Multi-material vibration reduction structure design method based on energy flow collaborative topological optimization

Abstract

The invention provides a multi-material vibration reduction structure design method based on energy flow collaborative topological optimization, and the method is based on an energy flow control topological optimization method of a Phipoting vector, and controls energy flow of a structure when the structure bears dynamic excitation by changing material distribution of a thin plate; as a physical quantity that measures the magnitude and direction of the energy flow at one point, the Phiinge vector is selected as an objective function of the optimization model, which has the advantage of being able to specify the direction of the energy flow, thereby more flexibly manipulating the energy flow. According to the method provided by the invention, the microstructures and the distribution of the microstructures on the anisotropic material thin plate can be effectively designed to control the energy flow direction.

Description

Technical Field

[0001] This application relates to a computer-aided design method, and more particularly to a design method for a shock-absorbing structure, which has broad application prospects in fields such as aerospace, construction, and engineering equipment. Background Art

[0002] Controlling the energy in a structure under dynamic loads can achieve functions such as impact resistance, vibration energy harvesting and damping, and fatigue life prediction. This holds broad application potential in aerospace, architectural design, and equipment protection. Common energy metrics typically include the total energy of the structure (or a specific portion) under steady-state or transient loads, such as total strain energy, energy dissipated by damping structures, and energy harvested by piezoelectric structures. These metrics are scalar quantities and refer to the energy stored or consumed by the structure. Furthermore, the energy flow within a structure is also a key area of ​​research interest. Relevant energy flow metrics include group velocity in wave dynamics, heat flux in thermodynamics, and the Poynting vector in electromagnetism and elasticity.

[0003] The Poynting vector is a physical quantity defined point by point that reflects the magnitude and direction of energy flow. It represents the power of external work rather than the energy stored or consumed. Therefore, the Poynting vector can reflect the direction of energy flow inside the structure and has a wide range of applications in underground structure detection and impact positioning. In addition, researchers have also explored other methods based on the Poynting vector. Rupp et al. used the Poynting vector to measure the effect of curved waveguides. Mann et al. derived the light wave transmission efficiency of metasurfaces based on the Poynting vector. Romano et al. proposed a strength measurement for thin plates and shells based on the Poynting vector. However, under dynamic loads, the response of the structure is very complex, and heuristic design cannot achieve good control of the Poynting vector. It is necessary to use inverse design such as optimization methods to design the material distribution of the structure.

[0004] Among the many optimization methods, topology optimization is considered to be a powerful method that can provide innovative and high-performance designs under complex and multiple constraints. It has been widely used to improve structural performance, such as improving structural stiffness, vibration attenuation, impact energy absorption, etc. In dynamic topology optimization, energy-based objective functions are more common. Olhoff and Du optimized the design of thin plate structures subjected to simple harmonic loads with the minimum dynamic flexibility as the goal. Zhang et al. optimized the design of thermoelastic structures using elastic strain energy as the optimization target. Yang et al. used the square of displacement as a constraint to obtain the optimization result of the minimum volume fraction. Zhao and Wang compared different optimization targets, including dynamic flexibility, average strain energy and square of displacement. Takezawa et al. proposed to use the imaginary part of the complex compliance of the damping structure under simple harmonic load as the optimization target.

[0005] Currently, there are only a few studies on optimizing the design of structures from the perspective of the Poynting vector. Jensen and Sigmund were the first to optimize the optical field using the Poynting vector as the optimization target. They optimized the design of photonic waveguides to obtain waveguides that can achieve 90° bends, and then applied this optimization method to the design of T-type optical waveguides. Later, Larsen et al. extended this method to the field of elastic waves, using the Poynting vector as the optimization target to perform dual-material optimization design of the macroscopic material distribution of thin plates, achieving energy focusing and annular flow. However, the above studies are all single-scale topological optimization of macroscopic material distribution. There is currently no dual-scale topological optimization method based on the Poynting vector.

[0006] The dual-scale concurrent topology optimization method takes the macroscopic structural unit as the microstructure, broadens the optimization design domain, and can obtain more diverse results using the limited design domain. Yan et al. proposed a concurrent topology optimization method based on the BESO method to obtain a design with minimum flexibility. Kato et al. decoupled the multi-scale structural analysis method and reduced the computational cost of topology optimization. Xia and Breitkopf considered the nonlinear problem introduced by dual-scale analysis and combined it with FE 2 Analytical methods and multi-scale optimization methods are used to obtain structures with maximum stiffness. Meng et al. took uncertainty into account and used multi-scale topology optimization methods to design multi-scale structures that can withstand thermal and mechanical loads. In dynamic optimization, dual-scale concurrent topology optimization methods have also been widely used. Liang and Du optimized acoustic vibration structures based on multi-material concurrent topology optimization methods to reduce vibration noise. Vicente et al. developed an effective concurrent topology optimization method to minimize the frequency response of hierarchical structures within a specified frequency range. Furthermore, Zhao et al. proposed a sensitivity decoupling method to improve the optimization efficiency in response to the time-consuming problem of dual-scale frequency response analysis. Since the Poynting vector is very sensitive to local material properties, the direction of local energy flow is manipulated through concurrent design of microstructures and macrostructures to achieve flexible energy flow control. Summary of the Invention

[0007] The purpose of this application is to develop an effective multi-material vibration damping structure design method based on energy flow collaborative topology optimization to achieve the control of energy flow in thin plates at a given frequency.

[0008] To achieve the above-mentioned objectives, the present application proposes a multi-material vibration damping structure design method based on energy flow collaborative topology optimization, which includes the following steps: using macro design variables to represent the distribution of multiple microstructures of the vibration damping structure, and using micro design variables to represent the distribution of micro units of each microstructure, wherein the distribution of the microstructure represents the macro structure of the multi-material vibration damping structure, and the distribution of the micro units represents the material distribution of the microstructure; through concurrent optimization of the micro design variables and the macro design variables, the distribution of the micro units in the microstructure and the distribution of the microstructure are synchronously iteratively updated to obtain the optimized design of the multi-material vibration damping structure; wherein the equivalent elastic matrix of each microstructure is obtained based on the homogenization method to achieve the optimization of the micro design variables, and the distribution of the microstructure is optimized to control the energy flow on the multi-material vibration damping structure to achieve the optimization of the macro design variables, wherein the Poynting vector is used as the objective function to control the energy flow on the multi-material vibration damping structure.

