Size control method in topological optimization and related device

Abstract

The invention discloses a size control method in topological optimization and a related device, and the method comprises the steps: firstly building a projection density field of a to-be-optimized engineering structure, then carrying out the size punishment and density punishment of the projection density field of the to-be-optimized engineering structure, obtaining a pseudo-density variable after the size and density punishment, and carrying out the size punishment and density punishment. And further performing finite element analysis, calculating response values of the target function and the constraint, and performing sensitivity analysis on the response values. And performing iterative optimization based on a sensitivity analysis result to obtain an optimization result. According to the method, an area which does not meet the size requirement is punished, so that the cost performance of materials in the area which violates the size requirement is low due to reduction of the material performance or increase of the material consumption, and structural forms which violate the size requirement are gradually eliminated in the iterative optimization process; therefore, finite element analysis on the structural configuration which does not meet the size requirement is avoided, the time consumption of the topological optimization process is reduced, and the topological optimization efficiency is improved.

Description

Technical Field

[0001] The present application relates to the technical field of engineering structure topology optimization, and in particular to a dimension control method and related devices in topology optimization. Background Art

[0002] Topology optimization is one of the important methods for lightweight structural design in engineering fields such as aerospace. Topology optimization based on density method is widely used in engineering structure design.

[0003] Currently, many engineering structure designs are limited by factors such as space and process, and therefore require a certain size range. Topology optimization dimensional control based on the density method provides effective technical support for lightweight design of engineering structures.

[0004] However, one related art dimensional control method applies minimum dimensional control after optimizing the configuration, which has limited ability to change the topology. Another related art implements dimensional control in the form of constraints, but the determination of dimensional control constraints does not require finite element analysis. This means that even when it is known that dimensional constraints are not met, time-consuming finite element analysis is still required, increasing the time cost of topology optimization. Summary of the Invention

[0005] In view of the above problems, this application provides a dimension control method and related device in topology optimization to solve at least some of the above problems. The specific solution is as follows:

[0006] A first aspect of the present application provides a size control method in topology optimization, comprising:

[0007] Establish the projection density field of the engineering structure to be optimized;

[0008] Perform size penalty and density penalty on the projected density field of the engineering structure to be optimized, and obtain a pseudo-density variable after size and density penalty;

[0009] Finite element analysis is performed based on pseudo-density variables after size and density penalties, and response values ​​of the objective function and constraints are calculated. A sensitivity analysis is performed on the response values, and based on the sensitivity analysis results, a topology optimization algorithm is used to iteratively optimize the design variable field of the optimization problem corresponding to the engineering structure to be optimized to obtain an optimization result of the engineering structure to be optimized.

[0010] In a possible implementation, performing size penalty control and density penalty on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after size and density penalties includes:

[0011] Performing entity size control on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after entity size penalty;

[0012] Density penalty is performed on the pseudo-density variable after the entity size penalty to obtain an elastic modulus corresponding to the engineering structure to be optimized, and the elastic modulus is obtained according to the pseudo-density variable after the entity size penalty and the density penalty.

[0013] In a possible implementation, performing physical size control on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after physical size penalty includes:

[0014] For the entity structure in the projected density field of the engineering structure to be optimized, the following formula is used to perform the minimum size penalty to obtain the pseudo-density variable after the minimum entity penalty:

[0015]

[0016] Among them, x Om represents the pseudo density variable obtained after implementing the minimum size penalty for the entity, x represents the projected density field, Y O,m Indicates the minimum size opening operation. Indicates that the expansion operation is performed on the search area with the minimum control size r1 of the entity. Indicates that the erosion operation is performed on the search area of ​​r1.

[0017] In a possible implementation, performing physical size control on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after physical size penalty includes:

[0018] For the entity structure in the projected density field of the engineering structure to be optimized, the following formula is used to perform the maximum size penalty to obtain the pseudo-density variable after the maximum entity penalty:

[0019]

[0020] Among them, x OM represents the result of a series of morphological operations with maximum size penalty for the entity, x represents the projected density field, Indicates that the circular search area with radius R2 is used for expansion operation. Indicates that an erosion operation is performed on a circular search area with a radius of R1, where R1 is the maximum control size of the entity and R1 is greater than R2.

[0021] In a possible implementation, performing density penalty on the pseudo-density variable after the entity size penalty to obtain the elastic modulus corresponding to the engineering structure to be optimized includes:

[0022] The maximum size penalty and the minimum size penalty are applied to the entities in the projected density field of the engineering structure to be optimized, and the unified expression of the elastic modulus of the unit after the density penalty is applied is as follows:

[0023]

[0024] Among them, E i represents the elastic modulus of the i-th unit of the engineering structure to be optimized, Represents the pseudo-density variable corresponding to the i-th unit, x Om,i It represents the pseudo density variable corresponding to the i-th unit after the minimum entity size penalty is applied to the projected density field. It represents the pseudo density variable of the i-th unit after the maximum entity size penalty is applied to the projected density field; pr is the density penalty coefficient (i.e., p above), pm is the minimum size penalty coefficient, pM is the maximum size penalty coefficient, pm and pM are both less than pr, and pMa is the parameter of the maximum size penalty, which is used to adjust the degree of its impact on the density penalty; E0 represents the elastic modulus when the material is full, E min Represents the minimum elastic modulus of the material.

[0025] In a possible implementation, performing size penalty control and density penalty on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after size and density penalties includes:

[0026] Performing hole size control on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after hole size penalty;

[0027] Density penalty is performed on the pseudo-density variable after the hole size penalty to obtain the material volume corresponding to each unit of the engineering structure to be optimized. The material volume of any unit is obtained according to the pseudo-density variable and unit volume after the size and density penalty corresponding to any unit.

[0028] In a possible implementation, performing physical size control on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after physical size penalty includes:

[0029] For the hole structure in the projected density field of the engineering structure to be optimized, the following formula is used to perform the minimum size penalty to obtain the pseudo-density variable after the minimum hole penalty:

[0030]

[0031] Among them, x Cm represents the pseudo-density variable obtained after implementing the minimum size penalty for holes, represents the projected density field, Indicates that the circular search area with radius r1 is expanded. Indicates that the erosion operation is performed in a circular search area with a radius of r1, where r1 is the minimum control radius of the hole.

[0032] In a possible implementation, performing physical size control on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after physical size penalty includes:

[0033] For the hole structure in the projected density field of the engineering structure to be optimized, the following formula is used to perform the maximum size penalty to obtain the pseudo-density variable after the maximum hole penalty:

[0034]

[0035] Among them, x CM Represents the value of the projected density field after the maximum hole size penalty series morphological operation is implemented, represents the projected density field, Indicates that the erosion operation is performed in a circular search area with a radius of R2. Indicates that a dilation operation is performed on a circular search area with a radius of R1, where R1 is the maximum controllable size of the hole and R1 is larger than R2.

[0036] In a possible implementation, performing density penalty on the pseudo-density variable after the hole size penalty to obtain the material volume corresponding to each unit of the engineering structure to be optimized includes:

[0037] After applying density penalty and maximum and minimum size penalties to the holes in the engineering structure to be optimized, the material volume of each unit is obtained as follows:

[0038]

[0039] Among them, v i represents the material volume of the i-th unit after the hole size penalty is applied to the projected density field, represents the volume of the i-th unit, represents the pseudo-density variable corresponding to the i-th unit; x Cm,i It represents the pseudo density variable corresponding to the i-th unit after the minimum hole size penalty is applied to the projected density field, (1-(1-x CM,i ) pMa ) pM It represents the pseudo-density variable corresponding to the i-th unit after the maximum hole size penalty is applied to the projected density field; pr is the density penalty coefficient, pm is the minimum size penalty coefficient, pM is the maximum size penalty coefficient, and pMa is the parameter of the maximum size penalty.

