A mobile morphing component optimization method incorporating topological variables

Abstract

The present application belongs to the technical field of structure topology optimization, and discloses a mobile deformable component optimization method with introduced topology variable. The method comprises the following steps: for a given design domain, constructing design variables of each component in the design domain, the design variables including geometry variables and topology variables; constructing topology description functions of each component with the design variables of each component, constructing a topology description function of an overall structure formed by connecting all components with the topology description functions of all components; constructing an optimization model with the minimum flexibility value of the overall structure as the target and the design variables as the optimization objects under the constraints of available volume and topology variable; initializing the design variables of each component, solving the optimization model and obtaining the optimal design variables of each component, so as to realize the optimization of the structure. Through the present application, the optimization efficiency is improved and the problem of component overlap in the optimized structure is solved.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of structural topology optimization, and more particularly relates to a mobile deformable component optimization method with a topology variable. BACKGROUND

[0002] In the field of structural optimization, topology optimization technology aiming to find the best material layout in a given design domain has now become an important part of the product innovation design platform, especially with the development of additive manufacturing technology, manufacturing-oriented topology optimization technology and its industrial applications have attracted widespread attention. At present, the most representative topology optimization methods include homogenization method, progressive structural optimization method, variable density method (such as SIMP), level set method, mobile deformable component (usually referred to as MMC) method, etc.

[0003] Compared with other optimization methods, the MMC method integrates shape, size and topology optimization, has a small number of design variables and is completely independent of the finite element analysis model (depending on the number of components in the design domain), and will not suddenly increase due to the encryption of finite element mesh partitioning, greatly saving the calculation cost; and its material is completely 0-1 distribution, without gray units, with clear boundaries, so it has developed rapidly in the past decade. In addition, we can easily extract the geometric size, position inclination angle and other information of the components, so this method can also seamlessly connect with commercial software such as CAD.

[0004] However, the current MMC method still has some problems, such as the phenomenon of component overlap in the optimized structure, which brings inconvenience to the subsequent processing of the results; and the optimization analysis of the hidden components in the optimization process is actually meaningless and wastes time. Therefore, a method is needed to avoid the phenomenon of component overlap in the optimized structure. SUMMARY

[0005] In view of the above defects or improvement needs of the prior art, the present application provides a mobile deformable component optimization method with a topology variable, which solves the problem of component overlap in the optimized structure.

[0006] To achieve the above-mentioned purpose, according to one aspect of the present application, a mobile deformable component optimization method with a topology variable is provided, which comprises the following steps:

[0007] For a given design domain, design variables of each component in the design domain are constructed, which include geometric variables and topology variables;

[0008] A topology description function of each component containing topology variables is constructed using the design variables of each component, a topology description function of the overall structure formed by connecting all components is constructed using the topology description functions of all components, and then a compliance function of the overall structure is obtained.

[0009] An optimization model is constructed with the minimum value of the flexibility function of the whole structure as the objective, the final value of the topology variable being either 0 or 1 as the constraint condition, and the design variable as the optimization object; the design variable of each component is initialized, and the optimization model is solved to obtain the optimal design variable of each component, thereby realizing the optimization of the components.

[0010] Further preferably, the topology description function of the topology variable is as follows:

[0011]

[0012] wherein

[0013]

[0014] wherein (x, y) is the node coordinate, L i is the half-length of the current component, θ i is the inclination angle of the current component, (x 0i , y 0i ) is the center position coordinate of the current component; β i is the topology variable of the current component, q is the penalty on the topology variable, and f(x') is the profile function of the component.

[0015] Further preferably, the flexibility function is as follows:

[0016]

[0017] wherein (x, y) is the node coordinate, D is the design variable of all components in the design domain, Ω D is the design domain, H = H(z) is the Heaviside function, φ s is the topology description function of the whole structure, Γ t is the Neumann boundary, f and t are the body force density at Ω i , i = 1,..., n and the surface traction at the Neumann boundary, respectively, and u is the displacement field.

[0018] Further preferably, the Heaviside function is as follows:

[0019]

[0020] wherein α is the void density to ensure the non-singularity of the whole stiffness matrix; ∈ is a parameter to control the regularization amplitude, and z is the independent variable.