[0009] In some embodiments, the equivalent elastic matrices of the different microstructures constituting the model of the vibration damping structure are weighted based on a discrete material optimization method to obtain the elastic matrix of the multi-material vibration damping structure; at the microscopic scale, a modified isotropic material method with a penalty term is used to interpolate the Young's modulus of each microscopic unit constituting each microstructure.

[0010] In some embodiments, a random topography function is used to provide an initial solution to the objective function.

[0011] In some embodiments, the initial solution is optimized multiple times and the optimization results are compared to screen the optimal solution of the objective function.

[0012] In some embodiments, the multi-material vibration damping structure is a thin plate, and the Poynting vector is a simple harmonic Poynting vector.

[0013] In some embodiments, the enforced symmetry constraints are achieved by rotating or flipping the density matrix.

[0014] In some embodiments, the objective function is an objective function based on a designated focal area, an objective function for shielding energy flow in a certain area, or an objective function for energy circulation.

[0015] In some embodiments, the number of microstructures constituting the model of the vibration damping structure is four.

[0016] In some embodiments, a gradient-based moving asymptote method is used to solve the optimization problem, giving the sensitivity of the optimization objective and constraints.

[0017] In some embodiments, the adjoint method is used to give the derivatives of the harmonic Poynting vector with respect to the design variables as sensitivities.

[0018] The present invention constructs optimization targets based on the Poynting vector and derives the macro-micro sensitivity of each optimization target; gives the initial optimization solution through a random morphology function to obtain more diverse optimization design results; obtains its equivalent elastic matrix based on a homogenization method at the micro level, and optimizes the distribution of the multiple microstructures at the macro level to control the energy flow on the thin plate.

[0019] In some embodiments, a relatively general optimization model is proposed, using a given Poynting vector as the optimization target. This optimization model can specify different targets as needed. In some embodiments, the material distribution of the anisotropic material sheet is optimized based on three different targets: energy focusing, energy flow exclusion zone, and energy circulation. For shielding targets, microstructures with directional maximum stiffness are used to control energy flow.

[0020] The method proposed in this application can effectively design microstructures and their distribution on anisotropic material sheets to control the direction of energy flow. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 It is a two-scale topology optimization diagram based on Poynting vector.

[0022] Figure 2 is the Poynting vector distribution on the thin plate.

[0023] Figure 3 Shown is the macro-micro topology of a dual-scale square plate.

[0024] Figure 4 Schematic diagram of three objective functions. Part 4(a) is the focusing objective function; Part 4(b) is the protection objective function; and Part 4(c) is the circulation objective function.

[0025] Figure 5 The spatial distribution of the density values ​​before truncating is shown, where part 5(a) shows an undulation similar to a hilly morphology, and part 5(b) is the initial structure after truncation. Figure 6 Schematic diagram of the symmetry of different microstructures, where part 6 (a) shows two microstructures that each meet the axial symmetry requirements. Figure 6 Part (b) shows a pair of microstructures that meet the mirror symmetry requirements.

[0026] Figure 7 The figure shows a thin plate with three edges clamped. A line load is applied to the left boundary of the plate. The excitation frequency is ω =2000Hz, the excitation direction is perpendicular to the thin plate.

[0027] Figure 8 Schematic diagram of the optimization result focusing at the center. Figure 8 Part (a) shows the macro and micro optimization results; Figure 8 Part (b) shows the amplitude distribution of the simple harmonic Poynting vector; Figure 8 Part (c) shows the displacement amplitude distribution; Figure 8 Part (d) shows the direction of the vector whose amplitude is greater than 30% of the maximum value in the simple harmonic Poynting vector on the thin plate; Figure 8 Part (e) shows the direction of the Poynting vector at the upper right corner of the thin plate, and the backflow phenomenon is within the black dotted box.

[0028] Figure 9 is a schematic diagram of the equivalent elastic matrix of the microstructure and the Poynting vector amplitude of a thin plate composed of a single microstructure. Figure 9 Part (a) is a thin plate completely covered with microstructure 1; Figure 9 Part (b) is a thin plate completely covered with microstructure 2; Figure 9 Part (c) shows a thin plate completely covered with microstructures 3 .

[0029] Figure 10 Optimization results with different focus positions as the target, Poynting vector distribution (amplitude and direction) and displacement amplitude distribution, the focus area is a 0.3m×0.3m area. Figure 10 Part (a) shows the optimization result when the center of the focus area is at (L / 3, L / 2); Figure 10 Part (b) shows the optimization result when the center of the focus area is at (L*2 / 3, L / 2); Figure 10 Part (c) shows the optimization result when the center point of the focus area is at (L / 2, L*2 / 3).

[0030] Figure 11 Schematic diagram of the optimization results for different focusing frequencies, with the strip-like structure marked with white dashed lines. Figure 11 Part (a) is the excitation frequency ω optimization =1800Hz optimization result; Figure 11 Part (b) is the excitation frequency ω optimization =2000Hz optimization result; Figure 11 Part (c) is the excitation frequency ω optimization =2200Hz optimization result; Figure 11 Part (d) considers three excitation frequencies simultaneously ω optimization1 =1800Hz, ω optimization2 =2000Hz, ω optimization3=2200Hz optimization result.

[0031] Figure 12 The following are the Poynting vector cloud diagrams of different optimization results at excitation frequencies of 1800Hz, 200Hz, and 2200Hz, as well as the sweep result diagram in the frequency range [1500Hz, 2500Hz].

[0032] Figure 13 Schematic diagram of the energy flow restricted area model.

[0033] Figure 14 The optimization results targeting the energy flow forbidden zone are shown. Figure 14 Part (a) shows the macroscopic distribution and microscopic material distribution of the optimization results. The microstructure is a 3×3 array, and the red box is a single microstructure. Figure 14 Part (b) is the amplitude cloud diagram of the simple harmonic Poynting vector of the optimization result; Figure 14 Part (c) is the displacement amplitude cloud map; Figure 14 Part (d) shows the Poynting vector direction distribution with the optimization results as the background; Figure 14 Part (e) shows the locally enlarged distribution of the Poynting vector direction.

