Improved dmo topology optimization method and analysis device subject to different material yield criteria

Abstract

The application discloses an improved DMO topology optimization method and analysis equipment obeying different material yield criteria, belongs to the technical field of engineering structure optimization, and comprises the following steps: establishing a topology optimization mathematical model by taking minimum flexibility as an objective function, different yield criterion stress constraints and cost constraints as constraint functions; rewriting interpolation of the DMO method based on an elastic tensor into element superposition method for element discretization stiffness interpolation; multiple stress values of each element of each material can be calculated by using separable stress interpolation scheme based on the element stacking method; performing finite element analysis and stress calculation on the structure, respectively giving von-Mises and Drucker-Prager yield criteria to steel and concrete materials, and using an MMA algorithm to iteratively calculate results to obtain an approximate optimal solution; cyclically iterating and updating element density and the above solving steps until a new topology optimization structure reaches a preset convergence condition. The application more accurately performs steel-concrete composite structure topology optimization, and the obtained topology optimization result has manufacturability.

Description

Technical Field

[0001] This invention belongs to the field of engineering structure optimization technology, specifically relating to an improved DMO topology optimization method and analysis equipment that conforms to different material yield criteria. Background Technology

[0002] Topology optimization, by adjusting material distribution and structural shape to meet design requirements and reduce material usage, is increasingly being applied in aerospace, mechanical design, vehicle engineering, and civil engineering. Bridge structures are subject to complex stresses and frequently face the problem of tensile cracking in concrete. Ensuring that bridge structures are protected from tensile stress during topology optimization design is a crucial issue. Therefore, to promote the application of topology optimization in steel-concrete composite structures, it is necessary to develop a topology optimization method and analysis equipment for steel-concrete composite structures.

[0003] Discretized interpolation models (DMOs) are a feasible method for handling topology optimization problems involving different materials. They assign materials based on their elastic modulus interpolation, assuming equal tensile and compressive properties. However, concrete exhibits inconsistent tensile and compressive properties, making this method unsuitable for topology optimization of steel-concrete composite structures. Therefore, it is necessary to assign von Mises and Drucker-Prager yield criteria to steel and concrete materials respectively, using these as stress constraint functions to control the materials within their yield surfaces, thereby achieving topology optimization for steel-concrete composite structures. Summary of the Invention

[0004] The purpose of this invention is to provide an improved DMO topology optimization method and analysis device that conforms to different material yield criteria. The von-Mises and Drucker-Prager yield criteria are respectively assigned to steel and concrete materials and used as their stress constraint functions to construct a theoretical model for topology optimization of steel-concrete composite structures that conforms to different material yield criteria.

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

[0006] An improved DMO topology optimization method that conforms to different material yield criteria includes the following steps:

[0007] 1) Define the design domain and establish a topology optimization mathematical model with minimum compliance as the objective function, different yield criterion stress constraints and total cost constraints as constraint functions, and assign initial element density as design variable to each element in the design domain;

[0008] 2) The DMO method based on elastic tensor interpolation is rewritten into element stacking method for element discretization stiffness interpolation, which allows multiple material sets to be placed side by side on the same element coordinates. In the stacked material sets, appropriate material items are retained to fill the element nodes through topology optimization. Multiple stress values ​​of each element of each material can be calculated using a separable stress interpolation scheme based on element stacking method.

[0009] 3) Perform finite element analysis and stress calculation on the structure, solve for the nodal stiffness matrix, nodal displacement matrix, and element stress matrix, filter and project the element density, make the design domain of discrete element stress continuous through qp relaxation, and aggregate discrete stress constraints into global constraints through the p-norm scheme. Assign von-Mises and Drucker-Prager yield criteria to steel and concrete materials respectively, calculate the sensitivity of objective function and constraint function to element density, and use the MMA algorithm to iteratively calculate the results to obtain the approximate optimal solution.

[0010] 4) Iterate through the cell density and the above solution steps until the new topology optimization structure reaches the preset convergence condition, and output the three-dimensional structure diagram, iteration curve and normalized stress diagram after topology optimization.

[0011] Optionally, in step 1), the topology optimization mathematical model is as shown in equation (1):

[0012]

[0013] In the formula, Minimize represents the objective function taking its minimum value, Subject to represents the constraint condition, C represents the structural compliance, and K, U, and F represent the structural compliance. ext These are respectively represented as the global stiffness matrix, displacement vector, and external load vector; g c Represented as the total cost constraint, NM VM NM represents the material number that obeys the von Mises stress yield criterion. DP This represents the number of materials that conform to the Drucker-Prager strength criterion. Let be the element density of the i-th candidate material that conforms to the von Mises stress yield criterion after filtering and projection. Let be the element density of the m-th candidate material that conforms to the Drucker-Prager strength criterion after filtering and projection. Let represent the cost coefficient of the i-th alternative material that conforms to the von Mises stress yield criterion. V represents the cost coefficient of the m-th alternative material that conforms to the Drucker-Prager strength criterion. e Expressed as unit volume, g c,max Represented as the total structural cost constraint; gi,max Let σ represent the safety factor for the i-th material that conforms to the von Mises stress yield criterion. i,max Let σ be the approximate maximum stress of the p-norm condensation of the i-th material that obeys the von Mises stress yield criterion. i,lim Let f represent the yield stress of the i-th material that obeys the von Mises stress yield criterion; m,max (DP) represents the DP criterion strength formula for the m-th material that conforms to the Drucker-Prager strength criterion after p-norm condensation; x represents the element density design variable.

