Thermal coupling multi-material topological optimization design method based on mworks

Abstract

The invention relates to the technical field, in particular to a thermal coupling multi-material topological optimization design method based on mworks. According to the technical scheme, the method comprises the steps that a design domain and load boundary conditions are defined in mWorks, and grids are divided; meanwhile, initial optimization parameters are defined, finite element analysis is carried out, and a target function, a constraint function and the sensitivity of corresponding units are calculated; when the number of iterations is greater than 1, averaging the current sensitivity information and the sensitivity information of the last iteration so as to carry out smoothing processing; sequentially updating the design variables by adopting a floating projection method according to the material sequence; if iteration convergence is carried out, smooth design is carried out on the structure, and if iteration convergence is not carried out, the steps 2-4 are repeated; and judging whether the smooth solution is converged or not, and if not, repeating the steps 2-5 until the program is strictly converged. Compared with a traditional method, the method has the advantages that one parameter is used for replacing all penalty coefficients, the dependence of a traditional unit base method on the penalty coefficients is overcome, the parasitic phenomenon is avoided, and a design result with better flexibility performance is obtained.

Description

Technical Field

[0001] The present invention relates to the technical field, and in particular to a thermal-mechanical coupling multi-material topology optimization design method based on mWorks. Background Art

[0002] As an art of "digging holes" in structures, topology optimization can innovatively design the topological configuration of structural materials under specific constraints to obtain the optimal performance of the optimization target. According to the different ways of defining design variables, topology optimization can be divided into two categories. The first category is the unit-based method with units as design variables, such as the variable density method (SIMP), the evolutionary structural optimization method / bidirectional evolutionary structural optimization method (ESO / BESO), and the binary structure topology optimization method (TOBS); the second category is the boundary-based method with structural boundaries as design variables, such as the level set method (LSM), the movable deformable component method (MMC), and the feature-driven method (FDO).

[0003] As a hot topic in the field of structural optimization, topology optimization has been successfully applied to many application fields such as machinery, construction, aerospace, and medical biology. However, in most studies, a given external mechanical load is usually used for topology optimization design, and the external load on the structure is not related to the structural topology. Typical engineering structures, such as aircraft engines, thermal engine equipment, and aerospace thermal protection systems, are not only affected by mechanical loads. In actual work, they are inevitably affected by temperature due to working heat and high-temperature working environments. As a result, due to the thermal-mechanical coupling effect, the structure is subjected to thermal loads and generates thermoelastic forces, which significantly affect the material properties.

[0004] At the same time, in order to meet the comprehensive index requirements of lightweight and multifunctional structures, topology optimization design has changed the topology optimization design mode of using a single material to multi-phase material topology optimization design. At present, the multi-material thermal coupling topology optimization methods based on the unit basis method are highly dependent on the penalty coefficient in the interpolation. However, as the actual requirements of engineering become higher and higher, the calculation of thermal coupling becomes more accurate, and the types of materials involved in the design increase, the introduction of a large number of penalty coefficients will be inevitable, and the selection of these coefficients will greatly increase the workload of the optimization model. If the selection is inappropriate, it will be difficult to obtain a clear structure, and even parasitic phenomena will occur.

[0005] To this end, we propose a thermo-mechanical coupling multi-material topology optimization design method to solve the existing problems. Summary of the invention

[0006] The purpose of the present invention is to propose a thermal-mechanical coupling multi-material topology optimization design method based on mWorks in order to solve the problems existing in the background technology.

[0007] To achieve the above-mentioned purpose, the present invention provides the following technical solutions: a thermomechanical coupling multi-material topology optimization design method based on mWorks, comprising the following six steps, the step 1: defining the design domain, load boundary conditions, and meshing in mWorks; defining the initial optimization parameters at the same time, the step 2: performing finite element analysis, calculating the objective function, constraint function and sensitivity of the corresponding unit; the step 3: when the number of iterations is greater than 1, averaging the current sensitivity information and the sensitivity information of the previous iteration for smoothing; the step 4: according to the material sequence, the floating projection method is used in mWorks to update the design variables in sequence; the step 5: if the iteration converges, the structure is smoothly designed, if not, repeating steps 2-4; the step 6: judging whether the smooth solution converges, if not, repeating steps 2-5 until the program strictly converges.

