A thin-walled component robustness optimization method considering size and material nondeterminism

Abstract

The application discloses a thin-walled component robustness optimization method considering size and material non-determinacy, relates to the technical field of computer-aided engineering and structural optimization design, and comprises the following steps: a structure finite element model is established, and the structure is discretized into four-node quadrilateral elements; design variables are defined as the thickness and elastic modulus of each element, and the fluctuation range of the uncertainty of geometric size and material properties is set; a double-layer optimization model is constructed, the outer layer takes the structure weight and flexibility as the objective function, and the SQP algorithm is used for optimization; the inner layer carries out finite element analysis, calculates the element stiffness matrix and sensitivity, and determines the worst uncertainty parameter combination according to the sign of the sensitivity; the structure stiffness matrix and the load vector are updated in real time in the iteration process, and the convergence of the algorithm is verified; the application adopts a double-layer iteration strategy, simultaneously considers the geometric size deviation and the basic material attribute dispersion problem commonly existing in engineering components, and improves the reliability of the structure in the uncertain environment.

Description

Technical Field

[0001] This invention relates to the fields of computer-aided engineering (CAE) and structural optimization design, specifically to a robustness optimization method for thin-walled components that takes into account the nondeterminism of dimensions and materials. Technical Background

[0002] In high-end manufacturing fields such as aerospace and mechanical equipment, lightweight and high-performance structural design is crucial. Traditional structural topology or dimensional optimization design is usually based on deterministic assumptions, which assume that the geometric dimensions (such as thickness) and material properties (such as elastic modulus) of the structure remain precisely unchanged during manufacturing and service. However, in actual engineering, due to machining tolerances, non-uniformity of material preparation processes, and changes in the service environment, structural parameters often exhibit unavoidable random fluctuations or range uncertainties. Ignoring these uncertainties may lead to a sharp decline in the performance of the optimized structure under actual operating conditions, or even failure. For example, a structure optimized based on nominal parameters may be very sensitive to thickness variations; once the local thickness becomes thinner due to manufacturing errors, its stiffness will fail to meet design requirements. Currently, there is considerable research on single-type uncertainties, but efficient optimization design methods that simultaneously consider the mixed uncertainties of geometric dimensions (such as unit thickness) and physical properties (such as elastic modulus) are still relatively scarce.

[0003] Existing solutions have significant limitations. While the traditional safety factor method is simple, it often results in overly bulky structures, failing to meet the ultra-lightweight requirements of fields like aerospace. Reliability optimization design based on probabilistic models (RBDO), although providing quantified reliability, typically requires precise knowledge of the parameter probability distributions. However, in practical engineering, due to scarce sample data, obtaining accurate distribution functions is difficult. Furthermore, RBDO usually relies on Monte Carlo simulations, with computational costs increasing exponentially with the number of uncertain parameters. On the other hand, non-probabilistic interval optimization based on convex models, while not requiring precise probability distributions, is inefficient when dealing with mixed uncertainties, particularly in finding the worst-case scenario in the inner layer of traditional nested algorithms. Relying solely on swarm intelligence algorithms like genetic algorithms requires re-invoking the finite element method for each iteration, leading to enormous computational overhead. Therefore, establishing an optimization method capable of simultaneously handling mixed geometric and material uncertainties and efficiently finding the worst-case scenario to ensure design robustness has significant engineering application value. Existing structural optimization methods often treat the thickness of thin-plate parts as an ideal uniform value. However, for thin plates produced using additive manufacturing or precision casting processes, edge effects and uneven cooling can lead to a heterogeneous distribution of the actual thickness across the entire field. Furthermore, the elastic modulus of high-performance composite materials is subject to random influences during processing, exhibiting typical range-specific uncertainties. Traditional single-variable uncertainty models cannot describe the coupled failure mechanism of geometric deviations and material discretization, making it difficult to control the failure rate of design results under extreme mixed conditions. Summary of the Invention

[0004] The solution to the above-mentioned technical problems is: a robust optimization method for thin-walled components that takes into account the uncertainty of size and material. Through a two-layer optimization strategy, design variables are optimized in the outer layer and worst-case analysis of uncertain parameters is performed in the inner layer, thereby achieving robust design of the structure.

