Energy finite element topological optimization method for medium-high frequency vibration curved surface design

Abstract

The invention discloses an energy finite element topological optimization method for medium-high frequency vibration curved surface design, which comprises the following steps of: discretizing a curved surface structure grid, researching the transmission condition of vibration energy among grid curved surfaces by applying an energy finite element method, and researching an energy transmission coefficient of a lower boundary of any coupling angle by adopting a wave theory; the method comprises the following steps: establishing an energy finite element model under a'unit independent grid ', establishing a relation between a curved surface reinforced structure and a plane reinforced structure by adopting a conformal mapping method by taking the length, width, center point coordinates and inclination angles of reinforcing ribs as design variables by adopting a movable deformation component method, and solving energy distribution by utilizing a projection mode; taking the energy flexibility of the curved surface structure as an optimization target, taking the material consumption of the reinforcing rib as a constraint, and applying a moving asymptote optimization method to obtain an optimized reinforcing rib layout; according to the method, energy distribution of various complex curved surface structures can be solved, an optimal design scheme of reinforcing rib distribution is obtained, the performance of the curved surface structures is improved, and the calculation amount is reduced.

Description

Technical Field

[0001] The present invention belongs to the technical field of high-frequency vibration structure optimization design, and particularly relates to an energy finite element topology optimization method for medium- and high-frequency vibration curved surface design. Technical Background

[0002] With the progress of technology, the working environment of mechanical engineering facilities, especially high-end mechanical equipment, is becoming increasingly complex. The operating speed of mechanical structures is getting faster and faster, and the medium- and high-frequency vibration problems are prominent. The curved surface stiffening structure is also an important and common part among them. How to effectively design the curved surface stiffening configuration and improve the performance under medium- and high-frequency vibration has become a difficult problem.

[0003] When the classical finite element method is faced with medium- and high-frequency large-size structures, the mesh size should be less than one-sixth of the mechanical wave wavelength, otherwise the calculation results will be inaccurate; this leads to an increase in mesh density and calculation amount. At the same time, the mode superposition problem existing in the high-frequency band will also lead to inaccurate calculation results. Establishing an analysis method from the perspective of energy can avoid the above problems, and thus the energy finite element method has begun to be taken seriously by researchers.

[0004] The energy field is a typical discontinuous physical field in the curved surface structure. In the analysis of discontinuous physical fields, the common finite element method usually adds nodes on both sides of the discontinuous interface to simulate the discontinuity of the physical field; in the topology optimization process, the discontinuous boundaries in the design domain are constantly changing, and the mesh must be continuously rebuilt, resulting in a large amount of calculation and low efficiency. Therefore, it is necessary to construct a new type of mesh that can adapt to any configuration change. For example, a patent application named "An Energy Finite Element Analysis Method Adapted to the Dynamic Change of the Wave Group Transmission Interface" (publication number CN111832205A) is applied to the plane structure, while the design of the combination of energy finite element and topology optimization in the curved surface structure is lacking.

[0005] Based on the above technical background, analysts currently urgently need a method that can realize the optimization design of medium- and high-frequency curved surface stiffening structures at a relatively low calculation cost; topology optimization has a high design freedom. Combining the energy finite element method with topology optimization can provide a new and effective solution for the optimization design of curved surface structures. Summary of the Invention

[0006] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide an energy finite element topology optimization method for medium- and high-frequency vibration curved surface design. By applying the energy finite element method, the energy finite element model is constructed under the "unit independent mesh", which can adapt to the discontinuous change of the energy density field in the topology optimization process, without mesh reconstruction, greatly reducing the amount of operation and improving the operation efficiency.

[0007] In order to achieve the above objectives, the technical solution adopted by the present invention is:

[0008] An energy finite element topology optimization method for medium and high frequency vibration curved surface design, comprising the following steps:

[0009] 1) Preprocessing:

[0010] 1.1) Finite element analysis: Define density, elastic modulus, and Poisson's ratio according to material properties, perform finite element analysis on the initial curved surface structure geometric model, and define the target optimization frequency according to the actual working conditions;

[0011] 1.2) Unit independent grid processing: Discretize the curved surface structure design domain into a grid composed of four-node rectangular elements, import the grid information (grid and node numbers, node coordinates) into the energy finite element model, and perform independent grid processing. The grid cells do not share nodes and boundaries, and each cell is numbered counterclockwise starting from the bottom left node to obtain a "unit independent grid" without common nodes;