[0034] Figure 15 Schematic diagram of macro-micro optimization results with different microstructure numbers. Figure 15 Part (a) shows the optimization results with two microstructures; Figure 15 Part (b) shows the optimization results with four microstructures; Figure 15 Part (c) shows the optimization results with six microstructures; Figure 15 Part (d) shows the optimization results with eight microstructures.

[0035] Figure 16 Schematic diagram of optimization results with different symmetry constraints. Figure 16 Part (a) shows the optimization results with symmetry constraints only at the micro level; Figure 16 Part (b) shows the optimization results with symmetry constraints only at the macro level; Figure 16 Part (c) shows the optimization results without any symmetry constraints at both macro and micro levels; Figure 16 Part (d) shows that there are no symmetry constraints at both the macro and micro levels, but the initial microscopic solution is the optimization result of a symmetric structure.

[0036] Figure 17 Schematic diagram of the energy circulation model.

[0037] Figure 18 Schematic diagram of the optimization results with energy circulation as the goal. Figure 18 (a) Macroscopic distribution and microscopic material distribution of the optimization results. The microstructure is a 3×3 array, and the red box is a single microstructure. Figure 18(b) Magnitude cloud diagram of the simple harmonic Poynting vector of the partial optimization result; Figure 18 (c) Local displacement amplitude cloud map; Figure 18 (d) Distribution of Poynting vector directions with the optimization results as background; Figure 18 (e) Partially enlarged distribution of the Poynting vector direction. DETAILED DESCRIPTION

[0038] The method of the present invention is described in detail below with reference to the accompanying drawings.

[0039] The following is a detailed introduction to the specific implementation methods of this application in three parts. The first part gives the Poynting vector, homogenization theory and finite element method; the second part gives the optimized interpolation model, optimization column formula and sensitivity analysis; the third part optimizes different optimization objectives and compares different optimization results.

[0040] 1. Thin plate energy flow based on Poynting vector

[0041] It is hoped that the energy flow can be controlled by performing dual-scale collaborative topology optimization design on the material distribution of the thin plate, such as Figure 1 To achieve this goal, we first need to introduce the relevant theories, including Poynting vector theory, finite element method and numerical homogenization.

[0042] 1.1 Poynting vector theory

[0043] The Poynting vector can characterize the direction and rate of energy flow in an electromagnetic field or an acoustic field. According to the derivation of Auld, the Poynting vector P( x , y , z , t ) is: (1) Where v is the velocity vector and T is the stress tensor. This article discusses the energy flow in thin plates, so we are only concerned with the energy flow within the surface: (2) in P x , P y are the in-plane Poynting vectors x ,y component, are the stress tensor components, is the velocity component. Integrate Equation (2) along the thickness direction of the thin plate , we can get: (3) in, are the Poynting vectors integrated along the thickness direction, M x , M y and M xy is the bending moment component, T x , T y is the shear force, w is the mid-surface deflection.

[0044] For a thin plate composed of anisotropic materials, its constitutive equation and shear force expression are: (4) Substituting equation (4) into equation (3), we can obtain the in-plane Poynting vector expression of the thin plate:

[0045]

[0046] (5) For simple harmonic vibration, the displacement at any time can be expressed as: (6) in is the complex amplitude of the simple harmonic oscillation, is the circular frequency of vibration, It represents the real part of the complex number, e is the base of the natural logarithm function, and i is the imaginary unit.

[0047] The total energy rate of the Poynting vector over one vibration period , according to formula (5):

[0048]

[0049]

[0050] (7)

[0051] The superscript * represents the complex conjugate, i is the imaginary unit, represents taking the real part of a complex number, is the complex amplitude of simple harmonic oscillation, and the subscripts x, y, xx, xy, yy, xxx, xxy, xyy, and yyy represent the order and direction of the derivative respectively. 11、D 12 、D 13 、D 21 、D 22 、D 23 、D 31 、D 32 、D 33 represent the elements in the equivalent elastic matrix respectively.

[0052] W is called the simple harmonic Poynting vector, and Equation (7) is the calculation expression of the simple harmonic Poynting vector of the thin plate. Note that it should be defined point by point.

[0053] 1.2 Finite element discretization

[0054] The finite element method is used to analyze steady-state dynamic problems. The general finite element dynamic equation is: (8) in is the load frequency, is the total mass matrix, is the overall damping matrix, is the overall stiffness matrix, is the loading matrix, is the displacement matrix. The damping matrix C is constructed based on Rayleigh damping: (9) The left side of formula (8) Together they are called the dynamic stiffness matrix S, and we have: (10) The finite element discrete expression of formula (7) is: (11) in , represents the simple harmonic Poynting vector component of each macro unit, d e is the macro unit displacement, and It is a local macro unit matrix constructed based on the elastic matrix and shape function:

[0055]

[0056] (12) Where N is the shape function, D ijis the material elastic matrix. Since the simple harmonic Poynting in Equation (7) is defined point by point, a point Gaussian integral is used here to replace the Poynting vector integral of the entire macro unit. For the Poynting vector integral of a specified area, its expression is: (13) in , is the Poynting vector integral of the specified region, and is the local macro unit matrix of the specified area and It is grouped in the same way as the overall stiffness matrix, that is, and Same dimension as the global stiffness matrix.

[0057] At the microscopic scale, the homogenization method is used to calculate the equivalent elastic matrix of each microstructure: (14) in is the equivalent elastic matrix, is the elastic matrix of the material, is the unit strain, is the characteristic strain, Y is the microstructure region. Based on the homogenization method of Xia and Breitkopf, a numerical homogenization method for the microstructure of thin plates is derived. Based on periodic boundary conditions and unit strain load, the static finite element equation is derived as: (15) The subscripts 1, 2, 3, and 4 represent the degrees of freedom of the corner nodes, internal nodes, left and lower boundary nodes, and right and upper boundary nodes, respectively. , represents the specified displacement of the four corner points, are the prescribed displacements of the right and upper boundary nodes, i.e. The nodes here refer to the nodes of the microstructure after finite element division.

