A method and system for multiphase filling kinetics topology optimization of functionally graded porous structures under transient loading

Abstract

The application relates to a kind of transient load under the function gradient porous structure of multiphase filling dynamics topology optimization method and system, belong to the topology optimization design field of function gradient porous structure.The technical problems to be solved by the application are: how to fully exploit design space, construct the efficient function gradient porous structure of multiphase filling dynamics topology optimization method, realize the optimal anti-vibration design of porous filling structure with high-quality transient dynamic performance.Technical points: based on characteristic orthogonal decomposition method, the reduced transient dynamic finite element equation of function gradient porous multiphase filling structure is constructed;the material interpolation model and volume constraint model of function gradient porous multiphase filling structure are constructed, and the transient dynamic multiphase filling topology optimization reduced model with the minimum dynamic compliance as the optimization objective is proposed;the reduced sensitivity time-domain accompanying equation is constructed, and the sensitivity of dynamic compliance and volume constraint is efficiently calculated;based on ADMM-MMA hybrid optimization algorithm, the optimal topology design of function gradient porous multiphase filling structure is efficiently solved and obtained.The application can maximize the optimal anti-vibration performance of function gradient porous filling structure, and is especially suitable for the structure lightweight anti-vibration design of large engineering equipment.

Description

Technical Field

[0001] This invention relates to the field of topology optimization design of functionally graded porous structures, and more specifically, to a method and system for multiphase filling dynamic topology optimization of functionally graded porous structures under transient loads. Background Technology

[0002] The rapid development of the aerospace industry has placed higher demands on the extreme performance of equipment, including ultra-lightweight, ultra-high load-bearing capacity, energy absorption, vibration reduction, and noise reduction. The dynamic topology optimization design of lightweight functionally graded porous structures (FJD) has become an important technical path to meet these performance requirements. Full-scale design methods utilize topology optimization to control the distribution of solid-phase materials within the neighborhood, generating FJD-filled structures. This method avoids scale separation problems, achieves an optimal overall design with consistent design and manufacturing performance, and ensures smooth connections between optimized substructures. However, existing research focuses on the dynamic topology optimization of single-phase materials, while the multiphase filling method of FJD can further expand the design space and improve the transient mechanical performance of porous structures. Furthermore, under transient loads, the topology optimization of FJD involves the coordinated solution of spatiotemporal response prediction and temporal adjoint equations. With a large number of elements and a small time step, large-scale iterative solutions and design variable updates consume significant computational costs, resulting in a significant reduction in computational efficiency.

[0003] Therefore, how to fully explore the design space, construct an efficient multiphase filling dynamic topology optimization method for functionally graded porous structures, and realize the optimal vibration-resistant design of porous filling structures with excellent transient dynamic performance is a key technical problem that urgently needs to be solved in the field of engineering equipment. Summary of the Invention

[0004] The technical problem to be solved by this invention is:

[0005] Existing design techniques do not implement topology optimization of functionally graded porous structures from the perspective of multiphase material filling, thus failing to maximize the optimal vibration resistance of the filled structures. Furthermore, topology optimization of functionally graded porous structures involves the coordinated solution of spatiotemporal response prediction and time-domain adjoint equations. Large-scale iterative solutions and design variable updates in topology optimization engineering consume significant computational costs, leading to the thorny problem of making optimization methods difficult to apply on a large scale in engineering.

[0006] To achieve the above objectives, in a first aspect, the present invention provides a multiphase filling dynamic topology optimization method for functionally graded porous structures under transient loads, comprising the following steps:

[0007] S101: Based on the characteristic orthogonal decomposition method, orthogonal basis is sought, and the matrix in the full-order transient dynamic finite element equation is projected into a low-order matrix, thereby forming the reduced-order transient dynamic finite element equation of the functionally graded porous multiphase filling structure.

[0008] S102: Construct a material interpolation model for functionally graded porous multiphase filled structures, establish local and global volume constraints on material usage, and propose a transient dynamic multiphase filled topology optimization reduction model with the goal of minimizing dynamic compliance.

[0009] S103: Based on the above transient dynamic multiphase filling topology optimization reduction model, a reduced-order sensitivity time-domain adjoint equation is constructed to efficiently calculate the dynamic compliance and volume constraint sensitivity of functionally graded porous multiphase filling structures.

[0010] S104: Construct an ADMM (Alternating Direction Multiplier Method)-MMA (Moving Asymptote Method) hybrid optimization algorithm to efficiently solve for the optimal topology design of functionally graded porous multiphase filled structures.

[0011] Secondly, a multiphase filling dynamic topology optimization system for functionally graded porous structures under transient loading includes:

[0012] The initialization module constructs the design domain of the functionally graded porous multiphase filled structure and initializes the design variables; it also pre-sets the local volume constraints and overall volume constraints of the multiphase material usage in the filled structure, the algorithm parameters of the Newmark time integration scheme, the parameters of the multiphase material interpolation model, and the parameters of the ADMM-MMA hybrid optimization algorithm.

[0013] Preprocessing module: The design domain of discrete functionally graded porous multiphase filled structures, defining displacement boundary conditions, initial conditions and applying transient dynamic loads; combined with the multiphase material interpolation model to calculate the stiffness, mass and damping matrices of the unit, and assembling them to form the full-order transient dynamic finite element equation of the functionally graded porous multiphase filled structure;

[0014] The dynamic analysis module employs the unconditionally stable Newmark time integration scheme to solve the full-order transient dynamic finite element equations of the functionally graded porous multiphase filled structure, obtaining the transient image matrix of the displacement field. An orthogonal basis is constructed by calculating the eigenvalue problem of the correlation matrix. Subsequently, a reduced-order transient dynamic finite element equation of the filled structure is formed using projection transformation, thereby calculating the approximate value of the dynamic compliance of the filled structure and the local and global volume constraints of the multiphase material.

[0015] The optimization solution module calculates the sensitivity information of the multiphase filling dynamic topology optimization model of the functionally graded porous structure under transient loads, updates the design variables based on the ADMM-MMA hybrid optimization algorithm, and determines whether the optimization iteration results meet the convergence criteria. If not, the optimization iteration continues until the convergence criteria are met.