[0005] The solution provided by this invention to the above-mentioned technical problems is: a method for optimizing the robustness of thin-walled components while taking into account dimensional and material indeterminism, comprising the following steps: S1. Establish the initial geometric model of the structure, mesh it, and obtain a finite element discrete model containing node coordinates and element connection relationships; S2. Define the optimization design variables and uncertainty parameters, set the thickness and elastic modulus of each element as design variables, and set their corresponding geometric uncertainty intervals and material uncertainty intervals; S3. Construct a two-layer optimization mathematical model. The outer layer optimization is used to minimize the structural weighted objective function, and the inner layer optimization is used to solve the worst structural response under the disturbance of uncertain parameters. Based on the current design variables and initial material and geometric parameters, assemble the stiffness matrices of each element to form the overall stiffness matrix, and establish the structural equilibrium equation in combination with external loads, providing a basis for the inner layer uncertainty analysis and structural response calculation. S4. In the inner loop, the nodal displacements are obtained through finite element calculation, and the sensitivity of the objective function to the uncertain parameters is calculated by the direct differential method based on the differential of the equilibrium equation. The worst parameter combination is determined according to the sign of the sensitivity. S5. Update the overall stiffness matrix and load vector using the updated worst-case parameter combination, solve the structural response, and return to the outer optimizer; S6. Determine whether the outer layer optimization has converged; if the change in the outer layer design variables or the change in the objective function meets the preset convergence threshold, output the optimal thickness distribution and elastic modulus contour plot; otherwise, return to S4, re-execute the worst-case search of the inner layer under the updated outer layer design variables, and repeat S4-S6 until the outer layer optimization converges.

[0006] Specifically, step S1 is implemented as follows: S11. Establish the structural macroscopic geometric parameters and node coordinate system; Define the length and height parameters of the structural physical design domain, and set the number of nodes (noNDX and noNDY) in the horizontal and vertical directions. Generate the global coordinate matrix (gcoord) for all nodes using bilinear interpolation or a regular mesh generation algorithm.

[0007] S12. Mesh discretization and initial geometric property calculation; The solution domain is discretized using four-node quadrilateral elements, and the element node connection matrix is ​​established.

[0008] Specifically, using bilinear functions Constructing a physical coordinate system With natural coordinate system The mapping relationship between them, where natural coordinates The shape function expressions for the four nodes are as follows:

[0009] For each unit, utilize Gaussian numerical integration method calculates the Jacobian determinant using shape functions and their derivatives. Then, by integration, the initial geometric area of ​​each unit can be obtained. :

[0010] Based on this, the mechanical properties of the element are further constructed. According to the geometric equations of elasticity, the strain vector within the element... With nodal displacement vector The relationship between strain and displacement is expressed through the strain-displacement matrix. Establish, that is .matrix It consists of the partial derivatives of the shape functions with respect to the global coordinates, and requires the inverse of the Jacobian matrix. Coordinate transformation calculations are performed. For plane stress problems, the material constitutive matrix is ​​introduced. :

[0011] in For Poisson's ratio. Finally, the element stiffness matrix... The explicit integral form is defined as:

[0012] Using the above formula, the uncertainty parameter (Actual thickness) and The (real modulus) is directly coupled into the construction process of the stiffness matrix, providing a computational basis for subsequent uncertainty response analysis and worst-case search.

[0013] Specifically, step S2 is implemented as follows: S21. Define the optimization design variable vector and boundary constraints; The thickness of each element in the finite element model With elastic modulus As independent design variables, construct a global design variable vector. Set the constraint range for the thickness variable. Constraint range of elastic modulus variable ; S22. Set the mixed uncertainty parameters and their fluctuation range; Define the uncertainty fluctuation range of geometric thickness (For example Uncertainty fluctuation range of material elastic modulus Introducing uncertain state variables and This describes the actual parameter deviation of each unit in the current design state:

[0014] Specifically, step S3 is implemented as follows:

[0015] S31. Construct the outer layer optimization model; Constructing a generalized material cost objective function Its physical meaning is the generalized structural weight that takes into account material properties. The calculation formula is as follows:

[0016] in, For material density, For unit area, Design variable (physical thickness) For design variables (physical elastic modulus).