[0058] Due to the special properties of thin plates, their expressions are:

[0059] (16) Where L and W are the length and width of the microstructure, and x and y are the node coordinates. The node displacement is obtained by solving equation (15) and substituted into the two-dimensional finite element expression of equation (14): (17) in That is, the unit node displacement, is the element stiffness matrix.

[0060] 2. Multi-microstructure macro-micro collaborative topology optimization

[0061] The energy flow on the thin plate is controlled by collaborative topology optimization. To this end, this section introduces the optimization-related theories, including collaborative optimization design variables, interpolation functions, objective functions and constraints, sensitivity, and random topology functions.

[0062] 2.1 Design variables and interpolation functions

[0063] Collaborative topology optimization design of dual-scale structures based on density method, Figure 3 The macro-micro topology of a dual-scale square plate is shown. The macro design variable is X, the micro design variable is x, and the total design variable s is:

[0064] (18) in m is the number of candidate microstructures, n and N are the number of microscopic design variables and the corresponding number of macroscopic design variables of a single microstructure.

[0065] At the microscale, the modified isotropic material with penalty (SIMP) method is used to interpolate the Young's modulus of each microelement constituting the microstructure: (19) in E 1 and E 2 are the Young's moduli of material 1 and material 2 respectively, p 1 is the micro penalty factor, It is obtained after density filtering of micro-design variables. The density filtering method refers to the work of Andreassen et al. Note that since the microstructure is filtered here, periodic boundary conditions need to be considered.

[0066] At the macroscale, according to the discrete material optimization (DMO) method proposed by Stegmann and Lund, the equivalent elastic matrices of different microstructures are weighted to obtain the macro unit, that is, the elastic matrix of the microstructure: (20) in It is i The equivalent elastic matrix of a microstructure, is the weighting coefficient, which is related to the macro design variable X ij The relationship is: (twenty one) in p 2 is the macro penalty factor. First, the macro design variables X ij Density filtering is performed to obtain , and then obtain according to the DMO interpolation method , and finally normalized to obtain .

[0067] 2.2 Objective Function and Constraints

[0068] By optimizing the material distribution on the thin plate, the distribution of the simple harmonic Poynting vector can be controlled, thereby manipulating the energy flow of the thin plate. Here is a general optimization formula for manipulating the simple harmonic Poynting vector, where the objective function can be adjusted to meet specific goals: (twenty two) In the above formula (22), ( a 2) Middle f is the objective function, which is constructed based on the simple harmonic Poynting vector, and the simple harmonic Poynting vector of multiple regions can be specified. a 4) The volume fraction of the material constraining the microscopic level is , ( a 5) Constrain the minimum volume fraction of each microstructure at the macro level to be at least.

[0069] Here we give three forms of objective functions, targeting the three design goals of energy focusing, shielding and circulation, such as Figure 4 shown.

[0070] Energy Focus: Figure 4 As shown in part (a), energy focusing maximizes the energy flow power in the specified focusing region. That is, when energy focusing is achieved in this region, the objective function is set to: (twenty three) It means that the horizontal harmonic Poynting vector passing through the focusing area is the largest. When the vertical energy flow power is required to be the largest, it can be directly Replace with .

[0071] Energy flow restricted area: When energy flow is required to bypass a certain area, making the area a restricted area for energy flow, the objective function is set as: (twenty four) Figure 4The shielded area in part (b) is the energy flow restricted area. a is a constant used to amplify the magnitude of the simple harmonic Poynting vector after square calculation. Equation (24) indicates that the energy is controlled to avoid flowing into the shielded area.

[0072] Energy circulation: When energy is required to flow in a circular manner within a specified area, the optimization objective is set to: (25) Here are multiple regions region1, region2, region3, region4, that is Figure 4 Regions 1, 2, 3, and 4 in part (c) are part of the circulation area, which is used to guide the flow of energy to form a cycle.

[0073] 2.3 Sensitivity analysis

[0074] The optimization problem is solved using a gradient-based moving asymptote method, and the sensitivity of the optimization objective and constraints is given. The objective function is constructed based on the simple harmonic Poynting vector. Here, the derivative of the simple harmonic Poynting vector with respect to the design variable is given using the adjoint method: (26) in s i are (macro or micro) design variables, d is the displacement vector, S is the overall dynamic stiffness matrix, is the adjoint vector, T is the symbol for transposing the matrix, Q is the characteristic matrix of the simple harmonic Poynting vector, and d* is the conjugate vector of the displacement vector d. The derivative of the overall stiffness matrix with respect to the macro-micro design variables is: (27) in( b 1) is the sensitivity of the simple harmonic Poynting vector to the macro design variables, ( b 2) is the derivative of the micro-design variable. At the macro level, according to Equation (21) and the chain rule: (28) in is the weighted coefficient of density filtering, and its expression is , r min is the filter radius, It is m Unit and n The distance of the unit.

[0075] For micro-design variables, according to formula (19): (29) in is the weighted coefficient for density filtering of microstructures, and its expression is the same as The difference is that it takes into account the periodic boundary conditions of the microstructure.

[0076] 2.4 Random shape function

[0077] The optimization problem of Equation (22) is an optimization problem with a large number of local solutions. In order to obtain a better optimization solution in the design space, the random morphological description function (RMDFs) proposed by Vel and Goupee can be used to give the initial solution of the microstructure: (30) in q is the number of Gaussian functions used, the amplitude ci and the point coordinates is randomly given, is the spatial length of the Gaussian function, and its calculation expression is: (31) in l is the side length of the microstructure. Based on (30), we can obtain the density field with spatially uneven distribution. In order to limit the range of the density field to Between, the function value is normalized: (32) Truncating the function values ​​yields the distribution of the two-phase material: (33) in is the cutoff value obtained based on the dichotomy method, so that the initial structure satisfies the material volume constraint requirement of formula (22). Based on the above method, a random material distribution can be obtained, such as Figure 5 shown. Figure 5 Part (a) shows the spatial distribution of the untruncated density values, which presents an undulation similar to the hilly topography. Figure 5 Part (b) is the initial structure after truncation. Therefore, compared with completely random, the random morphology function has the characteristics of both random distribution and local aggregation.

