Topological optimization method of phase transition super-elastic material based on nonlinear constitutive model

Abstract

The application discloses a phase transition super-elastic material topology optimization method based on a nonlinear constitutive model and belongs to the technical field of intelligent material structure design and multi-physical field optimization. The method comprises the following steps: establishing a super-elastic constitutive model containing a martensitic volume fraction to describe the coupling relationship between material phase transition behavior and mechanical properties; constructing a topology optimization model with the minimum structural flexibility as an objective and the volume fraction and a balance equation as constraints; adopting an additional super-elastic technology of dynamic reconstruction to solve the numerical instability problem in the optimization process; performing sensitivity analysis through an adjoint vector method; and obtaining an optimal topology configuration through iterative solution by using an optimal criteria method. The application can efficiently and stably realize the structure optimization of the phase transition super-elastic material, significantly improves the calculation efficiency and numerical stability of the optimization process, and is suitable for the structure design of intelligent materials such as shape memory alloys in the fields of aerospace, medical devices and the like.

Description

Technical Field

[0001] This invention belongs to the field of intelligent material structure design and multiphysics optimization technology, specifically involving a topology optimization method for hyperelastic materials with phase change characteristics (such as shape memory alloys). Background Technology

[0002] Hyperelastic materials with phase change properties (typically represented by shape memory alloys) exhibit irreplaceable advantages in scenarios requiring dynamic mechanical response adjustment, adaptive deformation, or extreme load-bearing conditions due to their unique phase change hyperelasticity, shape memory effect, and high energy absorption capacity. As high-end equipment continues to upgrade its structural requirements for "lightweight, high reliability, and high performance," higher demands are placed on the structural design precision and performance control capabilities of hyperelastic materials with phase change properties. Topology optimization, as a core technology for achieving efficient material distribution and on-demand performance matching, has become a key support for the structural design of hyperelastic materials.

[0003] While some progress has been made in the research of topology optimization for hyperelastic materials, several limitations remain in both technical implementation and practical application. Previous topology optimization techniques for hyperelastic materials have largely focused on optimizing single mechanical parameters (such as stiffness, strength, and energy absorption), neglecting the dynamic coupling relationship between material phase transformation behavior and mechanical properties. In nonlinear topology optimization of large-scale structures, nonlinear optimization problems often suffer from excessively long optimization times and enormous computational loads due to limitations in solution methods, resulting in very limited design space and unsatisfactory optimization results.

[0004] In summary, current topology optimization methods for hyperelastic materials have significant shortcomings in areas such as synergistic optimization of phase transition and mechanical properties, multiphysics coupling modeling, and timeliness, making it difficult to meet the application requirements of "lightweight, high reliability, and high performance" for hyperelastic material structures in aerospace, medical devices, and automotive industries. Therefore, developing an efficient topology optimization method that can synergistically consider the phase transition characteristics of materials has urgent practical significance and important engineering value. Summary of the Invention

[0005] To overcome the shortcomings of the existing technology, the purpose of this invention is to provide an efficient and stable method for topology optimization of phase change hyperelastic materials.

[0006] To achieve the above and other objectives, this invention proposes a topology optimization method for phase transition hyperelastic materials based on a nonlinear constitutive model, comprising:

[0007] Based on nonlinear solid mechanics theory, a hyperelastic constitutive model incorporating material phase transition information was established to capture the complex mechanical behavior of hyperelastic materials.

[0008] By introducing an internal variable (martensite volume fraction), a martensite volume fraction evolution equation is constructed to describe the relationship between stress and phase transformation state in hyperelastic materials.

[0009] A finite element analysis model of the hyperelastic material is established based on the aforementioned nonlinear constitutive relation.

[0010] Based on the aforementioned relationship, the expression for the external load on the node is derived, and the nonlinear equilibrium equation is further solved to obtain the structural displacement response of the hyperelastic material.

[0011] A topology optimization mathematical model for phase change hyperelastic materials is established, using the minimization of the flexibility of the hyperelastic material structure as the objective function and the volume fraction and residual vector of the equilibrium equation of the hyperelastic material structure as constraints.

[0012] The numerical instability caused by excessive deformation of low-density elements during the optimization process is solved by using the additional hyperelasticity technique of dynamic reconstruction.

[0013] The optimization objective function and constraint sensitivity information of the hyperelastic material structure are calculated based on the topology optimization mathematical model of the hyperelastic material. Sensitivity filtering technology is then used to correct the optimization objective function and the sensitivity information.

[0014] The topology optimization problem of the hyperelastic material is solved using the optimal criterion method, and it is determined whether the convergence condition of the optimal criterion method is met. If so, the optimal result is output.

