A structural topology optimization method under low carbon constraints

Abstract

The present invention discloses a structural topology optimization method under low-carbon constraints, which is characterized in that a topology optimization model is constructed based on a typical topology optimization method, with a carbon emission limit as a low-carbon constraint condition and structural flexibility as a target. By analyzing the correlation between the future global anthropogenic greenhouse gas emission reduction rate and the atmospheric CO2 concentration, a functional relationship between the greenhouse gas emission reduction rate and the set service year of the structure is established. Within the service life of the structure, the total carbon emission value of the structure during its service life is obtained by accumulating the greenhouse gas emission reduction rate and the initial carbon emission value. The maximum carbon emission value is set according to the target greenhouse gas emission reduction rate during the service life of the structure, and this is used as a low-carbon constraint condition for topology optimization to construct a topology optimization model under low-carbon constraints. The present invention establishes a connection between the greenhouse gas emission reduction rate and carbon emissions, analyzes and obtains the structural carbon emission limit and converts it into a low-carbon constraint condition, and obtains a structural optimization model with low carbon emissions while meeting the structural strength.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural optimization, and specifically relates to a structural topology optimization method under low-carbon constraints. Background Art

[0002] As a new development model, low-carbon development is considered an effective way to deal with global warming and energy crisis. How to take effective measures to reduce greenhouse gas emissions has become a hot topic in product design development.

[0003] Generally speaking, the better the structural layout of a component, the lower the carbon emissions generated during its lifecycle. Structural optimization design methods can achieve better product structural layouts and are increasingly being used for product optimization and improvement. Specifically, they can be categorized into size optimization, shape optimization, and topology optimization. Topology optimization is a design method that, within a given design space, reduces material usage and lowers manufacturing costs by adding specific constraints while still meeting the engineer's desired structural properties, such as strength, stiffness, and stability. This approach is crucial for the low-carbon, green production of products.

[0004] Currently, structural topology optimization mostly uses constraints such as volume fraction, mass, stress, displacement, and natural frequency to optimize and improve the structure, but this can lead to increased carbon emissions during manufacturing. If carbon emission reduction is incorporated into the topology optimization model during structural optimization, topologies that meet performance and mechanical requirements while also having low carbon emissions can be obtained. However, research specifically addressing this issue is limited. Summary of the Invention

[0005] The purpose of the present invention is to provide a structural topology optimization method under low-carbon constraints. The problems to be solved are: first, how to determine the structural carbon emission limit; second, how to add the carbon emission limit as a low-carbon constraint into the structural topology optimization.

[0006] In order to realize the above invention, a structural topology optimization method under low carbon constraints is proposed, with the goal of minimizing structural flexibility and the structural carbon emission limit as a constraint. The technical solution implementation steps are as follows:

[0007] Step 1: Define the target product structure working conditions and design domain, set boundary conditions, determine the structural stress points, and calculate the loads.

[0008] Step 2: Use the variable density method to establish a topology optimization model with the relative density of the unit as the design variable, the minimum flexibility of the structure as the optimization goal, and the structural carbon emission limit as the constraint condition.

[0009] Step 3: Discretize the initial design domain into nelx×nely finite element units, introduce intermediate density units through the variable density method, set the penalty factor and filter radius, and initialize the design variables.

[0010] Step 4: Use the improved solid isotropic material penalty model (SIMP) to calculate the elastic modulus of the unit, obtain the unit stiffness matrix, assemble the overall stiffness matrix of the structure, perform finite element analysis on the structure under the set boundary conditions and loads, and calculate the displacement of the unit node.

[0011] Step 5: Accumulate the carbon emissions of each finite element to obtain the overall carbon emissions of the structure, and solve the target flexibility function and the sensitivity of the overall carbon emission constraint of the structure to the design variables.

