CN116244992A - Pipeline system parameterized finite element modeling method based on uncoordinated entity units - Google Patents

Pipeline system parameterized finite element modeling method based on uncoordinated entity units Download PDF

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CN116244992A
CN116244992A CN202310162678.9A CN202310162678A CN116244992A CN 116244992 A CN116244992 A CN 116244992A CN 202310162678 A CN202310162678 A CN 202310162678A CN 116244992 A CN116244992 A CN 116244992A
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node
unit
pipeline
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coordinates
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季文豪
孙伟
王鑫
张秉杰
李晖
汪博
马辉
林君哲
韩清凯
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AECC Shenyang Engine Research Institute
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Abstract

The invention provides a pipeline system parameterized finite element modeling method based on a non-coordination entity unit, which comprises the following steps: acquiring a shape function of a main node and a shape function of an additional node, and eliminating the degree of freedom of the additional node in a static aggregation mode; correcting the coordination entity unit based on the degree of freedom of the additional node, so as to obtain a rigidity matrix and a quality matrix of the non-coordination entity unit; grid division is carried out on the pipeline system, and unit node coordinates of the pipeline straight line segment are obtained; obtaining a bending radius corresponding to a unit node of a bent arc section of the pipeline according to a bending radius and space geometric transformation relation at a bending center line of the pipeline, and further obtaining a unit node coordinate of the bent arc section of the pipeline; and obtaining a rigidity matrix and a quality matrix of the unit according to the unit node coordinates of the straight line section and the curved arc section of the pipeline, and combining to obtain a total rigidity matrix and a quality matrix, thereby completing modeling of a pipeline system. The invention can be suitable for any tubular pipeline system and can carry out rapid parameterized finite element modeling of the pipeline system.

Description

Pipeline system parameterized finite element modeling method based on uncoordinated entity units
Technical Field
The invention relates to the field of pipeline system dynamics modeling, in particular to a pipeline system parameterized finite element modeling method based on non-coordinated entity units.
Background
The pipeline system of the aeroengine is a transport channel of working media such as engine fuel oil, lubricating oil, air and the like, and is an important component of an engine energy supply system, so that the working reliability of the pipeline system directly influences the safety and the service life of the engine. In actual operation, the pipeline system is connected with the casing through the supporting structures such as the clamp and the bracket, so that exciting force generated by the rotor of the aero-engine can be transmitted to the pipeline through the casing, and vibration of the pipeline system is caused. In order to improve the stability and reliability of the pipeline system, the dynamics of the pipeline system is usually analyzed and designed, and the finite element method is always the most main method for modeling the dynamics of the pipeline system due to better applicability.
The finite element method needs to build a finite element model of a pipeline system, and most of the prior documents build the finite element model of the pipeline system based on Hypermesh or ANSYS platforms, so that the method has better effect in the aspect of analyzing single pipelines. However, when the system laying or topology optimization is performed on complex pipelines in the pipeline system of the aeroengine, the finite element model of the pipeline system needs to be updated continuously, at this time, the finite element model of the pipeline system is re-created based on the Hypermesh or ANSYS platform, data transmission and reading between different software need to be realized, and the modeling efficiency is extremely low.
Disclosure of Invention
According to the technical problem that the modeling efficiency of the pipeline system is low when the complex pipeline in the pipeline system of the aeroengine is subjected to system laying or topology optimization, the pipeline system parameterized finite element modeling method based on the uncoordinated entity unit is provided. The invention can be suitable for any tubular pipeline system and can carry out rapid parameterized finite element modeling of the pipeline system.
The invention adopts the following technical means:
a piping system parameterized finite element modeling method based on uncoordinated entity units, comprising:
acquiring a shape function of a main node and a shape function of an additional node, and eliminating the degree of freedom of the additional node in a static aggregation mode;
correcting the coordination entity unit based on the degree of freedom of the additional node, so as to obtain a rigidity matrix and a quality matrix of the non-coordination entity unit;
grid division is carried out on the pipeline system, and unit node coordinates of the pipeline straight line segment are obtained;
obtaining a bending radius corresponding to a node of a bent circular arc section of the pipeline according to the bending radius and the space transformation relation of the bending center line of the pipeline;
acquiring unit node coordinates of a curved arc section of the pipeline;
and acquiring a rigidity matrix and a quality matrix of the pipeline unit according to the unit node coordinates of the pipeline straight line segment and the unit node coordinates of the pipeline curved circular arc segment, and grouping to obtain a total rigidity matrix and a quality matrix, thereby completing pipeline system modeling.
