A multi-material thermo-elastic topology optimization method with different moduli in tension and compression

Abstract

The present invention belongs to the technical field of civil engineering and topology optimization, and particularly relates to a thermo-elastic topology optimization method for multi-material with different tensile and compressive moduli, comprising the following steps: Step 1: Establish a constitutive relation model of bi-modulus thermo-elasticity; Step 2: Apply the derivation of the self-consistent shear modulus to give the complete elastic matrix and numerical analysis of the thermo-elasticity problem of materials with different tensile and compressive moduli; Step 3: Construct a multi-material thermo-elastic optimization framework based on the moving deformable component method, and realize the structural configuration optimization through the update of geometric information. The present invention simplifies the geometric identification and post-processing of tension rods and compression rods. Since components are used as the structural elements and the same component represents the same material with different moduli, the positions of the compression rods and the tension rods can be directly selected and the tension-compression rod structure can be extracted, and the extraction difficulty of the post-processing can be greatly reduced compared with the existing methods.

Description

Technical Field

[0001] The present invention belongs to the technical fields of civil engineering and topology optimization, and particularly relates to a multi-material thermo-elastic topology optimization method with different tensile and compressive moduli. Background Art

[0002] The strut-and-tie model is an idealized model of the internal force flow in the structural concrete region. By finding a truss-like system composed of concrete struts, steel tie rods, and connecting nodes, a feasible solution for the load transfer mechanism can be given. As a logical extension of the truss model, the strut-and-tie model is based on the lower bound theorem of limit analysis. By ignoring the compatibility conditions and constructing a lower bound stress field that satisfies the equilibrium and failure criteria, a conservative estimate of the safe bearing capacity of reinforced concrete structures with discontinuities can be provided. Currently, it is recognized as the most reliable tool for dealing with discontinuous or disturbed regions.

[0003] The topology optimization method is an innovative and powerful structural design method. By constructing theoretical and numerical procedures, it is possible to find the optimal layout of a predetermined amount of material within the design domain. As the complexity of the designed structure increases, the challenge of generating a suitable strut-and-tie model also increases. This makes the strut-and-tie design guided by the topology optimization method the current research frontier and has made substantial progress in the past two decades. Topology optimization methods represented by the ground-based structure method, the evolutionary structural optimization method, the solid isotropic material with penalty method, etc. have been widely used. The optimization results reveal the load paths representing the force transfer mechanism as the design inspiration for tie rods and struts. The above research all shows great potential of the topology optimization method in the problem of designing strut-and-ties.

[0004] Temperature change is an important factor affecting the design of reinforced concrete structures in engineering. The thermal properties of concrete materials can affect the thermal conductivity, thermal diffusivity, and coefficient of thermal expansion, and may induce concrete cracking in severe cases. As a potential cause of structural failure, considering the thermal effect in the design of tension and compression bars can further improve the structural performance of reinforced concrete and reduce the failure risk. However, existing research mainly focuses on the structural response under pure mechanical loads, and the influence mechanism of temperature change and the resulting thermal environment on the design of tension and compression bars remains unclear. In addition, the material properties of the steel bars and concrete that make up the tension and compression bar structures are also different. The good compressive characteristics of concrete materials determine that they mainly represent the compression bar structure. Due to the low tensile strength, the concrete materials in the tension bar structure are extremely prone to excessive cracking, causing the structure to completely fail. Therefore, their distribution in the tensile area should be avoided, and steel bar materials should be used instead to strengthen the tension bar structure. However, there is currently a lack of research on introducing the tensile-compressive asymmetry characteristics of multiple materials into the topology optimization method to achieve the design of tension and compression bars. Finally, due to the lack of clear geometric information about compressive struts and tensile bars, the existing research results only provide design inspiration and require manual adjustment. This process is full of subjective intervention, which may hinder the realization of the optimal tension and compression bar design. Therefore, we propose a multi-material thermo-elastic topology optimization method with different moduli for tension and compression bars. Summary of the Invention