[0016] This invention provides a method and system for multiphase filling dynamic topology optimization of functionally graded porous structures under transient loads, which has the following beneficial technical effects:

[0017] This invention addresses the challenges of multiphase filling dynamics topology optimization for functionally graded porous structures (FJPs): First, existing design techniques do not implement topology optimization of FJPs from the perspective of multiphase material filling, thus failing to maximize the optimal vibration resistance of the filled structure. Second, topology optimization of FJPs involves the joint solution of spatiotemporal response prediction and temporal adjoint equations. The large-scale iterative solution and design variable updates in topology optimization engineering consume significant computational costs, leading to the thorny problem that optimization methods are difficult to apply on a large scale in engineering.

[0018] This invention introduces a multiphase filling mode for functionally graded porous structures (FJPs), expanding the potential of multiphase materials in dynamic design. It integrates the orthogonal decomposition method (ODM) into the transient dynamic topology optimization problem of the filled structure, efficiently obtaining a design topology with superior transient dynamic performance. This invention proposes a method and system for multiphase filling dynamic topology optimization of FJPs under transient loads, including: defining an initial design domain and algorithm parameters; constructing a reduced-order transient dynamic finite element equation for the FJP multiphase filled structure based on the ODM method; constructing a material interpolation model and a volume constraint model for the FJP multiphase filled structure; proposing a reduced-order transient dynamic multiphase filled topology optimization model with dynamic compliance minimization as the optimization objective; collaboratively constructing a reduced-order sensitivity time-domain adjoint equation to efficiently calculate the dynamic compliance and volume constraint sensitivity of the FJP multiphase filled structure; and efficiently solving for the optimal topology design of the FJP multiphase filled structure based on the ADMM-MMA hybrid optimization algorithm. The entire design process is completed based on full-scale finite element analysis, avoiding scale separation problems and ensuring smooth connections between optimized multiphase filled substructures.

[0019] This invention proposes a multiphase filling dynamics topology optimization method and system for functionally graded porous structures (FJPs). This method can form an optimal topology configuration with an "external open shell + internal self-supporting filling structure," reducing the need for self-supporting structures and facilitating the additive manufacturing process of FJPs. In particular, it maximizes the optimal vibration resistance design of FJPs. The method introduces a model reduction strategy using eigenvalue orthogonal decomposition, achieving a synergistic reduction in order between the transient dynamic equations and the time-domain dependent sensitivity adjoint equations. This significantly improves the computational efficiency of the transient dynamics topology optimization problem for FJPs. Therefore, this invention provides a highly efficient and agile transient dynamics design method for FJPs, enabling optimal filling design of multiphase materials and maximizing the optimal vibration resistance performance of FJPs. It is particularly suitable for the lightweight vibration resistance design of large-scale engineering equipment. Attached Figure Description

[0020] Figure 1 This is a flowchart of a multiphase filling dynamic topology optimization method for functionally graded porous structures under transient loads, according to the present invention.

[0021] Figure 2 This is a schematic diagram of the multiphase filling dynamic topology optimization system of a functionally graded porous structure under transient load according to the present invention.

[0022] Figure 3 For the cantilever beam problem under half-wave sinusoidal excitation,

[0023] (a) is the structural design domain in Embodiment 1 of the present invention;

[0024] (b) is the half-wave sinusoidal load in Embodiment 1 of the present invention.

[0025] Figure 4 For the load application time A schematic diagram of the topology optimization results for a cantilever beam filled with a single-phase material.

[0026] (a) is the topology optimization result without total volume constraints in Embodiment 1 of the present invention. ;

[0027] (b) The topology optimization result including total volume constraints in Embodiment 1 of the present invention. ;

[0028] (c) The iterative history of dynamic compliance and volume fraction without total volume constraint in Embodiment 1 of the present invention;

[0029] (d) is the iterative history of dynamic compliance and volume fraction under total volume constraints in Embodiment 1 of the present invention.

[0030] Figure 5 For the load application time A schematic diagram of the topology optimization results for a cantilever beam filled with two-phase materials.

[0031] (a) The topology optimization result of the non-porous filled cantilever beam in Embodiment 1 of the present invention. ;

[0032] (b) The topology optimization results of the porous filled cantilever beam in Embodiment 1 of the present invention. ;

[0033] (c) The iterative history of dynamic compliance and volume fraction in Embodiment 1 of the present invention;

[0034] (d) is the convergence history of the convergence index in Embodiment 1 of the present invention.

[0035] Figure 6 This is a schematic diagram of the topology optimization results of the cantilever beam under different load application times in Embodiment 1 of the present invention.

[0036] (a) The load application time in Embodiment 1 of the present invention The topology optimization results of cantilever beams filled with single-phase materials. ;

[0037] (b) The load application time in Embodiment 1 of the present invention The topology optimization results of the cantilever beam filled with two-phase materials. ;

[0038] (c) The load application time in Embodiment 1 of the present invention The topology optimization results of cantilever beams filled with single-phase materials. ;

[0039] (d) is the load application time in Embodiment 1 of the present invention. The topology optimization results of the cantilever beam filled with two-phase materials. .

[0040] Figure 7 For the load application time A schematic diagram of the topology optimization results for a cantilever beam filled with three-phase materials.

[0041] (a) The topology optimization result of the cantilever beam filled with three-phase material in Embodiment 1 of the present invention. ;

[0042] (b) The iterative history of dynamic compliance and volume fraction in Embodiment 1 of the present invention;

[0043] (c) is the convergence history of the convergence index in Embodiment 1 of the present invention. Detailed Implementation

[0044] This invention addresses the challenges of multiphase filling dynamics topology optimization for functionally graded porous structures (FJPCs): multiphase filling dynamic design and efficient computational transient optimization based on model reduction strategies. It proposes a multiphase filling dynamics topology optimization method and system for FJPCs under transient loads, achieving efficient optimal vibration-resistant design of the filled structure.