Further, the shape function of the master node is:
Figure BDA0004094737350000021
wherein ,
Figure BDA0004094737350000022
is the coordinate value at parent cell node i, +.>
Figure BDA0004094737350000023
The shape function of the additional node is:
N 9 (ξ)＝1-ξ 2 ，N 10 (η)＝1-η 2 ，N 11 (ζ)＝1-ζ 2
further, the stiffness matrix and mass matrix of the cell are obtained from the following calculations:
Figure BDA0004094737350000024
Figure BDA0004094737350000025
Figure BDA0004094737350000026
Figure BDA0004094737350000027
Figure BDA0004094737350000028
Figure BDA0004094737350000029
wherein ,Ke As a matrix of the stiffness of the cell,
Figure BDA0004094737350000032
the strain matrix is an additional strain matrix, B is a strain matrix corresponding to the main degree of freedom, and D is an elastic matrix; m is M e Is the mass matrix of the unit, ρ is the density, N is the matrix of shape functions, and V is the volume.
Further, grid division is performed on the pipeline system, and unit node coordinates of the pipeline straight line segment are obtained, including:
the 8 main nodes of the corresponding unit are respectively numbered as i, l, k, j, m, p, o and n, wherein i, l, p and m form one sector, j, k, o and n form another sector, for any unit, the node coordinates of the initial sectors can be considered as known, the node coordinates of the i, l, p and m sectors are selected as known, and then the node coordinates of the j, k, o and n can be expressed as
x j ＝x i +S L d x y j ＝y i +S L d y z j ＝z i +S L d z
x n ＝x m +S L d x y n ＝y m +S L d y z n ＝z m +S L d z
x k ＝x l +S L d x y k ＝y l +S L d y z k ＝z l +S L d z
x o ＝x p +S L d x y o ＝y p +S L d y z o ＝z p +S L d z
in the formula ,dx ,d y ,d z The direction vector of the straight line segment and the direction cosine of the x, y and z axes are respectively, and if the key point numbers at the two ends of the straight line segment are g and g+1, the direction vector d of the straight line segment is x ,d y ,d z Can be expressed as
Figure BDA0004094737350000031
In the method, in the process of the invention, |·| represents the 2-norm of the vector.
Further, obtaining a bending radius corresponding to a unit node of a bending arc section of the pipeline based on the bending radius at the bending center line of the pipeline comprises the following steps:
solving an included angle between a reference line of the arc section and a zero line, wherein the reference line of the arc section is a reference line with the bending radius equal to the bending center line;
and acquiring an included angle between the perpendicular bisector and the datum line based on the included angle between the datum line and the zero line of the arc section, and calculating the bending radius of the unit node of the bending arc part based on the included angle between the perpendicular bisector and the datum line.
Further, obtaining the unit node coordinates of the curved arc segment of the pipeline includes:
assuming that the arc section is obtained by bending straight line sections g (g+1) and (g+1) (g+2), taking the nth from the center arc line of the bent arc section as BE j The unit of the circle, taking unit nodes m and n as an example, describes a node coordinate solving method:
the coordinates of the node m are known, and the coordinates of the node n can be obtained by using a vector decomposition method, and specific components
Figure BDA0004094737350000041
Parallel to vector->
Figure BDA0004094737350000042
And component->
Figure BDA0004094737350000043
Then
∠Fmn＝κ j
in the formula ,κj Is beyond the center C i Perpendicular bisector of chord mn and arc section initial edge C i B included angle, expressed as
Figure BDA0004094737350000044
in the formula ,
Figure BDA0004094737350000045
the central angle corresponding to each unit of the bending section is set;
the coordinates of the intermediate conversion point F can be expressed as
Figure BDA0004094737350000046
Figure BDA0004094737350000047
Figure BDA0004094737350000048
The coordinates of the node n can be further obtained as
Figure BDA0004094737350000049
Figure BDA00040947373500000410
Figure BDA00040947373500000411
in the formula ,sC For the unit length of the arc segment,
Figure BDA00040947373500000412
and
Figure BDA00040947373500000413
Vectors respectively->
Figure BDA00040947373500000414
Direction angles relative to the x, y and z axes;
using
Figure BDA00040947373500000415
Direction angle of (2) instead of->
Figure BDA00040947373500000416
Wherein the coordinates of point B are according to the direction vector +.>
Figure BDA00040947373500000417
Obtained, C i The coordinates of the points are three planes (plane of the bending arc, the vector +.>
Figure BDA00040947373500000418
Plane as normal vector and passing E point by vector +.>
Figure BDA00040947373500000419
A plane that is a normal vector).
