Hyperelastic material deformation simulation method based on variational eigenvalue filtering and related device

Abstract

The application discloses a hyperelastic material deformation simulation method based on variational eigenvalue filtering and related equipment, and belongs to the technical field of computer physical simulation. In each Newton iteration step of the deformation simulation, the local Hessian matrix of each unit of the hyperelastic material is obtained for eigenvalue decomposition; for the existing negative eigenvalue, the gradient projection in the corresponding eigenvector direction is calculated, and the corresponding direction is classified into a reliable direction set or an unreliable direction set according to the sign of the gradient projection; the global system energy and the global Hessian matrix are used to calculate a global dimensionless factor; based on the gradient projection, the negative eigenvalue and the global dimensionless factor, the mixed coefficients corresponding to the two types of direction sets are determined respectively; the negative eigenvalue is modified by using the mixed coefficients, and the local positive definite Hessian matrix is reconstructed, and then a new global Hessian matrix is assembled to solve the Newton step length to update the material node position. The application realizes global dimensionless and adaptive filtering under the variational framework of joint optimization, and improves the solving efficiency of the complex hyperelastic material deformation simulation.

Description

Technical Field

[0001] This invention belongs to the fields of computer graphics, physical simulation, numerical optimization and finite element analysis, and specifically relates to a method and related equipment for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering. Background Technology

[0002] In the fields of computer graphics and virtual reality, deformation simulation of hyperelastic materials is one of the core technologies in physical simulation. This type of problem is typically modeled as a nonlinear optimization problem and is widely solved using the projective Newton method. The core operation of the projective Newton method is to correct the eigenvalues ​​of the element Hessian matrix to ensure the matrix's positive definiteness, thereby obtaining a reliable descent direction.

[0003] Currently, the mainstream eigenvalue correction methods include: (1) Eigenvalue clamping method: The negative eigenvalues ​​are set to a small positive number ε, while the positive eigenvalues ​​remain unchanged. This method discards negative curvature information, and the descent direction deviates from the direction of the minimum value. Under high Poisson's ratio and large deformation, the convergence speed is slow.

[0004] (2) Absolute value filtering method (Abs): The absolute value of the negative eigenvalue is taken and the negative curvature amplitude is retained. The stability is significantly improved in quasi-static high Poisson ratio simulation, but excessive damping may be introduced in the near convex region.

[0005] (3) Trust Region Adaptive Filtering: This method discretely switches between Clamp and Abs filtering based on the model fit. However, it has a delayed response and cannot make an immediate decision at the current step.

[0006] The above methods all lack joint optimization modeling of energy reduction and matrix correction costs, fail to provide an analytical adaptive eigenvalue correction strategy under a unified variational framework, and fail to unify the dimensions at a global scale, resulting in a mismatch between the correction strategy and the deformed state. The methods have low solution efficiency and slow energy reduction. Summary of the Invention

[0007] To address the problems existing in the prior art, this invention provides a method and related equipment for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering. The aim is to establish a variational framework that jointly optimizes energy reduction and matrix correction costs, and to unify physical dimensions at a global scale. This provides an analytical adaptive eigenvalue correction strategy, overcoming the shortcomings of existing technologies such as blindly discarding negative curvature information, introducing excessive damping in near-convex regions, and lag in discrete switching responses, which lead to mismatches between correction strategies and deformation states. This effectively solves convergence problems in complex scenarios such as high Poisson's ratio and large deformation, improves the solution efficiency of hyperelastic material deformation simulation, and accelerates the reduction of system energy.

[0008] To solve the above-mentioned technical problems, the present invention is achieved through the following technical solution: According to a first aspect of the present invention, a method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering is provided, comprising: Obtain the local Hessian matrix and local gradient of each unit of the hyperelastic material in the current Newton iteration step, as well as the global system energy and global Hessian matrix of the hyperelastic material; The local Hessian matrix is ​​subjected to eigenvalue decomposition to obtain eigenvalues ​​and corresponding eigenvectors. If there are negative eigenvalues ​​among the eigenvalues, the gradient projection in the direction of the eigenvector is calculated based on the local gradient and the eigenvector corresponding to the negative eigenvalue. Based on the sign of the gradient projection, the corresponding feature vector direction is classified into a reliable direction set or an unreliable direction set. Based on the global system energy and the global Hessian matrix, calculate the global dimensional unification factor for unifying physical dimensions. Based on the gradient projection, the negative eigenvalue, and the global dimensional unification factor, the reliable direction mixing coefficient corresponding to the reliable direction set and the unreliable direction mixing coefficient corresponding to the unreliable direction set are determined respectively. Using the reliable direction mixing coefficient and the unreliable direction mixing coefficient, the corresponding negative eigenvalues ​​in the local Hessian matrix are corrected respectively, and the positive definite Hessian matrix of each element is reconstructed based on the corrected eigenvalues ​​and the eigenvector. The positive definite Hessian matrices of each element are assembled into a new global Hessian matrix, and the Newton step size is determined based on the new global Hessian matrix to update the nodal positions of the hyperelastic material.

[0009] In one possible implementation of the first aspect, the step of calculating the gradient projection along the direction of the feature vector based on the local gradient and the feature vector corresponding to the negative eigenvalue, and classifying the corresponding feature vector direction into a reliable direction set or an unreliable direction set according to the sign of the gradient projection, includes: For any negative eigenvalue whose absolute value is greater than a preset small positive number, the transpose of its corresponding eigenvector is multiplied by the local gradient to obtain the gradient projection. If the gradient projection is greater than zero, the corresponding feature vector direction is assigned to the reliable direction set. If the gradient projection is not greater than zero, the corresponding feature vector direction is assigned to the unreliable direction set.

[0010] In one possible implementation of the first aspect, calculating the global dimensional unification factor for unifying physical dimensions based on the global system energy and the global Hessian matrix includes: Calculate the square of the Frobenius norm of the global Hessian matrix; The ratio of the global system energy to the square of the Frobenius norm is determined as the global dimensional unification factor.

[0011] In one possible implementation of the first aspect, determining the reliable direction mixing coefficient corresponding to the reliable direction set and the unreliable direction mixing coefficient corresponding to the unreliable direction set based on the gradient projection, the negative eigenvalue, and the global dimensional unification factor includes: If the set of reliable directions is not empty, calculate the ratio of the square of the gradient projection of each direction in the set of reliable directions to the negative number of the corresponding negative eigenvalue, and sum the ratios to obtain the first aggregation parameter of the reliable direction; calculate the square of the negative number of the corresponding negative eigenvalue of each direction in the set of reliable directions, and sum the squares to obtain the second aggregation parameter of the reliable direction; divide the first aggregation parameter of the reliable direction by twice the product of the global dimensionless factor and the second aggregation parameter of the reliable direction, and determine the cube root of the quotient as the optimal mixing coefficient of the reliable direction, and determine the reliable direction mixing coefficient based on the optimal mixing coefficient of the reliable direction; If the set of unreliable directions is not empty, calculate the ratio of the square of the gradient projection of each direction in the set of unreliable directions to the negative number of the corresponding negative eigenvalue, and sum the ratios to obtain the first aggregation parameter of the unreliable direction; calculate the square of the negative number of the corresponding negative eigenvalue of each direction in the set of unreliable directions, and sum the squares to obtain the second aggregation parameter of the unreliable direction; divide the first aggregation parameter of the unreliable direction by twice the product of the global dimensionless factor and the second aggregation parameter of the unreliable direction, and determine the square root of the quotient as the optimal mixing coefficient of the unreliable direction, and determine the mixing coefficient of the unreliable direction based on the optimal mixing coefficient of the unreliable direction.

[0012] In one possible implementation of the first aspect, the method further includes: obtaining the element energy of each element of the hyperelastic material in the current Newton iteration step; and calculating, based on the element energy and a preset safety factor, the reliable direction physical lower bound corresponding to the reliable direction set and the unreliable direction physical lower bound corresponding to the unreliable direction set, including: Obtain the first safety factor for the reliable direction and the second safety factor for the unreliable direction; According to the formula Calculate the physical lower bound of the reliable direction. ,in, The first aggregation parameter for the reliable direction. The first safety factor, The unit energy; According to the formula Calculate the physical lower bound of the unreliable direction. ,in, The first aggregation parameter for the unreliable direction. This is the second safety factor.

[0013] In one possible implementation of the first aspect, determining the reliable direction mixing coefficient based on the reliable direction optimal mixing coefficient and determining the unreliable direction mixing coefficient based on the unreliable direction optimal mixing coefficient includes: Select the larger value between the physical lower bound of the reliable direction and the optimal mixing coefficient of the reliable direction, compare the larger value with 1, and take the smaller value as the mixing coefficient of the reliable direction; Select the larger value between the physical lower bound of the unreliable direction and the optimal mixing coefficient of the unreliable direction, compare the larger value with 1, and take the smaller value as the mixing coefficient of the unreliable direction.

