A transient heat transfer topology optimization method and system for truss structures based on the isogeometric stiffness diffusion method

Abstract

A transient heat transfer topology optimization method and system for truss structures based on isogeometric stiffness diffusion method, which relates to the field of engineering structure topology optimization design, and is proposed to improve the thermal protection performance of truss structures in unsteady thermal environments. Project the heat conduction and heat capacity matrices of bar elements onto the weak material isogeometric background mesh to construct an isogeometric transient heat transfer finite element model of the truss structure; use an interior point method that combines linear search and trust region optimization strategies to update the iterative design variables and seek the optimal layout design of the truss structure; construct an isogeometric transient heat transfer finite element model of the truss structure; use the trust region-based moving asymptote method to update the iterative design variables and seek the optimal layout design of the truss structure; according to the minimum size constraints of parameters such as the cross-section and length of bar elements, reconstruct the optimal layout of the truss structure to obtain a truss structure design layout that is convenient for engineering applications. The present invention is used for the layout optimization design of truss structures under transient heat flux loads.

Description

Technical Field

[0001] The present invention relates to the field of engineering structure topology optimization design. More specifically, the present invention relates to a transient heat transfer topology optimization method and system for truss structures based on the isogeometric stiffness diffusion method. Background Art

[0002] Truss-type load-bearing structures have excellent mechanical properties such as high specific stiffness and specific strength, and are widely used in military equipment such as rockets, missiles, and hypersonic aircraft. The aerodynamic heat load acting on the surface of the above-mentioned equipment has characteristics such as short heating time and large acting intensity, and the heat transfer process has a significant transient effect. With the development of weapon equipment, the requirements for structural thermal protection and lightweight in harsh thermal environments are becoming increasingly demanding. Therefore, the thermal protection design of truss-type load-bearing structures considering the influence of transient heat transfer has become a key and challenging problem.

[0003] For the layout optimization problem of truss mechanisms, the topology optimization method of continuum structures and the base structure method are mainly used. However, the former needs to introduce a manufacturable configuration of the structure obtained based on skeleton extraction technology; the latter transforms the local optimization problem of the truss into a highly redundant member size optimization problem, and its optimization results are often too complex to be directly applied to engineering design.

[0004] The prior art with the document number DOI: 10.26991 / d.cnki.gdllu.2022.003469 discloses an efficient analysis and optimization method for load-bearing structures based on the isogeometric stiffness diffusion method. This prior art can only solve the topology optimization (i.e., layout optimization) problem of truss structures under static loads. The lightweight truss structures widely used in the aviation and aerospace fields usually serve in harsh unsteady thermal environments. However, in the existing research on the thermal protection topology optimization design of truss structures, transient heat transfer is usually equivalent to steady-state heat transfer, ignoring the time correlation of the heat dissipation process. The simplification of the above design thermal load results in the optimized structure not meeting the design requirements in the actual thermal environment. Especially under the action of short-time thermal shock loads, the topology optimization problem of truss structures considering transient heat transfer effects is more complex and difficult to achieve. In view of the unsteady heat transfer characteristics of truss structures in the aviation and aerospace fields, studying the topology optimization problem of truss structures under transient thermal loads is more in line with practical applications, but few people in the prior art have given a clear solution. Summary of the Invention

[0005] The technical problem to be solved by the present invention is:

[0006] In view of the above-mentioned problems, disadvantages and deficiencies, the present invention provides a transient heat transfer topology optimization method and system for truss structures based on the isogeometric stiffness diffusion method.

[0007] To achieve the above object, in a first aspect, the present invention provides a transient heat transfer topology optimization method for a truss structure based on the isogeometric stiffness diffusion method, which includes the following steps:

[0008] S101: Construct a design domain of the truss structure on the isogeometric background mesh of the weak material, set the thermal boundary conditions and apply transient heat flux loads, and initialize design variables such as the node positions and cross-sectional areas of the rod elements and algorithm parameters: the thermal material properties of the rods, the thermal material properties and size information of the isogeometric background mesh of the weak material, and the convergence parameters;

[0009] S102: Construct an isogeometric transient heat transfer finite element model of the truss structure and calculate the transient heat dissipation efficiency of the truss structure under the action of transient heat flux;

[0010] S103: Establish an optimization model for minimizing the transient heat dissipation efficiency of the truss structure, and combine the initialized design variables and algorithm parameters of the design domain to solve and obtain the optimal topology (layout) of the heat protection truss structure;

[0011] S104: Post-process the obtained optimal topology of the heat protection truss structure considering transient effects, reconstruct to obtain a clear design layout of the truss structure, and finally obtain the transient heat transfer topology design of the truss structure based on the isogeometric stiffness diffusion method.