[0012] 2) Construct the boundary conditions of the energy finite element model: Initialize the maximum number of iterations k max 、 Initialize the maximum allowable error ε for optimization termination, and initialize the upper limit M of material consumption upp , Define the loaded node numbers, sizes, and constrained node numbers under the unit independent grid according to the actual working conditions;

[0013]

[0014] F is the node load size, J is the Jacobian matrix, and π in (x, y) is the energy input at the node (x, y);

[0015] 3) Construct the level set description model:

[0016] 3.1) Initialize the stiffener layout: Adopt the movable deformation component method, describe the stiffener structure using components, and describe the stiffeners on the overall curved surface structure by overlapping components one by one. Initialize the component design variables according to the actual curved surface stiffening structure layout. The design variables include length, width, center point coordinates, and tilt angle. Complete the initial component layout and use the level set description function to complete the geometric description of the components in the plane structure space;

[0017]

[0018] In the formula: φ(x, y) is the level set of the component at the point (x, y); L, T, θ, x0, and y0 are the length, width, tilt angle, and center point coordinates of a component respectively;

[0019] 3.2) Conformal mapping: The conformal mapping method is adopted. The surface grid structure is used as the physical space, and the planar structure is used as the parametric space. On the basis of retaining the surface structure characteristics, a topological conformal mapping relationship is established between the physical space and the parametric space. The independently meshed elements in the physical space that have been obtained are transformed onto the parametric space. The geometric description of the stiffeners is carried out in the planar parametric space, and then the information of the elements is mapped back to the physical space through the topological conformal mapping relationship to complete the layout of the stiffeners on the surface structure;

[0020] 4) Project to construct an energy finite element calculation model: The level set description model of the stiffeners is projected onto the independently meshed elements by the density method to obtain the thickness of the stiffeners on each element. The material consumption of the stiffeners is calculated according to Equation (3) and used as the optimization constraint function;

[0021]

[0022] In the formula, M is the material consumption of the stiffeners, h is the element thickness (surface thickness + stiffener thickness), S is the element area, o is the element number, and n is the total number of elements;

[0023] 5) Construct an energy finite element model:

[0024] 5.1) Solve the element energy matrix and the boundary energy matrix: According to the energy finite element calculation model obtained by projection, after obtaining the thickness of each element, the element energy matrix K is calculated according to Equation (4). At the same time, according to the wave theory, the reflection coefficient and refraction coefficient between the coupled elements are calculated, and then the boundary energy matrix Q between the coupled elements is calculated according to Equation (5);

[0025]

[0026]

[0027] In the formula: K is the element stiffness matrix, c g is the group velocity of elastic wave propagation, including flexural wave f, longitudinal wave l, and shear wave s, η is the damping loss factor of the structure, e is the elastic wave energy density ignoring the near-field effect, N is the shape function, x and y are the global coordinate systems, Ω o is the element design domain, τ is the reflection coefficient, q is the element boundary energy flow, Q is the boundary energy matrix, Г o is the element boundary;

[0028] 5.2) Assemble the element energy matrix and the boundary energy matrix under the "independently meshed elements": The assembly rules of the independently meshed elements are constructed to obtain the total matrix form of the energy finite element model. Finally, the energy density distribution of the design domain is analyzed based on this, and the energy compliance of the surface stiffened structure is calculated according to Equation (6) and used as the optimization objective function;

[0029]

[0030] In the formula: J is the objective function: the energy flexibility of the curved surface stiffened structure, which reflects the magnitude of the vibration energy stored in the structure;

[0031] 6) Construct an optimization model: Adopt the moving asymptote method as the optimization method. According to the relational expressions of the design variables, objective function, and constraint function, define the sensitivities of the objective function and constraint function to the design variables respectively. Substitute the objective function, constraint function, and their sensitivities into the moving asymptote optimizer to achieve optimization convergence and iterative update of the design variables. When the number of iteration steps reaches the maximum number of iteration steps kmax, or the change in the objective function is lower than the maximum allowable error ε, the optimization terminates and outputs the optimized design result;

[0032]