[0012] Step 6: Construct the Lagrangian function of the target model so that the optimal solution satisfies the Kuhn-Tucker condition, i.e., the KT condition. Use the sensitivity filtering method to adjust the sensitivity of the target flexibility function and the low-carbon constraint condition. Use the optimization criterion method (OC method) to obtain the heuristic update format of the design variables and update the design variables.

[0013] Step 7: Determine whether the result meets the optimization convergence conditions. If not, go to step 4 and perform iterative calculations in sequence. If it meets the conditions, terminate the topology optimization process and obtain a topology optimization model that meets the low-carbon constraints.

[0014] Furthermore, the topology optimization model described in step 2 is:

[0015] find:x e ={x1,x2,x3,···,x n} T ∈Ω,e=1,2,···,n

[0016] min:C(x e )=U T KU

[0017] stg w ≤GWP max

[0018] KU=F

[0019] 0<x min ≤x e ≤1

[0020] Among them, x e is the relative density of the material unit; Ω is the design space; C(x e ) is the flexibility of the structure; U is the displacement vector of the structure; K is the stiffness matrix of the structure; F is the force vector of the structure; g wis the global warming potential of the structure; GWP max is the maximum global warming potential of the structure; x min It is the minimum value of the relative density of the material unit. Generally, a small value close to 0 is taken to prevent the unit stiffness matrix from being singular.

[0021] Furthermore, the structural carbon emission limit constraint function expression in step 2 is as follows:

[0022]

[0023] Where g0 is the initial carbon emission value of the structure; T d is the design service life of the structure; α is the greenhouse gas emission reduction rate.

[0024] Furthermore, the improved SIMP interpolation model described in step 4 is as follows:

[0025] E e =E e (x e )=E min +(x e ) p (E0-E min ), x e ∈(0,1]

[0026] Among them, E e (x e ) is the element stiffness matrix in the iterative process; E0 is the initial element stiffness matrix of the material; p is the penalty factor, which can speed up the convergence speed and make the intermediate element density tend to the values ​​at both ends (i.e. 0 or 1).

[0027] Furthermore, the calculation method for structural carbon emissions described in step 5 is as follows:

[0028]

[0029] Where x e,i is the relative density of the unit containing the i-th material in the structure; v e,i is the volume of the i-th material unit; ρ i is the density of the i-th material; E i is the carbon emission factor of the i-th material; n is the number of units; m is the number of material types.

[0030] Furthermore, the sensitivity of the structural carbon emission constraint to the design variables in step 5 is as follows:

[0031]

[0032] Furthermore, the sensitivity of the structural flexibility objective function to the design variables in step 5 is as follows:

[0033]

[0034] Furthermore, the Lagrangian function expression satisfying the KT condition constructed under the low carbon constraint in step 6 is as follows:

[0035]

[0036] Furthermore, the iterative format of the Lagrangian function solved in step 6 is obtained by the fixed point idea:

[0037] make Then there is

[0038] Furthermore, in step 6, the heuristic update format of the design variables obtained by the optimization criterion method (OC method) is as follows:

[0039]

[0040] Where k is the number of iterations; ξ is the damping factor, which can ensure the convergence of the results and is generally 0.5; m is the forward moving limit, which can improve the stability of the iterative process and is generally 0.1 to 0.3.

[0041] Furthermore, the optimization convergence condition in step seven is:

[0042]

[0043] Where ε is the convergence accuracy.

[0044] Compared with the prior art, the structural topology optimization method under low-carbon constraints provided by the present invention has the following beneficial effects:

[0045] (1) The present invention can achieve the goal of ensuring structural rigidity while reducing structural carbon emissions during the process of structural topology optimization;

[0046] (2) The carbon emission limit calculation method proposed in the present invention can be expanded to include functions when facing a product structure composed of more than one material. This allows for a better material layout to be obtained during topology optimization, which is beneficial for material conservation. Therefore, the present invention has a wide range of applications.