[0012] In a second aspect, the present invention provides a transient heat transfer topology optimization system for a truss structure based on the isogeometric stiffness diffusion method, which includes:

[0013] An initialization module that establishes an initial layout of the truss structure on the isogeometric background mesh, initializes design variables (including node coordinates and cross-sectional areas) of the rod elements and optimization algorithm parameters: the thermal conductivity material properties of the rods, the thermal conductivity material properties and size information of the isogeometric background mesh of the weak material, and the convergence parameters, and sets external loads and boundary conditions;

[0014] An isogeometric finite element analysis module that uses the isogeometric stiffness diffusion method to obtain the heat conduction and heat capacity diffusion matrices of the truss elements, assembles the heat conduction and heat capacity matrices of the truss structure according to the node numbers corresponding to the isogeometric background mesh of the weak material to form an isogeometric transient heat transfer finite element model of the truss structure. Solve the above model using an unconditionally stable generalized Crank–Nicolson finite difference scheme to calculate the transient heat dissipation efficiency of the truss structure;

[0015] An optimization solution module that calculates sensitivity information, uses the moving asymptote method based on the trust region to update the design variables of the rod elements, and determines whether the optimization iteration result satisfies the convergence criterion. If it does not satisfy, continue with the optimization iteration until the convergence criterion is met;

[0016] The post-processing module reconstructs the optimal layout of the truss structure according to the minimum size constraints of parameters such as the cross-section and length of the bar elements, so as to obtain a truss structure design layout convenient for engineering applications.

[0017] The present invention relates to a transient heat transfer topology optimization method and system for truss structures based on the isogeometric stiffness diffusion method, and has the following beneficial technical effects:

[0018] The present invention considers the problem of efficient heat conduction and lightweight design of truss structures under transient heat flux loads. Through the construction of a transient heat transfer analysis model and an optimization algorithm, the topology optimization of truss structures under transient heat flux loads is realized, so as to improve the thermal protection performance of truss structures in unsteady thermal environments and solve the problem of efficient heat conduction and lightweight design of truss structures under transient heat flux loads.

[0019] The method of the present invention projects the heat conduction and heat capacity matrices of bar elements onto a weak material isogeometric background grid through the principle of energy equivalence to construct an isogeometric transient heat transfer finite element model of the truss structure; based on the strategy of discrete-then-differentiate sensitivity analysis, the consistent sensitivity analysis of the time-space discrete transient heat transfer system is realized; the interior point method integrating the linear search and trust region optimization strategies is used to update the iterative design variables and seek the optimal layout design of the truss structure. Thus, a transient heat transfer topology optimization method for truss structures based on the isogeometric stiffness diffusion method is proposed, including: defining the initial design domain and algorithm parameters, constructing an isogeometric transient heat transfer finite element model of the truss structure; solving the above model by the unconditionally stable generalized Crank–Nicolson finite difference method, calculating the objective function, and obtaining the sensitivity values of the design variables based on the discrete-then-differentiate sensitivity analysis strategy; using the trust-region-based moving asymptotes method to update the iterative design variables and seek the optimal layout design of the truss structure; reconstructing the optimal layout of the truss structure according to the minimum size constraints of parameters such as the cross-section and length of the bar elements, and further obtaining a truss structure design layout convenient for engineering applications.

[0020] The present invention is used to solve the layout optimization design problem of truss structures under transient heat flux loads. Description of the Drawings

[0021] Figure 1 It is a flow chart of the transient heat transfer topology optimization method for truss structures based on the isogeometric stiffness diffusion method of the present invention;

[0022] Figure 2 It is a schematic diagram of the principle of the isogeometric stiffness diffusion method for transient heat transfer problems of the present invention;

[0023] Figure 3 It is a schematic structural diagram of a transient heat transfer topology optimization system for truss structures based on the isogeometric stiffness diffusion method of the present invention;

[0024] Figure 4 Among them, (a) is a schematic diagram of the design domain of Embodiment 1 of the present invention;

[0025] (b) is a schematic diagram of the transient heat flux load of Embodiment 1 of the present invention;

[0026] Figure 5 Among them, (a) is the optimized layout of Embodiment 1 of the present invention when the heat flux load acting time t f = 3 s;

[0027] (b) is the optimized layout of Embodiment 1 of the present invention when the heat flux load acting time t f = 30 s;

[0028] (c) is the optimized layout of Embodiment 1 of the present invention when the heat flux load acting time t f = 600 s;

[0029] Figure 6 Among them, (a) is the convergence curve of the optimization process of Embodiment 1 of the present invention when the heat flux load acting time t f = 3 s;

[0030] (b) is the convergence curve of the optimization process of Embodiment 1 of the present invention when the heat flux load acting time t f = 30 s;

[0031] (c) is the convergence curve of the optimization process of Embodiment 1 of the present invention when the heat flux load acting time t f = 600 s;

[0032] Figure 7 Among them, (a) is the optimized layout of Embodiment 1 of the present invention using the variable density method when the heat flux load acting time t f = 3 s;

[0033] (b) is the optimized layout of Embodiment 1 of the present invention using the variable density method when the heat flux load acting time t f = 30 s;

[0034] (c) is the optimized layout of Embodiment 1 of the present invention using the variable density method when the heat flux load acting time t f = 600 s;

[0035] (d) is the comparison of the optimized target values obtained by using the method of the present invention and the variable density method in Embodiment 1 of the present invention. Detailed implementation manners

[0036] The existing research work on truss - like discrete structures is based on steady - state heat conduction analysis to optimize the heat conduction path of materials. However, in actual engineering, the temperature field changes with time, that is, it has transient characteristics. Steady - state heat conduction ignores the historical information of the heat conduction process, and this information will have a significant impact on the topological results. Especially for the thermal protection design of structures under short - time thermal shock loads, the transient effect of heat conduction problems should be considered. In addition, the transient dynamic topology optimization problem of discrete structures will face complex sensitivity analysis and the problem of convergence difficulties caused by local optimal solution traps, making it difficult to obtain an effectively engineered truss - type load - bearing structure. To address the above problems, the present invention proposes an isogeometric stiffness diffusion method for transient heat transfer problems to achieve topological optimization (i.e., layout optimization) of the thermal protection performance of truss structures. The implementation process of the present invention is described below with reference to the accompanying drawings.

