Adaptive positive definite method of hessian matrix in projection newton method for hyperelastic material deformation simulation and related equipment

Abstract

The application discloses a self-adaptive positive definite method of a Hessian matrix in a projection Newton method for super-elastic material deformation simulation and related equipment. The method performs eigenvalue decomposition on a unit Hessian matrix, calculates a negative eigenvalue energy proportion, that is, a sum of absolute values of all negative eigenvalues divided by a sum of absolute values of all eigenvalues; determines a regularization strength parameter according to the proportion, and respectively determines a first correction coefficient of a negative eigenvalue positively correlated with the regularization strength parameter and a second correction coefficient of a positive eigenvalue negatively correlated with the regularization strength parameter; and reconstructs a positive definite Hessian matrix by correcting the negative eigenvalue by using the first correction coefficient and correcting the positive eigenvalue by using the second correction coefficient. The application directly quantifies a unit non-convex degree by using the negative eigenvalue energy proportion, realizes on-demand regularization, and balances a descending speed and numerical stability by independently processing the positive and negative eigenvalues.

Description

Technical Field

[0001] This invention belongs to the field of computer graphics and physical simulation, and specifically relates to an adaptive positive definiteness method and related equipment for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials. Background Technology

[0002] In the fields of computer graphics and physical simulation, the deformation simulation of hyperelastic materials (such as rubber, human soft tissue, and fabric) is often modeled as a nonlinear optimization problem and solved using the projective Newton method. The key to the projective Newton method lies in correcting the eigenvalues ​​of the element Hessian matrix to ensure its positive definiteness, thereby obtaining the descent direction.

[0003] For the problem of non-positive definite Hessian matrices, existing techniques generally employ eigenvalue clamping or absolute value filtering. Eigenvalue clamping sets negative eigenvalues ​​to a very small positive number while leaving positive eigenvalues ​​unchanged; absolute value filtering directly takes the absolute value of negative eigenvalues, preserving the negative curvature amplitude. To balance stability and convergence speed, some techniques propose adaptive filtering based on the trust region, discretely switching between clamping and absolute value filtering according to the model fit.

[0004] The aforementioned existing technologies share the following common drawbacks: On the one hand, regardless of clamping, absolute value filtering, or trust region switching, they all lack direct quantification of the current non-convexity of the element. The regularization strategy does not match the actual deformation state of the element, resulting in slow convergence or numerical instability in nonlinear deformation scenarios such as shearing, torsion, and bending. On the other hand, existing methods use a uniform correction rule for all eigenvalues ​​and fail to set mutually restraining independent correction coefficients for positive and negative eigenvalues, thus making it difficult to simultaneously meet the requirements of rapid descent and condition number control. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention provides an adaptive positive definiteness method and related equipment for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials. The purpose is to directly quantify the non-convexity of the unit Hessian matrix, so that the regularization intensity is adaptively adjusted according to the deformation instability risk, and independent correction coefficients that counterbalance each other are set for positive and negative eigenvalues, thereby improving both convergence speed and numerical stability under nonlinear deformation scenarios such as shear, torsion, and bending.

[0006] 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, an adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials is provided, comprising: Obtain the Hessian matrix of the current unit configuration, and perform eigenvalue decomposition on the Hessian matrix to obtain each eigenvalue and its corresponding eigenvector. Each eigenvalue includes positive eigenvalues ​​and negative eigenvalues. The proportion of negative eigenvalue energy is calculated based on each eigenvalue, and the proportion of negative eigenvalue energy is equal to the sum of the absolute values ​​of all negative eigenvalues ​​divided by the sum of the absolute values ​​of all eigenvalues. The regularization intensity parameter is determined based on the proportion of negative eigenvalue energy, and the regularization intensity parameter increases monotonically as the proportion of negative eigenvalue energy increases. A first correction coefficient for negative eigenvalues ​​and a second correction coefficient for positive eigenvalues ​​are determined based on the regularization strength parameter, wherein the first correction coefficient is positively correlated with the regularization strength parameter and the second correction coefficient is negatively correlated with the regularization strength parameter. Each negative eigenvalue is corrected using the first correction coefficient, and each positive eigenvalue is corrected using the second correction coefficient, resulting in corrected eigenvalues. The positive definite Hessian matrix is ​​reconstructed using the corrected eigenvalues ​​and the eigenvectors.

[0007] In one possible implementation of the first aspect, the calculation of the negative eigenvalue energy proportion based on the respective eigenvalues ​​specifically involves:

[0008] in, The proportion of negative eigenvalues; A negative eigenvalue is defined as a value less than 10 ... eigenvalues, Preset to small positive numbers; For all eigenvalues.

[0009] In one possible implementation of the first aspect, determining the regularization intensity parameter based on the proportion of negative eigenvalue energy specifically includes: The non-convex intensity coefficient is calculated based on the proportion of negative eigenvalue energy, and the calculation method is as follows:

[0010] in, It is a non-convex strength coefficient; Preset sensitivity parameters; The proportion of negative eigenvalues; The regularized strength parameter is calculated based on the non-convex strength coefficient, and the calculation method is as follows:

[0011] in, This is the regularization strength parameter; As the baseline regularization strength, It is a non-convex intensity influence factor.

[0012] In one possible implementation of the first aspect, the first correction coefficient is determined and clamped to the [0,1] interval in the following manner:

[0013] in, This is the first correction factor; The correction of the negative eigenvalues ​​is performed in the following manner:

[0014] in, These are the corrected eigenvalues; It is a negative eigenvalue; It is a preset small positive number.

[0015] In one possible implementation of the first aspect, the second correction coefficient is determined and clamped to the interval [0.5, 1] ​​as follows:

[0016] in, This is the second correction factor; The correction of the positive eigenvalues ​​is performed in the following manner:

[0017] in, These are the corrected eigenvalues; It is a positive eigenvalue; It is a preset small positive number.

[0018] In one possible implementation of the first aspect, before obtaining the Hessian matrix of the current unit configuration and performing eigenvalue decomposition on the Hessian matrix, a pre-judgment is also included: Check if the Hessian matrix satisfies ,in, Represent each diagonal element of the Hessian matrix. Preset to small positive numbers; If the condition is met, skip all subsequent steps and directly use the original Hessian matrix as the positive definite Hessian matrix.

[0019] In one possible implementation of the first aspect, after reconstructing the positive definite Hessian matrix, the method further includes: The positive definite Hessian matrices obtained from the reconstruction of each element are assembled into a global Hessian matrix, and the Newton step is solved. , obtain node increment .

[0020] One possible implementation of the first aspect also includes condition number control: Calculate the condition number of the reconstructed positive definite Hessian matrix. If the condition number exceeds a preset threshold, scale all eigenvalues ​​of the positive definite Hessian matrix as a whole.

[0021] 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 adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating deformation of hyperelastic materials.

[0022] 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 adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials.

[0023] According to a fourth aspect of the present invention, a computer program product is provided, which, when executed by a processor, implements the aforementioned adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating deformation of hyperelastic materials.