[0014] Optionally, in step 2), the element discretization stiffness interpolation is as shown in equation (2):

[0015] (2)

[0016] In the formula, k e The element stiffness matrix is ​​represented by NM after interpolation, where NM represents the material type, q represents the penalty factor for the isotropic interpolation model of solids, and k represents the interpolated element stiffness matrix. i It is represented as the product of the elastic tensor of the i-th material and the element stiffness matrix k0.

[0017] Optionally, in step 3), the method for cell density filtering and projection includes:

[0018] The formulas for the filtering function and the projection function, as well as the formulas for the sensitivity of the filtering function and the sensitivity of the projection function, are shown in equations (3)-(8):

[0019] (3)

[0020] (4)

[0021] (5)

[0022] (6)

[0023] (7)

[0024] (8)

[0025] In the formula, N e (R) is represented by unit x e The neighborhood within the filtering radius, R represents the filtering radius, the value of which is adjusted according to the finite element mesh size of the structure, and is not less than 3 element lengths; The density of the filtered cells is represented by H, where H is the weighting factor. Represented as a unit and unit The distance between the centroids; β represents the sharpness of the projection function, with a maximum value of 40; η represents the inflection point coefficient of the projection function, with a value of 0.5. This is expressed as the sensitivity of the filter unit density to the original unit density; This is expressed as the sensitivity of the projection cell density to the filter cell density.

[0026] Optionally, in step 3), the method of assigning von Mises and Drucker-Prager yield criteria to steel and concrete materials respectively by the qp relaxation and p-norm scheme includes:

[0027] The qp scheme establishes a relaxation calculation formula for Cauchy stress, an element von-Mises stress formula for materials conforming to the von-Mises yield criterion, and a Drucker-Prager strength calculation formula for materials conforming to the Drucker-Prager strength criterion. It uses the p-norm formula to aggregate the Mises stress and DP strength formulas, establishes an adaptive constraint scaling factor formula to correct the p-norm stress measurement, and establishes approximate maximum Mises stress and Drucker-Prager strength formulas, as shown in equations (9)-(22):

[0028] (9)

[0029] (10)

[0030] (11)

[0031] (12)

[0032] (13)

[0033] (14)

[0034] (15)

[0035] (16)

[0036] (17)

[0037] (18)

[0038] (19)

[0039] (20)

[0040] (twenty one)

[0041] (twenty two)

[0042] In the formula, Let qp represent the Cauchy stress tensor of the i-th material, and n represent the relaxation coefficient qp, with a value of 0.5. , and Represented as normal stress components, , and Represented as shear stress components, D represents the local normal stress component in the x-direction. e i Let B be the constitutive matrix of the i-th candidate material in the e-th finite element, and let B be the element deformation matrix. e Let J1 be the displacement matrix of the e-th element node; J2 be the first invariant of the stress tensor; σ be the second invariant of the stress tensor. e i,VM The apparent local Mises stress of the i-th candidate material is represented by the von-Mises yield criterion. Let f represent the element Drucker-Prager strength of the m-th candidate material conforming to the Drucker-Prager strength criterion, where α and β are two parameters determined by the envelope of the Drucker-Prager strength criterion. t and f c denoted as the uniaxial tensile strength and uniaxial compressive strength of materials that conform to the Drucker-Prager strength criterion, respectively; nele represents the total number of elements, and nelx, nely, and nelz represent the number of elements in the x-direction, y-direction, and z-direction, respectively. and These represent the maximum Mises stress and DP criterion equations for p-norm condensation, respectively, where p represents the p-norm with a value of 10; I represents the iteration number; and c... I i,VM and c I m,DP These are represented as the von Mises stress adaptive constraint scaling factor and the Drucker-Prager criterion adaptive constraint scaling factor, respectively, to correct p-norm stress measurements and compensate for the difference between the p-norm maximum stress and the approximate maximum stress; k I This is represented as a scaling factor, with a value of 0.5; σ i,max and f m,max (DP) represents the approximate maximum von Mises stress and Drucker-Prager strength formula after condensation correction, respectively.

[0043] Optionally, in step 3), the formulas for calculating the sensitivity of the objective function and constraint function to the element density are shown in equations (23)-(36):

[0044] (twenty three)

[0045] (twenty four)

[0046] (25)

[0047] (26)

[0048] (27)

[0049] (28)

[0050] (29)

[0051] (30)

[0052] (31)

[0053] (32)

[0054] (33)

[0055] (34)

[0056] (35)

[0057] (36)

