A novel method for constructing a hyperelastic constitutive model based on projected manifold

Abstract

The application discloses a novel super-elastic constitutive model construction method based on a projection manifold, comprising: inputting target super-elastic material properties, application working conditions and test types, and configuring construction parameters; obtaining a rotation tensor and an isochoric projection tensor based on deformation gradient decomposition, constructing a double-branch Frobenius norm polynomial distortion energy item, and introducing a scaling factor alpha to regulate the volume penalty stiffness, respectively forming compressible and incompressible strain energy functions; fitting material constants through standard mechanical tests, compiling a finite element subprogram to carry out simulation checking, and embedding a projection dynamics or alternating direction multiplier method framework to realize GPU parallel verification. The model can accurately describe the special mechanical behavior of porous materials, biological soft tissues and the like, can simultaneously adapt to traditional finite elements, projection dynamics and alternating direction multiplier method parallel frameworks, can balance simulation accuracy and calculation efficiency, and is suitable for high-precision engineering analysis and product certification scenes.

Description

Technical Field

[0001] This invention relates to the field of hyperelastic material mechanics modeling technology, specifically to a novel method for constructing hyperelastic constitutive models based on projected manifolds. Background Technology

[0002] With the development of lightweight design (such as automatic structural optimization), simulation-driven design (such as simulation of bio-soft materials in medical devices), flexible robot control systems, and digital twin systems involving large deformations, the demand for efficient simulation of hyperelastic materials is becoming increasingly urgent. In current engineering practice, especially for ultra-large-scale nonlinear finite element implicit analysis (such as contact and hyperelastic materials) and structural topology optimization problems, although parallel computing technology has made great progress, the core bottleneck of the algorithm itself—repeatedly reconstructing the global stiffness matrix and solving large-scale sparse linear systems (such as...)—remains unresolved. This inherently leads to numerous serial dependencies, irregular memory accesses, and logical branches in the computation process, making it difficult to fully utilize the parallel computing capabilities of GPUs as in artificial intelligence training. Therefore, the computational performance of such problems remains primarily limited by CPU single-core frequency, memory bandwidth, and PCIe transmission latency; even with GPU acceleration, the bottleneck often shifts to preprocessor construction and contact search, which still require high-frequency single-core CPU processing. In engineering scenarios requiring rapid iteration or real-time response, traditional finite element methods are no longer sufficient.

[0003] Highly parallelizable methods in computer graphics—Projective Dynamics (PD) and the Alternating Direction Method of Multipliers (ADMM)—fully utilize the parallel capabilities of GPUs to achieve extremely high computational efficiency. However, the traditional constitutive models used in parallel solution frameworks such as PD and ADMM are often significantly simplified, for example, retaining only the basic elastic response term. This results in models that can only accurately describe the linear elastic behavior within a small deformation range, lacking the ability to characterize key mechanical features such as nonlinear distortion and volumetric stiffness evolution under large deformation conditions in hyperelastic materials like rubber, biological soft tissues, and porous foams. Furthermore, these simplified constitutive models lack standardized and reproducible engineering calibration procedures, making it difficult to produce accuracy certification documents that meet the compliance review requirements of fields such as medical devices, flexible robots, and lightweight structure optimization. Therefore, they cannot be directly applied to engineering simulation analysis and product compliance certification processes that have stringent requirements for computational accuracy, result reliability, and data traceability. Summary of the Invention

[0004] The purpose of this invention is to propose a novel method for constructing a hyperelastic constitutive model based on projected manifolds. By constructing a hyperelastic constitutive model based on projected manifolds according to the material's compression characteristics and actual application conditions, the local solution steps of this constitutive model can be highly decoupled. This allows the model to be used not only for traditional finite element simulations but also as a unified constitutive modeling scheme, which can be embedded into two independent and efficient parallel computing frameworks: projective dynamics and the alternating direction multiplier method. This enables efficient and accurate numerical simulation of the large deformation behavior of hyperelastic materials. Simultaneously, a standardized tensile test calibration and multi-round simulation iterative optimization process are provided, significantly improving parallel computing efficiency while ensuring the accuracy of large deformation simulation, to meet the high-precision simulation and engineering certification needs of various hyperelastic materials.