[0024] Compared with the prior art, the present invention has at least the following beneficial effects: This invention, for the first time, quantifies the non-convexity of a cell as a scalar index by calculating the ratio of the sum of the absolute values ​​of negative eigenvalues ​​in the Hessian matrix of the cell to the sum of the absolute values ​​of all eigenvalues. This allows the regularization strength to be continuously and adaptively adjusted according to the risk of deformation instability, overcoming the slow convergence or numerical instability problems caused by the lack of direct quantification of the physical deformation state of the cell in existing technologies. Based on this, this invention determines two correction coefficients according to the regularization strength parameter. The coefficient positively correlated with the regularization strength is used to amplify negative eigenvalues ​​to retain descent direction information and accelerate convergence, while the coefficient negatively correlated with the regularization strength is used to moderately reduce positive eigenvalues ​​to improve the condition number, control the step size, and prevent oscillations. This strategy of independently handling positive and negative eigenvalues ​​simultaneously satisfies the requirements of rapid descent and stability, a balance that existing technologies cannot achieve by using a uniform rule for all eigenvalues.

[0025] Furthermore, when the unit Hessian matrix has no negative eigenvalues, the proportion of negative eigenvalue energy is zero. This method does not perform regularization and automatically degenerates into the traditional Newton method, ensuring optimal performance in positive definite scenarios.

[0026] Since the adaptive mechanism of this method relies on the spectral characteristics of the Hessian matrix itself, which include deformation state and material properties, it can achieve better convergence speed and numerical stability than existing technologies for both high and low Poisson's ratio materials under various nonlinear deformation scenarios such as shearing, torsion and bending. Attached Figure Description

[0027] 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.

[0028] Figure 1 The flowchart of an adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials is shown in this invention. Figure 2 Comparison of convergence speed with existing methods under shear deformation scenario (shear strain 0.3); Figure 3 Comparison of convergence speed with existing methods under shear deformation scenarios (shear strain 1.0); Figure 4 Comparison of convergence speed with existing methods under torsional deformation scenarios (relative torsional angle 0.1π); Figure 5 Comparison of convergence speed with existing methods under torsional deformation scenarios (relative torsional angle 0.25π); Figure 6 Comparison of convergence speed with existing methods under bending deformation scenarios (rotation angle 0.1π); Figure 7 Comparison of convergence speed with existing methods under bending deformation scenarios (rotation angle 0.75π); Figure 8 Comparison of convergence speed with existing methods under tensile deformation scenarios (axial length ratio 2.0); Figure 9 Comparison of convergence speed with existing methods under tensile deformation scenarios (axial length ratio 3.0); Figure 10 Comparison of convergence speed with existing methods under compression deformation scenarios (axial length ratio 0.9); Figure 11Comparison of convergence speed with existing methods under compression deformation scenarios (axial length ratio 0.5). Detailed Implementation

[0029] 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.

[0030] like Figure 1 As shown, this embodiment of the invention provides an adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials, specifically including the following steps: S1. Obtain the unit Hessian matrix and perform eigenvalue decomposition. S101. In each Newton iteration step of the deformation simulation, for each element, first obtain the Hessian matrix H of the current element configuration, where matrix H is an n×n symmetric matrix and n is the number of element degrees of freedom.

[0031] S102. As an optional fast judgment, each diagonal element of matrix H can be checked. Does it meet the following conditions:

[0032] in, The preset small positive number has a value range of 10. -10 Up to 10 -6 Recommended: 10 -8 If all diagonal elements satisfy the above equation, then set the correction flag to false and jump to step S7.

[0033] S103. If the condition is not met, the QR algorithm is used to perform eigenvalue decomposition on matrix H to obtain the eigenvalues ​​of matrix H. and the corresponding feature vector .

[0034] If no negative feature value exists, set the correction flag to false and proceed to step S7; if a negative feature value exists, set the correction flag to true.

[0035] S2, Calculate the proportion of negative eigenvalue energy. S201, Traverse all eigenvalues Calculate the proportion of negative eigenvalue energy This proportion equals the sum of the absolute values ​​of all negative eigenvalues ​​divided by the sum of the absolute values ​​of all eigenvalues, i.e.

[0036] in, It is a negative eigenvalue. That is, less than eigenvalues.

[0037] S3. Calculate the non-convex strength coefficient. S301, Calculate the non-convex strength coefficient

[0038] in, The preset sensitivity parameter has a value range of 2 to 6, with 4 being recommended. Non-convex intensity coefficient. The value range is from 0 to 1; when the negative eigenvalue has a higher energy percentage hour ,when hour .

[0039] S4. Determine the regularization strength parameter. S401. Calculate the regularization strength parameter according to the following formula.

[0040]

[0041] in, The baseline regularization strength is 1.6 to 2.4, with 2.0 being recommended. This is a non-convex intensity influence factor, with a value ranging from 1.0 to 3.0, and a recommended value of 2.0. When the non-convex intensity coefficient... hour ,when hour .

[0042] S5, Correcting negative eigenvalues S501, for each less than negative eigenvalues First, calculate the first correction factor. And clamp the coefficient to a closed interval between 0 and 1.

[0043] S502, then according to the formula The negative eigenvalues ​​are corrected to obtain the corrected eigenvalues. .

[0044] S6, Correcting positive eigenvalues S601, for each greater than positive eigenvalues First, calculate the second correction factor. And clamp the coefficient to a closed interval between 0.5 and 1.

[0045] S602, then according to the formula Correcting the positive eigenvalues ​​yields the corrected eigenvalues. .

[0046] S603. For each small eigenvalue, i.e., the absolute value does not exceed... eigenvalues, i.e. Directly set it as ,Right now .

[0047] S7, Reconstruction unit positive definite Hessian matrix S701. If the correction flag is true, then use the corrected eigenvalue. and the original eigenvector The positive definite Hessian matrix of the reconstructed unit: ,in The diagonal matrix formed by the corrected eigenvalues. Original feature vector The matrix formed. If the correction flag is false, then the element-positive definite Hessian matrix is ​​equal to the original Hessian matrix.

[0048] S8. Assemble the global Hessian matrix. S801. Assemble the corrected positive definite Hessian matrices of each element into a global Hessian matrix, and then solve for the Newton step. , obtain node increment .

[0049] The theoretical principles of this invention will be explained below.

[0050] negative eigenvalue energy ratio This directly reflects the non-convexity of the element's Hessian matrix and the extreme degree of element deformation. When When all eigenvalues ​​are positive, it indicates that the element is in an undeformed or mildly deformed state, and the energy landscape is convex. In this case, a conservative strategy, i.e., clamping filtering, should be adopted; when When the value approaches 1, negative eigenvalues ​​dominate, indicating that the cell is in a state of extreme deformation or near-flipping, and the energy landscape is highly non-convex. In this case, an aggressive strategy, namely absolute value filtering, should be adopted. This physical meaning provides a theoretical basis for the adaptive decision-making of this invention, enabling the filtering strategy to be adaptively adjusted according to the actual physical state of the cell.

[0051] Using exponential function Will Mapping to a closed interval between 0 and 1 has the following advantages: (1) When When smaller, right Sensitivity ensures early response; (2) when When it is large, Approaching 1 avoids over-amplification; (3) The exponential form ensures the smooth continuity of the function.