[0014] In one possible implementation of the first aspect, the step of using the reliable direction mixing coefficient and the unreliable direction mixing coefficient to correct the corresponding negative eigenvalues ​​in the local Hessian matrix, and reconstructing the positive definite Hessian matrix of each element based on the corrected eigenvalues ​​and the eigenvectors, includes: Calculate the ratio of the sum of the absolute values ​​of all negative eigenvalues ​​in the current unit to the sum of the absolute values ​​of all eigenvalues ​​to determine the proportion of negative eigenvalue energy, and calculate the positive eigenvalue shrinkage coefficient based on the proportion of negative eigenvalue energy. For the negative eigenvalues ​​in the reliable direction set, multiply their corresponding inverses by the reliable direction mixing coefficients to obtain the corrected eigenvalues; For the negative eigenvalues ​​in the unreliable direction set, multiply their corresponding inverses by the unreliable direction mixing coefficients to obtain the corrected eigenvalues; For positive eigenvalues ​​greater than the preset small positive number, multiply them by the positive eigenvalue shrinkage coefficient to obtain the corrected eigenvalues; For feature values ​​whose absolute value is not greater than the preset small positive number, they are corrected to the preset small positive number; A diagonal matrix is ​​constructed based on the corrected eigenvalues, and the positive definite Hessian matrix is ​​reconstructed by combining the eigenvectors of the local Hessian matrix.

[0015] In one possible implementation of the first aspect, the method further includes, prior to performing eigenvalue decomposition on the local Hessian matrix: Determine whether the absolute values ​​of the main diagonal elements of the local Hessian matrix are all greater than the sum of the absolute values ​​of the corresponding row's off-diagonal elements and the sum of a preset small positive number; If so, the local Hessian matrix is ​​determined to satisfy the strict diagonal dominance condition. The eigenvalue decomposition and correction steps are skipped, and the local Hessian matrix is ​​directly used as a positive definite Hessian matrix to participate in the assembly of the new global Hessian matrix.

[0016] According to a second aspect of the present invention, a computer device is provided, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the computer program, implements the aforementioned method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering.

[0017] According to a third aspect of the present invention, a computer-readable storage medium is provided, the computer-readable storage medium storing a computer program, which, when executed by a processor, implements the aforementioned method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering.

[0018] According to a fourth aspect of the present invention, a computer program product is provided, which, when executed by a processor, implements the aforementioned method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering.

[0019] Compared with the prior art, the present invention has at least the following beneficial effects: This invention provides a method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering. In each Newton iteration step, a global dimensional unification factor is calculated by acquiring the global system energy and the global Hessian matrix, achieving automatic unification of physical dimensions at the global scale. This effectively avoids the mismatch between the correction strategy and the actual deformation state caused by dimensional inconsistencies. Simultaneously, when negative eigenvalues ​​exist in the local Hessian matrix, this method calculates the gradient projection along the corresponding eigenvector direction and accurately classifies negative curvature directions into reliable or unreliable direction sets based on the sign of the gradient projection. This overcomes the defect of existing technologies that blindly discard negative curvature information, causing the descent direction to deviate from the minimum value. Furthermore, this method comprehensively utilizes gradient projection, negative eigenvalues, and the global dimensional unification factor, respectively targeting reliable directions... By determining matching mixing coefficients for the set of unreliable directions and using these mixing coefficients to differentiate and positively definitely reconstruct the corresponding negative eigenvalues ​​in the local Hessian matrix, an analytical adaptive eigenvalue correction strategy can be provided within a variational framework that jointly optimizes energy reduction and matrix correction costs when assembling the new global Hessian matrix and solving the Newton step. This solves the problem of lag in discrete switching response in existing technologies. This method not only fully preserves the negative curvature information in the reliable descent direction, overcoming the slow convergence problem under high Poisson's ratio and large deformation, but also provides precise constraints on the unreliable direction, effectively avoiding the risk of introducing excessive damping in the near-convex region. Thus, while ensuring the system's extremely high robustness, it improves the energy reduction rate and overall solution efficiency in the deformation simulation of hyperelastic materials. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the specific embodiments of the present invention, the drawings used in the description of the specific embodiments will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0021] Figure 1 This is a flowchart illustrating a method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering, according to an embodiment of the present invention. Figure 2 A comparison of convergence speeds under torsional deformation scenarios for materials with low Poisson's ratio (relative torsional angle 0.6π). Figure 3 A comparison of convergence speeds for materials with high Poisson's ratios under torsional deformation scenarios (relative torsional angle 0.8π). Figure 4 A comparison of convergence speeds under tensile deformation scenarios for materials with low Poisson's ratio (axial length ratio 2.0). Figure 5A comparison of convergence speeds under tensile deformation scenarios for materials with high Poisson's ratio (axial length ratio 3.0). Figure 6 A comparison of convergence speeds under bending deformation scenarios for materials with low Poisson's ratio (rotation angle 0.4π). Figure 7 This is a comparison of convergence speeds under bending deformation scenarios for materials with high Poisson's ratio (rotation angle 0.1π). Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0023] This invention proposes a method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering. This method is applied to computer equipment, correcting the mixing coefficients by analytically solving for the optimal eigenvalues ​​of the local Hessian matrix in the projective Newton method, and introducing a global dimensional unification factor, directional reliability classification, and differentiated physical energy constraints to achieve adaptive filtering.

[0024] like Figure 1 As shown, in each Newton iteration step of the simulation of hyperelastic material deformation, this embodiment specifically includes the following steps: S1. Acquire data and perform local Hessian matrix eigenvalue decomposition. In the current Newton iteration step of the deformation simulation, the local Hessian matrix of each element of the hyperelastic material is first acquired. Local gradient The unit energy of the current unit And the global system energy of the hyperelastic material under the current deformation state. and the global Hessian matrix The local Hessian matrix for Symmetric matrix, where This represents the number of degrees of freedom per unit.

[0025] As an optional, fast assessment, the local Hessian matrix is ​​examined before eigenvalue decomposition. Does it satisfy the strict diagonal dominance condition? Determine the elements on the main diagonal. Are all values ​​greater than the sum of the absolute values ​​of the non-diagonal elements in the corresponding row? With preset small positive numbers The sum of

[0026] in, The value range is 10 -10- Up to 10 -6 Recommended: 10 -8 .

[0027] If the strict diagonal dominance condition is met, the correction flag is set to false, the eigenvalue decomposition and correction steps are skipped, the local Hessian matrix is ​​directly used as the positive definite Hessian matrix to participate in the subsequent global assembly, and the process jumps to S10. If the conditions are not met, then the Jacobi iterative method or the QR algorithm is used to perform eigenvalue decomposition on the local Hessian matrix to obtain the eigenvalues. and the corresponding feature vector At this point, it is determined whether there are negative eigenvalues ​​among the eigenvalues. If they do not exist, the correction flag is set to false and the process jumps to S10; if they do exist, the correction flag is set to true.

[0028] S2, Calculate gradient projection and classify. For any absolute value greater than a preset small positive number negative eigenvalues Extract the negative value of the negative eigenvalue. .

[0029] Based on the acquired local gradient The eigenvector corresponding to the negative eigenvalue Multiplying the transpose of the eigenvector by the local gradient yields the gradient projection along the direction of the eigenvector. .

[0030] Subsequently, based on gradient projection The sign of the gradient projection is used to classify the corresponding feature vector direction: if the gradient projection Then the corresponding feature vector direction will be assigned to the reliable direction set. If gradient projection Then the corresponding eigenvector direction will be classified into the unreliable direction set. .

[0031] S3, Set the safety factor First safety factor for obtaining reliable direction And a second safety factor in unreliable directions Both have a range of values. For version 1.0, to ensure a balance between stability and convergence speed, the recommended parameters are: and .

[0032] S4. Calculate the global dimensional uniformity factor. Based on global system energy and the global Hessian matrix Calculate the global dimensional unification factor used to unify physical quantities. This ensures that the objective function has consistent dimensions, automatically carrying the physical dimensions [energy] / [force·length]. The specific calculation process is as follows: set up ; Calculate the square of the Frobenius norm of the global Hessian matrix. ; The ratio of the global system energy to the square of the aforementioned Frobenius norm is defined as the global dimensional unification factor. .

[0033] S5, Calculate the aggregation parameters of the unit group Aggregate calculations are performed on the parameters within each directional set: For reliable direction set Calculate the square of the gradient projection in each direction within the set. The opposite of the corresponding negative eigenvalue The ratios of the given values ​​are summed to obtain the first aggregation parameter for the reliable direction. Simultaneously, calculate the square of the negative eigenvalue corresponding to each direction within the set, which is then equal to the square of its opposite. The squares of each term are summed to obtain the second aggregation parameter for the reliable direction. .

[0034] For unreliable direction set Similarly, the first aggregation parameter for the unreliable direction is calculated. And the second aggregation parameter of the unreliable direction. .

[0035] S6. Calculate the variational optimal mixing coefficients. Analyze and solve for the variational optimal solution in different directions separately: If reliable direction set Non-empty, the first aggregation parameter of the reliable direction. Divide by the global dimension unification factor With the second aggregation parameter The product is doubled, and the cube root of the resulting quotient is determined as the optimal mixing coefficient for the reliable direction. ; If reliable direction set If empty, there is no reliable direction to process; If unreliable direction set Non-empty, the first aggregation parameter for unreliable directions. Divide by the global dimension unification factor The second aggregation parameter with unreliable direction The product is doubled, and the square root of the resulting quotient is determined as the optimal mixing coefficient for the unreliable direction. ; If unreliable direction set If empty, then there are no unreliable directions to process.

[0036] S7. Calculate the lower bound of the energy constraint. To avoid energy fluctuations caused by single-step corrections, based on the acquired unit energy... And the physical lower bound is calculated based on the first and second safety factors: According to the formula Calculate the physical lower bound of the reliable direction.

[0037] According to the formula Calculate the unreliable physical lower bound for the direction.

[0038] Specifically, the physical lower bound of the reliable directions corresponding to the set of reliable directions is determined by constraints. Transform into Solve the quadratic equation and take the positive root: .

[0039] Specifically, the lower bound of the unreliable direction physical boundary corresponding to the set of unreliable directions is determined by... have to: .

