Implementation method of integral transformation fracture phase field model of random-order polynomial cohesion law
By developing a finite element subroutine and utilizing polynomial fitting and integral transform techniques, we have achieved simulation calculations of the fracture phase field model of an arbitrary-order polynomial cohesion law using integral transform. This solves the endpoint singularity problem that is difficult to achieve in existing technologies and improves the accuracy and efficiency of the calculations.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- BEIJING INST OF TECH
- Filing Date
- 2026-01-19
- Publication Date
- 2026-05-01
AI Technical Summary
Existing integral transform phase field models are difficult to implement in general-purpose finite element software, which has difficulty in realizing the energy density eigenfunctions and the first two derivatives of polynomial cohesion laws of arbitrary order. Furthermore, there is the problem of endpoint singularity, which makes it difficult to directly realize fracture phase field models within the framework of general-purpose finite element software.
A finite element subroutine was developed to approximate the cohesion curve using a polynomial fitting method. The cohesive eigenfunctions of the cohesion law of arbitrary order polynomials were derived, and the analytical expression of the conjugate energy density eigenfunctions was obtained based on integral transform. The McLaurin series was used to remove endpoint singularities, and the expansion expressions of the target order polynomial energy density eigenfunctions and their first two derivatives were realized.
Simulation calculations of the integral transform fracture phase field model of polynomial cohesion law of arbitrary order were realized under the general finite element framework, which improved the accuracy and efficiency of the calculation and solved the endpoint singularity problem.
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Abstract
Description
Technical Field
[0001] This invention relates to the field of finite element analysis technology for phase field models, and in particular to a method for implementing an integral transform fracture phase field model based on the cohesion law of arbitrary order polynomials. Background Technology
[0002] Phase-field damage models are very popular in computational modeling of fracture problems. This popularity is mainly due to the ability of the fracture phase-field method to describe the initiation, propagation, bifurcation, or fusion of complex two-dimensional or three-dimensional crack configurations without requiring any phenomenological fracture criteria or crack tracking algorithms like the extended finite element method. In existing literature, most models are only applicable to brittle fracture, such as the widely used AT-1 and AT2 models. The simulation methods of these two phase-field models depend on the mesh scale; different mesh scales result in different simulation accuracies. Therefore, integral transform phase-field models have emerged. However, existing integral transform phase-field cohesive fracture models are difficult to implement in general-purpose finite element software, even when dealing with relatively simple exponential cohesion laws, let alone complex cohesion curves such as arbitrary-order polynomial cohesion laws. Finite element implementations require solving the energy density eigenfunctions and their first two derivatives of arbitrary-order polynomial cohesion laws. Furthermore, the integral transform phase-field method suffers from endpoint singularities; that is, when the phase field value is 0 or 1, the integration interval may degenerate to 0 or the integrand may exhibit singularities. Existing general-purpose finite element software cannot find the energy density eigenfunctions of cohesion laws of arbitrary-order polynomials, their first two derivatives, and solve the singularity problem, making it difficult to directly implement the integral transform phase field model within the framework of general-purpose finite elements.
[0003] Therefore, there is an urgent need to provide a method for implementing an integral transform fracture phase field model of an arbitrary-order polynomial cohesion law. Summary of the Invention
[0004] This invention provides a method for implementing an integral transform fracture phase-field model of an arbitrary-order polynomial cohesion law. The technical solution is as follows: On the one hand, a method for implementing an integral transform fracture phase field model of an arbitrary-order polynomial cohesion law is provided, which is applied to finite element analysis software. The method includes: Obtain a subroutine for solving the energy density eigenfunction expression of an arbitrary-order polynomial and its first two derivatives, and input the parametric equation of the cohesion law of the arbitrary-order polynomial and the target order into the subroutine; The cohesive eigenfunctions of the cohesion law of an arbitrary-order polynomial are derived using the subroutine. After obtaining the analytical expression of the energy density eigenfunction of the arbitrary-order polynomial conjugate with the cohesive eigenfunction based on the integral transform, the expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives are solved. The Maclaurin series is then used to determine the expansion expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives at the zero point to remove endpoint singularities. The expression for the energy density eigenfunction of the target polynomial and its first two derivatives, as well as the expansion expression of the energy density eigenfunction of the target polynomial and its first two derivatives at zero, are input into the solver used for the integral transform fracture phase-field model to obtain the simulation calculation results of the integral transform fracture phase-field model of the target polynomial cohesion law.
