A Structural Dynamics Topology Optimization Design Method Driven by Critical Stiffness

Abstract

The present invention discloses a structural dynamics topology optimization design method driven by critical stiffness. With the original design domain and optimization model as inputs, the multi-objective optimization model that couples static stiffness and natural frequency is relaxed into two consecutive single-objective topology optimization stages by using critical stiffness information. A design domain transformation process is carried out between the two-stage topology optimizations, and the optimization solutions of the two stages are combined as an approximate solution to the original problem. The present invention ensures the static stiffness performance of the optimization result, can effectively improve the natural frequency of the structure, and avoid resonance failure of parts under dynamic loads.

Description

Technical Field

[0001] The present invention belongs to the technical field of structural lightweight design, and particularly relates to a structural dynamics topology optimization design method driven by critical stiffness. Background Technique

[0002] As an important means of structural innovative design and structural lightweight design, the topology optimization design method can achieve the optimal distribution of materials under multi-load and multi-condition constraints, and effectively improve the material utilization rate. With the development of additive manufacturing technology, the preparation difficulties of complex structures in traditional manufacturing methods, especially structures such as special-shaped pipes and disordered porous structures, have been solved, greatly expanding the degree of freedom of structural design and providing new opportunities for structural lightweight design. Conducting research on the structural dynamics topology optimization design method for additive manufacturing has become the most potential method to improve the payload, flight range, and maneuverability of aviation and aerospace equipment.

[0003] The design of structural dynamics performance is a key issue that must be solved in the lightweight design process of complex structural products in the fields of aviation, aerospace, and ordnance. Restricted by traditional design concepts and manufacturing processes, the discontinuous design mode is extremely likely to interfere with and damage the optimality of the structure, limiting the optimization design space of the structure and making the improvement of product performance very limited. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a structural dynamics topology optimization design method driven by critical stiffness in view of the above-mentioned deficiencies of the prior art. Taking the optimization of the natural frequency in the main frequency-domain dynamics topology optimization as the starting point, the structural dynamics topology optimization design driven by critical stiffness is carried out, which can improve the natural frequency of the structure while ensuring the static stiffness.

[0005] To achieve the above technical purpose, the technical solution adopted by the present invention is as follows:

[0006] A structural dynamics topology optimization design method driven by critical stiffness, taking the original design domain and the optimization model as inputs, using the critical stiffness information to relax the original multi-objective optimization model that couples static stiffness and natural frequency into two consecutive single-objective topology optimization stages, performing design domain transformation processing between the two-stage topology optimization, and combining the optimization solutions of the two stages as an approximate solution to the original problem.

[0007] To optimize the above technical solution, the specific measures taken also include:

[0008] The above method includes:

[0009] Step 1: Input the original multi-objective topology optimization model that couples static stiffness and natural frequency, its design domain, and critical stiffness information;

[0010] Step 2: Based on the progressive structure optimization method, perform progressive optimization with static stiffness as the single optimization objective in the first stage. During the progressive optimization process, monitor the difference between the current stiffness and the critical stiffness. When the current stiffness is lower than the critical stiffness, the optimization terminates, and the current iteration step is rolled back to obtain an intermediate solution that can meet the critical stiffness constraint as the first-stage solution;

[0011] Step 3: Perform a Boolean subtraction operation between the original design domain and the first-stage solution to obtain the design domain for the second-stage topology optimization, and subtract the volume of the first-stage solution from the original volume constraint as the volume constraint for the second stage;

[0012] Step 4: Based on the progressive structure optimization method, perform progressive optimization with the natural frequency as the single optimization objective in the second stage on the design domain obtained from the Boolean operation. During the progressive optimization process, monitor the difference between the current volume information and the volume constraint of this stage. When the current volume is greater than the volume constraint of the original problem, roll back the current iteration step, and calculate the difference between the volume after rollback and the volume constraint of the current stage; use the calculated difference as the rejection ratio for the current iteration step to perform the last step of natural frequency optimization iteration;

[0013] Step 5: Read the optimization results of the first stage and the second stage, and perform a Boolean sum operation on them. Take the combined optimization result as the approximate solution of the original multi-objective topology optimization problem;

[0014] Step 6: Map the design variables back into the three-dimensional material space and perform post-processing to obtain the three-dimensional model of the dynamic optimization result.

