Method for optimizing shape of pipeline support structure based on isogeometric analysis

Abstract

The invention discloses a shape optimization method of a pipeline support structure based on isogeometric analysis, and belongs to the technical field of pipeline shape optimization. The problem that an existing pipeline support structure is poor in vibration reduction performance is solved. The method specifically comprises the following steps: step 1, identifying edge information of a pipeline support from an image containing the pipeline support, extracting a contour line from the edge information, and extracting a contour point from the contour line as a control point; parameterization description is carried out on the coordinates of the control points, and a two-dimensional plane parameterization model of the pipeline support is obtained; 2, taking the abscissa of a control point in the two-dimensional plane parameterized model as a design variable, and establishing a target function according to the design variable; step 3, establishing a shape optimization mathematical model based on the objective function and the system balance equation; and 4, solving the shape optimization mathematical model, and completing the shape optimization of the pipeline support structure according to the abscissa optimization result of each control point. The method can be applied to shape optimization of the pipeline support structure.

Description

Technical Field

[0001] The invention belongs to the technical field of pipeline shape optimization, and in particular relates to a shape optimization method for a pipeline support structure based on isogeometric analysis. Background Art

[0002] Pipeline structures are widely used in systems such as ships and aircraft. In ships, pipeline structures are mainly used to transport liquids and gases, including fuel lines, fresh water lines, sewage lines, compressed air lines, etc. These pipelines connect different systems on the ship, such as engines, generators, water treatment systems, etc. In aircraft, pipeline structures are also very important components. There are many different types of pipelines on aircraft, such as fuel lines, hydraulic lines, air conditioning lines, oxygen lines, etc. These pipelines connect different systems, such as engines, landing gears, flight control systems, etc. The bracket can absorb and reduce the vibration and impact generated by the pipeline during operation, reducing damage to the pipeline and its connectors. Therefore, the design and maintenance of the pipeline bracket structure are crucial for the normal operation of ships and aircraft.

[0003] Vibration has adverse effects on pipeline systems, equipment and structures. Vibration may cause pipeline fatigue, leakage or rupture, and equipment is also susceptible to damage. Strong vibration may cause people to slip, fall or lose balance. Vibration reduction can effectively reduce these risks, extend the service life of equipment, and protect the integrity of equipment and structures. However, the vibration reduction performance of existing pipeline support structures is still poor. Therefore, it is necessary to study the shape optimization vibration reduction technology of pipeline support structures to solve the current problems. Summary of the invention

[0004] The purpose of the present invention is to solve the problem of poor vibration reduction performance of the existing pipeline support structure, and to propose a shape optimization method of the pipeline support structure based on isogeometric analysis.

[0005] The technical solution adopted by the present invention to solve the above technical problems is: a shape optimization method of a pipeline support structure based on isogeometric analysis, the method specifically comprising the following steps:

[0006] Step 1: Identify edge information of the pipeline support from the image containing the pipeline support, extract contour lines from the edge information, and extract contour points from the contour lines as control points;

[0007] Then, the coordinates of the control points are parametrically described using the B-spline curve to obtain a two-dimensional plane parametric model of the pipeline support;

[0008] Step 2: taking the horizontal coordinates of the control points in the two-dimensional plane parametric model of the pipeline support as design variables, and then establishing an objective function based on the design variables;

[0009] Step 3: Establish a shape optimization mathematical model based on the objective function and system equilibrium equation;

[0010] Step 4: Solve the shape optimization mathematical model established in step 3 to obtain the horizontal coordinate optimization results of each control point, and complete the shape optimization of the pipeline support structure.

[0011] The beneficial effects of the present invention are:

[0012] When performing a parametric description of a two-dimensional cross-sectional model of a pipeline support, the present invention selects key points through a contour extraction method, thereby overcoming the limitations of the traditional finite element method in structural optimization. In isogeometric analysis, a smooth curve can be drawn through the information of the control points, thereby avoiding irregular shapes that do not conform to reality and reducing the details that need to be drawn. The shape of the curve can be adjusted by modifying the control points to more accurately control the shape of the curve. At the same time, by setting the optimization target of the objective function in the isogeometric analysis to reduce the displacement of the control points, in actual application, the method of the present invention can effectively reduce vibration of the pipeline structure and improve the performance of the pipeline. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 It is a flow chart of a shape optimization method of a pipeline support structure based on isogeometric analysis of the present invention;

