High-speed aircraft wing and rudder structure optimization design method based on extended multi-scale finite element method
By combining extended multi-scale finite element method with additive manufacturing topology optimization and porous microstructure sandwich layer design, the problems of long calculation cycle, poor practicality and low multi-scale matching degree in existing wing and rudder design are solved, and efficient wing and rudder structure optimization is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- DALIAN UNIV OF TECH
- Filing Date
- 2023-06-13
- Publication Date
- 2026-06-23
AI Technical Summary
Existing topology optimization methods fail to effectively consider additive manufacturing constraints, resulting in complex optimization results with low practicality. They also have low computational efficiency for porous microstructures, poor matching between macro and micro structures, and difficulty in achieving multi-scale coordinated design.
An extended multi-scale finite element method was adopted, combined with additive manufacturing topology optimization method. By adding process constraints and geometric limitations in topology optimization, force transmission path characteristics were extracted, and a porous microstructure sandwich layer was used for collaborative integrated design to optimize the topology stiffeners and micro-filling structure.
It improves the engineering practicality and computational efficiency of topology optimization results, realizes multi-scale matching of macro and micro structures, and provides a wing and rudder design scheme that is closer to engineering practice.
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Figure CN116842633B_ABST
Abstract
Description
Technical Field
[0001] This application belongs to the field of structural optimization technology, and relates to a method for optimizing the design of wing and rudder structures of high-speed aircraft based on the extended multi-scale finite element method. Background Technology
[0002] As a key component of high-speed aircraft, the wing rudder structure plays a crucial role in controlling the aircraft's attitude during flight. Given the extreme load conditions that high-speed flight wing rudders must withstand during flight, topological stiffening—micro-filling structures can effectively achieve lightweight and multi-functional designs for wing rudders while meeting harsh service environments and advanced performance requirements.
[0003] Topology optimization methods and additive manufacturing technology have flourished in recent years due to their significant engineering applications. Their organic integration can break through traditional size and shape optimization design, realizing the design and manufacturing of ultra-lightweight, multi-scale, and integrally efficient load-bearing structures. This makes the integrated design and manufacturing of topology stiffening and micro-filling structures for wing rudders possible. However, existing topology stiffening and micro-filling structure design methods have many drawbacks: First, for topology optimization, if the unique manufacturing constraints in the additive manufacturing process are not considered, the optimization result is often a complex conceptual design structure with low practical value; second, for porous microstructure filling layers, parametric microstructures suffer from problems such as large microstructure model size and low computational efficiency; finally, for the size optimization stage, existing methods still remain at the stage of first optimizing the solid structure and then micro-filling, making it difficult to achieve multi-scale matching and coordinated design of macrostructure and micro-filling structure.
[0004] To address the issue of low applicability of topology optimization results, it is advisable to incorporate process constraints into the topology optimization process, developing additive manufacturing structure topology optimization methods to obtain optimization results that more closely reflect engineering realities. For porous microstructures, the extended multi-scale finite element method (EPM) constructs basis functions numerically by solving local subproblems on elements. These basis functions accurately and effectively reflect the characteristics of porous microstructures, allowing for accurate solutions at the macroscopic level. Finally, to address the issue of macroscopic-microscopic matching, a porous microstructure core layer can be added during the dimensional optimization stage. The core layer and topological stiffeners can be designed collaboratively and integratedly, effectively resolving the matching problem. Summary of the Invention
[0005] This invention provides a wing and rudder design method based on the extended multi-scale finite element method, which solves the technical problems of long calculation cycle, poor multi-scale matching between macroscopic overall structure and microscopic filling structure, and poor practicality in the prior art.
[0006] The technical solution adopted by this invention to solve its technical problem is as follows:
[0007] A method for optimizing the design of wing and rudder structures of high-speed aircraft based on the extended multi-scale finite element method includes the following steps:
[0008] Step 1: Use additive manufacturing structure topology optimization methods to complete the structure topology optimization calculation based on multiple load conditions and boundary conditions; Step 1 includes the following steps:
[0009] Step 1.1: Based on the traditional wing rudder topology optimization method, which sets the skin and rudder shaft as non-design domains, a solid area of 5-10mm is reserved at the leading edge, trailing edge, leading tip, and bottom of the wing rudder and added to the non-design domain of topology optimization. The rudder core area other than the above areas is set as the design domain of topology optimization.