[0005] The purpose of the present invention is to provide a multi-material thermo-elastic topology optimization method with different moduli for tension and compression bars, which first considers the influence of the thermal environment on the design of tension and compression bars. To prevent premature failure or performance degradation of the structure, it is crucial to incorporate the thermal effect into the design. The proposed method can achieve the design of tension and compression bars under different temperature changes by focusing on analyzing the influence mechanism of the thermal environment on the design of tension and compression bars; selecting realistic reinforced concrete materials with different moduli. The obvious tensile and compressive tendencies exhibited by steel bars and concrete respectively. Introducing a multi-material thermo-elastic topology optimization framework with different moduli for tension and compression can accurately describe such problems, which is closer to engineering practice compared to most designs using single linear elastic materials; simplified geometric identification and post-processing of tension and compression bars. Since components are used as structural elements and the same component represents the same material with different moduli, the positions of compression bars and tension bars can be directly selected and the tension and compression bar structures can be extracted. Compared with existing methods, the extraction difficulty of post-processing can be significantly reduced.

[0006] The technical solutions adopted by the present invention are specifically as follows:

[0007] A multi-material thermo-elastic topology optimization method with different moduli for tension and compression bars, comprising the following steps:

[0008] Step 1: Establish a bi-modulus thermo-elastic constitutive relation model;

[0009] Step 2: Apply the derivation of the self-consistent shear modulus to give the complete elastic matrix and numerical analysis of the thermo-elastic problem of materials with different moduli for tension and compression;

[0010] Step 3: Construct a multi-material thermo-elastic optimization framework with different tensile and compressive moduli based on the moving deformable component method;

[0011] Step 301: Initialize the tensile and compressive bars using the multi-material thermo-elastic optimization framework of the moving deformable component method;

[0012] Step 302: Assemble the stiffness matrix using the completed elastic matrix iteration algorithm and analyze the structure with different moduli by equivalent temperature load;

[0013] Step 303: Determine the optimization formulation and calculate the analysis sensitivity;

[0014] Step 304: Update the design variables according to the sensitivity information;

[0015] Step 305: Check whether the convergence is qualified. If it is qualified, complete the extraction of the multi-material tensile and compressive structure, and realize the structural configuration optimization by updating the geometric information.

[0016] Preferably, in the said Step 1, the tensile / compressive state of the material point is determined according to the principal stress, and the thermo-elastic constitutive relation model of this kind of material is defined as:

[0017]

[0018] In the formula, σ i , ε i and κ i are the principal stress, the overall principal strain and the thermal strain in the i-th direction respectively; E + , ν + , E - , ν - are the elastic modulus and Poisson's ratio in the tensile and compressive states respectively; to satisfy the symmetry of the constitutive matrix and the system conservation, the parameter C 0 needs to satisfy:

[0019]

[0020] Preferably, in the said Step 2, the consistent shear modulus is deduced using the completed elastic matrix iteration algorithm and added to the constitutive relation in the main coordinate system; according to four different principal stress states, namely: ε p ={(e 1 , e 2 ) ∈ R 2 |(e 1 , e 2 ) ∈ S i}, i = 1, 2, 3, 4;

[0021]

[0022] For the bimodular thermoelastic problem in two - dimensional cases, the four completed constitutive matrices are respectively expressed as:

[0023]

[0024] where, e i = ε i - κ i , i = 1, 2; Since the elastic strain and stress tensors at any material point are always coaxial, using n 1 =(l 1 , m 1 ) T , n 2 =(l 2 , m 2 ) T to represent the principal coordinate vector of strain or stress, and transforming the above results into the global coordinate system, we can obtain:

[0025]

[0026] The transformation matrix can be expressed as:

[0027]

[0028] Based on the above results, the completed stiffness matrix and the equivalent thermal load at the k - th iteration step can be respectively expressed as:

[0029]

[0030] Preferably, in step 2, the analysis process of the iteration algorithm of the completed constitutive matrix for the bimodular thermoelastic structure is as follows:

[0031] Step 201: Select or corresponding to construct the stiffness matrix K (0) and F th (0) , and obtain the initial displacement U (0) ;

[0032] Step 202: Calculate the principal strain (k-1) at the q - th Gauss integration point of the e - th element from U

[0033] Step 203: Determine the completed constitutive matrix according to equations (2.6 - 2.9) and determine

[0034] according to equation (2.10) (k) Update Kth(k) ;

[0035] Step 205: Solve for K (k) U (k) = F + F th(k) Update U (k) ;

[0036] Step 206: Take tol as the tolerance to judge the iteration convergence. If ||U (k) - U (k-1 )|| / ||U (k-1 )|| ≤ tol, then end the iteration, and U (k-1) is the solution to this problem. Otherwise, k = k + 1 and go to Step 202.

