CN117010097A - Integral design method of tubular member - Google Patents
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Abstract
The invention provides an overall design method of a tubular member, wherein the method comprises the steps of constructing a tubular curved surface by using a B-spline curved surface based on a base function defined by modification, and avoiding a mode of splicing the curved surfaces; and a NURBS curve based on a modified defined basis function is used for rounding, a constraint condition of a linear equation set is applied to the head-tail part interval, and the mode of splicing the curve is avoided, so that the whole curve has good smoothness, the program implementation is simplified, and the calculation requirement is reduced. The invention can generate the initial tubular curved surface by only providing node vectors and times of the basis functions in two directions respectively, and modifies the part of the curved surface by adjusting the control point grids in the Euclidean space so as to realize the overall design of the tubular curved surface.
Description
Technical Field
The invention belongs to the field of geometric design, and particularly relates to an overall design method of a tubular member.
Background
There is a great need in many engineering fields for a tubular-like member design, in particular for the manufacture of aircraft engines, wherein a rather critical inlet duct is a typical tubular-like member. The air inlet channel plays a vital role in ensuring that an aircraft engine provides enough power, runs safely and normally for a long time, meets special performance requirements (stealth, supersonic speed and the like) of the aircraft, and often presents complex and changeable shapes. In the engineering design field, conic sections such as circles are important and are common. For example, a kerf at one end of an air inlet pipe of an airplane is circular, which has high requirements on the expression capability and the design accuracy of a design method.
In modern times, traditional geometric designs have been replaced with computer and aided design software. In the current Computer Aided Design (CAD) software, when representing a tubular-like member, a standard B-spline surface or NURBS surface is generally based, and a plurality of surfaces are spliced to form the tubular-like surface, which has the following problems:
1. the tubular-like members cannot be integrally represented, a splicing mode is needed, and the design is inconvenient, so that the design efficiency is reduced.
2. The overall smoothness cannot be ensured, and particularly at the joint of curved surfaces, additional constraint on the control point network is usually required to meet the smoothness requirement of the curved surfaces. Not only extra work is brought, but also the design accuracy is affected.
The existence of the problems also has negative effects on the subsequent numerical simulation of the tubular-like member and the planning of the processing path. Thus the methods in current computer aided design software are not ideal for geometric designs of tubular-like members.
Disclosure of Invention
The invention aims to provide an overall design technology of tubular components, which aims to solve the problems that the bottom layer representation based on the existing computer aided design software cannot carry out overall representation on the tubular components and cannot ensure the overall smoothness.
The technical solution for realizing the purpose of the invention is as follows:
a method of unitary design of a tubular member, comprising the steps of:
step 1, generating closed curves of inlet and outlet sections of tubular members: including non-circular closed curves or circular closed curves;
input: the times k of the generated closed curve and the node vector U; and (3) outputting: control point coordinates of the closed curve;
calculating expressions of the 0, …, k basis functions and the n-k-1, …, n-1 basis functions, respectively, and representing them as column vectors;
A. generating a non-circular closed curve by using a B spline:
transversely splicing column vectors of all the basis function expressions to serve as coefficient matrixes of homogeneous linear equation sets; solving the linear equation set to obtain a general solution; when the coordinates of a certain control point are adjusted, the undetermined coefficients in the general solution are changed according to the change quantity of the coordinates, so that the updated coordinates of other control points are calculated, and the cooperative control of the control points is completed; knowing the coordinates of the control points, and calculating a non-circular closed curve according to the definition of the B spline;
B. generating circular closed curve using NURBS curve
Transversely splicing column vectors of n-k-1, … and n-1 basis function expressions as coefficient matrixes of non-homogeneous linear equation sets, wherein each basis function corresponds to a weight factor; multiplying the column vector of each basis function expression by the corresponding weight to obtain a new set of vectors, transversely splicing the set of vectors into a coefficient matrix which is a full order matrix, moving the known quantity in the equation to the right as a constant term of a non-homogeneous linear equation set, solving the linear equation set to obtain a unique set of weight omega i I=n-k-1, …, n-1, the coordinates of the control point can be uniquely solved; knowing the coordinates of the control points, and calculating a circular closed curve according to the definition of NURBS splines;
step 2, generating a tubular curved surface:
input: number of times k in S direction 2 Sum node vector U 2 The method comprises the steps of carrying out a first treatment on the surface of the And (3) outputting: points on the curved surface;
firstly, converting the control point coordinates in the step 1 into three-dimensional coordinates and fixing the three-dimensional coordinates; for each row of control points divided by the inlet section and the outlet section in the T direction, calculating expressions of 0 th, … th, k th and n-k-1, … th, n-1 th basis functions respectively, representing the expressions as column vectors, and transversely splicing the column vectors into a matrix, thereby obtaining coefficient matrices of a plurality of homogeneous linear equation sets; respectively solving each homogeneous linear equation set to obtain a plurality of general solutions; when the coordinates of a certain control point in a certain column in the T direction are adjusted, according to the change quantity of the coordinates, the undetermined coefficients in the complete solution of the homogeneous linear equation set in the column are changed, so that the updated coordinates of other control points in the same column are calculated, and the cooperative control of the control points is completed; knowing the coordinates of the control points, calculating a tubular curved surface according to the definition of the curved surface, and then taking the points on the curved surface according to any precision and outputting the points.