[0015] For the original multi-objective topology optimization model described in Step 1 above, the objective function is the weighted coupling of the compliance representing static stiffness and a certain natural frequency, and there is a volume constraint with a lightweight requirement;

[0016] The optimization model is as follows:

[0017] find x1,x2,..,x n in Ω

[0018] min f=λ1C + λ2ω

[0019] s.t.K(x)U=F

[0020]

[0021] x i =x min or 1

[0022] where x iis the design variable of the mapping material unit, Ω is the original design domain, λ1 and λ2 are the weighting factors of multi-objective optimization, f is the minimization objective function of the optimization model, which consists of two terms: C is the compliance term representing the structural stiffness objective, and ω is the natural frequency term representing the dynamic objective. K is the structural stiffness matrix, U is the displacement matrix, F is the load vector, M is the mass matrix, is the modal eigenvector, υ ol is the original volume constraint, x min is the minimum value of the design variable specified to avoid singularity of the stiffness matrix.

[0023] In the above step 2, the progressive structural optimization method using the soft killing strategy is used to obtain better optimization ability and volume progressive process. And the progressive optimization with static stiffness as the single optimization objective based on the progressive structural optimization method reduces its own volume progressively, but the progressive termination condition is not to reach the volume constraint, but to monitor the stiffness information of the current iteration result during each progressive iteration process and compare it with the critical stiffness, and the optimization terminates after the progressive optimization until the current stiffness is lower than the critical stiffness.

[0024] In the above step 2, the static stiffness optimization model in the first stage is expressed as the following formula:

[0025] find[X1] = x1, x2,..., x n in Ω

[0026]

[0027] s.t. K(x)U = F

[0028]

[0029] x i = x min or 1

[0030] where υol1 is the volume after fallback, that is, the volume of the solution in the first stage, and [X1] is the set of design variables mapped by the material units included in the solution in the first stage.

[0031] In the above step 4, the design domain in the second stage of the natural frequency optimization based on the progressive structural optimization method is obtained by the Boolean subtraction of the original design domain and the solution in the first stage, which is a subset of the original design domain. The solution in the first stage is used as the non-design domain in the second stage optimization and still participates in the finite element modal operation during the iteration process.

[0032] In the above step 4, the progressive direction of the progressive structural optimization method is volume increasing, and the second stage natural frequency optimization model is as follows;

[0033]

[0034] Among them, [X2] is the set of design variables mapped by the material units included in the second-stage solution.

[0035] The present invention has the following beneficial effects:

[0036] The present invention can relax the original multi-objective topology optimization model of the static stiffness and natural frequency coupling by using the critical stiffness information into two continuous single-objective topology optimization models, and merge the solutions of the two single-objective optimizations as the approximate solution of the original dynamic multi-objective topology optimization problem, effectively improving the optimization efficiency while ensuring the optimization effect.

[0037] 1. The present invention relaxes the original multi-objective dynamic topology optimization model with the critical stiffness information, and takes the union of the two single-objective topology optimization solutions as the approximate solution of the original problem, avoiding the determination process of the weighting factor and the normalization process, and improving the optimization efficiency;

[0038] 2. The present invention relaxes the original multi-objective dynamic topology optimization model with the critical stiffness information, and completes the static stiffness optimization under the critical stiffness constraint in the first stage, ensuring the static stiffness performance of the optimization result;

[0039] 3. The present invention takes the maximization of the natural frequency as the optimization objective in the second stage, effectively improving the natural frequency of the structure and avoiding the resonance failure of the parts under dynamic loads. Description of the Drawings