[0037] Preferably, in the said Step 3, assuming that the current problem involves the optimal design of i types of solid materials, introduce i groups of components as structural elements, and each group of components has the same mechanical properties. If material m has a total of n m components, then the virtual region occupied by all i types of materials in the design domain can be expressed as:

[0038]

[0039] Based on the above idea, if the elastic modulus of the m-th type of material is set as E m , then the elastic modulus at any material point in the design domain can be interpolated and expressed as:

[0040]

[0041] Here H = H(x) is the Heaviside function. To implement numerical analysis, use the following regularized form of H(x):

[0042]

[0043] Specifically, for the two-phase material design of tension rod members and compression rod members in the design of tension and compression rods, that is, i = 2, Figure 2 shows the basic idea of this description method, and Equation (13) can be rewritten as:

[0044] E(x) = H(χ s,1 (x))E 1 + (1 - H(χ s,1 (x)))H(χ s,2 (x))E 2 (15).

[0045] Preferably, in the said Step 3, for the so-called surrogate material model, solve the two-dimensional plane problem on a fixed finite element mesh. From Equation (13), the element elastic modulus can be expressed in the following form:

[0046]

[0047] Preferably, in step 3, in the thermoelastic problem of the bi-modulus structure, for the tensile and compressive elastic moduli E + , ν + , E - , ν - adopting a surrogate material model, for the structural design of two-phase bi-modulus materials involved in the design of tension-compression bars, that is, i = 2, Equation (16) can be expressed as:

[0048]

[0049] Preferably, in step 3, the objective function of the thermoelastic problem is defined as minimizing the displacement at the applied force load point, and the optimization formulation can be expressed in the following form:

[0050]

[0051] s.t. K(u, D)u(D) = F + F th ;

[0052]

[0053] It can be seen from Equation (18) that the objective function is only related to the force load and the displacement response at the loading point, but the displacement response is affected by both the force load and the temperature change. Since the numerical analysis method of the bi-modulus structure still uses the completed elastic matrix iteration algorithm, the completed stiffness matrix K (k) and the equivalent thermal load F th(k) are constructed in the same way as Equations (10) and (11).

[0054] The technical effects achieved by the present invention are as follows:

[0055] In the present invention: the influence of the thermal-mechanical environment on the design of tension-compression bars is considered for the first time. In order to prevent the premature failure or performance degradation of the structure, it is crucial to incorporate the thermal effect into the design. The proposed method can achieve the design of tension-compression bars considering different temperature changes by focusing on analyzing the influence mechanism of the thermal-mechanical environment on the design of tension-compression bars.

[0056] In the present invention: the selection of different modulus reinforced concrete materials close to the actual situation. Steel bars and concrete respectively show obvious tensile and compressive tendencies. Introducing a multi-material tension-compression different modulus optimization framework can accurately describe such problems, which is closer to engineering practice compared with the design that mostly uses single linear elastic materials.

[0057] In the present invention: Simplified geometric identification and post - processing of tension and compression bars. Since components are used as structural elements and the same component represents the same material with different moduli, the positions of the compression bar and the tension bar can be directly selected and the tension - compression bar structure can be extracted, which can greatly reduce the extraction difficulty of post - processing compared with the existing methods. Brief Description of the Drawings

[0058] Figure 1 is a schematic diagram of the basic idea of the multi - material MMC method of the present invention;

[0059] Figure 2 is a schematic diagram of the basic idea of using the MMC method to describe two - phase materials in the present invention;

[0060] Figure 3 is the specific flowchart of step 3 of the present invention;

[0061] Figure 4 is a schematic diagram of a design example of thermal expansion of tension and compression bars in a classic single - bracket column of the present invention;

[0062] Figure 5 is the initial layout and optimization result of the MMC components of the present invention;

[0063] Figure 6 is the optimization design result of the tension and compression bars under temperature change in the present invention;

[0064] Figure 7 is a schematic diagram of a design example of thermal expansion of tension and compression bars in a simply - supported deep beam of the present invention;

[0065] Figure 8 is the topology optimization result and the corresponding tension - compression bar design generated by different L / D under different temperature changes in the present invention. Detailed Embodiment

[0066] In order to make the purpose and advantages of the present invention clearer, the present invention will be specifically described below in conjunction with embodiments. It should be understood that the following text is only used to describe one or several specific implementation manners of the present invention, and does not strictly limit the scope of protection specifically claimed by the present invention.

