Topology Optimization Method of Functionally Graded Structures Based on Elastic Hysteresis
Through the topology optimization method of functional gradient structure based on elastic hysteresis, the Helmholtz PDE equation and Heaviside function are used for optimization design, which solves the problem of large elastic hysteresis loss in functional gradient structures, and improves the stability and service life of the structure.
Patent Information
- Application Number
- CN202310294686.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-03-24
- Publication Date
- 2025-06-20
- Estimated Expiration
- 2043-03-24
AI Technical Summary
The prior art is difficult to effectively reduce the elastic hysteresis loss of functional gradient structures, resulting in the impact of structural stability and service life.
The topology optimization method of functional gradient structure based on elastic hysteresis is adopted. By defining hysteresis energy loss and design variables, the Helmholtz PDE equation and Heaviside function are used for optimization design to ensure that the structure has small relative density, high intensity and small hysteresis energy loss are small.
It realizes that while ensuring the continuity of the structural unit, reduces the hysteresis loss of the structure, improves the stability and service life of the structure, and obtains excellent mechanical properties.
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Figure CN116469490B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of structural topology optimization, and in particular to a functional gradient structural topology optimization method based on elastic hysteresis. Background Art
[0002] When an elastic element is loaded and unloaded within the elastic region, due to the strain lagging behind the stress, the loading curve and the unloading curve do not coincide to form a closed loop, which is called elastic hysteresis. For example, rubber material is the main component of a tire, with viscoelasticity and hysteresis. The rolling resistance of a wheel mainly comes from the internal elastic hysteresis loss, that is, the work done by the tire deformation cannot be fully recovered. Hysteresis is manifested in that the magnitude of the restoring force when the material is deformed by force is not only affected by the current deformation amount, but also related to the deformation and speed at the previous moment. Due to the existence of elastic hysteresis, the temperature of the component increases, which has a certain impact on the structure and performance. Therefore, reducing the elastic hysteresis loss of the structure is beneficial to improving the stability and service life of the structure.
[0003] A functionally graded structure is a structure with the characteristic of gradually changing performance. Using the topology optimization method to optimize the design of functionally graded (porous) structures has been widely applied in the fields of aerospace, military, nuclear energy, medical, automotive, architecture, etc. However, the current optimization methods are mainly applicable to single-region structural design, and there are problems such as non-connection or poor connectivity of different functional regions within the structure, and it is difficult to meet the design requirements of elastic multi-functional gradient structures, resulting in large hysteresis energy losses in the structure. Summary of the Invention
[0004] Aiming at the deficiencies of the prior art, the present invention provides a functional gradient structural topology optimization method and system based on elastic hysteresis, aiming to reduce the hysteresis loss of the structure while ensuring the continuity of the structural units.
[0005] The technical solution adopted by the present invention is as follows:
[0006] The present application provides a functional gradient structural topology optimization method based on elastic hysteresis, including:
[0007] Defining the hysteresis energy loss: establishing an elastic hysteresis model of the structure to be optimized, and calculating the hysteresis energy loss E of the structure l ;
[0008] Defining the design variables: dividing the structure to be optimized into several units, filtering the relative density ρ of each unit by using the Helmholtz PDE equation i to obtain the filtered relative density of the unit and projecting by using the Heaviside function to obtain the projected relative density of the unit The relative density ρi The material density of the structure The density ratio to the reference material, with the subscript i being the serial number of the element. According to Calculate the relative density of the overall structure
[0009] Discretize the design domain into several regions Ω, and define the density of the region as the local density
[0010] Using the relative density of the overall structure As the design variable, with the goal of minimizing the structural compliance C (structural strain energy), use the Helmholtz PDE equation to add corresponding constraints to the local density of each region At the same time, using the hysteretic energy loss E l And the volume as constraints, establish a multi-level functionally graded structure topology optimization model:
[0011]
[0012]
[0013] In the formula, U and K are the displacement vector and stiffness matrix of the overall structure respectively, Is the elastic modulus of the structure, u and k0 are the displacement vector and stiffness matrix of the element respectively; R is the filtering radius when filtering the relative density ρ i ▽ is the Laplace operator; Is the constraint radius of the region Ω, Is the local density of the j-th region Ω j ; α is the local density threshold; F is the load vector applied to the structure, F l Is the loading force, F u Is the unloading force, z is the vertical displacement, E * Is the set hysteretic energy loss threshold; V * Is the volume of the optimized structure, V is the initial volume of the structure, f is the set volume fraction; n and l are the total number of divided elements and the total number of regions respectively;
[0014] Iteratively solve the multi-level functionally graded structure topology optimization model to obtain the optimal design variables that meet the optimization goal, and obtain the optimal topological structure based on the optimal design variables.