[0040] S8. Determine the final mixing coefficient. By combining the optimal analytical solution and the physical security lower bound, the mixing coefficients for practical applications are determined: Select a reliable direction physical lower bound Optimal mixing coefficient with reliable direction The larger value in the set is selected and compared with 1. The smaller value is taken as the final mixing coefficient of the reliable direction set, thus determining the reliable direction mixing coefficient. ; Selecting an unreliable physical lower bound Optimal mixing coefficient with unreliable direction The larger value in the set is taken, and this larger value is compared with 1. The smaller value is taken as the final mixing coefficient of the unreliable direction set, that is, the unreliable direction mixing coefficient is determined. ; If the corresponding direction group is empty, skip it.

[0041] S9. Calculate the positive eigenvalue shrinkage coefficient. Calculate the ratio of the sum of the absolute values ​​of all negative eigenvalues ​​within the current cell to the sum of the absolute values ​​of all eigenvalues ​​to determine the proportion of negative eigenvalue energy. .

[0042] Based on this energy percentage Calculate the positive eigenvalue shrinkage coefficient .

[0043] S10, Eigenvalue Correction and Local Hessian Matrix Reconstruction If the aforementioned correction flag is true, then the corresponding negative eigenvalues ​​in the local Hessian matrix are adaptively corrected using the obtained reliable and unreliable direction mixing coefficients: For negative eigenvalues ​​in the reliable direction set, i.e. and Take its corresponding opposite number Mixing coefficient with reliable direction Multiply to obtain the corrected eigenvalues. ; For negative eigenvalues ​​in the unreliable direction set, i.e. and Take its corresponding opposite number Mixing coefficient with unreliable direction Multiply to obtain the corrected eigenvalues. ; For positive eigenvalues ​​greater than a preset small positive number Combine it with the positive eigenvalue shrinkage coefficient Multiply to obtain the corrected eigenvalues. ; For extremely small eigenvalues ​​whose absolute value is not greater than a preset small positive number Correct it to a preset small positive number. .

[0044] Finally, a diagonal matrix is ​​constructed based on the corrected eigenvalues. And combined with the original eigenvectors of the local Hessian matrix to form an orthogonal matrix According to the formula Reconstruct the positive definite Hessian matrix of each element.

[0045] If the correction flag is false. .

[0046] S11. Global Hessian matrix assembly and Newton step solution The reconstructed positive definite Hessian matrices of each element are assembled into a new global Hessian matrix. Solve the system of linear equations based on the new global Hessian matrix. (in For global gradient, (Using the Newton search direction), the Newton step size is determined through line search, thereby completing the update of the hyperelastic material node position for this iteration step, and continuing to the next Newton iteration step.

[0047] In the variational eigenvalue filtering-based deformation simulation method for hyperelastic materials provided in the above embodiments, the entire method, within a unified variational framework, possesses both theoretical completeness and physical rigor. Its feasibility and beneficial effects can be explained from the following theoretical perspectives: First, this embodiment establishes a variational framework for joint optimization of energy and matrix correction. Existing eigenvalue filtering methods typically only focus on local matrix positive definiteness, while this method, for the first time, incorporates the predicted energy decrease and matrix correction cost into a unified variational objective function. This is achieved by minimizing the predicted energy decrease and the matrix correction cost with a globally unified factor (i.e., minimizing...). This method takes minimizing the system energy fluctuations introduced by regularization as its criterion, which has clear physical meaning and reasonable optimization direction.

[0048] Secondly, this embodiment achieves globally unified dimensions and closed-form analytical solutions. Traditional methods are prone to failure under different materials, while this method automatically generates a globally unified dimension factor using the global system energy and global Hessian matrix, achieving automatic unification of physical dimensions without the need for manual setting of material or mesh parameters. Based on the unified dimensions, this method derives variational optimal solutions, namely, the optimal mixing coefficients in reliable directions are derived from a second-order approximation to a cube root analytical solution, and the optimal mixing coefficients in unreliable directions are derived from a first-order approximation to a square root analytical solution, making the calculation process simple and efficient.

[0049] Furthermore, this embodiment introduces directional reliability classification and differentiated physical lower bound constraints. By calculating the gradient projection and based on the sign of the gradient projection, negative curvature directions are distinguished into a set of reliable directions and a set of unreliable directions. For these two types of directions with different properties, the algorithm introduces physical lower bounds calculated based on element energy and first and second safety factors, respectively. This differentiated approach ensures that the update step size of reliable and unreliable directions is strictly physically constrained, enhancing the robustness of the algorithm under extreme deformation scenarios.

[0050] Thanks to the aforementioned joint optimization mechanism, this embodiment exhibits good convergence performance under various deformation scenarios: On the one hand, the algorithm requires fewer iterations and exhibits strong stability. Under tests involving deformation scenarios such as stretching, bending, and torsion, the convergence iteration count of the method in this embodiment is reduced by an average of 9.91% to 28.67% compared to existing Clamp, Abs, and TR methods. Simultaneously, the coefficients of variation in this embodiment are 64.51%, 86.88%, and 99.27% ​​of those of the Clamp, Abs, and TR methods, respectively, indicating more stable overall performance.

[0051] On the other hand, the algorithm has low single-step iteration energy and a rapid energy decrease. During the iterative solution process, this method can maintain low single-step energy and a fast energy decrease rate. This characteristic makes this method suitable not only for interactive real-time deformation simulation tasks with a fixed number of iterations, but also for non-interactive deformation simulation tasks where energy is the iteration termination threshold.

[0052] To further verify the effectiveness and advancement of the variational eigenvalue filtering-based hyperelastic material deformation simulation method provided in the above embodiments, the following detailed explanation is provided in conjunction with specific test verification scenarios and measured data.

[0053] The following test cases are conducted on the Ubuntu operating system to test the effectiveness of the analytical solution, global dimensional unification factor, directional reliability classification, and differentiated constraints in the proposed method, as well as its performance under different materials and deformation scenarios. The tests use the number of Newton iterations and the final energy of each iteration step as indicators to evaluate the algorithm's effectiveness and convergence speed and stability under different materials and deformation scenarios.

[0054] The specific test cases are as follows: (1) Test cases and results of the effectiveness of the global dimensional uniformity factor in the stretching scenario This test case evaluates the effectiveness of the proposed method's global dimensional uniformity factor in a tensile scenario. The test mesh uses a cuboid with 369 nodes and 990 tetrahedral elements. Material parameters are used for both high Poisson's ratio material (Young's modulus E = 100 MPa, Poisson's ratio ν = 0.495) and low Poisson's ratio material (Young's modulus E = 100 MPa, Poisson's ratio ν = 0.3). All eigenvalues ​​are corrected using this method, and the global dimensional uniformity factor is... Take constant 1 (non-dimensional) and (Physical unification of numerator and denominator dimensions, i.e., unified dimensions). The loading method is to fix the left side, i.e., X=0, and the right side, i.e., X= Apply positive horizontal displacement along the X-axis at the location ,in This is the original X-axis length. This refers to the X-axis length ratio. The tested axial length ratio. The values ​​were 1.0, 2.0, 2.5, 3.0, and 4.0, corresponding to axial tensile rates of 100%, 200%, 250%, 300%, and 400%, respectively. The recorded index was the number of Newton iterations. The number of iterations for different materials and dimensions is shown in Table 1.

[0055] Table 1. Test cases and results of the global dimensional uniformity factor under tensile deformation scenario.

[0056] Test results (see Table 1) show that, compared with the analytical solution without uniform dimensions, the analytical solution using uniform dimensions exhibits fewer iterations in all tensile test configurations, with the acceleration effect further enhancing as the stretch ratio increases, and the stability of the method is significantly improved. This advantage is even more pronounced in scenarios with large tensile deformation and for materials with high Poisson's ratio. Specific analysis is as follows: 1) For low Poisson's ratio materials, the average number of iterations without using a unified dimension is 10.2, while the average number of iterations with a unified dimension is 7.8, a reduction of approximately 23.5%. For high Poisson's ratio materials, the average number of iterations decreases from 48.4 to 23.4, a reduction of approximately 51.7%. The unified dimension method significantly reduces the number of iterations required for convergence, and the acceleration effect is more pronounced for high Poisson's ratio materials.

[0057] 2) The acceleration effect of the unified dimension method further increases with the increase of the stretch ratio. When the axial length ratio is 1.0, the number of iterations of the unified dimension method decreases by 0 and 2 times for low and high Poisson's ratio materials, respectively. When the axial length ratio increases to 4.0, the number of iterations of the unified dimension method can be reduced by 13 and 60 times for low and high Poisson's ratio materials, respectively. Overall, the number of iterations of the unified dimension method increases slowly with the axial length ratio, while it increases sharply when the unified dimension method is not used.

[0058] 3) Looking at the variation of the number of iterations with the axial length ratio, the fluctuations are drastic without a unified dimension: for low Poisson's ratio materials, the range is 15, the standard deviation is 5.91, and the coefficient of variation is 0.58; for high Poisson's ratio materials, the range is 87, the standard deviation is 27.96, and the coefficient of variation is 0.58. However, with a unified dimension, the range is only 2 for low Poisson's ratio materials, the standard deviation is 0.75, and the coefficient of variation is 0.10; for high Poisson's ratio materials, the range is 25, the standard deviation is 8.73, and the coefficient of variation is 0.37. The standard deviations of the unified dimension method for low and high Poisson's ratio materials are 11.9% and 31.1% of the former, respectively, and the coefficients of variation are 16.7% and 66.7% of the former, respectively. This indicates that its convergence stability is far superior to the method without a unified dimension, and this stability advantage is also more significant for high Poisson's ratio materials.