[0033] In the formula: x is both the geometric parameter of the component and the optimized design variable, including the center point coordinates, length, width, and inclination angle. j is the component number, and n com is the number of components. M(x) and M0 are the actual material usage and the upper limit of material usage respectively, and Ω x is the range of the design variables;

[0034] 7) Analysis and verification before and after the optimization of the curved surface stiffening configuration: Perform finite element analysis on the optimized design configuration obtained by energy finite element topology optimization and the initial configuration, compare the dynamic performance, and verify the effectiveness of the optimization method;

[0035] 8) Adaptability processing: Apply the energy finite element topology optimization method for the design of medium and high-frequency vibration curved surfaces. Complete the assembly of the overall matrix through the element-independent grid assembly rule, solve the energy distribution of the curved surface structure, and complete the topology optimization design.

[0036] Step 5.2) Construct an element-independent grid assembly rule, specifically: For the continuous boundaries that may exist in the curved surface structure, the coupling angle is 0 and the material is continuous. Adjacent nodes need to exchange elements to complete the recoupling after disassembling the nodes; For the boundaries with coupling angles or discontinuities, the disassembled nodes do not need to be recoupled.

[0037] Step 8) For the adaptability processing of the curved surface structure with complex multi-genus and closed curved surfaces in the topological configuration, it is necessary to adjust the element-independent grid assembly rule to adapt to the curved surface characteristics.

[0038] Compared with the prior art, the beneficial effects of the present invention are:

[0039] Since the present invention analyzes the energy response level inside the curved surface structure from the perspective of energy, it can reduce the number of meshes, solve the modal superposition problem, and obtain relatively accurate analysis results under medium and high frequency conditions. Combining with the topology optimization method with high design freedom, a new configuration of the curved surface stiffening structure is obtained, which improves the dynamic performance under medium and high frequency vibrations. To adapt to the discontinuous distribution characteristics of energy in the curved surface structure and the iterative evolution in the topology optimization process, "element-independent meshes" and the matrix assembly form under element-independent meshes are developed, avoiding mesh reconstruction, greatly reducing the computational amount, and accelerating the optimization efficiency. It provides an efficient and feasible solution for the optimization design of curved surface structures under medium and high frequency vibrations. Description of the Drawings

[0040] Figure 1 It is a schematic flow chart of an embodiment of the present invention.

[0041] Figure 2 It is a schematic diagram of the boundary conditions of an embodiment of the present invention.

[0042] Figure 3 It is a schematic diagram of the element-independent mesh division of an embodiment of the present invention.

[0043] Figure 4 It is a schematic diagram of the conformal mapping of an embodiment of the present invention.

[0044] Figure 5 It is a schematic diagram of the density method projection of an embodiment of the present invention.

[0045] Figure 6 It is a schematic diagram of the optimization iteration of an embodiment of the present invention. Detailed Embodiment

[0046] The present invention will be further described below in conjunction with the embodiments and the drawings.

[0047] Refer to Figure 1 , the energy finite element topology optimization method for the design of curved surfaces facing medium and high frequency vibrations includes the following steps:

[0048] 1) Preprocessing:

[0049] 1.1) Finite element analysis: In this embodiment, the structure of the design domain is a semi-cylindrical stiffening structure, which consists of a semi-cylindrical shell and stiffeners, both made of homogeneous aluminum material. The size parameters of the semi-cylindrical shell are φ200mm × 200mm, with a thickness of 5mm, and is strengthened by rectangular stiffeners with a thickness of 5mm. The four corners of the semi-cylindrical shell are fixed, and a high-frequency dynamic load is applied at the center of the upper surface, as Figure 2 shown; the elastic modulus of aluminum is taken as 7.1×10 10 N·m 2 , the Poisson's ratio is taken as 0.25, and the material density is taken as 2700kg / m 2; Perform finite element analysis on the initial surface structure geometric model. In this embodiment, 1000 Hz is selected as the target optimization frequency according to the frequency response curve;

[0050] 1.2) Element independent mesh processing: Discretize the semi-cylindrical shell design domain into a mesh composed of 40×40 four-node rectangular elements. Import the mesh information (mesh and node numbers, node coordinates) into the energy finite element model. To adapt to the discontinuous distribution characteristics of energy in the surface structure and the iterative evolution during the topology optimization process, the mesh is made independent. The mesh elements do not share nodes and boundaries. Each element is numbered counterclockwise starting from the bottom-left node to obtain a "unit independent mesh" without common nodes, which serves as the basis for the energy finite element model. As Figure 2 、 3 shown, the unit independent mesh can represent any configuration without mesh reconstruction;