[0015] Furthermore, use the Heaviside function to Perform projection to obtain the projected relative density of the element The projection function is as follows:
[0016]
[0017] In the formula, d is the projection parameter.
[0018] Furthermore, according to the relative density of the overall structure is calculated in the following way
[0019] In the formula, n is the total number of divided elements.
[0020] Furthermore, the elastic modulus of the structure is determined as follows:
[0021] The elastic modulus E(ρ i ) of the i-th element is determined by using the SIMP material interpolation method:
[0022]
[0023] In the formula, E0 is the elastic modulus of the material, and E min represents a constant to avoid singularity of the stiffness matrix k0; p is the penalty factor, which is used to penalize the intermediate density elements so that the relative density of the intermediate density elements tends to 0 or 1 as much as possible. The intermediate density elements are one or several elements with a density in the middle of the density value range;
[0024] The arithmetic mean of E(ρ i ) is calculated to obtain
[0025] Furthermore, an elastic hysteresis model of the structure to be optimized is established, and the hysteresis energy loss E l of the structure is calculated, including:
[0026] Taking the structure to be optimized as a nonlinear system, a differential equation of the nonlinear system is established:
[0027]
[0028] In the formula, F, m, z, are respectively the external force applied to the structure, the mass of the structure, the vertical displacement, the vertical velocity, and the vertical acceleration. C(z) and K(z) are respectively the composite damping coefficient and the composite elastic coefficient of the structure;
[0029] A parameter identification matrix is established through the measured data of the experiment, and the response loading curve and the unloading curve are fitted through the measured data of loading the external force and unloading the external force. E l is represented by the area enclosed by the loading curve and the unloading curve, that is:
[0030] E I = ∫|F l - F u |dz.
[0031] Further, the multi - level functional gradient structure topology optimization model is iteratively solved to obtain the optimal design variables that meet the optimization objectives, including:
[0032] Calculate the optimization objective C and the hysteresis energy loss E l The sensitivity of the design variables;
[0033] According to the sensitivity, the Moving Asymptote Method (MMA) is used to iteratively update the design variables to obtain the optimal design variables.
[0034] This application also provides a computer - readable storage medium, in which at least one instruction is stored, and the at least one instruction is loaded and executed by a processor to implement the functional gradient structure topology optimization method based on elastic hysteresis.
[0035] This application also provides a computer device, which includes a processor and a memory. At least one instruction is stored in the memory, and the instruction is loaded and executed by the processor to implement the functional gradient structure topology optimization method based on elastic hysteresis.
[0036] The beneficial effects of the present invention are as follows:
[0037] This application divides the design domain into multiple units, and uses the Helmholtz PDF equation to filter the density deformation of each unit to eliminate numerical instability problems. At the same time, different local constraint conditions are imposed on the discretized region, and the local constraints are integrated into the variable - density topology optimization method, achieving the effect of multi - region functional gradient microstructures, while maintaining good continuity between regions. The method of this application considers the hysteresis energy loss constraint to ensure that it is less than a specified threshold. The functional gradient structure designed by the method of this application has excellent mechanical properties such as a small relative density, high strength, and small hysteresis energy loss.
