CN111241672A - Antenna model load displacement measurement method of mixed basis functions - Google Patents
Antenna model load displacement measurement method of mixed basis functions Download PDFInfo
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Abstract
The invention discloses a method for measuring load displacement of an antenna model of a mixed basis function. Constructing a beam unit by using a NURBS basis function aiming at a beam, calculating a Bessel extraction operator of a basic unit, and coupling the beam-beam unit and the beam-shell unit; and establishing and sorting constraint equations, judging and setting displacement coefficient matrix setting according to the coupling relation among different basic units, fusing the displacement constraint equations into a system rigidity matrix of the whole geometric model by adopting a penalty function method, establishing a corrected potential energy functional considering the penalty function, and solving the final system balance equation to obtain the displacement of the antenna control vertex. The invention adopts different structural type components of the complex product of the spline basis function discrete array element antenna, thereby improving the flexibility of product model modeling.
Description
Technical Field
The invention relates to a complex product model load displacement measurement method, in particular to an antenna model load displacement measurement method of a mixed basis function.
Background
The antenna belongs to a complex industrial product formed by various structural types, has large size and comprises a shell, a beam, a rod and other structures. The geometric complexity of different structures is different, and the construction difficulty of geometric models is also different. For example, beam structures can often be simplified to express overall physical properties by their neutral layer curves, so NURBS is sufficient to construct isogeometric models of beams. Whereas the shell structure cannot usually be described with a single NURBS or Hermite surface patch, the T-spline is better suited to be used as an isogeometric basis function for the shell structure.
For the isogeometric models of antenna products composed of different structural members, coupling splicing (such as line units of beam structures, surface units of shell structures and the like) among basic units of various structural members needs additional processing, otherwise, grid units of various structural members cannot form a uniform whole, and accurate analysis on overall displacement of the antenna model cannot be carried out. The substitution method is that different structures of the antenna product are analyzed independently, and the connecting point of the support structure and the reflecting surface structure is used as a fixed supporting point of the transmitting surface structure, so that the degree of freedom of the fixed supporting point is limited, and the deformation of the support structure in the actual situation is ignored, so that the deviation between the displacement calculation result and the actual value of the transmitting surface of the antenna is not accurate enough.
Disclosure of Invention
In order to solve the problems in the background art, the invention provides a method for measuring the load displacement of geometric models such as antennas with mixed basis functions. The invention adopts different structural type components of the complex product of the spline basis function discrete array element antenna, thereby improving the flexibility and the measurement accuracy of the product model modeling.
The technical scheme adopted by the invention is as follows:
the antenna is divided into a support frame and a reflecting surface, wherein the support frame is formed by connecting a plurality of beams, and the reflecting surface is formed by connecting a plurality of shells; the method comprises the following steps:
(1) constructing a beam unit by using a NURBS basis function aiming at a beam, constructing a shell unit by using an unstructured T spline basis function aiming at a shell, and taking the beam unit and the shell unit as basic units; converting NURBS basis functions and unstructured T-spline basis functions into linear combinations of Bernstein polynomials defined in the [ -1,1] space by calculating Bezier extraction operators of the basic units, thus avoiding parameter mapping during Gaussian product calculation;
(2) coupling a beam-beam unit with a beam-shell unit, wherein the beam-beam unit is formed by connecting two beam units, and the beam-shell unit is formed by connecting a beam unit and a shell unit;
the following displacement constraint equations are established for the "beam-to-beam" unit coupling:
u1,1＝un,2
wherein u is1,1And un,2Respectively representing the control vertex P of one of the beam elements1 1And a control vertex P of another beam elementn 2The displacement vector of (2);
for the "beam-shell" unit coupling, the following displacement constraint equations are respectively established for tangency and intersection:
uI＝uJ(tangent)
Wherein u isIAnd uJThe displacement vectors of the control vertices on the beam element and the shell element, respectively, are indicated, the subscripts I and J indicate the number of the control vertices, respectively, p indicates the NURBS order of basis function, ue,jA displacement vector representing a control vertex corresponding to the shell element e, j represents a local number of the control vertex in the shell element e,