[0078] Since there are many local solutions to the optimization results, in order to obtain better optimization results, the same optimization problem can be optimized multiple times to obtain multiple optimization results. The initial solution of each time is determined by a random morphology function, and finally the example with the best optimization result is selected for analysis and comparison.

[0079] 2.5 Symmetry Constraints

[0080] In this application, the symmetry of the optimization results is ensured by mandatory symmetry constraints. In theory, if the influence of the initial solution is not considered, the gradient-based optimization solution algorithm will obtain an optimization result with the same symmetry as the optimization model. However, the initial solution given by the random morphology function does not meet the symmetry requirements and will introduce asymmetry in the initial stage of optimization. In addition, small perturbations in the numerical solution process will introduce asymmetry in the optimization process, which will lead to asymmetric optimized structures. To this end, mandatory symmetry constraints can be implemented by rotating or flipping the density matrix.

[0081] First, for the microstructure, it is expected to satisfy one of the two symmetries: axisymmetry and mirror symmetry, such as Figure 6 As shown. Figure 6 Part (a) Figure 6 Taking the four microstructures shown in part (b) as an example, Figure 6 The two microstructures in part (a) meet the requirements of axisymmetry, with the horizontal coordinate axis as the axis of symmetry. Mirror symmetry means that the two microstructures achieve mirror symmetry with the horizontal line as the mirror line, such as Figure 6 The two microstructures in part (b) are mirror images of each other.

[0082] At the macro level, the symmetry constraints are determined by the optimization problem. If symmetry is required, the symmetry at the macro level should be the same as the symmetry of the corresponding microstructure.

[0083] 3. Numerical Examples

[0084] In order to verify the proposed collaborative topology optimization algorithm based on the Poynting vector, specific optimization examples are given according to the different optimization objectives in Section 2.2. For all examples, the design domain is a square thin plate with a side length of L = 3m and a thickness of t = 0.05m. In the finite element analysis, the macrostructure mesh is divided into 100×100 and the microstructure mesh is also 100×100, both of which are calculated using thin plate elements. The two candidate materials for topology optimization are steel and polycarbonate. The material properties are shown in Table 1. The Ruili damping coefficient is α and β The values ​​are 1e-4 and 1e-5. At the micro level, the two candidate materials are constrained to account for 50% each. At the macro level, there are four microstructures in total, and the volume fraction of each microstructure is constrained to be no less than 10%. The penalty factors at the macro and micro levels are p micro = p macro =3, filter radius r micro = r macro =3.

[0085] Table 1 Material properties Material <![CDATA[Density (kg / m 3 )]]> Young's modulus (GPa) Poisson's ratio steel 7850 210 0.30 polycarbonate 1200 1.46 0.39 3.1 Energy Focusing

[0086] The models considered in this section are Figure 7 As shown, it is a three-side clamped thin plate. A line load is applied to the left boundary of the thin plate. The excitation frequency is ω = 2000 Hz, the excitation direction is perpendicular to the thin plate. In this section, the focus area is a 0.3m × 0.3m area at the center, and the goal is to maximize the horizontal energy flow through the focus area.

[0087] The optimization results are as follows Figure 8 As shown in Figure 8 (a), for example, the microstructure is displayed in a 3×3 array. From the macroscopic results, it is found that in the left area, that is, the part between the load boundary and the focus area, the microstructure distribution presents a "funnel" shape. From the Poynting vector amplitude in Figure (b), Figure 8 (c) The displacement amplitude and Figure 8 It can also be seen from the flow direction of the Poynting vector in part (d) that the energy is gathered into the focal area through the "funnel", especially in the yellow area (corresponding to microstructure 3) and the red area (corresponding to microstructure 4), where the energy flow is diverted. This is mainly due to the directionality of the microstructure. In the right area, the microstructure distribution is annular and strip-shaped, forming a structure similar to a "concave lens". This "concave lens" structure is conducive to changing the direction of energy propagation by using wave refraction. The local Poynting vector direction in the upper right corner is enlarged as shown in the figure below. Figure 8 As shown in part (e), it can be seen that the energy flows in the opposite direction in some areas, and even a circular flow of energy appears in the lower left corner, indicating that the structure of the annular strip can make the energy flow back after passing through the focal area, thereby increasing the energy flow power in the focal area.

[0088] In order to further illustrate the effect of microstructure on energy flow, three cases where the entire thin plate is covered with microstructures 1, 2 and 3 are calculated. The calculation results correspond to Figure 9 As shown. Microstructures 1 and 2 are both symmetrical structures, which leads to the symmetrical distribution of Poynting vectors on their respective thin plates. Figure 9 Part (a), Figure 9 The simple harmonic Poynting vector in part (b) is more focused toward the center, that is, the microstructure 2 guides the Poynting to propagate in the vertical direction, which means that the Poynting tends to propagate along the direction with the greatest stiffness. This characteristic is Figure 9This is also verified in part (c). Microstructure 3 has the highest stiffness at an angle of approximately 45°, which causes the harmonic Poynting vector to flow in this direction, resulting in an asymmetric Poynting vector flow on the thin plate. This example shows that in addition to the macroscopic material distribution, the microstructure also affects the flow of the Poynting vector, highlighting the necessity of collaborative topology optimization.

[0089] 3.1.1 Different focus positions

[0090] This section gives three optimization results specifying different optimization positions. Figure 10 (As shown. The focus area is 0.3m×0.3m, Figure 10 The center point of the focus area in part (a) is at [L / 3, L / 2]. Figure 10 (b) Part and Figure 10 The focusing centers of part (c) are at [L*2 / 3, L / 2] and [L / 2, L*2 / 3] respectively.

[0091] Optimization results and Figure 8 Similarly, there is a funnel-shaped structure in the front half of the thin plate that collects the input energy into the focal area, and a curved strip structure in the back half that guides the energy back to the focal area. From the Poynting vector diagram, it can be seen that the energy is correctly guided to the focal area and the backflow phenomenon near the focal area is more obvious, which increases the energy in the focal area and ultimately leads to an increase in displacement. Figure 8 (c) Partial Figure 10 In part (b), it can be seen that as the focusing position gradually moves away from the loading end, the focusing effect becomes relatively worse. This is mainly because as the distance increases, the damping loss increases, resulting in a decrease in energy flow.