[0005] To achieve the above-mentioned technical objectives, the technical solution adopted by the present invention is as follows:

[0006] A novel method for constructing hyperelastic constitutive models based on projected manifolds, the method comprising the following steps:

[0007] S1, Input relevant information about the target hyperelastic material to be modeled, including the material's compressibility properties, application conditions, and standard mechanical test type; Configure initial construction parameters according to the application conditions, including the polynomial number of the first distortion energy branch. The polynomial number of the second distortion energy branch and scaling factor , and For natural numbers, For constants, or based on deformation gradients The various expressions constructed correspond to the different volumetric deformation mechanical properties of the materials.

[0008] S2, the modeling logic is differentiated based on the compressibility properties of the target hyperelastic material, and is based on the deformation gradient. Decomposition yields rotation tensor and isochoric projection tensor Two types of projection tensors; construct the first distortion energy branch consisting of the Frobenius norm polynomial of the deformation gradient and the rotation tensor, and the second distortion energy branch consisting of the Frobenius norm polynomial of the right Cauchy Green tensor and the unit tensor, respectively, and superimpose the two distortion energy branches to obtain the distortion energy term.

[0009] For compressible hyperelastic materials, a distance penalty term is constructed based on the Frobenius norm of the deformation gradient and the isochoric projection tensor, and a scaling factor is introduced. By adjusting the volumetric penalty stiffness, a volumetric penalty energy sub-term is constructed. The distortion energy term and the volumetric penalty energy sub-term are superimposed to obtain the compressible hyperelastic strain energy function.

[0010] For incompressible hyperelastic materials, the volume remains constant by introducing Lagrange multipliers to construct constraint terms, and the incompressible hyperelastic strain energy function is obtained by superimposing only the distortion energy term.

[0011] S3: Process tensile specimens of the target hyperelastic material according to standard mechanical test types and conduct quasi-static tensile tests. Collect load and displacement data, and calculate the test force-displacement curve by combining the specimen geometry. Solve for all material constants of the strain energy function through inverse finite element fitting. If the fitting error of the material constants exceeds the preset error threshold, return to step S1 to readjust the number of polynomials in the first distortion energy branch. The polynomial number of the second distortion energy branch and scaling factor The strain energy function is iteratively updated until the fitting error of the material constants meets the standard, thus completing the calibration of the construction parameters;

[0012] S4: The strain energy function with completed structural parameter calibration is written as a finite element user subroutine, compiled, and then connected to the finite element simulation environment to conduct tensile simulation of specimens of the same specifications. The simulated force-displacement curve is extracted and compared with the experimental force-displacement curve. If the deviation between the two curves exceeds the preset deviation threshold, the process returns to step S1 and the polynomial quantity of the first distortion energy branch is readjusted. The polynomial number of the second distortion energy branch and scaling factor Repeat the parameter calibration and simulation verification process until the matching degree of the two curves meets the standard;

[0013] S5 embeds the subroutine into the parallel solution framework of projection dynamics or alternating direction multiplier method, and completes the large deformation simulation through local solution by GPU parallel unit to verify the simulation accuracy and real-time computing efficiency; in the local solution stage, only the rotation tensor and the isochoric projection tensor are solved in parallel separately; the stiffness matrix is ​​kept constant in the global solution stage; if the simulation accuracy or real-time computing efficiency does not meet the application requirements, return to step S1 to adjust the construction parameters and iteratively update the strain energy function until it meets the application requirements.

[0014] Further, in step S2, the compressible hyperelastic strain energy function is expressed as:

[0015]

[0016] in, and All are material constants. Used to characterize the distortion energy storage properties of compressible hyperelastic materials when the deformation gradient deviates from a pure rotating manifold. Used to characterize the tensile and shear nonlinear mechanical response of compressible hyperelastic materials when the right Cauchy Green tensor deviates from the undeformed state; The correlation coefficient is the bulk modulus. , and All results were obtained through standard tensile testing combined with inverse finite element fitting. is a scaling factor used to adjust the volumetric stiffness, the specific form of which depends on the mechanical behavior of the material during volumetric deformation. For deformation gradient; For the right Cauchy Green tensor; Denotes the Frobenius norm; For rotation tensors, by Singular value decomposition Determined as ; To and Unit tensors of the same dimension; For isochoric projection tensor, .

