Fluid pressure load floating projection topological optimization method based on mworks

Abstract

The invention relates to the technical field of fluid pressure loads, in particular to a fluid pressure load floating projection topological optimization method based on mworks. According to the technical scheme, the method comprises the following six steps: step 1, defining a design domain and load boundary conditions in MWorks, and dividing grids; the method comprises the following steps of: (1) carrying out iteration convergence and defining initial optimization parameters at the same time, (2) carrying out finite element analysis and calculating a target function, a constraint function and the sensitivity of a corresponding unit, (3) updating design variables in MWorks by adopting a floating projection method, (4) carrying out smooth design on the structure if iteration convergence is carried out, and (5) repeating the steps (2)-(3) if iteration convergence is not carried out. And step 5, judging whether the smooth solution is convergent or not, and if not, repeating the steps 2-4 until the program is strictly convergent. According to the method, a floating projection topological optimization method is adopted, the design is gradually pushed to 0 / 1 through beta iteration, all penalty coefficients are replaced with one parameter, the model is greatly simplified, and the workload is reduced for optimization design.

Description

Technical Field

[0001] The present invention relates to the technical field of fluid pressure loads, and particularly to a fluid pressure load floating projection topology optimization method based on mworks. Background Art

[0002] The research on fluid pressure loads has always attracted the attention of experts in various engineering fields. For example, such problems are encountered in various applications, such as civil and mechanical structures (aircraft, pumps, pressure vessels, ships, turbomachinery) under air, water, and / or snow loads, pneumatically or hydraulically actuated soft robots or compliant mechanisms, and pressure-loaded mechanical metamaterials.

[0003] It is worth noting that optimizing the shape, topology, and performance of a structure is directly related to the magnitude, position, and direction of the pressure load, and the magnitude, position, and direction of the pressure load vary with the design. For the topology optimization method of pressure loads, there is a problem that must be solved, namely the interface dependence problem existing at the coupling interface between the fluid and the solid.

[0004] Common methods are roughly divided into two types, namely the density-based method and the level set method. The density-based methods include: the interface tracking method, which can effectively identify the pressure boundary, but this method only considers the grid cells at the pressure boundary and has certain limitations; similarly, there is also a boundary identification method that uses a set of additional variables to predefine the pressure boundary, and the variables used will be optimized together with the design variables, which will greatly increase the computational amount.

[0005] In traditional element-based topology optimization methods (such as SIMP and BESO), the above post-processing seems inevitable for the final element-based solution, but the target performance of the obtained smooth design may deviate greatly from the optimized design. Compared with density-based topology optimization, the level set method can use implicit boundary descriptions to define pressure loads. On the other hand, due to boundary motion, the level set method tends to be more dependent on the initial design (van Dijk et al.). Gao et al. used a level set function (LSF) to represent the structure topology and overcome the difficulties related to boundary curve description in an efficient and robust manner. Xia et al. used the zero level sets of two LSFs to represent the free boundary and the pressure boundary respectively.

[0006] When performing topology optimization with existing element-based methods, it highly depends on the penalty in interpolation. When calculating the fluid pressure load model, a penalty coefficient needs to be introduced, which is not conducive to the expansion of the method. And in actual optimization, the penalty coefficient needs to be manually selected. When the selection is inappropriate, a clear configuration cannot be obtained or even non-convergence occurs.

[0007] Therefore, we propose a three-field floating projection topology optimization method based on mworks that depends on pressure loads to solve the existing problems. Summary of the Invention

[0008] The object of the present invention is to propose a three-field floating projection topology optimization method based on mworks for the problems existing in the background technology.

[0009] To achieve the above object, the present invention provides the following technical solution: A three-field floating projection topology optimization method based on mworks that depends on pressure loads, including the following six steps. Step 1: Define the design domain, load boundary conditions in MWorks, and divide the grid; at the same time, define the initial optimization parameters. Step 2: Perform finite element analysis, calculate the objective function, constraint function, and the sensitivity of the corresponding elements. Step 3: Use the floating projection method in MWorks to update the design variables. Step 4: If the iteration converges, perform a smooth design on the structure; if it does not converge, repeat steps 2-3. Step 5: Judge whether the smooth solution converges. If it does not converge, repeat steps 2-4 until the program converges strictly.