Further, the overall stiffness matrix and mass matrix are obtained from the following calculations:
Figure BDA0004094737350000051
Figure BDA0004094737350000052
wherein G is a conversion matrix of unit node freedom degree,
Figure BDA0004094737350000053
and
Figure BDA0004094737350000054
Respectively the v th i Stiffness matrix and mass matrix of individual entity units, n f Is the total number of units.
Compared with the prior art, the invention has the following advantages:
the finite element modeling method suitable for the pipeline system of any tubular shape provided by the invention has the advantages that grid division based on the finite element method has no grid dislocation and discontinuity phenomenon, and meanwhile, the model updating speed is higher, and the finite element model can be quickly created only by inputting tubular parameters. Therefore, the network separation time for realizing the pipeline system based on the Hypermesh or ANSYS platform is reduced, and a basic model is provided for the topology optimization of the dynamic pipeline system.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, and it is obvious that the drawings in the following description are some embodiments of the present invention, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a parameterized finite element modeling method for a pipeline system based on non-coordinated entity units.
Fig. 2 is a schematic diagram of a space 8-node uncoordinated entity unit according to an embodiment of the present invention, (a) is a straight unit, and (b) is a curved unit.
Fig. 3 is a schematic structural diagram of a pipeline system according to an embodiment of the present invention.
FIG. 4 is a schematic diagram of solving coordinates of a node of a straight line segment unit according to an embodiment of the present invention;
fig. 5 is a schematic diagram of solving a bending radius of a bending arc section unit node according to an embodiment of the present invention, (a) is an external schematic diagram, and (b) is a cross-sectional schematic diagram.
Fig. 6 is a schematic diagram of node coordinate solving of a curved arc segment unit according to an embodiment of the present invention.
Detailed Description
In order that those skilled in the art will better understand the present invention, a technical solution in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in which it is apparent that the described embodiments are only some embodiments of the present invention, not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the present invention without making any inventive effort, shall fall within the scope of the present invention.
As shown in fig. 1, the present invention provides a method for modeling a parameterized finite element of a pipeline system based on a non-coordinating entity unit, comprising:
s1, obtaining a shape function of a main node and a shape function of an additional node, and obtaining the degree of freedom of the additional node through a static aggregation mode.
S2, correcting the coordination entity unit based on the degree of freedom of the additional node, so that a rigidity matrix and a quality matrix of the unit are obtained.
The S1-S2 is mainly used for obtaining the rigidity matrix and the quality matrix of the space 8-node uncoordinated entity unit. Specifically:
in this embodiment, the space 8 node non-coordination entity unit is obtained by introducing 9 additional degrees of freedom based on a conventional space 8 node coordination entity unit. The static aggregation can be eliminated during the calculation of the rigidity matrix, so that the shearing locking effect of the coordination unit can be avoided, and the overall degree of freedom of the system is not increased.
The displacement of any point within a cell can be expressed as:
Figure BDA0004094737350000061
wherein u, v, w are displacement coordinate values of any point in the unit, N 1 ,···,N 11 Are all form functions, alpha 1 ，…，α 9 For the additional degree of freedom introduced;
Figure BDA0004094737350000062
is->
Figure BDA0004094737350000063
Displacement of three degrees of freedom of a single node.