[0040] In a possible implementation, performing a sensitivity analysis on the response value includes:

[0041] The derivative of the response value with respect to the design variable of the optimization problem of the engineering structure to be optimized is calculated, wherein the sensitivity of the response value after entity size and density penalty to the projected density field is calculated according to the following formula:

[0042]

[0043] in,

[0044]

[0045] Where f represents the response value after entity size and density penalty, represents the projected density field, x Om,i It represents the pseudo density variable corresponding to the i-th unit after the minimum entity size penalty is applied to the projected density field, x OM,i It represents the value corresponding to the i-th unit after the maximum entity size penalty series morphological operation on the projected density field, I is the unit matrix, X represents the input vector, G di (X) represents the derivative of the expansion operation, G er (X) represents the derivation of the corrosion operation, diag() represents the arrangement of elements along the diagonal to form a square matrix, represents the transposed matrix of the weight matrix, α is the aggregation parameter; r1 represents the minimum control size, R1 represents the maximum control size, and R2<R1.

[0046] In a possible implementation, performing a sensitivity analysis on the response value includes:

[0047] The sensitivity of the response value after hole size and density penalty to the projected density field is calculated according to the following formula:

[0048]

[0049] in,

[0050]

[0051] Where f represents the response value after entity size and density penalty, represents the projected density field, x Cm,i It represents the pseudo density variable corresponding to the i-th unit after the minimum hole size penalty is applied to the projected density field, x CM,i It represents the value corresponding to the i-th unit after the maximum hole size penalty series morphological operation on the projected density field, I is the unit matrix, X represents the input vector, G di (X) represents the derivative of the expansion operation, G er(X) represents the derivation of the corrosion operation, diag() represents the arrangement of elements along the diagonal to form a square matrix, represents the transposed matrix of the weight matrix, α is the aggregation parameter; r1 represents the minimum control size, R1 represents the maximum control size, R2<R1

[0052] A second aspect of the present application provides a size control device in topology optimization, comprising:

[0053] A projection density field establishment module is used to establish the projection density field of the engineering structure to be optimized;

[0054] The penalty module is used to perform size penalty control and density penalty on the projected density field of the engineering structure to be optimized, and obtain the pseudo-density variable after size and density penalty;

[0055] The iterative optimization module is used to perform finite element analysis to calculate the response values ​​of the objective function and constraints based on the pseudo-density variables after size and density penalties, and to perform sensitivity analysis on the response values. According to the sensitivity analysis results, a topology optimization algorithm is used to iteratively optimize the design variable field of the optimization problem corresponding to the engineering structure to be optimized to obtain the optimization result of the engineering structure to be optimized.

[0056] A third aspect of the present application provides an electronic device, comprising at least one processor and a memory connected to the processor, wherein:

[0057] The memory is used to store computer programs;

[0058] The processor is used to execute the computer program so that the electronic device can implement the size control method in topology optimization described in any one of the first aspects.

[0059] The fourth aspect of the present application provides a computer storage medium, characterized in that the storage medium carries one or more computer programs, and when the one or more computer programs are executed by an electronic device, the electronic device can implement the dimension control method in topology optimization as described in any one of the first aspects.

[0060] In a fifth aspect, the present application provides a computer program product comprising computer-readable instructions. When the computer-readable instructions are executed on an electronic device, the electronic device implements the dimension control method in topology optimization of the first aspect or any implementation of the first aspect.

[0061] The size control method in topology optimization provided by the present application first establishes a projected density field of the engineering structure to be optimized, and then performs size control on the projected density field of the engineering structure to be optimized, which may include a maximum / minimum entity size penalty to obtain a pseudo-density variable after the entity size penalty, and then performs a density penalty to obtain the elastic modulus of the unit, assembles the stiffness matrix based on the elastic modulus, and then calculates the objective function and related constraints based on the stiffness matrix. In addition, the maximum / minimum void size penalty can be performed on the projected density field to obtain the material volume corresponding to each unit after the void size penalty. A sensitivity analysis is then performed, and the design variable field is iteratively optimized through the topology optimization algorithm so that the target properties of the engineering structure are optimized. This method starts size control at the beginning of topology optimization, thereby avoiding the problem of difficulty in setting parameters when starting size control in the middle. At the same time, this method penalizes areas that do not meet the size requirements, causing their material properties to be reduced (when the entity size is controlled) or the material usage to increase (when the hole size is controlled), resulting in a lower cost-effectiveness of the materials in the areas that violate the size requirements. Structural forms that violate the size requirements are gradually eliminated during the iterative optimization process, thereby avoiding finite element analysis of structural configurations that do not meet the size requirements, reducing the time consumption of the topology optimization process, and improving the efficiency of topology optimization. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] The above and other features, advantages, and aspects of the various embodiments of the present disclosure will become more apparent with reference to the following detailed description in conjunction with the accompanying drawings. Throughout the drawings, the same or similar reference numerals represent the same or similar elements. It should be understood that the drawings are schematic and that the originals and elements are not necessarily drawn to scale.

[0063] Figure 1 A flowchart of a size control method in topology optimization provided in this application;

[0064] Figure 2 A schematic diagram of the search area in a topology optimization process provided in this application;

[0065] Figure 3 A schematic diagram showing a comparison before and after the minimum size penalty for an entity provided in this application;

[0066] Figure 4 A schematic diagram showing a comparison before and after the maximum size penalty for an entity provided in this application;

[0067] Figure 5 A schematic diagram showing a comparison before and after the minimum size penalty for holes provided in this application;

[0068] Figure 6 A schematic diagram showing a comparison before and after the maximum size penalty for holes provided in this application;

[0069] Figure 7 A schematic diagram of boundary conditions before topology optimization of an engineering structure provided in this application;

[0070] Figure 8 for Figure 7 Schematic comparison of the engineering structure shown after different topology optimizations;

[0071] Figure 9 Schematic diagram of boundary conditions before topology optimization of another engineering structure provided in this application;

[0072] Figure 10 for Figure 9 Schematic comparison of the engineering structure shown after different topology optimizations;

[0073] Figure 11 A schematic diagram of the structure of a size control device in topology optimization provided in this application;

[0074] Figure 12 This is a schematic diagram of the structure of an electronic device provided in this application. DETAILED DESCRIPTION

[0075] The following describes the embodiments of the present application in conjunction with the accompanying drawings. The terms used in the implementation methods of the present application are only used to explain the specific embodiments of the present application and are not intended to limit the present application.

[0076] The embodiments of the present application are described below in conjunction with the accompanying drawings. Those skilled in the art will appreciate that, with the development of technology and the emergence of new scenarios, the technical solutions provided in the embodiments of the present application are also applicable to similar technical problems.