[0067] A multi - material thermo - elastic topology optimization method for tension and compression with different moduli includes the following steps:

[0068] Step 1: Establish a constitutive relation model of thermo - elasticity with double moduli;

[0069] Step 2: Apply the derivation of the self - consistent shear modulus to give the complete elastic matrix and numerical analysis of the thermo - elastic problem of materials with different moduli in tension and compression;

[0070] Step 3: Construct a multi - material thermo - elastic optimization framework for tension and compression with different moduli based on the moving deformable component method;

[0071] Step 301: Initialize the tension-compression bar using the multi-material tension-compression different modulus thermo-elastic optimization framework with the moving deformable component method;

[0072] Step 302: Assemble the stiffness matrix using the complete elastic matrix iteration algorithm and analyze the different modulus structure with equivalent temperature loads;

[0073] Step 303: Determine the optimization formulation and calculate the analysis sensitivity;

[0074] Step 304: Update the design variables according to the sensitivity information;

[0075] Step 305: Check whether the convergence is qualified. If it is qualified, complete the extraction of the multi-material tension-compression structure and realize the structural configuration optimization through the update of geometric information.

[0076] In Step 1, according to the principal stress that should be adopted, determine the tensile / compressive state of the material point. The thermo-elastic constitutive relation model of this type of material is defined as:

[0077]

[0078] where σ i , ε i and κ i are the principal stress, the total principal strain, and the thermal strain in the i-th direction, respectively. E + , ν + , E - , ν - are the elastic modulus and Poisson's ratio in the tensile and compressive states, respectively. To satisfy the symmetry of the constitutive matrix and the system conservation, the parameter C 0 needs to satisfy:

[0079]

[0080] Preferably, in Step 2, use the complete elastic matrix iteration algorithm to derive the consistent shear modulus and add it to the constitutive relation in the main coordinate system; according to the four different principal stress states, namely: ε p ={(e 1 , e 2 ) ∈ R 2 |(e 1 , e 2 ) ∈ S i}, i = 1, 2, 3, 4;

[0081]

[0082] For the two-dimensional bimodulus thermo-elastic problem, the four complete constitutive matrices are respectively expressed as:

[0083]

[0084] where e i = ε i - κ i , i = 1, 2. Since the elastic strain and stress tensors at any material point are always coaxial, let n 1 = (l 1 , m 1 ) T , n 2 = (l 2 , m 2 ) T represent the principal coordinate vectors of strain or stress. Transforming the above results into the global coordinate system gives:

[0085]

[0086] The transformation matrix can be expressed as:

[0087]

[0088] Based on the above results, the completed stiffness matrix and equivalent thermal load at the k-th iteration step can be expressed as:

[0089]

[0090] Preferably, in step 2, the analysis process of the iterative algorithm for the completed constitutive matrix of the bimodulus thermoelastic structure is as follows:

[0091] Step 201: Select or corresponding to Construct the stiffness matrix K (0) and F th (0) , and obtain the initial displacement U (0) ;

[0092] Step 202: Calculate the principal strain (k-1) at the q-th Gauss integration point of the e-th element from U

[0093] Step 203: Determine the completed constitutive matrix according to equations (2.6 - 2.9) and determine

[0094] Step 204: Update K (k) and F th(k) according to equations (2.12 - 2.13);

[0095] Step 205: Solve K (k) U (k) = F + F th(k)Update U (k) ;

[0096] Step 206: Take tol as the tolerance to judge the iteration convergence. If ||U (k) - U (k-1) || / ||U (k-1) || ≤ tol, then end the iteration, and U (k-1) is the solution to this problem; otherwise, k = k + 1 and go to Step 202.

[0097] As Figure 1 shown, Figure 1 the basic idea of the multi - material MMC method. MMC is the moving deformable component method. In the figure, components of different colors represent materials with different mechanical properties; as Figure 1 shown, the MMC method is different from the classical framework established by implicit geometric descriptions. Components described by explicit geometric information are used as structural primitives. The position and geometric shape of each component are controlled by the central coordinates, half - length, half - width, and tilt angle. Through the update of geometric information (corresponding to the movement, deformation, disappearance, and overlap of components), structural configuration optimization is achieved. Compared with traditional implicit methods, the MMC method driven by explicit geometric parameters reduces the number of design variables and improves the solution efficiency. In the multi - material topology optimization method based on the MMC framework, the distribution of multiple materials can be described by the distribution of components of different - performance materials, so it naturally inherits all the characteristics of the MMC framework.