[0017] Further, in step S2, the incompressible hyperelastic strain energy function is expressed as:

[0018]

[0019] in, and All values ​​are material constants, obtained through standard tensile tests combined with inverse finite element fitting. Used to characterize the distortion energy storage properties of incompressible hyperelastic materials when the deformation gradient deviates from a pure rotating manifold. Used to characterize the tensile and shear nonlinear mechanical response of incompressible hyperelastic materials when the right Cauchy Green tensor deviates from the undeformed state; For deformation gradient; For the right Cauchy Green tensor; Denotes the Frobenius norm; For rotation tensors, by Singular value decomposition Determined as ; To and Unit tensors of the same dimension; ; It is the hydrostatic pressure.

[0020] Furthermore, in step S4, finite element simulation calculations are performed in Abaqus using the VUMAT or UMAT subroutine to verify the material parameters of the constitutive model.

[0021] Furthermore, when conducting mechanical tests in step S3 and simulations in step S4, specimens with identical specifications are used, and the loading method is either uniaxial tension or biaxial tension.

[0022] Furthermore, the target hyperelastic material includes, but is not limited to, rubber-like materials, porous elastomers, and biological soft tissue materials.

[0023] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0024] First, the novel hyperelastic constitutive model construction method based on projected manifold of the present invention relies on a universal testing machine to carry out standard mechanical tests, and combines the inverse finite element fitting process to realize batch calibration of material parameters. The strain energy function constructed by this method has a simple form and clear physical parameter meaning, supports advanced parameter calibration methods such as inverse finite element, and can systematically and uniquely determine all material constants from standard test data such as uniaxial and biaxial tests, avoiding the common problems of multiple solutions and overfitting in traditional high parameter models (such as high-order Ogden models), and reducing the calibration threshold in engineering applications.

[0025] Second, in the novel hyperelastic constitutive model construction method based on projected manifolds of the present invention, the scaling factor can be autonomously configured according to the properties of the material to be modeled during the process of constructing the volume penalty energy sub-term. This allows for flexible control over the evolution of the bulk modulus during deformation. For example, when taking... When, volume term It degenerates into relying solely on the contribution of material volume changes, at which point the model behavior is similar to that of the classical hyperelastic constitutive model; while when When taken as a constant, this volume term can enhance the volumetric stiffness of the material when it is distorted, giving the model an advantage that traditional hyperelastic constitutive models do not have when describing materials whose bulk modulus can be enhanced by distortion (such as porous materials and soft tissues such as biological tissues).

[0026] Third, the novel hyperelastic constitutive model construction method based on projective manifolds of this invention proposes a constitutive model that can be directly embedded into the PD or ADMM framework without the need to introduce additional auxiliary variables. Traditional hyperelastic constitutive models (such as Neo-Hookean, Mooney-Rivlin, and Ogden models) are unsuitable for PD solution processes due to their highly coupled energy forms. Furthermore, while some traditional models (such as the Ogden model) can be solved by introducing auxiliary variables in the independent parallel framework of ADMM, their local optimization steps still contain strong nonlinearities, resulting in extremely high iteration costs. In contrast, the local steps of the model in this invention within the ADMM framework can be transformed into analytical projection or low-order polynomial root finding, significantly improving computational efficiency. Its unified constitutive form can be flexibly deployed across various independent computational frameworks such as traditional finite element method, PD, and ADMM, achieving a single set of parameters applicable to multiple platforms.

[0027] Fourth, the novel hyperelastic constitutive model construction method based on projected manifold of the present invention, through a complete modeling process (material input - function construction - experimental calibration - finite element verification - parallel simulation verification), achieves a fitting accuracy comparable to that of classical hyperelastic constitutive models (such as Neo-Hookean and Mooney-Rivlin models) under standard loading conditions such as uniaxial and biaxial models, fully meeting the accuracy requirements of engineering analysis. Attached Figure Description

[0028] Figure 1 This is a schematic diagram of a uniaxial test specimen; the left figure is... Model specimen; the right figure is Model specimen;

[0029] Figure 2 for A schematic diagram comparing the results of model experiments and simulations;

[0030] Figure 3 for A schematic diagram comparing the results of model experiments and simulations;

[0031] Figure 4 A schematic diagram showing the force-displacement comparison curves of a bionic foot model simulation.