[0045] Combined with appendix Figures 1-7 Appendix 1 details the multiphase filling dynamics topology optimization method for functionally graded porous structures under transient loads, as described in this invention, so that the advantages and features of this invention can be more easily understood by those skilled in the art, thereby providing a clearer and more explicit definition of the scope of protection of this invention.

[0046] In a first aspect, the present invention provides a multiphase filling dynamic topology optimization method for functionally graded porous structures under transient loads, the flowchart of which is shown below. Figure 1 As shown, the main steps include:

[0047] S101: Based on the characteristic orthogonal decomposition method, orthogonal basis is sought, and the matrix in the full-order transient dynamic finite element equation is projected into a low-order matrix, thereby forming a reduced-order transient dynamic finite element equation for a functionally graded porous multiphase filling structure.

[0048] The elastodynamic behavior of the functionally graded porous multiphase filled structure is described by the motion control equations discretized using the finite element method, expressed as:

[0049]

[0050] In the formula: , , These represent the total mass, damping, and stiffness matrices of the infill structure, respectively. For time steps External load array, , , They are time steps The acceleration, velocity, and displacement arrays of the filled structure. The time step is the time at which the external load terminates.

[0051] Solving the above motion control equations using the Newmark method yields:

[0052]

[0053] In the formula: This is the time step increment.

[0054] By solving the full-order analysis model Obtain the displacement field in Spatial distribution at different times They are assembled to form a transient matrix. ,in The degrees of freedom of the finite element model after discretization are usually .

[0055] The characteristic orthogonal decomposition method, starting from the energy perspective, aims to find orthogonal basis. This ensures that all displacement snapshots of the above full-order model lie in orthogonal basis vectors. The sum of projected energy in the direction is maximized, and then... Construct a low-dimensional subspace that best uniformly approximates the solution space of the full-order model based on the basis.

[0056] The above-mentioned best uniform approximation problem can be transformed into solving the correlation matrix. The problem of obtaining eigenvalues, namely:

[0057]

[0058] In the formula: , Correlation matrix eigenvalues ​​and eigenvectors.

[0059] To extract the dominant deformation modes, i.e., the eigenmodes, that contribute most to the structural dynamic response throughout the entire time history, this can be achieved by truncating... The number of eigenvectors, selecting the minimum number of basis vectors. This makes the total energy of the low-dimensional subspace equal to the energy of the full-order solution space. Satisfy the following formula:

[0060]

[0061] The reduced basis vectors project the full-order solution space onto a lower-dimensional subspace, and the truncated basis vectors... ( Then, the displacement field of the infill structure can be approximately expressed as:

[0062]

[0063] In the formula: This is the correlation coefficient matrix, hereinafter referred to as... .

[0064] Substituting the above equation into the motion control equation and multiply by the left on both sides This yields the following reduced-order control equations:

[0065]

[0066]

[0067] In the formula: , , All are The matrix, the field variables to be solved are transformed into ,satisfy Therefore, the solution scale of the reduced-order analysis model is much smaller than that of the full-order model.

[0068] S102: Construct a material interpolation model for functionally graded porous multiphase filled structures, establish local and global volume constraints on material usage, and propose a transient dynamic multiphase filled topology optimization reduction model with the goal of minimizing dynamic compliance.

[0069] In this invention, step S102 may further include:

[0070] Step S102-1: Filter the design variables and construct a material interpolation model for a multiphase filled structure based on the Ersatz minimum positive number: volume and stiffness interpolation model. Use this model to calculate the global mass, stiffness and damping matrices in the full-order transient dynamic finite element equation.

[0071] To suppress the checkerboard pattern, the first Design variables corresponding to phase materials Weighted filtering is applied within the neighborhood of the cell, and the filtered design variables are... Represented as:

[0072]

[0073]

[0074] In the formula: This refers to the total number of multiple material types that include the void phase. , , Here is the filtering matrix. , Units Area and weight, The filter radius is Neighborhood unit The center (in terms of position vector) (representation) and unit The center (in terms of position vector) Euclidean distance (indicated by) This is the filter index.

[0075] In any filling Phase material unit Construct volume interpolation function abbreviated as Then we have:

[0076]

[0077]

[0078] In the formula: It is a positive number that is much less than 1. This is the smoothed Heaviside projection function. This is the projection sharpness parameter. This is the threshold value.

[0079] To ensure that the intermediate elastic modulus is adequately penalized, the multi-material stiffness interpolation function should be a convex function. Furthermore, the stiffness coefficients of the void material should be assigned extremely small positive numbers close to zero to avoid singularities in the stiffness matrix. Therefore, the multi-material stiffness interpolation function is expressed as:

[0080]

[0081] In the formula: As a penalty factor, For the first Stiffness coefficient of phase material, is the stiffness coefficient of the hollow material.

[0082] Based on the material interpolation model of the multiphase filled structure, the global mass, stiffness, and damping matrices in the full-order transient dynamic finite element equations are expressed as:

[0083]

[0084] In the formula: the damping matrix adopts the Rayleigh damping model. , These are mass damping and stiffness damping, respectively.

[0085] Step S102-2: Establish local volume constraints to control the local distribution density of multiphase materials in the neighborhood of the control unit; at the same time, collaboratively establish global volume constraints to accurately control the overall volume usage of multiphase materials.

[0086] To control the local distribution density of each phase material within the cell neighborhood, thereby enabling the filling design of multiphase materials, the following local volume constraint is introduced. (abbreviated as) ):

[0087]

[0088] In the formula: Encoding of the unit, Before removal Phase material, unit Remaining Volume fraction of phase material, This represents the upper limit of the local volume constraint. For unit Neighborhood radius The total number of contained units, i.e.

[0089]

[0090] The aforementioned local volume constraints need to be applied to every element. To reduce large-scale constraint calculations, a method is adopted. The norm condenses it into a single constraint as follows (abbreviated as) ):

[0091]

[0092] Where: penalty parameter After local constraint relaxation, each element The number of local volume constraints for phase materials is reduced to 1.