[0058] In the formula, This is expressed as the sensitivity of structural compliance to element projection density. It is represented as the element displacement matrix; ω represents the solid isotropic interpolation penalty term, and q represents the SIMP penalty coefficient, which takes the value of 3; This is expressed as the sensitivity of the element stiffness matrix to the element projected density. According to the chain rule, this term is derived from the sensitivity of the SIMP penalty term. and DMO stiffness interpolation sensitivity Multiply to obtain; This represents the sensitivity of the total structural cost to the element projection density; j represents the j-th element in the element column matrix. Let represent the sensitivity of the stress constraint function of the i-th material conforming to the von Mises yield criterion to the projected density of the subdomain elements of this material group. This term is derived from the sensitivity of the von Mises stress constraint function to the approximately maximum von Mises stress after condensation correction, according to the chain rule. Sensitivity of the approximate maximum von Mises stress after condensation correction to the element von Mises stress Sensitivity of element von Mises stress to Cauchy stress tensor and the sensitivity of the Cauchy stress tensor to element projected density Obtained by multiplication; Represented as the Cauchy stress tensor, it has 9 components in a three-dimensional element; L e This is represented as a global-local coordinate transformation matrix; for the cross-sensitivity of the approximate maximum von Mises stress to the projected density of other material subdomains after condensation correction, only the sensitivity of the Cauchy stress tensor to the element projected density is changed. change; Let represent the sensitivity of the stress constraint function of the m-th material conforming to the Drucker-Prager strength criterion to the projected density of the subdomain elements of this material group. This term is derived from the sensitivity of the Drucker-Prager stress constraint function to the element stress according to the Drucker-Prager criterion, based on the chain rule. Sensitivity of element Drucker-Prager criterion stress to Cauchy stress tensor and the sensitivity of the Cauchy stress tensor to element projected density The result is obtained by multiplying the products together.

[0059] An analytical apparatus employing the aforementioned improved DMO topology optimization method that conforms to different material yield criteria includes:

[0060] Input module: Used to input initial condition parameters for topology optimization, including the number of elements in the x-direction (nelx), y-direction (nely), z-direction (nelz), elastic modulus E of candidate materials, cost coefficient γ of each candidate material, and total structural cost constraint g. c,max The filter radius R and the yield strength σ of the material that obeys the von Mises yield criterion. i,max The uniaxial tensile strength f of materials that obey the Drucker-Prager strength criterion t and uniaxial compressive strength f c ;

[0061] The calculation module is used to perform finite element analysis and stress analysis on the structure, calculate the nodal stiffness matrix, nodal displacement matrix, and element stress matrix, make the design domain of discrete element stress continuous through qp relaxation, and aggregate discrete stress constraints into global constraints through the p-norm scheme. It also eliminates numerical problems such as checkerboard and grayscale phenomena through element density filtering and projection strategies, calculates the sensitivity of the objective function to element density, and the sensitivity of the cost constraint function and stress constraint function to design variables, and uses the MMA algorithm to iterate to obtain an approximate optimal solution.

[0062] Output module: Used to iteratively update the element density and solution steps until the new topology optimization structure reaches the preset convergence condition, and output the three-dimensional structure diagram, iteration curve and normalized stress diagram after topology optimization.

[0063] Compared with the prior art, the present invention has at least the following beneficial effects:

[0064] 1) By adding stress constraints in the iteration of topology optimization, the right-angle region is replaced with a circular arc region, which alleviates the stress concentration problem at the corner of the composite structure and effectively improves the stress performance of the structure. The optimized structure is more reasonable in engineering.

[0065] 2) By applying von-Mises and Drucker-Prager yield criteria stress constraints to materials with different properties, the optimization results obtained by this invention show that weak materials with lower elastic modulus are distributed in regions with lower stress levels, while materials with weak tensile properties are mainly distributed in the compression region. This indicates that this invention has the ability to develop the optimal distribution position of materials with asymmetric tensile and compressive strengths, and the theoretical model is applicable to the topology optimization design of steel-concrete composite structures.

[0066] 3) This invention can realize the topology optimization design of steel-concrete composite structures in a three-dimensional environment. The optimized structure has reasonable stress and manufacturability, which has important research and promotion significance. Attached Figure Description

[0067] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings:

[0068] Figure 1 This is a schematic diagram of the cell stacking method model used in this invention;

[0069] Figure 1 In the diagram, (a) represents candidate materials; (b) represents initial material mixing elements; (c) represents convergence result 1: blank elements; and (d) represents convergence result 2: single material elements.

[0070] Figure 2 A schematic diagram of the analysis device for the topology optimization method that conforms to different material yield criteria provided by the present invention;

[0071] Figure 3 A schematic diagram of the calculation process for the topology optimization method that follows different material yield criteria provided by the present invention;

[0072] Figure 4 The diagram shows the steel-concrete composite L-shaped beam structure to be optimized and its boundary conditions in Example 1.

[0073] Figure 5 The image shows the L-shaped beam structure after iteration using the improved DMO topology optimization method that follows different material yield criteria in Example 1.

[0074] Figure 6 This is an iterative graph of the objective function and cost constraint function in Example 1;

[0075] Figure 7 The stress diagram of the L-shaped beam semi-structure after topology optimization in Example 1;

[0076] Figure 7 In the diagram, (a) is the stress diagram of Q345 steel; (b) is the stress diagram of concrete. Detailed Implementation

[0077] The present invention will be further described below with reference to the embodiments and accompanying drawings. The accompanying drawings constitute a part of the present invention and, together with the embodiments of the present invention, are used to illustrate the principles of the present invention, but are not intended to limit the scope of the present invention.