[0005] On the other hand, an integral transform fracture phase field model realization device for polynomial cohesion of arbitrary order is provided, which is installed in finite element analysis software. The device includes: The acquisition unit is used to acquire a subroutine for solving the energy density eigenfunction expression of an arbitrary-order polynomial and its first two derivatives, and inputs the parametric equation of the cohesion law of the arbitrary-order polynomial and the target order into the subroutine. The solution unit is used to derive the cohesive eigenfunction of the cohesion law of an arbitrary-order polynomial using the subroutine, and after obtaining the analytical expression of the energy density eigenfunction of the arbitrary-order polynomial conjugate with the cohesive eigenfunction based on the integral transform, solve the expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives, and use the Maclaurin series to determine the expansion expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives at the zero point to remove endpoint singularities. The simulation unit is used to input the expression of the energy density eigenfunction of the target polynomial and its first two derivatives, as well as the expansion expression of the energy density eigenfunction of the target polynomial and its first two derivatives at zero points, into the solver for the integral transform fracture phase field model, so as to obtain the simulation calculation results of the integral transform fracture phase field model of the target polynomial cohesion law.
[0006] On the other hand, a computer device is provided, the computer device including a memory and a processor, the memory for storing computer programs, and the processor for executing the computer programs stored in the memory to implement the steps of the network anomaly detection method based on data analysis described above.
[0007] On the other hand, a computer-readable storage medium is provided, wherein a computer program is stored therein, and when the computer program is executed by a processor, it implements the steps of the network anomaly detection method based on data analysis described above.
[0008] On the other hand, a computer program product is provided, including a computer program that, when executed by a processor, implements the steps of the network anomaly detection method based on data analysis described above.
[0009] The technical solution provided by this invention can bring at least the following beneficial effects: A finite element subroutine was developed, capable of approximating the original cohesion curve with arbitrary precision based on polynomial fitting, deriving the cohesive eigenfunction of the cohesion law fitted by an arbitrary-order polynomial, and providing the analytical expression of the energy density eigenfunction of the arbitrary-order polynomial conjugate with the cohesive eigenfunction based on integral transform. On this basis, the expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives are solved. Furthermore, the Maclaurin series is used to determine the expansion expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives at zero points to remove endpoint singularities. This improves the theoretical framework of the finite element solution, enabling the simulation calculation of the fracture phase field model of the cohesion law of the target-order polynomial under the general finite element framework, greatly improving accuracy and computational efficiency. Attached Figure Description
[0010] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art 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 based on these drawings without creative effort.
[0011] Figure 1 This is a flowchart of a method for implementing an integral transform fracture phase field model of an arbitrary-order polynomial cohesion law according to an embodiment of the present invention. Figure 2 This is a schematic diagram of Cornelissen's cohesive intrinsic fracture law provided in an embodiment of the present invention; Figure 3 This is a normalized cohesion curve result diagram with a mesh size of 0.25 mm provided in an embodiment of the present invention; Figure 4 This is a normalized cohesion curve result diagram with a grid size of 0.1 mm provided in an embodiment of the present invention; Figure 5 This is a structural diagram of an apparatus for realizing an integral transform fracture phase field model of an arbitrary-order polynomial cohesion law according to an embodiment of the present invention. Figure 6 This is a hardware architecture diagram of a computer device provided in an embodiment of the present invention. Detailed Implementation
[0012] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are some embodiments of the present invention, but not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.
[0013] Please refer to Figure 1 This invention provides a method for implementing an integral transform fracture phase field model of an arbitrary-order polynomial cohesion law, applied to finite element analysis software. The method includes: Step 100: Obtain the subroutine for solving the energy density eigenfunction expression of an arbitrary-order polynomial and its first two derivatives, and input the parametric equation of the cohesion law of the arbitrary-order polynomial and the target order into the subroutine. Step 102: Use subroutines to derive the cohesive eigenfunctions of the cohesion law of arbitrary-order polynomials, and obtain the analytical expression of the energy density eigenfunctions of arbitrary-order polynomials conjugate with the cohesive eigenfunctions based on integral transform. Then, solve the expression of the energy density eigenfunctions of the target-order polynomial and its first two derivatives. Finally, use Maclaurin series to determine the expansion expression of the energy density eigenfunctions of the target-order polynomial and its first two derivatives at zero points to remove endpoint singularities. Step 104: Input the expression of the energy density eigenfunction of the target polynomial and its first two derivatives, as well as the expansion expression of the energy density eigenfunction of the target polynomial and its first two derivatives at zero, into the solver used for integral transform fracture phase field model to obtain the simulation calculation results of the integral transform fracture phase field model of the target polynomial cohesion law.