[0093] Local volume constraints limit the volume fraction of any material phase in the neighborhood of an element; however, they cannot precisely control the volume of this material phase used to fill the overall structure. Therefore, the following overall volume constraint for each material phase is introduced as a supplementary constraint:

[0094]

[0095] In the formula: To remove the previous structure After phase material, the remaining The total volume fraction of the phase material. for The upper limit, For positive numbers close to zero, in this invention .

[0096] Step S103-3: Evaluate the dynamic stiffness of the structure with the dynamic flexibility of the multiphase filled structure as the objective function, and establish a full-scale topology optimization and order reduction model of the functionally graded porous multiphase filled structure under transient load with the local distribution density of the material phase and the overall volume usage as the constraint functions.

[0097] The topology optimization mathematical formulation for minimizing the dynamic compliance of the functionally graded porous multiphase filled structure is as follows:

[0098]

[0099] In the formula: This is an estimate of the dynamic flexibility.

[0100] In conclusion, 2D finite element equations ( (to be reduced to) 2D finite element equations ( The dual variables of the topology optimization model—the Lagrange multipliers—must necessarily be reduced to the following order: Therefore, the motion control equations of the structure and the adjoint equations of sensitivity analysis are simultaneously reduced to low-dimensional equations, which helps to improve the efficiency of optimization iteration.

[0101] S103: Based on the above transient dynamic multiphase filling topology optimization reduction model, a reduced-order sensitivity time-domain adjoint equation is constructed to efficiently calculate the dynamic compliance and volume constraint sensitivity of functionally graded porous multiphase filling structures.

[0102] Based on the Newmark iterative equations in S101, the reduced-order motion control equations... ( The solution format for ) is:

[0103]

[0104] Substituting the above equation into the reduced-order motion control equation, we obtain the following residual equation. :

[0105]

[0106]

[0107]

[0108] In the formula: As a unit array, Increment step for time Extended load array, Increment step for time State matrix and It is a coefficient matrix.

[0109] At the initial moment, the residual equation Represented as:

[0110]

[0111]

[0112] The adjoint sensitivity analysis method employing a discretization-differentiation approach provides consistent sensitivity calculation results regardless of the number of time steps or the time integration scheme. Therefore, a residual weighting term is introduced into the objective function to enhance the result, constructing the following unified augmented Lagrangian function:

[0113]

[0114] In the formula: These are dual variables.

[0115] Objective function for design variables The sensitivity is:

[0116]

[0117] To eliminate the derivative terms of the state matrix with respect to the design variables, the dual variables must satisfy the following adjoint equation:

[0118]

[0119] The above adjoint equation can be solved by the following sequence equation, for Then we have:

[0120]

[0121] for Then we have:

[0122]

[0123]

[0124] Find the dual variables Then, the sensitivity of the objective function can be solved according to the following formula:

[0125]

[0126] Using the chain rule, the derivatives of local and global volume constraints with respect to the design variables are:

[0127]

[0128] S104: Construct an ADMM (Alternating Direction Multiplier Method)-MMA (Moving Asymptote Method) hybrid optimization algorithm to efficiently solve for the optimal topology design of functionally graded porous multiphase filled structures.

[0129] First, an initial feasible point is generated using the fast initial descent characteristic of ADMM (Alternating Direction Multiplier Method). Then, based on this, the final design result is obtained using MMA (Moving Asymptote Method), which has fine-grained convergence performance. Finally, a set of convergence indices is introduced. This is used to measure whether the optimization design has converged to a discrete solution, i.e.:

[0130]

[0131] If all design variables after filtering and projection processing or Then the convergence index ;otherwise, ,but .

[0132] Secondly, this invention provides a multiphase filling dynamics topology optimization system for functionally graded porous structures under transient loads, such as... Figure 2 As shown, it includes:

[0133] The initialization module constructs the design domain of the functionally graded porous multiphase filled structure and initializes the design variables; it also pre-sets the local volume constraints and overall volume constraints of the multiphase material usage in the filled structure, the algorithm parameters of the Newmark time integration scheme, the parameters of the multiphase material interpolation model, and the parameters of the ADMM-MMA hybrid optimization algorithm.

[0134] Preprocessing module: The design domain of discrete functionally graded porous multiphase filled structures, defining displacement boundary conditions, initial conditions and applying transient dynamic loads; combined with the multiphase material interpolation model to calculate the stiffness, mass and damping matrices of the unit, and assembling them to form the full-order transient dynamic finite element equation of the functionally graded porous multiphase filled structure;

[0135] The dynamic analysis module employs the unconditionally stable Newmark time integration scheme to solve the full-order transient dynamic finite element equations of the functionally graded porous multiphase filled structure, obtaining the transient image matrix of the displacement field. An orthogonal basis is constructed by calculating the eigenvalue problem of the correlation matrix. Subsequently, a reduced-order transient dynamic finite element equation of the filled structure is formed using projection transformation, thereby calculating the approximate value of the dynamic compliance of the filled structure and the local and global volume constraints of the multiphase material.

[0136] The optimization solution module calculates the sensitivity information of the multiphase filling dynamic topology optimization model of the functionally graded porous structure under transient loads, updates the design variables based on the ADMM-MMA hybrid optimization algorithm, and determines whether the optimization iteration results meet the convergence criteria. If not, the optimization iteration continues until the convergence criteria are met.

[0137] Example

[0138] Example 1

[0139] This invention addresses the multiphase filling dynamics topology optimization problem of a functionally graded porous cantilever beam structure under half-wave sinusoidal load, further verifying the effectiveness of the method.

[0140] Step S101: Based on the characteristic orthogonal decomposition method, seek the orthogonal basis, project the matrix in the full-order transient dynamic finite element equation into a low-order matrix, and thus form the reduced-order transient dynamic finite element equation of the functionally graded porous multiphase filling structure.

[0141] like Figure 3 As shown, the dimensions of the cantilever beam infill structure are: length is ,high ,thickness The left end of the design domain is fixed, and an amplitude of [value] is applied at the midpoint of the free edge on the right end. Half-wave sinusoidal load Load application time The Poisson's ratio of each phase material is... Both have mass density Young's modulus is , Material coefficient , , , The Young's modulus of the void phase is The Young's moduli of the other solid phases are respectively , , The mass, stiffness, and damping coefficients of each phase material are all... and Total number of response steps The initial design domain is discretized as follows: A plane stress element.