[0098] In Step 3, assume that the current problem involves the optimal design of i types of solid materials, and introduce i groups of components as structural primitives. Each group of components has the same mechanical properties (such as elastic modulus and Poisson's ratio). If material m has a total of n m components, then the virtual area occupied by all i types of materials in the design domain can be expressed as:

[0099]

[0100] Based on the above idea, if the elastic modulus of the m - th type of material is set to E m , then the elastic modulus at any material point in the design domain can be interpolated as:

[0101]

[0102] Here, H = H(x) is the Heaviside function. For numerical analysis, the following regularized form of H(x) is adopted:

[0103]

[0104] For the two - phase material design of tension members and compression members in the design of tension - compression bars, that is, i = 2, Figure 2The basic idea of the description method is shown. Equation (13) can be rewritten as:

[0105] E(x) = H(χ s,1 (x))E 1 +(1 - H(χ s,1 (x)))H(χ s,2 (x))E 2 (15).

[0106] As Figure 2 shown, Figure 2 Adopting the MMC method to describe the basic idea of two - phase materials. In step 3, for the so - called surrogate material model, a two - dimensional plane problem is solved on a fixed finite - element grid. From Equation (13), the element elastic modulus can be expressed in the following form:

[0107]

[0108] In step 3, in the thermo - elastic problem of a bimodular structure, for the tensile and compressive elastic moduli E + , ν + , E - , ν - Using the surrogate material model, for the structural design of two - phase bimodular materials involved in the design of tensile and compressive bars, that is, i = 2, Equation (16) can be expressed as:

[0109]

[0110] In step 3, the objective function of the thermo - elastic problem is defined as minimizing the displacement at the applied force load point. The optimization formulation can be expressed in the following form:

[0111]

[0112] s.t. K(u,D)u(D) = F + F th (u,D);

[0113]

[0114] As can be seen from Equation (18), the objective function is only related to the force load and the displacement response at the loading point, but the displacement response is affected by both the force load and the temperature change. Since the numerical analysis method for the bimodular structure still uses the completed elastic matrix iteration algorithm, the construction of the completed stiffness matrix K (k) and the equivalent thermal load F th(k) in the k - th iteration step is the same as in Equations (10) and (11).

[0115] In the present invention, it has to be said that, based on the above results, the numerical solution flow chart of the topology optimization problem of multi - phase bimodular thermo - elastic materials based on MMC is as Figure 3As shown. At the beginning of the optimization, multi-material components are arranged in the design domain and an initial design and assumed displacement response are specified. The true displacement response is calculated by the thermo-elastic complementary elastic matrix iteration algorithm. After sensitivity analysis, the moving asymptote method (MMA) is used to update the component design variables, and the next process is determined by checking the convergence. The subsequent numerical examples will prove the effectiveness of this optimization process.

[0116] It is worth mentioning that introducing the developed multi-phase and dual-modulus structural thermo-elastic topology optimization method into the design of tension and compression bars in reinforced concrete structures in engineering can provide more practical engineering practice guidance for this field. The relevant advantages are manifested in the following four aspects:

[0117] First, consider the influence of the thermal-mechanical environment on the design. The change in temperature will cause the expansion or contraction of the tension and compression bar structure. Compared with the case of applying pure force loads, the design problem considering the thermal-mechanical environment will be more complex but closer to engineering practice, and it should be fully considered during design to avoid premature failure or performance degradation of the structure.

[0118] Second, select materials closer to reality. The good compressive characteristics of concrete materials determine that it mainly represents the compression bar structure. Due to its low tensile strength, its distribution in the tensile area should be avoided, and steel bar materials are used to strengthen the tension bar structure. Therefore, steel bars and concrete respectively show the material characteristics of tending to be in tension and compression in the structure, showing the characteristics of different moduli in tension and compression.

[0119] Then, efficiently select the tension bars and compression bars. Since each group of components represents the same dual-modulus material, the tensile / compressive state of the optimized result components can be clearly obtained, realizing the efficient selection of tension bars and compression bars.