[0040] Figure 1 is the flow chart of the structural dynamic topology optimization design method driven by the critical stiffness of the present invention;

[0041] Figure 2 is the optimization process diagram of the three-dimensional cantilever beam of the present invention;

[0042] Figure 3 is the flow chart of the two-stage topology optimization implemented by the present invention;

[0043] Figure 4 is the example diagram of the design domain transformation of the present invention;

[0044] Figure 4 a is the initial design domain of the original multi-objective optimization problem; Figure 4 b is the distribution of the optimization solution in the design domain after the first-stage optimization is completed; Figure 4 c is the second-stage optimized design domain obtained by the Boolean operation of the original design domain and the first-stage optimization solution; Figure 4 d is the final merging of the first-stage optimization solution and the second-stage optimization solution. Detailed Embodiment

[0045] The following further describes the embodiments of the present invention in detail with reference to the drawings.

[0046] The present invention discloses a structural dynamics topology optimization design method driven by critical stiffness. Taking the original design domain and the optimization model as inputs, the original multi-objective optimization model that couples static stiffness and natural frequency is relaxed into two consecutive single-objective topology optimization stages by using critical stiffness information. A design domain transformation process is carried out between the two-stage topology optimizations, and the optimization solutions of the two stages are combined as an approximate solution to the original problem. Based on the characteristic that increasing the material stiffness of a static stiffness structure will not decrease, static stiffness optimization is used as the first optimization stage, and natural frequency optimization is used as the second optimization stage; the volume constraint of the original multi-objective optimization is realized through the design domain transformation in the first and second stage optimizations, and the natural frequency is increased while satisfying the critical stiffness constraint. The present invention can avoid the weighted factor problem of multi-objective topology optimization and realize the coupling of static stiffness and natural frequency in dynamic optimization; the union of the two single-objective topology optimization solutions is used as an approximate solution to the original problem, effectively improving the optimization efficiency while ensuring the optimization effect.

[0047] As Figure 1 shown, specifically including:

[0048] Step 1: Input the original multi-objective topology optimization model that couples static stiffness and natural frequency, its design domain, and critical stiffness information;

[0049] In the embodiment, for the original multi-objective topology optimization model described in Step 1, its objective function is the weighted coupling of the compliance representing static stiffness and a certain natural frequency, and there is a volume constraint for the lightweight requirement;

[0050] The optimization model is as follows:

[0051] find x1, x2,..., x n in Ω

[0052] min f = λ1C + λ2ω

[0053] s.t. K(x)U = F

[0054]

[0055] x i = x min or 1

[0056] where x i is the design variable mapping the material element, Ω is the original design domain, λ1 and λ2 are the weighted factors of the multi-objective optimization, f is the minimization objective function of the optimization model, which consists of two terms: C is the compliance term representing the structural stiffness objective, and ω is the natural frequency term representing the dynamic objective. K is the structural stiffness matrix, U is the displacement matrix, F is the load vector, M is the mass matrix, is the modal eigenvector, υol is the original volume constraint, xmin The minimum value of the design variable specified to avoid singularity of the stiffness matrix.

[0057] Step 2: Based on the progressive structure optimization method, perform the first-stage progressive optimization with static stiffness as the single optimization objective. During the progressive optimization process, monitor the difference between the current stiffness and the critical stiffness. When the current stiffness is lower than the critical stiffness, the optimization terminates, and the current iteration step is rolled back to obtain an intermediate solution that can satisfy the critical stiffness constraint as the first-stage solution.

[0058] In the embodiment, the progressive structure optimization method using the soft killing strategy is used in Step 2 to obtain better optimization ability and volume progressive process. And the progressive optimization with static stiffness as the single optimization objective based on the progressive structure optimization method reduces its own volume progressively, but the progressive termination condition is not to reach the volume constraint, but to monitor the stiffness information of the current iteration result during each progressive iteration and compare it with the critical stiffness. When the current stiffness is lower than the critical stiffness, the optimization terminates.