[0142] This embodiment considers the topology optimization design of functionally graded porous structures implemented under single-phase, two-phase, and three-phase solid material filling conditions.

[0143] The elastodynamic behavior of the functionally graded porous multiphase filled structure is described by the motion control equations discretized using the finite element method, expressed as:

[0144]

[0145] In the formula: , , These represent the total mass, damping, and stiffness matrices of the infill structure, respectively. For time steps External load array, , , They are time steps The acceleration, velocity, and displacement arrays of the filled structure. The time step is the time at which the external load terminates.

[0146] Solving the above motion control equations using the Newmark method yields:

[0147]

[0148] In the formula: This is the time step increment.

[0149] By solving the full-order analysis model Obtain the displacement field in Spatial distribution at different times They are assembled to form a transient matrix. ,in The degrees of freedom of the finite element model after discretization are usually .

[0150] The characteristic orthogonal decomposition method, starting from the energy perspective, aims to find orthogonal basis. This ensures that all displacement snapshots of the above full-order model lie in orthogonal basis vectors. The sum of projected energy in the direction is maximized, and then... Construct a low-dimensional subspace that best uniformly approximates the solution space of the full-order model based on the basis.

[0151] The above-mentioned best uniform approximation problem can be transformed into solving the correlation matrix. The problem of obtaining eigenvalues, namely:

[0152]

[0153] In the formula: , Correlation matrix eigenvalues ​​and eigenvectors.

[0154] To extract the dominant deformation modes, i.e., the eigenmodes, that contribute most to the structural dynamic response throughout the entire time history, this can be achieved by truncating... The number of eigenvectors, selecting the minimum number of basis vectors. This makes the total energy of the low-dimensional subspace equal to the energy of the full-order solution space. Satisfy the following formula:

[0155]

[0156] The reduced basis vectors project the full-order solution space onto a lower-dimensional subspace, and the truncated basis vectors... ( Then, the displacement field of the infill structure can be approximately expressed as:

[0157]

[0158] In the formula: This is the correlation coefficient matrix, hereinafter referred to as... .

[0159] Substituting the above equation into the motion control equation and multiply by the left on both sides This yields the following reduced-order control equations:

[0160]

[0161]

[0162] In the formula: , , All are The matrix, the field variables to be solved are transformed into ,satisfy Therefore, the solution scale of the reduced-order analysis model is much smaller than that of the full-order model.

[0163] S102: Construct a material interpolation model for functionally graded porous multiphase filled structures, establish local and global volume constraints on material usage, and propose a transient dynamic multiphase filled topology optimization reduction model with the goal of minimizing dynamic compliance.

[0164] Step S102-1: Filter the design variables and construct a material interpolation model for a multiphase filled structure based on the Ersatz minimum positive number: volume and stiffness interpolation model. Use this model to calculate the global mass, stiffness and damping matrices in the full-order transient dynamic finite element equation.

[0165] To suppress the checkerboard pattern, the first Design variables corresponding to phase materials Weighted filtering is applied within the neighborhood of the cell, and the filtered design variables are... Represented as:

[0166]

[0167]

[0168] In the formula: This refers to the total number of multiple material types that include the void phase. , , Here is the filtering matrix. , Units Area and weight, The filter radius is Neighborhood unit The center (in terms of position vector) (representation) and unit The center (in terms of position vector) Euclidean distance (indicated by) This is the filter index.

[0169] In any filling Phase material unit Construct volume interpolation function abbreviated as Then we have:

[0170]

[0171]

[0172] In the formula: It is a positive number that is much less than 1. This is the smoothed Heaviside projection function. This is the projection sharpness parameter. This is the threshold value.

[0173] To ensure that the intermediate elastic modulus is adequately penalized, the multi-material stiffness interpolation function should be a convex function. Furthermore, the stiffness coefficients of the void material should be assigned extremely small positive numbers close to zero to avoid singularities in the stiffness matrix. Therefore, the multi-material stiffness interpolation function is expressed as:

[0174]

[0175] In the formula: As a penalty factor, For the first Stiffness coefficient of phase material, is the stiffness coefficient of the hollow material.

[0176] Based on the material interpolation model of the multiphase filled structure, the global mass, stiffness, and damping matrices in the full-order transient dynamic finite element equations are expressed as:

[0177]

[0178] In the formula: the damping matrix adopts the Rayleigh damping model. , These are mass damping and stiffness damping, respectively.

[0179] Step S102-2: Establish local volume constraints to control the local distribution density of multiphase materials in the neighborhood of the control unit; at the same time, collaboratively establish global volume constraints to accurately control the overall volume usage of multiphase materials.

[0180] To control the local distribution density of each phase material within the cell neighborhood, thereby enabling the filling design of multiphase materials, the following local volume constraint is introduced. (abbreviated as) ):

[0181]

[0182] In the formula: Encoding of the unit, Before removal Phase material, unit Remaining Volume fraction of phase material, This represents the upper limit of the local volume constraint. For unit Neighborhood radius The total number of contained units, i.e.

[0183]

[0184] The aforementioned local volume constraints need to be applied to every element. To reduce large-scale constraint calculations, a method is adopted. The norm condenses it into a single constraint as follows (abbreviated as) ):

[0185]

[0186] Where: penalty parameter After local constraint relaxation, each element The number of local volume constraints for phase materials is reduced to 1.

[0187] Local volume constraints limit the volume fraction of any material phase in the vicinity of an element; however, they cannot precisely control the volume of this material phase used to fill the overall structure. Therefore, a global volume constraint for any material phase is introduced as a supplementary constraint.

[0188]

[0189] In the formula: To remove the previous structure After phase material, the remaining The total volume fraction of the phase material. for The upper limit, For positive numbers close to zero, in this invention .

[0190] Step S103-3: Evaluate the dynamic stiffness of the structure with the dynamic flexibility of the multiphase filled structure as the objective function, and establish a full-scale topology optimization and order reduction model of the functionally graded porous multiphase filled structure under transient load with the local distribution density of the material phase and the overall volume usage as the constraint functions.