[0038] Other features and advantages of the present invention will be described in the following specification, and, in part, will be obvious from the specification, or will be understood by implementing the present invention. BRIEF DESCRIPTION OF THE DRAWINGS
[0039] Figure 1 It is a flowchart of the method of the embodiment of the present invention.
[0040] Figure 2 It is a projection function curve graph under different projection function parameters of the embodiment of the present invention.
[0041] Figure 3 It is a structural schematic diagram of the design object of the embodiment of the present invention.
[0042] Figure 4 Schematic diagram of the topology optimization result of the design object of the embodiment of the present invention. Specific implementation manners
[0043] The following describes the specific implementation manners of the present invention with reference to the accompanying drawings.
[0044] See Figure 1 , the embodiment of the present application provides a topology optimization method for a functionally graded structure based on elastic hysteresis, including:
[0045] Define the hysteresis energy loss: establish an elastic hysteresis model of the structure to be optimized, and calculate the hysteresis energy loss E l , including:
[0046] Regard the structure to be optimized as a nonlinear system, and establish a differential equation of the nonlinear system:
[0047]
[0048] In the formula, F, m, z, are respectively the external force applied to the structure, the mass of the structure, the vertical displacement, the vertical velocity and the vertical acceleration, and C(z) and K(z) are respectively the composite damping coefficient and the composite elastic coefficient of the structure;
[0049] Establish a parameter identification matrix through the measured data of the experiment, and fit the response loading curve and the unloading curve through the measured data of applying the external force and unloading the external force. E l is represented by the area enclosed by the loading curve and the unloading curve, that is:
[0050] El = ∫|F l -F u |dz
[0051] Define the design variables: divide the structure to be optimized into several units, and use the Helmholtz PDE equation to filter the relative density ρ i of each unit to obtain the filtered relative density of the unit and use the Heaviside function to project to obtain the projected relative density of the unit The projection function is as follows:
[0052]
[0053] Among them, the relative density ρ i is the ratio of the material density of the structure to the density of the reference material, the subscript i is the serial number of the unit, and d is the projection parameter;
[0054] After filtering the design variables, gray elements are generated at the structural boundary, which may lead to problems such as unclear structural boundaries and poor manufacturability. Therefore, in this application, the Heaviside function is used to project the filtered physical variables to obtain an optimized structure with clear boundaries;
[0055] According to the relative density of the overall structure is calculated in the following way
[0056] where n is the total number of divided elements,
[0057] The design domain is discretely divided into several regions Ω, and the density of the region is defined as the local density
[0058] Taking the relative density of the overall structure as the design variable and aiming at minimizing the structural compliance C, the Helmholtz PDE equation is used to add corresponding constraints to the local density of each region. At the same time, taking the hysteretic energy loss E l and volume as constraints, a multi-level functionally graded structure topology optimization model is established:
[0059]
[0060]
[0061] In the above formula, U and K are the displacement vector and stiffness matrix of the overall structure respectively, is the elastic modulus of the structure, u and k0 are the displacement vector and stiffness matrix of the element respectively; R is the filtering radius when filtering the relative density ρ i ▽ is the Laplace operator; is the constraint radius of the region Ω, is the local density of the j-th region Ω j α is the local density threshold; F is the load vector applied to the structure, F l is the loading force, F u is the unloading force, z is the vertical displacement, E * is the set hysteretic energy loss threshold; V * is the volume of the optimized structure, V is the initial volume of the structure, f is the set volume fraction; n and l are the total number of divided elements and the total number of regions respectively;
[0062] Among them, the determination method of the elastic modulus of the structure is:
[0063] The SIMP material interpolation method is used to determine the elastic modulus E(ρ of the i-th elementi ):
[0064]