[0059] When the static stiffness single-objective topology optimization based on the progressive structure optimization method monitors that the stiffness of the current iteration step is lower than the critical stiffness, it indicates that the current volume cannot meet the critical stiffness constraint. Therefore, it is necessary to roll back to the previous iteration step to obtain an intermediate solution that can satisfy the critical stiffness constraint.

[0060] The static stiffness optimization model in the first stage is expressed as the following formula:

[0061] find [X1] = x1, x2,..., x n in Ω

[0062]

[0063] s.t. K(x)U = F

[0064]

[0065] x i = x min or 1

[0066] where υol1 is the volume after rolling back, that is, the volume of the first-stage solution, and [X1] is the set of design variables mapped by the material units included in the first-stage solution.

[0067] Step 3: Perform a Boolean subtraction operation on the original design domain and the first-stage solution to obtain the design domain for the second-stage topology optimization, and subtract the volume of the first-stage solution from the original volume constraint as the volume constraint for the second stage.

[0068] Step 3: The first-stage solution is obtained through volume progression and is a subset of the original design domain. The Boolean subtraction operation is performed between the original design domain and the first-stage solution to obtain the design domain for the second-stage topology optimization, thus ensuring that there is no intersection in the results of the two-stage topology optimization. The volume constraint of the original problem minus the volume of the first-stage solution is used as the volume constraint for the second stage, so that the union of the first-stage topology optimization solution and the second-stage topology optimization solution exactly satisfies the original volume constraint;

[0069] That is, the design domain Ω2 for the second stage is as follows:

[0070] Ω2 = Ω - [X1]

[0071] In the said Step 3, the material elements retained by the evolutionary structural optimization method are used as the solution, and the removed material elements (i.e., the design variables assigned x i = x min under the soft kill strategy) are not within the scope of the optimization solution.

[0072] Step 4: Based on the evolutionary structural optimization method, perform the second-stage progressive optimization with the natural frequency as the single optimization objective on the second-stage design domain Ω2 obtained by the Boolean operation. During the progressive optimization process, monitor the difference between the current volume information and the volume constraint of this stage. After the progressive optimization until the current volume is greater than the volume constraint of the original problem, roll back the current iteration step, and calculate the difference between the volume after rollback and the volume constraint of the current stage; use the calculated difference as the rejection ratio of the current iteration step to perform the last step of the natural frequency optimization iteration;

[0073] That is, in the said Step 4, a conformal constraint is imposed during the optimization process to ensure that the optimization result of the second stage is connected in the three-dimensional space with the optimization result of the first stage, guaranteeing that a single integral structure is obtained after the subsequent Boolean sum operation.

[0074] The implementation of the conformal constraint is completed through the gradual change of the second-stage design domain: The overall design domain assigned in the second stage is limited to the periphery of the optimization result of the first stage at the beginning of the iteration. As the material elements increase, the design domain around the solid material elements gradually opens up.

[0075] In the embodiment, the design domain for performing the natural frequency optimization based on the evolutionary structural optimization method in Step 4 is a subset of the original design domain. The solution of the first stage, as the non-design domain for the second-stage optimization, still participates in the finite element modal operation during the iteration process, thereby ensuring that the second-stage optimization solution can improve the natural frequency of the overall structure.

[0076] In Step 4, the progressive direction of the progressive structure optimization method is an increase in volume. When it is detected that the current volume is greater than the volume in the second stage, in order to ensure the volume constraint, the current iteration step is rolled back, and the difference between the volume after rollback and the volume constraint is calculated. The calculated volume difference is used as the rejection ratio of the iteration, so that the volume of the optimization result exactly meets the volume constraint.

[0077] And the natural frequency optimization model in the second stage is as follows;

[0078] find[X2] = x1, x2,..., x n in Ω

[0079] min ω

[0080]

[0081] x i = x min or 1

[0082] where [X2] is the set of design variables mapped by the material units included in the solution in the second stage.