[0052] In this invention, the determination of the correction coefficients is based on a weighted least squares variational problem. While ensuring Hessian positive definiteness, the distance between the correction matrix and the original matrix is ​​minimized, and a regularization term is used to accelerate the descent. This theoretical framework ensures the mathematical optimality of the correction coefficients.

[0053] This invention employs different hybrid strategies for positive and negative eigenvalues: negative eigenvalues ​​vary with the regularization strength parameter. As it increases, it amplifies, reaching up to the original absolute value; positive eigenvalues ​​increase with... The value is increased while moderately contracted, reaching as low as 50% of its original value. This design balances the condition number and descent rate, avoiding the dilemma that a single parameter cannot simultaneously satisfy the requirements for positive and negative eigenvalues. Because positive and negative eigenvalues ​​are processed independently, compared to the three existing methods—clamping filtering, absolute value filtering, and trust region-based adaptive filtering—this method is more effective in shear deformation scenarios. Figure 2 and Figure 3 Twisting and deforming scenes Figure 4 and Figure 5 And bending deformation scenarios Figure 6 and Figure 7 In nonlinear deformation scenarios, the energy decreases the most in a single iteration, and the convergence speed is the fastest; in scenarios with large stretching deformation, i.e. Figure 9 Under these conditions, the energy decrease in a single iteration is the greatest, and the convergence speed is the fastest; in scenarios with a relatively low compression ratio, i.e. Figure 10 Under these conditions, the energy decrease in a single iteration is the greatest, and the convergence speed is the fastest; in scenarios with a large compression ratio, i.e. Figure 11 Below, the stability is the best.

[0054] To better explain this invention, a more detailed explanation is provided below in conjunction with test cases and results.

[0055] The following test cases test the proposed method and three existing Hessian matrix regularization algorithms, namely Clamp, Abs, and Trust-region, on the Ubuntu operating system. The test cases measure the number of Newton iterations and the final energy of each iteration step under different materials and deformation scenarios, in order to evaluate the convergence speed and stability of the proposed method under different materials and deformation scenarios.

[0056] The specific test cases are as follows: (1) Test cases and results of high Poisson's ratio material under shear deformation scenario This test case is used to evaluate the convergence performance of the proposed method under shear deformation scenarios with high Poisson's ratio materials. The specific settings are as follows: The test mesh is a cuboid containing 369 nodes and 990 tetrahedral elements. The material parameters are high Poisson's ratio, with Young's modulus E = 100 MPa and Poisson's ratio ν = 0.495. 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 ​​were 0.3, 0.5, 0.8, 1.0, and 2.0. The recorded metrics were the number of Newton iterations and the final energy at each iteration step. The number of iterations for different algorithms is shown in Table 1, and some energy curves are shown in [Table 1]. Figure 2 and Figure 3 .

[0057] Table 1. Test cases and results for high Poisson's ratio materials under shear deformation scenarios.

[0058] (2) Test cases and results of low Poisson's ratio material under shear deformation scenario This test case is used to evaluate the convergence performance of the proposed method under shear deformation scenarios in low Poisson's ratio materials. The test mesh also uses a cuboid containing 369 nodes and 990 tetrahedral elements. The material parameters are low Poisson's ratio materials, with Young's modulus E = 100 MPa and Poisson's ratio ν = 0.3. The loading method is the same as in the test (1) above. The shear strain of the test The values ​​are 0.3, 0.5, 0.8, 1.0, and 2.0. The number of iterations for different algorithms is shown in Table 2.

[0059] Table 2 Test cases and results for low Poisson's ratio materials under shear deformation scenarios.

[0060] (3) Test cases and results of high Poisson's ratio materials under torsional deformation scenarios This test case evaluates the convergence performance of the proposed method under torsional deformation scenarios with high Poisson's ratio materials. The test mesh is a cuboid with 369 nodes and 990 tetrahedral elements. The material parameters are high Poisson's ratio, with Young's modulus E = 100 MPa and Poisson's ratio ν = 0.495. 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, 0.25π radians, 0.4π radians, 0.5π radians, and 0.8π radians, corresponding to relative torsion angles of 18 degrees, 45 degrees, 72 degrees, 90 degrees, and 144 degrees, respectively. The number of iterations for different algorithms is shown in Table 3, and some energy curves are shown in... Figure 4 and Figure 5 .

[0061] Table 3 Test cases and results for high Poisson's ratio materials under torsional deformation scenarios.

[0062] (4) Test cases and results of low Poisson's ratio material under torsional deformation scenario This test case is used to evaluate the convergence performance of the algorithm in a torsional deformation scenario with a low Poisson's ratio material. The test mesh is a cuboid with 369 nodes and 990 tetrahedral elements. The material parameters are low Poisson's ratio material, where Young's modulus E = 100 MPa and Poisson's ratio ν = 0.3. The loading method is the same as in test (4) above. The number of iterations for different algorithms is shown in Table 4.

[0063] Table 4. Test cases and results for low Poisson's ratio materials under torsional deformation scenarios.

[0064] (5) Test cases and results of high Poisson's ratio materials under bending deformation scenarios This test case evaluates the convergence performance of the proposed method under bending deformation scenarios in high Poisson's ratio materials. The test mesh is a cuboid with 369 nodes and 990 tetrahedral elements. The material parameters are high Poisson's ratio, with Young's modulus E = 100 MPa and Poisson's ratio ν = 0.495. The loading method is to fix the left side (X = 0) and the right side (X = ... A bending motion is applied at the point with the center of gravity as the center and the Z-axis as the rotation axis. The bending angles are 0.1π radians, 0.4π radians, 0.75π radians, and 0.8π radians, corresponding to rotation angles of 18 degrees, 72 degrees, 135 degrees, and 144 degrees, respectively. The number of iterations for different algorithms is shown in Table 5, and some energy curves are shown in... Figure 6 and Figure 7 .

[0065] Table 5. Test cases and results for high Poisson's ratio materials under bending deformation scenarios.

[0066] (6) Test cases and results of low Poisson's ratio material under bending deformation scenarios This test case is used to evaluate the convergence performance of the proposed method under bending deformation scenarios in low Poisson's ratio materials. The test mesh is a cuboid with 369 nodes and 990 tetrahedral elements. The material parameters are low Poisson's ratio material, where Young's modulus E = 100 MPa and Poisson's ratio ν = 0.3. The loading method is the same as in test (5) above. The number of iterations for different algorithms is shown in Table 6.

[0067] Table 6. Test cases and results for low Poisson's ratio materials under bending deformation scenarios.

[0068] (7) Test cases and results of high Poisson's ratio material under tensile deformation scenario This test case evaluates the convergence performance of the proposed method under tensile deformation scenarios in high Poisson's ratio materials. The test mesh is a cuboid with 369 nodes and 990 tetrahedral elements. The material parameters are high Poisson's ratio, with Young's modulus E = 100 MPa and Poisson's ratio ν = 0.495. The loading method is as follows: the left side (X = 0) is fixed, and the right side (X = ...) is fixed. 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 ​​are 2.0, 3.0, 3.5, and 4.0, corresponding to axial tensile rates of 100%, 200%, 250%, and 300%, respectively. The number of iterations for different algorithms is shown in Table 7, and some energy curves are shown in... Figure 8 and Figure 9 .