[0017] S32. Define constraints; Define the performance constraints of the structure under worst-case conditions. This mainly includes structural flexibility constraints (Compliance Constraints):

[0018] in It is the structural compliance under the worst-case combination of uncertainties calculated by the inner optimization loop. It is the maximum allowable flexibility value in the design.

[0019] Specifically, step S4 is implemented as follows:

[0020] S41. Assembly of finite element stiffness matrix under uncertain parameters; In the inner optimization loop, based on the current uncertainty coefficient and Update element properties and calculate the stiffness matrix of each element using numerical integration. And assemble the global stiffness matrix :

[0021] in The strain-displacement matrix is... The constitutive matrix of the material; S42. Sensitivity analysis and worst-case direction determination; Solve the equilibrium equations The nodal displacement vectors are obtained. Then, the structural equilibrium equations are differentiated with respect to the uncertainty parameters. Combining this with the functional relationship between the objective function and the nodal displacements, the sensitivity of the objective function to the uncertainty parameters is obtained. This sensitivity is calculated using the derivative of the element stiffness matrix with respect to the uncertainty parameters and the nodal displacement vectors, without requiring separate global finite element difference calculations for each uncertainty parameter.

[0022] Calculate the sensitivity of the objective function to the uncertainty parameter. :

[0023] in, The sensitivity of the thickness uncertainty parameter. The sensitivity of the modulus uncertainty parameter; Based on sensitivity The sign of the variable directly determines the worst-case uncertainty parameter. Value boundaries:

[0024] Specifically, the implementation method of step S5 is as follows: S51. System matrix reconstruction under worst-case conditions; When the inner optimization loop meets the convergence condition (i.e., it finds the uncertainty parameter that minimizes the structural performance), and After that, the code extracts the current value. and Using the Gaussian numerical integration method, the stiffness matrix of each element is recalculated based on the updated properties. And assembled to form a global stiffness matrix reflecting the worst-case working condition. This step ensures that the outer optimizer optimizes based on the weakest state of the structure, rather than the nominal state.

[0025] S52. Solving for the worst structural response; Apply displacement boundary conditions to the reconstructed system matrix and solve the linear equations to obtain the nodal displacement vectors under the worst-case condition. Subsequently, the objective function value required for outer layer optimization is calculated based on this displacement vector. (such as structural stiffness or weight) and constraint function values .

[0026] S53. Sensitivity analysis for outer layer optimization; To drive the outer SQP algorithm (fmincon) to perform gradient descent optimization, the objective function must be calculated with respect to the design variable (nominal thickness). and nominal modulus The derivative of the equilibrium equation. This invention is based on an analytical differentiation strategy of the equilibrium equation differential, which eliminates the need to solve other partial derivative equations. Instead, it directly utilizes the nodal displacement vectors obtained under the worst-case condition. Based on the expression of the objective function, the sensitivity of the objective function to the thickness variable of each element is calculated. and sensitivity to changes in elastic modulus The calculation formula is as follows:

[0027] Finally, the calculated function values ​​and gradient vectors (GradObj and GradConstr) are passed back to the outer optimizer for the next iteration update of the design variables.

[0028] Specifically, the outer optimizer uses the Sequential Quadratic Programming (SQP) algorithm for iterative optimization. SQP is one of the most effective gradient-based algorithms for solving this type of nonlinearly constrained optimization problem. In each iteration... First, based on the weighted objective function f(x) and the constraints G(x) (including worst-case compliance constraints and upper and lower bounds of variables), a Lagrange function is constructed. ,in These are Lagrange multipliers. Subsequently, the algorithm operates at the current point. The Lagrangian function is expanded using a second-order Taylor series, and the nonlinear constraints are linearized to a first-order linearization, constructing a quadratic programming (QP) subproblem to find the search direction for the design variables. The mathematical form of this subproblem is as follows:

[0029] in, and These are the first-order gradients of the objective function and the constraint function (such as robustness and flexibility constraints) at the current point, respectively. These gradient values ​​are precisely calculated using the direct differentiation method described in steps S4 and S53 above. For the positive definite approximation of the Hessian matrix of the Lagrange function, the BFGS formula is used to update the matrix based on the information from the previous iteration:

[0030] Solving this QP subproblem yields the optimal search direction. Then, the step size is determined by combining the line search strategy. Update design variables This allows for efficient approximation of the optimal solution that satisfies all uncertainties and performance constraints.