[0010] Step 1.2: Add demolding control as a geometric constraint to the topology optimization constraints; and add dimensional constraints to the topology optimization based on the overall dimensions of the wing and rudder structure. These generally include maximum member size constraints and minimum member size constraints, which can be used to limit the range of dimensional variations in the structure in the topology optimization results. The mathematical model for the geometric constraints is as follows:
[0011]
[0012] In the above formula: ρ i is the pseudo-density of the i-th element in the draft direction; n is the number of finite element elements in the design domain; L is the equivalent characteristic size of the void in the discrete element; L max It is the largest member size; L min It is the smallest member size; l d It is the average depth of the projection; S i It is the surface area of boundary element i.
[0013] Step 1.3: Topology optimization method based on the Variable Density Method (SIMP):
[0014] The basic idea of the variable density method is to artificially introduce a material with a relative density that varies between 0 and 1, and then assume that there is a certain correspondence between the relative density of the material and its physical properties (such as Young's modulus), expressing this correspondence through an explicit function of the relative density. As the most commonly used material interpolation model in the variable density method, the relative density x of the element material is defined. i With Young's elastic modulus E i The relational expression is:
[0015]
[0016] In the formula: E0 is the Young's modulus of the solid material (i.e., the material with a relative density of 1); E i(x) represents the interpolated elastic modulus; p is the penalty factor (usually 3), which is used to avoid a large number of intermediate density units in the optimization result, thus obtaining a clear material layout.
[0017] Since the finite element elements remain constant during topology optimization, the element stiffness matrix k i The change is only related to Young's modulus E i Regarding this, combining the finite element stiffness matrix calculation formula and the above equation, we can derive:
[0018]
[0019] In the formula: k0 is the element stiffness matrix of the solid material.
[0020] The design region Ω is divided into n finite element elements, and the design variable vector is X = [x1x2…x] n ] T The volume and physical properties of the structure are expressed as functions of X.
[0021] Based on the above, taking the maximum overall stiffness of the wing-rudder structure as the objective function, the mathematical model for topology optimization based on metal additive manufacturing under volume and geometric constraints is as follows:
[0022]
[0023] In the formula: v i V represents the volume of the i-th unit; V is the structural volume. The upper limit of the constraint volume is denoted by KU = F, which is the finite element equilibrium equation. K, U, and F are the overall stiffness matrix, overall displacement vector, and overall load vector of the structure, respectively.
[0024] Based on the mathematical model of topology optimization based on the variable density (SIMP) method, the wing rudder is subjected to topology optimization based on metal additive manufacturing, and the topology optimization results are obtained.
[0025] Step 2: Extract features from the force transmission path to complete the preliminary design of the topological stiffener configuration; Step 2 includes the following steps:
[0026] Step 2.1: The topology optimization results from Step 1 may contain isolated body features and broken branch features. Isolated body features are caused by the intermediate density units in the variable density method potentially resulting in isolated needle-like structures in the optimization results. Broken branch features are characterized by structures that are often not continuous or integrated in the optimization results. When extracting features from the force transmission path, the following criteria are used for the above features: For isolated body features, some isolated body features are removed because isolated bodies not only fail to improve the overall structural stiffness but also increase the structural weight; for broken branch features, the broken branch material is connected and extended to the frame, because broken branches are an important representation of the optimal force transmission path, and this method is used to improve overall stiffness and maintain aerodynamic shape.
[0027] Step 2.2: To obtain a topological stiffener design that conforms to engineering practice, the constraints of laser additive manufacturing processes also need to be considered. The constraint of closed cavities arises because when the structural model has closed holes, residual powder or support material inside these holes is difficult to remove, affecting the structure's performance. To address this constraint, powder removal holes are applied to the topological stiffeners and wing / rudder edges to minimize the performance impact of the closed cavity constraint. In addition, there is the constraint of overhang: the optimal forming angle of the stiffener structure should be 45° to the overall forming direction. To address this constraint, during the preliminary design stage of the topological stiffeners, ensuring that all stiffener structures have an angle greater than 45° with the forming direction provides self-support and effectively reduces material waste.
[0028] In summary, after fully considering the characteristics of the topology optimization results and the limitations of the process, the preliminary design of the topology stiffener configuration was completed.