[0078] In the description of this invention, "a plurality of" means two or more, unless otherwise expressly specified. The reference to "embodiment" herein means that a particular feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of the invention. The appearance of this phrase in various places in the specification does not necessarily refer to the same embodiment, nor is it a mutually exclusive, independent, or alternative embodiment. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described herein can be combined with other embodiments.

[0079] An improved DMO topology optimization method that conforms to different material yield criteria includes the following steps:

[0080] 1) Define the design domain and establish a topology optimization mathematical model with minimum compliance as the objective function, different yield criterion stress constraints and total cost constraints as constraint functions, and assign initial element density as design variable to each element in the design domain;

[0081] 2) The DMO method based on elastic tensor interpolation is rewritten into element stacking method for element discretization stiffness interpolation, which allows multiple material sets to be placed side by side on the same element coordinates. In the stacked material sets, appropriate material items are retained to fill the element nodes through topology optimization. Multiple stress values ​​of each element of each material can be calculated using a separable stress interpolation scheme based on element stacking method.

[0082] 3) Perform finite element analysis and stress calculation on the structure, solve for the nodal stiffness matrix, nodal displacement matrix, and element stress matrix, filter and project the element density, make the design domain of discrete element stress continuous through qp relaxation, and aggregate discrete stress constraints into global constraints through the p-norm scheme. Assign von-Mises and Drucker-Prager yield criteria to steel and concrete materials respectively, calculate the sensitivity of objective function and constraint function to element density, and use the MMA algorithm to iteratively calculate the results to obtain an approximate optimal solution.

[0083] 4) Iterate through the cell density and the above solution steps until the new topology optimization structure reaches the preset convergence condition, and output the three-dimensional structure diagram, iteration curve and normalized stress diagram after topology optimization.

[0084] The mathematical model for topology optimization is shown in equation (1) below:

[0085]

[0086] In the formula, Minimize represents the objective function being minimized, Subject to represents the constraints, C represents structural compliance, and K, U, and F represent the structural compliance. ext These are represented as the overall stiffness matrix, displacement vector, and external load vector, respectively. c Represented as the total cost constraint, NM VM NM represents the material number that obeys the von Mises stress yield criterion. DP This represents the number of materials that conform to the Drucker-Prager strength criterion. Let be the element density of the i-th candidate material that conforms to the von Mises stress yield criterion after filtering and projection. Let be the element density of the m-th candidate material that conforms to the Drucker-Prager strength criterion after filtering and projection. Let represent the cost coefficient of the i-th alternative material that conforms to the von Mises stress yield criterion. V represents the cost coefficient of the m-th alternative material that conforms to the Drucker-Prager strength criterion. e Expressed as unit volume, g c,max This is represented as the total structural cost constraint. i,maxLet σ represent the safety factor for the i-th material that conforms to the von Mises stress yield criterion. i,max Let σ be the approximate maximum stress of the p-norm condensation of the i-th material that obeys the von Mises stress yield criterion. i,lim Let f represent the yield stress of the i-th material that conforms to the von Mises stress yield criterion. m,max (DP) represents the DP criterion strength formula for the m-th material that follows the Drucker-Prager strength criterion after p-norm condensation, where x represents the element density design variable.

[0087] The element discretization stiffness interpolation method is shown in equation (2) below:

[0088] (2)

[0089] In the formula, k e The element stiffness matrix is ​​represented by NM after interpolation, where NM represents the material type, q represents the penalty factor for the solid isotropic interpolation model (SIMP), and k represents the interpolated element stiffness matrix. i It is represented as the product of the elastic tensor of the i-th material and the element stiffness matrix k0.

[0090] The element density needs to be filtered and projected onto the design variables. The density filtering formula is considered as the product of the coefficient matrix and the vector recording all element densities; the Heaviside projection function is both an extremum of the design variables and an effective constraint on the upper and lower bounds of the design variables, making the optimization results clear and reasonable. The formulas for the filtering function and the projection function, as well as the formulas for the sensitivity of the filtering function and the sensitivity of the projection function, are shown in the following equations (3)-(8):

[0091] (3)

[0092] (4)

[0093] (5)

[0094] (6)

[0095] (7)

[0096] (8)

[0097] In the formula, N e (R) is represented by unit x e The neighborhood within the filtering radius, R represents the filtering radius, the value of which is adjusted according to the finite element mesh size of the structure, and is generally not less than 3 element lengths; The density of the filtered cells is represented by H, where H is the weighting factor. Represented as a unit and unit The distance between the centroids; the parameter β represents the sharpness of the projection function. The value of β starts from a small value and is doubled after a certain number of iterations. The maximum value of β is set to 40. η represents the inflection point coefficient of the projection function, with a value of 0.5. This is expressed as the sensitivity of the filter unit density to the original unit density; This is expressed as the sensitivity of the projection unit density to the filter unit density;

[0098] The qp scheme establishes a relaxation calculation formula for Cauchy stress, an element von-Mises stress formula for materials that conform to the von-Mises yield criterion, and a Drucker-Prager strength calculation formula for materials that conform to the Drucker-Prager strength criterion. It uses the p-norm formula to aggregate the Mises stress and DP strength formulas, establishes an adaptive constraint scaling factor formula to correct the p-norm stress measurement, and establishes the approximate maximum Mises stress and Drucker-Prager strength formulas, as shown in the following equations (9)-(22):

[0099] (9)