[0059] (2) Test cases and results of the effectiveness of the global dimensional unification factor in the shearing scenario This test case evaluates the effectiveness of the proposed global dimensional uniformity factor in a shearing scenario. The specific settings are as follows: The test mesh is a cuboid containing 369 nodes and 990 tetrahedral elements. Material parameters are high Poisson's ratio material (Young's modulus E = 100 MPa, Poisson's ratio ν = 0.495) and low Poisson's ratio material (Young's modulus E = 100 MPa, Poisson's ratio ν = 0.3). The loading method is to fix the bottom surface (Y = 0) and the top surface (Y = ... Apply horizontal displacement ,in Shear strain. The shear strain measured. The values ​​are 0.1, 0.5, 1.0, 1.5, and 2.0. The recorded index is the number of Newton iterations. The number of iterations for this method under different materials and dimensions is shown in Table 2.

[0060] Table 2 Test cases and results of the global dimensional uniformity factor under shear deformation scenario.

[0061] The test results (see Table 2) show that, compared with the variational analytical solution without uniform dimensions, the analytical solution with uniform dimensions exhibits fewer iterations in all shearing test configurations, significantly reduces the number of steps required for convergence, and significantly improves the stability of the method.

[0062] The specific analysis is as follows: 1) For materials with low Poisson's ratio, the average number of iterations without using the unified dimension method is 7.6, while the number of iterations with the unified dimension method is 6.0, a reduction of approximately 21.1%; for materials with high Poisson's ratio, the average number of iterations decreases from 15.2 to 13.0, a reduction of approximately 14.5%. The unified dimension method reduces the number of iterations required for convergence.

[0063] 2) Looking at the variation of iteration number with shear strain, the variation is large without a unified dimension: the range is 6 and the standard deviation is 2.24 for low Poisson's ratio materials; the range is 20 and the standard deviation is 7.19 for high Poisson's ratio materials. However, with a unified dimension, the range is only 4 and the standard deviation is 1.41 for low Poisson's ratio materials; and the range is 16 and the standard deviation is 6.07 for high Poisson's ratio materials. The standard deviations of the unified dimension method on low and high Poisson's ratio materials are 63.6% and 84.7% of the former, respectively, indicating that its convergence stability is better than the method without a unified dimension.

[0064] (3) Test cases and results of the effectiveness of the global dimensional uniformity factor in bending scenarios This test case evaluates the effectiveness of the proposed global dimensional uniformity factor in bending scenarios. The test mesh uses a cuboid with 369 nodes and 990 tetrahedral elements. Material parameters are high Poisson's ratio material (Young's modulus E = 100 MPa, Poisson's ratio ν = 0.495) and low Poisson's ratio material (Young's modulus E = 100 MPa, Poisson's ratio ν = 0.3). The loading method is as follows: the left side (X = 0) is fixed, and the right side (X = ...) is fixed. A bend was applied at the point with the center of gravity as the point of rotation and the Z-axis as the axis of rotation. The bending angles were 0.1π radians, 0.25π radians, 0.4π radians, 0.6π radians, and 0.8π radians, corresponding to rotation angles of 18 degrees, 45 degrees, 72 degrees, 108 degrees, and 144 degrees, respectively. The number of Newton iterations was recorded. The number of iterations for different materials and dimensions is shown in Table 3.

[0065] Table 3 Test cases and results of the global dimensional uniformity factor under bending deformation scenario.

[0066] The test results (see Table 3) show that, compared with the variational analytical solution without uniform dimensions, the analytical solution with uniform dimensions exhibits fewer or the same number of iterations in all bending test configurations, and the stability of the method is significantly improved on low Poisson's ratio materials.

[0067] The specific analysis is as follows: 1) Without using the uniform dimension method, the average number of iterations on low Poisson's ratio materials is 10.6, while with the uniform dimension method it is 7.8, a reduction of approximately 26.4%; on high Poisson's ratio materials, the average number of iterations decreases from 25.4 to 19.4, a reduction of approximately 23.6%. The uniform dimension method significantly reduces the number of iterations required for convergence.

[0068] 2) Looking at the variation of the number of iterations with the axial length ratio, the variation is drastic for low Poisson's ratio materials without a unified dimension: the range is 6, the standard deviation is 2.33, and the coefficient of variation is 0.22. However, with a unified dimension, the range is only 2, the standard deviation is 0.75, and the coefficient of variation is 0.1. The unified dimension method has a standard deviation of 30.4% and a coefficient of variation of 50.0% of the former for low Poisson's ratio materials, indicating that its convergence stability is superior to the method without a unified dimension on low Poisson's ratio materials.

[0069] (4) Test cases and results of the effectiveness of the global dimensional uniformity factor in the torsion scenario This test case evaluates the effectiveness of the proposed analytical solution and globally unified dimensions in a torsional scenario. The test mesh is a cuboid with 369 nodes and 990 tetrahedral elements. Material parameters are high Poisson's ratio material (Young's modulus E = 100 MPa, Poisson's ratio ν = 0.495) and low Poisson's ratio material (Young's modulus E = 100 MPa, Poisson's ratio ν = 0.3). The loading method is to fix the bottom surface (Y = 0) and the top surface (Y = ... H A rotation angle about the Y-axis was applied. The tested rotation angles were 0.1π radians, 0.2π radians, 0.4π radians, 0.6π radians, and 0.8π radians, corresponding to relative torsion angles of 18 degrees, 36 degrees, 72 degrees, 108 degrees, and 144 degrees, respectively. The recorded index was the number of Newton iterations. The number of iterations for different materials and dimensions is shown in Table 4.

[0070] Table 4. Test cases and results of the global dimensional uniformity factor under torsional deformation scenario.

[0071] The test results (see Table 4) show that, compared with the variational analytical solution without uniform dimensions, the analytical solution with uniform dimensions exhibits fewer iterations in all torsion test configurations for high Poisson's ratio materials, significantly reduces the number of steps required for convergence, further enhances the acceleration effect as the torsion angle increases, and significantly improves the stability of the method.

[0072] The specific analysis is as follows: 1) Without using the uniform dimension method, the average number of iterations on high Poisson ratio materials is 23.4, while with the uniform dimension method it is 20.0, a reduction of approximately 14.5%. The uniform dimension method reduces the number of iterations required for convergence on high Poisson ratio materials.

[0073] 2) The acceleration effect of the unified dimension method further increases with the increase of the torsion angle on high Poisson's ratio materials. When the torsion angle is less than or equal to 0.4π, the number of iterations of the unified dimension method on high Poisson's ratio materials is almost the same as that without using a dimension. When the torsion angle increases to 0.6π, the number of iterations of the unified dimension method on high Poisson's ratio materials can be reduced by 9. The number of iterations of the unified dimension method increases slowly with the increase of the torsion angle, while it increases rapidly without using a unified dimension.

[0074] 3) Looking at the variation of the number of iterations with the axial length ratio, the fluctuations are drastic when no unified dimension is used on high Poisson's ratio materials: the range is 39, the standard deviation is 14.75, and the coefficient of variation is 0.63. However, after adopting a unified dimension, the range on high Poisson's ratio materials is 28, the standard deviation is 10.08, and the coefficient of variation is 0.50. The standard deviation of the unified dimension method on high Poisson's ratio materials is 68.7% of the former, and the coefficient of variation is 83.3% of the former, indicating that its convergence stability is better than the method without a unified dimension on high Poisson's ratio materials.

[0075] (5) Test cases and results of the effectiveness of directional reliability classification in shear scenarios This test case is used to evaluate the effectiveness of the proposed method's directional reliability classification in shear deformation scenarios. The test mesh, material parameters, loading method, and axial length ratio configuration are the same as in test case (2). When the algorithm does not classify the gradient projection direction of the negative eigenvector, it does not distinguish between reliable and unreliable directions and uniformly uses the analytical solution of the reliable direction; when using directional reliability classification, reliable and unreliable directions use their respective analytical solutions. The number of iterations of this method under different materials and different eigenvalue correction strategies is shown in Table 5.

[0076] Table 5. Test cases and results for directional reliability classification under shear scenarios.

[0077] The test results (see Table 5) show that, compared with the variational analytical solution without directional reliability classification, the analytical solution with directional reliability classification requires fewer iterations under the same iteration termination conditions in all shear test configurations on both high and low Poisson's ratio materials, and the stability of the method is significantly improved.

[0078] The specific analysis is as follows: 1) The average number of iterations for high Poisson's ratio materials without using the directional reliability classification method is 13.0, while it is 12.0 with the directional reliability classification method, a decrease of 7.7%. The average number of iterations for low Poisson's ratio materials without using the directional reliability classification method is 6.0, while it is 5.8 with the directional reliability classification method, a decrease of 3.3%. Using the directional reliability classification method reduces the number of iterations for both high and low Poisson's ratio materials.

[0079] 2) Looking at the variation of iteration number with shear strain, for high Poisson's ratio materials, the standard deviation is 6.07 without directional reliability classification; after using directional reliability classification, the standard deviation is 5.66, which is 93.3% of the former. For low Poisson's ratio materials, the standard deviation is 1.41 without directional reliability classification; after using directional reliability classification, the standard deviation is 1.17, which is 82.5% of the former. This indicates that its convergence stability is superior to the method without directional reliability classification and differential reduction for both high and low Poisson's ratio materials.