[0191] The topology optimization mathematical formulation for minimizing the dynamic compliance of the functionally graded porous multiphase filled structure is as follows:

[0192]

[0193] In the formula: This is an estimate of the dynamic flexibility.

[0194] In conclusion, 2D finite element equations ( (to be reduced to) 2D finite element equations ( The dual variables of the topology optimization model—the Lagrange multipliers—must necessarily be reduced to the following order: Therefore, the motion control equations of the structure and the adjoint equations of sensitivity analysis are simultaneously reduced to low-dimensional equations, which helps to improve the efficiency of optimization iteration.

[0195] S103: Based on the above-mentioned reduced-order transient dynamic multiphase filling topology optimization model, a reduced-order sensitivity time-domain adjoint equation is constructed to efficiently calculate the dynamic compliance and volume constraint sensitivity of functionally graded porous multiphase filling structures.

[0196] Based on the Newmark iterative equations in S101, the reduced-order motion control equations... ( The solution format for ) is:

[0197]

[0198] Substituting the above equation into the reduced-order motion control equation, we obtain the following residual equation. :

[0199]

[0200]

[0201]

[0202] In the formula: As a unit array, Increment step for time Extended load array, Increment step for time State matrix and It is a coefficient matrix.

[0203] At the initial moment, the residual equation Represented as:

[0204]

[0205]

[0206] The adjoint sensitivity analysis method employing a discretization-differentiation approach provides consistent sensitivity calculation results regardless of the number of time steps or the time integration scheme. Therefore, a residual weighting term is introduced into the objective function to enhance the result, constructing the following unified augmented Lagrangian function:

[0207]

[0208] In the formula: These are dual variables.

[0209] Objective function for design variables The sensitivity is:

[0210]

[0211] To eliminate the derivative terms of the state matrix with respect to the design variables, the dual variables must satisfy the following adjoint equation:

[0212]

[0213] The above adjoint equation can be solved by the following sequence equation, for Then we have:

[0214]

[0215] for Then we have:

[0216]

[0217]

[0218] Find the dual variables Then, the sensitivity of the objective function can be solved according to the following formula:

[0219]

[0220] Using the chain rule, the derivatives of local and global volume constraints with respect to the design variables are:

[0221] .

[0222] S104: Construct an ADMM (Alternating Direction Multiplier Method)-MMA (Moving Asymptote Method) hybrid optimization algorithm to efficiently solve for the optimal topology design of functionally graded porous multiphase filled structures.

[0223] First, an initial feasible point is generated using the fast initial descent characteristic of ADMM (Alternating Direction Multiplier Method). Then, based on this, the final design result is obtained using MMA (Moving Asymptote Method), which has fine-grained convergence performance. Finally, a set of convergence indices is introduced. This is used to measure whether the optimization design has converged to a discrete solution, i.e.:

[0224]

[0225] If all design variables after filtering and projection processing or Then the convergence index ;otherwise, ,but .

[0226] The specific optimization results are as follows: Figures 4-7 As shown in Appendix 1.

[0227] Figure 4 For the load application time The topology optimization results for a cantilever beam filled with a single-phase material. For example... Figure 4 As shown in 4.a and 4.c, using only local volume constraints, the volume fraction of the single-phase material is 0.48, which does not reach the upper limit of the constraint. This is because local volume constraints implicitly impose an upper limit on the overall volume constraint. However, by introducing volume constraints as a supplementary constraint, local volume constraints become secondary constraints, thus allowing for precise control of the overall material usage in the optimal design, satisfying... .like Figure 4 As shown in .b and 4.d, local volume constraints are no longer strictly guaranteed. Because the optimal topology using local volume constraints consumes more material, its dynamic flexibility is lower than that of the optimal topology that introduces additional global volume constraints.

[0228] Figure 5 For the load application time The topology optimization results and convergence history of a cantilever beam filled with two-phase materials are shown, where green represents strong materials and red represents weak materials. This is compared to traditional multi-material topology optimization (such as...). Figure 5 Unlike (as shown in .a), strong and weak phase materials are not concentrated in independent regions (e.g., Figure 5 Instead of being distributed in a slender, intersecting form (as shown in b), they are distributed in a crisscrossing pattern due to local volume constraints limiting the amount of material used in the vicinity of the unit cell. Compared to single-phase material-filled designs (such as...), Figure 4As shown in Figure .b), the strong material fibers are mainly interwoven inside the weak material crossbars, similar to fiber-reinforced composite materials, thus exhibiting lower dynamic flexibility and superior dynamic stiffness. Furthermore, the convergence index... , The value gradually decreases to 0, verifying the convergence of the proposed optimization algorithm.

[0229] Figure 6 For the load application time and The topology optimization results for the cantilever beam are shown below. Similar to the previous analysis, the dynamic stiffness of the two-phase material is superior to that of the single-phase material. Different working conditions In the optimal topology of the working condition, the crossbar elements tend to be distributed as free edges of a cantilever beam, which resists the enhanced vibration effect of the free end caused by rapid loading by increasing the inertial force.

[0230] Figure 7 For the load application time The optimization results and convergence history of the cantilever beam filled with three-phase materials are shown, where blue represents strong materials, yellow represents medium-strength materials, and red represents weak materials. Compared with the optimal topology filled with two-phase materials, the cross-bar element distribution of the three-phase material-filled configuration is relatively uniform, resulting in stronger load-bearing capacity. However, several jumps exist in the optimization process curve, which is due to changes in the sharpness parameter. When the density is doubled, the intermediate density value of the unit cells in the structure undergoes a sudden change. Using this... The purpose of the doubling strategy is to gradually transform intermediate density values ​​into strict 0-1 design results. Therefore, the proposed optimization framework exhibits good convergence.