[0092] 3.1.2 Different focusing frequencies

[0093] Different focusing frequencies will affect the optimization results, in addition to Figure 8 Optimized frequencies shown ω optimization =2000Hz optimization results, here are the optimization frequency ω optimization =1800Hz and ω optimization =2200Hz optimization results. In addition, the optimization objectives are adjusted to obtain the optimization results with good simple harmonic Poynting vector focusing effect at the above three frequencies. The optimization objectives of multiple frequencies are: The final optimization result is as follows Figure 11As shown. It can be seen that all the optimization results are similar to the previous optimization results, with a funnel-shaped front end and a concave lens back end. However, for the multi-frequency optimization results, the structure needs to achieve focusing at different frequencies, so the curved strip structure is compactly close together, unlike the single-frequency optimization results which have a clear strip structure with a certain interval (the strip structure is marked with a white dotted line). For the single-frequency optimization results, the higher the frequency, the more compact the strips are, as shown in Figure 11 As shown in part (a), ω optimization =1800Hz optimization results have 7 strip structures, and Figure 11 As shown in part (b), ω optimization =2000Hz optimization results have 7 half-strip structures, such as Figure 11 As shown in part (c), the strip structure on the far right is only half formed, and Figure 11 As shown in part (d), the structure with the largest number of 8 stripes ω optimization =2200Hz. This is because the higher the frequency, the shorter the wavelength, so a more compact strip structure is needed to influence the propagation of the wave and thus control the propagation of energy.

[0094] In order to compare the focusing effects of different optimization results at different frequencies, the excitation frequency ω excitation =1800Hz, 2000Hz and 2200Hz simple harmonic Poynting vector cloud diagram and sweep frequency analysis curve, such as Figure 12 As shown in the figure. The colorbar of the Poynting vector cloud is unified at the same frequency for easy comparison. For the optimization results of a single frequency, they all show obvious focusing effects at their corresponding optimization frequencies. In addition, the optimization frequency ω optimization The optimization results of 2000Hz and 2200Hz also have obvious focusing effects at the excitation frequencies of 1800Hz and 2000Hz respectively. This may be because the optimization frequencies are close. As for the optimization results of multiple frequencies, obvious focusing phenomena are shown at all three frequencies, and the second-ranked focusing effect is achieved at each excitation frequency, corresponding to Figure 12 The simple harmonic Poynting vector amplitude distribution, indicated by the green outline, is shown in the figure. This is also confirmed by the frequency sweep curve. Compared to the narrow effective focusing frequency range of the single-frequency optimization result, the multi-frequency optimization result demonstrates good focusing across a wider frequency range (1600Hz, 2200Hz). Therefore, this optimized structure is suitable for applications requiring focusing of waves across a wide frequency range.

[0095] 3.2 Energy flow restricted area

[0096] In this section, the objective function of shielding the energy flow in a certain area proposed by Equation (24) is considered. The size of the thin plate is the same as before, only the boundary conditions are changed to four-side clamped support, and the load acts on a 0.3m×0.3m square area in the center of the thin plate. The shielding area is a 0.18m×2.40m rectangular area located to the right of the load area, with the center coordinates of (1.5m, 1.8m). The specific model is as follows Figure 13 shown.

[0097] Figure 14 The optimization results targeting the energy flow forbidden zone are shown. Figure 14 In the optimization results shown in part (a), the red boxes indicate the areas subject to vertical loads, and the white boxes indicate the forbidden areas. From a microscopic perspective, the material distribution of the microstructures exhibits a distinct directional pattern. Microstructure 1 (dark blue) exhibits a columnar distribution along the vertical direction. This structure exhibits minimal stiffness in the horizontal direction, approximately equal to that of a soft material (polycarbonate), while achieving maximum stiffness in the horizontal direction, which is similar to that of a hard material (steel). As previously discussed, this directional stiffness helps guide wave propagation in the direction of maximum stiffness. Therefore, macroscopically, it is placed in the forbidden area to maximize shielding of horizontal energy transferred from the left. Microstructures 3 (yellow) and 4 (red) exhibit mirror-image distributions due to symmetry constraints, with maximum stiffness at approximately 45° and 135°, respectively. These two microstructures are placed on the left side of the plate to guide energy transmission. Microstructure 2 (green) exhibits high stiffness in both directions and is placed sparingly in the interstitial region to guide wave propagation.

[0098] contrast Figure 14 The simple harmonic Poynting amplitude distribution shown in part (b) and Figure 14 The displacement amplitude distribution diagram shown in part (c) clearly shows that there is an inconsistent distribution within the load application area (red box). The displacement amplitude is the largest at the center, while the Poynting vector has two maximum amplitude points, which shows that the Poynting vector and the displacement are not completely corresponding. According to formula (7), the Poynting vector is not only related to the displacement size, but also to the material properties. Therefore, the Poynting vector is not necessarily large where the displacement amplitude is large. Figure 14 (d) and Figure 14 From the Poynting vector direction distribution diagram and the local magnified image shown in (e), it can be seen that the optimized directional anisotropic microstructure directs more energy to the left area, especially in the upper left and lower left directions.

[0099] 3.2.1 Different numbers of microstructures

[0100] The number of microstructures will significantly affect the macroscopic microstructure distribution and have an impact on the optimization target. Here, we can use the energy flow restricted area as the objective function and specify different numbers of microstructures to obtain the corresponding optimization results. Figure 15 As shown, the number of microstructures is specified as 2, 4, 6, and 8, respectively.