[0100] (10)

[0101] (11)

[0102] (12)

[0103] (13)

[0104] (14)

[0105] (15)

[0106] (16)

[0107] (17)

[0108] (18)

[0109] (19)

[0110] (20)

[0111] (twenty one)

[0112] (twenty two)

[0113] In the formula, Let qp represent the Cauchy stress tensor of the i-th material, and n represent the relaxation coefficient qp, with a value of 0.5. , and Represented as normal stress components, , and Represented as shear stress components, This represents the local normal stress component in the x-direction. (D) e i Let B be the constitutive matrix of the i-th candidate material in the e-th finite element, and let B be the element deformation matrix. e Let J1 be the displacement matrix of the e-th element node; J2 be the first invariant of the stress tensor; σ be the second invariant of the stress tensor. e i,VM The apparent “local” Mises stress of the i-th candidate material is represented by the von-Mises yield criterion. Let f represent the element Drucker-Prager strength of the m-th candidate material conforming to the Drucker-Prager strength criterion, where α and β are two parameters determined by the envelope of the Drucker-Prager strength criterion. t and f c denoted as the uniaxial tensile strength and uniaxial compressive strength of materials that conform to the Drucker-Prager strength criterion, respectively; nele represents the total number of elements, and nelx, nely, and nelz represent the number of elements in the x-direction, y-direction, and z-direction, respectively. and These represent the maximum Mises stress and DP criterion equations for p-norm condensation, respectively, where p represents the p-norm with a value of 10; I represents the iteration number; and c... I i,VM and c I m,DP These are represented as the von Mises stress adaptive constraint scaling factor and the Drucker-Prager criterion adaptive constraint scaling factor, respectively, to correct p-norm stress measurements and compensate for the difference between the p-norm maximum stress and the approximate maximum stress; k I This is represented as a scaling factor to ensure that the scaling factor in each iteration is related to the historical data; its value is 0.5. i,max and f m,max (DP) represents the approximate maximum von Mises stress and Drucker-Prager strength formula after condensation correction, respectively.

[0114] In the above scheme, the formulas for the sensitivity of the objective function, the sensitivity of the cost constraint function, and the sensitivity of the stress constraint function are as shown in equations (23)-(36):

[0115] (twenty three)

[0116] (twenty four)

[0117] (25)

[0118] (26)

[0119] (27)

[0120] (28)

[0121] (29)

[0122] (30)

[0123] (31)

[0124] (32)

[0125] (33)

[0126] (34)

[0127] (35)

[0128] (36)

[0129] In the formula, This is expressed as the sensitivity of structural compliance to element projection density. It is represented as the element displacement matrix; ω represents the solid isotropic interpolation (SIMP) penalty term, and q represents the SIMP penalty coefficient, which takes the value of 3; This is expressed as the sensitivity of the element stiffness matrix to the element projected density. According to the chain rule, this term is derived from the sensitivity of the SIMP penalty term. and DMO stiffness interpolation sensitivity Multiply to obtain; This represents the sensitivity of the total structural cost to the element projection density; j represents the j-th element in the element column matrix. Let represent the sensitivity of the stress constraint function of the i-th material conforming to the von Mises yield criterion to the projected density of the subdomain elements of this material group. This term is derived from the sensitivity of the von Mises stress constraint function to the approximately maximum von Mises stress after condensation correction, according to the chain rule. Sensitivity of the approximate maximum von Mises stress after condensation correction to the element von Mises stress Sensitivity of element von Mises stress to Cauchy stress tensor and the sensitivity of the Cauchy stress tensor to element projected density Obtained by multiplication; Represented as the Cauchy stress tensor, it has 9 components in a three-dimensional element; L e This is represented as a global-local coordinate transformation matrix; for the cross-sensitivity of the approximate maximum von Mises stress to the projected density of other material subdomains after condensation correction, only the sensitivity of the Cauchy stress tensor to the element projected density is changed. change; Let represent the sensitivity of the stress constraint function of the m-th material conforming to the Drucker-Prager strength criterion to the projected density of the subdomain elements of this material group. This term is derived from the sensitivity of the Drucker-Prager stress constraint function to the element stress according to the Drucker-Prager criterion, based on the chain rule. Sensitivity of element Drucker-Prager criterion stress to Cauchy stress tensor and the sensitivity of the Cauchy stress tensor to element projected density The results are obtained by multiplying together; the cross sensitivity still follows the method of solving the sensitivity of von-Mises stress to the projected density of other material subdomains, that is, only the sensitivity of Cauchy stress tensor to the projected density of the element is changed.

[0130] An analysis device for an improved DMO topology optimization method that conforms to different material yield criteria includes an input module, a calculation module, and an output module;

[0131] Input module: Used to input initial condition parameters for topology optimization, including the number of elements in the x-direction (nelx), y-direction (nely), z-direction (nelz), elastic modulus E of candidate materials, cost coefficient γ of each candidate material, and total structural cost constraint g. c,max The filter radius R and the yield strength σ of the material that obeys the von Mises yield criterion. i,max The uniaxial tensile strength f of materials that obey the Drucker-Prager strength criterion t and uniaxial compressive strength fc wait;

[0132] The calculation module is used to perform finite element analysis and stress analysis on the structure, calculate the nodal stiffness matrix, nodal displacement matrix, and element stress matrix, make the design domain of discrete element stress continuous through qp relaxation, and aggregate discrete stress constraints into global constraints through the p-norm scheme. It also eliminates numerical problems such as checkerboard and grayscale phenomena through element density filtering and projection strategies, calculates the sensitivity of the objective function to element density, and the sensitivity of the cost constraint function and stress constraint function to design variables, and uses the MMA algorithm to iterate to obtain an approximate optimal solution.

