, system, article, or apparatus that comprises a list of steps or elements is not necessarily limited to those elements but may include other steps or elements not expressly listed or inherent to such process, method, article, or apparatus.
Spatially relative terms, such as "above … …," "above … …," "upper surface at … …," "above," and the like, may be used herein for ease of description to describe one device or feature's spatial location relative to another device or feature as illustrated in the figures. It will be understood that the spatially relative terms are intended to encompass different orientations in use or operation in addition to the orientation depicted in the figures. For example, if the device in the figures is turned over, elements described as "above" or "over" other devices or structures would then be oriented "below" or "beneath" the other devices or structures. Thus, the exemplary term "above … …" may include both orientations of "above … …" and "below … …". The device may also be positioned in other different ways, such as rotated 90 degrees or at other orientations, and the spatially relative descriptors used herein interpreted accordingly.
In the above detailed description, reference is made to the accompanying drawings, which form a part hereof. In the drawings, like numerals typically identify like components unless context indicates otherwise. The illustrated embodiments described in the detailed description, drawings, and claims are not meant to be limiting. Other embodiments may be utilized, and other changes may be made, without departing from the spirit or scope of the subject matter presented herein.
The above is only a preferred embodiment of the present application, and is not intended to limit the present application, but various modifications and variations can be made to the present application by those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application should be included in the protection scope of the present application.

Claims (6)

1. An interpolation method based on smoothness magnitude and candidate template point number, which is characterized by comprising the following steps:
acquiring a flow field region to be simulated, and splitting the flow field region to be simulated to obtain a plurality of structural grids;
determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates;
solving a Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through the target interpolation template;
wherein: the calculation formula of the candidate templates is as follows:
wherein: j represents a node index, h represents a grid spacing, U j Representing the physical quantity at node j, U j+1/2 Representing the actual physical quantity at the half node S k Representing the k-th candidate template,representing the approximate physical quantity at the half node obtained using the kth candidate template;
the calculation method of the smoothness measurement factor comprises the following steps:
wherein: IS k Representing a smoothness metric factor on the kth candidate template, U j ′,U j ″,And the 1-5 order derivatives of the physical quantity at the j node are respectively represented.
2. The method of claim 1, wherein the obtaining the flow field region to be simulated and splitting the flow field region to be simulated to obtain a plurality of structural grids comprises:
under a calculation coordinate system, solving each dimension of each structural grid based on a finite difference method of the structural grids to obtain a plurality of structural grids, wherein the structural grids comprise real solving points and virtual solving points, the values of the real solving points are determined according to the current initial value conditions, and the values of the virtual solving points are determined according to the edge value conditions.
3. The method of claim 2, wherein solving the Navier-Stokes equation with the target interpolation template using a non-linearly weighted higher order accuracy format comprises:
determining the position of a solution point and the position of a flux point corresponding to each structural grid according to the structural grids;
obtaining a physical quantity value at a half node by adopting a nonlinear weighted high-order precision interpolation method according to the physical quantity value at the node;
determining the physical quantity value of a half node according to the physical quantity number of the node by adopting a nonlinear weighted high-order precision interpolation method;
and determining the numerical flux at the node according to the physical quantity value of the node and the physical quantity value of the half node.
4. The method of claim 3, wherein the interpolation precision for the first number of candidate templates is a first order smoothness metric factor and the interpolation precision for the global second number of candidate templates is a second order smoothness metric factor, comprising:
determining the first number as M and the second number as N;
the smoothness metric factor corresponding to the M-point candidate template is determined according to the sum of squares of the difference operators of the M-2 order derivative and the M-1 order derivative.
5. An interpolation device based on smoothness magnitude and number of candidate templates, the device comprising:
the acquisition module is used for acquiring a flow field region to be simulated, and splitting the flow field region to be simulated to obtain a plurality of structural grids;
the determining module is used for determining a target interpolation template, wherein the target interpolation template at least comprises a first number of candidate templates and b global second number of candidate templates, and a and b are natural numbers larger than 0; the interpolation precision corresponding to the first number of candidate templates is a first-order smoothness metric factor, and the interpolation precision corresponding to the global second number of candidate templates is a second-order smoothness metric factor; the smoothness measurement factor is used for measuring parameters of the smoothness degree of the candidate templates;
the calculation module is used for solving a Navier-Stokes equation by adopting a nonlinear weighted high-order precision format through the target interpolation template;
wherein: the calculation formula of the candidate templates is as follows:
wherein: j represents a node index, h represents a grid spacing, U j Representing the physical quantity at node j, U j+1/2 Representing the actual physical quantity at the half node S k Representing the k-th candidate template,representing the approximate physical quantity at the half node obtained using the kth candidate template;
the calculation method of the smoothness measurement factor comprises the following steps:
wherein: IS k Representing a smoothness metric factor on the kth candidate template, U j ′,U j ″,And the 1-5 order derivatives of the physical quantity at the j node are respectively represented.
6. The apparatus of claim 5, wherein the means for obtaining is configured to:
under a calculation coordinate system, solving each dimension of each structural grid based on a finite difference method of the structural grids to obtain a plurality of structural grids, wherein the structural grids comprise real solving points and virtual solving points, the values of the real solving points are determined according to the current initial value conditions, and the values of the virtual solving points are determined according to the edge value conditions.
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Citations (9)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107273593A (en) * 2017-06-01 2017-10-20 北京航空航天大学 A kind of turbulence model and its method for building up predicted for High Mach number intense shock wave flow field Aerodynamic Heating
KR102101389B1 (en) * 2019-10-25 2020-04-16 서울대학교산학협력단 The method for numerical simulation using the well-balanced scheme and high order for the shallow water equation in the numerical analysis
CN112100835A (en) * 2020-09-06 2020-12-18 西北工业大学 High-efficiency high-precision numerical simulation method suitable for complex flow
CN112163312A (en) * 2020-08-17 2021-01-01 空气动力学国家重点实验室 Method for carrying out numerical simulation on compressible flow problem through high-order WENO format reduction
CN112214869A (en) * 2020-09-03 2021-01-12 空气动力学国家重点实验室 Improved high-order nonlinear space discrete method for solving Euler equation
CN114077802A (en) * 2021-11-22 2022-02-22 西安交通大学 Particle modeling method using shape function interpolation to replace kernel function approximation
CN114638173A (en) * 2022-01-25 2022-06-17 中国空气动力研究与发展中心计算空气动力研究所 High-order nonlinear shock wave capturing space dispersion method
CN115510717A (en) * 2022-10-14 2022-12-23 中国人民解放军国防科技大学 High-precision shock wave capturing method based on finite difference method
CN116415518A (en) * 2023-03-08 2023-07-11 北京理工大学 WENO-based high-resolution high-precision shock wave intermittent capturing method