[0101] For convenience, the narrow area between the shielding area and the load area is called the connection area. The optimization results mainly guide the energy flow by arranging microstructures with directional maximum stiffness on the connection area. Figure 15 (a) Microstructure 2, Figure 15 Microstructures 3 and 4 in part (c) have the maximum stiffness in the vertical direction, and they are placed in the connection area to prevent energy from flowing horizontally to the shielding area. Figure 15 (b) Part and Figure 15 Part (d) makes more use of microstructures with oblique maximum stiffness to guide energy propagation, e.g. Figure 15 (b) Microstructure 3 and Microstructure 4, Figure 15 The direction of maximum stiffness of microstructures 7 and 8 in part (d) is approximately perpendicular to the direction of energy flow, which greatly attenuates the energy flow to the shielding area. Except for the connection area, in the external area (i.e., the left side of the load area and the right side of the shielding area), although the structure appears to be a strip distribution similar to the focusing case on a macroscopic level, it is blocked by the connection area, which prevents energy from returning and increases energy loss. This feature is Figure 15 (a) Part and Figure 15 It is most obvious in part (c).

[0102] The Poynting vector amplitude of the shielded area is shown in Table 2. Since the load is on the left side of the shielded area, most of the energy enters the shielded area in a nearly horizontal direction. W x Much greater than W y In addition, among the four optimization results with different numbers of microstructures, Figure 15 Part (c) m =6) has the lowest simple harmonic Poynting vector, Figure 15 Part (b) m =4) and Figure 15 (c) is closer, and Figure 15 Part (a) m =2) and Figure 15 Part (d) m =8) is much higher than the first two, which shows that the more microstructures there are, the better. And because the initial solution is determined by the random morphology function, the optimized structure has a certain degree of randomness, so it is not possible to directly conclude thatm =6 is the best case.

[0103] Table 2. Poynting vector amplitude in the shielded area Example Figure 15 (a) Part (=2) Figure 15 (b) Part (=4) Figure 15 (c) Part (=6) Figure 15 (d) part (=8) <![CDATA[ W x (J)]]> <![CDATA[2.3224×10 -8 ]]> <![CDATA[9.5062×10 -9 ]]> <![CDATA[9.2604×10 -9 ]]> <![CDATA[1.8408×10 -8 ]]> <![CDATA[ W y (J)]]> <![CDATA[2.1960×10 -10 ]]> <![CDATA[-1.8286×10 -10 ]]> <![CDATA[2.4945×10 -10 ]]> <![CDATA[3.2808×10 -10 ]]> (J) <![CDATA[2.3225×10 -8 ]]> <![CDATA[9.5080×10 -9 ]]> <![CDATA[9.2638×10 -9 ]]> <![CDATA[1.8411×10 -8 ]]> 3.2.2 Different symmetries

[0104] During the optimization process, symmetry will significantly affect the optimization results. Section 2.2 provides the symmetry requirements for micro and macro scales. In summary, at the micro level, microstructures 1 and 2 are forced to be symmetrical in the vertical direction, while microstructures 3 and 4 are mirror-symmetric with the horizontal axis as the symmetry axis. This is true in all previous examples. At the macro level, the macro distribution corresponding to each microstructure also satisfies the corresponding symmetry. However, when the symmetry of the optimization model conflicts with the macro symmetry, the macro symmetry is abandoned and only the micro symmetry is required, such as Figure 10 (c) The case of asymmetric focusing.

[0105] In this section, we observe the influence of symmetry on the results by relaxing the requirements of macroscopic or microscopic symmetry. It is worth noting that in order to highlight the influence of symmetry on the structure, the initial solutions in this section are all Figure 14 The optimization results are the same, that is, taking it as a control example, the final optimization results are as follows Figure 16 The Poynting vector of the shielded area is shown in Table 3. Figure 16 Part (a) is the optimization result that only requires macroscopic symmetry. Figure 16 Part (b) only requires microscopic symmetry, while Figure 16 There is no symmetry constraint in part (c). Finally, in order to observe the effect of the symmetry of the initial solution, an optimization result based on a symmetric initial solution without symmetry constraints is given. Figure 16 (d). From the optimization results, Figure 16 Part (b), Figure 16 Part (d) and comparison example Figure 14 The results of are very similar. Their similarity lies in that the initial solutions are the same and symmetrical, so it is reasonable to get similar optimization results when the numerical fluctuations are ignored. Figure 16 (a) Part and Figure 16 (c) Their initial solutions are the same but asymmetric. In the optimization results, the microstructures are very similar, which is due to the same initial solution. However, in the macroscopic view, the distribution of the microstructures is very different. Figure 16 (c) has significant asymmetry, which is caused by the asymmetry of the microstructure at the beginning of the optimization. Figure 16 Part (a) looks more beautiful due to the macroscopic symmetry imposed. However, according to Table 3, although Figure 16Part (a) also has a structure that guides energy, but its shielding effect is the worst, about twice that of the other examples. Figure 16 (a) is the opposite, with similar microstructures Figure 16 Part (c) shows a good shielding effect, which is very close to the best optimization result, especially in the horizontal direction, where the minimum simple harmonic Poynting vector is achieved. This shows that asymmetric microstructures can also achieve better optimization effects through asymmetric distribution. Figure 16 (b) Part and Figure 16 The values ​​of the simple harmonic Poynting vectors in part (d) are exactly the same, which is reasonable for the two extremely similar optimized structures. Figure 14 The results are very close. Therefore, microstructure and macro distribution complement each other and need to work together to achieve the best effect.

[0106] Table 3. Poynting vector amplitudes in the shielded region of the optimization results with different symmetry constraints. Example Figure 16 Part (a) Figure 16 Part (b) Figure 16 Part (c) Figure 16 Part (d) <![CDATA[ W x (J)]]> <![CDATA[2.0382×10 -8 ]]> <![CDATA[9.4357×10 -9 ]]> <![CDATA[7.5588×10 -9 ]]> <![CDATA[9.4357×10 -9 ]]> <![CDATA[ W y (J)]]> <![CDATA[-3.1270×10 -10 ]]> <![CDATA[6.0770×10 -10 ]]> <![CDATA[5.7112×10 -9 ]]> <![CDATA[6.0770×10 -10 ]]> <![CDATA[ W y (J)]]> <![CDATA[-3.1270×10 -10 ]]> <![CDATA[6.0770×10 -10 ]]> <![CDATA[5.7112×10 -9 ]]> <![CDATA[6.0770×10 -10 ]]> 3.3 Energy Circulation

[0107] In this section, we consider the objective function of energy circulation proposed by Equation (24). In the previous focusing example, the phenomenon of energy backflow occurred, which helped to improve the degree of energy concentration. Inspired by this phenomenon, in this section, the energy is confined to an annular area on the thin plate by guiding the energy to form an annular flow. The annular area consists of four rectangular areas, each of which is 0.9m long and 0.21m wide. The vertical distance from the center of the thin plate to each rectangular area is 0.48m. The specific model is as follows: Figure 17 shown.