[0051] 2) Construct the boundary conditions of the energy finite element model: Initialize the maximum number of iterations k max to be 1000, initialize the maximum allowable error ε for optimization termination to be 1e -6 、initialize the upper limit M of material consumption upp to be 40%. According to the actual working conditions, fix 6 elements at the four corner boundaries and apply loading at the center point in proportion. The loading node numbers are 1559, 1562, 3240, 3241, as Figure 2 shown;

[0052]

[0053] where F is the load size on the node, J is the Jacobian matrix, and π in (x,y) is the energy input at the node (x,y);

[0054] 3) Construct the level set description model:

[0055] 3.1) Initialize the layout of stiffeners: Using the movable deformation component method, describe the stiffener structure using components. Stack and overlap the components to describe the stiffeners on the overall surface structure. According to the stiffened structure of the semi-cylindrical shell, the stiffener components with orthogonally distributed initial structure materials are evenly distributed on the entire surface of the semi-cylindrical shell. The component length is 80 mm, the width is 10 mm, and there are a total of 32 design optimization components. Complete the initial component layout and use the level set description function to complete the geometric description of the components in the plane space;

[0056]

[0057] where: φ(x,y) is the level set of the component at the point (x,y); L, T, θ, x0, y0 are the length, width, inclination angle, and center point coordinates of a component respectively;

[0058] 3.2) Conformal mapping: Using the conformal mapping method, with the semi-cylindrical shell as the physical space and the planar structure as the parameter space, a topological conformal mapping relationship is established between the physical space and the parameter space while retaining the surface structure characteristics. The independently meshed elements obtained in the physical space are transformed onto the parameter space, and the geometric description of the stiffeners is carried out in the planar parameter space. Then, through the topological conformal mapping relationship, the information of the elements is mapped back to the physical space to complete the layout of the stiffeners on the semi-cylindrical shell, as Figure 4 shown;

[0059] 4) Projecting to construct the energy finite element calculation model: The level set description model of the stiffeners in the planar parameter space is projected onto the independently meshed elements using the density method. The thicknesses of the unit stiffeners can be obtained as 5mm, 3.75mm, 2.5mm, 1.25mm, and 0mm respectively according to the ratio of the projection density, which serves as the basis for the energy finite element calculation model, as Figure 5 shown. Calculate the material consumption of the stiffeners according to Equation (3) as the optimization constraint function;

[0060]

[0061] In the formula, M is the material consumption of the stiffeners, h is the unit thickness (semi-cylindrical shell thickness + stiffener thickness), S is the unit area, o is the unit number, and n is the total number of units 40×40;

[0062] 5) Construct the energy finite element model:

[0063] 5.1) Solve the element energy matrix and the boundary energy matrix: According to the energy finite element calculation model obtained by projection, after obtaining the thicknesses of each element, calculate the element energy matrix K according to Equation (4). At the same time, according to the wave theory, calculate the reflection coefficient and refraction coefficient between the coupled elements, and then calculate the boundary energy matrix Q between the coupled elements according to Equation (5);

[0064]

[0065]

[0066] In the formula: K is the element stiffness matrix, c g is the group velocity of elastic wave propagation, including flexural wave f, longitudinal wave l, and shear wave s, η is the damping loss factor of the structure, e is the elastic wave energy density ignoring the near-field effect, N is the shape function, x and y are the global coordinate systems, Ω o is the unit design domain, τ is the reflection coefficient, q is the unit boundary energy flow, Q is the boundary energy matrix, and Г o is the unit boundary;

[0067] 5.2) Assemble the element energy matrix and the boundary energy matrix under the "independent unit grid": For the continuous boundary in the semi-cylindrical shell (the coupling angle is 0 and the material is continuous axially), adjacent nodes need to exchange elements to complete the recoupling after disassembling the nodes. For the boundary with a coupling angle or discontinuity (there is a coupling angle in the circumferential direction of the semi-cylindrical shell), the disassembled boundary nodes are coupled through the boundary energy matrix without the need for recoupling and assembly. In this way, the assembly rules under the independent unit grid of the element are constructed to obtain the total matrix form of the energy finite element model. Finally, the energy density distribution of the design domain is analyzed based on this, and the energy flexibility of the stiffened semi-cylindrical shell structure is calculated according to Equation (6) as the optimization objective function;