[0047] (3) The present invention adopts the optimization criterion method (OC method) widely used in the field of topology optimization for optimization solution calculation, and uses the improved SIMP interpolation model, which has a faster iteration speed, is conducive to later expansion and promotion, and has a wide range of applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a flow chart of a structural topology optimization method under low-carbon constraints of the present invention.

[0049] Figure 2 This is an example of the present invention: a schematic diagram of the initial structure, boundary conditions and load application points of a rear anti-collision beam of an automobile.

[0050] Figure 3 The topological structure diagram of the optimized rear anti-collision beam of an automobile obtained based on the present invention is shown in FIG.

[0051] Figure 4 This is a schematic diagram showing how the flexibility of the rear anti-collision beam structure of an automobile changes with the number of iterations during the optimization process obtained based on the present invention. DETAILED DESCRIPTION

[0052] The present invention will be further described in detail below with reference to the accompanying drawings and specific implementations:

[0053] like Figure 1 As shown, the present invention discloses a structural topology optimization method under low carbon constraints, comprising the following steps:

[0054] Step 1: Define the target product structure working conditions and design domain, set boundary conditions, determine the structural stress points, and calculate the loads.

[0055] Step 2: Use the variable density method to establish a topology optimization model with the relative density of the unit as the design variable, the minimum flexibility of the structure as the optimization goal, and the structural carbon emission limit as the constraint condition.

[0056] The structural topology optimization model under the low-carbon constraint is:

[0057]

[0058] Among them, x e is the relative density of the material unit; Ω is the design space; C(x e ) is the flexibility of the structure; U is the displacement vector of the structure; K is the stiffness matrix of the structure; F is the force vector of the structure; g w is the global warming potential of the structure; GWP max is the maximum global warming potential of the structure; x min It is the minimum value of the relative density of the material unit. Generally, a small value close to 0 is taken to prevent the unit stiffness matrix from being singular.

[0059] During the service life of the structure, the total carbon emission value of the structure during its service life is obtained by accumulating the greenhouse gas emission reduction rate and the initial carbon emission value. The maximum carbon emission value is set according to the target greenhouse gas emission reduction rate during the service life of the structure, which serves as the low-carbon constraint condition for topology optimization. The structural carbon emission limit constraint function expression is as follows:

[0060]

[0061] Where g0 is the initial carbon emission value of the structure; T d is the design service life of the structure; α is the greenhouse gas emission reduction rate.

[0062] Based on a large number of greenhouse gas mitigation scenarios investigated by Working Group III of the IPCC Fifth Assessment, and projections of the relationship between future global anthropogenic greenhouse gas emission reduction rates and atmospheric CO2 concentrations, it is assumed that carbon emission limits are proportional to greenhouse gas emission reduction rates and change deterministically and linearly over time. Based on this assumption, the formula for calculating the global greenhouse gas emission reduction rate is proposed as follows:

[0063]

[0064] Where Y is the set service year of the structure.

[0065] Step 3: Discretize the initial design domain into nelx×nely finite element units, introduce intermediate density units through the variable density method, set the penalty factor and filter radius, and initialize the design variables.

[0066] Step 4: Use the improved solid isotropic material penalty model (SIMP) to calculate the elastic modulus of the unit, obtain the unit stiffness matrix, assemble the overall stiffness matrix of the structure, perform finite element analysis on the structure under the set boundary conditions and loads, and calculate the displacement of the unit node.

[0067] The improved SIMP interpolation model is as follows:

[0068] E e =E e (x e )=E min +(x e ) p (E0-E min ), x e ∈(0,1] (4)

[0069] Among them, E e (x e ) is the element stiffness matrix in the iterative process; E0 is the initial element stiffness matrix of the material; p is the penalty factor, which can speed up the convergence speed and make the intermediate element density tend to the values ​​at both ends (i.e. 0 or 1).

[0070] Step 5: Accumulate the carbon emissions of each finite element to obtain the overall carbon emissions of the structure, and solve the target flexibility function and the sensitivity of the overall carbon emission constraint of the structure to the design variables.