The specific expression of the above-mentioned various shape functions is:
Figure BDA0004094737350000071
in the formula ,
Figure BDA0004094737350000072
form function as master node, ++>
Figure BDA0004094737350000073
As a function of the shape of the additional node.
Figure BDA0004094737350000074
Is->
Figure BDA0004094737350000075
Respectively +.>
Figure BDA0004094737350000076
The coordinate values of the individual nodes in the local coordinate system ζηζ are determined from the graph of the isoparametric elements as shown in fig. 2.
Further obtaining the shape function matrix of the degree of freedom of the main node as
N＝[N 1 I 3×3 N 2 I 3×3 N 3 I 3×3 N 4 I 3×3 N 5 I 3×3 N 6 I 3×3 N 7 I 3×3 N 8 I 3×3 ] (3)
The shape function matrix with additional degrees of freedom is
Figure BDA0004094737350000077
I.e. the displacement interpolation function can be expressed as
Figure BDA0004094737350000078
wherein ,qe As a main degree of freedom vector alpha e Is an additional degree of freedom vector.
The cell strain is
Figure BDA0004094737350000079
in the formula Be Is a main strain matrix, which can be expressed as
Figure BDA00040947373500000713
in the formula
Figure BDA00040947373500000710
Can be expressed as
Figure BDA00040947373500000711
Additional strain matrix
Figure BDA00040947373500000712
Can be expressed as +.>
Figure BDA0004094737350000081
The node displacement and strain are brought into the potential functional and the differential of the functional is made to be 0, so that the following can be obtained:
Figure BDA0004094737350000082
in the formula ,
Figure BDA0004094737350000083
and
Figure BDA0004094737350000084
Can be respectively expressed as
Figure BDA0004094737350000085
Wherein D is an elastic matrix.
From equation 2 of equation (10), an additional degree of freedom can be obtained
Figure BDA0004094737350000086
Further from the first equation of equation (10), the stiffness matrix of the uncoordinated entity unit can be obtained as
Figure BDA0004094737350000087
The quality matrix of the uncoordinated entity unit is
Figure BDA0004094737350000088
Where ρ is the density of the tube, N is the shape function, and V is the volume.
The use of jacobian matrix J in solving the strain matrix of a subunit, can be expressed in particular as
Figure BDA0004094737350000091
The Jacobian matrix J can be expressed as
Figure BDA0004094737350000092
It is not difficult to find that the jacobian matrix can be obtained only by knowing the node coordinates of the actual units. The Jacobian matrix is mainly used for solving a strain matrix and further solving a stiffness matrix and a quality matrix.
S3, grid division is carried out on the pipeline system, and unit node coordinates of the pipeline straight line segment are obtained.
Taking the pipeline shown in fig. 3 as an example, the pipeline is subjected to grid division, and a network division rule is specified as follows: the circumferential direction is ranked counterclockwise from inside to outside, and the result is shown in fig. 4. 8 master nodes of corresponding units are numbered i, l, k, j, m, p, o, n, wherein i, l, p, m, and j, k, o, n respectively form a sector, the initial sector node coordinates of any unit can be considered as known, the node coordinates of i, l, p, m sectors are selected to be known, and the node coordinates of j, k, o, n can be expressed as
Figure BDA0004094737350000101
in the formula ,dx ,d y ,d z The direction vector of the straight line segment and the direction cosine of the x, y and z axes are respectively. If the key point numbers at the two ends of the straight line segment are g and (g+1), the direction vector d of the straight line segment x ,d y ,d z Can be expressed as
Figure BDA0004094737350000102
In the method, in the process of the invention, |·| represents the 2-norm of the vector.
S4, obtaining the bending radius of the bending center line of the pipeline, and obtaining the bending radius corresponding to the unit node of the bending arc section of the pipeline based on the bending radius of the bending center line of the pipeline.
For any space pipeline, the bending radius of the pipeline at the bending center line can be extracted, but the bending radius corresponding to the unit node of the bending circular arc section is not equal to the bending radius of the bending center line, so that the bending radius of the unit node of the bending circular arc section of the pipeline needs to be solved.
Here with the bent arc section C of fig. 3 1 To illustrate the method of solving the bending radius, a generalized expression is then given.