[0014] In this embodiment of the invention, a finite element subroutine was developed, which can approximate the original cohesion curve with arbitrary precision based on the polynomial fitting method, derive the cohesive eigenfunction of the cohesion law of arbitrary order polynomial fitting, and give the analytical expression of the energy density eigenfunction of the arbitrary order polynomial conjugate with the cohesion eigenfunction based on integral transform. On this basis, the expression of the energy density eigenfunction of the target order polynomial and its first two derivatives are solved, and the expansion expression of the energy density eigenfunction of the target order polynomial and its first two derivatives at zero points is determined by McLaurin series to remove endpoint singularities. This improves the theoretical framework of finite element solution, enabling the simulation calculation of the fracture phase field model of the cohesion law of the target order polynomial under the general finite element framework, greatly improving accuracy and computational efficiency.
[0015] The following description Figure 1 The execution method of each step is shown.
[0016] For step 100: In this embodiment, since the fracture phase field method includes the equilibrium equation of the displacement field of the solid material and the constraint equilibrium equation of the crack phase field, there is no directly available module in the software. In order to solve the above problem, a subroutine based on user-defined units is developed to solve the expression of the energy density eigenfunction and its first two derivatives.
[0017] In solving phase-field models, some complex cohesion curves exist, such as the cohesion curve of concrete. However, this cohesion relationship cannot be directly expressed in the form of cohesion eigenfunctions and needs to be characterized by polynomial fitting. Therefore, it is necessary to develop a general-purpose finite element software subroutine to solve the expression of the target-order polynomial energy density eigenfunction and its first two derivatives, so as to input it into the solver of the phase-field model for simulation calculation of complex cohesion curves.
[0018] Based on the empirical cohesion law governing the failure of concrete under tensile loads, a schematic diagram of Cornelissen's cohesive intrinsic fracture law is shown below. Figure 2 As shown, the Cornelissen cohesion law is introduced as an example, that is, a 6th-order polynomial is used to fit the Cornelissen cohesion law. The expression is: in, , It is a normalized destruction of the opening displacement. It is the crack opening displacement at the final tensile failure. These are two fitting coefficients. The Cornlissen cohesion law is fitted using the parametric equations of an arbitrary-order polynomial cohesion law. Since the target order is 6, the fitting coefficients of the parametric equations of the arbitrary-order polynomial cohesion law can be calculated as follows: Input the parametric equations, target order, and fitting coefficients of the cohesion law of arbitrary order polynomials into the subroutine at each order.
[0019] Regarding step 102: In some implementations, the step "derives the cohesive eigenfunction of the cohesion law of an arbitrary-order polynomial using a subroutine, and obtains the analytical expression of the energy density eigenfunction of the arbitrary-order polynomial conjugate with the cohesive eigenfunction based on the integral transform" includes steps S1-S6: S1, obtain the definitions of equivalent crack opening displacement and equivalent cohesive stress, in order to define the parametric equations of the cohesive eigenvalue law.
[0020] In step S1, for real-world materials, the cohesion law can typically be expressed in the form of parametric equations. Establishing these parametric equations provides the necessary condition for establishing the relationship between the cohesion law and energy density. Figure 2 The cohesive intrinsic fracture law shown can be mathematically expressed as the equivalent crack opening displacement. and equivalent cohesive stress Mapping relationship between (in (The symbol represents the mapping) equivalent crack opening displacement and equivalent cohesive stress The definition of is: (1) in, These are the normalized crack surface normal opening displacement, tangential opening displacement, normal opening stress, and tangential opening stress, which can be expressed as: (2) here, These represent the normal opening displacement, tangential opening displacement, normal opening stress, and tangential opening stress of the crack surface, respectively. These are the tensile strength and shear strength of the material, respectively. The characteristic length of cohesive deformation of a material can be defined as: (3) in, It represents the fracture energy of a material, characterizing the material's ability to resist fracture.
[0021] For a typical softening curve, the mapping relationship is... Since it is monotonically decreasing, the parametric equations of the cohesive eigenvalue law can be defined as follows: (4) in, It is a parameter. It is a monotonically increasing function, that is, a cohesive eigenfunction.