[0080] (6) Test cases and results of the effectiveness of directional reliability classification in compression scenarios This test case is used to evaluate the effectiveness of the proposed method's reliability classification under compressive deformation scenarios. The test mesh is a cube containing 792 nodes and 2911 tetrahedral elements. Material parameters are high Poisson's ratio material (Young's modulus E = 100 MPa, Poisson's ratio ν = 0.495) and low Poisson's ratio material (Young's modulus E = 100 MPa, Poisson's ratio ν = 0.3). The loading method is to fix the bottom surface (Y = 0) and the top surface (Y = ... Apply uniaxial displacement ,in The height is the original cube height. The axial length ratio is the compression ratio. The tested axial length ratios were 0.9, 0.8, 0.7, 0.6, and 0.5, corresponding to axial compression rates of 10%, 20%, 30%, 40%, and 50%, respectively. The number of iterations for this method under different materials and different eigenvalue correction strategies is shown in Table 6.

[0081] Table 6. Test cases and results for directional reliability classification in compression scenarios.

[0082] The test results (see Table 6) show that, compared with the analytical solution that does not use directional reliability classification, the analytical solution that distinguishes directional reliability classification exhibits fewer iterations under all compressed test configurations, significantly reduces the number of steps required for convergence, and also improves the stability of the method.

[0083] 1) For materials with low Poisson's ratio, the average number of iterations without using the directional reliability classification method is 12.0, while with the directional reliability classification method it is 11.4, a decrease of approximately 5.0%. For materials with high Poisson's ratio, the average number of iterations decreases from 10.6 to 10.2, a decrease of approximately 3.8%. Using the directional reliability classification method reduces the number of iterations required for convergence.

[0084] 2) Looking at the variation of iteration number with axial length ratio, the fluctuation is larger without the directional reliability classification method: the range is 14 and the standard deviation is 4.96 for high Poisson's ratio materials; the range is 26 and the standard deviation is 9.82 for low Poisson's ratio materials. After using directional reliability classification, the range is 13 and the standard deviation is 4.53 for high Poisson's ratio materials; the range is 24 and the standard deviation is 9.05 for low Poisson's ratio materials. The standard deviations of the directional reliability classification method on high and low Poisson's ratio materials are 91.3% and 92.1% of the former, respectively, indicating that its convergence stability is better than the method without directional reliability classification.

[0085] (7) Test cases and results of the effectiveness of directional reliability classification in torsion scenarios This test case is used to evaluate the effectiveness of the proposed method's directional reliability classification in torsional deformation scenarios. The test mesh, material parameters, loading method, and axial length ratio configuration are the same as in test case (4). The recorded metric is the number of Newton iterations. The number of iterations for this method under different materials and different eigenvalue correction strategies is shown in Table 7.

[0086] Table 7. Orientation Reliability Classification: Test Cases and Results in Torsion Scenarios

[0087] The test results (see Table 7) show that, compared with the analytical solution that does not use directional reliability classification, the analytical solution that distinguishes directional reliability classification exhibits fewer iterations in all torsion test configurations, significantly reduces the number of steps required for convergence, and also improves the stability of the method.

[0088] 1) For materials with low Poisson's ratio, the average number of iterations without using the directional reliability classification method is 9.6, while with the directional reliability classification method it is 9.20, a reduction of approximately 4.2%. For materials with high Poisson's ratio, the average number of iterations decreases from 20 to 19.2, a reduction of approximately 4.0%. Using the directional reliability classification method reduces the number of iterations required for convergence.

[0089] 2) Looking at the variation of iteration number with axial length ratio, the fluctuations are larger without the directional reliability classification method: the range is 28 and the standard deviation is 10.08 for high Poisson's ratio materials; the range is 14 and the standard deviation is 5.12 for low Poisson's ratio materials. After using directional reliability classification, the range is 25 and the standard deviation is 9.06 for high Poisson's ratio materials; the range is 12 and the standard deviation is 4.40 for low Poisson's ratio materials. The standard deviations of the directional reliability classification method on high and low Poisson's ratio materials are 89.9% and 85.9% of the former, respectively, indicating that its convergence stability is better than the method without directional reliability classification.

[0090] (8) Test cases and results of the effectiveness of directional reliability classification in tensile scenarios This test case is used to evaluate the effectiveness of the proposed method's directional reliability classification in tensile deformation scenarios. The test mesh, material parameters, loading method, and axial length ratio configuration are the same as in test case (1). The number of iterations of this method under different materials and different eigenvalue correction strategies is shown in Table 5.

[0091] Table 8. Test cases and results for directional reliability classification under tensile scenarios.

[0092] The test results (see Table 8) show that, compared with the variational analytical solution without directional reliability classification, the analytical solution using directional reliability classification exhibits significantly improved stability under the same iteration termination conditions in all tensile test configurations on both high and low Poisson's ratio materials. Detailed analysis follows: Looking at the variation of iteration number with shear strain, for high Poisson's ratio materials, the standard deviation is 8.73 without directional reliability classification; after using directional reliability classification, the standard deviation is 8.57, which is 98.1% of the former. For low Poisson's ratio materials, the standard deviation is 0.75 without directional reliability classification; after using directional reliability classification, the standard deviation is 0.63, which is 84.5% of the former. This indicates that its convergence stability is better than the method without directional reliability classification and differential constraints for both high and low Poisson's ratio materials.

[0093] (9) Test cases and results of the effectiveness of difference constraints in the torsion scenario of high Poisson's ratio materials This test case is used to evaluate the effectiveness of the differential constraint of this method in torsional deformation scenarios of high Poisson's ratio materials. The test mesh, material parameters, loading method, and torsional angle configuration are the same as those in test case (4). This test case compares the performance of the algorithm with and without energy lower bound constraints in reliable and unreliable directions, respectively. The number of iterations is shown in Table 9.

[0094] Table 9. Test cases and results of difference constraints in torsion scenarios of high Poisson's ratio materials.

[0095] The test results show (see Table 9) that the difference constraint affects the average number of iterations and the standard deviation of the algorithm. When the difference constraint is used, the average number of iterations of the algorithm is less and the standard deviation is smaller.

[0096] 1) For materials with high and low Poisson ratios, the average number of iterations is 19.2 without using difference constraints; with difference constraints, the average number of iterations is 17.8, a decrease of approximately 7.3%. Difference constraints have an impact on the average number of iterations of the algorithm; the algorithm has fewer iterations when difference constraints are used.

[0097] 2) On high Poisson's ratio materials, the standard deviation of the number of iterations is 9.06 without using difference constraints; with difference constraints, the standard deviation is 8.38, which is 92.3% of the former. Difference constraints affect the standard deviation of the algorithm; the smaller the standard deviation of the algorithm, the better.

[0098] (10) Test cases and results of the effectiveness of difference constraints in shear scenarios of high Poisson's ratio materials This test case is used to evaluate the effectiveness of the differential constraints of this method under shear deformation scenarios of high Poisson's ratio materials. The test mesh, material parameters, loading method, and shear strain configuration are the same as in test case (2). This test case compares the performance of the algorithm with and without energy lower bound constraints in the reliable and unreliable directions, respectively. The number of iterations is shown in Table 10.

[0099] Table 10 Test cases and results of difference constraints in high Poisson's ratio material shearing scenarios.

[0100] Test results (see Table 10) show that the difference constraint affects the average number of iterations and the standard deviation of the algorithm. When the difference constraint is used, the average number of iterations of the algorithm is less and the standard deviation is smaller.

[0101] 1) For materials with high and low Poisson ratios, the average number of iterations is 12.0 without using difference constraints; with difference constraints, the average number of iterations is 11.2, a decrease of approximately 6.7%. Difference constraints affect the average number of iterations of the algorithm; the algorithm has fewer iterations when difference constraints are used.

[0102] 2) On high Poisson's ratio materials, the standard deviation of the number of iterations is 5.66 without using difference constraints; with difference constraints, the standard deviation is 4.96, which is 87.7% of the former. Difference constraints affect the standard deviation of the algorithm; the smaller the standard deviation of the algorithm, the better.

[0103] (11) Test cases and results of the effectiveness of difference constraints in tensile scenarios of high Poisson's ratio materials This test case is used to evaluate the effectiveness of the differential constraints of this method under tensile deformation scenarios of high Poisson's ratio materials. The test mesh, material parameters, loading method, and axial length ratio configuration are the same as those in test case (1). This test case compares the performance of the algorithm with and without energy lower bound constraints in reliable and unreliable directions, respectively. The number of iterations is shown in Table 11.

[0104] Table 11 Test cases and results of differential constraints in high Poisson's ratio material tensile scenarios.

[0105] Test results (see Table 11) show that the difference constraint affects the average number of iterations and the standard deviation of the algorithm. When the difference constraint is used, the algorithm has fewer average iterations and a smaller standard deviation.

[0106] 1) For materials with high and low Poisson ratios, the average number of iterations is 23.4 without using difference constraints; with difference constraints, the average number of iterations is 22.0, a decrease of approximately 6.0%. Difference constraints affect the average number of iterations of the algorithm; the algorithm has fewer iterations when difference constraints are used.

[0107] 2) On high Poisson's ratio materials, the standard deviation of the number of iterations is 8.57 without using difference constraints; with difference constraints, the standard deviation is 7.24, which is 83.7% of the former. Difference constraints affect the standard deviation of the algorithm; the smaller the standard deviation of the algorithm, the better.