[0069] Table 7 Test cases and results for high Poisson's ratio materials under tensile deformation scenarios.

[0070] (8) Test cases and results of low Poisson's ratio material under tensile deformation scenarios This test case is used to evaluate the convergence performance of the proposed method under tensile deformation scenarios in low Poisson's ratio materials. The test mesh is a cuboid with 369 nodes and 990 tetrahedral elements. The material parameters are low Poisson's ratio materials, where Young's modulus E = 100 MPa and Poisson's ratio ν = 0.3. The loading method is the same as in the test (7) above. The number of iterations for different algorithms is shown in Table 8.

[0071] Table 8 Test cases and results for low Poisson's ratio materials under tensile deformation scenarios.

[0072] (9) Test cases and results of high Poisson's ratio materials under compressive deformation scenarios This test case evaluates the convergence performance of the proposed method under compressive deformation scenarios with high Poisson's ratio materials. The test mesh is a cube containing 792 nodes and 2911 tetrahedral elements. The material parameters are high Poisson's ratio, with Young's modulus E = 100 MPa and Poisson's ratio ν = 0.495. The loading method is to fix the bottom surface (Y = 0) and the top surface (Y = ... Apply uniaxial displacement ,in The original cube height, The axial length ratio is the compression ratio. The tested axial length ratios were 0.9, 0.8, and 0.5, corresponding to axial compression ratios of 10%, 20%, and 50%, respectively. The number of iterations for different algorithms is shown in Table 9, and some energy curves are shown in... Figure 10 and Figure 11 .

[0073] Table 9. Test cases and results for high Poisson's ratio materials under compressive deformation scenarios.

[0074] (10) Test cases and results of low Poisson's ratio material under compressive deformation scenarios This test case is used to evaluate the convergence performance of the proposed method under compressive deformation scenarios with low Poisson's ratio materials. The test mesh is a cube containing 792 nodes and 2911 tetrahedral elements. The material parameters are low Poisson's ratio, with Young's modulus E = 100 MPa and Poisson's ratio ν = 0.3. The loading method is the same as in the tests described above. The number of iterations for different algorithms is shown in Table 10.

[0075] Table 10 Test cases and results for low Poisson's ratio materials under compressive deformation scenarios.

[0076] The experimental results for each test case are analyzed as follows: (1) Analysis of test case results under shear deformation scenario of high Poisson's ratio material See Table 1. Figure 2 and Figure 3 The test results for shear deformation of high Poisson's ratio materials show that this method has the fewest iterations, the fastest overall convergence speed, the best stability, the lowest single-step iteration energy, and the fastest energy decrease rate among all test cases for shear deformation of high Poisson's ratio materials (especially for large shear deformation). Specifically: 1) For each value of shear strain γ (0.3, 0.5, 0.8, 1.0, 2.0), the number of iterations required by this method is the smallest.

[0077] 2) When the shear strain γ increases from 0.3 to 2.0, the average number of iterations for this method is 9.0, while the average number of iterations for clamping filtering is 16.6, which is 84.4% higher than this method; the average number of iterations for absolute value filtering is 11.6, which is 28.9% higher than this method; and the average number of iterations for trust region filtering is 10.4, which is 15.6% higher than this method. This indicates that the method proposed in this invention has the fastest overall convergence speed.

[0078] 3) The number of iterations increases the slowest and most stable with shear deformation. When γ increases from 0.3 to 2.0, the number of iterations for clamping filtering increases from 8 to 29, an increase of 21, with a standard deviation of 7.94; the number of iterations for absolute value filtering increases from 7 to 17, an increase of 10, with a standard deviation of 3.44; the number of iterations for trust region filtering increases from 6 to 16, an increase of 10, with a standard deviation of 3.44; the number of iterations for this method increases from 6 to 13, an increase of 7, the slowest increase, with a standard deviation of 2.45, the smallest among all methods. This indicates that this method is the least sensitive to the increase of shear strain and has the best stability.

[0079] 4) The advantages are most obvious under large shear deformation. When γ = 0.3, the number of iterations for this method is 6, while the number of iterations for clamping filtering is 8, which is 33.3% higher than this method; the number of iterations for absolute value filtering is 7, which is 16.7% higher than this method; and the number of iterations for trust region filtering is 6, which is the same as this method. When γ = 2.0, the number of iterations for this method is 13, while the number of iterations for clamping filtering is 29, which is 123.1% higher than this method; the number of iterations for absolute value filtering is 17, which is 30.8% higher than this method; and the number of iterations for trust region filtering is 16, which is 23.1% higher than this method. This shows that this method has the fewest iterations and the fastest convergence speed under all shear deformation test cases, and its advantages are more obvious under large shear deformation.

[0080] 5) Minimum energy for single-step iteration (see...) Figure 2 and Figure 3 When γ equals 0.3 and 1.0, the energy of this method after each iteration is lower than that of the other three methods, and the energy at the end of the iteration is basically the same as that of the other three methods.

[0081] 6) The energy decreases the fastest (see...) Figure 2 and Figure 3 Regardless of whether the shear strain is small (γ equals 0.3) or high (γ equals 1.0), the slope of the energy curve of this method is the largest, indicating that its convergence speed is significantly better than the other three methods. The convergence speed of absolute value filtering and trust region filtering is the second fastest, while the convergence speed of clamping filtering energy is the slowest, and its convergence speed in the high shear strain scenario is significantly lower than that in the low shear strain scenario.

[0082] (2) Analysis of test case results under shear deformation scenario of low Poisson's ratio material As can be seen from the test results of shear deformation of low Poisson's ratio materials (see Table 2), the method proposed in this invention has the fewest iterations, the slowest increase in the number of iterations with shear strain, the fastest overall convergence speed, and the best stability in all test cases of shear deformation scenarios of low Poisson's ratio materials.

[0083] Specifically: 1) Minimum number of iterations across all test configurations. For each shear strain γ value (0.3, 0.5, 0.8, 1.0, 2.0), this method requires the fewest iterations.

[0084] 2) The number of iterations increases the slowest with shear strain, resulting in the fastest overall convergence speed and the best stability. When γ increases from 0.3 to 2.0, the number of iterations for this method, absolute value filtering, and trust region filtering increases from 4 to 6, an increase of 2, with a mean of 4.8 (minimum) and a standard deviation of 0.75 (minimum). The clamping filter shows the largest increase in the number of iterations, increasing from 4 to 7, an increase of 3, with a mean of 5.2 and a standard deviation of 1.17.

[0085] (3) Analysis of test case results under torsional deformation scenario of high Poisson's ratio material Test results of torsional deformation of high Poisson's ratio materials (see Table 3) Figure 4 and Figure 5 As can be seen, the method proposed in this invention has the fewest iterations, the fastest overall convergence speed, the best stability, the lowest energy per iteration step, and the fastest energy decrease speed in all test cases of torsional deformation scenarios of high Poisson's ratio materials (especially test cases with medium and large relative torsional angles).