[0065] In the formula, E0 is the elastic modulus of the material, and E min represents a constant to avoid singularity of the stiffness matrix k0, and in this embodiment, it takes 0.001; p is a penalty factor used to penalize the intermediate density elements so that the relative density of the intermediate density elements tends to 0 or 1 as much as possible. The intermediate density elements are one or several elements with a density in the middle section of the density value range;
[0066] Calculate the arithmetic mean of E(ρ i ), and obtain That is:
[0067] where i is the element number and n is the total number of divided elements;
[0068] Iteratively solve the multi-level functionally graded structure topology optimization model to obtain the optimal design variables that meet the optimization objectives, including:
[0069] Calculate the optimization objective C and the hysteresis energy loss E l for the sensitivity of the design variables:
[0070]
[0071] In the above formula, ρ is the initial relative density of the element, is the relative density after filtered projection, is the filtered relative density of the element. See Figure 2 for the projection function curve graph when the projection parameter d takes different values, which has symmetry. The horizontal and vertical coordinates are the relative densities before and after projection respectively. In the horizontal coordinate interval [-1, 1], this projection function projects the elements with a density value greater than 0 to 1 and the elements with a density value less than 0 to 0. The value of d will affect the iteration speed. The smaller d is, the stricter the projection division is, the clearer the result is, and the slower the iteration speed is. In order to obtain a clear structure and accelerate the iteration speed at the same time, the value of d gradually decreases as the number of iteration steps increases;
[0072] According to the sensitivity, use the moving asymptote method MMA to iteratively update the design variables to obtain the optimal design variables, and obtain the optimal topology structure based on the optimal design variables.
[0073] In order to generate a microstructure with gradually changing performance in the design area in this application, by adding different local density constraints in each area and driven by physical performance, the optimal solution is obtained to calculate different constraint radii at different positions, so as to achieve the effect of constructing different functionally graded microstructures in different areas.
[0074] The method of the present application will be further described below with specific examples.
[0075] As Figure 3 shown, the design object of this example is a bracket structure made of a linearly elastic material, structural steel, and its topology optimization design is carried out. Specifically, when implementing, COMSOL Multiphysics is used for finite element analysis and joint debugging is carried out through Matlab programming.
[0076] Specifically, create a model in COMSOL Multiphysics, define the design domain, load, and constraint conditions, introduce design variables using the PDF module, then initialize the mesh, store the model file, call Matlab to open the model, expand the mesh, solve for the initial values, and then iteratively calculate the objective function, constraint function, and sensitivity to the design variables of the model, and use the MMA algorithm to solve until convergence. The output optimized design structure is as Figure 4 shown. It can be proved from the optimized results that the structure inside the material has a complex geometric shape, the functional levels are more abundant, and the structures between regions still maintain good continuity.
[0077] Among them, the design domain is discretized using a hexagonal mesh of 100×100×100, the local constraint radius is 2r < R < 4r, where r is the mesh size, the initial value of the projection parameter d is 0.5, and its value gradually decreases with the number of iteration steps. The bottom of the design domain is fixed, a unit load is applied to its two side surfaces, the material elastic modulus E = 1, and the Poisson's ratio v = 0.3.
[0078] The embodiment of the present application also provides a computer-readable storage medium, in which at least one instruction is stored, and the at least one instruction is loaded and executed by a processor to implement the topology optimization method of the functional gradient structure based on elastic hysteresis.
[0079] The embodiment of the present application also provides a computer device, which includes a processor and a memory. At least one instruction is stored in the memory, and the instruction is loaded and executed by the processor to implement the topology optimization method of the functional gradient structure based on elastic hysteresis.
[0080] Those of ordinary skill in the art can understand that the above are only the preferred embodiments of the present invention and are not used to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, for those skilled in the art, they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.