[0108] (12) Test cases and results of the effectiveness of difference constraints in bending scenarios of high Poisson's ratio materials This test case is used to evaluate the effectiveness of the differential constraints of this method in the bending deformation scenario of a high Poisson's ratio material. The test mesh, material parameters, loading method, and test corner configuration are the same as those in test case (3). This test case compares the performance of the algorithm with and without energy lower bound constraints in the reliable and unreliable directions, respectively. The number of iterations is shown in Table 12.

[0109] Table 12 Test cases and results of difference constraints in bending scenarios of high Poisson's ratio materials.

[0110] Test results (see Table 13) show that the difference constraint affects the average number of iterations and the standard deviation of the algorithm. When the difference constraint is used, the average number of iterations of the algorithm is less and the standard deviation is smaller.

[0111] 1) For materials with high and low Poisson ratios, the average number of iterations is 20.2 without using difference constraints; with difference constraints, the average number of iterations is 19.2, a decrease of approximately 5.0%. Difference constraints affect the average number of iterations of the algorithm; the algorithm has fewer iterations when difference constraints are used.

[0112] 2) On high Poisson's ratio materials, the standard deviation of the number of iterations is 6.43 without using difference constraints; with difference constraints, the standard deviation is 5.95, which is 92.2% of the former. Difference constraints affect the standard deviation of the algorithm; the smaller the standard deviation of the algorithm, the better.

[0113] (13) Test cases and results of this method and existing methods in a torsion scenario This test case is used to evaluate the performance of this method under torsional deformation scenarios in materials with high and low Poisson's ratios. The test mesh, material parameters, loading method, and relative torsion angle configuration are the same as in test case (4). The number of iterations for different algorithms is shown in Tables 13 and 14. Some energy curves are shown in Table 14. Figure 2 and Figure 3 .

[0114] Table 13 Test cases and results of this method and existing methods in torsion scenarios of high Poisson's ratio materials.

[0115] Table 14 Test cases and results of this method and existing methods in torsion scenarios of low Poisson's ratio materials.

[0116] The test results are shown in Tables 13 and 14. Figure 2 and Figure 3 This method exhibits the fewest average iterations and the best stability in test cases under torsion scenarios; it also demonstrates the lowest single-step iteration energy and the fastest energy decrease rate on both high and low Poisson's ratio materials. Detailed analysis follows: 1) For high Poisson's ratio materials, the average number of iterations for clamping filtering is 24.6, for absolute value filtering it is 26.0, and for trust region filtering it is 20.2. Our proposed method has an average of 17.8 iterations, representing reductions of 27.6%, 31.5%, and 11.9% compared to the previous three methods, respectively. For low Poisson's ratio materials, the average number of iterations for clamping filtering is 9.4, for absolute value filtering it is 12.8, and for trust region filtering it is 10.0. Our proposed method has an average of 8.8 iterations, representing reductions of 6.4%, 31.3%, and 12.0% compared to the previous three methods, respectively. Our proposed method has fewer average iterations for both high and low Poisson's ratio materials than the three existing methods.

[0117] 2) Looking at the variation of iteration number with torsion angle, for high Poisson's ratio materials, the standard deviation of clamping filter is 14.93, the standard deviation of absolute value filter is 13.05, the standard deviation of trust region filter is 9.77, and the standard deviation of our method is 8.38, which are 56.1%, 64.2%, and 85.8% of the other three methods, respectively. For low Poisson's ratio materials, the standard deviation of clamping filter is 5.12, the standard deviation of absolute value filter is 8.63, the standard deviation of trust region filter is 5.25, and the standard deviation of our method is 3.97, which are 77.5%, 46.0%, and 75.6% of the other three methods, respectively. Our method demonstrates better stability than the three existing methods for both high and low Poisson's ratio materials.

[0118] 3) From the perspective of the logarithm of the energy curve (see...) Figure 2 and Figure 3 When the relative torsion angles are 0.6π (for low Poisson's ratio materials) and 0.8π (for high Poisson's ratio materials), the energy curve of this method is lower than that of the other three methods, and the energy after each iteration is less than that of the other three methods.

[0119] 4) From the slope of the energy curve (see...) Figure 2 and Figure 3 The slope of the clamping filter energy curve is the smallest, followed by the absolute value filter and trust region filter energy curves. The slope of the energy curve of this method is the largest, and the energy decreases the fastest, especially in the first 4 iterations.

[0120] (14) Test cases and results of this method and existing methods in bending scenarios This test case is used to evaluate the performance of this method in bending deformation scenarios of high Poisson's ratio and low Poisson's ratio materials. The test mesh, material parameters, loading method, and test rotation configuration are the same as in test case (3). The number of iterations for different algorithms is shown in Tables 15 and 16, and some energy curves are shown in Tables 15 and 16. Figure 6 and Figure 7 .

[0121] Table 15 Test cases and results of this method and existing methods in bending scenarios of high Poisson's ratio materials.

[0122] Table 16 Test cases and results of this method and existing methods in bending scenarios of low Poisson's ratio materials.

[0123] The test results are shown in Tables 15 and 16. Figure 6 and Figure 7This method has the fewest average iterations in test cases of curved scenarios and is the most stable on high Poisson's ratio materials; on both high and low Poisson's ratio materials, it has the lowest single-step iteration energy and the fastest energy decrease rate.

[0124] The specific analysis is as follows: 1) For high Poisson's ratio materials, the average number of iterations for clamping filtering is 23.4, for absolute value filtering it is 27.8, and for trust region filtering it is 22.6. Our proposed method has an average of 19.2 iterations, representing reductions of 17.9%, 30.9%, and 15.0% compared to the previous three methods, respectively. For low Poisson's ratio materials, the average number of iterations for clamping filtering is 8.2, for absolute value filtering it is 8.4, and for trust region filtering it is 8.0. Our proposed method has an average of 7.8 iterations, representing reductions of 4.9%, 7.1%, and 2.5% compared to the previous three methods, respectively. Our proposed method has fewer average iterations for both high and low Poisson's ratio materials than the three existing methods.

[0125] 2) Looking at the variation of iteration number with torsion angle, for high Poisson's ratio materials, the standard deviation of clamping filter is 7.34, absolute value filter is 9.02, trust region filter is 7.23, and our method's standard deviation is 5.95, representing 81.0%, 65.9%, and 82.3% of the first three methods, respectively. For low Poisson's ratio materials, the standard deviation of clamping filter is 0.40, absolute value filter is 2.06, trust region filter is 1.67, and our method's standard deviation is 0.75, representing 36.3% and 44.7% of the absolute value filter and trust region filter methods, respectively. For both high and low Poisson's ratio materials, our method demonstrates better stability than the three existing methods. For low Poisson's ratio materials, our method outperforms both absolute value filtering and trust region filtering.

[0126] 3) From the perspective of the logarithm of the energy curve (see...) Figure 6 and Figure 7 When the low Poisson's ratio material undergoes a deformation of 0.4π and the high Poisson's ratio material undergoes a deformation of 0.1π, the energy curve of this method is lower than that of the other three methods, and the energy after each iteration is less than that of the other three methods.

[0127] 4) From the slope of the energy curve (see...) Figure 6 and Figure 7 When the low Poisson's ratio material undergoes a deformation of 0.4π and the high Poisson's ratio material undergoes a deformation of 0.1π, the slope of the clamping filter energy curve is the smallest, followed by the absolute value filter and the trust region filter energy curves. The slope of the energy curve of this method is always the largest, and the energy decreases the fastest.

[0128] (15) Test cases and results of this method and existing methods in the stretching scenario This test case is used to evaluate the performance of the proposed method under tensile deformation scenarios in materials with high and low Poisson's ratios. The test mesh, material parameters, loading method, and axial length ratio configuration are the same as in test case (1). The number of iterations for different algorithms is shown in Tables 17 and 18, and some energy curves are shown in Tables 17 and 18. Figure 4 and Figure 5 .

[0129] Table 17 Test cases and results of this method and existing methods in tensile scenarios for high Poisson's ratio materials.

[0130] Table 18 Test cases and results of this method and existing methods in tensile scenarios for materials with low Poisson's ratio.

[0131] The test results are shown in Tables 17 and 18. Figure 4 and Figure 5 In test cases involving high Poisson's ratio materials in tensile scenarios, this method has the fewest average iterations and exhibits better stability than clamping filtering and absolute value filtering. In test cases involving low Poisson's ratio materials, the average iteration count is comparable to clamping filtering, both being less than absolute value filtering and trust region filtering, and the method demonstrates the best stability. Furthermore, in both high and low Poisson's ratio materials, the method exhibits the lowest single-step iteration energy and the fastest energy decay rate. Detailed analysis follows: 1) On high Poisson's ratio materials, the average number of iterations for clamping filtering is 44.0, for absolute value filtering it is 33.4, for trust region filtering it is 23.2, and for our method it is 22.0, representing reductions of 50.0%, 34.1%, and 5.2% respectively compared to the other three methods. On low Poisson's ratio materials, the average number of iterations for clamping filtering is 7.6, for absolute value filtering it is 9.8, for trust region filtering it is 8.8, and for our method it is 8.0, representing reductions of 18.4% and 9.1% respectively compared to absolute value filtering and trust region filtering. Our method has fewer average iterations on high Poisson's ratio materials than the three existing methods, and on low Poisson's ratio materials, the average number of iterations is comparable to clamping filtering and less than absolute value filtering and trust region filtering.