[0077] The terms "first", "second", etc. in the specification and claims of the present application and the above-mentioned drawings are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequential order. It should be understood that the terms used in this way can be interchangeable under appropriate circumstances, and this is merely a way of distinguishing the objects of the same attributes when describing them in the embodiments of the present application. In addition, the terms "including" and "having" and any of their variations are intended to cover non-exclusive inclusions, so that the process, method, system, product or equipment comprising a series of units need not be limited to those units, but may include other units that are not clearly listed or inherent to these processes, methods, products or equipment.

[0078] Before introducing the solutions provided by the embodiments of the present application in detail, a brief introduction to the relevant technical terms is given first:

[0079] Topology optimization is a structural optimization method that aims to achieve specific design goals by adjusting the material distribution within a design area. The basic idea is to remove or retain material within a given design space so that the structure achieves optimal performance while satisfying all constraints (such as mass and volume). Common structural responses include mass, volume, strain energy, and nodal displacements. Strain energy and nodal displacement responses are often used as objective functions in optimization problems, while mass and volume are often used as constraints.

[0080] In order to solve the problems existing in the size control method in the topology optimization of the related art, the present application provides a size control method in the topology optimization based on the density method, which realizes size control in a penalty manner, penalizes the areas that do not meet the size requirements, reduces the material performance or increases the material usage, resulting in a low cost performance of the materials in the areas that violate the size requirements. In the iterative optimization process, the structural forms that violate the size requirements are gradually eliminated, thereby obtaining a topology optimization result that meets the size requirements. For example, the corresponding structure can be selected for size control according to the actual needs of the engineering structure. For example, size control can be performed only on the solid structure, or only on the hole structure, or both the solid structure and the hole structure can be separately controlled. In addition, size control includes maximum size control and minimum size control, that is, size control includes maximum entity size control, minimum entity size control, maximum hole size control, and minimum hole size control. At least one of them can be selected for size control according to the actual optimization needs of the engineering structure.

[0081] The following will be combined Figure 1 The size control method in topology optimization provided by the embodiment of the present application is described in detail. Figure 1 As shown, the method may include the following steps:

[0082] S101, establishing a projection density field of the engineering structure to be optimized.

[0083] The engineering structure to be optimized refers to a functional structure that needs to be structurally optimized, and can be any engineering structure. This application does not impose any special restrictions on this.

[0084] In topology optimization, finite element analysis is often used to iteratively optimize engineering structures. Finite element analysis is to divide the engineering structure to be optimized into finite element meshes to obtain several units. Each unit in a mesh represents a small volume or area.

[0085] This embodiment is based on the topology optimization three-field method framework of the density method. The topology optimization three-field method includes the design variable field X, the filter density field and the projected density field (which can be simply referred to as the projection field).

[0086] Set the number of design variables equal to the number of grid elements and assign initial values ​​to obtain the initial value of the structural optimization problem, that is, the design variable field.

[0087] For example, the design variable field is denoted as x, where x=[x1,x2,...,x i ,...,x n ] T , x i represents the i-th design variable.

[0088] The design variable field x is filtered by density to obtain the filtered density field It is calculated as follows:

[0089]

[0090] Among them, x in the formula is the design variable field, W flt represents the filter matrix, It's W flt The element in row i and column j in Represents the volume of the i-th unit in the engineering structure to be optimized, ω i,j represents the weight of the jth unit to the ith unit, r flt Indicates the pre-set filter radius, ||loc i -loc j ||2 represents the center distance between the jth unit and the ith unit. loc i Represents the center position of the i-th unit.

[0091] Filter density field The projection density field is obtained by projection function processing For example, the Heaviside projection function can be used to filter the density field Processing to obtain the projected density field In topology optimization, the projected density field Usually expressed as: Projected density field Each value in represents the pseudo-density variable of the corresponding cell, for example, It represents the pseudo-density variable corresponding to the i-th unit, and its calculation formula can be expressed as:

[0092]

[0093] The parameter β in formula 2 represents the steepness of the projection function. In the three-field method for topology optimization, η is usually set to 0.5. The calculation formula can be simplified as:

[0094] S102, performing entity size penalty control on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after entity size penalty.

[0095] In the embodiment of the present application, a minimum entity size penalty and a maximum entity size penalty are respectively performed on the projected density field of the engineering structure to be optimized.

[0096] The size penalty method can be implemented using dilation, erosion, opening, and closing operations based on mathematical morphology. Its essence is to perform maximum value operations (such as minimum value operation min() and maximum value operation max()) in the local area of ​​the binary image.

[0097] A dilation operation dilates an object in an original image, increasing its area and thickness. This operation involves selecting a structuring element and sliding it across the image. This operation results in expanding the target area, filling holes, or connecting adjacent objects.

[0098] Erosion involves "eroding" an object in the original image, removing surrounding noise and small details. Similar to dilation, it involves selecting a structuring element and sliding it across the image. This operation is used to shrink the target area, remove small noise, or separate stuck objects.

[0099] The opening operation is a fundamental operation in mathematical morphology. It smoothes object boundaries and removes small particles of noise by first eroding and then dilating an image. This operation is used to smooth structural contours, remove small protrusions and isolated noise, and preserve the main shape.

[0100] The closing operation first dilates the original image and then erodes it to fill holes, smooth edges, and connect them. This operation is used to fill small holes, connect adjacent areas, and smooth boundaries.

[0101] For density variables that change continuously in the closed interval [0,1], the average form of the aggregation function KS (Kreisselmeier–Steinhauser) can be used to approximate the maximum value function. For example, the expression for the maximum value in the local area after using this method is as follows:

[0102]

[0103] Among them, x k represents the pseudo-density variable of the k-th unit, represents the unit volume of the kth unit, y i represents the maximum value after condensation corresponding to the i-th unit, Ω irepresents the search area of ​​the i-th unit set by the size control, α is the aggregation parameter, and usually takes a large value, such as 500, when there is no numerical problem.

[0104] The expression for calculating the local minimum value using the aggregation function is as follows:

[0105]

[0106] y in Equation 4 i represents the minimum value after aggregation corresponding to the i-th unit. Other parameters are the same as those in Formula 3 and are not repeated here.

[0107] Among them, the expansion operation is equivalent to the operation of taking the maximum value, and the erosion operation is equivalent to the operation of taking the minimum value. The calculation formula of the expansion operation and the erosion operation can be obtained by taking the maximum value of the above-mentioned aggregation function. In order to simplify the expression, the expansion operation (Y di (X)) and erosion operation (Y er (X)) can be represented by the following matrix:

[0108]

[0109] The opening operation can be expressed as Y O (X) = Y di (Y er (X)), the closing operation can be expressed as Y C (X) = Y er (Y di (X)).

[0110] Where X is the independent variable and Y is the output vector. Both X and Y are n×1 column vectors, n is the number of pseudo-density variables, and W LC is the corresponding weight matrix organized by size control region, which is n×n, where Represents the element in row i and column j. i,j The expression is as follows:

[0111]

[0112] Among them, Ω i is the local search area, which can be circular, such as Figure 2 The circular area 10 in the figure represents the search area, and the squares 11 represent the units into which the engineering structure is divided. The radius of the search area and half of the maximum / minimum size to be controlled are shown. Of course, in other embodiments, the search area can also be other shapes, such as an ellipse, a rectangle, etc. The embodiments of this application are described using a circular search area as an example.