[0068]

[0069] In the formula: J is the objective function: the energy flexibility of the curved surface stiffened structure, which reflects the magnitude of the vibration energy stored in the structure;

[0070] 6) Construct the optimization model: Use the moving asymptote method as the optimization method. According to the relationship between the design variables, the objective function, and the constraint function, define the sensitivities of the objective function and the constraint function to the design variables respectively. Substitute the objective function, the constraint function, and their sensitivities into the moving asymptote optimizer to achieve optimization convergence and iterative update of the design variables. When the number of iteration steps reaches the maximum number of iterations kmax, or the change in the objective function is lower than the maximum allowable error ε, the optimization terminates and outputs the optimized design result. As Figure 6 shown, the material usage of the structure changes from the original 50% volume ratio to 40% during the optimization iteration process, and the objective function, that is, the energy flexibility, is significantly reduced;

[0071]

[0072] In the formula: x is both the geometric parameter of the component and the optimization design variable, including the center point coordinates, length, width, and inclination angle. j is the component number, and n com is the number of components. Take the energy flexibility as an index to measure the dynamic performance of the structure, and take reducing the energy flexibility of the structure as the design goal. M(x) and M0 are the actual material usage and the upper limit of the material usage respectively, and Ω x is the range of the design variables;

[0073] 7) Analysis and verification of the semi-cylindrical shell stiffened structure before and after optimization: Perform finite element analysis on the optimized design configuration and the initial configuration of the semi-cylindrical shell stiffened structure obtained by energy finite element topology optimization, and compare the dynamic performance (displacement response) under the applied load. The maximum vibration deformation displacements of the original structure and the optimized structure are 11.49×10 -3 mm and 8.18×10 - 3mm, and the average vibration deformation displacements are 3.54×10 -3 mm and 2.96×10 -3 mm respectively; compared with the original structure, the maximum vibration deformation displacement of the optimized structure is reduced by 28.8%, and the average vibration deformation displacement is reduced by 16.4%. Finally, the main distribution of the optimized structure material is on the path from the vibration-loaded area to the fixed constraint. This structure enables the vibration energy to be more fully attenuated, thereby improving the dynamic performance of the entire structure;

[0074] 8) Adaptability treatment: Apply the energy finite element topology optimization method for the design of medium and high frequency vibration surfaces. The overall matrix can be assembled through the unit independent grid assembly rule, and the energy distribution of the stiffened semi-cylindrical shell structure can be solved to complete the topology optimization design. For complex surface structures with complicated topology configurations (such as multi-genus and closed surfaces), it is necessary to adjust the unit independent grid assembly rule to adapt to the surface characteristics.