[0029] Step 3: Using the extended multi-scale finite element method, calculate the multi-scale numerical basis functions of the porous microstructure sandwich layer to achieve rapid modeling of the porous microstructure sandwich layer structure; Step 3 includes the following steps:
[0030] Step 3.1: To complete the topological stiffening—microstructure design of the wing rudder, it is necessary to use the extended multi-scale finite element method to perform computational analysis on the porous microstructure. Based on the geometric envelope dimensions of the wing rudder structure and combined with design objectives (such as displacement, stress, and natural frequencies), a porous microstructure (such as lattice, honeycomb, etc.) filler is selected, and its envelope size is defined. A porous microstructure unit cell is selected as a subdomain of the macroscopic element, and the multi-scale basis functions are determined numerically, as shown in the formula:
[0031]
[0032] In the formula, N is the multi-scale basis function matrix of the coarse mesh element; u is the displacement vector of all fine mesh nodes on the sub-mesh; is the displacement vector of the coarse mesh element node.
[0033] Step 3.2: Using the multi-scale basis functions of the coarse mesh elements, obtain the total strain energy and equivalent stiffness matrix of the coarse mesh elements, as shown in the formula:
[0034]
[0035] Among them, K e Ω is the conventional stiffness matrix of any fine-grid element e within the coarse-grid; e Represents a coarse mesh region; u e B is the nodal displacement vector of element e; e and D e These are the strain-displacement matrix and material property matrix of element e, respectively; t is the thickness of the planar element; is the total number of micro-elements on the submesh; G e Π is the transformation matrix for the fine mesh element e, which represents the nodal displacement mapping relationship between the fine mesh element e and the corresponding coarse element; e K represents the elastic strain energy of element e; E It is the equivalent stiffness matrix of the coarse mesh element.
[0036] In summary, the overall element stiffness matrix of the structure can be derived, as shown in the formula:
[0037]
[0038] Where K is the overall stiffness matrix of the structure. For the matrix assembly operator, M is the total number of coarse mesh elements in the structure, and K E,i The equivalent stiffness matrix K of the coarse mesh element can be obtained. e To obtain. F is the macroscopic displacement vector of the structure. ext This represents the load vector applied to nodes at the macroscopic scale.
[0039] Step 3.3: Implement collaborative computation between the Extended Multiscale Finite Element Method (EPF-FEM) and the Finite Element Method (FEM). Since the coarse mesh required by the EPF-FEM to simplify the calculation of porous microstructures has a different resolution than the full-scale refined mesh used for the remaining parts, it is necessary to connect these two different types of elements to achieve collaborative computation between the EPF-FEM and the FEM. That is, the "master-slave" relationship of displacement is handled at the common nodes of these two types of elements.
[0040] The master-slave constraint equations between the coarse mesh elements and the stiffened structural mesh of the porous microstructure sandwich layer are as follows:
[0041]
[0042] In the above formula: K * Master-slave constraint stiffness matrix, T eThe transition matrix represents the relationship between a general finite element mesh and its adjacent coarse mesh elements.
[0043] Step 4: Perform integrated design of the porous microstructure sandwich layer and topological stiffeners, optimize the stiffening dimensions, microstructure, and dimensional parameters of the wing-rudder structure, achieve lightweight design of the wing-rudder structure, and complete the reconstruction of the 3D CAD model. Step 1 includes the following steps:
[0044] Step 4.1: Based on the structural characteristics of the wing rudder topology stiffening-micro-filling structure, dimensional parameters are used as design variables. Considering the ease of modification of shell elements in parametric modeling, shell element thickness is chosen to simulate dimensional design parameters. In the wing rudder topology stiffening-micro-filling structure, there are three key dimensional parameters: upper and lower skin thicknesses; border width; and topology stiffening width. These three parameters are selected as variables for the integrated design of the porous microstructure sandwich layer and topology stiffeners. In the actual manufacturing scheme of the wing rudder, the skin thickness and border width exhibit a consistent characteristic, while the topology stiffeners can be designed with a gradually changing width. Therefore, in the optimization results, the skin thickness and border width should maintain uniform values, while the stiffening width can present a gradually changing value based on the optimization iteration results. Thus, the design variables are as follows:
[0045] t = (t1, t2, t3) T
[0046]
[0047] In the formula: t n Here, t1 represents the thickness of the shell element, which is a design variable. In practical engineering, t1 represents the skin thickness; t2 represents the width of the border; and t3 is a discrete variable. The set consists of variables representing different shell element thicknesses, where t3 represents the width of stiffeners at different locations.
[0048] Step 4.2: Based on the geometric envelope dimensions of the wing-rudder structure, combined with the geometric characteristics of the force transmission path in the topology optimization process, and considering the actual manufacturing requirements, determine the upper and lower limits and initial values of each design variable. The upper and lower limits of each design variable are used as geometric constraints, as shown below:
[0049] st:a n ≤t n ≤b n
[0050] In the formula: a n and b n This indicates that the upper and lower limits are obtained for the nth design variable.