[0120] Finally, simplify the post-processing of the optimization results. The optimization results obtained based on the SIMP method need to further identify the truss-like structure and the connection point positions and perform shape optimization, which increases the design difficulty of the tension and compression bar model. Under the MMC optimization framework, the tension and compression bar model uses straight and overlapping connected components as the structural elements, which can effectively reduce the difficulty of extracting the truss-like structure.

[0121] For the present invention, it should be noted that according to the characteristics of the tension and compression bar members, the topology optimization method involving two-phase dual-modulus materials can be directly adopted to design the tension-compression bar model. In this section, two types of classical tension-compression bar design problems are studied, with particular attention paid to the influence of temperature changes on the structure, and the relevant advantages of the proposed optimization method for such problems are verified. It must be pointed out that most of the previous classical tension-compression bar design works have adopted the minimum compliance with volume constraint as the design goal. In practical engineering, if the temperature influence is considered, the structural compliance is not intuitive and difficult to accurately detect. To improve this problem, the design goal of the following numerical examples is updated to minimize the displacement at the applied force load point. As an index with clear physical meaning, it directly focuses on the performance at the structure loading point and is more operable in engineering practice, easy to measure and

[0122] In this embodiment, for example, in the design example of the tension-compression bar of a single bracket column, such as Figure 4 shown

[0123] Table 1 Dual-modulus material coefficients of steel bars and concrete

[0124]

[0125]

[0126] As a typical discontinuous region, this example aims to achieve the design of the tension-compression bar of the single bracket column structure as Figure 4 shown. The two ends of the column are fixed, the thickness δ = 0.3m, and the concentrated force F = 500KN. The dual-modulus characteristics of concrete and steel bars have been clearly stated, and the relevant parameter settings are shown in Table 1. The set tensile to compressive modulus ratio (i.e., ) is to respectively reflect the better compressive / tensile performance of concrete and steel bars. The design goal is to minimize the displacement at the loading point (Mathematical formulation2). To balance the truss-like design tendency of the tension-compression bar and the generation ability of the structural detail branches, Considering the actual temperature difference, the optimal material distribution is sought at ΔT = -20K, 0K, 20K.

[0127] Taking ΔT = 20K as an example, as Figure 5 shown, a final optimization result with an objective function value of 576.74, a volume fraction of V 1 / |D| = 7.5%, and V 2 / |D| = 7.5% is obtained. During the extraction process of the optimization result, since each component is explicitly represented, the coordinate and length information of the optimization result can be accurately identified. Since each component represents only one type of dual-modulus material, the components show a clear tendency to be in tension or compression, and the tension and compression bars can be directly obtained through Figure 5The topological optimization results shown are efficiently selected. It is worth mentioning that the thickness information of the optimized result components can provide important dimensional references for the production and manufacturing of actual tension and compression bar models.

[0128] Figure 6 The topological optimization results and the extracted final tension and compression bar models under different temperature changes were collected. They correspond to the optimization results for ΔT = -20K, 0K, and 20K respectively. When ΔT = 0K, the results are similar to those obtained in existing studies. Positive temperature changes will result in a newly generated concrete compression bar with a gradually increasing width in the compression area at the lower end of the column, which forms a complete compression bar force transmission path from the loading point to the constraint. As ΔT increases, the compression bar at the loading point expands upward. The complete compression bar force transmission path and more compression bar branches help enhance the structural compressive capacity to resist deformation and make full use of the upward thermal expansion effect at the loading point to reduce the downward displacement. At the same time, as ΔT increases, the constraint positions of the tension bar at the upper end and the compression bar at the lower end of the column gradually shift to the left, and the angle formed with the vertical direction gradually becomes larger. This is to reduce the vertical component of the thermal expansion effect to reduce the influence of the displacement at the loading point.

[0129] In this embodiment, for example, in the design example of a simply supported deep beam tension and compression bar, as Figure 7 shown,

[0130] This section aims to study the influence of temperature changes on the design of the optimal tension and compression bar model for a simply supported deep beam. As Figure 7 shown, for beams with different span-depth ratios, the beam thickness δ = 0.25m, the concentrated force F = 1200KN, and the material parameters of the selected concrete and steel bars are shown in Table 1. The design goal is to minimize the displacement at the loading point, and the volume constraint seeks the optimal distribution of concrete and steel bar materials at ΔT = -30K, 0K, and 30K. The beam depth D in all cases is 1m.