[0086] Specifically: 1) Minimum number of iterations across all test configurations. For every relative torsion angle value (from 0.1π to 0.8π), this method requires the fewest iterations.

[0087] 2) This method has the fewest average iterations and the fastest overall convergence speed. When the relative torsion angle ranges from 0.1π to 0.8π, the average number of iterations for this method is 17.2. The average number of iterations for clamping filtering (22.0) is 27.9% higher, the average number of iterations for absolute value filtering (25.4) is 47.7% higher, and the average number of iterations for trust region filtering (18.6) is 8.1% higher. This indicates that this method has the best average convergence speed and the fastest overall convergence speed.

[0088] 3) The number of iterations increases most slowly with the relative torsion angle, exhibiting the best stability. When the relative torsion angle increases from 0.1π to 0.8π, the clamping filter increases its iteration count from 7 to 43 (an increase of 36), with a standard deviation of 11.87; the absolute value filter increases from 9 to 45 (an increase of 36), with a standard deviation of 12.01; the trust region filter increases from 8 to 31 (an increase of 23), with a standard deviation of 7.74; while our method increases from 7 to 29 (the slowest increase of 22), with a standard deviation of 7.33 (the most stable). This indicates that our method is insensitive to the increase in relative torsion angle and exhibits the best stability.

[0089] 4) The advantages are most obvious under medium and large relative torsion angles. When the relative torsion angle is 0.1π, the number of iterations of this method is 7, the number of iterations of clamping filter (7) is the same, and the number of iterations of absolute value filter (9) is 28.6% more; when the relative torsion angle is 0.25π, the number of iterations of this method is 13, the number of iterations of clamping filter (16) is 23.1% more, and the number of iterations of absolute value filter (18) is 38.5% more; when the relative torsion angle is 0.8π (greater than 0.25π), the number of iterations of this method is 29, the number of iterations of clamping filter (43) is 48.3% more, and the number of iterations of absolute value filter (45) is 55.2% more. This shows that under medium and large relative torsion angles (0.25π and above), the advantage of this method in terms of convergence speed is more obvious than that under small relative torsion angles (0.1π).

[0090] 5) Minimum energy for single-step iteration (see...) Figure 4 and Figure 5 When the relative twist angle is 0.1π and 0.25π, the energy of this method after each iteration is lower than that of the other three methods, and the energy at the end of the iteration is basically the same as that of the other three methods.

[0091] 6) The energy decreases the fastest (see...) Figure 4 and Figure 5 At a relative torsion angle of 0.1π (small torsion angle), the slope of the energy curve of this method is similar to that of the other three methods. However, at a relative torsion angle of 0.25π (slightly larger than 0.1π, a medium torsion angle), the slope of the energy curve of this method is significantly greater than that of the other three methods, indicating that its advantage in energy reduction speed gradually becomes more prominent as the torsion angle increases. The convergence speed of absolute value filtering and trust region filtering is the next fastest, while the convergence speed of clamping filtering is the slowest, and its convergence speed in the scenario with a medium relative torsion angle (0.25π) is significantly lower than that in the scenario with a small relative torsion angle (0.1π).

[0092] (4) Analysis of test case results under torsional deformation scenario of low Poisson's ratio material As shown in Table 4, the test results of torsional deformation of low Poisson's ratio materials show that the proposed method has the fewest iterations, the slowest increase in the number of iterations with the relative torsion angle, the fastest overall convergence speed, and the best stability in all test cases of torsional deformation scenarios of low Poisson's ratio materials.

[0093] Specifically: 1) Minimum number of iterations across all test configurations. This method minimizes the number of iterations for every relative torsion angle (from 0.1π to 0.8π).

[0094] 2) The number of iterations increases slowest with the relative torsion angle, resulting in the fastest overall convergence speed and best stability. When the relative torsion angle increases from 0.1π to 0.8π, the number of iterations for clamping filtering increases from 4 to 18, a slow increase of 14, with a mean of 9.2 and a standard deviation of 4.83; the number of iterations for absolute value filtering increases from 4 to 28, a slow increase of 24, with a mean of 12.4 and a standard deviation of 8.36; the number of iterations for trust region filtering increases from 4 to 19, a slow increase of 15, with a mean of 9.8 and a standard deviation of 5.04; the number of iterations for the method proposed in this invention increases from 4 to 18, a slow increase of 14 (tied for slowest), with a mean of 9.0 (fastest overall) and a standard deviation of 4.82 (most stable).

[0095] (5) Analysis of test case results under bending deformation scenario of high Poisson's ratio material Test results of high Poisson's ratio materials under bending deformation scenarios (see Table 5) Figure 6 and Figure 7 As can be seen, this method has the best overall performance in the bending deformation scenario of high Poisson's ratio materials. In terms of convergence speed, stability and robustness, it is comparable to clamping filtering and significantly better than absolute value filtering and clamping filtering. In terms of energy, the descent speed is the best overall.

[0096] Specifically: 1) Minimum number of iterations across all test configurations. For every rotation angle value (from 0.1π to 0.8π), this method has the minimum number of iterations, and at a rotation angle of 0.4π, it is tied for the minimum with clamping filtering.

[0097] 2) The average number of iterations for all test cases is the fewest, and the overall convergence speed is the fastest. When the rotation angle increases from 0.1π to 0.8π, the average number of iterations for this method is 21.0; clamping filtering (26.5) has 26.2% more, absolute value filtering (29.0) has 38.1% more, and trust region filtering (24.0) has 14.3% more.

[0098] 3) The number of iterations increases slowly with the rotation angle, indicating good stability. When the rotation angle increases from 0.1π to 0.8π: the number of iterations for clamping filtering decreases from 26 to 16, then increases to 35 (an increase of 9), with a standard deviation of 6.87 (the lowest); the number of iterations for trust region filtering increases from 15 to 34 (an increase of 19), with a standard deviation of 7.45; the number of iterations for absolute value filtering increases from 18 to 41 (an increase of 23), with a standard deviation of 9.03; the number of iterations for our method increases from 13 to 31 (an increase of 18), with a standard deviation of 7.04 (the second lowest, only after clamping filtering). This shows that our method changes relatively smoothly with the increase of the rotation angle, and its stability is close to that of clamping filtering (the lowest standard deviation, the most stable), outperforming trust region filtering and absolute value filtering, which have larger fluctuations in the number of iterations.

[0099] 4) Minimum energy for single-step iteration (see...) Figure 6 and Figure 7 When the rotation angle is 0.1π and 0.75π, the energy of this method after the 4th iteration is lower than that of the other three methods, and the energy at the end of the iteration is basically the same as that of the other three methods. In application scenarios where the number of iterations is fixed and greater than 4, this method has the fastest energy decrease among the three methods.