[0071] The structural carbon emission calculation formula is:

[0072]

[0073] Where x e,i is the relative density of the unit containing the i-th material in the structure; v e,i is the volume of the i-th material unit; ρ i is the density of the i-th material; E i is the carbon emission factor of the i-th material; n is the number of units; m is the number of material types.

[0074] The sensitivity of the topology optimization model is solved as follows:

[0075] For load, since the load is applied from the outside, it is related to x e It does not matter, the load on x e The sensitivity is:

[0076]

[0077] Structural carbon emissions to x e The sensitivity is:

[0078]

[0079] For the displacement and stiffness constraints of the structure KU=F, both sides are simultaneously e Find the partial derivative and the result is as follows:

[0080]

[0081] The overall stiffness matrix of the structure is symmetrical, so F T =U T K T =U T K, both sides are simultaneously x e Find the partial derivative and the result is as follows:

[0082]

[0083] Multiply the left side of equation (8) by U T , multiply the right side of formula (9) by U, and the results are shown in formulas (10) and (11):

[0084]

[0085]

[0086] The overall flexibility of the structure is expressed as shown in formula (1), and both sides simultaneously e Find the partial derivative, and the result is shown in formula (12):

[0087]

[0088] Substituting equations (10) and (11) into equation (12), we can obtain the structural flexibility dependence of x according to the improved SIMP interpolation model: e The sensitivity is:

[0089]

[0090] Step 6: Construct the Lagrangian function of the target model so that the optimal solution satisfies the Kuhn-Tucker condition, i.e., the KT condition. Use the sensitivity filtering method to adjust the sensitivity of the target flexibility function and the low-carbon constraint condition. Use the optimization criterion method (OC method) to obtain the heuristic update format of the design variables and update the design variables.

[0091] The Lagrangian function of the topology optimization model is:

[0092]

[0093] Among them, λ g ,λ1, is the Lagrange multiplier (where λ1 is a vector and the rest are scalars).

[0094] The Lagrangian function at the extreme point The expression that satisfies the KT condition is:

[0095]

[0096] For the above formula, we discuss the following cases:

[0097] (1) When x min ≤x e When ≤1, the conditions in the formula can be obtained

[0098] (2) When x e When >1, we can get

[0099] (3) When x e <x min When , we can get

[0100] From the above discussion, when the relative density of the material is within the value range, that is, x min ≤x e ≤1, then there is Substituting it into formula (15), the result is as follows:

[0101]

[0102] Substituting equations (6), (8) and (13) into equation (16), we get the following results:

[0103]

[0104] The above formula is based on the idea of ​​fixed point iteration, and the result of converting it into an explicit form is as follows:

[0105]

[0106] make

[0107] Then there is

[0108] The heuristic update format of the design variables obtained by the optimization criterion method (OC method) is as follows:

[0109]

[0110] Where k is the number of iterations; ξ is the damping factor, which ensures the convergence of the results; and m is the forward moving limit, which improves the stability of the iterative process.

[0111] During the iteration, the Lagrange multiplier λ of the low carbon constraint is calculated using the bisection method. g , the calculation formula is as follows:

[0112]

[0113] Where k is the number of iterations; It is the intermediate variable of the Lagrange multiplier in the iterative process; and For the upper and lower limits of the Lagrange multiplier in the iteration process under KT conditions, set It is usually a larger value.

[0114] make when When when When in, is the structural carbon emissions during the iteration process.

[0115] Step 7: Determine whether the result meets the optimization convergence conditions. If not, go to step 4 and perform iterative calculations in sequence. If it meets the conditions, terminate the topology optimization process and obtain a topology optimization model that meets the low-carbon constraints.