[0037] Combined with the attached Figures 1 to 7 drawings, the transient heat transfer topology optimization of truss structures based on the isogeometric stiffness diffusion method described in the present invention is elaborated in detail, so that the advantages and features of the present invention can be more easily understood by those skilled in the art, thereby making a clearer and more definite definition of the protection scope of the present invention.

[0038] In the first aspect, the present invention provides transient heat transfer topology optimization of truss structures based on the isogeometric stiffness diffusion method, and its flowchart is as Figure 1 shown, mainly including the following steps:

[0039] Step S101: Construct the design domain of the truss structure on the isogeometric background grid of weak materials, set the thermal boundary conditions and apply transient heat flux loads, and initialize the design variables such as the node positions and cross - sectional areas of the rod elements, as well as the algorithm parameters: the thermal material properties of the rods, the material properties and size information of the isogeometric background grid of weak materials, and the convergence parameters.

[0040] In the present invention, an initial layout of the truss structure is established on the isogeometric background grid of weak materials, the thermal boundary conditions and transient heat flux loads are applied, and design variables and initial values are assigned to any rod element, and the algorithm parameters are set.

[0041] Step S102: Construct an isogeometric transient heat transfer finite - element model of the truss structure and calculate the transient heat dissipation efficiency of the truss structure under transient heat flux.

[0042] In the present invention, step S102 can be further included:

[0043] Step S102 - 1: Based on the principle of energy equivalence, project the heat conduction and heat capacity matrices of the rod elements onto the isogeometric background grid of weak materials to form an isogeometric transient heat transfer finite - element model of the truss structure.

[0044] Different from the steady-state heat conduction problem, the transient topology optimization problem of a truss structure needs to consider the influence of the time correlation of the heat transfer process. Therefore, how to incorporate the heat conduction and heat capacity properties of the truss structure into the geometric finite element model such as the transient heat transfer of the discrete system is the key to this invention. As Figure 2 shown, based on the principle of energy equivalence, the heat conduction matrix and heat capacity matrix of the bar element are diffused onto the geometric background grid of the weak material, and then there is:

[0045]

[0046] In the formula: and are the heat conduction and heat capacity diffusion matrices obtained by projecting the i-th bar element onto the geometric background grid of the weak material respectively, and are the element heat conduction and heat capacity matrices of the i-th bar element in the global coordinate system respectively, T ei 、 are the temperature and temperature time derivative arrays of the geometric background grid element respectively, T rei 、 are the temperature and temperature time derivative arrays of the bar element respectively.

[0047] Combined with the temperature coupling relationship T rei =N T T ei , the heat conduction and heat capacity diffusion matrices obtained by projecting the bar element onto the background grid are respectively:

[0048]

[0049] In the formula: N T is the shape function matrix.

[0050] Assemble the heat conduction and heat capacity diffusion matrices of the bar element on the geometric background grid of the weak material to obtain the overall heat conduction matrix K T and the heat capacity matrix C T , then there is:

[0051]

[0052] In the formula: K Tp 、C Tp represent the global heat conduction matrix and heat capacity matrix of the geometric background grid of the weak material respectively, and n is the total number of bar elements;

[0053] The geometric transient heat transfer finite element equation of the truss structure is:

[0054]

[0055] In the formula: Tt are the temperature time derivative and the temperature array of all control points of the isogeometric background mesh at the t-th time step, respectively, and P t is the heat flux load array at the t-th time step, and N t is the total number of time steps for the dynamic event.

[0056] Step S102-2: According to the stability condition, construct an unconditionally stable generalized Crank–Nicolson finite difference scheme to solve the isogeometric transient heat transfer finite element model of the truss structure constructed in claim 3, and calculate the transient heat dissipation efficiency of the truss structure.

[0057] For the isogeometric transient heat transfer finite element equation of the truss structure, which is solved by an unconditionally stable generalized Crank–Nicolson finite difference scheme, there is:

[0058]

[0059] Where: θ is the generalized Crank–Nicolson algorithm parameter. To ensure the unconditional stability of the algorithm, set θ = 1, and Δτ is the time increment step.

[0060] By using an unconditionally stable generalized Crank–Nicolson finite difference scheme to solve the isogeometric transient heat transfer finite element equation of the truss structure, the transient heat transfer dissipation efficiency of the following truss structure can be calculated:

[0061]

[0062] Where: (x i , y i ), (x i+1 , y i+1 ), and A i are the coordinates and cross-sectional area of node i and node i + 1 of the i-th bar element, and S i = {x i , y i , x i+1 , y i+1 , A i} is the design variable attached to the i-th bar element.

[0063] Step S103: Establish an optimization model for minimizing the transient heat dissipation efficiency of the truss structure, and combine the initial design variables and algorithm parameters of the design domain to solve and obtain the optimal topology (layout) of the heat-resistant truss structure.

[0064] In the present invention, step S103 may further include:

[0065] Step S103-1: Establish an optimization model for the problem of minimizing the transient heat dissipation efficiency of the truss structure, solve the adjoint sensitivity analysis equation based on discrete-then-differentiate, and calculate the sensitivity of the transient heat dissipation efficiency of the truss structure to volume constraints.