[0021] Further preferably, the topology description function of the whole structure is as follows:

[0022] φ s(x, y) = max(φ1,..., φ n )

[0023] where (x, y) is the node coordinate, n is the total number of components in the design domain, φ1,..., φ n represent the topological description function of the 1st to nth component, respectively.

[0024] Further preferably, the optimization model is as follows:

[0025] Find D = ((D 1 ) T ,..., (D i ) T ,..., (D n ) T ) T , u(x, y)

[0026]

[0027] where (x, y) is the node coordinate, n is the total number of components in the design domain, D i is the design variable vector of the ith component, D is the design variable summary of all components in the design domain, C is the compliance, Ω D is the design domain, H = H(z) is the Heaviside function, φ s is the topological description function of the overall structure, Γ t and Γ u are the Neumann boundary and Dirichlet boundary, respectively, f and t are the body force density at Ω i , i = 1,..., n and the surface traction at the Neumann boundary, u and v are the displacement field and test function, respectively, ε is the second-order linear strain tensor, is the upper bound of the available volume of solid material, β i is the topological variable of the ith component, μ is the upper limit of the topological variable constraint, is the allowable set of design variables, is the prescribed displacement on the Dirichlet boundary.

[0028] Further preferably, the calculation of the optimization model adopts the method of pseudo-material model finite element analysis, in which the Young's modulus E e of the finite element grid element is calculated as follows:

[0029]

[0030] where E is the Young's modulus of the material, H is the Heaviside function, i = 1,..., 4 is the value of the global structural topology descriptor function at the four nodes of element e, i.e. φ s .

[0031] Further preferably, in solving the optimization model, an optimization algorithm is used to update the design variables, wherein the update of the design variables is realized by calculating the sensitivity of the objective function and the constraint function, and then combining the moving asymptote method.

[0032] Further preferably, the sensitivity of the objective function is as follows:

[0033]

[0034] wherein C is the objective function, i.e. the structural flexibility, a is any design variable; u is the displacement field, E is the Young's modulus of the material, NE represents the total number of elements in the structure, H = H(z) is the Heaviside function, i = 1,..., 4 is the value of the global structural topology descriptor function at the four nodes of element e, i.e. φ s , k s is the stiffness matrix of element e corresponding to i = 1,..., 4 and E = 1.

[0035] The sensitivity of the constraint function is as follows:

[0036]

[0037]

[0038] wherein V is the volume constraint function, a is any design variable, NE is the total number of elements in the structure, H = H(z) is the Heaviside function, i = 1,..., 4 is the value of the global structural topology descriptor function at the four nodes of element e, i.e. φ s ; g β is the topology variable constraint function, β i is the topology variable of the i-th component.

[0039] Further preferably, the termination condition of the optimization iteration is as follows:

[0040] change = max{|D old -D new |}≤ε or loop≥maxiter

[0041] wherein change is the maximum value of the absolute value of the difference between the design variables of the last two consecutive iterations, D old is the design variable value of the last iteration, D newis the design variable value of the current iteration, epsilon is the lower limit value of change, loop is the current iteration number, and maxiter is the upper limit value of the iteration number.

[0042] Overall, compared with the prior art, the above technical solutions conceived by the present application have the following beneficial effects:

[0043] 1. The present application introduces a topological variable when constructing a component design variable, and introduces a topological variable when constructing a topological description function. Excess components in the optimization process gradually tend to zero due to their topological variables, rather than being hidden in large components, avoiding optimization analysis of hidden components in the optimization process and improving optimization efficiency. At the same time, the final optimized structure is composed of only necessary components, avoiding the phenomenon of component overlap.

[0044] 2. In the present application, a topological variable constraint condition is introduced when constructing an optimization model to avoid the existence of intermediate values of the topological variable, ensuring that the topological variable values of each component in the optimization result are either 0 or 1.

[0045] 3. The present application uses a pseudo-material model for finite element analysis, which can obtain the Young's modulus of the element according to the topological description function value of the overall structure of the element's four nodes, improving the calculation efficiency.