[0116] The optimization convergence condition is:

[0117]

[0118] Reference Figure 2-Figure 4 The invention is further described by taking a certain car rear anti-collision beam as an example. The anti-collision beam is an important part of the car body structure. It can directly protect the safety of the driver and passengers in a collision. Therefore, it has high strength requirements and consumes a lot of materials. A large amount of carbon emissions will be generated during its manufacturing process. The initial structure, boundary conditions and load action position of the car rear anti-collision beam are as follows: Figure 2 As shown in the figure, the front beam is now optimized. The initial structural design space is 120cm*12cm. There are two fixed ends on the lower side. The upper midpoint is subjected to a vertical downward concentrated force of 1kN. The overall structure of the rear anti-collision beam of the car is made of aluminum alloy with a Poisson's ratio of 0.3, an elastic modulus of 70GPa, and a density of 2.7×10 3 kg / m 3 The carbon emission factor is 16.56 kg CO2eq / kg. For carbon emission calculations, the beam width is 15 cm, and the design service life of the structure is set to 30 years. The design space of the beam is discretized into 1440 four-node rectangular elements. The penalty factor p during the iteration process is 3, the sensitivity filter radius is 1.5, the damping factor is 0.5, and the forward shift factor is 0.2.

[0119] Figure 3 This is the structural diagram of the anti-collision beam after optimization according to the constraint conditions. Compared with the original structure, it can be seen that the optimized structure better caters to the overall actual force and fixed constraints, and the material distribution is more reasonable.

[0120] Figure 4 The curve of the structural flexibility of the anti-collision beam changing with the number of iterations during the iteration process under the low carbon constraint is shown in Figure 2. Figure 4 The structural flexibility of the anti-collision beam decreased from 2602.44 N·m to a final value of 635.65 N·m, a reduction of approximately 75.60%. The carbon emissions of the original solid anti-collision beam were 965.78 kg CO₂eq. The carbon emissions limit for the anti-collision beam structure was set to no more than 526.45 kg CO₂eq, a reduction of approximately 45.49% compared to the original structure. The final result converged to 526.35 kg CO₂eq, satisfying the low-carbon constraint and meeting the expected requirements.

[0121] Based on SIMP-based structural topology optimization, this paper sets a carbon emission limit for the structure using greenhouse gas reduction rates and incorporates this limit into the topology optimization method as a low-carbon constraint. Through sensitivity analysis and combined with an optimization criterion approach, an iterative format for the topology optimization model under this low-carbon constraint was developed. MATLAB software was used to implement topology optimization improvements for an automotive rear bumper beam example.

[0122] It will be easily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A structural topology optimization method under low carbon constraints, characterized in that: The method is implemented as follows: Step 1: Define the target product structure working conditions and design domain, set boundary conditions, determine the structural stress points and calculate the loads; Step 2: Using the variable density method, the relative density of the unit is used as the design variable, the minimum flexibility of the structure is used as the optimization goal, and the carbon emission limit of the structure is used as the constraint condition to establish a topology optimization model; Step 3: Discretize the initial design domain into nelx × nely finite element elements, introduce intermediate density elements through the variable density method, set the penalty factor and filter radius, and initialize the design variables; Step 4: Use the improved solid isotropic material penalty model to calculate the elastic modulus of the unit, obtain the unit stiffness matrix, assemble the overall stiffness matrix of the structure, perform finite element analysis on the structure under the set boundary conditions and loads, and calculate the displacement of the unit node; Step 5: Accumulate the carbon emissions of each finite element to obtain the overall carbon emissions of the structure, and solve the target flexibility function and the sensitivity of the overall carbon emission constraint of the structure to the design variables; Step 6: Construct the Lagrangian function of the target model so that the optimal solution satisfies the Kurtak condition, i.e., the KT condition. Use the sensitivity filtering method to adjust the sensitivity of the target flexibility function and the low-carbon constraint condition. Use the optimization criterion method to obtain the heuristic update format of the design variables and update the design variables. Step 7: Determine whether the result meets the optimization convergence condition. If not, go to step 4 and perform iterative calculations in sequence. If it meets the condition, terminate the topology optimization process and obtain a topology optimization model that meets the low-carbon constraint. The topology optimization model is: in, is the relative density of the material unit; To design space; is the flexibility of the structure; is the displacement vector of the structure; is the stiffness matrix of the structure; is the force vector of the structure; is the carbon emission value of the structure; is the maximum global warming potential of the structure to be set; It is the minimum value of the relative density of the material unit. Taking a small value close to 0 can prevent the unit stiffness matrix from being singular. The calculation method for structural carbon emissions is as follows: Where, The structure contains Relative density of the material unit; For the The volume of the material unit; For the Density of the material; For the Carbon emission factors of the materials; is the number of units; is the number of material types.