[0231] As shown in Table 1, to verify the effectiveness of the order reduction strategy of the orthogonal characteristic decomposition method, the load application time was used as the benchmark. For example, the full-order optimization model and the reduced-order optimization model obtained the same optimal topology configuration and the objective function value was consistent. Compared with the average iteration time of a single iteration in the optimization process, the topology optimization method based on the reduced-order strategy can significantly shorten the computation time and improve the computation efficiency. Table 1 compares the optimization results of the full-order topology optimization model and the reduced-order topology optimization model in Embodiment 1 of the present invention (load application time). ).

[0232] Table 1

[0233]

[0234] The results of Embodiment 1 of this invention demonstrate that the proposed multiphase filling topology optimization method can form an optimal topology configuration with an "external open shell + internal self-supporting filling structure," reducing the need for self-supporting structures and facilitating the additive manufacturing process of multiphase functionally graded porous structures. In particular, the model reduction strategy using the eigenmode decomposition method maintains the same computational accuracy as the full-order topology optimization model while significantly improving computational efficiency. Therefore, the algorithm of this invention effectively solves the multiphase filling dynamics design problem of functionally graded porous structures under transient excitation and exhibits excellent scalability for large-scale engineering applications.

[0235] The present invention has been described in detail above with reference to specific embodiments and exemplary examples. However, these descriptions should not be construed as limiting the present invention. Those skilled in the art will understand that various equivalent substitutions, modifications, or improvements can be made to the technical solutions and embodiments of the present invention without departing from the spirit and scope of the present invention, and all such modifications and improvements fall within the scope of the present invention.

Claims

1. A method for multiphase filling dynamic topology optimization of functionally graded porous structures under transient loads, characterized in that, Includes the following steps: S101: Based on the characteristic orthogonal decomposition method, orthogonal basis is sought, and the matrix in the full-order transient dynamic finite element equation is projected into a low-order matrix, thereby forming the reduced-order transient dynamic finite element equation of the functionally graded porous multiphase filling structure. S102: Construct a material interpolation model for functionally graded porous multiphase filled structures, establish local and global volume constraints on material usage, and develop a transient dynamic multiphase filled topology optimization reduction model with the goal of minimizing dynamic compliance. S103: Based on the above transient dynamic multiphase filling topology optimization reduction model, a reduced-order sensitivity time-domain adjoint equation is constructed to efficiently calculate the dynamic compliance and volume constraint sensitivity of functionally graded porous multiphase filling structures. S104: Construct a hybrid optimization algorithm of alternating direction multiplier method (ADMM) and moving asymptote method (MMA) to efficiently solve for the optimal topology design of functionally graded porous multiphase filled structures.

2. The multiphase filling dynamics topology optimization method for functionally graded porous structures under transient loads as described in claim 1, characterized in that, Step S101 includes: The elastodynamic behavior of the functionally graded porous multiphase filled structure is described by the motion control equations discretized using the finite element method, expressed as: In the formula: , , These represent the total mass, damping, and stiffness matrices of the infill structure, respectively. For time step External load array, , , They are time steps The acceleration, velocity, and displacement arrays of the filled structure. The time step is the time at which the external load terminates; Solving the above motion control equations using the Newmark method yields: In the formula: Increment for time step; By solving the full-order analysis model Obtain the displacement field in Spatial distribution at different times They are assembled to form a transient matrix. ,in The degrees of freedom of the finite element model after discretization are usually ; The characteristic orthogonal decomposition method, starting from the energy perspective, aims to find orthogonal basis. This ensures that all displacement snapshots of the above full-order model lie in orthogonal basis vectors. The sum of projected energy in the direction is maximized, and then... Construct a low-dimensional subspace that best uniformly approximates the solution space of the full-order model based on the basis; The above-mentioned best uniform approximation problem can be transformed into solving the correlation matrix. The problem of obtaining eigenvalues, namely: In the formula: , Correlation matrix eigenvalues ​​and eigenvectors; To extract the dominant deformation modes, i.e., the intrinsic modes, that contribute the most to the structural dynamic response throughout the entire time history, truncation can be used. The number of eigenvectors, selecting the minimum number of basis vectors. This makes the total energy of the low-dimensional subspace equal to the energy of the full-order solution space. Satisfy the following formula: The reduced basis vectors project the full-order solution space onto a lower-dimensional subspace, and the truncated basis vectors... ( Then, the displacement field of the infill structure can be approximately expressed as: In the formula: This is the correlation coefficient matrix, hereinafter referred to as... ; Substituting the above equation into the motion control equation and multiply by the left on both sides This yields the following reduced-order control equations: In the formula: , , All are The matrix, the field variables to be solved are transformed into ,satisfy Therefore, the solution scale of the reduced-order analysis model is much smaller than that of the full-order model.

3. The multiphase filling dynamics topology optimization method for functionally graded porous structures under transient loads according to claim 1 or 2, characterized in that, Step S102 includes: Step S102-1: Filter the design variables and construct a material interpolation model for a multiphase filled structure based on the Ersatz minimum positive number: volume and stiffness interpolation model, and use this model to calculate the global mass, stiffness and damping matrices in the full-order transient dynamic finite element equations. Step S102-2: Establish local volume constraints to control the local distribution density of multiphase materials in the neighborhood of the control unit; at the same time, collaboratively establish global volume constraints to accurately control the overall volume usage of multiphase materials. Step S103-3: Evaluate the dynamic stiffness of the structure with the dynamic flexibility of the multiphase filled structure as the objective function, and establish a full-scale topology optimization and order reduction model of the functionally graded porous multiphase filled structure under transient load with the local distribution density of the material phase and the overall volume usage as the constraint functions.