[0100] 5) The energy decreases relatively quickly (see...) Figure 6 and Figure 7 At a rotation angle of 0.1π (small rotation angle), the energy curve of this method decreases smoothly, with a slope significantly greater than the other three methods; the energy curves of the trust region method and the absolute value method decrease smoothly; the energy curve of the clamping filter is not smooth, and the rate of decrease slows significantly from the 3rd iteration. At a rotation angle of 0.75π (large rotation angle), the energy curve of this method is smooth, and the slope is generally greater than the other three methods; in the first 10 iterations, the slope of this method is significantly greater than that of the trust region filter, and then the slope is basically the same as that of the trust region filter; the energy curve of the clamping filter is not smooth, and the rate of decrease slows significantly from the 5th iteration, and the slope is also significantly lower than that of this method. This indicates that, at different rotation angles, the energy decrease rate of this method is generally better than the other three methods.

[0101] (6) Analysis of test case results under bending deformation scenario of low Poisson's ratio material The test results under bending deformation scenarios of low Poisson's ratio materials (see Table 6) show that the proposed method has the fewest iterations and the number of iterations increases slowly with the relative torsion angle in all test cases of bending deformation of low Poisson's ratio materials, resulting in the best overall performance and good stability.

[0102] Specifically: 1) Minimum number of iterations across all test configurations. For each rotation angle value (0.1π, 0.4π, 0.75π, and 0.8π), this method has the minimum number of iterations. At a rotation angle of 0.1π, this method has the same minimum number of iterations as absolute value filtering and trust region filtering; at rotation angles of 0.75π and 0.8π, this method has the same minimum number of iterations as clamping filtering.

[0103] 2) The number of iterations increases slowly with the relative torsion angle, resulting in the best overall performance and good stability. When the rotation angle increases from 0.1π to 0.8π, the number of iterations for clamping filtering increases by 1 from 8 to 9, with a mean of 8.25 and a standard deviation of approximately 0.43 (the lowest); the number of iterations for absolute value filtering increases by 6 from 6 to 12, with a mean of 9.25 and a standard deviation of approximately 2.36; the number of iterations for trust region filtering increases by 6 from 6 to 12, with a mean of 8.75 and a standard deviation of approximately 1.92; the number of iterations for our method increases by 3 from 6 to 9, with a mean of 7.25 (the lowest) and a standard deviation of approximately 1.3 (the second lowest). This indicates that our method has the lowest average value and the best overall performance; clamping filtering is the most stable, followed by our method; absolute value filtering and trust region filtering exhibit the most significant fluctuations.

[0104] (7) Analysis of test case results under tensile deformation scenario of high Poisson's ratio material Test results of high Poisson's ratio material under tensile deformation scenarios (see Table 7) Figure 8 and Figure 9 As can be seen, the method proposed in this invention has the fastest overall convergence speed in all test cases of tensile deformation of high Poisson's ratio materials; it has good stability, second only to trust region filtering; in the case of large tensile scenarios, the convergence speed advantage is more obvious, the single-step iteration energy is smaller, and the energy decreases faster.

[0105] Specifically: 1) This method has the fewest or second-fewest iterations across all test configurations. With an axial length ratio of 2.0, this method has the second-fewest iterations, second only to clamping filtering; at other larger axial length ratios (3.0, 3.5, 4.0), this method has the fewest iterations.

[0106] 2) The method proposed in this invention has the fewest average iterations and the fastest overall convergence speed under all test configurations. The average number of iterations of the proposed method is 20.5, which is 70.7% higher than that of clamping filtering (35.0), 45.1% higher than that of absolute value filtering (29.75), and 6.1% higher than that of trust region filtering (21.75).

[0107] 3) The advantages are most obvious in the case of large stretching. When the axial length ratio is 3.0, the number of iterations of this method is 17, the number of iterations of clamping filter (36) is 111.8% higher, the number of iterations of absolute value filter (27) is 58.8% higher, and the number of iterations of trust region filter (19) is 11.8% higher; when the axial length ratio is 4.0, the number of iterations of this method is 26, the number of iterations of clamping filter (42) is 61.5% higher, the number of iterations of absolute value filter (42) is 61.5% higher, and the number of iterations of trust region filter (27) is 3.8% higher.

[0108] 4) The number of iterations increases slowly and is relatively stable with the axial length ratio. The number of iterations for clamping filtering increased from 12 to 50 (dropping back to 42 at a length ratio of 4.0), an increase of 38, with a standard deviation of 14.18; the number of iterations for absolute value filtering increased from 19 to 42 (decreasing slightly at a length ratio of 3.5 before rebounding), an increase of 23, with a standard deviation of 8.29; the number of iterations for trust region filtering increased from 14 to 27, an increase of 13 (the smallest), with a standard deviation of 5.54 (the smallest); the number of iterations for this method increased from 13 to 26, an increase of 13 (tied for the smallest), with a standard deviation of 5.68 (the second smallest), second only to trust region filtering. This indicates that the number of iterations for this method increases the slowest with increasing tensile deformation, making the method relatively stable.

[0109] 5) The energy of a single iteration is relatively small (see...) Figure 8 and Figure 9 Especially under large stretching. When the axial length ratio is 2.0, the energy of this method after each iteration is lower than that of absolute value filtering and trust region filtering, and slightly higher than that of clamping filtering. The energy at the end of the iteration is basically the same as that of the other three methods. When the axial length ratio is 3.0, the energy of this method after almost each iteration is lower than that of the other three methods. The energy at the end of the iteration is basically the same as that of the other three methods. Only in the first 3 iterations is the energy slightly higher than that of clamping filtering.

[0110] 6) The energy decreases relatively quickly (see...) Figure 8 and Figure 9 Especially in scenarios with large stretching, the energy decay rate is optimal overall. When the axial length ratio is 2.0 (stretch rate 100%), the slope of the energy curve of this method is greater than that of the trust region method and the absolute value method, and less than that of the clamping filter in the first 7 iterations. When the axial length ratio is 3.0 (stretch rate 200%), the slope of the energy curve of this method is generally greater than the other three methods, only slightly lower than that of the clamping filter in the first 3 iterations. This indicates that as the stretching increases, the energy decay rate advantage of this method gradually becomes more prominent, with the absolute value filter and trust region filter showing the next fastest decay rates. The clamping filter, on the other hand, shows the opposite: its energy decay rate is fastest at an axial length ratio of 2.0, but the slowest overall decay rate at an axial length ratio of 3.0.

[0111] (8) Analysis of test case results under tensile deformation scenario of low Poisson's ratio material As shown in Table 8, the test results of the low Poisson's ratio material under tensile deformation scenarios show that the proposed method has the fewest iterations, the fastest overall convergence speed, and the best stability across all test cases of low Poisson's ratio material tensile deformation.

[0112] Specifically: 1) Minimum number of iterations across all test configurations. For each stretch ratio value (2.0, 3.0, 3.5, 4.0), this method consistently requires the fewest iterations. At a stretch ratio of 2.0, this method has the same number of iterations as the other three methods, tying for the fewest. At stretch ratios of 3.0, 3.5, and 4.0, this method's number of iterations is the same as clamping filtering, also tying for the fewest.