[0015] Compared with the prior art, the beneficial effects of the present invention are as follows: based on the theory of nonlinear solid mechanics, an internal variable (martensite volume fraction) is introduced to establish a hyperelastic constitutive model containing material phase transformation information, which serves as the basis for the finite element analysis model. The structural displacement response of the phase transformation hyperelastic material is obtained by solving the nonlinear equilibrium equation. The numerical instability problem in the optimization process is solved by the additional hyperelastic technique of dynamic reconstruction, thereby enabling the topological configuration of the phase transformation hyperelastic material to be obtained more efficiently, stably, and accurately. Attached Figure Description

[0016] Figure 1 This is a flowchart of the topology optimization method for phase transition hyperelastic materials based on a nonlinear constitutive model in an embodiment of the present invention.

[0017] Figure 2 This is a schematic diagram of the design domain, load, and boundary conditions of the cantilever beam structure in an embodiment of the present invention.

[0018] Figure 3 The results show the topology optimization of the cantilever beam structure made of phase change hyperelastic material in the embodiments of the present invention.

[0019] Figure 4This is a schematic diagram of the bridge structure design domain, load, and boundary conditions in an embodiment of the present invention.

[0020] Figure 5 The results show the topology optimization of the bridge structure using phase change hyperelastic materials in this embodiment of the invention. Detailed Implementation

[0021] This invention proposes a topology optimization method for phase transition hyperelastic materials based on a nonlinear constitutive model. This method is an efficient topology optimization method for smart materials, and it takes the structural flexibility of the hyperelastic material as the objective function.

[0022] To consider the dynamic coupling relationship between material phase transformation behavior and mechanical properties, this invention establishes a nonlinear constitutive model containing an internal variable (martensite volume fraction) based on nonlinear solid mechanics theory. To address the numerical instability caused by excessive deformation of low-density elements during optimization, this invention employs an additional hyperelastic technique based on dynamic reconstruction to ensure the stability of the optimization process. The optimality criterion method is used as the optimization algorithm. Considering the large computational cost required to solve the objective function sensitivity using direct methods, this invention solves for the objective function sensitivity using adjoint vectors.

[0023] The method of the present invention will now be described in detail with reference to specific examples and accompanying drawings.

[0024] 1. Nonlinear constitutive model

[0025] Hyperelastic materials exhibiting austenitic-martensite phase transformation characteristics exhibit significantly nonlinear material properties due to the coupled influence of multiple factors, including ambient temperature, stress state, and phase transformation hysteresis. To account for the dynamic coupling between phase transformation behavior and mechanical properties, this invention introduces an internal variable (martensite volume fraction) to construct a phase transformation driving force function to describe the relationship between stress and phase transformation state in hyperelastic materials, thereby establishing a nonlinear constitutive model incorporating phase transformation information. Specifically:

[0026] 1.1 Free Energy Density Function of Hyperelastic Materials

[0027] This invention selects Green-Lagrange strain ε as the strain metric, and utilizes the right Cauchy-Green deformation tensor C = F. T F, the free energy density function of the two phases can be expressed as ψ = ψ(T, C), and the stress metric is the second Piola-Kirchhoff stress tensor. or The two invariants of the right Cauchy-Green deformation tensor can be expressed as:

[0028]

[0029] Where λ iIt is the main stretch.

[0030] The free energy density functions for austenite and martensite phases can be modeled as follows:

[0031]

[0032] in and It is the free energy that is only related to temperature, and a and m are material constants.

[0033] To establish the relationship between stress and phase transformation, the martensite volume fraction ξ is introduced to characterize the austenite-martensite phase transformation state. The free energy density function ψ should be expressed as a function of the deformation gradient and the volume fractions of the austenite and martensite phases, thus yielding...

[0034] ψ=(1-ξ)ψ A +ξψ M +kξ(1-ξ)

[0035] Where ξ is the volume fraction of martensite (i.e., phase change), and its value is between 0 (complete austenite) and 1 (complete martensite). kξ(1-ξ) is used to consider the interaction between the austenite phase and the martensite phase, such as the energy change caused by the interfacial energy and dislocation of the two phases during the phase transformation process. k is the interaction coefficient, which reflects the intensity of the interaction between the two phases.

[0036] 1.2 Phase Change Driving Force Function

[0037] The total energy functional in a nonlinear system can be expressed as:

[0038]

[0039] Where f is the force per unit volume of the object, u is the displacement vector field, t is the traction force on the boundary, and Ω is the material volume. This represents the volume boundary.