[0010] Preferably, Step 1, namely model establishment and parameter setting, includes design domain definition and grid division, material property definition, load and constraint conditions, and initial parameter setting.

[0011] Preferably, the design domain definition and grid division include defining the shape, size, and boundary conditions of the design domain in MWorks and performing grid division. The grid type and quantity 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 participating in the optimization, including elastic modulus, flow coefficient, drainage coefficient, and density.

[0012] Preferably, the load and constraint conditions include defining the pressure load boundary received by the design domain and the boundary constraints. The initial parameter setting includes setting the initial optimization parameters, including the filtering radius rmin and the minimum value of the design variable xmin.

[0013] Preferably, Step 2, namely finite element analysis, performs topology optimization finite element analysis, discretizes the design domain into finite elements, and establishes a topology optimization model. Perform fluid pressure load coupling analysis in mworks to calculate the field distributions such as displacement, stress, and pressure of the design domain. The objective function of the model is to minimize compliance, and the volume constraint is converted into a constraint condition in the optimization model.

[0014] Furthermore, the establishment of the topology optimization model in Step 2 is as follows:

[0015]

[0016] s.t: Ap = 0

[0017] Ku = F = -Tp

[0018]

[0019] 0 ≤ x e ≤ 1

[0020] where f 0 (u, x e ) is the structural flexibility, K and u are the global stiffness matrix and displacement vector respectively. A, T, F, and p are the global flow matrix, transformation matrix, nodal force vector, and pressure vector respectively. In addition, V f and V * are the material volume fraction and volume constraint respectively. Note that all mechanical equilibrium equations are satisfied under the small deformation assumption.

[0021] Preferably, in step 2, the adjoint matrix method is used to calculate the sensitivities of the objective function and constraint conditions with respect to the design variables. During the iteration process, the sensitivities are smoothed to improve the stability and convergence of the algorithm.

[0022] Furthermore, in step 2, the step of calculating the sensitivities of the topology optimization problem uses the adjoint matrix method to solve the sensitivities of the objective function and constraint conditions. The sensitivity analysis of the objective function is as follows:

[0023]

[0024] By the chain rule of differentiation:

[0025] The sensitivity analysis of the objective function is as follows:

[0026] By the chain rule of differentiation:

[0027] Preferably, in step 3, i.e., the design variable update: the floating projection method is used to update the design variables in MWorks.

[0028] Preferably, the floating projection method includes iteratively updating the design variables using the MMA optimization method, adopting a filtering scheme to modify the design variables into a specified form to simulate the 0 / 1 constraint of the design variables, gradually pushing the design variables towards 0 / 1 to simulate the 0 / 1 constraint of the design variables, repeating the finite element analysis, sensitivity calculation, and design variable update until the convergence condition is reached, applying implicit multiple floating projection constraints, and further modifying the design variables to strengthen the quasi-discrete constraint of the design variables.

[0029] Furthermore, in step 3, the design variables are sequentially updated using the floating projection method in MWorks as follows:

[0030] Iteratively update the design variables according to the MMA optimization method

[0031] Then, modify the design variables using a filtering scheme as:

[0032] Centered on element e, specify a radius r min Draw a circle with this radius to generate a filter support domain N C . Where w ej is the filter weight factor based on the distance r ej between element e and element j, and is defined as:

[0033]

[0034] where r min is the filtering 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. To strengthen the quasi-discrete constraints of the design variables, implicit multiple floating projection constraints are imposed, and the design parameters are further modified as:

[0035]

[0036] Here, η = 0.5, β 1 > 0. Apply the implicit filter and floating projection constraints to the design field and further update through the following formula

[0037]

[0038] The floating threshold th is determined by . β > 0 defines the strictness of the 0 / 1 constraints of the design variables. β starts from a small positive value, such as 10 -6 , which means that initially no 0 / 1 constraints are imposed.

[0039] Furthermore, the maximum change in the design variables is less than 0.01 and the total number of iterations for each β should be greater than 20 and less than 100 to achieve the element-based convergent solution for the current β:

[0040]

[0041] 20 ≤ β < 100.