[0051] Step 4.3: In the collaborative integrated design process, the porous microstructure sandwich layer is treated as a non-design domain. The equivalent stiffness matrix obtained in Step 3 is used to calculate the porous microstructure sandwich layer. The design variables are determined in Step 4.1, with the minimum overall volume of the wing and rudder as the objective function. The maximum displacement meeting the design requirements is set as one of the constraints, and the collaborative integrated design is performed in conjunction with the geometric constraints in Step 4.2.
[0052] In summary, the mathematical model for the integrated design of the porous microstructure sandwich layer and the topological reinforcing ribs is as follows:
[0053]
[0054] In the formula: V is the structural volume; V i The volume of the i-th unit is represented by the design variable t. n The function; KU=F is the finite element equilibrium equation, where K, U and F are the overall stiffness matrix, overall displacement vector and overall load vector of the structure, respectively; This is the maximum displacement limit.
[0055] The above mathematical model is used to perform a collaborative integrated design of the wing and rudder. Based on the results, the three-dimensional model of the wing and rudder topology stiffening-micro-filling optimization scheme is reconstructed and checked. If it meets the design objectives, the design is output; if it does not meet the design objectives, the collaborative integrated design process is repeated.
[0056] The beneficial effects of this invention are:
[0057] Firstly, regarding the topology optimization stage, the traditional method is used to set the "boundary" portion as a new non-design domain, and process constraints are added as geometric constraints, making the force transmission path shown in the optimization results clearer. Secondly, in the feature extraction stage of the force transmission path, the constraints of laser additive manufacturing process and the characteristics such as broken branches in the topology optimization results are fully considered to obtain a preliminary design of the topology stiffener configuration that conforms to engineering practice. Then, to address the problem of large computational scale and low efficiency of porous microstructure sandwich layers, the extended multi-scale finite element method is used to achieve rapid modeling of porous microstructure sandwich layer structures, simplifying the computation process and improving computational efficiency. Finally, to solve the scale separation problem between solid optimization and micro-filling in existing wing and rudder design methods, the porous microstructure sandwich layer is introduced as a non-design domain during the size optimization process, and the porous microstructure sandwich layer and topology stiffener are designed collaboratively and integratedly. This effectively solves the scale separation problem between solid optimization and micro-filling.
[0058] This invention can obtain wing and rudder structure optimization results applicable to engineering practice, making the optimization results, which previously only provided conceptual guidance for wing and rudder structure designers, more closely aligned with engineering practice. By using the extended multi-scale finite element method combined with the integrated design method of porous microstructure sandwich layer and topological stiffener, the invention improves both computational efficiency and effectiveness and practicality, and has significant guiding significance for the wing and rudder design of future supersonic aircraft. Attached Figure Description
[0059] Figure 1 This is a flowchart of the wing and rudder design method based on the extended multi-scale finite element method of this invention;
[0060] Figure 2 This is the initial wing-rudder model of a specific embodiment of the present invention;
[0061] Figure 3 This is the topology optimization result of a specific embodiment of the present invention;
[0062] Figure 4 This is the basic configuration of the topological stiffener in a specific embodiment of the present invention;
[0063] Figure 5 This is the result of the collaborative and integrated design of a specific embodiment of the present invention;
[0064] Figure 6 This is the optimal design result of a specific embodiment of the present invention. Detailed Implementation
[0065] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0066] Its flowchart is as follows Figure 1 As shown.
[0067] The wing geometry used as an illustrative embodiment is as follows: Figure 2 As shown. The wing-rudder's dimensions are: leading edge 500mm, wing-rudder base 600mm, and control surface area approximately 200,000 mm². 2 The loading method involves coupling the upper and lower rudder surfaces to the pressure core, with a load of 10000N. The boundary condition is that the rudder shaft is fixed. The specific steps are as follows:
[0068] Step 1.1: Based on its external dimensions, and using the traditional wing-rudder topology optimization method to define the leading edge, skin, and rudder shaft as non-design domains, add approximately 10mm of solid area to the trailing edge, leading tip, and bottom of the wing-rudder into the non-design domains. Define the rudder core area, excluding the aforementioned areas, as the design domain for topology optimization.