Compared with the prior art, the invention has the remarkable advantages that:
(1) According to the method, when the curve is rounded, constraint conditions of the equation set are applied to the head-tail part intervals, a splicing mode is not needed, so that the overall smoothness of the curve is good, and the circle can be truly represented;
(2) The constraint conditions in the method are linear equation sets, so that the realization of a program is facilitated, the calculation requirement of a computer is reduced, and the method has high efficiency.
Drawings
FIG. 1 is a flow chart of the present invention for generating a tubular curved surface.
FIG. 2 is a graph of the results of the present invention for generating a B-spline closed curve (as an inlet or outlet cross-section).
FIG. 3 is a graph of the results of the present invention in creating an intermediate curved surface between inlet and outlet sections.
FIG. 4 is a graph of the results of rounding NURBS curves of the present invention.
Detailed Description
The invention is further described with reference to the drawings and specific embodiments.
With reference to fig. 1-4, the present embodiment provides, in part, a method for integrally designing a tubular member, including:
step 1, generating a closed curve (inlet section or outlet section of tubular member) by using B-spline or NURBS spline
The closed curve contains two cases: A. a non-circular closed curve, B, a circular closed curve;
A. generating non-circular closed curves using B-splines
A1, supposing spline curves of k times and n control points, the node vector is
U＝[u 0 ,…,u k ,…,u e ,…,u n ,…,u n+k ]
Wherein u is 0 ＝…＝u k ＝0，u n ＝…＝u n+k ＝1，u e Is the e-th node.
When k is<e<n is 0<u e <1；
Involves [ u ] k ,u k+1 ) The basis function of the interval is N i,k (u) (i=0, …, k), related to [ u ] n-1 ,u n ) The basis function of the interval is N i,k (u)(i＝n-k-1,…,n-1)
Wherein N is i,k (u) is the ith kth basis function.
The B spline basis function is:
wherein N is i,p (u) is the ith p-th basis function.
The B spline curve is:
wherein P is i Is the coordinates (including the abscissa and the ordinate) of the ith control point.
A2, pair N i,k (u) (i=n-k-1, …, n-1) performs the following coordinate transformation
(1) When u is k+1 -u k ＝u n -u n-1 At this time, let u' =u+u n-1 -u k (u∈[u k ,u k+1 ))
(2) When u is k+1 -u k ≠u n -u n-1 In this case, the stretching transformation is performed to make
Wherein x is i And y i Respectively representing the abscissa and ordinate of the ith control point of the curve.
A.3 due to the interval [ u ] i ,u i+1 ]On the basis function N i,k (u) can be expressed as
That is, the basis function is represented by a column vector of k+1 dimensions:
wherein a is i,k The coefficient of the kth term of the ith basis function.