4. The multiphase filling dynamics topology optimization method for a functionally graded porous structure under transient load as described in claim 3, characterized in that, Step S102-1 includes: To suppress the checkerboard pattern, the first Design variables corresponding to phase materials Weighted filtering is applied within the neighborhood of the cell, and the filtered design variables are... Represented as: In the formula: This refers to the total number of multiple material types that include the void phase. , , Here is the filtering matrix. , Units Area and weight, The filter radius is Neighborhood unit The center (in terms of position vector) (representation) and unit The center (in terms of position vector) Euclidean distance (indicated by) The filter index; In any filling Phase material unit Construct volume interpolation function abbreviated as Then we have: In the formula: It is a positive number that is much less than 1. This is the smoothed Heaviside projection function. This is the projection sharpness parameter. For threshold; To ensure that the intermediate elastic modulus is adequately penalized, the multi-material stiffness interpolation function should be a convex function; for the stiffness coefficient of the void material, a very small positive number close to zero should be assigned to avoid singularities in the stiffness matrix; the multi-material stiffness interpolation function is expressed as: In the formula: As a penalty factor, For the first Stiffness coefficient of phase material, The stiffness coefficient of the cavitary material; Based on the material interpolation model of the multiphase filled structure, the global mass, stiffness, and damping matrices in the full-order transient dynamic finite element equations are expressed as: In the formula: the damping matrix adopts the Rayleigh damping model. , These are mass damping and stiffness damping, respectively.

5. The multiphase filling dynamics topology optimization method for a functionally graded porous structure under transient load as described in claim 4, characterized in that, Step S102-2 includes: To control the local distribution density of each phase within the cell neighborhood, thereby enabling the filling design of multiphase materials, the following local volume constraint is introduced. (abbreviated as) ): In the formula: The encoding of the unit, Before removal Phase material, unit The remaining Volume fraction of phase material, This represents the upper limit of the local volume constraint. For unit Neighborhood radius The total number of contained units, i.e. The aforementioned local volume constraints need to be applied to each element to reduce large-scale constraint calculations. The norm condenses it into a single constraint as follows (abbreviated as) ): Where: penalty parameter After local constraint relaxation, each element The number of local volume constraints for phase materials is reduced to 1. Local volume constraints limit the volume fraction of any material phase in the vicinity of an element; however, they cannot precisely control the volume of this material phase used to fill the overall structure. Therefore, a global volume constraint for any material phase is introduced as a supplementary constraint. In the formula: To remove the previous structure After phase material, the remaining The total volume fraction of the phase material. for The upper limit, For positive numbers close to zero, in this invention .

6. A multiphase filling dynamics topology optimization method for a functionally graded porous structure under transient load, as described in claim 4 or 5, characterized in that... Step S102-3 includes: The topology optimization mathematical formulation for minimizing the dynamic compliance of the functionally graded porous multiphase filled structure is as follows: In the formula: This is an estimate of the dynamic flexibility; In conclusion, 2D finite element equations ( (to be reduced to) 2D finite element equations ( The dual variables of the topology optimization model—the Lagrange multipliers—must necessarily be reduced to the following order: Therefore, the motion control equations of the structure and the adjoint equations of sensitivity analysis are simultaneously reduced to low-dimensional equations, which helps to improve the efficiency of optimization iteration.

7. A multiphase filling dynamics topology optimization method for a functionally graded porous structure under transient load as described in claim 1 or 6, characterized in that, Step S103 includes: Based on the Newmark iterative equations in S101, the reduced-order motion control equations... ( The solution format for ) is: Substituting the above equation into the reduced-order motion control equation, we obtain the following residual equation. : In the formula: As a unit array, Increment step for time Extended load array, Increment step for time State matrix and It is a coefficient matrix; At the initial moment, the residual equation Represented as: The adjoint sensitivity analysis method, employing a discretization-differentiation approach, provides consistent sensitivity calculation results regardless of the number of time steps or the time integration scheme. A residual weighting term is introduced into the objective function to enhance sensitivity, resulting in the following unified augmented Lagrangian function: In the formula: As dual variables; Objective function for design variables The sensitivity is: To eliminate the derivative terms of the state matrix with respect to the design variables, the dual variables must satisfy the following adjoint equation: The above adjoint equation can be solved by the following sequence equation, for Then we have: for Then we have: Find the dual variables Then, the sensitivity of the objective function can be solved according to the following formula: Using the chain rule, the derivatives of local and global volume constraints with respect to the design variables are: 。 8. The multiphase filling dynamics topology optimization method for a functionally graded porous structure under transient load as described in claim 7, characterized in that, Step S104 includes: First, an initial feasible point is generated using the fast initial descent characteristic of the Alternating Direction Multiplier Method (ADMM). Then, based on this, the Moving Asymptote Method (MMA), with its fine convergence performance, is employed to obtain the final design result. Finally, a set of convergence indices is introduced. This is used to measure whether the optimization design has converged to a discrete solution, i.e.: If all design variables after filtering and projection processing or Then the convergence index ;otherwise, ,but .

9. A multiphase filling dynamic topology optimization system for functionally graded porous structures under transient loads, characterized in that, The system is implemented based on the method of any one of claims 1 to 8, and includes the following program modules: The initialization module constructs the design domain of the functionally graded porous multiphase filled structure and initializes the design variables; it also pre-sets the local volume constraints and overall volume constraints of the multiphase material usage in the filled structure, the algorithm parameters of the Newmark time integration scheme, the parameters of the multiphase material interpolation model, and the parameters of the ADMM-MMA hybrid optimization algorithm. Preprocessing module: The design domain of discrete functionally graded porous multiphase filled structures, defining displacement boundary conditions, initial conditions and applying transient dynamic loads; combined with the multiphase material interpolation model to calculate the stiffness, mass and damping matrices of the unit, and assembling them to form the full-order transient dynamic finite element equation of the functionally graded porous multiphase filled structure; The dynamic analysis module uses the unconditionally stable Newmark time integration scheme to solve the full-order transient dynamic finite element equations of the functionally graded porous multiphase filled structure and obtain the transient matrix of the displacement field. An orthogonal basis is constructed by calculating the eigenvalues ​​of the correlation matrix. Then, a reduced-order transient dynamic finite element equation for the filled structure is formed by using projection transformation. This equation is used to calculate the approximate dynamic compliance of the filled structure and the local and global volume constraints of the multiphase material. The optimization solution module calculates the sensitivity information of the multiphase filling dynamic topology optimization model of the functionally graded porous structure under transient loads, updates the design variables based on the ADMM-MMA hybrid optimization algorithm, and determines whether the optimization iteration results meet the convergence criteria. If not, the optimization iteration continues until the convergence criteria are met.

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