[0008] Preferably, step 1, namely model establishment and parameter setting, includes design domain definition and meshing, material property definition, load and constraint conditions and initial parameter setting. The design domain definition and meshing include defining the shape, size and boundary conditions of the design domain in mWorks, and meshing. The type and number of meshes should be selected according to the specific problem to ensure calculation accuracy and efficiency. The material property definition includes defining the properties of multiple materials involved in the optimization, including elastic modulus, thermal expansion coefficient, thermal conductivity, and density.

[0009] Preferably, the load and constraint conditions include defining the mechanical load and thermal load on the design domain, as well as the mass constraint, and the initial parameter setting includes setting the initial optimization parameters, including the filter radius rmin, the minimum value of the design variable xmin, the mass constraint M f .

[0010] Preferably, step 2 is finite element analysis: performing topology optimization finite element analysis, discretizing the design domain into finite elements, and establishing a topology optimization model, using finite element software to perform thermal-mechanical coupling analysis, calculating the displacement, stress, temperature and other field distributions of the design domain, and the model objective function is to minimize flexibility, and convert mass constraints, thermal balance equations, thermal load vectors, etc. into constraints in the optimization model.

[0011] Furthermore, the topology optimization finite element analysis described in step 2 specifically includes: discretizing the design domain into N finite elements according to meshing, and the initial volumes of different materials satisfy the mass constraints as a whole, that is,

[0012]

[0013] Where M is the number of materials, V e is the actual volume of unit e, x e，iis the i-th material design variable, ρ i is the density of the ith material.

[0014] Furthermore, the topology optimization model established in step 2 is as follows:

[0015] Min.:c(x i，1 , x i，2 , ..., x i,N )=F T U

[0016] St:KU=F m +F th

[0017] K th T=Q

[0018]

[0019] 1≥x e,1 ≥x e，2 …≥x e，i …≥x e,N ≥0

[0020] 0≤x e，i ≤1 when x e，i ∈x i

[0021] (i=1,2,...,N materials; e=1,2,...M units)

[0022] Where c is the flexibility, F T represents the transpose of the matrix, U is the displacement, F m is the mechanical load, F th is the thermoelastic load, where M f is the mass constraint, K represents the stiffness matrix, K th represents the thermal stiffness matrix, T is the temperature field, and Q is the thermal load vector.

[0023] Preferably, in step 3, the sensitivity of the objective function and the constraint conditions to the design variables is calculated using the adjoint matrix method, and during the iteration process, the sensitivity is smoothed to improve the stability and convergence of the algorithm.

[0024] Furthermore, the step of calculating the sensitivity of the topology optimization problem described in step 3 uses the adjoint matrix method to solve the sensitivity of the objective function and the constraints, and the sensitivity analysis of the objective function is:

[0025]

[0026] Objective function nested formula: c(x i , T(x i,U(x i ))=F T (x i , T(x i )U(x i )

[0027] Heat balance equation residual: R th ≡K th (x i )T(x i )-Q(x i )=0

[0028] R m ≡K(x i )U(x i )-F(x i , T(x i ))=0

[0029] The adjoint method is used to construct the augmented objective function:

[0030] The nested formula of F is expressed as an equation: F(x i , T(x i ))=F m +F th (x i , T(x i ))

[0031] The sensitivity of the augmented objective function is:

[0032] By using the chain rule, the three items in the sensitivity of the augmented objective function can be written as:

[0033]

[0034] Then the augmented objective function sensitivity can be written as:

[0035] To avoid the computationally expensive implicit derivatives, the corresponding terms can be eliminated in cases where the adjoint variable is the solution of the adjoint system defined in the equation:

[0036] F T +λ T K=0

[0037]

[0038] The solution of the adjoint system is obtained: λ=-K -1 F=-U

[0039]

[0040] Where Q adj A virtual heat load vector representing the adjoint problem.