[0042] Preferably, in step 4, i.e., the smoothing design: convert the design variables into a level set function to describe the topological configuration of the structure, smooth the extracted design to obtain a more continuous structure topology, analyze the optimization result, evaluate the performance and reliability of the structure. If the iteration converges, perform a smoothing design on the structure, which is the level set function of the unit basis solution represented by the variable, and project the extracted design back to the fixed grid to calculate the volume fraction of the elastic material in each unit.

[0043] Furthermore, the specific method of the smoothing design in step 4 is as follows:

[0044] Once the convergence criterion for a given β is satisfied, check whether the solution of the original quasi-discrete topology optimization problem is acceptable. Construct a series of level set functions of the unit basis designs represented by the design variables.

[0045] Preferably, in step 5, determine whether strict convergence is achieved by calculating the difference between the extracted smoothing design objective function and the optimized design objective function.

[0046] Furthermore, the specific method for determining whether strict convergence is achieved in step 5 is as follows:

[0047] Perform a finite element analysis again to calculate the extracted smoothing design objective function f s (v e ). The difference between the objective function of the unit basis solution and the smoothing design solution can be expressed as:

[0048]

[0049] Once the extracted smoothing design on the surface is close to the optimized design, the entire optimization stops.

[0050] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0051] The present invention has the following beneficial effects:

[0052] Simplify the model and reduce the workload: By using the implicit floating projection constraint, all penalty coefficients are replaced by one parameter, greatly simplifying the model, reducing the workload of manually selecting penalty coefficients, and improving the stability and convergence of the algorithm;

[0053] Obtain a smooth optimization result: By converting the design variables into a level set function and performing smoothing, a more continuous structure topology can be obtained, avoiding the mutations and non-smooth boundaries that occur in traditional methods;

[0054] Improve the optimization efficiency: The method of the present invention has higher computational efficiency and can obtain the optimization result faster;

[0055] In summary, the three - field floating projection topology optimization method used in the present invention creatively integrates smoothed codes into the topology optimization method, enabling the obtained optimized configuration to have a better smooth edge. The β - iteration gradually pushes the design towards 0 / 1, and uses a single parameter to replace all penalty coefficients, greatly simplifying the model and reducing the workload for optimization design. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 FIG. is a schematic structural diagram of an internal pressure structure optimized by the method of the present invention;

[0057] Figure 2 FIG. is an iteration diagram of an internal pressure structure optimized by the method of the present invention;

[0058] Figure 3 FIG. is the element - based solution and its smoothed solution of an internal pressure structure optimized by the method of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0059] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0060] Embodiment 1

[0061] As Figures 1 - 3 shown, the pressure - load - dependent three - field floating projection topology optimization method proposed by the present invention has the following specific steps:

[0062] Step 1: Model establishment and parameter setting

[0063] Design domain definition and mesh generation: Use MWorks software to define a rectangular design domain with dimensions of 200 mm × 100 mm;

[0064] Establish a structure with an internal pressure of 1 bar at the bottom and zero pressure at the remaining boundaries, where the dimensions in the x and y directions are Lx = 200 mm and Ly = 100 mm respectively. Fixed constraints are applied to the bottom on the left and right sides of the design domain, the plane thickness is 1 mm, and square meshes with a width of 1 mm are used under plane stress conditions;

[0065] Material property definition:

[0066] Define two materials: poro - elastic material and solid material;

[0067] Solid material: The Young's modulus and Poisson's ratio are selected according to the actual material;

[0068] Poroelastic material: The Young's modulus and Poisson's ratio are set to 3×10 9 Nm-2 and 0.30 respectively.

[0069] Load and constraint conditions:

[0070] The bottom of the design domain is subjected to an internal pressure of 1 bar, and fixed constraints are applied to the bottoms of the left and right sides;

[0071] Initial parameter settings:

[0072] The volume fraction is set to 0.25, representing the proportion of solid material in the optimized structure, and the filtering radius is set to to control the smoothness of the design variables.