representing the basis functions corresponding to the intersections of the shell elements and the beam elements,
is the parameter coordinate of the intersection point of the shell unit and the beam unit;
(3) collating constraint equation information
Arranging the displacement constraint equation into the following form, and then judging and setting a displacement coefficient matrix according to the coupling relation between different basic units to set C1And C2：
(C1-C2)u＝0
Where u represents the displacement vector of all N control vertices of the entire isogeometric model, C1、C2Displacement coefficient matrixes respectively representing control vertexes of the basic units;
(4) a penalty function method is adopted to fuse the displacement constraint equation into a system stiffness matrix K of the whole geometric model, namely, the constraint equation information is fused into the system stiffness matrix; establishing a modified potential energy functional Π taking into account a function with penalties*：
Wherein S is a penalty parameter of a penalty function method, u represents the displacement of a control vertex, and F represents the load applied to the isogeometric model of the antenna;
and (3) solving a minimum value for the potential energy functional, and ordering:
the final system equilibrium equation is found below:
[K+S(C1-C2)T(C1-C2)]u＝F
K*u＝F
K*＝K+S(C1-C2)T(C1-C2)
wherein, K*Expressing a system stiffness matrix K of the isogeometric model after the displacement constraint equation is fused, and expressing matrix transposition by T;
the coupling of the beam to the beam and the beam to the shell only changes the rigidity matrix of the system, and the rigidity matrix K after the change*Is still symmetrical, has the same dimension as the original rigidity matrix and gives a balanced mode to the final systemThe model solving brings great benefits and convenience.
And solving the final system balance equation to obtain the displacement u-K of the antenna control vertex*)-1F。
In the step (3), the setting of the displacement coefficient matrix is judged according to the coupling relationship between different basic units, specifically:
for the "Beam-Beam" coupling case, C1、C2And respectively representing displacement coefficient matrixes of control vertexes corresponding to end points of the two beam units:
wherein, C1、C2All the matrixes are 3 multiplied by 3N-order matrixes, E is a three-order identity matrix, and I, J respectively represents the numbers of the control vertexes of the two beam units;
for the case of "beam-shell" tangent coupling, i.e. beam element and shell element are tangent, C1，C2And displacement coefficient matrixes respectively representing control vertexes corresponding to the end points of the beam unit and the shell unit:
wherein, C1、C2All the matrixes are 3 multiplied by 3 Nth-order matrixes, E is a third-order identity matrix, and I, J respectively represents the numbers of control vertexes of the beam units and the shell units;
for the case of "beam-shell" cross coupling, i.e. the beam element and the shell element intersect, C1，C2Respectively representing the displacement coefficient matrixes of the control vertexes corresponding to the end points of the beam unit and the shell unit at which the intersection point of the beam unit and the shell unit is located：
Wherein, J1,…,J(p+1)2The number of each control vertex of the shell unit, which represents the intersection point of the beam unit and the shell unit;
representing the basis functions corresponding to the intersections of the shell elements and the beam elements.
In the step (4), the final system balance equation is solved to obtain the displacement u ═ K of the control vertex*)-1F。
The antenna is specifically an active phased array antenna.
The invention fuses the equal-geometry beam-shell units based on different basis functions into the same whole according to the steps (2) and (3).
According to the invention, the load application region is subjected to appropriate numerical integration processing according to the step (4), and the load application of any region of the isogeometric analysis is realized.
The invention has the beneficial effects that:
the invention couples the basic units of different basis functions into a whole, realizes the integral analysis of different structures of the antenna model and realizes the accurate measurement of the integral displacement of the antenna product.
The invention provides a method for accurately constructing geometric models such as antennas with mixed basis functions into a whole, which is used for measuring load displacement of the geometric models such as antennas with mixed basis functions and is characterized in that strong coupling between Euler-Bernoulli (Euler-Bernoulli) beam units in three-dimensional space based on NURBS basis functions is constructed, two coupling conditions between the beam units and Kirchhoff-Love (Kirchoff-Love) shell units based on unstructured T splines are considered, coupling information of the basic units with different basis functions is fused, rigidity matrix construction of the geometric models such as the mixed basis functions is realized, integral analysis of different structures of the antenna models is realized, accurate measurement of integral displacement of antenna products is finally realized, and accuracy is improved.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Fig. 2 is a schematic diagram of the structure of the antenna of the embodiment.