[0041] By eliminating the implicit terms, the final expression for the sensitivity is obtained:

[0042]

[0043] The three items can be written as:

[0044]

[0045] Furthermore, in step 3, when the number of iteration steps is greater than 1, the sensitivity is smoothed, which is specifically expressed as:

[0046]

[0047] Preferably, step 4, namely, updating the design variables, is as follows: according to the material order, the design variables are updated using the floating projection method in mWorks, the design variables are gradually pushed to 0 / 1, the 0 / 1 constraints of the design variables are simulated, and the finite element analysis, sensitivity calculation and design variable update are repeated until the convergence conditions are reached.

[0048] Furthermore, in step 4, the design variables are updated in sequence using the floating projection method in mWorks as follows:

[0049] The optimality criterion equation can be expressed as:

[0050] in

[0051] Update the design variables according to the steepest descent principle:

[0052] Then use the filtering scheme to modify the design variables to:

[0053] Among them, r ej Indicates the e th The distance between the center of the first element and the center of the jth element. ej ) is the quality factor, defined as:

[0054]

[0055] Among them, r min is the filter radius specified by the user. In the floating projection method, implicit floating projection constraints are used to simulate the 0 / 1 constraints of the design variables. In order to strengthen the quasi-discrete constraints of the multi-material design variables, implicit multiple floating projection constraints are applied, and the multi-material design parameters are further modified as follows:

[0056]

[0057] Where β>0 controls the steepness of the Heaviside function, and the design is gradually pushed to 0 / 1 by iterating the value of β during the optimization process. i , can be easily determined by ensuring that the sum of the design variables before and after the projection remains unchanged, that is, We then update the multi-material design variables first and then by enforcing the inequalities given in Eq. The upper and lower bounds of the multi-material design variables are defined by:

[0058]

[0059] In the numerical implementation, the design variables are updated after projection according to the following equations:

[0060]

[0061] Where δ is the moving limit, and δ = 0.02 is used in this paper. Then, Will be used as The upper bound of Will be used as The upper bound of , and so on. Meanwhile, the normalized Lagrange multiplier can be easily determined by the corresponding mass constraint according to the double section method or the Newton-Raphson method. The updated design variables will be used for the next finite element analysis. The steepness of the Heaviside function starts from a small positive value, which means that the 0 / 1 constraints of the design variables are almost ignored. Once the current solution converges, it increases, which indicates that stricter 0 / 1 constraints of the design variables should be applied for the optimization. The given convergence criterion is defined as:

[0062]

[0063] Preferably, step 5 is smooth design: converting multi-material design variables into level set functions to describe the topological configuration of the structure, smoothing the extracted design to obtain a more continuous structural topology, analyzing the optimization results, evaluating the performance and reliability of the structure, and if the iteration converges, smoothing the structure. The smooth design constructs a series of level set functions of the element-based design represented by the multi-material design variables, and projects the extracted design back to the fixed grid to calculate the volume fraction of the poroelastic material in each unit.

[0064] Furthermore, the specific method of smooth design in step 5 is:

[0065] Once the convergence criterion for a given β is satisfied, we should check whether the solution to the original quasi-discrete topology optimization problem is desirable. We can construct a series of level set functions for element-based design represented by multi-material design variables. First, x e，i (e=1, 2, ..., M) linearly interpolated to the entire design domain as x i (x, y), and then construct the i-th level set function The smooth interface between material i and material i-1 can be represented by the zero level set as:

[0066]

[0067] Here, material 0 represents empty, i.e. filled with air medium. The extracted design is then projected back to the fixed grid to calculate the volume fraction of the poroelastic material in each cell. The threshold is determined by a bi-segmentation method to ensure

[0068] Furthermore, the method for determining whether strict convergence is achieved in step 6 is specifically as follows:

[0069] Perform another finite element analysis to calculate the extracted smooth design objective function C(v e,i ). The difference in the objective function between the optimized element-based design and the smooth design can be expressed as:

[0070]

[0071] Once the smooth design extracted from the surface is close to the optimized design, the entire optimization stops.

