Stress response assessment method for pipeline systems based on beam element and solid element coupling

Abstract

This invention provides a method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements, comprising: obtaining the stiffness and mass matrices of beam elements and the stiffness and mass matrices of incompatible solid elements; dividing the constrained region of the pipeline system into solid meshes and the unconstrained region into beam element meshes; obtaining the stiffness matrix of the interface-coupled elements based on the displacement relationship of the interface-coupled elements; obtaining the element stiffness and mass matrices of the pipeline system; obtaining the nodal displacement response of the pipeline system based on the equation of motion of the pipeline system; solving the stress response at the element nodes by stress grinding, and analyzing whether the maximum stress response of the pipeline system is within a preset reasonable response range. If it is within the reasonable response range, the stress response design of the pipeline system is judged to be reasonable; otherwise, it is judged to be unreasonable.

Description

Technical Field

[0001] This invention relates to the field of reduced-scale modeling of pipeline system dynamics, and more particularly to a method for evaluating the stress response of pipeline systems based on the coupling of beam elements and solid elements. Background Technology

[0002] Aero-engine piping systems are subjected to various excitations during operation, including aerodynamic excitation, pump-source excitation, and rotor imbalance. These external excitations can increase pipe stress, leading to defects such as pipe rupture, deformation, or weld cracks. Therefore, appropriate methods are needed to analyze and assess the stress response of the piping under external excitations during the design phase to improve the design reliability of the piping system. In stress analysis, the finite element method (FEM) is typically used to solve the stress response of the structure. This method primarily uses solid or shell elements to construct a refined model of the structure, allowing for a relatively accurate determination of the stress response at any part of the structure. However, the refined model has a large number of degrees of freedom, resulting in large stiffness and mass matrices, leading to low computational efficiency and making it difficult to provide effective data support for piping system optimization. Summary of the Invention

[0003] To address the aforementioned technical problem of insufficient effective data support for pipeline system optimization, this invention provides a method for evaluating the stress response of pipeline systems based on the coupling of beam elements and solid elements. This invention primarily utilizes beam elements and incompatible solid elements to jointly simulate the pipeline system. A finite element model of the pipeline system is created based on the displacement constraint equations of the beam and solid elements, thereby obtaining the stress response of the pipeline system and subsequently acquiring the improved stress at the element nodes, thus achieving pipeline system optimization.

[0004] The technical means employed in this invention are as follows:

[0005] A method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements, comprising:

[0006] The stiffness matrix and mass matrix of the beam element are obtained based on the three-dimensional Timoshenko beam element.

[0007] The characteristics of the pipeline solid part are simulated by using spatial 8-node non-coordinated solid elements, thereby obtaining the stiffness matrix and mass matrix of the non-coordinated solid elements;

[0008] The pipeline system is meshed, including dividing the constrained area of ​​the pipeline system into solid meshes and the unconstrained area of ​​the pipeline system into beam element meshes;

[0009] The connection between the beam element and the solid element node sharing the same cross section is realized by constructing an interface coupling element, and the stiffness matrix of the interface coupling element is obtained based on the displacement relationship of the interface coupling element.

[0010] The element stiffness matrix and mass matrix of the pipeline system are obtained based on the stiffness matrix and mass matrix of beam elements, the stiffness matrix and mass matrix of non-coordinated solid elements, and the stiffness matrix of interface coupling elements.

[0011] A basic excitation is applied to the pipeline system, and the nodal displacement response of the pipeline system is obtained based on the system's equation of motion.

[0012] Based on the nodal displacement response of the pipeline system, the stress response of any element at the element node can be obtained by stress grinding through the Gauss integration point.

[0013] Determine whether the stress response of any element at the Gauss integration point is within the preset reasonable response range. If it is within the reasonable response range, the stress response design of the pipeline system is considered reasonable; otherwise, it is considered unreasonable.

[0014] Furthermore, the stiffness matrix and mass matrix of the non-coordinated solid element are obtained according to the following calculations:

[0015]

[0016] Among them, K e B is the stiffness matrix of the non-conforming solid element, B is the strain matrix corresponding to the principal degrees of freedom, and D is the elasticity matrix. M is the strain matrix corresponding to the additional degrees of freedom. e Let ρ be the mass matrix of the non-coordinated solid element, N be the element shape function matrix, and V be the volume.

[0017] Furthermore, the stiffness matrix of the interface coupling unit is obtained according to the following calculation:

[0018]

[0019] Where K2 is the stiffness matrix of the interface coupling element, κ is the penalty factor, and G is the transformation matrix between nodal degrees of freedom and global degrees of freedom. c This is the displacement constraint matrix.