[0066] The problem of minimizing the transient heat dissipation efficiency of the truss structure is defined as:

[0067]

[0068]

[0069]

[0070]

[0071] In the formula: Briefly denoted as J0, J1 is the volume constraint, l i is the length of the i-th bar element, V max is the maximum allowable volume usage of the truss structure, x min and x max are the variation ranges of the x-direction coordinates of the bar element nodes, y min and y max are the variation ranges of the y-direction coordinates of the bar element nodes. The above coordinate ranges depend on the background grid size.

[0072] For the transient heat transfer optimization problem, the order of differentiation and discretization may affect the calculation accuracy of the time-domain sensitivity; the traditional "differentiate-first-then-discretize" adjoint variable method will cause consistency errors in sensitivity calculation; to avoid the above problems, the adjoint variable method based on "discrete-then-differentiate" is used to implement the sensitivity analysis of the transient heat transfer problem. Thus, the Lagrange multiplier λ t is introduced to construct the augmented function of the original objective function. According to the chain rule of differentiation, the sensitivity of the transient heat dissipation efficiency of the truss structure is expressed as:

[0073]

[0074] The Lagrange multiplier λ t can be obtained by solving the adjoint equation, and its calculation formula is:

[0075] For t = N t :

[0076]

[0077] For t = N t -1, L, 0:

[0078]

[0079] Given that the heat conduction and heat capacity matrices of the geometric background grid of weak materials and the like are independent of the design variables, the following is satisfied:

[0080]

[0081] The first-order partial derivative of the heat conduction and heat capacity diffusion matrices of the bar element with respect to S i is:

[0082]

[0083]

[0084] The sensitivities of the heat conduction and heat capacity matrices of the bar element with respect to A i , x i and y i are:

[0085]

[0086]

[0087]

[0088] In the formula: k c and c are the heat conduction coefficient and heat capacity of the truss material respectively. Similarly, the sensitivities of the heat conduction matrix and heat capacity matrix of the bar element with respect to x i+1 and y i+1 can be obtained.

[0089] The sensitivity calculation formula of the bar element length parameter with respect to the node coordinate variable is as follows:

[0090]

[0091]

[0092] In addition, it is necessary to calculate the sensitivity of the volume constraint with respect to the design variable as dG / dS i , which is expressed in components as:

[0093] dG / dx i = A i dl i / dx i , dG / dy i = A i dl i / dy i

[0094] dG / dx i+1 = A i dl i / dx i+1, dG / dy i+1 = A i dl i / dy i+1

[0095] dG / dA i = l i

[0096] Step S103-2: Transform the original transient heat dissipation efficiency minimization problem into an unconstrained problem, and use the trust-region-based moving asymptote method to update the design variables until the convergence criterion is satisfied or the maximum number of iterations is reached.

[0097] The original transient heat dissipation efficiency minimization problem is transformed into the following unconstrained problem, so there is:

[0098]

[0099]

[0100] In the formula: J(s il , Λ, μ) is the objective function of the above unconstrained problem, s il is the component form of the design variable S i , and there is {s il} l＝1～5 = S i = {x i , y i , x i+1 , y i+1 , A i}, Λ is the multiplier, μ is the penalty parameter, and l = max[J1, -Λ / μ].

[0101] Estimate the objective function of the above unconstrained problem using the moving asymptote function, so there is:

[0102]

[0103] In the formula: is the estimation function of J(s il , Λ, μ) at the k-th iteration, is the moving asymptote function for estimating J j (j = 0, 1), Λ k is the value of the multiplier Λ at the k-th iteration, μ k is the value of the penalty parameter μ at the k-th iteration, is the update step size of s il within the trust region.

[0104] In order to effectively control the moving asymptote function for J j(j = 0, 1) The estimation accuracy is introduced with a trust region radius in which is expressed as:

[0105]

[0106]

[0107]

[0108]

[0109] In the formula: is the k-th iteration value of the trust region radius, represents represents and s il is the upper and lower limit values of s il .

[0110] To test the effectiveness of the design variable update step size , the following merit function needs to be calculated:

[0111]

[0112]

[0113] In the formula: χ k is the value of the merit function χ at the k-th iteration, and Λ k+1 is the value of the multiplier Λ at the (k + 1)-th iteration.

[0114] If χ k > 0, is a valid value, update the design variable and the penalty parameter, then we have:

[0115]

[0116]

[0117] In the formula: μ k+1 is the value of the penalty parameter μ at the (k + 1)-th iteration, is the k-th iteration of the measurement J which is a metric parameter for the deviation from the volume constraint.

[0118] If χ k ≤ 0, is an invalid value, then the trust region radius needs to be updated. Execute the above loop process until the convergence criterion is met or the maximum number of iterations is reached.

[0119] S104. Post-process the dynamically optimal topology of the obtained truss structure, reconstruct to obtain a clear design layout of the truss structure, and finally obtain the transient heat transfer topology design of the truss structure based on the isogeometric stiffness diffusion method.

[0120] If the optimized cross-sectional area A < A min or l < l min , then delete the corresponding bar element to obtain a clear design layout of the final truss structure. Generally, select A min = 0.005A max (the maximum cross-sectional area of the bar in the optimal layout), l min = 0.001l max (the maximum length of the bar in the optimal layout), then delete the corresponding bar element to obtain a clear design layout of the final truss structure.