Taking the abscissa of any control point as an example, the constraint equation can be described as
This is a homogeneous linear system of equations and has an infinite number of solutions, the general solution of which is in the form of
Wherein the k+1 z-dimensional vectors on the right side of the equal sign are a basis solution of the homogeneous linear equation set, satisfying z=2 (k+1), i.e. the dimensions of all vectors on both sides of the equal sign are the same.
CP 0 ,…,CP k The total k+1 coefficients are undetermined coefficients, referred to herein as control parameters, and by adjusting these parameters, different solutions of the system of equations can be derived from the underlying solution system.
If CP is selected j (0.ltoreq.j.ltoreq.k) is used as the current control parameter, and the user tries to move the h (0.ltoreq.h.ltoreq.k or n-k-1.ltoreq.h.ltoreq.n-1) control point so that the abscissa thereof is x from the original one h Becomes x h ' i.e. Δx=x h ′-x h Then the control parameter CP is updated j Is CP j ', satisfy
Wherein c j,h Is the value in the underlying solution.
I.e.
The abscissa of all control points should be updated as follows:
regarding the ordinate of the control point, a constraint equation identical in form to the abscissa can be obtained, and the same procedure can be performed to cooperatively adjust the ordinate of all the constrained control points.
The coordinates of the control points are known, and a non-circular closed curve can be calculated according to the definition of the B-spline.
Fig. 2 shows a 3-degree B-spline closed curve containing 9 control points.
B. Generating circular closed curve using NURBS curve
B.1, firstly defining a group of rational B-splines of k times and n control points, defining node vectors
U＝[u 0 ,…,u k ,…,u e ,…u n-k-1 ,…,u n-1 ,u n ,…,u n+k ]
Wherein u is 0 ＝…＝u k ＝0，u n-k-1 ＝…＝u n-1 ＝1，u n ＝…＝u n+k 。
u e Is the e-th node. When k is<e<n-k-1, 0<u e <1；
The coordinate axes of the three-dimensional space are denoted by X, Y, ω. Three-dimensional homogeneous coordinate representation weighted control point
D i ＝[ω i x i ，ω i y i ，ω i ]When u is E [ u ] k ,u k+1 ) And u E [ u ] n-1 ,u n ) In the above, when the curve is projected onto the plane where ω=1, two curves where u is located in the two sections overlap each other. Knowing the curve on u.epsilon.0, 1), the control point and the corresponding weight, defining the curve
Wherein u is a curve parameter, N i,k (u) is the ith kth B-spline basis function, d i For the ith circle control point, x i And y i Respectively the abscissa, omega of the control point i Is the weight factor omega of the ith control point 0 ＝1，ω n-1 ＝1， θ is the central angle corresponding to the curve.
At this time, the liquid crystal display device,the defined curve has been closed end to form a circle.
B.2, in order to make the curve defined by the formula (8) still circular, writing the curve into the following form:
and the local support of the spline is obtained:
at this time, in order to ensure better smoothness of the circular curve, only the second half spline curve is required
And the two parts are overlapped with the defined circle. Condition transformation into u is located at [ u ] k ,u k+1 ]And [ u ] n-1 ,u n ]When the values of the denominators of the curves are consistent in the interval, the molecular parts are only required to be equal.
B.3, involve [ u ] k ,u k+1 ]The basis function of the interval is N 0,k (u),…,N i,k (u),…N k,k (u) relates to [ u ] n-1 ,u n ) The basis function of the interval is N n-k-1,k (u)、…、N n-1,k (u)。
A set of constrained linear equations is listed:
wherein 1 in (u+1) is u 0 To u n-1 Of (1) =u, i.e. n-1 -u 0 ＝1-0。
The corresponding coefficient satisfying the condition that the homogeneous term is 0, and the equation (14) is examined,
since the curve on u e [0, 1) is known, the control point coordinates and the corresponding weights, the set of equations is derived:
wherein a is 0,0 ,a 0,1 ,…,a 0,k Reference to A.3, b 0 ,…,b k A constant that is combined to a known quantity of the equation.
From the properties of B-spline curves, matrixIs a full order matrix, so omega can be uniquely solved i (i＝n-k-1,…,n-1)。
Omega to be solved i (i=n-k-1, …, n-1) is substituted into equations (12), (13), and the same results are obtained with respect to x i ，y i (i=n-k-1, …, n-1).