[0072] Compared with the prior art, the present invention has the following beneficial effects:

[0073] The existing unit-based method is highly dependent on the penalty in interpolation when performing topology optimization. When calculating the multi-material thermal coupling model, it is necessary to introduce penalty coefficients for different materials and different loads at the same time. This means that when the number of materials and loads increases, the penalty coefficient will increase exponentially. In actual optimization, each penalty coefficient needs to be selected manually. If the selection is inappropriate, a clear configuration cannot be obtained and even parasitic phenomena may occur. The present invention adopts a floating projection topology optimization method, uses β iteration to gradually push the design to 0 / 1, and replaces all penalty coefficients with one parameter, which greatly simplifies the model and reduces the workload for optimizing the design. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 It is a schematic diagram of the structure of the rectangular double clamping structure optimized by the method of the present invention;

[0075] FIG2 is a comparison of the present method and other traditional unit-based methods in optimizing the model under thermal-mechanical coupling; Figure 2-1Optimization results of SIMP interpolation method when T1 = 1℃; Figure 2-2 Optimization results of SIMP interpolation method when T1 = 3℃; Figure 2-3 Optimization results of the floating projection method;

[0076] Figure 3 It is an iterative diagram of a rectangular double clamping structure of a double material optimized by the method of the present invention;

[0077] Figure 4 The method of the present invention is used to optimize the unit basis solution and the smooth solution of the rectangular double clamping structure of double materials. DETAILED DESCRIPTION

[0078] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention. In order to better understand the present invention, the present invention is further described below in conjunction with the drawings, but the embodiments of the present invention are not limited to this.

[0079] Embodiment 1

[0080] like Figure 1-Figure 4 As shown, the present invention proposes a thermal-mechanical coupling multi-material topology optimization design method based on mWorks, and the specific steps are as follows:

[0081] Step 1: Establish a topological optimization design model of a rectangular double clamping structure, using a 120x80 grid, with the left and right sides fixed, a force Fa = -1500N in the middle of the bottom, the upper boundary temperature is T0 = 0 °C, the bottom temperature is T1 = 1 °C, the model length is L = 1200mm, the height is h = 800mm, the thickness is 10mm, a square grid with a width of 10mm is used, and the filter radius r min =3, volume fraction is 0.4. For the convenience of comparison, the material parameters remain unchanged, the elastic modulus of the material E1 = 30000, the thermal expansion coefficient k1 = 1, the thermal expansion coefficient α1 = 12e-6, and the Poisson's ratio is 0.4. To simplify the calculation, air is defined as material 0.

[0082] Step 2: Construct a thermal-mechanical coupling topology optimization mathematical model:

[0083] Min.:c(x i，1 , x i，2 , ..., x i，N )=F T U

[0084] St:KU=Fm +F th

[0085] K th T=Q

[0086]

[0087] 1≥x e,1 ≥x e，2 …≥x e,i …≥x e,N ≥0

[0088] 0≤x e，i ≤1 when x e,i ∈x i

[0089] (i=1,2,...,N materials; e=1,2,...M units)

[0090] Where c is the flexibility, F T represents the transpose of the matrix, U is the displacement, F m is the mechanical load, F th is the thermoelastic load, where M f is the mass constraint, K represents the stiffness matrix, K th represents the thermal stiffness matrix, T is the temperature field, and Q is the thermal load vector.

[0091] Step 3: The model is calculated using the floating projection algorithm of the present invention, and the simp interpolation method in the traditional unit basis method is also used for comparison.

[0092] As shown in Figure 2, the traditional unit-based method can only produce a clear design configuration when the penalty coefficient is appropriate. When the temperature changes, the penalty coefficient needs to be reselected. However, the method in the present invention can be directly applied to different temperature fields and can present a clear configuration.