[0133] Output module: Used to iteratively update the element density and the above solution steps until the new topology optimization structure reaches the preset convergence condition, and output the three-dimensional structure diagram, iteration curve and normalized stress diagram after topology optimization.

[0134] An improved DMO topology optimization method that conforms to different material yield criteria, using a cell stacking method model, such as... Figure 1 (a)- Figure 1 As shown in (d), the model can guarantee that von-Mises and Drucker-Prager yield criteria stress constraints are assigned to materials with different properties, such as steel and concrete, respectively, in topology optimization.

[0135] like Figure 2 As shown, an analysis device using the improved DMO topology optimization method that follows different material yield criteria is provided above. It includes an input module, a calculation module, and an output module.

[0136] Input module: Used to input initial condition parameters for topology optimization, including the number of elements in the x-direction (nelx), y-direction (nely), z-direction (nelz), elastic modulus E of candidate materials, cost coefficient γ of each candidate material, and total structural cost constraint g. c,max The filter radius R and the yield strength σ of the material that obeys the von Mises yield criterion. i,max The uniaxial tensile strength f of materials that obey the Drucker-Prager strength criterion t and uniaxial compressive strength f c wait;

[0137] The calculation module is used to perform finite element analysis and stress analysis on the structure, calculate the nodal stiffness matrix, nodal displacement matrix, and element stress matrix, make the design domain of discrete element stress continuous through qp relaxation, and aggregate discrete stress constraints into global constraints through the p-norm scheme. It also eliminates numerical problems such as checkerboard and grayscale phenomena through element density filtering and projection strategies, calculates the sensitivity of the objective function to element density, and the sensitivity of the cost constraint function and stress constraint function to design variables, and uses the MMA algorithm to iterate to obtain an approximate optimal solution.

[0138] Output module: Used to iteratively update the element density and the above solution steps until the new topology optimization structure reaches the preset convergence condition, and output the three-dimensional structure diagram, iteration curve and normalized stress diagram after topology optimization.

[0139] Example 1

[0140] The L-shaped beam is optimized using the improved DMO topology optimization method according to different material yield criteria described in this invention. The specific steps include:

[0141] 1) The program execution order in the embodiment is as follows: Figure 3 As shown, in step 1), a design domain is defined and a topology optimization mathematical model is established with minimum compliance as the objective function and different yield criteria stress constraints and volume constraints as constraint functions. An initial element density is assigned as a design variable to each element within the design domain, and relevant topology optimization parameters are determined. In step 2), the DMO method's interpolation based on the elastic tensor is rewritten as an element stacking method for discretized stiffness interpolation, allowing multiple material sets to be juxtaposed on the same element coordinate system. Appropriate material items are retained in the stacked material sets to fill element nodes. Multiple stress values ​​for each element of each material can be calculated using a separable stress interpolation scheme based on the element stacking method. In step 3), a finite element analysis is performed on the structure. The process involves stress calculation, solving for the nodal stiffness matrix, nodal displacement matrix, and element stress matrix, filtering and projecting the element density, making the design domain of discrete element stress continuous through qp relaxation, and aggregating discrete stress constraints into global constraints using the p-norm scheme. von Mises and Drucker-Prager yield criteria are applied to steel and concrete materials respectively. The sensitivity of the objective function and constraint function to element density is calculated, and the MMA algorithm is used to iteratively calculate the results to obtain an approximate optimal solution. In step 4), the element density and the above solution steps are iteratively updated until the new topology optimization structure reaches the preset convergence condition. The resulting 3D structure diagram, iteration curve, and normalized stress diagram are output.

[0142] 2) The steel-concrete composite L-shaped beam structure and boundary conditions to be optimized in the embodiment are as follows: Figure 4As shown in the figure; the structural dimensions are set to 1m×0.4m×1m, 1 is the consolidated region with dimensions of 0.4m×0.4m; 2 is the external load with a magnitude of 204kN, and the area of ​​action is the length of 3 units.

[0143] 3) The L-shaped beam structure after iteration using the improved DMO topology optimization method that follows different material yield criteria in the embodiment is as follows: Figure 5 As shown in the figure; 1 is Q345 steel with an elastic modulus of 206 GPa, a yield strength of 345 MPa, and a cost coefficient of 1; 2 is concrete material with an elastic modulus of 40 GPa, an ultimate tensile strength of 10 MPa, an ultimate compressive strength of 100 MPa, and a cost coefficient of 0.3.

[0144] The iterative graphs of the objective function and cost constraint function calculated using the method described in this embodiment are as follows: Figure 6 As shown, the stress diagram of the L-shaped beam semi-structure after topology optimization, calculated using the method described in this embodiment, is as follows: Figure 7 As shown, the results indicate that the method of the present invention has the ability to develop the optimal allocation position of materials with asymmetric tensile and compressive strength, and its theoretical model is applicable to the topology optimization design of steel-concrete composite structures.