[0020] Furthermore, the element stiffness matrix and mass matrix of the piping system are obtained through the following calculations:

[0021] K = K s +K b +K c

[0022] M = M s +M b

[0023] Where K is the element stiffness matrix of the pipeline system, K sK is the stiffness matrix of the non-conforming solid element. b K is the stiffness matrix of the beam element. c M is the coupling stiffness matrix, and M is the element mass matrix of the piping system. s M is the mass matrix of the non-coordinated entity unit. b Let be the mass matrix of the beam element.

[0024] Furthermore, the nodal displacement response of the pipeline system is obtained based on the following calculations:

[0025]

[0026] Where x is the nodal displacement response of the pipeline system, F is the external load, ω is the external excitation frequency, α and β are the Rayleigh damping coefficients, and ω is the external load. j Let Φ be the natural frequency of the j-th order. m This is the first m-order regularization matrix obtained from modal analysis of the finite element model of the pipeline system. This corresponds to the j-th mode shape.

[0027] Furthermore, the stress response of any element at the Gauss integration point is obtained according to the following calculation:

[0028] σ n =DB n u n

[0029] Where, σ n Let B be the stress at the Gaussian integration point, D be the elasticity matrix, and B be the stress at the Gaussian integration point. n Let u be the strain matrix corresponding to the Gaussian integration point. n This represents the displacement response of the element node.

[0030] Furthermore, the stress response at the element nodes is obtained through stress smoothing, including: introducing an improved stress solution through stress smoothing, which is obtained from the stress at the Gauss integration point, specifically expressed as:

[0031]

[0032]

[0033] in, i = 1, 2, ..., 8 represent the stress improvement values ​​of 8 nodes in an element. i = 1, 2, ..., 8 represent the stress solutions at the Gauss integration points of a single element obtained by the finite element method.

[0034] Compared with the prior art, the present invention has the following advantages:

[0035] 1. This invention provides a method for rapid evaluation of the stress response of a pipeline system. The pipeline system is jointly simulated using beam elements and non-coordinated solid elements. A finite element model of the pipeline system is created based on the displacement constraint equations of the beam elements and solid elements, thereby obtaining the improved stress at the element nodes and providing effective data support for the optimization of the pipeline system.

[0036] 2. In this invention, most of the pipes in the pipeline system are simulated using beam elements, which reduces the degree of freedom of the pipeline system and thus increases the computational efficiency. The more refined the mesh of the pipeline system, the more superior the method proposed in this invention becomes. Attached Figure Description

[0037] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0038] Figure 1 This is a flowchart of a method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements, according to the present invention.

[0039] Figure 2 The diagram shows a beam element and an 8-node non-coordinated solid element provided in the embodiment of the present invention. (a) is a solid element and (b) is a beam element.

[0040] Figure 3 The diagram below shows a network of the pipeline system provided in an embodiment of the present invention. (a) is a three-dimensional model of the space pipe, (b) is a solid model of the space pipe, and (c) is a solid model of the space pipe beam.

[0041] Figure 4 The diagram shows a coupling unit provided in an embodiment of the present invention. (a) shows the coupling between the solid and the beam, and (b) shows the single-point coupling constraint. Detailed Implementation

[0042] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0043] like Figure 1 As shown, this invention provides a method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements, comprising:

[0044] S1. Obtain the stiffness matrix and mass matrix of the beam element based on the three-dimensional Timoshenko beam element.

[0045] This embodiment uses a three-dimensional Timoshenko beam element; see the element diagram below. Figure 2 (b) Its stiffness matrix and mass matrix can be obtained from the following literature: Chai Q, Zeng J, Ma H, et al. A dynamic modeling approach for nonlinear vibration analysis of the L-type pipeline system with clamps. Chinese Journal of Aeronautics, 2020, 33(12): 3253-3265.

[0046] S2. The characteristics of the pipeline solid part are simulated by using a non-coordinated solid element with 8 spatial nodes, thereby obtaining the stiffness matrix and mass matrix of the non-coordinated solid element.

[0047] This embodiment uses an 8-node non-coordinated solid element to simulate the characteristics of the pipeline solid part. The element diagram is shown below. Figure 2 As shown in (a), the shape function corresponding to the degree of freedom of this element is

[0048]

[0049] In the formula (ξ) i ,η i ,ζ i ) is the coordinate value at node i of the parent element, and the value is either 1 or -1.