[0131] Figure 8 The optimized results and the extracted tension and compression bar models for L / D = 2, 3, and 4 are shown. The blue components represent concrete, and the red components represent steel bars. Through an extraction process similar to that of the single corbel column tension and compression bar design example, Figure 8 the extracted tension and compression bar models are directly given and can be efficiently selected based on the topological optimization results. Structurally, the topological optimization results for different L / D cases all achieve a layout form with a triangular component system as the basic structural unit, which is consistent with the existing tension and compression bar design without considering temperature changes. In all three cases, the steel bar material accounts for only a small amount, indicating that when the total material usage is certain, more concrete needs to be allocated to improve the structural stiffness and reduce the displacement at the loading point. It is worth noting that when the temperature change is extended to ΔT = -30K and 30K, the corresponding optimized configurations do not differ significantly from those at ΔT = 0K, and the material usage does not change significantly either.

[0132] F in the optimization results of Table 2 T u m and F T u th Calculation result comparison

[0133]

[0134] To understand why temperature change has no obvious effect on the design of the tension-compression bar model of a simply supported beam with different span-depth ratios, it can be considered from the perspective of the design objective. The displacement response in the objective function can be split into the sum of the displacement responses of the mechanical part and the thermal part. Therefore, the objective function can also be split into two terms: F T u m and F T u th , where u m is the displacement response of the current structure under pure force load, and u th is the displacement response of the current structure under the steady-state temperature field. Taking ΔT = 30K as an example, the calculation results of F T u m and F T u th in the three cases are respectively summarized in Table 2. It can be found that F T u m >>F T u th , proving that the influence of temperature change on this simply supported beam is very limited. This is because when the structure is simply supported and only the temperature change is added, due to the uniform thermal expansion of the structure, the total strain of the structure is only related to the thermal strain, that is, ε = κ. And in engineering, the thermal expansion coefficient of the material and the thermal strain κ caused by the temperature change are small, which will lead to the fact that the thermo-elastic optimization design is approximately equivalent to the optimization design under pure force load. When a structure with fixed-end constraints is only subjected to temperature change, due to the limitation of the constraints, the total strain of the structure will be related not only to the thermal strain. Even if the thermal strain κ is small, u th under the steady-state temperature field may have a greater impact on the design objective. For example, in the example of a single bracket column, Figure 6 The optimized configuration shown in F T u m = 410.92, F T u th = -328.73. In this case, the influence of temperature change on the optimized configuration cannot be ignored, and the finally derived tension-compression bar configurations change greatly with temperature. In practical engineering, the design of tension-compression bars considering temperature change can be approximately degraded to the design of tension-compression bars under pure force load under simply supported constraints, and this phenomenon is jointly determined by the design objective and the constraint conditions.

[0135] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention. The structures, devices, and operation methods not specifically described and explained in the present invention are implemented according to the conventional means in the art without special explanation and limitation.

Claims

1. A multi-material thermoelastic topology optimization method with different tensile and compressive moduli, characterized by: The following steps are involved: Step 1: Establish a dual-modulus thermoelastic constitutive relationship model; Step 2: Apply the derivation of the self-consistent shear modulus to give the completed elastic matrix and numerical analysis of the thermoelastic problem of materials with different tensile and compressive moduli; the completed stiffness matrix and equivalent thermal load of the kth iteration step can be expressed as: Step 3: Construct a multi-material thermoelastic optimization framework with different moduli in tension and compression based on the mobile deformable component method; Step 301: Initialize the tension and compression rods using a multi-material tension and compression different modulus thermoelastic optimization framework of a mobile deformable component method; Step 302: using a complete elastic matrix iteration algorithm to assemble a stiffness matrix and perform equivalent temperature load analysis on structures with different moduli; Step 303: Determine the optimized column formula and calculate the analytical sensitivity; Step 304: updating the design variables according to the sensitivity information; Step 305: Check whether the convergence is qualified. If it is qualified, the extraction of the multi-material tension-compression structure is completed, and the structural configuration optimization is realized by updating the geometric information; In step 3, it is assumed that the current problem involves the optimization design of i types of solid materials, and i groups of components are introduced as structural primitives. Each group of components has the same mechanical properties. If material m has n m components, the virtual area occupied by all i-type materials in the design domain can be expressed as: If the elastic modulus of the mth material is set to E m , then the elastic modulus at any material point in the design domain can be expressed by interpolation as: Here H = H(x) is the Heaviside function. In order to implement numerical analysis, the following regularized form of H(x) is used: For the two-phase material design of the tension rod and compression rod in the tension and compression rod design, i.e., i = 2, equation (13) can be rewritten as: E(x)=H(x s,1 (x))E1+(1-H(x s,1 (x)))H(x s,2 (x))E2 (15); In step 3, the objective function of the thermoelastic problem is defined as minimizing the displacement at the point where the force load is applied. The optimization formula can be expressed as follows: min f2=F T u(D); s.t.K(u,D)u(D)=F+F th ; It can be seen from formula (18) that the objective function is only related to the force load and the displacement response at the loading point, but the displacement response is affected by the force load and temperature change. Since the numerical analysis method of the dual-modulus structure still uses the completed elastic matrix iterative algorithm, the completed stiffness matrix K of the kth iteration step is (k) and equivalent heat load F th(k) The construction of is the same as equations (10) and (11).