[0014] Figure 2 It is an overall schematic diagram of the pipeline structure of the present invention;

[0015] Figure 3 This is a perspective view of the rubber encapsulation of the pipeline structure of the present invention;

[0016] Figure 4 It is a schematic diagram of the main pipeline structure of the present invention;

[0017] Figure 5 This is a schematic diagram of the pipeline support structure of the present invention;

[0018] Figure 6 A two-dimensional cross-sectional model diagram of the pipeline support structure before optimization of the present invention;

[0019] Figure 7 This is a two-dimensional cross-sectional model diagram of the optimized pipeline support structure of the present invention. DETAILED DESCRIPTION

[0020] Specific implementation method 1: Combination Figure 1 The present embodiment describes a shape optimization method for a pipeline support structure based on isogeometric analysis, and the method specifically comprises the following steps:

[0021] Step 1: Identify edge information of the pipeline support from the image containing the pipeline support, extract contour lines from the edge information, and extract contour points from the contour lines as control points;

[0022] Then, the coordinates of the control points are parametrically described using the B-spline curve to obtain a two-dimensional plane parametric model of the pipeline support;

[0023] Step 2: taking the horizontal coordinates of the control points in the two-dimensional plane parametric model of the pipeline support as design variables, and then establishing an objective function based on the design variables;

[0024] Step 3: Establish a shape optimization mathematical model based on the objective function and system equilibrium equation;

[0025] Step 4: Solve the shape optimization mathematical model established in step 3 to obtain the horizontal coordinate optimization results of each control point, and complete the shape optimization of the pipeline support structure.

[0026] Specific implementation method 2: This implementation method is different from specific implementation method 1 in that, in step 1, edge information of the pipeline support is identified using an edge detection algorithm.

[0027] The other steps and parameters are the same as those in the first embodiment.

[0028] Specific implementation method three: This implementation method is different from specific implementation methods one or two in that the two-dimensional plane parameterized model of the pipeline support is specifically:

[0029]

[0030] Among them, S(u,v) is the two-dimensional plane parameterized model of the pipeline support, Pi,j is the coordinate of the control point, R i,j (u,v) is the basis function, i=0,1,…,n, j=0,1,…,m.

[0031] The other steps and parameters are the same as those in the first or second embodiment.

[0032] This embodiment uses B-spline curves in geometric modeling, and the obtained two-dimensional model of the pipeline support has high smoothness and high flexibility, and the shape of the curve can be adjusted by modifying the control points.

[0033] Specific implementation method 4: This implementation method is different from any one of the specific implementation methods 1 to 3 in that the objective function is specifically:

[0034] f=I T u(x)

[0035] Where f is the objective function, I represents a vector containing only 0 and 1, and the length of vector I is equal to the number of control points, the superscript T represents the transpose, and u(x) represents the vector composed of the displacement of each control point.

[0036] The other steps and parameters are the same as those in Specific Embodiments 1 to 3.

[0037] This implementation method establishes the objective function in the isogeometric analysis based on the displacement of the selected control points, which can reduce the degree of freedom of the optimization problem and thus reduce the complexity of the calculation; the optimization goal is to reduce the displacement and thus suppress the vibration.

[0038] Specific implementation method 5: This implementation method is different from the specific implementation methods 1 to 4 in that the specific form of the shape optimization mathematical model is:

[0039] min f=I T u(x)

[0040] stKu(x)=F

[0041] S≤S *

[0042] x min ≤x i ≤x max

[0043] Among them, Ku(x)=F is the system equilibrium equation, F is the vector composed of the forces applied at each control point, K is the stiffness matrix in the system equilibrium equation, S is the area of ​​the two-dimensional plane parameterized model of the pipeline support, S * is the upper limit of the area, x i is the horizontal coordinate of the ith control point, x min is the lower limit of the horizontal coordinate of the ith control point, x max is the upper limit of the horizontal coordinate of the ith control point.

[0044] The other steps and parameters are the same as those in Specific Embodiments 1 to 4.

[0045] The process of establishing a two-dimensional plane parameterized model and deriving the system equilibrium equation Ku(x)=F in the present invention both belongs to the isogeometric analysis process.