[0083] Step 5: Read the optimization results of the first stage and the second stage, and perform a Boolean sum operation on them. The combined optimization result is used as an approximate solution to the original multi-objective topology optimization problem, as follows:

[0084] [X] = [X1] ∪ [X2]

[0085] Since there is no intersection between the optimization results of the two stages after the design domain transformation, the union [X] of the optimization results exactly meets the volume constraint of the original problem and can be used as an approximate solution to the original problem.

[0086] Step 6: Map the design variable result [X] back into the three-dimensional material space and perform post-processing to obtain the three-dimensional model of the dynamic optimization result.

[0087] Embodiment

[0088] Taking a three-dimensional cantilever beam as an example, the structural changes in the optimization process are as Figure 2 shown.

[0089] Among them, the topology optimization of the two stages is realized based on the progressive structure optimization method of the soft kill strategy to find the critical stiffness by using the volume progressive process; in the design domain transformation and material unit mapping, the remaining material units are used as the solution, and the removed material units (i.e., the material units assigned x under the soft kill strategy) i = x minThe design variables) do not fall within the scope of the optimal solution; during the optimization process, conformal constraints are imposed to ensure that the optimization results of the second stage are connected in three-dimensional space with those of the first stage, guaranteeing that a single integral structure is obtained after the Boolean sum operation.

[0090] Figure 3 This is the flowchart of the two-stage single-objective dynamic optimization driven by critical stiffness for the present invention. Figure 4 This is an example diagram of the design domain transformation for the present invention. Figure 4 a is the initial design domain of the original multi-objective optimization problem; after the first-stage optimization, the optimal solutions are distributed in the design domain as shown in Figure 4 b; the second-stage optimized design domain is obtained through the Boolean operation of the original design domain and the optimal solutions of the first stage, as shown in Figure 4 c; finally, the optimal solutions of the first stage and the second stage are merged, as shown in Figure 4 d.

[0091] The present invention can use critical stiffness information to relax the original multi-objective topology optimization model that couples static stiffness and natural frequency into two consecutive single-objective topology optimization models, and merge the solutions of the two single-objective optimizations as an approximate solution to the original dynamic multi-objective topology optimization problem, effectively improving the optimization efficiency while ensuring the optimization effect.

[0092] 1. The present invention relaxes the original multi-objective dynamic topology optimization model with critical stiffness information, and takes the union of the two single-objective topology optimization solutions as the approximate solution to the original problem, avoiding the process of determining the weighting factor and the normalization process, and improving the optimization efficiency.

[0093] 2. The present invention relaxes the original multi-objective dynamic topology optimization model with critical stiffness information, and completes the static stiffness optimization under the critical stiffness constraint in the first stage, ensuring the static stiffness performance of the optimization results.

[0094] 3. The present invention takes the maximization of the natural frequency as the optimization objective in the second stage, effectively improving the natural frequency of the structure and avoiding resonance failure of the parts under dynamic loads.

[0095] The above are only the preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions falling within the concept of the present invention belong to the protection scope of the present invention. It should be noted that for those of ordinary skill in the art, several improvements and refinements made without departing from the principle of the present invention should be regarded as within the protection scope of the present invention.