[0132] 2) Looking at the variation of iteration number with torsion angle, for high Poisson's ratio materials, the standard deviation of clamping filter is 22.02, absolute value filter is 10.40, trust region filter is 5.74, and our method's standard deviation is 7.24, representing reductions of 32.9% and 69.6% compared to the previous two methods, respectively. For low Poisson's ratio materials, the standard deviation of clamping filter is 1.02, absolute value filter is 2.32, trust region filter is 1.72, and our method's standard deviation is 0.63, representing reductions of 62.0%, 27.3%, and 36.8% compared to the previous three methods, respectively. For both high and low Poisson's ratio materials, our method demonstrates better stability than clamping filter and absolute value filter; for low Poisson's ratio materials, our method demonstrates better stability than clamping filter, absolute value filter, and trust region filter.

[0133] 3) From the perspective of the logarithm of the energy curve (see...) Figure 4 and Figure 5 When the low Poisson's ratio material undergoes deformation with an axial length ratio of 2.0, except for step 1, the energy curve of this method is lower than that of the other three methods; when the high Poisson's ratio material undergoes deformation with an axial length ratio of 3.0, the energy curve of this method is also lower than that of the other three methods. The energy of this method is lower than that of the other three methods after almost every iteration.

[0134] 4) From the slope of the energy curve (see...) Figure 4 and Figure 5 When the low Poisson's ratio material undergoes deformation with an axial length ratio of 2.0, the slope of the absolute value filtering energy curve is the smallest, followed by the trust region filtering energy curve. The slopes (steepness) of the energy curves of this method and the clamping filter are the largest. When the high Poisson's ratio material undergoes deformation with an axial length ratio of 3.0, the slope of the clamping filter energy curve is the smallest, followed by the absolute value filtering energy curve, followed by the trust region filtering energy curve. The slope of the energy curve of this method is consistently the largest, especially in the first 7 iterations, where the energy decreases the fastest.

[0135] In summary: Based on the test cases and results of the global dimension unification factor of this method in various test scenarios (see test cases (1)-(4)), compared with the analytical solution using constant dimensions, the analytical solution using the global dimension unification factor has the following advantages: 1) In the tensile deformation scenario, it has fewer iterations, better stability, and the acceleration effect is further enhanced as the stretching ratio increases, with a significant improvement in stability. This advantage is even more pronounced in the large tensile deformation scenario and in high Poisson's ratio materials; 2) In the shear deformation scenario, it has fewer iterations and better stability; 3) In the bending deformation scenario, it has fewer iterations, and the stability is significantly improved in low Poisson's ratio materials; 4) In the torsional deformation scenario, in high Poisson's ratio materials, it has fewer iterations, the acceleration effect is further enhanced as the torsional angle increases, and the stability of the method is significantly improved.

[0136] From the test cases and results of the directional reliability classification of this method in various test scenarios (see test cases (5)-(8)), compared with the analytical solution that does not distinguish directional reliability, the analytical solution using the directional reliability classification is: 1) In the shear deformation, compression deformation and torsional deformation scenarios, the high Poisson's ratio material and the low Poisson's ratio material have fewer iterations and better stability; (2) In the tensile deformation scenario, the high Poisson's ratio material and the low Poisson's ratio material have better stability.

[0137] From the test cases and results of the differentiated constraints of this method in various test scenarios (see test cases (9)-(12)), compared with the analytical solution without energy constraints, the analytical solution with energy constraints has fewer average iterations, faster convergence speed and better method stability in the torsional deformation, shear deformation, tensile deformation and bending deformation scenarios of high Poisson's ratio materials.

[0138] Based on the test cases and results of this method and the three existing methods in various test scenarios (see test cases (13)-(15)), this method: 1) has the fewest average iterations and the best stability in torsional deformation scenarios of high Poisson's ratio and low Poisson's ratio materials; 2) has the fewest average iterations in bending deformation scenarios of high Poisson's ratio and low Poisson's ratio materials, and the best stability in bending deformation scenarios of high Poisson's ratio materials; 3) in tensile deformation scenarios, it has the fewest average iterations on high Poisson's ratio materials and the stability of the method is better than clamping filtering and absolute value filtering, and the average iterations on low Poisson's ratio materials are less than absolute value filtering and trust region filtering, and the stability is the best. 4) Under the three deformation scenarios of torsion, tension, and bending, the total average number of iterations for clamping filtering is approximately 19.53, for absolute value filtering it is approximately 19.70, and for trust region filtering it is approximately 15.47. Our method has the fewest total average iterations, approximately 13.93, representing reductions of 28.67%, 29.27%, and 9.91% respectively compared to the other three methods. 5) Under the three deformation scenarios of torsion, tension, and bending, the total standard deviation of the number of iterations for clamping filtering is approximately 7.33, for absolute value filtering it is approximately 12.98, and for trust region filtering it is approximately... The coefficient of variation (COP) of this method is 8.92, and the total standard deviation is approximately 7.97, which is 46.02%, 61.45%, and 89.43% of the previous three methods, respectively. 6) Under the three deformation scenarios of torsion, tension, and bending, the COP of the clamping filter iterations is approximately 0.89, the COP of the absolute value filter iterations is approximately 0.66, the COP of the trust region filter iterations is approximately 0.58, and the COP of this method is approximately 0.57, which is 64.51%, 86.88%, and 99.27% ​​of the previous three methods, respectively, indicating that this method has the most stable overall performance under various scenarios.

[0139] From the energy decrease curve (see) Figures 2-7 From the perspective of the present invention, the method proposed in this invention has the minimum single-step iteration energy and the fastest energy decrease rate in twisting, bending and stretching scenarios. Therefore, it is not only applicable to interactive application scenarios with a fixed number of iterations, such as object deformation simulation in computer graphics, but also applicable to non-interactive application scenarios with energy as the iteration termination threshold, such as object deformation simulation in movies.

[0140] In summary, the global unified dimension, directional reliability classification, and differentiated constraints of the variational analytical solution proposed in this invention are effective under different scenarios and materials. In most scenarios with high Poisson's ratio materials and low Poisson's ratio materials, it outperforms the three existing filtering methods and can be applied to deformation simulation of complex objects and scenarios.

[0141] To adapt to the computing power of different computer hardware architectures and the differentiated needs of various specific deformation scenarios, in addition to the above basic embodiments, the present invention also provides the following optional specific implementation methods and parameter configuration schemes: In the first alternative implementation (basic parameter configuration example), for most conventional hyperelastic material deformation simulations, a recommended combination of basic parameters is used to obtain stable performance: a preset small positive number is set. When setting the safety factor, let the first safety factor in the reliable direction be... The second safety factor in the unreliable direction ; In calculating the global dimensional uniformity factor At that time, according to the formula = Calculation, where molecules Use the current global system energy directly.

[0142] In the second alternative implementation (gradient-dependent globally unified dimensional embodiment), an alternative logic for calculating the globally unified factor is provided. Specifically, the globally unified factor... The basic formula remains the same. However, the difference lies in the settings. That is, the global gradient based on the current iteration step. The square of the L2 norm and the global Hessian matrix The ratio of norms is used to determine The numerical value. This implementation introduces the unbalanced force (gradient) information of the global system, providing another dynamic adjustment dimension based on the system residual state for dimensional unification.

[0143] In the third alternative implementation (a simplified implementation with globally unified dimensions), to further reduce the computational overhead of system-level scalar aggregation, the globally unified dimension factor is directly applied. Set to 1 or other preset constant. This implementation is suitable for specific physical simulation scenarios where the dimensions of the objective function are naturally close to the same order of magnitude, or where the sensitivity to the material's own parameters is relatively low.

[0144] In the fourth alternative implementation (GPU parallel acceleration embodiment), the algorithm flow of the present invention is specifically configured to adapt to the parallel computing architecture of a graphics processing unit (GPU). Under this architecture, the global dimensional uniformity factor... The calculation is performed only once at the beginning of each Newton iteration step by the host computer, and then broadcast to all processing units; while the reliable directional mixing coefficients are calculated locally for each unit. Unreliable directional mixing coefficient and positive eigenvalue shrinkage coefficient The computation and eigenvalue correction steps are configured to be executed independently and in complete parallel by multiple processing cores. This implementation eliminates computational data dependencies between units, maximizing the utilization of the GPU's high concurrency throughput.

[0145] In the fifth optional implementation (including an accelerated implementation with pre-judgment), to reduce unnecessary matrix operation overhead, a pre-judgment logic for eigenvalues ​​is added after obtaining eigenvalues ​​in step S1 and before performing gradient projection classification in step S2. Specifically, the minimum eigenvalue of the local Hessian matrix is ​​extracted. Determine the smallest eigenvalue. Is it greater than the preset small positive number? If the value is greater than 0, it indicates that the deformation state of the current element is in a strictly local positive definite stable region. In this case, the calculation process of the mixing coefficient and shrinkage coefficient in steps S2 to S9 is skipped directly, the original local Hessian matrix is ​​kept unchanged, and it is directly output to step S11 for global assembly. This implementation can reduce unnecessary computational overhead when the object undergoes small deformation or rigid body translation (small deformation scenario).

[0146] In the sixth optional implementation (simplified real-time simulation embodiment), a simplified correction scheme for interactive real-time simulation is provided. When performing step S9 to calculate the positive eigenvalue shrinkage coefficient, it does not rely on the dynamic statistics of the eigenvalue energy ratio, but directly uses a preset constant formula for shrinkage, specifically setting the positive eigenvalue shrinkage coefficient. This implementation method eliminates the steps of summing the absolute values ​​of all eigenvalues ​​and calculating the ratios, making it suitable for real-time simulation scenarios with high computational real-time requirements.