[0050] The shape functions corresponding to the additional degrees of freedom are

[0051] N9(ξ)=1-ξ 2 N 10 (η)=1-η 2 N 11 (ζ)=1-ζ 2 (2)

[0052] Then the shape function matrix N of the element node and the shape function matrix of the additional degrees of freedom They can be represented as follows:

[0053] N = [N1I] 3×3 N2I 3×3 N3I 3×3 N4I 3×3 N5I 3×3 N6I 3×3N7I 3×3 N8I 3×3 (3)

[0054]

[0055] The principal strain matrix corresponding to the master node can then be expressed as:

[0056]

[0057] In the formula It can be represented as

[0058]

[0059] The strain matrix corresponding to the additional degrees of freedom can be expressed as:

[0060]

[0061] According to the principle of minimum potential energy, we can obtain:

[0062]

[0063] In the formula, and They can be represented as follows:

[0064]

[0065] Where D is the elasticity matrix.

[0066] By eliminating additional degrees of freedom through static condensation, the stiffness matrix of the incompatible element can be obtained as follows:

[0067]

[0068] The mass matrix of an incoordinating entity can be represented as:

[0069]

[0070] Where M e Let ρ be the mass matrix of the non-coordinated solid element, N be the element shape function matrix, and V be the volume.

[0071] S3. Mesh the piping system. Solid meshes are created for the constrained regions of the piping system to obtain a refined finite element model. For the unconstrained regions, beam element meshes are created. The resulting mesh model for this example is as follows: Figure 3 As shown in the figure. In this application, the clamp support area or fixed end of the system is defined as the constrained area, and other areas are defined as the unconstrained area.

[0072] S4. The connection between the beam element and the solid element node sharing the same cross section is realized by constructing an interface coupling element, and the stiffness matrix of the interface coupling element is obtained based on the displacement relationship of the interface coupling element.

[0073] In this implementation, because the number of degrees of freedom of beam element nodes and solid element nodes is different, interface coupling elements are introduced to connect beam elements and solid element nodes that share the same cross section. Specifically, as follows... Figure 4 As shown. Specifically includes:

[0074] S401. Solving for the displacement relationship of coupled elements. Here, we choose to analyze the displacement constraint relationship of coupled elements on section A. When using beam elements to simulate the pipeline, the nodes of the beam elements are located on section A, and the node displacement vector is...

[0075] q b =[u b v b w b φ x φ y φ z ] T (12)

[0076] When the node translates along the x, y, and z directions, the translational displacement of any point on section A can be expressed as:

[0077] Δu p =u b Δv p =v b Δw p =w b (13)

[0078] When the cross section rotates about the x-axis, the displacement changes of node p in the y and z directions can be expressed as:

[0079] Δv x =r p [cos(α+φ x

[14] -cosα

[0080] Δw x =r p [sin(α+φ x

[15]

[0081] In the formula, α is the position angle of node p; r p Let p be the radius of node p relative to the node of the beam element.

[0082] When the cross section rotates about the y-axis, the displacement changes of the node in the x and z directions can be expressed as:

[0083] Δuy =r pz sinφ y (16)

[0084] Δw z =r pz (cosφ y -1) (17)

[0085] In the formula r pz For r p The projection on the z-axis is specifically as follows:

[0086] r pz =r p sinα (18)

[0087] When the cross section rotates about the z-axis, the displacement changes of the node in the x and y directions can be expressed as:

[0088] Δu z =-r py sinφ z (19)

[0089] Δv z =-r py (1-cosφ z (20)

[0090] In the formula r py For r p The projection on the y-axis is specifically as follows:

[0091] r py =r p cosα (21)

[0092] The displacement of node p can then be expressed as

[0093] u = u b +r pz sinφ y -r py sinφ z (twenty two)

[0094] v = v b +r p [cos(α+φ x )-cosα]-r py (1-cosφ z ) (twenty three)

[0095] w = w b +r p [sin(α+φ x )-sinα]+r pz (cosφy -1) (24)

[0096] Analyzing equations (22) to (24), it can be found that the nodal displacements of beam elements and solid elements have a typical nonlinear relationship, and the transformation matrix for the displacement relationship between the two cannot be obtained. Here, the nonlinear terms in equations (22) to (24) are expanded using Taylor expansion, specifically as follows:

[0097]

[0098] In the formula, “o” represents a higher-order infinitesimal. By omitting the higher-order infinitesimal terms in equation (25) and substituting them into equations (22) to (24), we can obtain...