[0040] To find the equilibrium state, the principle of minimum potential energy is applied, requiring the first variation of the energy function to be zero, i.e., δW = 0. This involves W with respect to the displacement field u and the phase variable. The change. This change can be represented as:

[0041]

[0042] Where f is the force per unit volume of the object, u is the displacement vector field, and t is the traction force on the boundary.

[0043] This change leads to a set of Euler-Lagrange equations. For displacement u, we obtain...

[0044]

[0045] Stress can be expressed as:

[0046]

[0047] The martensite volume fraction satisfies:

[0048]

[0049] The negative value is defined as the phase transition driving force. f q When the value is greater than 0, austenite will transform into martensite, f q When <0, martensite will transform into austenite, f q When the phase transition value is 0, the phase transition state is stable.

[0050] The hyperelastic constitutive model, which incorporates material phase transformation information, is composed of stress-strain relations and the evolution equation of martensite volume fraction. It describes the mechanical behavior and phase transformation process of phase transformation hyperelastic materials under stress.

[0051] Compared to constitutive models that do not include internal variables, the nonlinear constitutive model in this invention can clearly describe the dynamic coupling relationship between material phase transformation behavior and mechanical properties when applied to optimization algorithms.

[0052] In practical applications, it is necessary to determine various material parameters through experiments based on the specific characteristics of phase change hyperelastic materials (such as shape memory alloys) in order to accurately predict the phase change behavior and complex mechanical response of the material under different stress conditions.

[0053] 2. Topology Optimization

[0054] 2.1 Optimization Model

[0055] The optimal material configuration that minimizes compliance is sought within a given design domain. The optimization problem can be expressed as:

[0056]

[0057] Where x is the element density as a design variable, and C is the objective function to be minimized. V is the relative density, and V is the actual volume fraction of the material. * Let P be the external force vector, and f be the specified material volume fraction. int For internal nodal force vectors, This is the residual force vector.

[0058] Using the SIMP method, the free energy density of element i is penalized, and the free energy density of element i can be expressed as:

[0059]

[0060] Where ψ0 is the free energy density of the perfectly solid material, and p is the penalty factor used to penalize intermediate densities during the optimization process. The unit relative density.

[0061] 2.2 Nonlinear Finite Element Model

[0062] The tangent elastic tensor D describes the rate of change of stress with strain, and linearizes the nonlinear relationship piecewise, i.e., ΔS = D:Δe. The tangent elastic tensor D can be calculated based on the free energy and a given deformation state, and its expression is:

[0063]

[0064] The equilibrium equations are rewritten as a set of nonlinear equations in a standard finite element program:

[0065]

[0066] The tangent stiffness matrix K of the nonlinear system T It can be represented as:

[0067]

[0068] In the formula B N B represents the nonlinear part of the strain-displacement matrix. G ∑ represents the linear part of the strain-displacement matrix, and ∑ is the block matrix representation of the second Piola-Kirchhoff stress S.

[0069] Nonlinear equations can be solved by iteratively calculating linear equations using the Newton-Raphson iteration method. Let U be the global displacement vector in the k-th iteration. (k) The global displacement vector in subsequent iterations can be updated as follows:

[0070] U (k+1) =U (k) +ΔU (k)

[0071] Where ΔU (k) The displacement update increment is obtained by solving the following system of linear equations:

[0072] K T (U (k) )ΔU (k) =-R(u (k) )

[0073] 2.3 Numerical stabilization techniques

[0074] To address the numerical instability caused by material and geometric nonlinearity during the optimization process, this invention proposes using a dynamic reconstruction-based additional hyperelastic technique to ensure the stability of the optimization process.

[0075] In practice, the original material unit (phase change material) and the additional material unit are overlapped and share a set of nodes and node displacements to reconstruct a new material unit.

[0076] Furthermore, this invention directly uses reconstructed material elements for finite element analysis, ensuring that the number of elements to be calculated within the design domain is consistent with that without using numerical stabilization methods. Compared to traditional additional hyperelasticity techniques, the dynamically reconstructed additional hyperelasticity technique used in this invention can significantly reduce solution time while addressing numerical instability issues.

[0077] The free energy density function of the Yeoh model is:

[0078]

[0079] Where C 10 C 20 Here, K1 and K2 are the bulk moduli of the material. During the finite element analysis, these moduli will be adjusted to effective values ​​that suppress numerical instability based on the logarithmic equivalent strain of the elements. 20 The adaptive adjustment standard is:

[0080]

[0081] Where ε von For logarithmic equivalent change, ε * To adjust the threshold, it is set to 0.3, and the superscript k is the number of iterations in the finite element analysis.