[0069] Step 1.2: Add the stamping manufacturing constraint from the demolding control as one of the geometric constraints in the topology optimization constraints; and add member size constraints based on the overall dimensions of the wing and rudder structure, generally including: maximum member size constraint and minimum member size constraint, which can be used to limit the range of structural size variations in the topology optimization results. The maximum member size is 50mm, and the minimum member size is 10mm. The specific mathematical model of the geometric constraints is as follows:
[0070]
[0071] In the formula: ρ i is the pseudo-density of the i-th element in the draft direction; n is the number of elements; L is the equivalent characteristic size of the discrete element void; l d It is the average depth of the projection; S i It is the surface area of boundary element i.
[0072] Step 1.3: Use a topology optimization method based on the Variable Density Method (SIMP):
[0073] The basic idea of the variable density method is to artificially introduce a material with a relative density that varies between 0 and 1, and then assume that there is a certain correspondence between the relative density of the material and its physical properties (such as Young's modulus), expressing this correspondence through an explicit function of the relative density. As the most commonly used material interpolation model in the variable density method, the relative density x of the element material is defined. i With Young's elastic modulus E i The relational expression is:
[0074]
[0075] In the formula: E0 is the Young's modulus of the solid material (i.e., the material with a relative density of 1); E i (x) represents the interpolated elastic modulus; p is the penalty factor (usually 3), which avoids a large number of intermediate density elements in the optimization result, thus obtaining a clear material layout. Since the finite element elements remain constant during the topology optimization process, the element stiffness matrix k... i The change is only related to Young's modulus E i Regarding this, combining the finite element stiffness matrix calculation formula and the above equation, we can derive:
[0076]
[0077] In the diagram: k0 is the element stiffness matrix of the solid material.
[0078] If the design region Ω is divided into n finite element elements, then the design variable vector is X = [x1x2…x...]. n ] T The volume and physical properties of the structure can be expressed as a function of X.
[0079] Based on the above, taking the maximum overall stiffness of the wing-rudder structure as the objective function, and the volume fraction after constraint optimization as 20% of the initial volume, the mathematical model for topology optimization of the continuum structure under volume and geometric constraints is as follows:
[0080]
[0081] In the formula: v i V represents the volume of the i-th unit; V is the structural volume. The upper limit of the constraint volume is defined; KU = F is the finite element equilibrium equation, where K, U, and F are the overall stiffness matrix, overall displacement vector, and overall load vector of the structure, respectively. The obtained topology optimization results are as follows: Figure 3 As shown.
[0082] Step 2.1: When extracting features from the force transmission path, the following criteria are used to address the isolated body features and broken branch features that may appear in the topology optimization results: Remove some isolated body features. Since the intermediate density unit of the variable density method may cause isolated body features in the optimization results, it will not only fail to improve the overall structural stiffness, but also increase the structural weight, which does not meet the lightweight design criteria; Connect the broken branch feature materials. In the wing and rudder topology optimization results, the structure is often not continuous and integrated, and the broken branch is an important representation of its optimal force transmission path. Therefore, it is necessary to extend it to the frame to improve the overall stiffness and maintain the aerodynamic shape.
[0083] Step 2.2: Furthermore, the forming angle of the topological stiffener structure is constrained by the SLM process. The optimal forming angle of the stiffener structure should be 45° to the overall forming direction to achieve self-support and effectively reduce material waste. Regarding... Figure 3 The topology optimization results shown extend the broken branches and remove some isolated features, ultimately completing the preliminary design of the topological stiffener configuration, as follows: Figure 4 As shown.
[0084] Step 3.1: Based on the fact that the wing rudder's thickest point is approximately 40mm and its narrowest point is approximately 5mm, the most widely used BCC lattice is selected as the filling lattice configuration. The lattice cell size is 20mm, and the rod diameter is 1mm. A lattice unit cell is selected as a subdomain of the macroscopic unit, and the multi-scale basis functions are determined using numerical methods, as shown in the formula:
[0085]
[0086] In the formula, N is the multi-scale basis function matrix of the coarse mesh element; u is the displacement vector of all fine mesh nodes on the sub-mesh; is the displacement vector of the coarse mesh element node.
[0087] Step 3.2: Using the multi-scale basis functions of the coarse mesh element, obtain the equivalent stiffness matrix and total strain energy of the coarse mesh element, as shown in the formula:
[0088]
[0089] Among them, K e Ω is the conventional stiffness matrix of any fine-grid element e within the coarse-grid; e Represents a coarse mesh region; u e B is the nodal displacement vector of element e; e and D e These are the strain-displacement matrix and material property matrix of element e, respectively; t is the thickness of the planar element; is the total number of micro-elements on the submesh; G e Π is the transformation matrix for the fine mesh element e, which represents the nodal displacement mapping relationship between the fine mesh element e and the corresponding coarse element; e K represents the elastic strain energy of element e; E It is the equivalent stiffness matrix of the coarse mesh element.