2. The method for optimizing the thermoelastic topology of multiple materials with different tensile and compressive moduli according to claim 1, characterized in that: In step 1, the tensile / compressive state of the material point is determined according to the principal stress, and the thermoelastic constitutive relationship model of this type of material is defined as: In the formula, σ i , ε i and κ i are the principal stress, overall principal strain and thermal strain in the i-th direction respectively; E + ,ν + ,E - ,ν - are the elastic modulus and Poisson's ratio in tension and compression respectively; in order to satisfy the symmetry of the constitutive matrix and the conservation of the system, the parameter C0 needs to satisfy:

3. The method for optimizing the thermoelastic topology of multiple materials with different tensile and compressive moduli according to claim 1, characterized in that: In step 2, a consistent shear modulus is derived using the complete elastic matrix iteration algorithm and added to the constitutive relation of the principal coordinate system; according to four different principal stress states, namely: ε p ={(e1,e2)∈R 2 |(e1,e2)∈S i },i=1,2,3,4; For the two-dimensional dual-modulus thermoelastic problem, the four complete constitutive matrices are expressed as: In the formula, e i =ε i -κ i ,i=1,2; Since the elastic strain and stress tensors of any material point are always coaxial, use n1=(l1,m1) T ,n2=(l2,m2) T The principal coordinate vector representing strain or stress, transforming the above results to the global coordinate system yields: Transformation Matrix It can be expressed as: Based on the above results, the completed stiffness matrix and equivalent thermal load of the kth iteration step can be expressed as:

4. The method for optimizing the thermoelastic topology of multiple materials with different tensile and compressive moduli according to claim 1, characterized in that: In step 2, the analysis process of the iterative algorithm for the completed constitutive matrix of the dual-modulus thermoelastic structure is as follows: Step 201: Select or Corresponding Construct the stiffness matrix K (0) and F th(0) , get the initial displacement U (0) ; Step 202: From U (k-1) Calculate the principal strain at the qth Gaussian integration point of the eth element Step 203: Determine the complete constitutive matrix according to equations (2.6-2.9) And according to formula (2.10) Step 204: Update K according to equation (2.12-2.13) (k) and F th(k) ; Step 205: Solving for K (k) U (k) =F+F th(k) Update U (k) ; Step 206: Take tol as the tolerance to determine the convergence of the iteration. If ||U (k) -U (k-1) || / ||U (k-1) ||≤tol then the iteration ends, U (k-1) is the solution to the problem, otherwise k=k+1, go to step 202.

5. The method for optimizing the thermoelastic topology of multiple materials with different tensile and compressive moduli according to claim 1, characterized in that: In step 3, the so-called proxy material model solves the two-dimensional plane problem on a fixed finite element grid. According to formula (13), the unit elastic modulus can be expressed as follows:

6. The method for optimizing the thermoelastic topology of multiple materials with different tensile and compressive moduli according to claim 1, characterized in that: In step 3, in the thermoelastic problem of the dual-modulus structure, the tensile and compressive elastic moduli E + ,ν + ,E - ,ν - Using the proxy material model, for the structural design of two-phase dual-modulus materials involved in the tension and compression rod design, that is, i = 2, equation (16) can be expressed as:

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