[0031] Specifically, the implementation method of step S6 is as follows: S61. Outer layer optimization convergence judgment; Determine the norm of the change in design variables between two consecutive iterations Is it less than the preset tolerance? S62. Post-processing for result visualization; If convergence is achieved, calculate the black and white thresholds for the optimal thickness distribution. The topological configuration diagram of the element thickness distribution is plotted; simultaneously, using pseudo-color mapping technology, color cloud maps of the element elastic modulus distribution and thickness-elastic modulus element co-distribution cloud maps are plotted. Furthermore, based on the optimized thickness distribution vector t and the threshold... Determine the configuration of this region. If this region... The area is determined to be a non-load-bearing void and rendered with a white background; if the unit thickness is... The region is determined to be an effective structural configuration, and the optimized elastic modulus value corresponding to the unit is extracted. The color is mapped to the corresponding color according to the preset color mapping table and then filled.

[0032] The beneficial effects of this invention are as follows: This invention realizes an efficient robustness design algorithm for structures based on a two-layer iterative optimization strategy. First, based on finite element theory, it proposes a computational method for robustness optimization design of thin-walled components that takes into account both manufacturing dimensions and material uncertainties. It connects macroscopic geometric dimensions (thickness) and basic material properties (elastic modulus) through an outer-layer design optimization and an inner-layer worst-case analysis. Furthermore, by combining the Sequential Quadratic Programming (SQP) algorithm and sensitivity analysis based on direct differentiation, it directly locks the worst-case boundary of uncertain parameters by using the positive and negative signs of sensitivity. It provides an efficient robustness solution strategy by updating design variables through outer-layer optimization and updating the overall stiffness matrix in real time through an inner-layer optimization loop. The outer loop is applied to the gradient optimization of the weighted objective function of the structure, and the inner loop is applied to the prediction of the worst-case response of the structure caused by uncertain parameters. It also employs hybrid parametric modeling to address the common geometric manufacturing tolerances (thickness variation) and material property discrepancies (modulus fluctuation) in engineering components. A robust optimization design method based on sensitivity-driven approach is proposed using a two-layer nested architecture. It uses the idea of ​​"worst-case approximation" to solve the problem that traditional deterministic optimization results are prone to failure under parameter fluctuations. This enables high-reliability structural design that considers geometric-material mixed uncertainties. The optimization results are visually displayed through thickness black-and-white topology diagrams, modulus color cloud diagrams, and collaborative distribution cloud diagrams, which accelerates product design iteration. Attached Figure Description

[0033] Figure 1 This is a flowchart of the structural robustness bilayer optimization design method for the mixed uncertainty of geometry and materials of the present invention; Figure 2 This is a schematic diagram of the initial finite element mesh generation of the structure; Figure 3 This is a schematic diagram of the optimized unit thickness distribution (black and white topology diagram); Figure 4 This is a schematic diagram of the optimized unit elastic modulus distribution (color cloud map); Figure 5This is a schematic diagram of the optimized thickness-elastic modulus unit cooperative distribution (color cloud map); Figure 6 It is a convergence history curve of the objective function during the optimization process. Detailed Implementation

[0034] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the 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.

[0035] like Figure 1 As shown, a robustness optimization method for thin-walled components that takes into account both dimensional and material indeterminism includes the following steps: S1. Establish the initial geometric model of the structure, mesh it, and obtain a finite element discrete model containing node coordinates and element connection relationships; the specific implementation method of step S1 is as follows: S11. Establish the structural macroscopic geometric parameters and node coordinate system; Define the length and height parameters of the structural physical design domain, and set the number of nodes in the horizontal and vertical directions, noNDX and noNDY, respectively. Use bilinear interpolation or a regular mesh generation algorithm to generate the global coordinate matrix gcoord for all nodes. In a preferred embodiment of the invention, the design domain length is set to 40 units and the height to 20 units, and it is discretized into... There are a total of 1200 four-node elements. The initial value of the thickness design variable is set to 0.01, and the initial value of the elastic modulus is normalized to 2.0 (corresponding to the code). The range of uncertainty fluctuations is set as follows: .