At this time, the expression (12) becomes:
wherein c 0 ,…,c k A constant that is combined to a known quantity of the equation.
Can solve x i (i＝n-k-1,…,n-1)。
(13) The formula becomes:
wherein d is 0 ,…,d k A constant that is combined to a known quantity of the equation.
Can solve y i (i＝n-k-1,…,n-1)。
Knowing the coordinates of the control points, a complete circular closed curve can be calculated from p (u) defined by equation (8).
Step 2, generating a B-spline tubular surface (generating an intermediate surface on the basis of the step 1)
2.1 times k of basis function in given T direction 1 Node vector
And the number k of basis functions in the S direction 2 Node vector
In total, 4 parameters are used in the node vector, n and m are the number of control points in the T direction and the S direction respectively. The T-direction and S-direction here refer to the two directions of the control point grid, respectively.
Note that: node vector U 1 Sum of times k 1 Should be consistent with the node vector U and the number k used in step 1.
2.2 for any control point (x i ,y i ) Convert it into (x) i ,y i 0) for any control point (x) on the outlet cross section generated in step 1 i ,y i ) Convert it into (x) i ,y i 0), wherein l is the distance of the inlet section from the outlet section.
2.3 for a given node vector U 1 And U 2 Taking the parameter u E [ u ] 0 ,u n+k ),v∈[u′ 0 ,u′ m+k ) Then the B-spline surface is defined as
Wherein P is a point on the B-spline surface corresponding to the parameter (u, v); p (P) ij Is the coordinates of the control points on the control point grid with index i in the T direction and index j in the S direction;represents the ith k in the T direction 1 The secondary basis function (i)＝0,1,...,n-1)；/>Represents the jth k in the S direction 2 Secondary basis function (j=0, 1,) m-1.
2.4 generating a tubular surface, applying a constraint of a linear equation set to the B-spline surface generated above in the T direction, letting
Wherein if itThen->
Otherwise
Thus there is
The constraint equation is relatively complex. To simplify the constraint equation, the constraints are enforced so that
Thereby converting the constraint equation into the condition of generating the B spline closed curve in the step 1, namely for the j-th row of control points (the number of the control points is n) in the direction of the control point grid T, the constraint equation is that
The adjustment of the control point grid can be realized by following the mode completely consistent with the step A.3, thereby changing the shape of the tubular curved surface.
After the control point grid is adjusted, the points on the tubular curved surface are calculated and output with arbitrary precision according to the formula (18).
The resulting curved surface effect is shown in fig. 3.
The design of the tubular curved surface is finished, and the corresponding tubular member is produced according to actual needs.
Step 3, verifying the micromanipulation of the round-like B spline curve
Only the microminiaturization of the basis functions needs to be examined.
Let C (U) be defined as the node vector u= [0 ], k ,u k+1 ,...,1]a B-spline curve of n control points,
where there are n+k+1 nodes in the node vector U.
Order the
And deriving the data to obtain:
in B-spline definition, the type 0/0 definition value is 0, thereby
In the above
In the defined spline, the node weight is defined as 1, whereby the spline number is subtracted by 1 once each time the spline is derived. Until the derivation is to the primary spline, no derivation can be performed.
Examining the spline curve, which consists of more than 4 times of splines and has a node weight of 1, at least 3 times of conduction is performed.