[0113] The minimum size penalty (or simply minimum entity penalty) for entities in the projected density field is to perform an opening operation on the circular search area of ​​the projected density field with the minimum radius r1, which can weaken the performance of units at small structures in the projected density field. The minimum entity penalty for the projected density field can be expressed as follows:

[0114]

[0115] Among them, x Om represents the pseudo density variable obtained after implementing the minimum size penalty for the entity, x Om The subscript O indicates the processing of entities, and the subscript m indicates the minimum size control. For example, Figure 3 This is a one-dimensional schematic diagram of the minimum entity penalty process, where the minimum size control radius r1 = 0.5. Curve 1 represents the projected density field, with the horizontal axis representing the position (or unit) in space and the vertical axis representing the pseudo-density variable corresponding to each position (or unit). Curve 2 shows the change in pseudo-density for each unit after the entity is eroded. The dashed line in Curve 2 represents the original projected density field for each unit, and the solid line represents the projected density field after the erosion operation. Curve 3 shows the projected density field after the entity is eroded and then dilated. The dashed line in Curve 3 represents the original projected density field for each unit, and the solid line represents the projected density field after the minimum size penalty is applied to the entity. Comparing the dashed and solid lines in Curve 3 shows that after the minimum size penalty is applied to the entities, the larger entities remain unchanged, while the smaller entities experience a reduction in material. This means that the smaller entities provide weaker performance with the same material, resulting in a low cost-effectiveness.

[0116] The maximum size penalty (also referred to as the maximum entity penalty) is implemented for entities in the projected density field. This is done by dilating the projected density field with a circular search area of ​​smaller radius R2, then eroding the circular search area of ​​maximum radius R1, and then dilating the circular search area of ​​smaller radius R2. In this case, R2 is equivalent to an implicit parameter for local porosity. The corresponding calculation process can be expressed as follows:

[0117]

[0118] Among them, x OM The subscript O indicates the processing of entities, and the subscript M indicates the maximum size control.

[0119] Then, x OM Attenuation is performed according to the power law (controlled by the parameter pMa), and the maximum entity size penalty term is obtained by subtracting the attenuated value from 1, that is, (1-x OM,i pMa ), Represents the pseudo-density variable corresponding to the i-th unit after the maximum entity size penalty is applied to the projected density field. Among them, pMa is the parameter of the maximum size penalty, which is used to adjust its penalty degree for intermediate density.

[0120] In addition, it should be noted that the minimum size penalty term can be regarded as x OM 、x CM With projection field The element-by-element ratio of The presence of 0 in will cause numerical problems, so the penalty term for the minimum entity size is not listed separately.

[0121] When the value of R2 is very small, the area after the corrosion operation is often a line or a point (a row or a unit). The intuitive visual effect of penalizing this area is "ditching" or "punching". If the minimum hole penalty control is also applied at this time, the small gaps created by "ditching" or "punching" will trigger the minimum hole penalty, which will cause a penalty conflict in the optimization problem. To avoid such conflicts, R2>r0 can be set, where r0 represents the minimum size radius of the hole penalty.

[0122] For example, Figure 4 This is a one-dimensional diagram of the maximum entity penalty process, with a maximum size control radius of R1 = 1.5 and a smaller radius parameter of R2 = 0.2. Curve 1 in this figure represents the projected density field, curve 2 represents the projected density field after the entity is dilated with a circular search area of ​​radius R2, curve 3 represents the projected density field after the dilation operation of curve 2 is eroded with a circular search area of ​​radius R1, curve 4 represents the projected density field after the erosion operation of curve 3 is dilated with a circular search area of ​​radius R2, and curve 5 represents the corresponding projected density field after the maximum entity size penalty is performed, i.e. The corresponding curve graph. Figure 4 The meanings of the solid and dashed lines in each curve are the same as Figure 3 The same, no further description here.

[0123] Depend on Figure 4 It can be seen that after the maximum size penalty is applied to the entities, the smaller entities do not change, while holes appear in the middle of the larger entities, which means that there is material in the middle of the larger entities but no performance is provided.

[0124] S103, performing density penalty on the pseudo-density variables after the entity size penalty, to obtain an assembly stiffness matrix of a finite element model corresponding to the engineering structure to be optimized.

[0125] The size penalty and density penalty for the entity are essentially penalties for the elastic modulus.

[0126] Density penalty refers to a method that optimizes structural performance by adjusting material distribution through the introduction of a penalty factor. Common density penalty methods include the Solid Isotropic Material Penalty Model (SIMP) and the Material Property Interpolation Model (RAMP). These methods optimize material distribution by defining a penalty factor, p, that correlates element density with the material's elastic modulus or other unquantified properties. The penalty factor, p, gradually eliminates low-density materials during the optimization process while retaining high-density materials, thereby achieving the goal of optimizing the structure.

[0127] Taking the SIMP method as an example, the penalty factor p assigns a very small elastic modulus to elements with a density close to zero during the optimization process. This makes these elements more likely to deform when subjected to stress, and they are therefore considered invalid and eliminated. Conversely, the material in high-density areas maintains a higher elastic modulus and is retained after optimization.

[0128] The elastic modulus refers to the ratio of stress to strain in a material during its elastic deformation phase, and is used to measure a material's ability to resist elastic deformation. The larger the elastic modulus, the stiffer the material, and the smaller the elastic deformation under the same stress.

[0129] The expression for imposing only SIMP density penalty without imposing entity size penalty is as follows:

[0130]

[0131] Where p is the density penalty coefficient (i.e., density penalty factor), usually p=3, E0 represents the elastic modulus when the material is full (without any penalty), E min Represents the minimum elastic modulus of the material, usually set to a very small value, such as (1e-9)*E0.

[0132] In the embodiment of the present application, the unified expression of the elastic modulus after applying the maximum and minimum entity size penalties on the basis of the SIMP density penalty is as follows:

[0133]

[0134] Where pr is the density penalty coefficient (i.e., p above), pm is the minimum size penalty coefficient, and pM is the maximum size penalty coefficient. Both pm and pM are smaller than pr. pMa is the parameter of the maximum size penalty, which is used to adjust the degree of its impact on the density penalty. Represents the pseudo density variable corresponding to the i-th unit. Om,i It represents the pseudo density variable corresponding to the minimum entity size penalty of the i-th unit, Represents the pseudo-density variable corresponding to the maximum entity size penalty of the i-th unit.

[0135] Usually pr=pm=pM=3, pMa=3, and finally the following expression is obtained:

[0136] E i =x Om,i 3 (1-x OM,i 3 ) 3 (E0-E min )+E min (11)

[0137] Then, the elastic modulus after size and density penalty is used to assemble the stiffness matrix corresponding to the finite element model. For example, the unit stiffness matrix can be calculated according to the unit type (such as three-dimensional tetrahedron, hexahedron, two-dimensional triangle, quadrilateral, etc.), and the corresponding elements are superimposed according to the node degree of freedom number to form the overall stiffness matrix K of the structure, whose unit is N / m.

[0138] In topology optimization, the elastic modulus is a key parameter in the stiffness matrix. Specifically, the elements in the stiffness matrix reflect the stiffness characteristics of the structure in different directions, and these stiffness characteristics are directly affected by the elastic modulus of the material. Therefore, changes in the elastic modulus will lead to changes in the stiffness matrix, which in turn affects the overall stiffness of the structure.

[0139] S104, performing a hole size penalty on the projected density field of the engineering structure to be optimized, and obtaining a volume after the hole size penalty.