Analysis of the bent arc segment C of FIG. 5 1 It is easy to find out that the arc section C 1 Lying in a plane formed by straight segments 1-2 and 3-4 which can be seen as a clockwise rotation by an angle α from the plane xoz, the corresponding bend radius of the bent arc segment cell nodes can be seen in fig. 5.
It can be seen from fig. 5 that, because of the existence of the tubular space transformation angle, the zero line of the bending radius deviates from the datum line of the pipe section by an angle alpha, and the bending radius of the nodes i, l, k, j, m, p, o, n can be expressed as
Figure BDA0004094737350000103
in the formula θb The included angle between the perpendicular bisector and the datum line can be expressed as
Figure BDA0004094737350000111
Phi and lambda are the position angles of the cell nodes i and m, respectively, and can be expressed as
φ＝n e ψ (21)
λ＝(n e +1)ψ (22)
Wherein psi is the central angle corresponding to each unit, n e Numbering the units relative to the reference line.
The formula of the formula (19) is generalized to solve the bending radius of the unit node of any arc section, and the bending arc of the C section is arbitrarily taken, so that the included angle between the datum line and the zero line of the arc section is
Figure BDA00040947373500001111
Is that
Figure BDA0004094737350000112
in the formula ,
Figure BDA0004094737350000113
is formed by straight line segment (n c -1)n c And straight line segment n c (n c +1) plane A c And straight line segment n c (n c+1) and (nc +1)(n c +2) plane A c+1 Can be defined by the normal vector v of two planes p1 and vp2 Obtained has a size of
Figure BDA0004094737350000114
It should be noted that the number of the components,
Figure BDA0004094737350000115
not only the size but also the direction, specifically defined as the plane A as seen from the pipeline starting end along the pipeline direction c Clockwise rotation results in plane A c+1 Then->
Figure BDA0004094737350000116
Is positive; conversely, if plane A c Anticlockwise rotation to obtain plane A c+1 Then->
Figure BDA0004094737350000117
Is negative.
And solving the bending radius of the bending arc part unit node. The bending radius of the unit node of the bending arc part can be expressed as
Figure BDA0004094737350000118
in the formula ,θ'b The included angle between the perpendicular bisector and the datum line can be expressed as
Figure BDA0004094737350000119
It should be noted that the number of the substrates,
Figure BDA00040947373500001110
plane xoz and plane A 2 An included angle between the two. In addition, for any pipeline, "±" in the table means that the bending directions are different, specifically, when the included angle between the normal vector of the plane where the arc is located and the y-axis direction of the overall coordinate system is smaller than 90 degrees, "+", otherwise "-".
S5, obtaining the unit node coordinates of the curved arc section of the pipeline.
Taking an arbitrary arc section as shown in fig. 6 as an example, it is assumed that the arc section is obtained by bending straight line sections g (g+1) and (g+1) (g+2), and the central arc of the arc section after bending is BE. From any one of the nth j The unit of the circle, the unit nodes m and n are taken as an illustration of a method for solving the node coordinates. The coordinates of the node m are known, and the coordinates of the node n can be obtained by using a vector decomposition method, and specific components
Figure BDA0004094737350000121
Parallel to vector->
Figure BDA0004094737350000122
And component->
Figure BDA0004094737350000123
Then
∠Fmn＝κ j (27)
in the formula ,κj Is beyond the center C i Perpendicular bisector of chord mn and arc section initial edge C i B, can be expressed as
Figure BDA0004094737350000124
in the formula ,
Figure BDA0004094737350000125
and (5) corresponding central angles of each unit of the bending section.
The coordinates of the intermediate conversion point F can be expressed as
Figure BDA0004094737350000126
The coordinates of the node n can be further obtained as
Figure BDA0004094737350000127
in the formula ,sC Is the unit length of the arc section;
Figure BDA0004094737350000128
and
Figure BDA0004094737350000129
Vectors respectively->
Figure BDA00040947373500001210
Direction angles relative to the x, y and z axes. Because of->
Figure BDA00040947373500001211
Parallel to->
Figure BDA00040947373500001212
Thus can use +.>
Figure BDA00040947373500001213
Direction angle of (2) instead of->
Figure BDA00040947373500001214
Is a direction angle of (a). Wherein the coordinates of point B can be determined from the direction vector +.>
Figure BDA00040947373500001215
Obtained, C i The coordinates of the points can be three planes (the plane in which the bending arc is located)Point B is crossed by vector->
Figure BDA00040947373500001216
Plane as normal vector and passing E point by vector +.>
Figure BDA00040947373500001217
A plane that is a normal vector).