[0046] 4. The Heaviside function used in the present application is improved based on the modification of the component topological description function, which shifts the overall function image to the right by one unit length, adapting to the current component topological description model. BRIEF DESCRIPTION OF DRAWINGS

[0047] Figure 1 is a flowchart of a mobile deformable component optimization method introducing a topological variable constructed according to the preferred embodiment of the present application;

[0048] Figure 2 is a schematic diagram of the geometric variable of a component constructed according to the preferred embodiment of the present application;

[0049] Figure 3 is a structural schematic diagram of an optimization model constructed according to the preferred embodiment of the present application;

[0050] Figure 4 is a structural schematic diagram of component initialization in a design domain constructed according to the preferred embodiment of the present application;

[0051] Figure 5 is the final result obtained by optimizing according to the traditional MMC method;

[0052] Figure 6 is the final result obtained by optimizing the mobile deformable component optimization method introducing a topological variable constructed according to the preferred embodiment of the present application. DETAILED DESCRIPTION

[0053] In order to make the objectives, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and should not be used to limit the present application. In addition, the technical features involved in the various embodiments of the present application described below can be combined with each other as long as they do not conflict with each other.

[0054] As shown in Figure 1 , a mobile variable component optimization method introducing a topological variable, the method comprising the following steps:

[0055] S1: introducing a topological variable in the classical mobile variable component (referred to as MMC for short) method, the value of the topological variable varies between 0 and 1, and the topological variable is 0, which represents the disappearance of the component, and the topological variable is 1, which represents the existence of the component. The design variables constructed by the present application include the topological variable and the geometric variables of the components in the classical MMC method, the geometric variables are used to control the size, shape and position of the components, and the topological variables are used to control the existence and non-existence of the components;

[0056] As shown in Figure 2 , a component with a width that changes quadratically is used, and each component is given a topological variable β i , at this time, for the i-th component in the design domain, the design variable is D i =(x 0i , y 0i , L i , t 1i , t 2i , t 3i , θ i , β i ) T , which includes geometric variables and topological variables.

[0057] S2: constructing a topological description function for describing the components using the design variables, and constructing a topological description function of the overall structure by taking the union of the topological description functions of all components through a Boolean operation; and constructing a function of flexibility using the topological description function of the overall structure.

[0058] (1) constructing the topological description function of each component

[0059] In the classical MMC method, a topology description function (TDF) similar to a level set function is used to describe the geometry of components: for a component in the design domain, the TDF value inside the component is greater than 0, the TDF value on the edge of the component is equal to 0, and the TDF value outside the component is less than 0. The component contour is obtained by cutting at the height of 0.

[0060] In order to make the component with a topology variable of 0 disappear, the cutting plane cannot be the 0 level plane after the introduction of the topology variable, and the TDF of the component needs to be partially modified: the original TDF is added by 1, and a topology variable β i and a penalty q are introduced. After the modification, for any ith component in the design domain, the TDF is defined as:

[0061]

[0062] wherein

[0063]

[0064] In the formula, (x, y) represents the node coordinates, L i represents the half length of the current component, θ i represents the inclination angle of the current component, (x 0i , y 0i ) represents the center position coordinates of the current component; β i represents the topology variable of the current component, and q is the penalty for the topology variable, so that it is better driven to 0 or 1; f(x') is the contour function of the component, which is defined as follows for a component with quadratic width variation:

[0065]

[0066] In the formula, t 1i , t 2i , and t 3i respectively represent the half width of the left end, the right end, and the middle of the current component. At this time, for any ith component in the design domain, the TDF value inside the component is greater than 1, the TDF value on the edge of the component is equal to 1, and the TDF value outside the component is less than 1, that is:

[0067]

[0068] In the formula, (x, y) represents the node coordinates, Ω i represents the area occupied by the ith component, and Ω D represents the design domain.

[0069] (2) Calculate the topology description function of the overall structure formed by all components

[0070] φ s (x, y) = max(φ1,..., φ n n), where n is the total number of components in the design domain, so that the TDF of the overall structure is obtained:

[0071]

[0072] where (x, y) denotes the node coordinates, Ω s denotes the region occupied by the overall structure, Ω D denotes the design domain. At this time, cutting is performed at the height 1, and the components with the topological variable 0 will disappear because their TDF becomes 0 and cannot be cut.