[0140] Furthermore, a hole size penalty, such as a minimum hole size penalty and a maximum hole size penalty, can be applied to the projected density field to limit the size of the holes. The purpose of the minimum hole size penalty is to avoid generating holes that are too small, so that the generated holes are not too small, avoiding manufacturing difficulties or performance problems caused by too small a size. The purpose of the maximum hole size penalty is to prevent the generation of holes that are too large, and a maximum hole size penalty can be set. This helps to control the hole size in the design area so that the optimization results have reasonable structure and performance in practical applications. Among them, the hole size penalty will affect the material volume, that is, the essence of the hole size penalty is to penalize the material volume.

[0141] In an exemplary embodiment, the minimum hole size penalty is to close the circular search area of ​​the projected density field with a minimum radius r1, so that the small holes in the projected density field can occupy more material. Its expression is as follows:

[0142]

[0143] Among them, x Cmrepresents the pseudo-density variable after applying the minimum hole size penalty, where x Cm The subscript C indicates the treatment of holes, and the subscript m indicates the minimum size control. Indicates that the erosion operation is performed in a circular search area with a radius of Indicates that a dilation operation is performed on a circular search area with a radius of r1.

[0144] For example, Figure 5 A one-dimensional diagram of the minimum size penalty process for holes in engineering structures. In this example, the minimum size control radius r1 = 0.5. Curve 1 represents the projected density field, Curve 2 represents the projected density field after erosion with the minimum size control radius r1, and Curve 3 represents the projected density field after dilation of Curve 2.

[0145] Depend on Figure 5 It can be seen that after the minimum size penalty is implemented for holes, the larger holes do not change, while the material usage of smaller holes increases, which means that smaller holes take up more material and have a low cost-effectiveness.

[0146] The maximum hole size penalty is to perform an erosion operation on a circular search area with a smaller radius R2 in the projected density field, then perform an expansion operation on a circular search area with a maximum radius R1, and then perform an erosion operation on a circular area with a smaller radius R2. This allows larger holes in the projected density field to occupy more material. In this case, R2 is equivalent to an implicit parameter of the local solidity rate. The corresponding calculation process is as follows and its expression is as follows:

[0147]

[0148] Among them, x CM represents the result of applying a series of morphological operations with a maximum hole size penalty, where x CM The subscript C indicates the treatment of holes, and the subscript M indicates the maximum size control. Indicates that the erosion operation is performed in a circular search area with a radius of R2. Indicates that a dilation operation is performed on a circular search area with a radius R1, where R2 is smaller than R1.

[0149] For example, Figure 6This is a one-dimensional diagram of the maximum size penalty process for holes in engineering structures. The maximum size control radius R1 = 1.5, and the smaller radius parameter R2 = 0.2. In this figure, curve 1 projects the density field, curve 2 represents the projected density field after the hole is eroded with a radius of R2, curve 3 represents the projected density field after the result of the erosion operation (i.e., the processing result shown in curve 2) is expanded with a radius of R1, curve 4 represents the projected density field after the result of the expansion operation (i.e., the processing result shown in curve 3) is eroded with a radius of R2, and curve 5 represents the corresponding projected density field after the maximum hole size penalty is applied, i.e. Corresponding effect.

[0150] Depend on Figure 6 It can be seen that after the maximum size penalty is applied to the holes in the engineering structure, the smaller holes do not change, while the material usage of the larger holes increases, which means that larger holes will take up more material and have a low cost-effectiveness.

[0151] The hole size penalty affects the volume of the structure, which is usually used as a constraint, i.e., a volume constraint. The unified expression for the volume after applying the SIMP density penalty and the maximum and minimum hole size penalties is as follows:

[0152]

[0153] Among them, v i represents the material volume of the i-th unit after the hole size penalty, Represents the volume of the i-th unit. pr is the density penalty coefficient, pm is the minimum size penalty coefficient, pM is the maximum size penalty coefficient, and pMa is the parameter of the maximum size penalty, which is used to adjust the degree of its impact on the density penalty. Cm,i Represents the pseudo-density variable corresponding to the i-th unit after the minimum hole size penalty is applied to the projected density field, Represents the pseudo-density variable corresponding to the i-th unit 2 after the maximum hole size penalty is applied to the projected density field.

[0154] In topology optimization, constraints are usually expressed in terms of linear volumes, i.e., pr = pm = pM = 1, pMa = 3. The final expression is as follows:

[0155]

[0156] Furthermore, you can use v i Calculate the volume constraint of the topology optimization problem, that is, for all v i Perform the summation.

[0157] It should be noted that during the topology optimization process of the same engineering structure, it may be necessary to perform both maximum / minimum solid size control and maximum / minimum hole size control, that is, to control all four dimensions simultaneously, or it may be necessary to perform only at least one of the four dimensions. In other words, it may be necessary to perform both S102 and S104, or only one of them.

[0158] S105 , after calculating the objective function and the response values ​​of the constraints and performing sensitivity analysis, the design variable field is iteratively optimized using a topology optimization algorithm to obtain an optimized result of the engineering structure.

[0159] The objective function is a function corresponding to the target attribute. For example, the target attribute includes but is not limited to at least one of the following: overall stiffness, heat transfer capacity, deformation capacity, stress level, etc.

[0160] For example, taking the target attribute being the overall stiffness (i.e., overall compliance) of an engineering structure as an example, the overall stiffness of the engineering structure can be represented by compliance. The lower the compliance of the engineering structure, the higher the stiffness of the structure, and the higher the compliance, the lower the stiffness of the structure.

[0161] For example, the optimization formula for the corresponding topology optimization problem with compliance as the goal is as follows:

[0162]

[0163] 0≤x i ≤1

[0164] The objective function is: C = F T u, the goal is to minimize the overall compliance C of the engineering structure. Where C represents the overall compliance of the engineering structure, u is the displacement vector of each unit of the engineering structure under the action of the above load, and F represents the load vector of the engineering structure.

[0165] Where K represents the overall stiffness matrix (N / m), the subscript g of K indicates that a minimum entity size penalty or a minimum hole size penalty is applied, and the subscript m indicates that a material property penalty is applied. g represents the material volume after the minimum entity or hole size penalty, v e is the column vector volume of the unit volume, Represents the upper limit of the volume ratio constraint. ||||1 represents the 1 norm, which means the sum of the absolute values ​​of the elements of the column vector, which is the sum() function for volume. ||v g ||1 represents the sum of the material volumes of each unit after penalty, ||v e ||1 represents the sum of the volumes of all units.

[0166] Sensitivity analysis involves optimizing the values ​​of design variables to improve design performance by analyzing their impact on the objective function and constraints. Its core approach is to quantify the sensitivity of design variables to the objective function and constraints. Specifically, by calculating the derivatives of the objective function and constraints with respect to the design variables, it is possible to assess how the objective function and constraints change as the design variables change.

[0167] Sensitivity analysis is to obtain the derivatives of the structural response values ​​(such as compliance, strain energy, node displacement, etc.) with respect to the design variables, that is, to perform chain derivation. In the nested layers of chain derivation, it is necessary to calculate the derivatives of the elastic modulus with respect to the projected density field after size and density penalties, as well as the derivatives of the volume with respect to the projected density field after size and density penalties. The other derivation processes are the same as those in the prior art. The following will focus on the derivation process of the elastic modulus with respect to the projected density field after size and density penalties, as well as the derivation process of the volume with respect to the projected density field after size and density penalties.