[0121] In a second aspect, the present invention provides a truss structure dynamic topology optimization system based on the isogeometric stiffness and mass diffusion method, as Figure 3 shown, which includes:

[0122] An initialization module that establishes an initial layout of the truss structure on the isogeometric background grid, initializes the design variables of the bar elements (including node coordinates and cross-sectional area) and optimization algorithm parameters: the thermal conductivity material properties of the bars, the thermal conductivity material properties and size information of the isogeometric background grid of the weak material, and the convergence parameters, and sets the external load and boundary conditions;

[0123] An isogeometric finite element analysis module that applies the isogeometric stiffness diffusion method to obtain the heat conduction and heat capacity diffusion matrices of the truss elements, assembles the heat conduction and heat capacity matrices of the truss structure according to the node numbers corresponding to the isogeometric background grid of the weak material, and forms an isogeometric transient heat transfer finite element model of the truss structure. Solve the above model using an unconditionally stable generalized Crank–Nicolson finite difference scheme to calculate the transient heat dissipation efficiency of the truss structure;

[0124] An optimization solution module that calculates the sensitivity information, uses the moving asymptote method based on the trust region to update the design variables of the bar elements, and determines whether the optimization iteration result meets the convergence criterion. If it does not meet the criterion, continue the optimization iteration until the convergence criterion is met;

[0125] A post-processing module that reconstructs the optimal layout of the truss structure according to the minimum size constraints of parameters such as the cross-section and length of the bar elements to obtain a truss structure design layout convenient for engineering applications. Embodiment

[0126] Example 1

[0127] Step S101. Construct the design domain of the truss structure on the geometric background grid of the weak material, set the thermal boundary conditions and apply the transient heat flux load, and initialize the design variables such as the node positions and cross-sectional areas of the rod elements and the algorithm parameters, (as Figure 4 shown).

[0128] As Figure 4 shown, the structural dimensions of the design domain are: length L = 20 mm, thickness h = 1 mm. The heat transfer between the square plate and the environment is ignored, that is, the four sides are set as adiabatic boundaries. The thermal conductivity of the material kc = 10 W / (m·K), and the specific heat capacity c = 10 6 J / (m 3 ·K). Temperature constraints t0 = 0 °C are set at the four corner points of the square plate, and a step heat flux q(t) = 0.1 W (0 ≤ t ≤ t f ) is applied at the geometric center. The heating times t f are 3 s, 30 s, and 600 s respectively. The number of sampling points for the time-domain response The maximum number of iterations is selected as 500, the upper limit of the truss structure volume is 60, and the background grid size is 1 mm.

[0129] Step S102. Construct the isogeometric transient heat transfer finite element model of the truss structure and calculate the transient heat dissipation efficiency of the truss structure under the action of the transient heat flux.

[0130] Step S102-1. Based on the principle of energy equivalence, project the heat conduction and heat capacity matrices of the rod elements onto the isogeometric background grid of the weak material to form the isogeometric transient heat transfer finite element model of the truss structure.

[0131] Based on the principle of energy equivalence, diffuse the heat conduction and heat capacity matrices of the rod elements onto the isogeometric background grid of the weak material, then there are:

[0132]

[0133] In the formula: and are the heat conduction and heat capacity diffusion matrices obtained by projecting the i-th rod element onto the isogeometric background grid of the weak material respectively, and are the element heat conduction and heat capacity matrices of the i-th rod element in the global coordinate system respectively, T ei , are the temperature and temperature time derivative arrays of the isogeometric background grid element respectively, T rei , are the temperature and temperature time derivative arrays of the rod element respectively.

[0134] Combined with the temperature coupling relationship T rei = N T Tei The heat conduction and heat capacity diffusion matrices obtained by projecting the rod element onto the geometric background mesh of the weak material are respectively:

[0135]

[0136] where: N T is the non-uniform rational B-spline function matrix.

[0137] Assembling the heat conduction and heat capacity diffusion matrices of the rod element on the geometric background mesh of the weak material, the overall heat conduction matrix K T and the heat capacity matrix C T are obtained, and then:

[0138]

[0139] where: K Tp , C Tp represent the global heat conduction matrix and heat capacity matrix of the geometric background mesh of the weak material, and n is the total number of rod elements;

[0140] The isogeometric transient heat transfer finite element equation of the truss structure is:

[0141]

[0142] where: T t are respectively the temperature time derivative and temperature array of all control points of the isogeometric background mesh at the t-th time step, P t is the heat flux load array at the t-th time step, N t is the total time step of the dynamic event.

[0143] Step S102-2: Use the generalized Crank–Nicolson finite difference method to solve the above model and calculate the transient heat dissipation efficiency of the truss structure.

[0144] For the isogeometric transient heat transfer finite element equation of the truss structure, solved by the unconditionally stable generalized Crank–Nicolson finite difference scheme, we have:

[0145]

[0146] where: θ is the generalized Crank–Nicolson algorithm parameter. To ensure the unconditional stability of the algorithm, θ = 1, and Δτ is the time increment step.

[0147] By using the unconditionally stable generalized Crank–Nicolson finite difference scheme to solve the isogeometric transient heat transfer finite element equation of the truss structure, the transient heat transfer dissipation efficiency of the following truss structure can be calculated.