[0090] In summary, the overall element stiffness matrix of the structure can be derived, as shown in the formula:
[0091]
[0092] In the formula, K is the overall stiffness matrix of the structure. For the matrix assembly operator, M is the total number of coarse mesh elements in the structure, and K e,i The equivalent stiffness matrix K of the coarse mesh element can be obtained. e To obtain. F is the macroscopic displacement vector of the structure. ext This represents the load vector applied to nodes at the macroscopic scale.
[0093] Step 3.3: Implement co-calculation between the Extended Multiscale Finite Element Method (EPF-FEM) and the Finite Element Method (FEM). Since the coarse mesh required by the EPF-FEM to simplify the calculation of porous microstructures has a different resolution than the full-scale refined mesh used for the remaining parts, it is necessary to connect these two different types of elements to achieve co-calculation between the EPF-FEM and the FEM. That is, the "master-slave" relationship of displacement is handled at the common nodes of these two types of elements.
[0094] The master-slave constraint equations between the coarse mesh elements and the stiffened structural mesh of the porous microstructure sandwich layer are as follows:
[0095]
[0096] In the formula: Master-slave constraint stiffness matrix, T e The transition matrix represents the relationship between a general finite element mesh and its adjacent coarse mesh elements.
[0097] Step 4.1: Based on the structural characteristics of the wing rudder topology stiffening-micro-filling structure, dimensional parameters are used as design variables. Considering the ease of modification of shell elements in parametric modeling, shell element thickness is chosen to simulate dimensional design parameters. In the wing rudder topology stiffening-micro-filling structure, there are three key dimensional parameters: upper and lower skin thicknesses; border width; and topology stiffening width. These three parameters are selected as variables for the integrated design of the lattice sandwich layer and topology stiffeners. In the actual manufacturing scheme of the wing rudder, the skin thickness and border width exhibit a consistent characteristic, while the topology stiffeners can be designed with a gradually changing width. Therefore, in the optimization results, the skin thickness and border width should maintain uniform values, while the stiffening width can present a gradually changing value based on the optimization iteration results. Thus, the design variables are as follows:
[0098] t = (t1, t2, t3) T
[0099]
[0100] In the formula: t n Here, t1 represents the thickness of the shell element, which is a design variable. In practical engineering, t1 represents the skin thickness; t2 represents the width of the border; and t3 is a discrete variable. The set consists of variables representing different shell element thicknesses, where t3 represents the width of stiffeners at different locations.
[0101] Step 4.2: Based on the geometric dimensions of the wing and rudder structure, combined with the geometric characteristics of the force transmission path in the topology optimization process, and considering the actual manufacturing requirements, determine the upper and lower limits and initial values of each design variable. The skin thickness is set to 0.7–1.2 mm, the frame width to 1–5 mm, and the stiffener width to 2–10 mm. The upper and lower limits of each design variable are used as geometric constraints, as shown below:
[0102]
[0103] Step 4.3: In the collaborative integrated design process, the porous microstructure sandwich layer is treated as a non-design domain. The equivalent stiffness matrix obtained in Step 2 is used to calculate the porous microstructure sandwich layer. Design variables are determined in Step 4.1. The objective function is to minimize the overall volume of the wing and rudder, with the maximum displacement meeting the design requirement of 12mm as a constraint. The collaborative integrated design is then performed in conjunction with the geometric constraints from Step 4.2.
[0104] In summary, the mathematical model for the integrated design of the lattice sandwich layer and the topological stiffener is as follows:
[0105]
[0106] In the formula: V is the structural volume; V i The volume of the i-th unit is represented by the design variable t. n The function; KU=F is the finite element equilibrium equation, where K, U and F are the overall stiffness matrix, overall displacement vector and overall load vector of the structure, respectively; Maximum displacement limit. The results of the collaborative integration are as follows: Figure 5 As shown.
[0107] according to Figure 5 The results show that the actual parameter values of each design variable were determined, namely, the width of the wing rudder bottom and leading edge edge is 3mm, the trailing edge width is 3mm, the skin thickness is 0.9mm, and the stiffening width is designed in a gradient form, with the widest point at the bottom being 8mm and the narrowest point at the top being 2mm. Based on the above design variable values, the wing rudder topology stiffening-micro-filling structure design was completed, as follows: Figure 6 As shown. After verification, it meets the design objectives.