[0036] S12. Mesh discretization and initial geometric property calculation; like Figure 2 As shown, the solution domain is discretized using four-node quadrilateral elements, and the element node connection matrix is ​​established.

[0037] Specifically, using bilinear functions Constructing a physical coordinate system With natural coordinate system The mapping relationship between them, where natural coordinates The shape function expressions for the four nodes are as follows:

[0038] For each unit, utilize Gaussian numerical integration method calculates the Jacobian determinant using shape functions and their derivatives. Then, by integration, the initial geometric area of ​​each unit can be obtained. :

[0039] Based on this, the mechanical properties of the element are further constructed. According to the geometric equations of elasticity, the strain vector within the element... With nodal displacement vector The relationship between strain and displacement is expressed through the strain-displacement matrix. Establish, that is .matrix It consists of the partial derivatives of the shape functions with respect to the global coordinates, and requires the inverse of the Jacobian matrix. Coordinate transformation calculations are performed. For plane stress problems, the material constitutive matrix is ​​introduced. :

[0040] in For Poisson's ratio. Finally, the element stiffness matrix... The explicit integral form is defined as:

[0041] Using the above formula, the uncertainty parameter (Actual thickness) and The (real modulus) is directly coupled into the construction process of the stiffness matrix, providing a computational basis for subsequent uncertainty response analysis and worst-case search.

[0042] S2. Define the optimization design variables and uncertainty parameters, set the thickness and elastic modulus of each element as design variables, and set their corresponding geometric uncertainty intervals and material uncertainty intervals; S21. Define the optimization design variable vector and boundary constraints; The thickness of each element in the finite element model With elastic modulus As independent design variables, construct a global design variable vector. Set the constraint range for the thickness variable. Constraint range of elastic modulus variable ; S22. Set the mixed uncertainty parameters and their fluctuation range; Define the uncertainty fluctuation range of geometric thickness (For example Uncertainty fluctuation range of material elastic modulus Introducing uncertain state variables and This describes the actual parameter deviation of each unit in the current design state:

[0043] S3. Construct a two-layer optimization mathematical model. The outer layer optimization is used to minimize the structural weighted objective function, and the inner layer optimization is used to solve the worst structural response under the disturbance of uncertain parameters. Based on the current design variables and initial material and geometric parameters, assemble the stiffness matrices of each element to form the overall stiffness matrix, and establish the structural equilibrium equation in combination with external loads, providing a basis for the inner layer uncertainty analysis and structural response calculation. S31. Construct the outer layer optimization model; Constructing a generalized material cost objective function Its physical meaning is the generalized structural weight that takes into account material properties. The calculation formula is as follows:

[0044] in, For material density, For unit area, Design variable (physical thickness) For design variables (physical elastic modulus).

[0045] S32. Define constraints; Define the performance constraints of the structure under worst-case conditions. This mainly includes structural flexibility constraints (Compliance Constraints):

[0046] in It is the structural compliance under the worst-case combination of uncertainties calculated by the inner optimization loop. It is the maximum allowable flexibility value in the design.

[0047] S4. In the inner optimization loop, the nodal displacements are obtained through finite element calculation, and the sensitivity of the objective function to the uncertain parameters is calculated by the direct differential method based on the differential of the equilibrium equation. The worst parameter combination is determined according to the sign of the sensitivity. S41. Assembly of finite element stiffness matrix under uncertain parameters; In the inner loop, based on the current uncertainty coefficient and Update element properties and calculate the stiffness matrix of each element using numerical integration. And assemble the global stiffness matrix :

[0048] in The strain-displacement matrix is... The constitutive matrix of the material; S42. Sensitivity analysis and worst-case direction determination; Solve the equilibrium equations The nodal displacement vectors are obtained. Then, the structural equilibrium equations are differentiated with respect to the uncertainty parameters. Combining this with the functional relationship between the objective function and the nodal displacements, the sensitivity of the objective function to the uncertainty parameters is obtained. This sensitivity is calculated using the derivative of the element stiffness matrix with respect to the uncertainty parameters and the nodal displacement vectors, without requiring separate global finite element difference calculations for each uncertainty parameter.