[0046] Specific implementation method 6: This implementation method is different from any one of the specific implementation methods 1 to 5 in that the stiffness matrix K in the system equilibrium equation is:

[0047]

[0048] Among them, B is the strain matrix, D is the elastic matrix, and Ω is the integration domain.

[0049] The other steps and parameters are the same as those in Specific Implementation Methods 1 to 5.

[0050] Specific implementation method seven: This implementation method is different from any one of specific implementation methods one to six in that the moving asymptote optimization algorithm is used to solve the shape optimization mathematical model established in step 3.

[0051] The other steps and parameters are the same as those in Specific Embodiments 1 to 6.

[0052] Example

[0053] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, but they are not intended to be limiting of the present invention. Figure 1 As shown, the method of this embodiment specifically includes the following steps:

[0054] Step 1: Use an edge detection algorithm to identify edge information of the pipe support in the image containing the pipe support, then extract a contour line from the edge information using a contour extraction method, and extract contour points from the contour line as control points. The control points can be selected according to actual conditions, and the selected control points can be selected as long as they can represent the contour line;

[0055] Then, the coordinates of the control points are parametrically described using the B-spline curve to obtain a two-dimensional plane parametric model of the pipeline support;

[0056] The two-dimensional plane parameterized model of the pipeline support is specifically:

[0057]

[0058] Where Pi,j is the coordinate of the control point, N i,p (u) is the node vector U = [u0,u1,…un + p +1 ] is a B-spline basis function of degree p on N j,q (v) is the node vector V = [v0, v1, ...v n+q+1 ] is a B-spline basis function of degree q on R i,j (u,v) is the basis function, i=0,1,…,n, j=0,1,…,m, i and j represent two directions of the plane parameterized model, and a control point is determined by the two directions. n is the dimension in the i direction, and m is the dimension in the j direction.

[0059] Step 2: taking the horizontal coordinates of the control points in the two-dimensional plane parametric model of the pipeline support as design variables, and then establishing an objective function based on the design variables;

[0060] To evaluate the shock absorption effect, the objective function is defined as the displacement of a control point under the condition of applying a specific force. The objective function is specifically:

[0061] f=I T u(x)

[0062] Where f is the objective function, I represents a vector containing only 0 and 1, and the length of vector I is equal to the number of control points, the superscript T represents the transpose, and u(x) represents the vector composed of the displacement of each control point.

[0063] Each control point has a corresponding objective function. Therefore, for the objective function corresponding to the current control point, in the vector I of the objective function, only the element corresponding to the current control point is 1, and the elements corresponding to the other control points are 0.

[0064] Step 3: Establish a shape optimization mathematical model based on the objective function and the system equilibrium equation; the specific form of the shape optimization mathematical model is:

[0065] min f=I T u(x)

[0066] stKu(x)=F

[0067] S≤S *

[0068] x min ≤x i ≤x max

[0069] Among them, Ku(x)=F is the system equilibrium equation, F is the vector composed of the forces applied at each control point (only the position corresponding to the current control point has the actual applied force value, and the element values ​​of the positions corresponding to other control points are 0), K is the stiffness matrix in the system equilibrium equation, S is the area of ​​the two-dimensional plane parameterized model of the pipeline support, S * is the upper limit of the area, x i is the horizontal coordinate of the ith control point, x min is the lower limit of the horizontal coordinate of the ith control point, x max is the upper limit of the horizontal coordinate of the ith control point.

[0070] The stiffness matrix K in the system equilibrium equation is specifically:

[0071]

[0072] Among them, B is the strain matrix, D is the elastic matrix, and Ω is the integration domain.

[0073] As for the method of solving the integral function, since it is difficult to derive an analytical expression for the integrand in the integral formula of the isoparametric stiffness matrix, the calculation of the isoparametric unit uses numerical integration to find the approximate value of the integral. In order to reduce the number of calculation points, Gaussian numerical integration is often used.

[0074] Step 4: Use the moving asymptote optimization algorithm to solve the shape optimization mathematical model, obtain the horizontal coordinate optimization results of each control point, and complete the shape optimization of the pipeline support structure.