[0073] (3) Construction of the compliance function of the overall structure

[0074] The relationship of the compliance function is as follows:

[0075]

[0076] where (x, y) denotes the node coordinates, D denotes the design variables of all components in the design domain, Ω D denotes the design domain, H = H(z) is the Heaviside function, φ s is the topological description function of the overall structure, Γ t denotes the Neumann boundary, f and t are the body force density at Ω i , i = 1,..., n and the surface traction at the Neumann boundary, respectively, and u is the displacement field.

[0077] S3: Definition of the structure optimization problem: the design objective is to minimize the compliance of the structure, and the design constraints include the volume constraint of the overall structure and the topological variable constraint;

[0078] A penalty is imposed on the topological variable β i to make it tend to 0 or 1 faster, so as to eliminate redundant components in the optimization process. However, from the optimization results, this penalty cannot guarantee the "black and white" of the components, and the optimized β i may converge to an intermediate value between 0 and 1. Therefore, a topological variable constraint g β is introduced in the optimization to prevent the intermediate value of β i . The constraint is defined as follows:

[0079]

[0080] where the upper limit μ is a small positive number, n is the total number of components in the design domain, and β itopology variable of the ith component. This constraint is handled with a continuation strategy: the constraint starts with a large upper bound, e.g., μ = 10; when the value of this constraint changes little, i.e., satisfies the following equation, the upper bound becomes a small positive number, e.g., μ = 0.01.

[0081]

[0082] where tolerance δ g = 10%, k is the current iteration number, and the specific values of μ and δ g can be adjusted according to the actual situation. Considering the compliance minimization problem under the constraints of available volume and topology variable, the optimization model can be expressed as:

[0083] Find D = ((D 1 ) T ,..., (D i ) T ,..., (D n ) T ) T , u((x, y))

[0084]

[0085] where (x, y) is the node coordinate, n is the total number of components in the design domain, D i is the design variable vector of the ith component, D represents the design variables of all components in the design domain, C represents the compliance, Ω D represents the design domain, H = H(z) is the Heaviside function, φ s is the topology description function of the overall structure, Γ t and Γ u represent the Neumann boundary and Dirichlet boundary respectively, f and t are the body force density at Ω i , i = 1,..., n and the surface traction at the Neumann boundary, u and v are the displacement field and the corresponding test function defined on Ω , , is the fourth-order isotropic elastic tensor of the material constituting the ith component, where E is the Young's modulus and v is the Poisson's ratio, and δ represent the fourth-order and second-order unit tensors respectively, is the Kronecker product, and the symbol ε represents the second-order linear strain tensor, is the upper bound of the available volume of solid material artificially specified, β i is the topology variable of the ith component, μ is the upper limit of the topology variable constraint artificially specified, is the allowable set of design variables, is the prescribed displacement on the Dirichlet boundary, for simplicity we set

[0086] S4: solving the optimization model

[0087] (1) The design domain is uniformly discretized by four-node bilinear quadrilateral elements, and a pseudo-material model is used for finite element analysis to improve the calculation efficiency; in this embodiment, the design domain is meshed, and the number of rectangular grid elements is 80x40, and the number of element nodes is 81x41.

[0088] As Figure 3 shown, the optimization problem of minimizing the flexibility of a plane cantilever beam structure with concentrated load is taken as an example to explain the present application. The initial structure is set in the given 2m x 1m rectangular design domain Ω D , the degrees of freedom of the left side area are fixed, and a vertical downward concentrated force f = 1N is applied at the midpoint of the right side, and the design goal is to minimize the flexibility of the structure under the constraint of available solid materials.

[0089] Like most topology optimization methods, four-node bilinear quadrilateral elements are used to uniformly discretize the design domain. In order to improve the calculation efficiency, a pseudo-material model is used for finite element analysis, once the TDF values of the four nodes of the element are known, the Young's modulus of the element can be interpolated as:

[0090]

[0091] In the formula, E is the Young's modulus of the material, H = H(z) is the Heaviside function, i = 1,..., 4 is the integral structure topology description function value (i.e. φ s ) on the four nodes of the element e. In order to realize numerically, as a common practice in the literature, the Heaviside function is often regularized as H ∈ (z):

[0092]

[0093] Where ∈ is a parameter that controls the regularization amplitude; α is a small positive number to ensure the non-singularity of the overall stiffness matrix. But due to the modification of the component TDF in the foregoing, the above Heaviside function also needs to be modified:

[0094]

[0095] Compared to the old Heaviside function, the new Heaviside function graph has shifted to the right by one unit. This change is reasonable because in the classic MMC method, the TDF value is 0 at component edges, and the TDF value range corresponding to the edge transition region is φ. i ∈[-∈,∈]; In the improved method, the TDF value at the component edge is 1, and the TDF value range corresponding to the edge transition area is .