[0108] The final optimization results are as follows Figure 18 As shown, it should be noted that the symmetry only guarantees the microscopic symmetry. Figure 18 The optimized result shown in part (a) is a spiral structure, while the microstructure primarily exhibits horizontal (microstructure 2) and vertical (microstructures 3 and 4) distributions. The latter two types of microstructures are the primary drivers of energy flow control: horizontal circulation regions 2 and 4 are primarily composed of microstructure 2, while vertical circulation regions 1 and 3 are primarily composed of microstructures 3 and 4. This means that the direction of maximum stiffness of the microstructures aligns with the specified energy flow direction. This distribution effectively guides energy propagation and increases the amplitude of the harmonic Poynting vector. Figure 18 (b) Part and Figure 18 The harmonic Poynting vector amplitude and displacement amplitude shown in part (c) both indicate that the optimized structure achieves energy circulation, and Figure 18 (d) Part and Figure 18 The direction of the harmonic Poynting vector shown in part (e) indicates that energy circulates within the specified area.

[0109] The numerical examples described above demonstrate that, at the macroscopic level, optimization results often utilize stripe-like structures to control energy flow. This leads to energy reflux near the focal region in focusing optimization results. For circulation optimization results, the spiral stripe structure is the primary factor guiding energy circulation. Optimization results at different focusing positions still exhibit a funnel-like and stripe-like structure, with energy reflux near the focal region, enhancing focusing efficiency. The focusing position should be considered when designing the corresponding design. Designs for different focusing positions exhibit similar characteristics. Regarding the excitation frequency, a higher excitation frequency reduces the distance between stripes, resulting in more stripe structures on the plate. There is no need to constrain the excitation frequency. This illustrates how different excitation frequencies can lead to variations in the final design. Increasing the number of microstructures does not necessarily lead to better optimization results; rather, it significantly increases design variables, leading to reduced efficiency. The number of microstructures is determined to be four. Symmetry constraints ensure that the microstructures are coordinated with the macrostructure distribution. Microstructures exhibit directional stiffness, and different angles of directional stiffness correspond to different macroscopic distributions. That is, the directional stiffness of the microstructure is coordinated with the macroscopic distribution to achieve the specified goal.

[0110] Although the embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention without departing from the principles and purpose of the present invention.

Claims

1. A multi-material vibration damping structure design method based on energy flow collaborative topology optimization, characterized by: Using macro design variables to represent the distribution of multiple microstructures of the vibration damping structure, and using micro design variables to represent the distribution of micro units of each microstructure, wherein the distribution of the microstructures represents the macrostructure of the multi-material vibration damping structure, and the distribution of the micro units represents the material distribution of the microstructure; By concurrently optimizing the microscopic design variables and the macroscopic design variables, a synchronous iterative update of the distribution of the microscopic units in the microstructure and the distribution of the microstructure is achieved to obtain an optimized design of the multi-material vibration damping structure; The equivalent elastic matrix of each microstructure is obtained based on the homogenization method to achieve the optimization of microscopic design variables, and the distribution of the microstructure is optimized to control the energy flow on the multi-material vibration damping structure to achieve the optimization of macroscopic design variables. The Poynting vector is used as the objective function to control the energy flow on the multi-material vibration damping structure.

2. The multi-material vibration damping structure design method based on energy flow collaborative topology optimization according to claim 1 is characterized in that: weighting the equivalent elastic matrices of different microstructures constituting the model of the vibration damping structure based on a discrete material optimization method to obtain the elastic matrix of the multi-material vibration damping structure; At the microscopic scale, the Young's modulus of each microscopic element constituting each of the microstructures is interpolated using a modified isotropic material method with a penalty term.

3. The multi-material vibration damping structure design method based on energy flow collaborative topology optimization according to claim 1 is characterized in that: A random shape function is used to give an initial solution to the objective function.

4. The multi-material vibration damping structure design method based on energy flow collaborative topology optimization according to claim 3 is characterized by: The initial solution is optimized multiple times and the optimization results are compared to screen the optimal solution of the objective function.

5. The multi-material vibration damping structure design method based on energy flow collaborative topology optimization according to claim 1 is characterized in that: The Poynting vector is a simple harmonic Poynting vector W, defined as W=(Wx, Wy); ； ; Among them, the superscript * represents the complex conjugate, i is the imaginary unit, represents the real part of a complex number. is the complex amplitude of simple harmonic oscillation, and the subscripts x, y, xx, xy, yy, xxx, xxy, xyy, and yyy represent the order and direction of the derivative respectively. 11 、D 12 、D 13 、D 21 、D 22 、D 23 、D 31 、D 32 、D 33 represent the elements in the equivalent elastic matrix respectively.

6. The multi-material vibration damping structure design method based on energy flow collaborative topology optimization according to claim 5 is characterized in that: Enforce symmetry constraints by rotating or flipping the density matrix.

7. The multi-material vibration damping structure design method based on energy flow collaborative topology optimization according to claim 1 is characterized in that: The objective function is an objective function based on a designated focus area, an objective function for shielding energy flow in a certain area, or an objective function for energy circulation.

8. The multi-material vibration damping structure design method based on energy flow collaborative topology optimization according to claim 1 is characterized in that: There are four types of microstructures constituting the model of the vibration damping structure.

9. The multi-material vibration damping structure design method based on energy flow collaborative topology optimization according to claim 1 is characterized in that: The optimization problem is solved by a gradient-based moving asymptote method, and the sensitivities of the optimization objective and constraints are given.

10. The multi-material vibration damping structure design method based on energy flow collaborative topology optimization according to claim 9, characterized in that: The derivatives of the simple harmonic Poynting vector with respect to the design variables are given as sensitivities using the adjoint method.

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