[0032] Figure 5 This is a schematic diagram of the simulation cloud map results for a bionic foot model; the left image is... Model contour plot results; the right figure shows the contour plot results of the traditional Neo-Hookean model;

[0033] Figure 6 This is a schematic diagram showing the comparison of simulation calculation efficiency curves. Detailed Implementation

[0034] The embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0035] This invention discloses a novel method for constructing a hyperelastic constitutive model based on projected manifolds, the method comprising the following steps:

[0036] S1, Input relevant information about the target hyperelastic material to be modeled, including the material's compressibility properties, application conditions, and standard mechanical test type; Configure initial construction parameters according to the application conditions, including the polynomial number of the first distortion energy branch. The polynomial number of the second distortion energy branch and scaling factor , and For natural numbers, For constants, or based on deformation gradients Various expressions (such as the Frobenius norm) are constructed, and different expression forms correspond to the different volumetric deformation mechanical properties of materials.

[0037] S2, the modeling logic is differentiated based on the compressibility properties of the target hyperelastic material, and is based on the deformation gradient. Decomposition yields rotation tensor and isochoric projection tensor Two types of projection tensors are used; a first distortion energy branch consisting of the Frobenius norm polynomial of the deformation gradient and the rotation tensor and a second distortion energy branch consisting of the Frobenius norm polynomial of the right Cauchy Green tensor and the unit tensor are constructed respectively. The distortion energy term is obtained by superimposing the two distortion energy branches.

[0038] For compressible hyperelastic materials, a distance penalty term is constructed based on the Frobenius norm of the deformation gradient and the isochoric projection tensor, and a scaling factor is introduced. By adjusting the volumetric penalty stiffness, a volumetric penalty energy sub-term is constructed. Superimposing the distortion energy term and the volumetric penalty energy sub-term yields the compressible hyperelastic strain energy function, expressed as:

[0039] (1)

[0040] in, and All are material constants. Used to characterize the distortion energy storage properties of compressible hyperelastic materials when the deformation gradient deviates from a pure rotating manifold. Used to characterize the tensile and shear nonlinear mechanical response of compressible hyperelastic materials when the right Cauchy Green tensor deviates from the undeformed state; The correlation coefficient is the bulk modulus. , and All results were obtained through standard tensile testing combined with inverse finite element fitting. is a scaling factor used to adjust the volumetric stiffness, the specific form of which depends on the mechanical behavior of the material during volumetric deformation. For deformation gradient; For the right Cauchy Green tensor; Denotes the Frobenius norm; For rotation tensors, by Singular value decomposition Determined as ; To and Unit tensors of the same dimension; For isochoric projection tensor, .

[0041] For incompressible hyperelastic materials, by introducing Lagrange multipliers to construct constraint terms to maintain volume invariance, and only superimposing the distortion energy term, the incompressible hyperelastic strain energy function is obtained, expressed as:

[0042] (2)

[0043] in, and All values ​​are material constants, obtained through standard tensile tests combined with inverse finite element fitting. Used to characterize the distortion energy storage properties of incompressible hyperelastic materials when the deformation gradient deviates from a pure rotating manifold. Used to characterize the tensile and shear nonlinear mechanical response of incompressible hyperelastic materials when the right Cauchy Green tensor deviates from the undeformed state; For deformation gradient; For the right Cauchy Green tensor; Denotes the Frobenius norm; For rotation tensors, by Singular value decomposition Determined as ; To and Unit tensors of the same dimension; ; It is the hydrostatic pressure (Lagrange multiplier), used to maintain a constant volume.

[0044] S3: Process tensile specimens of the target hyperelastic material according to standard mechanical test types and conduct quasi-static tensile tests (such as uniaxial or biaxial tensile tests). Collect load and displacement data, and calculate the test force-displacement curve by combining the specimen geometry. Solve for all material constants of the strain energy function through inverse finite element fitting. If the fitting error of the material constants exceeds the preset error threshold, return to step S1 to readjust the number of polynomials in the first distortion energy branch. The polynomial number of the second distortion energy branch and scaling factor The strain energy function is iteratively updated until the fitting error of the material constants meets the standard, thus completing the calibration of the construction parameters.

[0045] S4. The strain energy function with completed structural parameter calibration is written as a finite element user subroutine, compiled, and then connected to the finite element simulation environment (e.g., finite element simulation in Abaqus via VUMAT or UMAT subroutines) to perform tensile simulation of specimens of the same specifications. The simulated force-displacement curve is extracted and compared with the experimental force-displacement curve. If the deviation between the two curves exceeds the preset deviation threshold, return to step S1 and readjust the polynomial number of the first distortion energy branch. The polynomial number of the second distortion energy branch and scaling factor Repeat the parameter calibration and simulation verification process until the matching degree of the two curves meets the standard.