[0075] In the process of solving the moving asymptote optimization algorithm, it is necessary to perform sensitivity analysis on the structure, which includes calculating the sensitivity of the objective function and the sensitivity of the constraint conditions. Sensitivity analysis is an indicator used to evaluate the sensitivity of the objective function to the disturbance of the design variables. The sensitivity value represents the degree to which the model target is affected by the design variable. The objective function is obtained by the structural sensitivity analysis, and the constraint condition is the derivative of the design variable.

[0076] The present invention adopts an analytical method to perform sensitivity analysis and calculation on the structure, and the method for solving the sensitivity matrix is ​​as follows:

[0077] (1) According to the objective function f, its sensitivity can be expressed as:

[0078]

[0079] Therefore, only du / dx i is an unknown term. The specific form of the displacement derivative is obtained by solving the derivative of the discrete system equilibrium equation Ku = F, as follows:

[0080]

[0081]

[0082] Get the objective function f for the design variable x i The first-order derivative of is:

[0083]

[0084] Next, the sensitivity of the total stiffness matrix to the design variables is calculated. The total stiffness matrix is ​​assembled from single stiffness matrices, so the derivatives of the single stiffness matrix to the design variables are derived first.

[0085] The single rigid matrix is ​​expressed as follows:

[0086]

[0087] (2) According to the constraint function, its sensitivity can be expressed as:

[0088]

[0089] in, is the area of ​​the parameter unit, |J k | is the Jacobian of the unit.

[0090] The sensitivity calculation is ultimately transformed into finding B′ and |J|′, where the matrix B consists of the derivatives of the basis functions with respect to the physical coordinates.

[0091] Among them, the strain matrix B is expressed as:

[0092]

[0093] The shape function N in finite element analysis uses R as the NURBS basis function in isogeometric analysis, and nen represents the number of basis functions;

[0094] Define the matrix G to store the derivatives of all basis functions with respect to physical coordinates, and define the matrix Store the derivatives of all basis functions with respect to the parameter coordinates in the following form:

[0095]

[0096]

[0097] The above two matrices can be connected by the chain rule of integration, that is, And the Jacobian matrix is ​​given by the matrix and x, the form is Where x is the physical point coordinate matrix, so the final representation of the G′ matrix is:

[0098] G′=-Gx′G

[0099] Similarly, the derivative of the Jacobian matrix determinant |J′ can also be converted into a form related to G and x′ as follows:

[0100]

[0101] Finally, the calculation results of each sensitivity are determined by the derivative of the physical coordinates with respect to the design variables. According to the coordinate mapping relationship, the physical coordinates are composed of the physical control point coordinates and the basis functions, which are expressed as follows:

[0102]

[0103] Since the design variable is the control point that controls the external geometry of the model, and its value corresponds to an element in the matrix P, only the matrix P is related to the design variable in the above formula, and the final form of its derivative is:

[0104]

[0105] like Figure 2 is the overall schematic diagram of the pipeline structure, such as Figure 3This is a perspective view of the rubber encapsulation of the pipeline structure of the present invention, showing the overall structure of the support structure pipeline, which is mainly composed of a ring, a support structure and filling rubber; the inner diameter and outer diameter of the ring part are respectively poured with rubber, thereby forming the internal and external frames of the pipeline; the support structure is distributed in the middle of the two rings in an annular direction in n numbers; the filling rubber is tightly combined with the external frame to enhance the stability of the entire pipeline; the pipeline is axially arranged in m periods, and there are m unit cell structures arranged along the axial direction of the pipeline.

[0106] like Figure 4 This is a schematic diagram of the main body of the pipeline structure of the present invention. The main body of the pipeline structure has m unit cell structures arranged along the axial direction of the pipeline, and n support structures are distributed in an annular direction between two circular rings.

[0107] Figure 5 Schematic diagram of the pipeline support structure of the present invention. The two-dimensional cross-sectional shape of the pipeline support structure can be clearly seen from the schematic diagram of the support structure.