Claims
1. A topology optimization method for functionally graded structures based on elastic hysteresis, characterized in that, Including: Define the hysteretic energy loss: Establish an elastic hysteresis model for the structure to be optimized and calculate the hysteretic energy loss E of the structure l ; Define the design variables: Divide the structure to be optimized into several elements, and use the Helmholtz PDE equation to filter the relative density ρ of each element i to obtain the filtered relative density of the elements and use the Heaviside function to perform projection to obtain the projected relative density of the elements The relative density ρ i is the ratio of the material density of the structure to the density of the reference material. The subscript i is the element number. Calculate the overall relative density of the structure according to The design domain is discretely divided into several regions Ω, and the density of the region is defined as the local density Taking the relative density of the overall structure as the design variable, aiming at minimizing the structural flexibility C, adding corresponding constraints to the local density of each region by using the Helmholtz PDE equation, while taking the hysteretic energy loss E l and volume as constraints, a multi-level functionally graded structure topology optimization model is established: In the formula, U and K are the displacement vector and stiffness matrix of the overall structure respectively, is the elastic modulus of the structure, u and k0 are the displacement vector and stiffness matrix of the element respectively; R is the filtering radius when filtering the relative density ρ i is the filtering radius when filtering the relative density ρ is the Laplace operator; is the constraint radius of the region Ω, is the local density of the j-th region Ω j α is the local density threshold; F is the load vector applied to the structure, F l is the loading force, F u is the unloading force, z is the vertical displacement, E * is the set hysteretic energy loss threshold; V * is the volume of the optimized structure, V is the initial volume of the structure, f is the set volume fraction; n and l are the total number of divided elements and the total number of regions respectively; Iteratively solve the multi-level functionally graded structure topology optimization model to obtain the optimal design variables that meet the optimization objectives, and obtain the optimal topology structure based on the optimal design variables.
2. The topology optimization method for functionally graded structures based on elastic hysteresis according to claim 1, characterized in that, Use the Heaviside function to perform projection to obtain the relative density of the projected element The projection function is as follows: In the formula, d is the projection parameter.
3. The topology optimization method for functionally graded structures based on elastic hysteresis according to claim 1, characterized in that, According to calculate the relative density of the overall structure in the following manner In the formula, n is the total number of divided units.
4. The topology optimization method for functionally graded structures based on elastic hysteresis according to claim 1, characterized in that, The elastic modulus of the structure The determination method is as follows: The elastic modulus E(ρ of the i-th element is determined by the SIMP material interpolation method i ): where E0 is the elastic modulus of the material, and E min represents a constant to avoid singularity of the stiffness matrix k0; p is a penalty factor used to penalize the intermediate density elements so that the relative density of the intermediate density elements tends to 0 or 1 as much as possible. The intermediate density elements are one or several elements with a density in the middle section of the density value range. Calculate the arithmetic mean of E(ρ i ) to obtain 5. The topology optimization method for functionally graded structures based on elastic hysteresis according to claim 1, characterized in that, Establish an elastic hysteresis model of the structure to be optimized and calculate the hysteresis energy loss E of the structure l , including: Regarding the structure to be optimized as a nonlinear system, establish the differential equation of the nonlinear system: where F, m, z, are respectively the external force applied to the structure, the mass of the structure, the vertical displacement, the vertical velocity, and the vertical acceleration, and C(z) and K(z) are respectively the composite damping coefficient and the composite elastic coefficient of the structure; Establish a parameter identification matrix based on the data measured through experiments, and fit the response loading curve and unloading curve with the measured data of applying external force and unloading external force, E l Represented by the area enclosed by the loading curve and the unloading curve, that is: E l = ∫|F l - F u |dz。 6. The topology optimization method for functionally graded structures based on elastic hysteresis according to claim 1, characterized in that, Iteratively solve the multi-level functionally graded structure topology optimization model to obtain the optimal design variables that meet the optimization objectives, including: Calculate the optimization objective C and the hysteresis energy loss E l The sensitivity of the design variables; According to the sensitivity, use the moving asymptote method MMA to iteratively update the design variables to obtain the optimal design variables.
7. A computer-readable storage medium, characterized in that, At least one instruction is stored in the storage medium, and the at least one instruction is loaded and executed by a processor to implement the functionally graded structure topology optimization method based on elastic hysteresis as described in any one of claims 1 to 6.
8. A computer device, characterized in that, The computer device includes a processor and a memory. At least one instruction is stored in the memory, and the instruction is loaded and executed by the processor to implement the functionally graded structure topology optimization method based on elastic hysteresis as described in any one of claims 1 to 6.