[0046] S5. The subroutine is embedded into the projection dynamics PD or ADMM parallel solution framework. Large deformation simulation is completed through local solution by GPU parallel units to verify simulation accuracy and real-time computing efficiency. In the local solution stage, only the rotation tensor and the isochoric projection tensor are solved in parallel separately. The stiffness matrix is ​​kept constant in the global solution stage. If the simulation accuracy or real-time computing efficiency does not meet the application requirements, the process returns to step S1 to adjust the construction parameters and iteratively update the strain energy function until it meets the application requirements.

[0047] Example

[0048] For ease of explanation, the combinations of frequencies below will be abbreviated and numbered with superscripts. Superscript indicates compressible. This represents incompressible. For example... Indicates to retain only A distortion energy and A compressible model; Indicates reservation and Each once and A compressible model; This indicates that the two distortion energies are retained and The compressible model; the meaning of the numbering for the incompressible model is similar.

[0049] For the general form of compressible hyperelastic material (1), the following specific example can be constructed:

[0050] When the distortion energy in equation (1) is only retained One item, and take At that time, the model is visualized as

[0051] Example 1: (3)

[0052] When the distortion energy in equation (1) is simultaneously preserved and One of each, and take At that time, the model is visualized as

[0053] Example 2: (4)

[0054] In Examples 1 and 2 (Formulas (3) and (4)), Since it is a constant, the volume-dependent energy density is directly derived from... and Control. Due to This depends not only on the volume ratio It also depends on the magnitude of the deformation gradient. It is directly proportional. Therefore, in regions of large distortion, even if the volume change is small, this energy will increase proportionally. The volume constraint penalty is significantly enhanced by the increase of distortion, meaning that the volume constraint penalty is strengthened by the distortion. This characteristic of volume-distortion coupling is consistent with the mechanical response of many porous materials (such as foamed metals and polyurethane foams) and biological soft tissues (such as liver, brain tissue, and cartilage).

[0055] When the distortion energy in equation (1) is simultaneously preserved and One item each, and take At that time, the model is visualized as

[0056] Example 3: (5)

[0057] At this point, the stiffness of the volume term is independent of distortion, enabling more flexible volume response modulation, a characteristic similar to that of the volume term in the traditional hyperelastic constitutive model.

[0058] For the general form of incompressible hyperelastic materials (2), the following specific examples can be constructed:

[0059] When the distortion energy in equation (2) is only retained When one item is used, the model is visualized as

[0060] Example 4: (6)

[0061] When the distortion energy in equation (2) is simultaneously preserved and When each item is represented, the model is visualized as follows:

[0062] Example 5: (7)

[0063] To verify the broad applicability of the model, uniaxial tensile calibration tests were conducted on two typical hyperelastic materials: Example 1 ( The model uses conventional nitrile rubber, and the specimen preparation standard follows GB / T 528-2009; Example 2 ( The model used Ecoflex™ series platinum-catalyzed silicone rubber, and the specimen preparation standard referred to ASTM D638-2014. The geometric models of the two test specimens are as follows: Figure 1 As shown.

[0064] The test was conducted using a quasi-static uniaxial tensile method on a standard general-purpose mechanical testing machine, with a loading rate set at 1.5 mm / min. Displacement was measured by the travel distance of the upper and lower grippers, and the force signal was obtained from the tensile load through high-precision force sensors mounted on the grippers. To prevent slippage, the grippers used toothed clamps with anti-slip strips to secure the specimen holding end.

[0065] The parameters of each model were obtained by fitting the experimental data.

[0066] for Model: , Nitrile rubber is a nearly incompressible material. Therefore, a larger one is adopted. It punishes changes in volume severely.

[0067] for Model: , , This silicone rubber can also be considered as an approximately incompressible material, with a bulk modulus much greater than its shear modulus; therefore, a larger value is also taken. The value is used to maintain near-incompressible properties.