[0049] Calculate the sensitivity of the objective function to the uncertainty parameter. :

[0050] in, The sensitivity of the thickness uncertainty parameter. The sensitivity of the modulus uncertainty parameter; Based on sensitivity The sign of the variable directly determines the worst-case uncertainty parameter. Value boundaries:

[0051] S5. Update the overall stiffness matrix and load vector using the updated worst-case parameter combination, solve the structural response, and return to the outer optimizer; S51. System matrix reconstruction under worst-case conditions; When the inner optimization loop meets the convergence condition (i.e., it finds the uncertainty parameter that minimizes the structural performance), and After that, the code extracts the current value. and Using the Gaussian numerical integration method, the stiffness matrix of each element is recalculated based on the updated properties. And assembled to form a global stiffness matrix reflecting the worst-case working condition. This step ensures that the outer optimizer optimizes based on the weakest state of the structure, rather than the nominal state.

[0052] S52. Solving for the worst structural response; Apply displacement boundary conditions to the reconstructed system matrix and solve the linear equations to obtain the nodal displacement vectors under the worst-case condition. Subsequently, the objective function value required for outer layer optimization is calculated based on this displacement vector. (such as structural stiffness or weight) and constraint function values .

[0053] S53. Sensitivity analysis for outer layer optimization; To drive the outer SQP algorithm (fmincon) to perform gradient descent optimization, the objective function must be calculated with respect to the design variable (nominal thickness). and nominal modulus The derivative of the equilibrium equation. This invention is based on an analytical differentiation strategy of the equilibrium equation differential, which eliminates the need to solve other partial derivative equations. Instead, it directly utilizes the nodal displacement vectors obtained under the worst-case condition. Based on the expression of the objective function, the sensitivity of the objective function to the thickness variable of each element is calculated. and sensitivity to changes in elastic modulus The calculation formula is as follows:

[0054] Finally, the calculated function values ​​and gradient vectors (GradObj and GradConstr) are passed back to the outer optimizer for the next iteration update of the design variables.

[0055] Specifically, the outer optimizer uses a Sequential Quadratic Programming (SQP) algorithm for iterative optimization. SQP is one of the most effective gradient-based algorithms for solving this type of nonlinearly constrained optimization problem. In each iteration... First, based on the weighted objective function f(x) and the constraints G(x) (including worst-case compliance constraints and upper and lower bounds of variables), a Lagrange function is constructed. ,in These are Lagrange multipliers. Subsequently, the algorithm operates at the current point. The Lagrangian function is expanded using a second-order Taylor series, and the nonlinear constraints are linearized to a first-order linearization, constructing a quadratic programming (QP) subproblem to find the search direction for the design variables. The mathematical form of this subproblem is as follows:

[0056] in, and These are the first-order gradients of the objective function and the constraint function (such as robustness and flexibility constraints) at the current point, respectively. These gradient values ​​are precisely calculated using the direct differentiation method described in steps S4 and S53 above. For the positive definite approximation of the Hessian matrix of the Lagrange function, this embodiment uses the BFGS formula to update based on the iteration information from the previous step:

[0057] Solving this QP subproblem yields the optimal search direction. Then, the step size is determined by combining the line search strategy. Update design variables This allows for efficient approximation of the optimal solution that satisfies all uncertainties and performance constraints.

[0058] S6. Determine whether the outer layer optimization has converged; if the change in the outer layer design variables or the change in the objective function meets the preset convergence threshold, output the optimal thickness distribution and elastic modulus contour plot; otherwise, return to S4, re-execute the worst-case search of the inner layer under the updated outer layer design variables, and repeat S4-S6 until the outer layer optimization converges.

[0059] S61. Outer layer optimization convergence judgment; Determine the norm of the change in design variables between two consecutive iterations Is it less than the preset tolerance? S62. Post-processing for result visualization; If convergence occurs, plot the convergence history curve as follows: Figure 6 The figure shows the black and white thresholds for the optimal thickness distribution, and the calculation is performed. , drawing as Figure 3 The topological configuration diagram of the element thickness distribution is shown; simultaneously, using pseudo-color mapping technology, a diagram is drawn as follows. Figure 4 The color contour plot shows the element elastic modulus distribution. Further combining the thickness distribution results with the elastic modulus distribution results, the thickness and elastic modulus of each element within the effective structural region are jointly characterized, and plotted as shown below. Figure 5 The thickness-elastic modulus unit collaborative distribution cloud map is shown.