[0096] (2) Calculate the sensitivity of the objective function to the design variables, calculate the sensitivity of the constraint function to the design variables, update the design variables based on the above sensitivity, and solve the optimization design problem;

[0097] The design variable D is updated by combining the objective function, constraint function, objective function sensitivity, and constraint function sensitivity with the moving asymptotic method to obtain the updated design variable D. new ; Using the updated design variable D new The updated TDF of each component is obtained, and thus the updated overall structure is obtained.

[0098] The objective function is structural compliance, and its sensitivity to any design variable 'a' can be written as:

[0099]

[0100] Where K is the overall stiffness matrix of the structure, k s For corresponding The element stiffness matrix for i = 1, ..., 4 and E = 1. In the formula, the symbol NE represents the total number of elements in the structure. It can be easily calculated because φ s (x) is an explicit function of a. However, in this invention, to enhance the generality of the code, we use... finite difference quotient (i.e. To approximate the calculation Due to φ s (x) is an explicit function of a, and finite difference operations are performed only at the cell level, therefore computation Very fast. Numerical examples show that this method handles the problem well.

[0101] In addition, we have the sensitivity of the volume constraint function to any design variable a:

[0102]

[0103] In the formula, NE represents the total number of elements in the structure, and H = H(z) is the Heaviside function. i = 1, …, 4 represents the value of the overall structure topology description function on the four nodes of the unit e (i.e. φ s ).

[0104] According to the topology variable constraint calculation formula, the sensitivity of the topology variable constraint function to the topology variable is:

[0105]

[0106] In the formula, β i is the topology variable of the i th component.

[0107] (3) When the updated structure satisfies the optimization termination condition, the optimization ends and the required optimization structure is obtained.

[0108] The optimization termination condition is set as:

[0109] change = max{|D old -D new |}≤ε or loop≥maxiter

[0110] Wherein, change is the maximum value of the absolute value of the difference between the design variables in the last two consecutive iterations, D old represents the design variable value of the last iteration, D new represents the design variable value of the current iteration, ε represents the lower limit value of change, loop represents the current iteration number, and maxiter represents the upper limit value of the iteration number. In this embodiment, ε is 0.25%, and maxiter is the upper limit value of the iteration number, which is 1000 here.

[0111] The initial design is shown in Figure 4 , and the 16 components are evenly distributed in the design domain. The cantilever beam optimization result obtained by the method provided by the present application is shown in Figure 6 , and the compliance value is 74.461 after 155 iterations, and the optimization structure is composed of only 8 components, and the computer time is 20.8172 seconds. As a comparison, the cantilever beam optimization result obtained by the classical MMC method is shown in Figure 5 , and the compliance value is 74.680 after 244 iterations, but the optimization structure contains 16 components, and the computer time is 27.7380 seconds. Compared with the classical MMC method, the MMC method provided by the present application introduces the topology variable, which improves the optimization efficiency while obtaining the best material distribution, so that the optimization structure is composed of only the necessary components, and there is no redundant component, avoiding the phenomenon of component overlap in the optimization result.