[0147] In another embodiment of the present invention, a computer device is provided, comprising a processor and a memory. The memory stores a computer program, which includes program instructions. The processor executes the program instructions stored in the computer storage medium. The processor may be a Central Processing Unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. It is the computing and control core of the terminal, suitable for implementing one or more instructions, specifically suitable for loading and executing one or more instructions in the computer storage medium to achieve a corresponding method flow or corresponding function. The processor described in this embodiment of the present invention can be used for the operation of a method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering.

[0148] In another embodiment of the present invention, a storage medium is provided, specifically a computer-readable storage medium (Memory), which is a memory device in a computer device used to store programs and data. It is understood that the computer-readable storage medium here can include both the built-in storage medium in the computer device and extended storage media supported by the computer device. The computer-readable storage medium provides storage space that stores the terminal's operating system. Furthermore, the storage space also stores one or more instructions suitable for loading and execution by a processor. These instructions can be one or more computer programs (including program code). It should be noted that the computer-readable storage medium here can be Random Access Memory (RAM) or non-volatile memory, such as at least one disk storage device. The processor can load and execute one or more instructions stored in the computer-readable storage medium to implement the corresponding steps of the above embodiment regarding a method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering.

[0149] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, optical storage, etc.) containing computer-usable program code.

[0150] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0151] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0152] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0153] This invention also provides a computer program product for executing any of the above-described methods for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering. Since the computer program product provided by this invention belongs to the same inventive concept as the above-described method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering, it possesses all the advantages of the above-described method. Therefore, the beneficial effects of the computer program product provided by this invention will not be elaborated upon here.

[0154] In this invention, the terms "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., refer to a specific feature, structure, material, or characteristic described in connection with that embodiment or example, which is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0155] Finally, it should be noted that the above-described embodiments are merely specific implementations of the present invention, used to illustrate the technical solutions of the present invention, and not to limit them. The scope of protection of the present invention is not limited thereto. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that any person skilled in the art can still modify or easily conceive of changes to the technical solutions described in the foregoing embodiments within the scope of the technology disclosed in the present invention, or make equivalent substitutions for some of the technical features; and these modifications, changes, or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should all be covered within the scope of protection of the present invention.

Claims

1. A method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering, characterized in that, include: Obtain the local Hessian matrix and local gradient of each unit of the hyperelastic material in the current Newton iteration step, as well as the global system energy and global Hessian matrix of the hyperelastic material; The local Hessian matrix is ​​subjected to eigenvalue decomposition to obtain eigenvalues ​​and corresponding eigenvectors. If there are negative eigenvalues ​​among the eigenvalues, the gradient projection in the direction of the eigenvector is calculated based on the local gradient and the eigenvector corresponding to the negative eigenvalue. Based on the sign of the gradient projection, the corresponding feature vector direction is classified into a reliable direction set or an unreliable direction set. Based on the global system energy and the global Hessian matrix, calculate the global dimensional unification factor for unifying physical dimensions. Based on the gradient projection, the negative eigenvalue, and the global dimensional unification factor, the reliable direction mixing coefficient corresponding to the reliable direction set and the unreliable direction mixing coefficient corresponding to the unreliable direction set are determined respectively. Using the reliable direction mixing coefficient and the unreliable direction mixing coefficient, the corresponding negative eigenvalues ​​in the local Hessian matrix are corrected respectively, and the positive definite Hessian matrix of each element is reconstructed based on the corrected eigenvalues ​​and the eigenvector. The positive definite Hessian matrices of each element are assembled into a new global Hessian matrix, and the Newton step size is determined based on the new global Hessian matrix to update the nodal positions of the hyperelastic material.

2. The method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering according to claim 1, characterized in that, The step of calculating the gradient projection along the direction of the feature vector based on the local gradient and the feature vector corresponding to the negative eigenvalue, and classifying the corresponding feature vector direction into a reliable direction set or an unreliable direction set according to the sign of the gradient projection, includes: For any negative eigenvalue whose absolute value is greater than a preset small positive number, the transpose of its corresponding eigenvector is multiplied by the local gradient to obtain the gradient projection. If the gradient projection is greater than zero, the corresponding feature vector direction is assigned to the reliable direction set. If the gradient projection is not greater than zero, the corresponding feature vector direction is assigned to the unreliable direction set.

3. The method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering according to claim 1, characterized in that, The calculation of the global dimensional unification factor for unifying physical dimensions based on the global system energy and the global Hessian matrix includes: Calculate the square of the Frobenius norm of the global Hessian matrix; The ratio of the global system energy to the square of the Frobenius norm is determined as the global dimensional unification factor.

4. The method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering according to claim 2, characterized in that, The step of determining the reliable direction mixing coefficients corresponding to the reliable direction set and the unreliable direction mixing coefficients corresponding to the unreliable direction set based on the gradient projection, the negative eigenvalues, and the global dimensional unification factor includes: If the set of reliable directions is not empty, calculate the ratio of the square of the gradient projection of each direction in the set of reliable directions to the negative number of the corresponding negative eigenvalue, and sum the ratios to obtain the first aggregation parameter of the reliable direction; calculate the square of the negative number of the corresponding negative eigenvalue of each direction in the set of reliable directions, and sum the squares to obtain the second aggregation parameter of the reliable direction; divide the first aggregation parameter of the reliable direction by twice the product of the global dimensionless factor and the second aggregation parameter of the reliable direction, and determine the cube root of the quotient as the optimal mixing coefficient of the reliable direction, and determine the reliable direction mixing coefficient based on the optimal mixing coefficient of the reliable direction; If the set of unreliable directions is not empty, calculate the ratio of the square of the gradient projection of each direction in the set of unreliable directions to the negative number of the corresponding negative eigenvalue, and sum the ratios to obtain the first aggregation parameter of the unreliable direction; calculate the square of the negative number of the corresponding negative eigenvalue of each direction in the set of unreliable directions, and sum the squares to obtain the second aggregation parameter of the unreliable direction; divide the first aggregation parameter of the unreliable direction by twice the product of the global dimensionless factor and the second aggregation parameter of the unreliable direction, and determine the square root of the quotient as the optimal mixing coefficient of the unreliable direction, and determine the mixing coefficient of the unreliable direction based on the optimal mixing coefficient of the unreliable direction.

5. The method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering according to claim 4, characterized in that, The method further includes: obtaining the element energy of each element of the hyperelastic material in the current Newton iteration step; and calculating, based on the element energy and a preset safety factor, the physical lower bound of the reliable direction corresponding to the reliable direction set and the physical lower bound of the unreliable direction corresponding to the unreliable direction set, including: Obtain the first safety factor for the reliable direction and the second safety factor for the unreliable direction; According to the formula Calculate the physical lower bound of the reliable direction. ,in, The first aggregation parameter for the reliable direction. The first safety factor, The unit energy; According to the formula Calculate the physical lower bound of the unreliable direction. ,in, The first aggregation parameter for the unreliable direction. This is the second safety factor.

6. The method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering according to claim 5, characterized in that, The step of determining the reliable direction mixing coefficient based on the reliable direction optimal mixing coefficient, and determining the unreliable direction mixing coefficient based on the unreliable direction optimal mixing coefficient, includes: Select the larger value between the physical lower bound of the reliable direction and the optimal mixing coefficient of the reliable direction, compare the larger value with 1, and take the smaller value as the mixing coefficient of the reliable direction; Select the larger value between the physical lower bound of the unreliable direction and the optimal mixing coefficient of the unreliable direction, compare the larger value with 1, and take the smaller value as the mixing coefficient of the unreliable direction.

7. A method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering according to any one of claims 1 to 6, characterized in that, The step of using the reliable direction mixing coefficient and the unreliable direction mixing coefficient to correct the corresponding negative eigenvalues ​​in the local Hessian matrix, and reconstructing the positive definite Hessian matrix of each element based on the corrected eigenvalues ​​and the eigenvectors, includes: Calculate the ratio of the sum of the absolute values ​​of all negative eigenvalues ​​in the current unit to the sum of the absolute values ​​of all eigenvalues ​​to determine the proportion of negative eigenvalue energy, and calculate the positive eigenvalue shrinkage coefficient based on the proportion of negative eigenvalue energy. For the negative eigenvalues ​​in the reliable direction set, their corresponding inverses are multiplied by the reliable direction mixing coefficients to obtain the corrected eigenvalues; For the negative eigenvalues ​​in the unreliable direction set, multiply their corresponding inverses by the unreliable direction mixing coefficients to obtain the corrected eigenvalues; For positive eigenvalues ​​greater than the preset small positive number, multiply them by the positive eigenvalue shrinkage coefficient to obtain the corrected eigenvalues; For feature values ​​whose absolute value is not greater than the preset small positive number, they are corrected to the preset small positive number; A diagonal matrix is ​​constructed based on the corrected eigenvalues, and the positive definite Hessian matrix is ​​reconstructed by combining the eigenvectors of the local Hessian matrix.

8. The method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering according to claim 1, characterized in that, Before performing eigenvalue decomposition on the local Hessian matrix, the method further includes: Determine whether the absolute values ​​of the main diagonal elements of the local Hessian matrix are all greater than the sum of the absolute values ​​of the corresponding row's off-diagonal elements and the sum of a preset small positive number; If so, the local Hessian matrix is ​​determined to satisfy the strict diagonal dominance condition. The eigenvalue decomposition and correction steps are skipped, and the local Hessian matrix is ​​directly used as a positive definite Hessian matrix to participate in the assembly of the new global Hessian matrix.

9. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements a method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering as described in any one of claims 1 to 8.

10. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements a method for simulating the deformation of hyperelastic materials based on variational eigenvalue filtering as described in any one of claims 1 to 8.

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