[0060] Simultaneously, based on the optimized thickness distribution vector t and the threshold... Determine the configuration of this region. If this region... The area is determined to be a non-load-bearing void and rendered with a white background; if the unit thickness is... The region is determined to be an effective structural configuration, and the optimized elastic modulus value corresponding to the unit is extracted. Based on a preset color mapping table, it is mapped to the corresponding color and filled to form... Figure 3 Optimized thickness-elastic modulus unit co-distribution color cloud map.

[0061] The above description is not intended to limit the present invention in any way. Although the present invention has been disclosed through the above embodiments, it is not intended to limit the present invention. Any person skilled in the art can make changes or modifications to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A method for optimizing the robustness of thin-walled components while considering dimensional and material indeterminism, characterized in that, Includes the following steps: S1. Establish the initial geometric model of the structure, mesh it, and obtain a finite element discrete model containing node coordinates and element connection relationships; S2. Define the optimization design variables and uncertainty parameters, set the thickness and elastic modulus of each element as design variables, and set their corresponding geometric uncertainty intervals and material uncertainty intervals; S3. Construct a two-layer optimization mathematical model. The outer layer optimization is used to minimize the structural weighted objective function, and the inner layer optimization is used to solve the worst structural response under the disturbance of uncertain parameters. Based on the current design variables and initial material and geometric parameters, assemble the stiffness matrices of each element to form the overall stiffness matrix, and establish the structural equilibrium equation in combination with external loads, providing a basis for the inner layer uncertainty analysis and structural response calculation. S4. In the inner layer optimization, the nodal displacements are obtained through finite element calculation, the uncertainty parameters are differentiated based on the equilibrium equation, the sensitivity of the objective function to the uncertainty parameters is calculated, and the worst parameter combination is determined according to the sign of the sensitivity. S5. Update the overall stiffness matrix and load vector using the worst-case parameter combination, solve for the structural response and return to the outer optimizer; S6. Determine whether the outer layer optimization has converged; if the change in the outer layer design variables or the change in the objective function meets the preset convergence threshold, output the optimal thickness distribution and elastic modulus contour plot; otherwise, return to S4, re-execute the worst-case search of the inner layer under the updated outer layer design variables, and repeat S4-S6 until the outer layer optimization converges.

2. The method for optimizing the robustness of thin-walled components while considering dimensional and material indeterminism according to claim 1, characterized in that, Step S1 includes: S11. Establish the structural macroscopic geometric parameters and node coordinate system; Define the length and height parameters of the structural physical design domain, and set the number of nodes noNDX and noNDY in the horizontal and vertical directions; generate the global coordinate matrix gcoord for all nodes using bilinear interpolation or a regular mesh generation algorithm. S12. Mesh discretization and initial geometric property calculation; The solution domain is discretized using four-node quadrilateral elements, and an element node connection matrix is ​​established. Specifically, using bilinear functions Constructing a physical coordinate system With natural coordinate system The mapping relationship between them, where natural coordinates The shape function expressions for the four nodes are as follows: in, , , and These are the four nodes of the quadrilateral; For each element, the Jacobian determinant is calculated using Gaussian numerical integration, through shape functions and their derivatives. Then, by integration, the initial geometric area of ​​each unit can be obtained. : Based on this, the mechanical properties of the element are further constructed; according to the geometric equations of elasticity, the strain vector within the element is... With nodal displacement vector The relationship between strain and displacement is expressed through the strain-displacement matrix. Establish, that is ;matrix It consists of the partial derivatives of the shape functions with respect to the global coordinates, and requires the inverse of the Jacobian matrix. Perform coordinate transformation calculations; for plane stress problems, introduce the material constitutive matrix. : in Poisson's ratio, Represented as the true elastic modulus of the material, the final element stiffness matrix... The explicit integral form is defined as: in To determine the true thickness, the uncertainty parameter is determined using the above formula. and This is directly coupled into the construction process of the stiffness matrix, providing a computational basis for subsequent uncertainty response analysis and worst-case search.