Claims

1. A structural dynamics topology optimization design method driven by critical stiffness, characterized in that, Taking the original design domain and the optimization model as inputs, using the critical stiffness information, the original multi-objective optimization model that couples the static stiffness and the natural frequency is relaxed into two consecutive single-objective topology optimization stages. A design domain transformation process is carried out between the two-stage topology optimizations, and the optimization solutions of the two stages are combined as an approximate solution to the original problem. The method includes: Step 1: Input the original multi-objective topology optimization model that couples the static stiffness and the natural frequency, its design domain, and the critical stiffness information; Step 2: Based on the evolutionary structural optimization method, perform the first-stage evolutionary optimization with the static stiffness as the single optimization objective. During the evolutionary optimization process, monitor the difference between the current stiffness and the critical stiffness. After the evolutionary optimization until the current stiffness is lower than the critical stiffness, the optimization is terminated, and the current iteration step is rolled back to obtain an intermediate solution that can satisfy the critical stiffness constraint as the first-stage solution; Step 3: Perform a Boolean subtraction operation on the original design domain and the first-stage solution to obtain the design domain for the second-stage topology optimization, and subtract the volume of the first-stage solution from the original volume constraint as the volume constraint for the second stage; Step 4: Based on the evolutionary structural optimization method, perform the second-stage evolutionary optimization with the natural frequency as the single optimization objective on the design domain obtained by the Boolean operation. During the evolutionary optimization process, monitor the difference between the current volume information and the volume constraint for the second stage. After the evolutionary optimization until the current volume is greater than the volume constraint of the original problem, roll back the current iteration step, and calculate the difference between the volume after rollback and the volume constraint for the current stage; Use the calculated difference as the rejection ratio for the current iteration step to perform the last step of the natural frequency optimization iteration; Step 5: Read the optimization results of the first stage and the second stage, and perform a Boolean sum operation on them. Take the combined optimization result as an approximate solution to the original multi-objective topology optimization problem; Step 6: Map the design variables back into the three-dimensional material space and perform post-processing to obtain the three-dimensional model of the dynamic optimization result.

2. A structural dynamics topology optimization design method driven by critical stiffness according to claim 1, characterized in that For the original multi-objective topology optimization model described in Step 1, its objective function is the weighted coupling of the compliance representing the static stiffness and a certain natural frequency, and there is a volume constraint for the lightweight requirement; The optimization model is as follows: where x i is the design variable of the mapping material unit, Ω is the original design domain, λ1 and λ2 are the weighting factors of multi-objective optimization, f is the minimization objective function of the optimization model, which consists of two terms: C is the compliance term representing the structural stiffness objective, ω is the natural frequency term representing the dynamic objective, K is the structural stiffness matrix, U is the displacement matrix, F is the load vector, M is the mass matrix, is the modal eigenvector, vol is the original volume constraint, x min is the minimum value of the design variable specified to avoid singularity of the stiffness matrix.

3. A structural dynamics topology optimization design method driven by critical stiffness according to claim 1, characterized in that In Step 2, the evolutionary structural optimization method using the soft kill strategy is used to obtain better optimization ability and volume evolution process. And the evolutionary optimization with the static stiffness as the single optimization objective based on the evolutionary structural optimization method gradually reduces its own volume in an evolutionary manner. However, the evolutionary termination condition is not to reach the volume constraint, but to monitor the stiffness information of the current iteration result during each evolutionary iteration and compare it with the critical stiffness. After the evolutionary optimization until the current stiffness is lower than the critical stiffness, the optimization is terminated.

4. A structural dynamics topology optimization design method driven by critical stiffness according to claim 1, characterized in that In Step 2, the static stiffness optimization model for the first stage is expressed as the following formula: where vol1 is the volume after rollback, that is, the volume of the first-stage solution, and [X1] is the set of design variables mapped by the material elements included in the first-stage solution.

5. A structural dynamics topology optimization design method driven by critical stiffness according to claim 1, characterized in that In step 4, the second-stage design domain for natural frequency optimization based on the progressive structural optimization method is obtained by Boolean subtraction of the original design domain and the solution of the first stage. It is a subset of the original design domain. The solution of the first stage, as the non-design domain for the second-stage optimization, still participates in the finite element modal calculation during the iteration process.

6. The structural dynamics topology optimization design method driven by critical stiffness according to claim 1, characterized in that In step 4, the progressive direction of the progressive structural optimization method is towards volume increase, and the second-stage natural frequency optimization model is as follows: where [X2] is the set of design variables mapped by the material elements included in the second-stage solution.

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