[0148]

[0149] In the formula: (x i , y i ), (x i+1 , y i+1 ), and A i are the coordinates and cross-sectional areas of node i and node i + 1 of the i-th bar element, and S i = {x i , y i , x i+1 , y i+1 , A i} is the design variable attached to the i-th bar element.

[0150] Step S103: Establish an optimization model for minimizing the transient heat dissipation efficiency of the truss structure, and solve it by combining the initial design variables and algorithm parameters of the design domain to obtain the optimal topology (layout) of the heat-resistant truss structure.

[0151] Step S103-1: Establish an optimization model for minimizing the transient heat dissipation efficiency of the truss structure, solve the adjoint sensitivity analysis equation based on discrete-then-differentiate, and calculate the sensitivity of the transient heat dissipation efficiency and volume constraint of the truss structure.

[0152] The problem of minimizing the transient heat dissipation efficiency of the truss structure is defined as:

[0153]

[0154]

[0155]

[0156]

[0157] In the formula: Briefly denoted as J0, J1 is the volume constraint, l i is the length of the i-th bar element, V max is the maximum allowable volume usage of the truss structure, x min and x max are the ranges of the x-direction coordinates of the bar element nodes, and y min and y max are the ranges of the y-direction coordinates of the bar element nodes. The above coordinate ranges depend on the background grid size.

[0158] The sensitivity analysis of the transient heat transfer problem is implemented using the adjoint variable method based on "discrete-then-differentiate". Thus, the Lagrange multiplier λ tConstruct the augmented function of the original objective function. According to the chain rule of differentiation, the sensitivity of the transient heat dissipation efficiency of the truss structure is expressed as:

[0159]

[0160] The Lagrange multiplier λ t can be obtained by solving the adjoint equation, and its calculation formula is:

[0161] For t = N t :

[0162]

[0163] For t = N t -1, L, 0:

[0164]

[0165] Given that the heat conduction and heat capacity matrices of the geometric background grid such as weak materials are independent of the design variables, then it satisfies:

[0166]

[0167] The first-order partial derivatives of the heat conduction and heat capacity diffusion matrices of the rod element with respect to S i are:

[0168]

[0169]

[0170] The sensitivities of the heat conduction and heat capacity matrices of the rod element with respect to A i , x i and y i are:

[0171]

[0172]

[0173]

[0174] In the formula: k c and c are the heat conduction coefficient and heat capacity of the truss material respectively. Similarly, the sensitivities of the heat conduction matrix and heat capacity matrix of the rod element with respect to x i+1 and y i+1 can be obtained.

[0175] The calculation formula for the sensitivity of the rod element length parameter with respect to the node coordinate variable is as follows:

[0176]

[0177]

[0178] In addition, it is necessary to calculate the sensitivity of the volume constraint with respect to the design variable as dG / dS i , which in component form is:

[0179] dG / dx i = A i dl i / dx i , dG / dy i = A i dl i / dy i

[0180] dG / dx i+1 = A i dl i / dx i+1 , dG / dy i+1 = A i dl i / dy i+1

[0181] dG / dA i = l i

[0182] Step S103-2: Transform the original transient heat dissipation efficiency minimization problem into an unconstrained problem, and use the trust-region-based moving asymptote method to update the design variables until the convergence criterion is satisfied or the maximum number of iterations is reached.

[0183] The original transient heat dissipation efficiency minimization problem is transformed into the following unconstrained problem, so there is:

[0184]

[0185]

[0186] In the formula: J(s il , Λ, μ) is the objective function of the above unconstrained problem, s il is the component form of the design variable S i , and there is {s il} l＝1～5 = S i = {x i , y i , x i+1 , y i+1 , A i}, Λ is the multiplier, μ is the penalty parameter, and l = max[J1, -Λ / μ].

[0187] Estimate the objective function of the above unconstrained problem using the moving asymptote function, so there is:

[0188]

[0189] Wherein: is the estimation function of J(s il , Λ, μ) at the k-th iteration, is the moving asymptote function for estimating J j (j = 0, 1), Λ k is the value of the multiplier Λ at the k-th iteration, μ k is the value of the penalty parameter μ at the k-th iteration, is s il 's update step size within the trust region.

[0190] In order to effectively control the estimation accuracy of the moving asymptote function for J j (j = 0, 1), a trust region radius is introduced in and is expressed as:

[0191]

[0192]

[0193]

[0194]

[0195] Wherein: is the value of the trust region radius at the k-th iteration, represents represents and s il is s il 's upper and lower limit values.

[0196] In order to test the effectiveness of the design variable update step size , the following value function needs to be calculated:

[0197]

[0198]

[0199] Wherein: χ k is the value of the value function χ at the k-th iteration, Λ k+1 is the value of the multiplier Λ at the (k + 1)-th iteration.

[0200] If χ k > 0, If it is a valid value, update the design variables and penalty parameters, then we have:

[0201]

[0202]

[0203] where: μ k+1 is the value of the penalty parameter μ at the (k + 1)-th iteration, is the k-th iteration of measuring J is the metric parameter deviating from the volume constraint.

[0204] If χ k ≤ 0, is an invalid value, then the trust region radius needs to be updated. Execute the above loop process until the convergence criterion is met or the maximum number of iterations is reached.