[0108] The above-described embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.
Claims
1. A method for optimizing the design of wing and rudder structures of high-speed aircraft based on the extended multi-scale finite element method, characterized in that, Includes the following steps: Step 1: Use the additive manufacturing structure topology optimization method to complete the structure topology optimization calculation based on multiple load conditions and boundary conditions; Step 2: Extract features from the force transmission path to complete the preliminary design of the topological stiffener configuration; Step 3: Use the extended multi-scale finite element method to calculate the multi-scale numerical basis functions of the porous microstructure sandwich layer, so as to realize the rapid modeling of the porous microstructure sandwich layer structure. Step 4: Conduct a collaborative integrated design of the porous microstructure sandwich layer and the topological stiffeners, optimize the stiffener dimensions, microstructure and dimensional parameters of the wing rudder structure, realize the lightweight design of the wing rudder structure and complete the reconstruction of the three-dimensional CAD model; Step 4 includes the following steps: Step 4.1: Select three parameters—top and bottom skin thickness, border width, and topological stiffener width—as variables for the integrated design of the porous microstructure sandwich layer and the topological stiffener. The design variables are as follows: In the formula: This is a design variable, representing the thickness of the shell element; in practical engineering... Represents skin thickness; Represents the width of the border; For discrete variables The set consists of variables representing different shell element thicknesses, i.e. Represents the width of reinforcement at different locations; Step 4.2: Based on the geometric envelope dimensions of the wing-rudder structure, combined with the geometric characteristics of the force transmission path in the topology optimization process, and considering the actual manufacturing requirements, determine the upper and lower limits and initial values of each design variable; use the upper and lower limits of each design variable as geometric constraints, specifically: In the formula: This indicates that the upper and lower limits are obtained for the nth design variable; Step 4.3: In the collaborative integrated design process, the porous microstructure sandwich layer is regarded as the non-design domain of the collaborative integrated design. In the calculation process, the equivalent stiffness matrix obtained in Step 3 is used to calculate the porous microstructure sandwich layer. The design variables are determined by Step 4.1, with the minimum overall volume of the wing and rudder as the objective function. The maximum displacement that meets the design requirements is set as one of the constraints, and the collaborative integrated design is carried out in combination with the geometric constraints in Step 4.
2. In summary, the model for the integrated design of porous microstructure sandwich layer and topological stiffener is as follows: In the formula: For structural volume; Indicates the first The volume of each unit is a design variable. The function; The finite element equilibrium equations are... and These are the overall structural stiffness matrix, overall displacement vector, and overall load vector, respectively. Maximum displacement limit; The above model is used to perform a collaborative integrated design of the wing and rudder. Based on the results, the three-dimensional model of the wing and rudder topology stiffening-micro-filling optimization scheme is reconstructed and verified. If it meets the design objectives, the design is output; if it does not meet the design objectives, the collaborative integrated design process is repeated.
2. The method for optimizing the design of wing and rudder structures of high-speed aircraft based on the extended multi-scale finite element method according to claim 1, characterized in that, Step 1 includes the following steps: Step 1.1: Based on the traditional wing rudder topology optimization method, which sets the skin and rudder shaft as non-design domains, a certain distance of solid area is reserved at the leading edge, trailing edge, leading tip, and bottom of the wing rudder and added to the non-design domain of topology optimization. The rudder core area other than the above areas is set as the design domain of topology optimization. Step 1.2: Add demolding control as a geometric constraint to the topology optimization constraints; and add dimensional constraints to the topology optimization based on the overall dimensions of the wing and rudder structure to limit the range of dimensional changes in the structure in the topology optimization results. Step 1.3: Topology optimization method based on SIMP (Simplified Method for Variable Density): Combining the finite element stiffness matrix calculation formula and the relative density of the element material With Young's modulus The relational expression yields: In the formula: The element stiffness matrix of the solid material; As a penalty factor, its function is to avoid the existence of a large number of intermediate density units in the optimization results and to obtain a clear material layout; Design area Divided into There are finite element elements, and the design variable vector is... The volume and physical properties of the structure are expressed as The function; Based on the above, taking the maximum overall stiffness of the wing-rudder structure as the objective function, the mathematical model for topology optimization based on metal additive manufacturing under volume and geometric constraints is as follows: In the formula: It is the first The pseudo density of each element in the draft direction; It is the number of finite element elements in the design domain; It is the equivalent characteristic size of the void in the discrete unit; It is the largest member size; It is the smallest member size; Indicates the first The volume of each unit; For structural volume; To constrain the upper limit of volume; The finite element equilibrium equations are... and These are the overall structural stiffness matrix, overall displacement vector, and overall load vector, respectively. It is the average depth of the projection; It is a boundary unit Surface area; Based on the mathematical model of SIMP topology optimization based on the variable density method, the wing rudder is optimized using metal additive manufacturing to obtain the topology optimization results.