[0168] Among them, the chain derivation of the objective function based on the three-field method is as follows:

[0169]

[0170] Among them, f represents the response value, x is the pseudo density variable, is the filtered density field, Is the projected density field. Where diag() means arranging the elements diagonally to form a square matrix, the matrix G H The elements in can also be expressed as follows:

[0171]

[0172] The derivatives of dilation, erosion, opening, and closing operations are as follows:

[0173]

[0174] Where X represents the input vector, G di (X) represents the derivative of the expansion operation, G er (X) represents the derivative of the corrosion operation, represents the transposed matrix of the weight matrix, and α is the aggregation parameter.

[0175] In order to achieve the chain derivation of uniform penalty for size density, we can remember:

[0176]

[0177] You can x Om 、x OM Treated as independent variables, P as a multivariate function Perform chain derivation and write in vector form as follows:

[0178]

[0179] The projection field of each independent variable can be obtained (in, ) is:

[0180]

[0181] Wherein, I in Formula 22 represents the identity matrix.

[0182] The response value f is usually a scalar, where the response value associated with the density penalty can be expressed as a variable f(P) about P, ​​or can be viewed as a variable about ψ(t),Y O,m (t),Y O,M (t), its expression is f(ψ(t),Y O,m (t),Y O,M (t)).

[0183] Among them, the derivative of P with respect to its independent variables, that is, The derivatives with respect to the independent variables are as follows:

[0184]

[0185] Then f(ψ(t),Y O,m (t),Y O,M The derivatives of (t)) with respect to the independent variables of P and the expressions assembled into column vectors are as follows:

[0186]

[0187] Combining Equation 22 with chain derivation, we can obtain the sensitivity of the structural response to the projected density field after the entity size and density penalty:

[0188]

[0189] According to formula 22, And I is the identity matrix, so, In addition, G O,m (t) and G O,M (t) See the expression in Equation 22.

[0190] Similarly, the chain derivation of the maximum and minimum hole size penalties is similar to the derivation of the structural response with respect to the projected density field, as follows:

[0191] In order to achieve the chain derivation of uniform penalty for size density, we can remember:

[0192]

[0193] Will x Cm 、x CM As independent variables, Q as a multivariate function Perform chain derivation and write in vector form as follows:

[0194]

[0195] The projection field of each independent variable can be obtained The derivative of is:

[0196]

[0197] The response value f related to the hole size penalty (i.e., volume penalty) can be viewed as a variable f(Q) about Q, or as a variable about ψ(t), Y C,m (t), Y C,M (t) variable f(ψ(t),Y C,m (t),Y C,M (t)).

[0198] According to formula 26, the derivatives of Q with respect to its independent variables are as follows:

[0199]

[0200] Then f(ψ(t),Y C,m (t),Y C,M The derivatives of (t)) with respect to the independent variables of Q and the form of assembling them into column vectors are as follows:

[0201]

[0202] Combining Equation 28 with chain derivation, we can obtain the sensitivity of the structural response value (i.e., material volume) after the hole size penalty to the projected density field as follows:

[0203]

[0204] Furthermore, the sensitivity analysis process described above can be used to obtain information about the sensitivity of the objective function / volume with respect to the design variables and optimize the design variables. Based on this sensitivity information, the topology optimization algorithm can more effectively adjust the design variables, reducing the number of iterations and improving computational efficiency. After optimizing the design variables, the above steps S101 to S105 are repeated, and finally, after determining that the iterations have converged, the optimal results for the design variables are output.

[0205] In one possible implementation, convergence is determined when the change in the target property after a predetermined number of consecutive iterative optimizations is less than a threshold. Alternatively, convergence can be determined based on changes in the pseudo-density variable of the cell. This is not further detailed here.

[0206] The size control method in topology optimization provided in this embodiment first obtains the projected density field of the engineering structure to be optimized, and then performs size control on the projected density field of the engineering structure to be optimized, which may include the maximum / minimum entity size penalty to obtain the pseudo-density variable after the entity size penalty, and then performs density penalty to obtain the elastic modulus of the unit, assembles the stiffness matrix based on the elastic modulus, and then calculates the objective function and related constraints according to the stiffness matrix. In addition, the maximum / minimum void size penalty can be performed on the projected density field to obtain the material volume corresponding to each unit after the void size penalty. Then, a sensitivity analysis is performed, and the design variable field is iteratively optimized through the topology optimization algorithm so that the target properties of the engineering structure are optimized. This method starts size control at the beginning of topology optimization, thereby avoiding the problem of difficulty in setting parameters when starting size control in the middle. At the same time, this method penalizes areas that do not meet the size requirements, causing their material properties to be reduced (when the entity size is controlled) or the material usage to increase (when the hole size is controlled), resulting in a lower cost-effectiveness of the material in the area that violates the size requirements. Therefore, in the iterative optimization process, the structural forms that violate the size requirements are gradually eliminated, thereby achieving a topology optimization result that meets the size requirements, thereby avoiding finite element analysis of structural configurations that do not meet the size requirements, reducing the time consumption of the topology optimization process, and improving the efficiency of topology optimization.

[0207] Taking the optimization goal of minimizing compliance as an example, the effect of applying size control is analyzed based on a specific example:

[0208] The example uses the International System of Units and performs topology optimization design in an area with a length of 300 and a height of 150. The entire area is discretized into 300*150 quadrilateral elements, and the constraint design domain volume ratio is 0.55. The material properties E=1, Poisson's ratio μ=0.3, E min =1e-9.

[0209] The penalty coefficients pr, pm, pM and parameter pMa are set as described above, that is, pr=pm=pM=3 and parameter pMa=3 are taken in the maximum / minimum entity size penalty; pr=pm=pM=1 and pMa=3 are taken in the maximum / minimum hole size penalty.

[0210] Optimized filter radius r flt , maximum entity control radius R1, minimum entity control radius r1, maximum hole control radius R0, minimum hole control radius r0. For the specific values ​​of these parameters, please refer to the values ​​marked in the optimization result diagram.

[0211] Calculation example 1:

[0212] like Figure 7 The boundary conditions are shown in the figure. A symmetry constraint is applied on the left, a vertical displacement constraint is applied to the lower right corner, and a concentrated force F = 0.5 is applied to the upper left corner. To avoid boundary effects at the boundaries of the search area, the natural grid is expanded and set to holes (i.e., no material).

[0213] in, Figure 8 (1) is the optimization result without size control. Figure 8 (2) is the optimization result after applying the maximum entity control, and the maximum entity control radius R1 = 11.25. Figure 8 (3) is the optimization result of applying maximum and minimum entity control at the same time, where the maximum entity control radius R1 = 11.25 and the minimum entity control radius r1 = 5.4. Figure 8 The black area 31 represents a solid part, and the white area 32 represents a hole.

[0214] Calculation example 2:

[0215] Boundary conditions such as Figure 9 As shown in the figure, a fixed constraint is applied to the lower side and a uniform force is applied to the upper side, with the sum of the forces being F = 1. The top and bottom layers of elements are set as non-design domains.