Fig. 3 is a schematic diagram of the antenna and its external load at a 45 degree operating angle.
Fig. 4 is a schematic view of 3 types of antenna mount beam sections.
FIG. 5 is a schematic diagram comparing a geometric model such as a mixed basis function with a conventional finite element analysis model.
Fig. 5(a) is a geometric model diagram such as a mixing basis function of an antenna.
Fig. 5(b) is a finite element analysis model diagram of the antenna.
FIG. 6 is a displacement cloud obtained by geometric analysis such as mixed basis functions.
FIG. 7 is a schematic diagram of a geometric model such as an antenna reflector for electromagnetic analysis.
Fig. 8 is a 2D gain pattern on the phi-0 degree polarization plane.
Detailed Description
The invention is further explained below in connection with an antenna model.
The examples of the invention are as follows:
take the antenna shown in fig. 2 as an example. The distribution of electromagnetic amplitude and phase of the aperture of the antenna can be changed by precision errors generated by the antenna structure under the action of external force such as self weight, wind load and the like, a red dotted line in the figure represents a deformed reflecting surface, and the shape error delta finally influences the far-field electrical property of the antenna.
(1) The truss and the reflecting surface of the antenna structure are respectively constructed by Euler-Bernoulli beam elements based on NURBS and Kirchoff-Love shell elements based on unstructured T-splines, as shown in figure 3, the yellow part is a main reflecting surface, and the blue part is an antenna support. And analyzing the surface error of the reflecting surface antenna under the action of gravity and wind load of 20m/s at a working angle of 45 degrees.
The beam-beam unit coupling method is used for coupling connection between antenna support beam structures, and the beam-shell unit coupling method is used for coupling connection between the antenna support beam structures and antenna reflecting surfaces.
(2) The antenna support beam structure comprises a total of 3 beam cross-sectional forms, each as shown in fig. 4.
(3) The maximum displacement of the reflector antenna under the gravity and wind load and the error of the reflector surface are calculated by adopting a mixed basis function geometric analysis and the like and a traditional finite element analysis method respectively, for example, fig. 5 shows a mixed basis function geometric model and the like and a traditional finite element analysis model respectively.
(4) Taking different mesh thickness degrees, and comparing the convergence condition of the calculation results of the geometric analysis of the mixed basis function and the like and the traditional finite element analysis along with the change of the number of the nodes, as shown in table 1, wherein N represents the number of control vertexes of the geometric analysis and the number of unit nodes of the traditional finite element analysis.
TABLE 1 Convergence analysis of maximum displacement and profile error
In order to achieve the same computational accuracy, geometric analysis of mixed basis functions and the like only uses less than one fifth of the computational scale. Fig. 6 is a displacement cloud diagram obtained by geometric analysis such as mixed basis function.
(5) And (3) taking points which are fine enough on the isogeometric model to enable the minimum distance to meet the requirement of less than 0.3 time of wavelength required by electromagnetic performance analysis, generating a triangular mesh model containing displacement information by using the points as shown in FIG. 7, and analyzing the influence of the deformation error of the reflecting surface on the gain.
(6) The 2D gain directional diagram of the deformed reflecting surface on the phi 0-degree polarization plane is obtained through calculation, as shown in fig. 8, after deformation, the antenna gain is also changed, and a gain diagram obtained through calculation results of geometric analysis models such as a mixed basis function is similar to a gain diagram obtained through a traditional finite element method of a finest grid, so that the problem of error amplification caused by geometric discrete errors is greatly solved.