[0073] Step 2: Perform topology optimization finite element analysis, discretize the design domain into finite elements, and establish a topology optimization model. Conduct a fluid pressure load coupling analysis in mworks, calculate the field distributions of displacement, stress, pressure, etc. in the design domain. The objective function of the model is to minimize compliance, and the volume constraint is converted into a constraint condition in the optimization model.

[0074] The establishment of the topology optimization model in Step 2 is as follows:

[0075]

[0076] s.t: Ap = 0

[0077] Ku = F = -Tp

[0078]

[0079] 0 ≤ x e ≤ 1

[0080] where f 0 (u, x e ) is the structural compliance, K and u are the global stiffness matrix and displacement vector respectively. A, T, F, and p are the global flow matrix, transformation matrix, nodal force vector, and pressure vector respectively. In addition, V f and V * are the material volume fraction and volume constraint respectively. Note that all mechanical equilibrium equations are satisfied under the small deformation assumption.

[0081] In Step 2, the adjoint matrix method is used to calculate the sensitivities of the objective function and constraint conditions with respect to the design variables. During the iteration process, the sensitivities are smoothed to improve the stability and convergence of the algorithm.

[0082] The steps for calculating the sensitivities of the topology optimization problem in Step 2 use the adjoint matrix method to solve the sensitivities of the objective function and constraint conditions. The sensitivity analysis of the objective function is as follows:

[0083]

[0084] By the chain rule of differentiation:

[0085] The sensitivity analysis of its objective function is:

[0086] By the chain rule of differentiation:

[0087] Step 3: Update the design variables using the floating projection method in MWorks.

[0088] The floating projection method includes iteratively updating the design variables using the MMA optimization method, adopting a filtering scheme to modify the design variables into a specified form to simulate the 0 / 1 constraint of the design variables, gradually pushing the design variables towards 0 / 1 to simulate the 0 / 1 constraint of the design variables, repeating the finite element analysis, sensitivity calculation, and design variable update until the convergence condition is reached, applying the implicit multiple floating projection constraints, and further modifying the design variables to strengthen the quasi-discrete constraint of the design variables.

[0089] The specific process of updating the design variables in turn using the floating projection method in MWorks in Step 3 is as follows:

[0090] Iteratively update the design variables according to the MMA optimization method

[0091] Then adopt a filtering scheme to modify the design variables to:

[0092] With element e as the center and a specified radius r min Draw a circle with this radius to generate a filter support domain of N C . Where w ej is the filter weight factor based on the distance r ej between element e and element j, and is defined as:

[0093]

[0094] where r min is the filtering radius specified by the user. In the floating projection method, the implicit floating projection constraint is used to simulate the 0 / 1 constraint of the design variables. To strengthen the quasi-discrete constraint of the design variables, the implicit multiple floating projection constraints are applied, and the design parameters are further modified to:

[0095]

[0096] where η = 0.5, β 1 > 0, apply the implicit filter and floating projection constraints to the design field, and further update through the following formula

[0097]

[0098] The floating threshold th is determined by β > 0 defines the strictness of the 0 / 1 constraint of the design variable. β starts from a small positive value, such as 10 -6 , which means that the 0 / 1 constraint is not imposed initially.

[0099] The maximum change of the design variable is less than 0.01 and the total number of iterations for each β should be greater than 20 and less than 100, then the cell-based convergent solution for the current β can be achieved:

[0100]

[0101] 20 ≤ β < 100.

[0102] Repeat steps 2 and 3 until the convergence condition is met.

[0103] Step 4 converts the design variable into a level set function to describe the topological configuration of the structure, smooths the extracted design to obtain a more continuous structure topology, analyzes the optimization result, evaluates the performance and reliability of the structure. If the iteration converges, perform a smooth design on the structure, the level set function of the cell-based solution represented by the variable, and project the extracted design back to the fixed grid to calculate the volume fraction of the elastic material in each cell.

[0104] The specific method of the smooth design in step 4 is as follows:

[0105] Once the convergence criterion for a given β is met, check whether the solution of the original quasi-discrete topology optimization problem is acceptable. Construct a series of level set functions of the cell-based designs represented by the design variables.

[0106] In step 5, it is judged whether the strict convergence is achieved by calculating the difference between the extracted smooth design objective function and the optimized design objective function.