Claims (6)
1. A method of integrally designing a tubular member, comprising the steps of:
step 1, generating closed curves of inlet and outlet sections of tubular members: including non-circular closed curves or circular closed curves;
input: the times k of the generated closed curve and the node vector U; and (3) outputting: control point coordinates of the closed curve;
calculating expressions of the 0, …, k basis functions and the n-k-1, …, n-1 basis functions, respectively, and representing them as column vectors;
A. generating a non-circular closed curve by using a B spline:
transversely splicing column vectors of all the basis function expressions to serve as coefficient matrixes of homogeneous linear equation sets; solving the linear equation set to obtain a general solution; when the coordinates of a certain control point are adjusted, the undetermined coefficients in the general solution are changed according to the change quantity of the coordinates, so that the updated coordinates of other control points are calculated, and the cooperative control of the control points is completed; knowing the coordinates of the control points, and calculating a non-circular closed curve according to the definition of the B spline;
B. generating circular closed curve using NURBS curve
Transversely splicing column vectors of n-k-1, … and n-1 basis function expressions as coefficient matrixes of non-homogeneous linear equation sets, wherein each basis function corresponds to a weight factor; multiplying the column vectors of each basis function expression by the corresponding weight to obtain a new set of vectors, transversely stitching the set of vectors into a coefficient matrix, the matrix being a full order matrix,moving the known quantity in the equation to the right as a constant term of a non-homogeneous linear equation set, solving the linear equation set to obtain a set of unique weight omega i I=n-k-1, …, n-1, uniquely solving for the coordinates of the control point; knowing the coordinates of the control points, and calculating a circular closed curve according to the definition of NURBS splines;
step 2, generating a tubular curved surface:
input: number of times k in S direction 2 Sum node vector U 2 The method comprises the steps of carrying out a first treatment on the surface of the And (3) outputting: points on the curved surface;
firstly, converting the control point coordinates in the step 1 into three-dimensional coordinates and fixing the three-dimensional coordinates; for each row of control points divided by the inlet section and the outlet section in the T direction, calculating expressions of 0 th, … th, k th and n-k-1, … th, n-1 th basis functions respectively, representing the expressions as column vectors, and transversely splicing the column vectors into a matrix, thereby obtaining coefficient matrices of a plurality of homogeneous linear equation sets; respectively solving each homogeneous linear equation set to obtain a plurality of general solutions; when the coordinates of a certain control point in a certain column in the T direction are adjusted, according to the change quantity of the coordinates, the undetermined coefficients in the complete solution of the homogeneous linear equation set in the column are changed, so that the updated coordinates of other control points in the same column are calculated, and the cooperative control of the control points is completed; knowing the coordinates of the control points, calculating a tubular curved surface according to the definition of the curved surface, and then taking the points on the curved surface according to any precision and outputting the points.
2. The method of overall design of a tubular member according to claim 1, wherein the constraint equation is descriptive of a homogeneous system of linear equations when generating the non-circular closed curve:
wherein p is i X, the abscissa representing the ith control point of the curve i Or the ordinate y i ，a i,k The coefficient of the kth term of the ith basis function.
3. The method of overall design of a tubular member according to claim 1, wherein the system of non-homogeneous linear equations for the weights when generating a circular closed curve is:
wherein a is k,k Coefficient, ω, of the kth term of the kth basis function n-1 Weight of the n-1 th control point, b 0 ,…,b k A constant that is combined to a known quantity of the equation.
4. The method of integral design of a tubular member according to claim 1, wherein the system of non-homogeneous linear equations about the abscissa of the control points when generating a circular closed curve is:
wherein a is k,k Coefficient, ω, of the kth term of the kth basis function n-1 Is the weight factor of the n-1 control point, p n-1 The coordinates of the n-1 th control point of the curve (may be the abscissa x i May also be the ordinate y i )，c 0 ,…,c k A constant that is combined to a known quantity of the equation.
5. The method of claim 1, wherein the plurality of homogeneous sets of linear equations in step 2 are:
wherein P is ij Is the coordinates of the control points on the control point grid with index i in the T direction and index j in the S direction;represents the ith k in the T direction 1 Secondary basis function, i=0, 1,..n-1; u is a parameter in the T direction; u (u) ′ Is a parameter subjected to coordinate transformation in the T direction.
6. The method of integral design of a tubular member according to claim 1, further comprising step 3 of verifying the micropowders of the round-like B-spline curve:
wherein C (U) is defined as the node vector u= [0 ], k ,u k+1 ,...,1]b-spline curves of n control points, with nodes n+k+1 in node vector U, P i Is the coordinates of the ith control point. C (C) ′ (u) deriving u for it;
defining a node weight as 1 in a defined spline curve, thereby deriving the spline curve once each time, and subtracting 1 from the spline times; until the derivation is to the primary spline, no derivation can be performed.
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