S6, acquiring a rigidity matrix and a quality matrix of the pipeline unit according to the unit node coordinates of the pipeline straight line segment and the unit node coordinates of the pipeline curved circular arc segment, and combining to obtain a total rigidity matrix and a quality matrix, thereby completing modeling of a pipeline system.
Specifically, the overall stiffness matrix and mass matrix are:
Figure BDA0004094737350000131
wherein G is a conversion matrix of unit node freedom degree,
Figure BDA0004094737350000132
and
Figure BDA0004094737350000133
Respectively the v th i Stiffness matrix and mass matrix of individual entity units, n f Is the total number of units.
In the topological optimization process of the pipeline system, when the pipeline system is changed in shape, the steps S3-S6 are re-executed to obtain the rigidity matrix and the quality matrix of any single pipeline system.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention, and not for limiting the same; although the invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical scheme described in the foregoing embodiments can be modified or some or all of the technical features thereof can be replaced by equivalents; such modifications and substitutions do not depart from the spirit of the invention.

Claims (7)

1. A method for parameterized finite element modeling of a pipeline system based on uncoordinated entity units, comprising:
acquiring a shape function of a main node and a shape function of an additional node, and eliminating the degree of freedom of the additional node in a static aggregation mode;
correcting the coordination entity unit based on the degree of freedom of the additional node, so as to obtain a rigidity matrix and a quality matrix of the non-coordination entity unit;
grid division is carried out on the pipeline system, and unit node coordinates of the pipeline straight line segment are obtained;
obtaining a bending radius corresponding to a node of a bent circular arc section of the pipeline according to the bending radius and the space transformation relation of the bending center line of the pipeline;
acquiring unit node coordinates of a curved arc section of the pipeline;
and obtaining the rigidity matrix and the quality matrix of the unit according to the unit node coordinates of the straight line section of the pipeline and the unit node coordinates of the curved circular arc section of the pipeline, and combining to obtain the overall rigidity matrix and the quality matrix, thereby completing the modeling of the pipeline system.
2. The method for modeling a parameterized finite element of a pipeline system based on non-coordinated entity units according to claim 1, wherein the shape function of the master node is:
Figure FDA0004094737340000011
wherein ,
Figure FDA0004094737340000012
is parent unit node +>
Figure FDA0004094737340000013
Coordinate value of (I/O)>
Figure FDA0004094737340000014
The shape function of the additional node is:
N 9 (ξ)＝1-ξ 2 ，N 10 (η)＝1-η 2 ，N 11 (ζ)＝1-ζ 2
3. the method for modeling a parameterized finite element of a piping system based on uncoordinated solid units of claim 1, wherein the stiffness matrix and the mass matrix of the unit are obtained according to the following calculation:
Figure FDA0004094737340000015
Figure FDA0004094737340000021
Figure FDA0004094737340000022
Figure FDA0004094737340000023
Figure FDA0004094737340000024
Figure FDA0004094737340000025
wherein ,Ke The stiffness matrix of the unit is represented by an additional strain matrix, B is represented by a strain matrix corresponding to a main degree of freedom, and D is represented by an elastic matrix; m is M e Is the mass of the unitThe quantity matrix, ρ is the density, N is the shape function matrix, and V is the volume.