[0068] To verify the accuracy of the proposed constitutive model, the experimental process was reproduced in Abaqus finite element software with the same dimensions and boundary conditions. One end of the model was fixed with constraints, while the other end was subjected to displacement loads consistent with those in the experiment. The proposed constitutive relation was implemented through the VUMAT subroutine in the simulation calculation. Figure 2 , Figure 3The force-displacement curves from simulation and experiment for the two models are presented for comparison. It can be seen that the proposed model can effectively characterize the nonlinear mechanical response of hyperelastic materials over a wide deformation range, and the results agree well with the experimental results.

[0069] To examine the model's universality and computational efficiency advantages, a complex biomimetic foot model was further used for comparative simulations. Under the premise of completely identical mesh, boundary conditions, loading history, and solver settings, both the classic Neo-Hookean model and the model proposed in this invention were used. Compressible model (i.e., Equation (3)) Simulation was performed using two models. The overall force-displacement curves obtained from the two models are shown below. Figure 4 The trends are highly consistent, both exhibiting similar nonlinear characteristics. Further observation of the plantar stress distribution cloud maps from the two simulations reveals this. Figure 5 The observed stress distribution patterns are largely consistent, validating the computational rationality of the proposed model. Furthermore, Figure 6 The cumulative total computation time of the two models was compared: when simulating to the same target physics time, the Neo-Hookean model took approximately [time missing]. The time taken is seconds, while the model of this invention only takes about [time missing]. The total time consumption is reduced by approximately 63%, which fully demonstrates the computational cost advantage of this invention within the traditional finite element analysis framework.

[0070] The above tests have demonstrated the accuracy and efficiency of the constitutive model. Furthermore, the efficiency of the model can be further enhanced when combined with the PD and ADMM algorithms.

[0071] Based on the compressible hyperelastic constitutive equation For example, the following PD algorithm framework can be constructed:

[0072] The target simulation object is discretized into an unstructured mesh, and the entire mesh is composed of... It consists of nodes, and is sampled at discrete time points. Perform time evolution. The system state changes from... Decision, among which and The vertices are respectively at Position and velocity at any given time. The computational flow of the PD algorithm within a single time step is as follows:

[0073] Step 1: ,in For the mass matrix of the discrete system, External force; Let the time step be; , ;

[0074] Step 2 (Local Solution): Fix the current iterative solution Solving each element locally and ;

[0075] Step 3 (Global Solution): Fix each element and Solving the global linear system yields new values: ;make and order ;

[0076] Repeat steps 2 and 3 until... Less than the threshold, after convergence, we obtain ;

[0077] Step 4: Update Speed .

[0078] Since the local solution in step 2 can be parallelized, and the optimization problem in step 3 is essentially a linear solution problem with an invariant stiffness matrix, PD does not need to continuously update the tangent stiffness matrix and repeatedly solve it in the nonlinear deformation simulation iteration, unlike the traditional hyperelastic nonlinear finite element method. Therefore, its computational efficiency is significantly better than the latter.

[0079] Although preferred embodiments of this application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of this application.

[0080] Obviously, those skilled in the art can make various modifications and variations to this application without departing from the spirit and scope of this application. Therefore, if such modifications and variations fall within the scope of the claims of this application and their equivalents, this application also intends to include such modifications and variations.