[0216] Figure 10 (1) is the optimization result without size control, Figure 10 (2) is the optimization result after applying the maximum hole size control, where the maximum hole control radius R0=9. Figure 10 (3) is the optimization result of applying maximum and minimum hole size control at the same time, where the maximum hole control radius R0 = 9 and the minimum hole control radius r0 = 5.4. Figure 10 The black area 41 represents a solid part, and the white area 42 represents a hole.

[0217] The above describes a size control method in topology optimization provided by an embodiment of the present application. The following describes an apparatus for executing the above method.

[0218] See also Figure 11 , Figure 11 This is a schematic diagram of the structure of a size control device in topology optimization provided in an embodiment of the present application. Figure 11 As shown, the device includes:

[0219] A projection density field establishing module 101 is used to establish a projection density field of the engineering structure to be optimized;

[0220] A penalty module 102 is used to perform size penalty control and density penalty on the projected density field of the engineering structure to be optimized, and obtain a pseudo-density variable after size and density penalties;

[0221] The iterative optimization module 103 is used to perform finite element analysis to calculate the response values ​​of the objective function and constraints based on the pseudo-density variables after size and density penalties, perform sensitivity analysis on the response values, and iteratively optimize the design variable field of the optimization problem corresponding to the engineering structure to be optimized using a topology optimization algorithm based on the sensitivity analysis results to obtain an optimization result of the engineering structure to be optimized.

[0222] The above modules are specifically used to execute the corresponding steps in the above method embodiment, which will not be described in detail here.

[0223] An electronic device is also provided in an embodiment of the present application. Figure 12 , which shows a schematic diagram of the structure of an electronic device suitable for implementing the embodiments of the present application. The electronic device in the embodiments of the present application may include but is not limited to fixed terminals such as mobile phones, laptops, PDAs (personal digital assistants), PADs (tablet computers), desktop computers, etc. Figure 12 The electronic device shown is merely an example and should not limit the functions and scope of use of the embodiments of the present application.

[0224] like Figure 12 As shown, the electronic device may include a processing device (e.g., a central processing unit, a graphics processing unit, etc.) 201, which can perform various appropriate actions and processes according to a program stored in a read-only memory (ROM) 202 or a program loaded from a storage device 208 into a random access memory (RAM) 203. When the electronic device is powered on, the RAM 203 also stores various programs and data required for the operation of the electronic device. The processing device 201, the ROM 202, and the RAM 203 are connected to each other via a bus 204. An input / output (I / O) interface 205 is also connected to the bus 204.

[0225] Typically, the following devices may be connected to the I / O interface 205: an input device 206 including, for example, a touch screen, a touchpad, a keyboard, a mouse, a camera, a microphone, an accelerometer, a gyroscope, etc.; an output device 207 including, for example, a liquid crystal display (LCD), a speaker, a vibrator, etc.; a storage device 208 including, for example, a memory card, a hard disk, etc.; and a communication device 209. The communication device 209 may allow the electronic device to communicate with other devices wirelessly or by wire to exchange data. Although Figure 12 The electronic device is shown with various devices, but it should be understood that it is not required to implement or possess all of the devices shown. More or fewer devices may be implemented or possessed instead.

[0226] An embodiment of the present application also provides a computer program product including computer-readable instructions. When the computer-readable instructions are executed on an electronic device, the electronic device implements any of the dimension control methods in topology optimization provided in the embodiments of the present application.

[0227] A computer-readable storage medium is also provided in an embodiment of the present application. The storage medium carries one or more computer programs. When the one or more computer programs are executed by an electronic device, the electronic device can implement any size control method in topology optimization provided in an embodiment of the present application.

[0228] It should also be noted that the device embodiments described above are merely illustrative, wherein the units described as separate components may or may not be physically separate, and the components displayed as units may or may not be physical units, that is, they may be located in one place, or they may be distributed across multiple network units. Some or all of the modules may be selected according to actual needs to achieve the purpose of the present embodiment. In addition, in the drawings of the device embodiments provided in this application, the connection relationship between the modules indicates that there is a communication connection between them, which can be specifically implemented as one or more communication buses or signal lines.

[0229] Through the description of the above embodiments, those skilled in the art can clearly understand that the present application can be implemented by means of software plus necessary general hardware, and of course can also be implemented by special hardware including application-specific integrated circuits, special CPUs, special memories, special components, etc. In general, all functions performed by computer programs can be easily implemented with corresponding hardware, and the specific hardware structures used to implement the same function can also be diverse, such as analog circuits, digital circuits or special circuits, etc. However, for the present application, software program implementation is a better implementation method in most cases. Based on this understanding, the technical solution of the present application is essentially or the part that contributes to the prior art can be embodied in the form of a software product, which is stored in a readable storage medium, such as a computer's floppy disk, USB flash drive, mobile hard disk, ROM, RAM, magnetic disk or optical disk, etc., and includes a number of instructions to enable a computer device (which can be a personal computer, training equipment, or network equipment, etc.) to execute the methods described in each embodiment of the present application.

[0230] In the above embodiments, all or part of the embodiments may be implemented by software, hardware, firmware, or any combination thereof. When implemented by software, all or part of the embodiments may be implemented in the form of a computer program product.

[0231] The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, the process or function described in the embodiment of the present application is generated in whole or in part. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable devices. The computer instructions can be stored in a computer-readable storage medium, or transmitted from one computer-readable storage medium to another computer-readable storage medium. For example, the computer instructions can be transmitted from a website, a computer, a training device or a data center by wired (e.g., coaxial cable, optical fiber, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) mode to another website, a computer, a training device or a data center. The computer-readable storage medium can be any available medium that a computer can store or a data storage device such as a training device, a data center, etc. that includes one or more available media integrations. The available medium can be a magnetic medium, (e.g., a floppy disk, a hard disk, a tape), an optical medium (e.g., a DVD), or a semiconductor medium (e.g., a solid-state drive (SSD)).

Claims

1. A size control method in topology optimization, characterized in that: include: Establish the projection density field of the engineering structure to be optimized; Perform size penalty and density penalty on the projected density field of the engineering structure to be optimized, and obtain a pseudo-density variable after size and density penalty; Finite element analysis is performed based on pseudo-density variables after size and density penalties, and response values ​​of the objective function and constraints are calculated. A sensitivity analysis is performed on the response values, and based on the sensitivity analysis results, a topology optimization algorithm is used to iteratively optimize the design variable field of the optimization problem corresponding to the engineering structure to be optimized to obtain an optimization result of the engineering structure to be optimized.

2. The method according to claim 1, characterized in that The method of performing size penalty control and density penalty on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after size and density penalty includes: Performing entity size control on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after entity size penalty; Density penalty is performed on the pseudo-density variable after the entity size penalty to obtain an elastic modulus corresponding to the engineering structure to be optimized, and the elastic modulus is obtained according to the pseudo-density variable after the entity size penalty and the density penalty.

3. The method according to claim 2, characterized in that The physical size control of the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after physical size penalty includes: For the entity structure in the projected density field of the engineering structure to be optimized, the following formula is used to perform the minimum size penalty to obtain the pseudo-density variable after the minimum entity penalty: Among them, x Om represents the pseudo-density variable obtained after implementing the minimum size penalty for the entity, represents the projected density field, Y O,m Indicates the minimum size opening operation. Indicates that the expansion operation is performed on the search area with the minimum control size r1 of the entity. Indicates that the erosion operation is performed on the search area of ​​r1.