2. The structural topology optimization method under low-carbon constraints according to claim 1, characterized in that: The low-carbon constraint construction method is as follows: (1) During the service life of the structure, the total carbon emission value of the structure during its service life is obtained by accumulating the greenhouse gas emission reduction rate and the initial carbon emission value. The maximum carbon emission value is set according to the target greenhouse gas emission reduction rate during the service life of the structure, which is used as the low-carbon constraint condition for topology optimization. The calculation formula is as follows: in, is the initial carbon emission value of the structure; is the design service life of the structure; is the greenhouse gas emission reduction rate; (2) Based on a large number of greenhouse gas mitigation scenarios investigated by Working Group III of the IPCC Fifth Assessment, and the prediction of the corresponding relationship between the future global anthropogenic greenhouse gas emission reduction rate and atmospheric CO2 concentration, it is assumed that the carbon emission limit is proportional to the greenhouse gas emission reduction rate and changes deterministically and linearly over time. Based on the assumption, the calculation formula for the global greenhouse gas emission reduction rate is proposed as follows: in, is the greenhouse gas emission reduction rate; Set the useful life of the structure.

3. The structural topology optimization method under low carbon constraints according to claim 2, characterized in that: The improved solid isotropic material penalty model is as follows: ， in, is the element stiffness during the iteration process; is the initial element stiffness of the material; A very small stiffness assigned to the void region can prevent the generation of a stiffness singular matrix; is a penalty factor that can speed up the convergence and make the density of the intermediate units tend to the values ​​at the two ends, where the values ​​at the two ends are 0 or 1.

4. The structural topology optimization method under low carbon constraints according to claim 3, characterized in that: The sensitivity of the structural carbon emission constraint to the design variables is as follows: The sensitivity of the structural flexibility objective function to the design variables is as follows: 。 5. The structural topology optimization method under low-carbon constraints according to claim 4, characterized in that: The Lagrangian function expression that satisfies the KT condition under the low carbon constraint is as follows: in, , , , is the Lagrange multiplier, where is a vector, and the rest are scalars.

6. The structural topology optimization method under low carbon constraints according to claim 5, characterized in that: During the iteration process, the Lagrange multiplier of the low carbon constraint is calculated using the bisection method. , the calculation formula is as follows: in is the number of iterations; It is the intermediate variable of the Lagrange multiplier in the iterative process; and are the upper and lower limits of the Lagrange multiplier in the iterative process under KT conditions, Calculate carbon emissions during the iteration of the optimization model season ,when When ;when When ,in is the initial carbon emission value of the structure.

7. The structural topology optimization method under low carbon constraints according to claim 6, characterized in that: The iterative format of the Lagrangian function obtained by the fixed point idea is: ,make , then .

8. The structural topology optimization method under low carbon constraints according to claim 7, characterized in that: The heuristic update format of the design variables obtained by the optimization criterion method is as follows: in, is the number of iterations; is the damping factor, which can ensure the convergence of the results and is 0.5; It is the positive movement limit, which can improve the stability of the iterative process and is set to 0.1~0.

3.

9. The structural topology optimization method under low carbon constraints according to claim 8, characterized in that: The optimization convergence condition is: in, is the convergence accuracy.

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