[0145] The above description is merely a specific embodiment of the present invention, and the scope of protection of the present invention is not limited thereto. Any transformations or substitutions that can be conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. An improved DMO topology optimization method that conforms to different material yield criteria, characterized in that, Includes the following steps: 1) Define the design domain and establish a topology optimization mathematical model with minimum compliance as the objective function, different yield criterion stress constraints and total cost constraints as constraint functions, and assign initial element density as design variable to each element in the design domain; 2) The DMO method based on elastic tensor interpolation is rewritten into element stacking method for element discretization stiffness interpolation, which allows multiple material sets to be placed side by side on the same element coordinates. In the stacked material sets, appropriate material items are retained to fill the element nodes through topology optimization. Multiple stress values ​​of each element of each material can be calculated using a separable stress interpolation scheme based on element stacking method. 3) Perform finite element analysis and stress calculation on the structure, solve for the nodal stiffness matrix, nodal displacement matrix, and element stress matrix, filter and project the element density, make the design domain of discrete element stress continuous through qp relaxation, and aggregate discrete stress constraints into global constraints through the p-norm scheme. Assign von-Mises and Drucker-Prager yield criteria to steel and concrete materials respectively, calculate the sensitivity of objective function and constraint function to element density, and use the MMA algorithm to iteratively calculate the results to obtain an approximate optimal solution. 4) Iterate through the cell density and the above solution steps until the new topology optimization structure reaches the preset convergence condition, and output the three-dimensional structure diagram, iteration curve and normalized stress diagram after topology optimization. In step 1), the topology optimization mathematical model is shown in equation (1): In the formula, Minimize represents the objective function taking its minimum value, Subject to represents the constraint condition, C represents the structural compliance, and K, U, and F represent the structural compliance. ext These are respectively represented as the global stiffness matrix, displacement vector, and external load vector; g c Represented as the total cost constraint, NM VM NM represents the material number that obeys the von Mises stress yield criterion. DP This represents the number of materials that conform to the Drucker-Prager strength criterion. Let be the element density of the i-th candidate material that conforms to the von Mises stress yield criterion after filtering and projection. Let be the element density of the m-th candidate material that conforms to the Drucker-Prager strength criterion after filtering and projection. Let represent the cost coefficient of the i-th alternative material that conforms to the von Mises stress yield criterion. V represents the cost coefficient of the m-th alternative material that conforms to the Drucker-Prager strength criterion. e Expressed as unit volume, g c,max Represented as the total structural cost constraint; g i,max Let σ represent the safety factor for the i-th material that conforms to the von Mises stress yield criterion. i,max Let σ be the approximate maximum stress of the p-norm condensation of the i-th material that obeys the von Mises stress yield criterion. i,lim Let f represent the yield stress of the i-th material that obeys the von Mises stress yield criterion; m,max (DP) represents the DP criterion strength formula for the m-th material that obeys the Drucker-Prager strength criterion after p-norm condensation; x represents the element density design variable. In step 2), the element discretization stiffness interpolation is as shown in equation (2): （2）， In the formula, k e The element stiffness matrix is ​​represented by NM after interpolation, where NM represents the material type, q represents the penalty factor for the isotropic interpolation model of solids, and k represents the interpolated element stiffness matrix. i It is represented as the product of the elastic tensor of the i-th material and the element stiffness matrix k0.

2. The improved DMO topology optimization method according to claim 1, which follows different material yield criteria, is characterized in that... In step 3), the method for cell density filtering and projection includes: The formulas for the filtering function and the projection function, as well as the formulas for the sensitivity of the filtering function and the sensitivity of the projection function, are shown in equations (3)-(8): （3）， （4）， （5）， （6）， （7）， （8）， In the formula, N e (R) is represented by unit x e The neighborhood within the filtering radius, R represents the filtering radius, the value of which is adjusted according to the finite element mesh size of the structure, and is not less than 3 element lengths; The density of the filtered cells is represented by H, where H is the weighting factor. Represented as a unit and unit The distance between the centroids; β represents the sharpness of the projection function, with a maximum value of 40; η represents the inflection point coefficient of the projection function, with a value of 0.

5. This is expressed as the sensitivity of the filter unit density to the original unit density; This is expressed as the sensitivity of the projection cell density to the filter cell density.

3. The improved DMO topology optimization method and analysis equipment according to claim 1, which follows different material yield criteria, is characterized in that... In step 3), the method of assigning von Mises and Drucker-Prager yield criteria to steel and concrete materials respectively using the qp relaxation and p-norm scheme includes: The qp scheme establishes a relaxation calculation formula for Cauchy stress, an element von-Mises stress formula for materials conforming to the von-Mises yield criterion, and a Drucker-Prager strength calculation formula for materials conforming to the Drucker-Prager strength criterion. It uses the p-norm formula to aggregate the Mises stress and DP strength formulas, establishes an adaptive constraint scaling factor formula to correct the p-norm stress measurement, and establishes approximate maximum Mises stress and Drucker-Prager strength formulas, as shown in equations (9)-(22): （9）， （10）， （11）， （12）， （13）， （14）， （15）， （16）， （17）， （18）， （19）， （20）， （21）， （22）， In the formula, Let qp represent the Cauchy stress tensor of the i-th material, and n represent the relaxation coefficient qp, with a value of 0.