[0093] Step 4: Change the properties of the two materials and test the performance of the thermal-mechanical coupling multi-material model. The elastic modulus of the two materials are E1=0.8, E2=1; the thermal expansion coefficients are k1=1, k2=0.75; the thermal expansion coefficients are α1=24e-6, α2=12e-6; the material densities are ρ1=0.5, ρ2=1, and the external force Fa=1 and the temperature T1=200. The iterative curve of the model is as follows: Figure 3 As shown, the final unit basis solution and its smooth solution are as follows Figure 4 As shown in the figure, the model is proved to be applicable to multi-material thermal-mechanical coupling model.

[0094] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

Claims

1. A thermomechanical coupling multi-material topology optimization design method based on mWorks, comprising the following six steps, characterized in that: The step 1: defines the design domain, load boundary conditions, and meshes in mWorks; defines initial optimization parameters at the same time; the step 2: performs finite element analysis, calculates the objective function, constraint function, and sensitivity of corresponding units; the step 3: when the number of iterations is greater than 1, averages the current sensitivity information and the sensitivity information of the previous iteration for smoothing; the step 4: updates the design variables in sequence in mWorks using a floating projection method according to the material sequence; the step 5: if the iteration converges, performs smooth design on the structure; if not, repeats steps 2-4; the step 6: determines whether the smooth solution converges; if not, repeats steps 2-5 until the program strictly converges.

2. The mWorks-based thermomechanical coupling multi-material topology optimization design method according to claim 1, characterized in that: The step 1 is model establishment and parameter setting, which includes design domain definition and mesh division, material property definition, load and constraint conditions and initial parameter setting. The design domain definition and mesh division include defining the shape, size and boundary conditions of the design domain in mWorks, and performing mesh division. The type and number of meshes should be selected according to the specific problem to ensure calculation accuracy and efficiency. The material property definition includes defining the properties of multiple materials involved in the optimization, including elastic modulus, thermal expansion coefficient, thermal conductivity, and density.

3. The mWorks-based thermomechanical coupling multi-material topology optimization design method according to claim 2 is characterized by: The load and constraint conditions include defining the mechanical load and thermal load on the design domain, as well as mass constraints. The initial parameter setting includes setting initial optimization parameters, including filter radius, minimum value of design variables, and mass constraints.

4. The mWorks-based thermomechanical coupling multi-material topology optimization design method according to claim 1, characterized in that: The step 2 is finite element analysis: perform topology optimization finite element analysis, discretize the design domain into finite elements, and establish a topology optimization model. Use finite element software to perform thermal-mechanical coupling analysis, calculate the displacement, stress, temperature and other field distributions of the design domain, and minimize flexibility. The model objective function is to minimize flexibility, and convert mass constraints, thermal balance equations, thermal load vectors, etc. into constraints in the optimization model.

5. The mWorks-based thermomechanical coupling multi-material topology optimization design method according to claim 1, characterized in that: In step 3, the sensitivity of the objective function and the constraint conditions to the design variables is calculated using the adjoint matrix method. During the iteration process, the sensitivity is smoothed to improve the stability and convergence of the algorithm.

6. The mWorks-based thermomechanical coupling multi-material topology optimization design method according to claim 1, characterized in that: The step 4 is the updating of design variables: according to the material order, the design variables are updated by using the floating projection method in mWorks, the design variables are gradually pushed to 0 / 1, the 0 / 1 constraint of the design variables is simulated, and the finite element analysis, sensitivity calculation and design variable updating are repeated until the convergence condition is reached.

7. The mWorks-based thermomechanical coupling multi-material topology optimization design method according to claim 1, characterized in that: The step 5 is smooth design: converting the multi-material design variables into level set functions to describe the topological configuration of the structure, smoothing the extracted design to obtain a more continuous structural topology, analyzing the optimization results, evaluating the performance and reliability of the structure, and if the iteration converges, smoothing the structure. The smooth design constructs a series of level set functions of the element-based design represented by the multi-material design variables, and projects the extracted design back to the fixed grid to calculate the volume fraction of the poroelastic material in each unit.

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