[0113] 2) The number of iterations increases slowest with the relative torsion angle, resulting in the fastest overall convergence speed and best stability. When the stretching ratio increases from 2.0 to 4.0, the clamping filter increases its iteration count by 2 (minimum) from 6 to 8, with a mean of 7.25 (minimum) and a standard deviation of 0.83 (minimum); the absolute value filter increases its iteration count by 5 (minimum) from 6 to 11, with a mean of 9.00 and a standard deviation of 1.87; the trust region filter increases its iteration count by 4 (minimum) from 6 to 10, with a mean of 8.25 and a standard deviation of 1.48; the method in this case increases its iteration count by 2 (tied for minimum) from 6 to 8, with a mean of 7.25 (tied for minimum) and a standard deviation of 0.83 (tied for minimum). This method has the same minimum mean and variance as the clamping filter, indicating that its convergence speed and stability are superior to both the absolute value filter and the trust region filter.

[0114] (9) Analysis of test case results under compression deformation scenario of high Poisson's ratio material Test results of high Poisson's ratio materials under compression deformation scenarios (see Table 9) Figure 10 and Figure 11 As can be seen, the method proposed in this invention has a smaller number of iterations and better overall performance in all test cases of compression deformation of high Poisson's ratio materials. The number of iterations decreases more slowly and is more stable as the axial length ratio decreases. The single-step iteration energy is smaller and the energy decrease rate is faster. When the compression deformation is small, the number of iterations is the fewest, the single-step iteration energy is the smallest, and the energy decrease rate is the fastest.

[0115] Specifically: 1) Under all test configurations, the method requires the fewest iterations when the compression deformation is small; when the compression deformation is large, the number of iterations is second only to clamping filtering. When the axial length ratio is 0.9 and 0.8, this method requires the fewest iterations; when the axial length ratio is 0.5, this method requires the second fewest iterations, second only to clamping filtering.

[0116] 2) The average number of iterations is relatively small, second only to clamp filtering, and the overall performance is good. The average number of iterations of this method is 0.67. The average number of iterations of absolute value filtering (14.33) is 34.3% higher, the average number of iterations of trust region filtering (12.0) is 12.5% ​​higher, and the average number of iterations of clamp filtering (9.67) is 9.4% lower. Therefore, the average number of iterations of this method is the second smallest, second only to clamp filtering.

[0117] 3) The number of iterations increases slowly and is relatively stable as the axial length ratio decreases. The clamping filter's iteration count increases by 10 (minimum) from 6 to 16, with a standard deviation of 4.5 (minimum); the absolute value filter's iteration count increases by 23 (minimum) from 6 to 29, with a standard deviation of 10.4; the trust region filter's iteration count increases by 16 (minimum) from 6 to 22, with a standard deviation of 7.12; the method in this case increases by 15 (second smallest) from 5 to 20, with a standard deviation of 6.65 (second smallest), second only to the trust region filter. This indicates that the number of iterations in this method increases slowly with increasing compressive deformation, making the method relatively stable.

[0118] 4) The energy of a single iteration is relatively small (see...) Figure 10 and Figure 11 When the axial length ratio is 0.9, the energy of this method after each iteration is lower than that of the other three methods, and the energy at the end of the iteration is basically the same as that of the other three methods; when the axial length ratio is 0.5, the energy of this method after almost each iteration is lower than that of absolute value filtering and trust region filtering, and slightly higher than that of clamping filtering.

[0119] 5) The energy decreases relatively quickly, with the fastest energy decrease occurring during small compression deformation (see...). Figure 10 For larger compressive deformations, the energy decrease rate is the second best (see...). Figure 11 When the axial length ratio is 0.9 (compression rate of 10%), the slope of the energy curve of this method is greater than that of the other three methods; when the axial length ratio is 0.5 (compression rate of 50%), the slope of the energy curve of this method is greater than that of absolute value filtering and trust region filtering, and second only to clamp filtering.

[0120] (10) Analysis of test case results under compression deformation scenario of low Poisson's ratio material The test results under the compression deformation scenario of low Poisson's ratio material (see Table 10) show that the method has fewer iterations in all test cases of compression deformation of low Poisson's ratio material, the number of iterations increases the slowest as the axial length ratio decreases, the overall convergence speed is faster, and the stability is better; the number of iterations is the fewest when the compression deformation is small.

[0121] Specifically: 1) Under all test configurations, the number of iterations is the fewest when the compression deformation is small; the number of iterations is less when the compression deformation is large (see Table 10). When the stretch ratio is 0.9 and 0.8, the number of iterations of this method is the fewest; when the stretch ratio is 0.5, the number of iterations of this method is less, only greater than that of clamping filtering.

[0122] 2) The number of iterations increases most slowly with decreasing stretching ratio, resulting in faster overall convergence and better stability. The mean and variance of the number of iterations are both small, indicating good stability. When the stretching ratio decreases from 0.9 to 0.5, the number of iterations for clamping filtering increases from 4 to 20, with a mean of 9.67 (minimum) and a standard deviation of 7.93 (minimum); the number of iterations for absolute value filtering increases from 4 to 51, with a mean of 20.33 and a standard deviation of 23.64; the number of iterations for trust region filtering increases from 4 to 33, with a mean of 14.33 and a standard deviation of 14.52 (second smallest); and the number of iterations for this method increases from 4 to 32, with a mean of 13.67 (second smallest) and a standard deviation of 14.52 (tied for second smallest). This demonstrates that this method has good convergence speed and stability, second only to clamping filtering and superior to absolute value filtering and trust region filtering.

[0123] In summary: Based on the test cases and results of high Poisson's ratio materials under various test scenarios, this method exhibits the following characteristics: 1) In shear, torsion, and bending scenarios, it has the fewest iterations, the fastest overall convergence speed, and the best stability (especially in scenarios with large shear deformation, medium and large torsion deformation), the lowest single-step iteration energy, and the fastest energy decrease rate; 2) In stretching scenarios, it has the fastest overall convergence speed, fewer iterations (second only to clamping filtering), good stability (second only to trust region filtering), lower single-step iteration energy (higher than clamping filtering when stretching is small), and a faster energy decrease rate (iteration). (Initially slower than clamping filter); when the stretch length ratio is particularly large, it has the fewest iterations, the smallest single-step iteration energy, and the fastest energy decrease rate; 3) In compression scenarios, it has fewer iterations (second only to clamping filter), faster overall convergence speed (second only to clamping filter), better stability (second only to clamping filter), smaller single-step iteration energy (higher than clamping filter when compression deformation is large), and faster energy decrease rate (slower than clamping filter when compression deformation is large); when compression deformation is small, it has the fewest iterations, the smallest single-step iteration energy, and the fastest energy decrease rate.

[0124] Based on the test cases and results of low Poisson's ratio materials under various test scenarios, this method has the fewest iterations, the fastest overall convergence speed, and the best stability in shear, torsion, and tension scenarios; in bending scenarios, it has the fewest iterations, the fastest overall convergence speed, and good stability (second only to clamping filtering); in compression scenarios, it has fewer iterations (second only to clamping filtering), faster overall convergence speed (second only to clamping filtering), and good stability (second only to clamping filtering); and it has the fewest iterations when the compression deformation is small.

[0125] From the energy curve, the method proposed in this invention has the minimum single-step iteration energy and the fastest energy decrease rate in scenarios of shearing, twisting, bending, large stretching and small compression. Therefore, it is not only suitable for interactive application scenarios with a fixed number of iterations, such as object deformation simulation in computer graphics, but also for non-interactive application scenarios with energy as the iteration termination threshold, such as object deformation simulation in movies.