[0107] The specific method of judging whether the strict convergence is achieved in step 5 is as follows:

[0108] Perform another finite element analysis to calculate the extracted smooth design objective function f s (v e ). The difference between the objective functions of the cell-based solution and the smooth design solution can be expressed as:

[0109]

[0110] Once the smooth design extracted from the surface is close to the optimized design, the whole optimization stops.

[0111] Finally, the iteration curve of the model is asFigure 2 As shown, the final unit fundamental solution and its smoothed solution are as Figure 3 shown, and the model is proven to be applicable to the fluid pressure load model.

[0112] The above specific embodiments are merely several preferred embodiments of the present invention. Based on the technical solution of the present invention and the relevant revelations of the above embodiments, those skilled in the art can make various alternative improvements and combinations to the above specific embodiments.

[0113] For those skilled in the art, it is obvious that the present invention is not limited to the details of the above exemplary embodiments, and can be implemented in other specific forms without departing from the spirit or basic characteristics of the present invention. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-limiting. The scope of the present invention is defined by the appended claims rather than the above description. Therefore, it is intended to embrace all changes falling within the meaning and scope of the equivalent elements of the claims in the present invention.

Claims

1. A floating projection topology optimization method for fluid pressure load based on mWorks includes the following six steps: step 1: defining the design domain, load boundary conditions, and meshing in MWorks; and defining initial optimization parameters at the same time; step 2: performing finite element analysis, calculating the objective function, constraint function, and sensitivity of the corresponding unit; step 3: updating the design variables using the floating projection method in MWorks; step 4: if the iteration converges, the structure is smoothly designed; if not, steps 2-3 are repeated; step 5: judging whether the smooth solution converges; if not, steps 2-4 are repeated until the program is strictly converged.

2. The mWorks-based fluid pressure load floating projection topology optimization method according to claim 1 is characterized by: 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.

3. The mWorks-based fluid pressure load floating projection topology optimization method according to claim 2 is characterized in that: 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, flow coefficient, drainage coefficient, and density.

4. The mWorks-based fluid pressure load floating projection topology optimization method according to claim 2 is characterized in that: The load and constraint conditions include defining the pressure load boundary of the design domain and the boundary constraint, and the initial parameter setting includes setting the initial optimization parameters, including the filter radius rmin and the minimum value xmin of the design variable.

5. The mWorks-based fluid pressure load floating projection topology optimization method according to claim 1 is characterized by: The step 2 is finite element analysis, which performs topology optimization finite element analysis, discretizes the design domain into finite elements, and establishes a topology optimization model. Fluid pressure load coupling analysis is performed in mworks to calculate the displacement, stress, pressure and other field distributions of the design domain. The model objective function is to minimize flexibility, and the volume constraints are converted into constraints in the optimization model.

6. The mWorks-based fluid pressure load floating projection topology optimization method according to claim 5 is characterized by: In step 2, 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.

7. The mWorks-based fluid pressure load floating projection topology optimization method according to claim 1 is characterized by: The step 3 is design variable updating: updating the design variables using the floating projection method in MWorks.

8. The mWorks-based fluid pressure load floating projection topology optimization method according to claim 7 is characterized in that: The floating projection method includes iteratively updating the design variables using the MMA optimization method, modifying the design variables to a specified form using a filtering scheme to simulate the 0 / 1 constraints of the design variables, gradually pushing the design variables to 0 / 1 to simulate the 0 / 1 constraints of the design variables, repeating finite element analysis, sensitivity calculation and design variable updating until convergence conditions are reached, applying implicit multiple floating projection constraints, and further modifying the design variables to strengthen the quasi-discrete constraints of the design variables.

9. The mWorks-based fluid pressure load floating projection topology optimization method according to claim 1 is characterized by: The step 4 is smooth design: converting the 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 level set function of the unit basis solution represented by the variables, and projecting the extracted design back to the fixed grid to calculate the volume fraction of the elastic material in each unit.

10. The mWorks-based fluid pressure load floating projection topology optimization method according to claim 1, characterized in that: In step 5, whether strict convergence is achieved is determined by calculating the difference between the extracted smooth design objective function and the optimized design objective function.

Images