[0099]

[0100] From equation (26), the nodal displacement relationship of the i-th coupled element can be obtained as follows:

[0101]

[0102] In the formula Let be the displacement vector of the solid element node in the coupled element, specifically

[0103]

[0104] T i The nodal displacement transformation matrix of the coupled element can be specifically expressed as:

[0105]

[0106] Substituting equation (29) into equation (27) yields the displacement constraint equations for the coupled elements, which can be combined into matrix C. i The elements in the matrix are represented as

[0107]

[0108] S402. Solve for the stiffness matrix of the coupled element. From the displacement constraint equation of equation (30), the stiffness matrix of the coupled element can be obtained using the constraint variational principle.

[0109]

[0110] In the formula, κ is the penalty factor, which is taken as κ=max(diag(K) s G is the transformation matrix between the nodal degrees of freedom and the total degrees of freedom. c The displacement constraint matrix obtained from equation (30) can be expressed as follows:

[0111]

[0112] S5. Obtain the element stiffness matrix and mass matrix of the pipeline system based on the stiffness matrix and mass matrix of beam elements, the stiffness matrix and mass matrix of non-coordinated solid elements, and the stiffness matrix of interface-coupled elements.

[0113] In this embodiment, the element stiffness matrix and mass matrix of the pipeline system are obtained and solved in the following manner.

[0114]

[0115] In the formula, K is the element stiffness matrix of the pipeline system, K s K is the stiffness matrix of the non-conforming solid element. b K is the stiffness matrix of the beam element. c M is the coupling stiffness matrix, and M is the element mass matrix of the piping system. s M is the mass matrix of the non-coordinated entity unit. b This represents the mass matrix of the beam element. The subscript "s" represents the solid element finite element model, and "b" represents the beam element finite element model.

[0116] S6. Apply basic excitation to the pipeline system and obtain the nodal displacement response of the pipeline system based on the equation of motion of the pipeline system.

[0117] In this embodiment, a basic excitation with an amplitude of 1g is applied to the pipeline system in the z-direction. The equation of motion for the pipeline system is then:

[0118]

[0119] In the formula, M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the pipeline system, respectively. The damping matrix can be expressed using Rayleigh damping as follows:

[0120]

[0121] E z The inertial force indicator vector for the tube based on the Timoshenko beam element is 1 when the degree of freedom is in the z-direction, and 0 otherwise; üg is the basic acceleration excitation, which can be expressed as

[0122]

[0123] Modal analysis was performed on the obtained finite element model of the pipeline system to obtain the regularization matrix Φ of the first m orders. m Select Φ m If the column vectors in equation (34) form the modal space, then the displacement vector in equation (34) can be expressed as:

[0124]

[0125] In the formula, x represents the nodal displacement response of the pipeline system, F represents the external load, ω represents the external excitation frequency, α and β are the Rayleigh damping coefficients, and ω is the external excitation frequency. j Let Φ be the natural frequency of the j-th order. m This is the first m-order regularization matrix obtained from modal analysis of the finite element model of the pipeline system. This corresponds to the j-th mode shape.

[0126] S7. Based on the nodal displacement response of the pipeline system, the stress response of any element at the Gauss integration point can be obtained using the Gauss integration point. Furthermore, the stress response at the element node can be obtained by using the stress smoothing method.

[0127] The displacement response of the system can be obtained using equation (37). Furthermore, the stress response of any element at the Gauss integration point can be obtained using the Gauss integration point, specifically expressed as follows:

[0128] σ n =DB n u n (38)

[0129] Where, σ n Let B be the stress at the Gaussian integration point, D be the elasticity matrix, and B be the stress at the Gaussian integration point. n Let u be the strain matrix corresponding to the Gaussian integration point. n This represents the displacement response of the element node.

[0130] It is not difficult to see that the stress response at the Gauss integration point can be obtained using equation (38), but the stress response at the element node cannot be obtained. Therefore, an improved stress solution is introduced here by stress smoothing. The stress solution needs to be obtained from the finite element method. Satisfying the weighted least squares principle, i.e.

[0131]

[0132] The stress improvement value here can be obtained using nodal stress interpolation, i.e.

[0133]

[0134] Substitute equation (40) into equation (39) and then... The first variation is 0, which gives us

[0135]

[0136] The nodal stresses can be obtained by solving equation (41) using Gauss integration, and the improved stress solution can be obtained by further using equation (40). For the incompatible element in this paper, at the Gauss integration point, there is... Substituting the Gauss integral points into the shape function of the incompatible element yields the stress improvement values ​​for the 8 nodes of an element.