Claims

1. The energy finite element topology optimization method for medium and high frequency vibration surface design is characterized by: The following steps are involved: 1) Pre-treatment: 1.1) Finite element analysis: Define density, elastic modulus, and Poisson's ratio based on material properties, perform finite element analysis on the initial surface structure geometry model, and define the target optimization frequency based on actual working condition analysis; 1.2) Unit independent mesh processing: discretize the surface structure design domain into a mesh composed of four-node rectangular units, import the mesh information (mesh and node numbers, node coordinates) into the energy finite element model, and process the mesh independently. The mesh units do not share nodes and boundaries. Each unit is numbered counterclockwise starting from the lower left corner node to obtain a "unit independent mesh" without common nodes. 2) Construct boundary conditions of energy finite element model: Initialize the maximum number of iterations k max , the maximum allowable error ε of initialization optimization termination, the upper limit of initialization material consumption M upp , define the loaded node number, size and constraint node number under the unit independent grid according to the actual working conditions; F is the node load size, J is the Jacobian matrix, π in (x,y) is the energy input at the node (x,y); 3) Constructing the level set description model: 3.1) Initialize the rib layout: Use the movable deformable component method to describe the rib structure using components, and use the superposition of components to describe the ribs on the overall curved surface structure. Initialize the component design variables according to the actual curved surface reinforcement structure layout. The design variables include length, width, center point coordinates and inclination angle. Complete the initial component layout, and use the level set description function to complete the geometric description of the component in the plane structure space. Where: φ(x,y) is the level set of the component at point (x,y); L, T, θ, x0, y0 are the length, width, tilt angle and center point coordinates of a component respectively; 3.2) Conformal mapping: A conformal mapping method is used, with the surface mesh structure as the physical space and the plane structure as the parameter space. On the basis of retaining the characteristics of the surface structure, a topological conformal mapping relationship is established between the physical space and the parameter space. The obtained physical space unit independent mesh is converted to the parameter space, and the geometric description of the reinforcement is performed in the plane parameter space. Then, the unit information is mapped back to the physical space through the topological conformal mapping relationship to complete the reinforcement layout on the surface structure. 4) Projection to construct the energy finite element calculation model: The level set description model of the stiffener is projected onto the unit independent grid using the density method to obtain the thickness of the stiffener on each unit. The material consumption of the stiffener is calculated according to formula (3) as the optimization constraint function; In the formula, M is the amount of reinforcement material, h is the unit thickness (surface thickness + reinforcement thickness), S is the unit area, o is the unit number, and n is the total number of units; 5) Construct energy finite element model: 5.1) Solving the unit energy matrix and boundary energy matrix: According to the energy finite element calculation model obtained by projection, after obtaining the thickness of each unit, the unit energy matrix K is calculated according to formula (4), and at the same time, the reflection coefficient and refraction coefficient between the coupled units are calculated according to the wave theory, and then the boundary energy matrix Q between the coupled units is calculated according to formula (5); Where: K is the unit stiffness matrix, c g is the group velocity of elastic wave propagation, including bending wave f, longitudinal wave l and shear wave s, η is the damping loss factor of the structure, e is the elastic wave energy density ignoring the near-field effect, N is the shape function, x and y are the global coordinate system, Ω o is the unit design domain, τ is the reflection coefficient, q is the unit boundary energy flow, Q is the boundary energy matrix, Г o is the unit boundary; 5.2) Assemble the unit energy matrix and boundary energy matrix under "unit independent grid": Construct the unit independent grid assembly rules to obtain the total matrix form of the energy finite element model, and finally analyze the energy density distribution of the design domain based on it, and calculate the energy flexibility of the curved surface reinforced structure according to formula (6) as the optimization objective function; Where: J is the objective function: the energy flexibility of the curved reinforced structure reflects the magnitude of the vibration energy stored in the structure; 6) Constructing the optimization model: Using the moving asymptote method as the optimization method, according to the relationship between the design variables and the objective function and constraint function, define the sensitivity of the objective function and constraint function to the design variables respectively, bring the objective function, constraint function and their sensitivity into the moving asymptote optimizer, realize the optimization convergence and iterative update of the design variables, when the number of iterations reaches the maximum number of iterations kmax, or the change of the objective function is lower than the maximum allowable error ε, the optimization is terminated and the optimization design result is output; Where: x is both the geometric parameter of the component and the optimization design variable, including the center point coordinates, length, width and tilt angle, j is the component number, n is com is the number of components, M(x) and M0 are the actual material usage and the upper limit of material usage respectively, Ω x is the design variable range; 7) Analysis and verification of the surface reinforcement configuration before and after optimization: Perform finite element analysis on the optimized design configuration obtained by energy finite element topology optimization and the initial configuration, compare the dynamic performance, and verify the effectiveness of the optimization method; 8) Adaptive processing: Apply the energy finite element topology optimization method for medium and high frequency vibration surface design, complete the assembly of the overall matrix through the unit independent grid assembly rules, solve the energy distribution of the surface structure, and complete the topology optimization design.

2. The method according to claim 1, characterized in that: Step 5.2) Construct unit independent grid assembly rules, specifically: for continuous boundaries that may exist in the surface structure, the coupling angle is 0 and the material is continuous, and adjacent nodes need to exchange elements to complete the re-coupling after the nodes are disassembled; for boundaries with coupling angles or discontinuities, the disassembled nodes do not need to be re-coupled.

3. The method according to claim 1, characterized in that: Step 8) For the adaptive processing of surface structures with complex topological configurations such as multi-grids and closed surfaces, the unit independent grid assembly rules need to be adjusted to adapt to the surface characteristics.

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