[0108] First, NURBS splines are used to establish a two-dimensional model of the pipe support cross section, such as Figure 6 The 2D cross-sectional model of the pipe support structure before optimization is shown in Figure 1. The 7 control points on the left side of the cross section are used as optimization design variables for shape optimization. The two direction nodes are U = {00 0111} and The times are p=2 and q=2, Young's modulus E=180Gpa, and Poisson's ratio μ=0.3. The geometric model of the bracket section with 3×7 control points is obtained. The upper and lower edges are fixed, and a uniform load F=10N is applied on the left side. In order to maintain the reasonable and effective design of the bracket structure, the x and y coordinates of the control points of the cross-sectional design variables cannot exceed plus or minus 10% of their original positions. The cross-sectional diagram of the bracket after shape optimization is shown in the figure below. Figure 7 As shown in the figure, after calculation, the lateral displacement of the objective function control point of the section with the initial design was 0.4651 mm after being subjected to uniform load, while the displacement of the control point calculated after optimization was 0.4071 mm, a reduction of 12%.

[0109] In summary, compared with the original pipeline, the optimized pipeline of the embodiment of the present invention has the outstanding feature of optimizing the shape of the two-dimensional cross-section of the pipeline support structure through isogeometric analysis, which can effectively suppress the vibration amplitude of the pipeline.

[0110] The above calculation examples of the present invention are only used to explain the calculation model and calculation process of the present invention in detail, and are not intended to limit the implementation methods of the present invention. For ordinary technicians in the relevant field, other different forms of changes or modifications can be made based on the above description. It is impossible to list all the implementation methods here. All obvious changes or modifications derived from the technical solution of the present invention are still within the scope of protection of the present invention.

Claims

1. A shape optimization method for a pipeline support structure based on isogeometric analysis, characterized in that: The method specifically comprises the following steps: Step 1: Identify edge information of the pipeline support from the image containing the pipeline support, extract contour lines from the edge information, and extract contour points from the contour lines as control points; Then, the coordinates of the control points are parametrically described using the B-spline curve to obtain a two-dimensional plane parametric model of the pipeline support; Step 2: taking the horizontal coordinates of the control points in the two-dimensional plane parametric model of the pipeline support as design variables, and then establishing an objective function based on the design variables; Step 3: Establish a shape optimization mathematical model based on the objective function and system equilibrium equation; Step 4: Solve the shape optimization mathematical model established in step 3 to obtain the horizontal coordinate optimization results of each control point, and complete the shape optimization of the pipeline support structure.

2. The shape optimization method of a pipeline support structure based on isogeometric analysis according to claim 1, characterized in that: In step 1, edge detection algorithm is used to identify edge information of the pipeline support.

3. The shape optimization method of a pipeline support structure based on isogeometric analysis according to claim 2, characterized in that: The two-dimensional plane parameterized model of the pipeline support is specifically: Among them, S(u,v) is the two-dimensional plane parameterized model of the pipeline support, P i,j is the coordinate of the control point, R i,j (u,v) is the basis function, i=0,1,…,n, j=0,1,…,m.

4. The shape optimization method of a pipeline support structure based on isogeometric analysis according to claim 3 is characterized in that: The objective function is specifically: f=I T u(x) Where f is the objective function, I represents a vector containing only 0 and 1, and the length of vector I is equal to the number of control points, the superscript T represents the transpose, and u(x) represents the vector composed of the displacement of each control point.

5. The shape optimization method of a pipeline support structure based on isogeometric analysis according to claim 4, characterized in that: The specific form of the shape optimization mathematical model is: min f=I T u(x) stKu(x)=F S≤S * x min ≤x i ≤x max Among them, Ku(x)=F is the system equilibrium equation, F is the vector composed of the forces applied at each control point, K is the stiffness matrix in the system equilibrium equation, S is the area of ​​the two-dimensional plane parameterized model of the pipeline support, S * is the upper limit of the area, x i is the horizontal coordinate of the ith control point, x min is the lower limit of the horizontal coordinate of the ith control point, x max is the upper limit of the horizontal coordinate of the ith control point.

6. The shape optimization method of a pipeline support structure based on isogeometric analysis according to claim 5, characterized in that: The stiffness matrix K in the system equilibrium equation is: K=∫ Ω B T DBdΩ Among them, B is the strain matrix, D is the elastic matrix, and Ω is the integration domain.

7. The shape optimization method of a pipeline support structure based on isogeometric analysis according to claim 6, characterized in that: The moving asymptote optimization algorithm is used to solve the shape optimization mathematical model established in step 3.

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