Claims (4)
1. A load displacement measurement method of an antenna model of a mixed basis function is characterized in that aiming at an isogeometric model of an antenna, a line is divided into a support frame and a reflecting surface, the support frame is formed by connecting a plurality of beams, and the reflecting surface is formed by connecting a plurality of shells; the method comprises the following steps:
(1) constructing a beam unit by using a NURBS basis function aiming at a beam, constructing a shell unit by using an unstructured T spline basis function aiming at a shell, and taking the beam unit and the shell unit as basic units; converting NURBS basis function and unstructured T-spline basis function into linear combination of Bernstein polynomial defined in [ -1,1] space by calculating Bessel extraction operator of basic unit, thus avoiding parameter mapping during Gaussian product calculation;
(2) coupling a beam-beam unit with a beam-shell unit, wherein the beam-beam unit is formed by connecting two beam units, and the beam-shell unit is formed by connecting a beam unit and a shell unit;
the following displacement constraint equations are established for the "beam-to-beam" unit coupling:
u1,1＝un,2
wherein u is1,1And un,2Respectively representing the control vertex P of one of the beam elements1 1And a control vertex P of another beam elementn 2The displacement vector of (2);
for the "beam-shell" unit coupling, the following displacement constraint equations are respectively established for tangency and intersection:
uI＝uJ
wherein u isIAnd uJThe displacement vectors of the control vertices on the beam element and the shell element, respectively, are indicated, the subscripts I and J indicate the number of the control vertices, respectively, p indicates the NURBS order of basis function, ue,jA displacement vector representing a control vertex corresponding to the shell element e, j represents a local number of the control vertex in the shell element e,
representing the basis functions corresponding to the intersections of the shell elements and the beam elements,
is the parameter coordinate of the intersection point of the shell unit and the beam unit;
(3) collating constraint equation information
Arranging the displacement constraint equation into the following form, and then judging and setting a displacement coefficient matrix according to the coupling relation between different basic units to set C1And C2：
(C1-C2)u＝0
Where u represents the displacement vector of all N control vertices of the entire isogeometric model, C1、C2Displacement coefficient matrixes respectively representing control vertexes of the basic units;
(4) a penalty function method is adopted to fuse the displacement constraint equation into a system rigidity matrix K of the whole geometric model; establishing a modified potential energy functional Π taking into account a function with penalties*：
Wherein S is a penalty parameter of a penalty function method, u represents the displacement of a control vertex, and F represents the load applied to the isogeometric model of the antenna;
and (3) solving a minimum value for the potential energy functional, and ordering:
the final system equilibrium equation is found below:
[K+S(C1-C2)T(C1-C2)]u＝F
K*u＝F
K*＝K+S(C1-C2)T(C1-C2)
wherein, K*Expressing a system stiffness matrix K of the isogeometric model after the displacement constraint equation is fused, and expressing matrix transposition by T;
and solving the final system balance equation to obtain the displacement u-K of the antenna control vertex*)-1F。
2. The method of claim 1, wherein the method comprises the following steps: in the step (3), the setting of the displacement coefficient matrix is judged according to the coupling relationship between different basic units, specifically:
for the "Beam-Beam" coupling case, C1、C2And respectively representing displacement coefficient matrixes of control vertexes corresponding to end points of the two beam units:
wherein, C1、C2All the matrixes are 3 multiplied by 3N-order matrixes, E is a three-order identity matrix, and I, J respectively represents the numbers of the control vertexes of the two beam units;
for the case of "beam-shell" tangent coupling, i.e. beam element and shell element are tangent, C1，C2And displacement coefficient matrixes respectively representing control vertexes corresponding to the end points of the beam unit and the shell unit:
wherein, C1、C2All are 3X 3N-order matrixes, E is a three-order identity matrix, I, J respectively represents beam unitsAnd the number of the shell unit control vertex;
for the case of "beam-shell" cross coupling, i.e. the beam element and the shell element intersect, C1，C2Respectively representing the displacement coefficient matrixes of the control vertexes corresponding to the end points of the beam unit and the shell unit, wherein the intersection point of the beam unit and the shell unit is located:
wherein, J1,…,J(p+1)2The number of each control vertex of the shell unit, which represents the intersection point of the beam unit and the shell unit;
representing the basis functions corresponding to the intersections of the shell elements and the beam elements.
3. The method of claim 1, wherein the method comprises the following steps: in the step (4), the final system balance equation is solved to obtain the displacement u ═ K of the control vertex*)-1F。
4. The method of claim 1, wherein the method comprises the following steps: the antenna is specifically an active phased array antenna.
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