[0082] The free energy density function of the reconstructed unit can be expressed as:

[0083]

[0084] The tangent elastic tensor changes to

[0085]

[0086] 2.4 Sensitivity Analysis

[0087] This invention uses the optimal criterion method as the optimization solution method. Sensitivity analysis of the objective function is necessary to determine the derivative of the objective function with respect to given design variables, which is essential for solving optimization problems using the optimal criterion method. Considering the large computational cost required to solve the sensitivity of the objective function using direct methods, this invention uses the adjoint vector method.

[0088] Objective function C with respect to design variable x jThe sensitivity can be expressed as:

[0089]

[0090] in Let be the differential of the objective function with respect to relative density. Using the adjoint vector method, the sensitivity of the extended objective function to relative density after introducing the adjoint vector is:

[0091]

[0092] Pick The above formula can be rewritten as:

[0093]

[0094] in It is an internal nodal force. It is the internal force of the phase change material. It is the internal force of the added material.

[0095] 2.5 Optimize the process

[0096] Step S1: Divide the design domain into a mesh and apply corresponding boundary conditions based on the mesh node coordinates and degree-of-freedom coordinates. Given the initial design variable x for each element.

[0097] Step S2: Based on nonlinear solid mechanics theory, a nonlinear constitutive model of the hyperelastic material is established by introducing martensite volume fraction. First, the external load is input into the material design domain (Step S21); according to nonlinear solid mechanics theory, the global tangential stiffness matrix of the material is assembled and the element stress is calculated (Step S22); the martensite volume fraction is updated according to the phase transformation driving force function to obtain the phase transformation hyperelastic material properties under the current input (Step S23); additional material parameters are initialized to ensure the positive definiteness of the global tangential stiffness matrix (Step S24); the global tangential stiffness matrix of the material element is assembled and reconstructed (Step S25).

[0098] Step S3: Construct and solve the nonlinear finite element model, use the rate of change of stress with strain to linearize the nonlinear relationship piecewise, and solve the nonlinear equilibrium equation of the hyperelastic material by Newton's iteration method to obtain the structural displacement response of the hyperelastic material under external load.

[0099] Step S4: Calculate the sensitivity of the objective function. Use the adjoint vector method to solve the sensitivity of the objective function to the design variables, so as to identify the influence of the design variables on the objective function and guide the optimization process.

[0100] Step S5: Iteratively solve the optimization problem using the optimality criterion method, and set the stopping criterion as follows: the optimization process will terminate when the maximum difference between the design variables of two adjacent iteration steps is less than a preset value, or the number of iteration steps is greater than a preset value; if the stopping criterion is not met, update the design variables and return to step 2 to continue solving until the condition is met. For a detailed flowchart of the topology optimization implementation, see [link to flowchart]. Figure 1 .

[0101] 3 Calculation examples

[0102] 3.1 Topology Optimization of Two-Dimensional Cantilever Beam Structures

[0103] To verify the feasibility of the topology optimization method for hyperelastic materials based on a nonlinear constitutive model, a two-dimensional cantilever beam structure is used as an example to demonstrate the effectiveness and computational efficiency of the method.

[0104] The geometry and boundary conditions of the cantilever beam are as follows: Figure 2 As shown in (a) and (b), the cantilever beam is 240 mm long and 80 mm high, and is divided into 19,200 two-dimensional four-node elements. The external load P will be uniformly distributed on the central node (a) or the lowermost node (b), and the external load P is set to 1 N.

[0105] The optimal configuration obtained by solving the structural optimization problem for two load distributions, such as... Figure 3 (a) and Figure 3 As shown in (b).

[0106] 3.2 Topology Optimization of Three-Dimensional Bridge Structures

[0107] To verify the feasibility of the topology optimization method for hyperelastic materials based on a nonlinear constitutive model under different inputs, a three-dimensional bridge structure is used as an example to demonstrate the effectiveness of the method under different boundary conditions.

[0108] The geometric dimensions and boundary conditions of the bridge are as follows: Figure 4 As shown. The bridge is 72mm long, 20mm wide, and 24mm high, and is divided into 72*20*24 three-dimensional eight-node hexahedral elements. The external load P will be uniformly distributed on the top surface of the bridge, and the external load P is set to 500N.

[0109] The optimal configuration obtained by solving the bridge structure optimization problem, such as Figure 5 As shown.

[0110] Through the above steps and embodiments, the topology optimization method for phase change hyperelastic materials based on nonlinear constitutive models can ensure an efficient and stable optimization process when optimizing the structure of such smart materials. It can systematically explore all possibilities of hyperelastic material structural design and find the optimal design under given boundary conditions and constraints within the design domain.