5. , and Represented as normal stress components, , and Represented as shear stress components, D represents the local normal stress component in the x-direction. e i Let B be the constitutive matrix of the i-th candidate material in the e-th finite element, and let B be the element deformation matrix. e Let J1 be the displacement matrix of the e-th element node; J2 be the first invariant of the stress tensor; σ be the second invariant of the stress tensor. e i,VM The apparent local Mises stress of the i-th candidate material is represented by the von-Mises yield criterion. Let f represent the element Drucker-Prager strength of the m-th candidate material conforming to the Drucker-Prager strength criterion, where α and β are two parameters determined by the envelope of the Drucker-Prager strength criterion. t and f c denoted as the uniaxial tensile strength and uniaxial compressive strength of materials that conform to the Drucker-Prager strength criterion, respectively; nele represents the total number of elements, and nelx, nely, and nelz represent the number of elements in the x-direction, y-direction, and z-direction, respectively. and These represent the maximum Mises stress and DP criterion equations for p-norm condensation, respectively, where p represents the p-norm with a value of 10; I represents the iteration number; and c... I i,VM and c I m,DP These are represented as the von Mises stress adaptive constraint scaling factor and the Drucker-Prager criterion adaptive constraint scaling factor, respectively, to correct p-norm stress measurements and compensate for the difference between the p-norm maximum stress and the approximate maximum stress; k I This is represented as a scaling factor, with a value of 0.5; σ i,max and f m,max (DP) represents the approximate maximum von Mises stress and Drucker-Prager strength formula after condensation correction, respectively.

4. The improved DMO topology optimization method and analysis equipment according to claim 1, which follows different material yield criteria, is characterized in that... In step 3), the formulas for calculating the sensitivity of the objective function and constraint function to the element density are shown in equations (23)-(36): （23）， （24）， （25）， （26）， （27）， （28）， （29）， （30）， （31）， （32）， （33）， （34）， （35）， （36）， In the formula, This is expressed as the sensitivity of structural compliance to element projection density. It is represented as the element displacement matrix; ω represents the solid isotropic interpolation penalty term, and q represents the SIMP penalty coefficient, which takes the value of 3; This is expressed as the sensitivity of the element stiffness matrix to the element projected density. According to the chain rule, this term is derived from the sensitivity of the SIMP penalty term. and DMO stiffness interpolation sensitivity Multiply to obtain; This represents the sensitivity of the total structural cost to the element projection density; j represents the j-th element in the element column matrix. Let represent the sensitivity of the stress constraint function of the i-th material conforming to the von Mises yield criterion to the projected density of the subdomain elements of this material group. This term is derived from the sensitivity of the von Mises stress constraint function to the approximately maximum von Mises stress after condensation correction, according to the chain rule. Sensitivity of the approximate maximum von Mises stress after condensation correction to the element von Mises stress Sensitivity of element von Mises stress to Cauchy stress tensor and the sensitivity of the Cauchy stress tensor to element projected density Obtained by multiplication; Represented as the Cauchy stress tensor, it has 9 components in a three-dimensional element; L e This is represented as a global-local coordinate transformation matrix; for the cross-sensitivity of the approximate maximum von Mises stress to the projected density of other material subdomains after condensation correction, only the sensitivity of the Cauchy stress tensor to the element projected density is changed. change; Let represent the sensitivity of the stress constraint function of the m-th material conforming to the Drucker-Prager strength criterion to the projected density of the subdomain elements of this material group. This term is derived from the sensitivity of the Drucker-Prager stress constraint function to the element stress according to the Drucker-Prager criterion, based on the chain rule. Sensitivity of element Drucker-Prager criterion stress to Cauchy stress tensor and the sensitivity of the Cauchy stress tensor to element projected density The result is obtained by multiplying the products together.

5. An analytical apparatus employing the improved DMO topology optimization method according to any one of claims 1-4, which conforms to different material yield criteria, characterized in that, include: Input module: Used to input initial condition parameters for topology optimization, including the number of elements in the x-direction (nelx), y-direction (nely), z-direction (nelz), elastic modulus E of candidate materials, cost coefficient γ of each candidate material, and total structural cost constraint g. c,max The filter radius R and the yield strength σ of the material that obeys the von Mises yield criterion. i,max Uniaxial tensile strength f of materials that obey the Drucker-Prager strength criterion t and uniaxial compressive strength f c ; The calculation module is used to perform finite element analysis and stress analysis on the structure, calculate the nodal stiffness matrix, nodal displacement matrix, and element stress matrix, make the design domain of discrete element stress continuous through qp relaxation, and aggregate discrete stress constraints into global constraints through the p-norm scheme. It also eliminates numerical problems such as checkerboard and grayscale phenomena through element density filtering and projection strategies, calculates the sensitivity of the objective function to element density, and the sensitivity of the cost constraint function and stress constraint function to design variables, and uses the MMA algorithm to iterate to obtain an approximate optimal solution. Output module: Used to iteratively update the element density and solution steps until the new topology optimization structure reaches the preset convergence condition, and output the three-dimensional structure diagram, iteration curve and normalized stress diagram after topology optimization.

Images