Patent Citations (9)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107273593A (en) * 2017-06-01 2017-10-20 北京航空航天大学 A kind of turbulence model and its method for building up predicted for High Mach number intense shock wave flow field Aerodynamic Heating
KR102101389B1 (en) * 2019-10-25 2020-04-16 서울대학교산학협력단 The method for numerical simulation using the well-balanced scheme and high order for the shallow water equation in the numerical analysis
CN112163312A (en) * 2020-08-17 2021-01-01 空气动力学国家重点实验室 Method for carrying out numerical simulation on compressible flow problem through high-order WENO format reduction
CN112214869A (en) * 2020-09-03 2021-01-12 空气动力学国家重点实验室 Improved high-order nonlinear space discrete method for solving Euler equation
CN112100835A (en) * 2020-09-06 2020-12-18 西北工业大学 High-efficiency high-precision numerical simulation method suitable for complex flow
CN114077802A (en) * 2021-11-22 2022-02-22 西安交通大学 Particle modeling method using shape function interpolation to replace kernel function approximation
CN114638173A (en) * 2022-01-25 2022-06-17 中国空气动力研究与发展中心计算空气动力研究所 High-order nonlinear shock wave capturing space dispersion method
CN115510717A (en) * 2022-10-14 2022-12-23 中国人民解放军国防科技大学 High-precision shock wave capturing method based on finite difference method
CN116415518A (en) * 2023-03-08 2023-07-11 北京理工大学 WENO-based high-resolution high-precision shock wave intermittent capturing method

Non-Patent Citations (7)

* Cited by examiner, † Cited by third party
Title
New nonlinear weights for improving accuracy and resolution of weighted compact nonlinear scheme;Zhenguo Yan 等;Computers & Fluids;第127卷;226-240 *
Weighted compact nonlinear hybrid scheme based on a family of mapping functions for aeroacoustics problem;Yu Peixun 等;RESEARCH ARTICLE;第34卷(第4期);1-12 *
二维非结构网格生成及其数值模拟;朱君 等;南京航空航天大学学报(第02期);159-163 *
加权紧致非线性格式在热化学非平衡流数值模拟中的应用;葛明明 等;国防科技大学学报;第38卷(第05期);163-169 *
基于高精度方法的气动光学效应数值模拟;孙喜万;中国优秀博士学位论文全文数据库 工程科技Ⅱ辑(第02期);C031-79 *
高超声速湍流直接数值模拟技术;李新亮;航空学报;第36卷(第01期);147-158 *
高阶精度耗散加权紧致非线性格式;邓小刚;中国科学A辑(第12期);1104-1117 *

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