[0137]

[0138] In the formula, a, b, c, d can be expressed as

[0139]

[0140] The improved stress at the element node can be obtained by using equation (42). For common nodes, the element stress can be processed.

[0141] S8. Determine whether the maximum stress response of the pipeline system is within the preset reasonable response range. If it is within the reasonable response range, the stress response design of the pipeline system is deemed reasonable; otherwise, it is deemed unreasonable.

[0142] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements, characterized in that, include: The stiffness matrix and mass matrix of the beam element are obtained based on the three-dimensional Timoshenko beam element. The characteristics of the pipeline solid part are simulated by using spatial 8-node non-coordinated solid elements, thereby obtaining the stiffness matrix and mass matrix of the non-coordinated solid elements; The pipeline system is meshed, including dividing the constrained area of ​​the pipeline system into solid meshes and the unconstrained area of ​​the pipeline system into beam element meshes; The connection between the beam element and the solid element node sharing the same cross section is realized by constructing an interface coupling element, and the stiffness matrix of the interface coupling element is obtained based on the displacement relationship of the interface coupling element. The element stiffness matrix and mass matrix of the pipeline system are obtained based on the stiffness matrix and mass matrix of beam elements, the stiffness matrix and mass matrix of non-coordinated solid elements, and the stiffness matrix of interface coupling elements. A basic excitation is applied to the pipeline system, and the nodal displacement response of the pipeline system is obtained based on the system's equation of motion. Based on the nodal displacement response of the pipeline system, the stress response of any element at the Gauss integration point can be obtained by using the Gauss integration point, and the stress response at the element node can be obtained by stress smoothing. Determine whether the maximum stress response of the pipeline system is within the preset reasonable response range. If it is within the reasonable response range, the stress response design of the pipeline system is considered reasonable; otherwise, it is considered unreasonable.

2. The method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements according to claim 1, characterized in that, The stiffness and mass matrices of the non-conforming solid elements are obtained from the following calculations: M e =∫ V ρN T NdV Among them, K e B is the stiffness matrix of the non-conforming solid element, B is the strain matrix corresponding to the principal degrees of freedom, and D is the elasticity matrix. M is the strain matrix corresponding to the additional degrees of freedom. e Let ρ be the mass matrix of the non-coordinated solid element, N be the element shape function matrix, and V be the volume.

3. The method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements according to claim 1, characterized in that, The stiffness matrix of the interface coupling unit is obtained according to the following calculation: Where K2 is the stiffness matrix of the interface coupling element, κ is the penalty factor, and G is the transformation matrix between nodal degrees of freedom and global degrees of freedom. c This is the displacement constraint matrix.

4. The method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements according to claim 1, characterized in that, The element stiffness matrix and mass matrix of the piping system are obtained from the following calculations: K=K s +K b +K c M=M s +M b Where K is the element stiffness matrix of the pipeline system, K s K is the stiffness matrix of the non-conforming solid element. b K is the stiffness matrix of the beam element. c M is the coupling stiffness matrix, and M is the element mass matrix of the piping system. s M is the mass matrix of the non-coordinated entity unit. b Let be the mass matrix of the beam element.

5. The method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements according to claim 1, characterized in that, The nodal displacement response of the piping system is obtained from the following calculations: Where x is the nodal displacement response of the pipeline system, F is the external load, ω is the external excitation frequency, α and β are the Rayleigh damping coefficients, and ω is the external excitation frequency. j Let Φ be the natural frequency of the j-th order. m The first m regularization matrices are obtained from modal analysis of the finite element model of the pipeline system. This corresponds to the j-th mode shape.

6. The method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements according to claim 1, characterized in that, The stress response of any element at the Gauss integration point is obtained from the following calculation: s n =DB n you n Where, σ n Let B be the stress at the Gaussian integration point, D be the elasticity matrix, and B be the stress at the Gaussian integration point. n Let u be the strain matrix corresponding to the Gaussian integration point. n This represents the displacement response of the element node.

7. The method for evaluating the stress response of a pipeline system based on the coupling of beam elements and solid elements according to claim 6, characterized in that, The stress response at element nodes is obtained through stress smoothing, including: introducing an improved stress solution through stress smoothing, which is obtained from the stress at the Gauss integration point, specifically expressed as follows: in, i = 1, 2, ..., 8 represent the stress improvement values ​​of 8 nodes in an element. These represent the stress solutions at the Gauss integration point of a single element obtained through the finite element method.

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