3. The method for optimizing the robustness of thin-walled components while considering dimensional and material indeterminism according to claim 1, characterized in that, Step S2 includes: S21. Define the optimization design variable vector and boundary constraints; The thickness of each element in the finite element model With elastic modulus As independent design variables, construct a global design variable vector. Set the constraint range for the thickness variable. Constraint range of elastic modulus variable ; S22. Set the mixed uncertainty parameters and their fluctuation range; Define the uncertainty fluctuation range of geometric thickness Uncertainty fluctuation range of material elastic modulus Introducing uncertain state variables and This describes the actual parameter deviation of each unit in the current design state: 。 4. The robustness optimization method for thin-walled components that takes into account both dimensional and material indeterminism, as described in claim 1, is characterized in that... Step S3 includes: S31. Construct the outer layer optimization model; Constructing a generalized material cost objective function Its physical meaning is the generalized structural weight that takes into account material properties, and the calculation formula is as follows: in, For material density, For unit area, Physical thickness It is the physical elastic modulus. S32. Define constraints; Define the performance constraints of the structure under worst-case conditions, including structural flexibility constraints: in It is the structural compliance under the worst-case combination of uncertainties calculated by the inner optimization loop. It is the maximum allowable flexibility value in the design.

5. The method for optimizing the robustness of thin-walled components while considering dimensional and material indeterminism according to claim 1, characterized in that, Step S4 includes: S41. Assembly of finite element stiffness matrix under uncertain parameters; In the inner optimization loop, based on the current uncertainty coefficient and Update element properties and calculate the stiffness matrix of each element using numerical integration. And assemble the global stiffness matrix : in The strain-displacement matrix is... The constitutive matrix of the material; S42. Sensitivity analysis and worst-case direction determination; Solve the equilibrium equations The nodal displacement vectors are obtained; then the structural equilibrium equations are differentiated with respect to the uncertainty parameters, and the sensitivity of the objective function to the uncertainty parameters is obtained by combining the functional relationship between the objective function and the nodal displacements. The sensitivity is obtained by calculating the derivative of the element stiffness matrix with respect to the uncertainty parameters and the nodal displacement vectors, without having to perform global finite element difference calculations for each uncertainty parameter separately. Calculate the sensitivity of the objective function to the uncertainty parameter. : in, The sensitivity of the thickness uncertainty parameter. The sensitivity of the modulus uncertainty parameter; Based on sensitivity The sign of the variable directly determines the worst-case uncertainty parameter. Value boundaries: 。 6. The method for optimizing the robustness of thin-walled components while considering dimensional and material indeterminism according to claim 1, characterized in that, The specific implementation method of step S5 is as follows: S51. System matrix reconstruction under worst-case conditions; When the inner optimization loop meets the convergence condition, it means that the uncertainty parameter that minimizes the structural performance has been found. and Then, extract the data at this time. and ; The stiffness matrix of each element is recalculated using the Gaussian numerical integration method based on the updated properties. And assembled to form a global stiffness matrix reflecting the worst-case working condition. ; S52. Solving for the worst structural response; Apply displacement boundary conditions to the reconstructed system matrix and solve the linear equations to obtain the nodal displacement vectors under the worst-case condition. ; Calculate the objective function value required for outer layer optimization based on this displacement vector. and constraint function values ; S53. Sensitivity analysis for outer layer optimization; Specifically, by differentiating the structural equilibrium equations with respect to the design variables, the relationship between nodal displacements and the design variables is obtained. Then, combining the expressions for the objective function and constraint functions, the sensitivity of the objective function to the thickness variables of each element is calculated. and sensitivity to changes in elastic modulus The calculation formula is as follows: The calculated function values ​​and gradient vectors are passed back to the outer optimizer for the next iteration update of the design variables; the outer optimizer uses a sequential quadratic programming algorithm for iterative optimization.

7. The method for optimizing the robustness of thin-walled components while considering dimensional and material indeterminism according to claim 1, characterized in that, Step S6 includes: S61. Outer layer optimization convergence judgment; Determine the norm of the change in design variables between two consecutive iterations Is it less than the preset tolerance? S62. Post-processing for result visualization; If the convergence condition is met, the optimized thickness distribution and elastic modulus distribution results will be output.

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