[0112] Those skilled in the art can easily understand that the above description is only the preferred embodiment of the present application, and is not intended to limit the present application. Any modification, equivalent replacement and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for optimizing a mobile deformable component by incorporating topological variables, characterized in that, The method includes the following steps: For a given design domain, design variables are constructed for each component within that design domain, including geometric and topological variables. The topology description function of each component containing topology variables is constructed using the design variables of each component. The topology description function of the whole structure formed by connecting all components is constructed using the topology description function of all components. Then, the flexibility function of the whole structure is obtained. An optimization model is constructed with the goal of minimizing the overall structure's flexibility function value, and the constraint that the final value of the topology variable is either 0 or 1. The design variables are the optimization objects. The design variables of each component are initialized, and the optimization model is solved to obtain the optimal design variables of each component, thereby achieving component optimization. The component contains the following topological description function for topological variables: in in,( x, y ) are the node coordinates. It is half the length of the current component. It is the tilt angle of the current component. () represents the center coordinates of the current component; It is the topology variable of the current component. q It is a penalty for topological variables. It is the component's outline function; The optimization model is as follows: in,( x, y ) are the node coordinates. n It is the total number of all components within the design domain. It is the design variable vector of the i-th component. It is a summary of design variables for all components within the design domain. C It's about softness. It is a design domain. H=H(z) It's the Heaviside function. It is the topological description function of the overall structure. and These are the Neumann boundary and the Dirichlet boundary, respectively. f and t They are in The body density at the point and the surface traction at the Neumann boundary, u and v These are the displacement field and the test function, respectively. It is a second-order linear strain tensor. It is the upper limit of the usable volume of solid materials. It is the topology variable of the i-th component. It is the upper limit of topological variable constraints. It is the permissible set of design variables. It is the specified displacement on the Dirichlet boundary. It is the fourth-order isotropic elastic tensor of the material constituting the i-th component. .

2. The method for optimizing a mobile deformable component by incorporating topological variables as described in claim 1, characterized in that, The compliance function is as follows: in,( x, y ) are the node coordinates. It is a summary of design variables for all components within the design domain. It is a design domain. H=H (z) It's the Heaviside function. It is the topological description function of the overall structure. It is the Neumann boundary. f and t They are in The body density at the point and the surface traction at the Neumann boundary, u It is a displacement field.

3. The method for optimizing a mobile deformable component by incorporating topological variables as described in claim 2, characterized in that, The Heaviside function is as follows: in, The void density is used to ensure the nonsingularity of the overall stiffness matrix; It is a parameter that controls the magnitude of regularization. z It is the independent variable.

4. A method for optimizing a mobile deformable component by incorporating topological variables as described in claim 1 or 2, characterized in that, The topological description function of the overall structure is as follows: in,( x, y ) are the node coordinates. n It is the total number of all components within the design domain. Representing the 1st to the 2nd n Topology description functions for each component.

5. The method for optimizing a mobile deformable component by incorporating topological variables as described in claim 1, characterized in that, The optimization model is calculated using a pseudo-material model finite element analysis method, where the Young's modulus of the finite element mesh element is... The calculation formula is as follows: Where E is the Young's modulus of the material. It's the Heaviside function. It is a unit e The overall structural topology description function values ​​at the four nodes, i.e. .

6. The method for optimizing a mobile deformable component by incorporating topological variables as described in claim 1, characterized in that, When solving the optimization model, an optimization algorithm is used to update the design variables. The update of the design variables is achieved by calculating the sensitivity of the objective function and the constraint function, and then combining the moving asymptote method.

7. The method for optimizing a mobile deformable component by incorporating topological variables as described in claim 6, characterized in that, The formula for the sensitivity of the objective function is as follows: in, C It is the objective function, namely structural flexibility. a Design variables arbitrarily; u It is a displacement field, and E is the Young's modulus of the material. NE This represents the total number of units in the structure. H=H(z) It's the Heaviside function. It is a unit e The overall structural topology description function values ​​on the four nodes (i.e.) ), For corresponding The element stiffness matrix with E=1; The sensitivity of the constraint function is as follows: in, V It is a volume constraint function. a For arbitrary design variables, NE It is the total number of units in the structure. H=H(z) It's the Heaviside function. It is a unit e The overall structural topology description function values ​​at the four nodes, i.e. ; It is a topological variable constraint function. It is the first i Topology variables of each component.

8. The method for optimizing a mobile deformable component by incorporating topological variables as described in claim 6, characterized in that, The termination condition for the optimization iteration is as follows: in, change It is the maximum absolute value of the difference between the design variables in the two most recent consecutive iterations. These are the design variable values ​​from the previous iteration. It is the design variable value of the current iteration. yes change The lower limit value, loop It is the current iteration number. maxiter It is the upper limit of the number of iterations.

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