[0205] Step S104: Post-process the optimal topology of the heat-proof truss structure considering transient effects, reconstruct to obtain a clear design layout of the truss structure, and finally obtain the transient heat transfer topology design of the truss structure based on the isogeometric stiffness diffusion method.

[0206] If the optimized cross-sectional area A < A min or l < l min , then delete the corresponding bar element to obtain a clear design layout of the final truss structure. Select A min = 0.005A max (the maximum cross-sectional area of the bar in the optimal layout), l min = 0.001l max (the maximum length of the bar in the optimal layout), then delete the corresponding bar element to obtain a clear design layout of the final truss structure.

[0207] The specific optimization results are as Figures 5 to 7 shown.

[0208] Figure 5 These are the optimization results of the method of the present invention. When t f = 3s (as in Figure 5 (a)), the heat conduction is in the initial stage, and the truss is mainly distributed in the diagonal direction near the heat source, and the bars with smaller cross-sectional dimensions face the adiabatic boundaries around. When t f = 30s (as in Figure 5 (b)), the distribution of the truss is affected by the temperature constraints at the four corners of the square plate, showing a diagonal radial divergence distribution pattern. When t f = 600s (as in Figure 5 (c)), the heat conduction of the truss reaches a steady state. Thus, the transient heat transfer effect directly affects the optimized layout of the truss structure.

[0209] Figure 6 corresponding to Figure 5 the iterative process of topology optimization, where the transient heat dissipation efficiency and volume constraint both converge stably to the final values.

[0210] Figure 7 This is the topology optimization result of the truss structure using the variable density method in Example 1 of the present invention. Comparing Figure 5 with Figure 7 , under the same boundary conditions and material usage constraint conditions, for the above three design conditions, the present invention method and the variable density method present similar layout optimization results, but the former produces a clearer and more manufacturable geometric configuration, especially the target performance is significantly better than the latter (as shown in Figure 7 (d)). Thus, the truss layout optimization method based on the isogeometric stiffness diffusion method can obtain a heat protection design scheme with better performance, and the truss layout optimization result does not require complex feature extraction and can be directly used for engineering design.

[0211] The result of Example 1 of the present invention shows that compared with the variable density method, the method of the present invention can obtain a clearer truss layout and better target performance, thus verifying the superiority of the algorithm of the present invention for the truss dynamic layout optimization problem.

[0212] The present invention has been described in detail above in combination with specific embodiments and exemplary examples, but these descriptions should not be construed as limiting the present invention. Those skilled in the art understand that without departing from the spirit and scope of the present invention, various equivalent substitutions, modifications or improvements can be made to the technical solutions and their implementation manners of the present invention, and all of these fall within the scope of the present invention.

Claims

1. A transient heat transfer topology optimization method for truss structures based on the equal geometric stiffness diffusion method, characterized in that: The steps include: S101: Construct the design domain of the truss structure on the geometric background mesh of weak materials, set thermal boundary conditions and apply transient thermal flow loads, and initialize design variables such as node positions and cross-sectional areas of the bar elements and algorithm parameters: thermal material properties of the bar, material properties and size information of the geometric background mesh of weak materials, and convergence parameters; S102: Construct an isogeometric transient heat transfer finite element model of the truss structure and calculate the transient heat dissipation efficiency of the truss structure under the action of transient heat flux; S103: Establish an optimization model for minimizing the transient heat dissipation efficiency of the truss structure, combine the initialization design variables and algorithm parameters of the design domain, and solve to obtain the optimal topology of the heat-resistant truss structure; S104: Post-process the optimal topology of the heat-resistant truss structure considering transient effects, reconstruct and obtain a clear design layout of the truss structure, and finally obtain the transient heat transfer topology design of the truss structure based on the equal geometric stiffness diffusion method; Step S102 includes: Step S102-1, based on the energy equivalence principle, project the heat conduction and heat capacity matrix of the rod unit onto the isogeometric background grid of the weak material to form an isogeometric transient heat transfer finite element model of the truss structure; Step S102-2, using the generalized Crank–Nicolson finite difference method to solve the above model and calculate the transient heat dissipation efficiency of the truss structure; The step S102-1 includes: Based on the energy equivalence principle, the heat conduction matrix and heat capacity matrix of the rod unit are diffused to the geometric background grid of weak materials, and the temperature coupling relationship T between the rod unit and the geometric background grid of weak materials is combined. rei =N T T ei , then the heat conduction and heat capacity diffusion matrices obtained by projecting the rod unit onto the geometric background grid of weak materials are: Where: and are the heat conduction and heat capacity diffusion matrices obtained by projecting the i-th rod element onto the geometric background grid of weak material, and are the unit heat conduction and heat capacity matrices of the i-th rod unit in the global coordinate system, N T is the non-uniform rational B-spline function matrix, T ei and T rei They are the temperature arrays of the geometric background grid cells and rod cells of weak materials; The heat conduction and heat capacity diffusion matrices of the rod unit are assembled on the geometric background grid of weak materials to obtain the overall heat conduction matrix K T and the heat capacity matrix C T , then: Where: K Tp , C Tp represents the global heat transfer matrix and heat capacity matrix of the geometric background grid of weak materials, n is the total number of rod elements; The isogeometric transient heat transfer finite element equation of the truss structure is: Where: T t are the temperature time derivatives and temperature arrays of all control points of the geometric background grid of weak materials at the tth time step, P t is the heat flux load array at the tth time step, N t is the total time step of the dynamics event; Step S102-2 includes: According to the stability condition, an unconditionally stable generalized Crank–Nicolson finite difference scheme is constructed to solve the isogeometric transient heat transfer finite element model of the truss structure constructed in claim 3 and calculate the transient heat dissipation efficiency of the truss structure; The step S103 includes: Step S103-1, establishing an optimization model for minimizing the transient heat dissipation efficiency of the truss structure, solving the adjoint sensitivity analysis equation based on first discretization and then differentiation, and calculating the sensitivity of the transient heat dissipation efficiency of the truss structure to the volume constraint; Step S103-2, converting the original transient heat dissipation efficiency minimization problem into an unconstrained problem, and using a trust region-based moving asymptote method to update the design variables until the convergence criterion is met or the maximum number of iterations is reached; The step S103-1 includes: The problem of minimizing the transient heat dissipation efficiency of the truss structure is defined as: Where: Δτ is the time increment, is the transient heat dissipation efficiency of the truss structure (abbreviated as J0), S i ={x i ,y i ,x i+1 ,y i+1 ,A i } is the design variable attached to the i-th bar element, where (x i ,y i ) and (x i+1 ,y i+1 ) are the coordinates of nodes i and i+1 of the i-th rod element, J1 is the volume constraint, A i is the cross-sectional area of ​​the i-th rod element, l i is the length of the i-th rod unit, V max is the maximum volume allowed for the truss structure, x min and x max is the coordinate variation range of the rod unit node along the x direction, y min and max is the coordinate range of the rod unit node along the y direction, and the above coordinate range depends on the background grid size; For transient heat transfer optimization problems, the order of differentiation and discretization may affect the calculation accuracy of time domain sensitivity; the traditional "differentiation first-then discretization" adjoint variable method will cause consistency errors in sensitivity calculations; in order to avoid the above problems, the adjoint variable method based on "discretization first-then differentiation" is used to implement sensitivity analysis of transient heat transfer problems; the Lagrange multiplier λ is introduced t The augmented function of the original objective function is constructed. According to the chain derivation rule, the sensitivity of the transient heat dissipation efficiency of the truss structure is expressed as: Lagrange multiplier λ t It can be obtained by solving the adjoint equation, and its calculation formula is: For t = N t : For t = N t -1,…,0:

2. The transient heat transfer topology optimization method for truss structure based on equal geometric stiffness diffusion method according to claim 1 is characterized in that: The corresponding step S103-2 includes: The problem of minimizing the efficiency of transient heat dissipation in the prime mover is transformed into the following unconstrained problem: Where: J(s il ,Λ,μ) is the objective function of the above unconstrained problem, s il is the design variable S i The component form of {s il } l＝1～5 =S i ={x i ,y i ,x i+1 ,y i+1 ,A i }, Λ is the multiplier, μ is the penalty parameter, l = max[J1, -Λ / μ]; Using the moving asymptote function to estimate the objective function of the above unconstrained problem, we have: Where: For the kth iteration, for J(s il ,Λ,μ) estimation function, To estimate J j The moving asymptote function, Λ k is the value of the multiplier Λ at the kth iteration, μ k is the value of the penalty parameter μ at the kth iteration, For il The update step size within the trust region; In order to effectively control the moving asymptote function For J j The estimation accuracy of The trust region radius is introduced in ; then, the design variable update step size is checked by the value function The effectiveness of If it is valid, update the design variables; Design variables il The update formula is: Where: and For il The values ​​at the k+1th and kth iterations, and s il For il The upper and lower limits of If the update step size If it is invalid, the trust region radius is updated based on the trust region framework, and the above cycle is executed until the convergence criterion is met or the maximum number of iterations is reached.

3. A transient heat transfer topology optimization system for truss structures based on the equal geometric stiffness diffusion method, the system being used to implement the method described in claim 1 or 2, characterized in that: Includes the following program modules: Initialization module, establishes the initial layout of the truss structure on the isogeometric background grid, initializes the design variables of the bar unit and the optimization algorithm parameters: the thermal conductivity material properties of the bar, the thermal conductivity material properties and size information of the isogeometric background grid of the weak material, and the convergence parameters, sets the external loads and boundary conditions; initializes the design variables of the bar unit including the node coordinates and the cross-sectional area; The isogeometric finite element analysis module uses the isogeometric stiffness diffusion method to obtain the heat conduction and heat capacity diffusion matrix of the truss unit. The heat conduction and heat capacity matrix of the truss structure is assembled according to the node numbers corresponding to the isogeometric background grid of the weak material, forming an isogeometric transient heat transfer finite element model of the truss structure. The unconditionally stable generalized Crank–Nicolson finite difference scheme is used to solve the above model and calculate the transient heat dissipation efficiency of the truss structure. The optimization solution module calculates the sensitivity information, uses the moving asymptote method based on the trust region, updates the design variables of the rod unit, and determines whether the optimization iteration result meets the convergence criterion. If not, the optimization iteration continues until the convergence criterion is met. The post-processing module reconstructs the optimal layout of the truss structure according to the minimum size constraints of parameters such as the cross-section and length of the rod unit, so as to obtain a truss structure design layout that is convenient for engineering applications.

4. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program, and the computer program is configured to implement the steps of the transient heat transfer topology optimization method for a truss structure based on the equal geometric stiffness diffusion method described in any one of claims 1 to 2 when called by a processor.

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