Claims

1. A novel method for constructing a hyperelastic constitutive model based on projected manifolds, characterized in that, The method includes the following steps: S1, Input relevant information about the target hyperelastic material to be modeled, including the material's compressibility properties, application conditions, and standard mechanical test type; Configure initial construction parameters according to the application conditions, including the polynomial number of the first distortion energy branch. The polynomial number of the second distortion energy branch and scaling factor , and For natural numbers, For constants, or based on deformation gradients The various expressions constructed correspond to the different volumetric deformation mechanical properties of the materials. S2, the modeling logic is differentiated based on the compressibility properties of the target hyperelastic material, and is based on the deformation gradient. Decomposition yields rotation tensor and isochoric projection tensor Two types of projection tensors; construct the first distortion energy branch consisting of the Frobenius norm polynomial of the deformation gradient and the rotation tensor, and the second distortion energy branch consisting of the Frobenius norm polynomial of the right Cauchy Green tensor and the unit tensor, respectively, and superimpose the two distortion energy branches to obtain the distortion energy term. For compressible hyperelastic materials, a distance penalty term is constructed based on the Frobenius norm of the deformation gradient and the isochoric projection tensor, and a scaling factor is introduced. By adjusting the volumetric penalty stiffness, a volumetric penalty energy sub-term is constructed. The distortion energy term and the volumetric penalty energy sub-term are superimposed to obtain the compressible hyperelastic strain energy function. For incompressible hyperelastic materials, the volume remains constant by introducing Lagrange multipliers to construct constraint terms, and the incompressible hyperelastic strain energy function is obtained by superimposing only the distortion energy term. S3: Process tensile specimens of the target hyperelastic material according to standard mechanical test types and conduct quasi-static tensile tests. Collect load and displacement data, and calculate the test force-displacement curve by combining the specimen geometry. Solve for all material constants of the strain energy function through inverse finite element fitting. If the fitting error of the material constants exceeds the preset error threshold, return to step S1 to readjust the number of polynomials in the first distortion energy branch. The polynomial number of the second distortion energy branch and scaling factor The strain energy function is iteratively updated until the fitting error of the material constants meets the standard, thus completing the calibration of the construction parameters; S4: The strain energy function with completed structural parameter calibration is written as a finite element user subroutine, compiled, and then connected to the finite element simulation environment to conduct tensile simulation of specimens of the same specifications. The simulated force-displacement curve is extracted and compared with the experimental force-displacement curve. If the deviation between the two curves exceeds the preset deviation threshold, the process returns to step S1 and the polynomial quantity of the first distortion energy branch is readjusted. The polynomial number of the second distortion energy branch and scaling factor Repeat the parameter calibration and simulation verification process until the matching degree of the two curves meets the standard; S5 embeds the subroutine into the parallel solution framework of projection dynamics or alternating direction multiplier method, and completes the large deformation simulation through local solution by GPU parallel unit to verify the simulation accuracy and real-time computing efficiency; in the local solution stage, only the rotation tensor and the isochoric projection tensor are solved in parallel separately; the stiffness matrix is ​​kept constant in the global solution stage; if the simulation accuracy or real-time computing efficiency does not meet the application requirements, return to step S1 to adjust the construction parameters and iteratively update the strain energy function until it meets the application requirements.

2. The method for constructing a novel hyperelastic constitutive model based on projected manifolds according to claim 1, characterized in that, In step S2, the compressible hyperelastic strain energy function is expressed as: in, and All are material constants. Used to characterize the distortion energy storage properties of compressible hyperelastic materials when the deformation gradient deviates from a pure rotating manifold. Used to characterize the tensile and shear nonlinear mechanical response of compressible hyperelastic materials when the right Cauchy Green tensor deviates from the undeformed state; The correlation coefficient is the bulk modulus. , and All results were obtained through standard tensile testing combined with inverse finite element fitting. is a scaling factor used to adjust the volumetric stiffness, the specific form of which depends on the mechanical behavior of the material during volumetric deformation. For deformation gradient; For the right Cauchy Green tensor; Denotes the Frobenius norm; For rotation tensors, by Singular value decomposition Determined as ; To and Unit tensors of the same dimension; For isochoric projection tensor, .

3. The method for constructing a novel hyperelastic constitutive model based on projected manifolds according to claim 1, characterized in that, In step S2, the incompressible hyperelastic strain energy function is expressed as: in, and All values ​​are material constants, obtained through standard tensile tests combined with inverse finite element fitting. Used to characterize the distortion energy storage properties of incompressible hyperelastic materials when the deformation gradient deviates from a pure rotating manifold. Used to characterize the tensile and shear nonlinear mechanical response of incompressible hyperelastic materials when the right Cauchy Green tensor deviates from the undeformed state; For deformation gradient; For the right Cauchy Green tensor; Denotes the Frobenius norm; For rotation tensors, by Singular value decomposition Determined as ; To and Unit tensors of the same dimension; ; It is the hydrostatic pressure.

4. The method for constructing a novel hyperelastic constitutive model based on projected manifolds according to claim 1, characterized in that, In step S4, finite element simulation calculations are performed in Abaqus using the VUMAT or UMAT subroutine to verify the material parameters of the constitutive model.

5. The method for constructing a novel hyperelastic constitutive model based on projected manifolds according to claim 1, characterized in that, When conducting mechanical tests in step S3 and simulations in step S4, specimens with identical specifications are used, and the loading method is either uniaxial tension or biaxial tension.

6. The method for constructing a novel hyperelastic constitutive model based on projected manifolds according to claim 1, characterized in that, The target hyperelastic material includes, but is not limited to, rubber-like materials, porous elastomers, and biological soft tissue materials.

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