3. The method for optimizing the design of wing and rudder structures of high-speed aircraft based on the extended multi-scale finite element method according to claim 2, characterized in that, In step 1.1, a certain distance refers to a solid area of 5-10mm.
4. The method for optimizing the design of wing and rudder structures of high-speed aircraft based on the extended multi-scale finite element method according to claim 2, characterized in that, In step 1.2, the dimensional constraints include maximum member dimensional constraints and minimum member dimensional constraints. The mathematical model of the geometric constraints is as follows: 。 5. The method for optimizing the design of wing and rudder structures of high-speed aircraft based on the extended multi-scale finite element method according to claim 1, characterized in that, Step 2 includes the following steps: Step 2.1: For the topology optimization results of Step 1, there may be isolated body features and broken branch features; the isolated body feature is: due to the intermediate density unit of the variable density method, isolated needles may appear in the optimization results; the broken branch feature is: the structure in the optimization results is often not continuous and integral; when extracting features of the force transmission path, the following criteria are used for the above features: remove some isolated body features, connect the broken branch feature material, and extend it to the border. Step 2.2: To obtain a topological stiffener design that conforms to engineering practice, it is also necessary to consider the constraints of laser additive manufacturing process; for the constraint of closed cavity, powder outlet holes are applied on the topological stiffener and the wing / rudder frame to minimize the performance impact caused by the closed cavity constraint; for the constraint of overhang, in the preliminary design stage of the topological stiffener, it is ensured that the angle between all stiffening structures and the forming direction is greater than 45°. In summary, after fully considering the characteristics of the topology optimization results and the limitations of the process, the preliminary design of the topology stiffener configuration was completed.
6. The method for optimizing the design of wing and rudder structures of high-speed aircraft based on the extended multi-scale finite element method according to claim 1, characterized in that, Step 3 includes the following steps: Step 3.1: Based on the design objectives and wing / rudder shape, select a porous microstructure unit cell model as the sub-mesh of the macroscopic unit, and determine the multi-scale basis functions using numerical methods, as shown in the formula: In the formula, The multi-scale basis function matrix of the coarse grid cells; This represents the displacement vector of all fine mesh nodes on the submesh; is the displacement vector of the nodes of the coarse mesh element; Step 3.2: Using the multi-scale basis functions of the coarse mesh elements, obtain the total strain energy and equivalent stiffness matrix of the coarse mesh elements, as shown in the formula: in It is the conventional stiffness matrix of any fine mesh element e within the coarse mesh; Represents a coarse grid region; It is the nodal displacement vector of the fine mesh element e; These are the strain-displacement matrix and material property matrix of the fine mesh element e, respectively, and t is the thickness of the planar element; This represents the total number of micro-units on the subgrid; Let be the transformation matrix of the fine mesh element e, which represents the nodal displacement mapping relationship between the fine mesh element e and the corresponding coarse element; Let e be the elastic strain energy of the fine mesh element e; It is the equivalent stiffness matrix of the coarse mesh element; In summary, the overall element stiffness matrix of the structure is derived, as shown in the formula: in, K Here is the overall stiffness matrix of the structure. Assemble operators for matrices. M The total number of coarse grid cells in the structure. Equivalent stiffness matrix of coarse mesh elements Seek; Let be the macroscopic displacement vector of the structure. This represents the load vector applied to nodes at the macroscopic scale. Step 3.3: Establish the master-slave constraint equations between the coarse mesh elements and the stiffened structural mesh of the porous microstructure sandwich layer to achieve collaborative calculation between the extended multi-scale finite element method and the finite element method; the master-slave constraint equations between the coarse mesh elements and the stiffened structural mesh of the porous microstructure sandwich layer are as follows: In the above formula: Master-slave constraint stiffness matrix The transition matrix represents the relationship between a general finite element mesh and its adjacent coarse mesh elements.