[0111] Nonlinear constitutive models that consider the dynamic coupling relationship between material phase transformation behavior and mechanical properties have significant advantages over constitutive models that do not consider material phase transformation characteristics. For example, they can capture material properties more accurately in practical engineering applications, and can obtain the optimal topological configuration of hyperelastic materials by combining them with optimization algorithms.

[0112] The dynamic reconstruction-based hyperelasticity technique used in this invention can effectively solve the numerical instability problem in the optimization process. By establishing a nonlinear finite element model with reconstruction elements, the model solution can have higher computational efficiency and numerical stability.

[0113] Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific examples, and various details in this specification can be modified and changed based on different viewpoints and applications without departing from the spirit of the present invention.

[0114] The above embodiments are merely illustrative of the principles and effects of the present invention and are not intended to limit the invention. Any person skilled in the art can modify the above embodiments without departing from the spirit and scope of the invention. Therefore, the scope of protection of the present invention should be as set forth in the claims.

Claims

1. A topology optimization method for phase transition hyperelastic materials based on a nonlinear constitutive model, characterized in that, Includes the following steps: Step S1, establish a nonlinear constitutive model: based on nonlinear solid mechanics theory, martensite volume fraction is introduced as an internal variable to construct a hyperelastic constitutive model containing material phase transformation information. The model includes stress-strain relationship and martensite volume fraction evolution equation. Step S2, constructing a finite element analysis model: Based on the nonlinear constitutive model, a finite element analysis model of the phase change hyperelastic material is established, and the structural displacement response is obtained by solving the nonlinear equilibrium equation; Step S3, establish a topology optimization mathematical model: with the minimization of structural compliance as the objective function and the material volume fraction and equilibrium equation residuals as constraints, construct a topology optimization model; Step s4, numerical stability processing: The additional hyperelasticity technique of dynamic reconstruction is used to suppress the numerical instability of low-density elements during the optimization process; Step S5, Sensitivity Analysis: The adjoint vector method is used to calculate the sensitivity of the objective function to the design variables; Step S6, Optimization Solution: Iteratively solve the topology optimization model using the optimal criterion method until the convergence condition is met, and output the optimal topology configuration.

2. The method according to claim 1, characterized in that, In step S1, the value of martensite volume fraction ξ ranges from 0 to 1, where 0 represents a fully austenitic phase and 1 represents a fully martensite phase.

3. The method according to claim 1, characterized in that, In step S1, the free energy density function of the hyperelastic constitutive model is expressed as: ψ=(1-ξ)ψ A +xψ M +kξ(1-ξ) Where, ψ A and ψ M denoted as the free energy density functions for the austenitic and martensitic phases, respectively, and k is the interaction coefficient.

4. The method according to claim 1, characterized in that, In step S4, the dynamic reconstruction of the additional hyperelasticity technique uses the Yeoh model as the additional material model, and its free energy density function is: Among them, C 10 C 20 K1 and K2 are material parameters, I1 is the first invariant of the right Cauchy-Green deformation tensor, and J is the rate of volume change.

5. The method according to claim 4, characterized in that, The parameter C in the Yeoh model 20 Based on the unit logarithmic equivalent dynamic adjustment, the adjustment formula is as follows: Where, ε von For logarithmic equivalent change, ε * The threshold value is 0.

3.

6. The method according to claim 1, characterized in that, In step s5, the sensitivity of the objective function to the design variables is calculated using the adjoint vector method, specifically through the following formula: Among them, K T Let P be the tangent stiffness matrix, and f be the external load vector. int This represents the force vector at the internal nodes.

7. The method according to claim 1, characterized in that, In step S6, the convergence condition is that the maximum change in the design variable in adjacent iteration steps is less than a preset threshold, or the number of iterations reaches a preset maximum value.

8. The method according to claim 1, characterized in that, The method is applicable to the structural optimization design of hyperelastic materials with phase change properties, such as shape memory alloys.

9. The method according to claim 1, characterized in that, The topology optimization model uses the SIMP method for material interpolation, and the element free energy density is expressed as: in, Let p be the relative density of the unit cell, p be the penalty factor, and ψ0 be the free energy density of the perfectly solid material.

10. The method according to claim 1, characterized in that, The nonlinear equilibrium equation is solved using the Newton-Raphson iterative method, and the displacement vector is updated in each iteration as follows: U (k+1) = U (k) +ΔU (k) ΔU(k) ​​is obtained by solving the following system of linear equations: K T (AT (k) )ΔU (k) =-R(U (k) )。

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