[0126] In summary, this invention outperforms the other three methods in most scenarios involving high and low Poisson's ratio materials, and can be applied to deformation simulation of complex objects and scenes.

[0127] Based on the above basic implementation methods, those skilled in the art can select the following preferred implementation methods according to actual application scenarios.

[0128] Example 1: Implementation of Basic Parameters This embodiment follows the basic implementation steps described above, with specific parameter settings as follows: baseline regularization strength. The value is set to 2.0, representing the non-convex intensity influence factor. Set the sensitivity parameter to 2.0. Take 4.0, preset as a small positive number. Pick This parameter configuration is suitable for most hyperelastic material deformation simulation scenarios.

[0129] Example 2: Implementation method with pre-judgment This embodiment adds a preliminary judgment step before calculating the non-convex strength coefficient. Specifically, it first calculates the minimum eigenvalue of the element Hessian matrix. If the minimum eigenvalue is greater than a preset small positive number If the calculation of the nonconvex intensity coefficient, regularized intensity parameter, corrected negative eigenvalue, and corrected positive eigenvalue is skipped (steps S3 to S6), the original Hessian matrix is ​​used directly; if the minimum eigenvalue is less than or equal to a preset small positive number... And non-convex strength coefficient Less than the non-convex strength coefficient threshold If the calculation of the regularization intensity parameter, the correction of negative eigenvalues, and the correction of positive eigenvalues ​​are skipped (steps S4 to S6), the original Hessian matrix is ​​used directly. This implementation is suitable for scenarios with small deformations and can reduce unnecessary computational overhead.

[0130] Example 3: Implementation method with condition number control This embodiment adds a condition number control step after determining the regularization strength parameter. Specifically, it calculates the condition number of the corrected Hessian matrix, which is the ratio of the largest eigenvalue to the smallest eigenvalue. . If condition number Then, all feature values ​​are scaled as a whole: This implementation method is suitable for extreme deformation scenarios and can improve numerical stability.

[0131] Example 4: Implementation method including online parameter adjustment This embodiment adds an online parameter adjustment step after each Newton iteration. Specifically, it calculates the trust region ratio. .when At that time, Update to 0.9 times the original value, that is ,when At that time, Updated to 1.02 times the original value, that is and restrictions This implementation combines the advantages of real-time spectral information and historical optimization feedback information.

[0132] Example 5: Simplified Implementation of Real-Time Simulation This embodiment uses a simplified formula to determine the first and second correction coefficients, specifically: , The influence of weights is ignored. This implementation method is suitable for real-time simulation scenarios with high computational speed requirements.

[0133] 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 to operate an adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials.

[0134] 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 adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials in the above embodiments.

[0135] 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.

[0136] 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.

[0137] 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.

[0138] 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.

[0139] This invention also provides a computer program product for executing any of the above-described adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials. Since the computer program product provided by this invention belongs to the same inventive concept as the above-described adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials, the computer program product provided by this invention possesses all the advantages of the above-described adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials. Therefore, the beneficial effects of the computer program product provided by this invention will not be elaborated upon here.

[0140] 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.

[0141] 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. An adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials, characterized in that, include: Obtain the Hessian matrix of the current unit configuration, and perform eigenvalue decomposition on the Hessian matrix to obtain each eigenvalue and its corresponding eigenvector. Each eigenvalue includes positive eigenvalues ​​and negative eigenvalues. The proportion of negative eigenvalue energy is calculated based on each eigenvalue, and the proportion of negative eigenvalue energy is equal to the sum of the absolute values ​​of all negative eigenvalues ​​divided by the sum of the absolute values ​​of all eigenvalues. The regularization intensity parameter is determined based on the proportion of negative eigenvalue energy, and the regularization intensity parameter increases monotonically as the proportion of negative eigenvalue energy increases. A first correction coefficient for negative eigenvalues ​​and a second correction coefficient for positive eigenvalues ​​are determined based on the regularization strength parameter, wherein the first correction coefficient is positively correlated with the regularization strength parameter and the second correction coefficient is negatively correlated with the regularization strength parameter. Each negative eigenvalue is corrected using the first correction coefficient, and each positive eigenvalue is corrected using the second correction coefficient, resulting in corrected eigenvalues. The positive definite Hessian matrix is ​​reconstructed using the corrected eigenvalues ​​and the eigenvectors.

2. The adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials according to claim 1, characterized in that, The calculation of the negative eigenvalue energy ratio based on each eigenvalue is specifically as follows: in, The proportion of negative eigenvalues; A negative eigenvalue is defined as a value less than 10 ... eigenvalues, Preset to small positive numbers; For all eigenvalues.

3. The adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials according to claim 1, characterized in that, The step of determining the regularization intensity parameter based on the proportion of negative eigenvalue energy specifically includes: The non-convex intensity coefficient is calculated based on the proportion of negative eigenvalue energy, and the calculation method is as follows: in, It is a non-convex strength coefficient; Preset sensitivity parameters; The proportion of negative eigenvalues; The regularized strength parameter is calculated based on the non-convex strength coefficient, and the calculation method is as follows: in, This is the regularization strength parameter; As the baseline regularization strength, It is a non-convex intensity influence factor.

4. The adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials according to claim 3, characterized in that, The first correction coefficient is determined and clamped to the [0,1] interval in the following manner: in, This is the first correction factor; The correction of the negative eigenvalues ​​is performed in the following manner: in, These are the corrected eigenvalues; It is a negative eigenvalue; It is a preset small positive number.

5. The adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials according to claim 3, characterized in that, The second correction factor is determined and clamped to the [0.5, 1] ​​interval as follows: in, This is the second correction factor; The correction of the positive eigenvalues ​​is performed in the following manner: in, These are the corrected eigenvalues; It is a positive eigenvalue; It is a preset small positive number.

6. The adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials according to claim 1, characterized in that, Before obtaining the Hessian matrix of the current unit configuration and performing eigenvalue decomposition on the Hessian matrix, a pre-judgment is also included: Check if the Hessian matrix satisfies ,in, Represent each diagonal element of the Hessian matrix. Preset to small positive numbers; If the condition is met, skip all subsequent steps and directly use the original Hessian matrix as the positive definite Hessian matrix.

7. The adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials according to claim 1, characterized in that, After reconstructing the positive definite Hessian matrix, the following is also included: The positive definite Hessian matrices obtained from the reconstruction of each element are assembled into a global Hessian matrix, and the Newton step is solved. , obtain node increment .

8. The adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials according to claim 1, characterized in that, It also includes condition number control: Calculate the condition number of the reconstructed positive definite Hessian matrix. If the condition number exceeds a preset threshold, scale all eigenvalues ​​of the positive definite Hessian matrix as a whole.

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 an adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials, 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 an adaptive positive definite method for the Hessian matrix in the projective Newton method for simulating the deformation of hyperelastic materials, as described in any one of claims 1 to 8.

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