4. The method for modeling parameterized finite elements of a pipeline system based on uncoordinated entity units of claim 1, wherein the step of meshing the pipeline system to obtain unit node coordinates of straight line segments of the pipeline comprises the steps of:
the 8 master nodes of the corresponding unit are respectively numbered as i, l, k, j, m, p, o and n, wherein i, l, p and m form one sector, j, k, o and n form another sector, for any unit, the node coordinates of the initial sectors can be considered as known, the node coordinates of the i, l, p and m sectors are selected as known, and then the node coordinates of the j, k, o and n can be expressed as
x j ＝x i +S L d x y j ＝y i +S L d y z j ＝z i +S L d z
x n ＝x m +S L d x y n ＝y m +S L d y z n ＝z m +S L d z
x k ＝x l +S L d x y k ＝y l +S L d y z k ＝z l +S L d z
x o ＝x p +S L d x y o ＝y p +S L d y z o ＝z p +S L d z
in the formula ,dx ,d y ,d z The direction vector of the straight line segment and the direction cosine of the x, y and z axes are respectively, and if the key point numbers at the two ends of the straight line segment are g and g+1, the direction vector d of the straight line segment is x ,d y ,d z Can be expressed as
Figure FDA0004094737340000026
In the method, in the process of the invention, |·| represents the 2-norm of the vector.
5. The method for modeling a parameterized finite element of a pipeline system based on a non-coordinated entity unit according to claim 1, wherein the step of obtaining the bending radius corresponding to the unit node of the bent arc segment of the pipeline based on the bending radius at the bending center line of the pipeline comprises the steps of:
solving an included angle between a reference line of the arc section and a zero line, wherein the reference line of the arc section is a reference line with the bending radius equal to the bending center line;
and acquiring an included angle between the perpendicular bisector and the datum line based on the included angle between the datum line and the zero line of the arc section, and calculating the bending radius of the unit node of the bending arc part based on the included angle between the perpendicular bisector and the datum line.
6. The method for modeling a parameterized finite element of a pipeline system based on uncoordinated solid units of claim 1, wherein obtaining the coordinates of the unit nodes of the curved arc segment of the pipeline comprises:
assuming that the arc section is obtained by bending straight line sections g (g+1) and (g+1) (g+2), taking the nth from the center arc line of the bent arc section as BE j The unit of the circle, taking unit nodes m and n as an example, describes a node coordinate solving method:
the coordinates of the node m are known, and the coordinates of the node n can be obtained by using a vector decomposition method, and specific components
Figure FDA0004094737340000031
Parallel to vector->
Figure FDA0004094737340000032
And component->
Figure FDA0004094737340000033
Then
∠Fmn＝κ j
in the formula ,κj Is beyond the center C i Perpendicular bisector of chord mn and arc section initial edge C i B included angle, expressed as
Figure FDA0004094737340000034
in the formula ,
Figure FDA0004094737340000035
the central angle corresponding to each unit of the bending section is set;
the coordinates of the intermediate conversion point F can be expressed as
Figure FDA0004094737340000036
Figure FDA0004094737340000037
Figure FDA0004094737340000038
The coordinates of the node n can be further obtained as
Figure FDA0004094737340000039
Figure FDA00040947373400000310
Figure FDA00040947373400000311
in the formula ,sC For the unit length of the arc segment,
Figure FDA0004094737340000041
and
Figure FDA0004094737340000042
Vectors respectively->
Figure FDA0004094737340000043
Direction angles relative to the x, y and z axes;
using
Figure FDA0004094737340000044
Direction angle of (2) instead of->
Figure FDA0004094737340000045
Wherein the point B is a point generated by bending the straight line segment g (g+1) and the coordinates thereof are +_according to the direction vector>
Figure FDA0004094737340000046
Obtained, C i The coordinates of the points are obtained by using the intersection points of three planes which are the planes where the bending circular arcs are respectively, and the points B are crossed by vectors +.>
Figure FDA0004094737340000047
Plane as normal vector and passing E point by vector +.>
Figure FDA0004094737340000048
Is the plane of the normal vector.
7. The method for modeling a parameterized finite element of a piping system based on uncoordinated solid units of claim 1, wherein the overall stiffness matrix and mass matrix are obtained according to the following calculations:
Figure FDA0004094737340000049
Figure FDA00040947373400000410
wherein G is a conversion matrix of unit node freedom degree,
Figure FDA00040947373400000411
and
Figure FDA00040947373400000412
Respectively the v th i Stiffness matrix and mass matrix of individual entity units, n f Is the total number of units. />
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