4. The method according to claim 2, characterized in that The physical size control of the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after physical size penalty includes: For the entity structure in the projected density field of the engineering structure to be optimized, the following formula is used to perform the maximum size penalty to obtain the pseudo-density variable after the maximum entity penalty: Among them, x OM Represents the result of a series of morphological operations after applying the maximum size penalty to the entity. represents the projected density field, Indicates that the circular search area with radius R2 is used for expansion operation. Indicates that an erosion operation is performed on a circular search area with a radius of R1, where R1 is the maximum control size of the entity and R1 is greater than R2.

5. The method according to any one of claims 2 to 4, characterized in that Performing density penalty on the pseudo-density variable after the entity size penalty to obtain the elastic modulus corresponding to the engineering structure to be optimized, including: The maximum size penalty and the minimum size penalty are applied to the entities in the projected density field of the engineering structure to be optimized, and the unified expression of the elastic modulus of the unit after the density penalty is applied is as follows: Among them, E i represents the elastic modulus of the i-th unit of the engineering structure to be optimized, Represents the pseudo-density variable corresponding to the i-th unit, x Om,i It represents the pseudo density variable corresponding to the minimum entity size penalty of the i-th unit, It represents the pseudo-density variable after the maximum entity size penalty of the i-th unit; pr is the density penalty coefficient (i.e., p above), pm is the minimum size penalty coefficient, pM is the maximum size penalty coefficient, pm and pM are both less than pr, pMa is the parameter of the maximum size penalty, which is used to adjust the degree of its impact on the density penalty; E0 represents the elastic modulus when the material is full, E min Represents the minimum elastic modulus of the material.

6. The method according to claim 1, characterized in that The method of performing size penalty control and density penalty on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after size and density penalty includes: Performing hole size control on the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after hole size penalty; Density penalty is performed on the pseudo-density variable after the hole size penalty to obtain the material volume corresponding to each unit of the engineering structure to be optimized. The material volume of any unit is obtained according to the pseudo-density variable and unit volume after the size and density penalty corresponding to any unit.

7. The method according to claim 6, characterized in that The physical size control of the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after physical size penalty includes: For the hole structure in the projected density field of the engineering structure to be optimized, the following formula is used to perform the minimum size penalty to obtain the pseudo-density variable after the minimum hole penalty: Among them, x Cm represents the pseudo-density variable obtained after implementing the minimum size penalty for holes, represents the projected density field, Indicates that the circular search area with radius r1 is expanded. Indicates that the erosion operation is performed in a circular search area with a radius of r1, where r1 is the minimum control radius of the hole.

8. The method according to claim 6, characterized in that The physical size control of the projected density field of the engineering structure to be optimized to obtain a pseudo-density variable after physical size penalty includes: For the hole structure in the projected density field of the engineering structure to be optimized, the following formula is used to perform the maximum size penalty to obtain the pseudo-density variable after the maximum hole penalty: Among them, x CM represents the result of a series of morphological operations after applying the maximum size penalty to the holes. represents the projected density field, Indicates that the erosion operation is performed in a circular search area with a radius of R2. Indicates that a dilation operation is performed on a circular search area with a radius of R1, where R1 is the maximum controllable size of the hole and R1 is larger than R2.

9. The method according to any one of claims 6 to 8, characterized in that Performing density penalty on the pseudo-density variable after the hole size penalty to obtain the material volume corresponding to each unit of the engineering structure to be optimized, including: After applying density penalty and maximum and minimum size penalties to the holes in the engineering structure to be optimized, the material volume of each unit is obtained as follows: Among them, v i represents the material volume of the i-th unit after the hole size penalty is applied to the projected density field, represents the volume of the i-th unit, represents the pseudo-density variable corresponding to the i-th unit; x Cm,i It represents the pseudo density variable corresponding to the i-th unit after the minimum hole size penalty is applied to the projected density field, (1-(1-x CM,i ) pMa ) pM It represents the pseudo-density variable corresponding to the i-th unit after the maximum hole size penalty is applied to the projected density field; pr is the density penalty coefficient, pm is the minimum size penalty coefficient, pM is the maximum size penalty coefficient, and pMa is the parameter of the maximum size penalty.

10. The method according to claim 1, characterized in that Perform sensitivity analysis on the response values, including: The derivative of the response value with respect to the design variable of the optimization problem of the engineering structure to be optimized is calculated, wherein the sensitivity of the response value after entity size and density penalty to the projected density field is calculated according to the following formula: Among them, Y O,m (t)=[x Om,i T , Y O,M (t)=[x OM,i T , ​​ Where f represents the response value after entity size and density penalty, represents the projected density field, x Om,i It represents the pseudo density variable corresponding to the i-th unit after the minimum entity size penalty is applied to the projected density field, x OM,i It represents the value corresponding to the i-th unit after the maximum entity size penalty series morphological operation on the projected density field, I is the unit matrix, X represents the input vector, G di (X) represents the derivative of the expansion operation, G er (X) represents the derivation of the corrosion operation, diag() represents the arrangement of elements along the diagonal to form a square matrix, represents the transposed matrix of the weight matrix, α is the aggregation parameter; r1 represents the minimum control size, R1 represents the maximum control size, and R2<R1.

11. The method according to claim 1, characterized in that Perform sensitivity analysis on the response values, including: The sensitivity of the response value after hole size and density penalty to the projected density field is calculated according to the following formula: Among them, Y C,m (t)=[x Cm,i [[ID=k]] T ,Y C,M (t)=[x CM,i T , It should be noted that there may be some inaccuracies or incompleteness in the original text, especially the expressions like "[x" which seem not well - formed. This translation is based on the best understanding of the given content.​ Where f represents the response value after entity size and density penalty, represents the projected density field, x Cm,i It represents the pseudo density variable corresponding to the i-th unit after the minimum hole size penalty is applied to the projected density field, x CM,i It represents the value corresponding to the i-th unit after the maximum hole size penalty series morphological operation on the projected density field, I is the unit matrix, X represents the input vector, G di (X) represents the derivative of the expansion operation, G er (X) represents the derivation of the corrosion operation, diag() represents the arrangement of elements along the diagonal to form a square matrix, represents the transposed matrix of the weight matrix, α is the aggregation parameter; r1 represents the minimum control size, R1 represents the maximum control size, and R2<R1.

12. A size control device in topology optimization, characterized in that: include: A projection density field establishment module is used to establish the projection density field of the engineering structure to be optimized; The penalty module is used to perform size penalty control and density penalty on the projected density field of the engineering structure to be optimized, and obtain the pseudo-density variable after size and density penalty; The iterative optimization module is used to perform finite element analysis to calculate the response values ​​of the objective function and constraints based on the pseudo-density variables after size and density penalties, and to perform sensitivity analysis on the response values. According to the sensitivity analysis results, a topology optimization algorithm is used to iteratively optimize the design variable field of the optimization problem corresponding to the engineering structure to be optimized to obtain the optimization result of the engineering structure to be optimized.

13. An electronic device, characterized in that: comprising at least one processor and a memory connected to the processor, wherein: The memory is used to store computer programs; The processor is used to execute the computer program so that the electronic device can implement the size control method in topology optimization according to any one of claims 1 to 11.

14. A computer storage medium, characterized in that The storage medium carries one or more computer programs, which, when executed by an electronic device, enable the electronic device to implement the dimension control method in topology optimization as described in any one of claims 1 to 11.

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