[0022] S2. Based on the regularized variational method, determine the expression for the Griffith surface energy to determine the energy functional of the quasi-elastic body.
[0023] In step S2, we first introduce the integral transform phase-field fracture model, considering a quasi-elastic body, denoted as... ,in Let be the real space (which can be understood as the actual space occupied), and let be the dimension of the elastic body. Here, we only consider one-dimensional or two-dimensional problems. The elastic body has Dirichlet boundary conditions. Neumann boundary conditions The mathematical relationship between the two is as follows: , Let represent the crack surface. According to the regularized variational method, the Griffith surface energy can be approximately expressed as: (5) in, It is the crack density function, with the phase field value as the independent variable. and its gradient , This indicates the damage status of the material.
[0024] Without loss of generality, the energy functional of quasi-elastic bodies It has the following forms: (6) Wherein represents the displacement of the quasi-elastic body , It is the elastic potential energy density, expressed as the strain tensor. and phase field The function, The simple form, which does not involve any decomposition of positive or negative energy, is: in, It is a fourth-order elastic tensor. Let be the energy degradation function. It has a unified form: (8) here, These are constants related to material strength: (9) These are the elastic modulus and shear modulus of the material. This is an intrinsic length parameter, which is associated with the subsequent numerical implementation.
[0025] In equation (8) This is called the energy density eigenfunction, which is related to the crack density function: (10) S3. The analytical derivation of the one-dimensional rod is carried out. The parametric equation of the cohesive eigenfunction law is substituted into the energy functional of the quasi-elastic body to obtain the conjugate relationship between the cohesive eigenfunction and the energy density eigenfunction.
[0026] In step S3, by analytically deriving the one-dimensional rod, substituting formulas (1)-(4) into the energy functional formula (6), we can obtain the conjugate relationship between the cohesive eigenfunction and the energy density eigenfunction: (11) The above equation yields the cohesive eigenfunction. With energy density eigenfunction The relationship between these is the first equation of the Feng-Li integral transform; however, what is usually obtained from experiments is the target cohesion law, and thus the cohesive eigenfunctions. Given the energy density eigenfunction Unknown.
[0027] S4. Perform an integral transformation on the conjugate relation to obtain the expression for the energy density eigenfunction.
[0028] In step S4, an integral transformation is performed on the above equation (11) to transform the unknown into... The expression for the energy density eigenfunction is obtained as follows: (12) This equation is the second equation of the Feng-Li integral transform; therefore, once the cohesion law is determined, the energy density eigenfunctions are known. Combining the theoretical formulas above, It is the only unknown function. However, calculating the above formula generally requires certain skills. This approach focuses on the calculation of complex cohesion laws.
[0029] S5: Obtain the parametric equation of the cohesion law of the input polynomial of arbitrary order, and substitute it into the parametric equation of the cohesion eigenvalue law to obtain the cohesion eigenfunction of the cohesion law of arbitrary order polynomial.
[0030] Some complex cohesion laws correspond to It may be extremely complex, even unrepresentable as an elementary function. A general and feasible approach is presented to approximate the original eigenfunction with arbitrary precision. First, the original complex cohesion law is approximated using multi-order polynomials. Then, the energy density eigenfunction of the cohesion law for arbitrary-order polynomials is given. .
[0031] The parametric equation for the cohesion law of an arbitrary-order polynomial is defined as follows: (13) In the formula, N is the order of the polynomial cohesion law. denoted as the fitting coefficient, and n as the degree of the polynomial.
[0032] The aforementioned polynomial cohesion law can be a fit to a complex cohesion law or a direct fit to experimental data. These fitting coefficients should satisfy the following constraints: (14) It is the crack opening displacement corresponding to a tensile stress of 0. This is to ensure that the opening displacement reaches the target value when the crack fully expands. , In order to ensure that The crack opening displacement is 0. This is to ensure that the fitted fracture energy matches the actual fracture energy of the material. Consistent.
[0033] Substituting the parametric equation of the cohesion law of arbitrary order polynomials (Equation 13) into the parametric equation of the cohesion eigenvalue law (Equation 4), we obtain the cohesion eigenfunctions of the cohesion law of arbitrary order polynomials: (15) S6. Substituting the cohesive eigenfunctions of the cohesion law of an arbitrary-order polynomial into the expression for the energy density eigenfunctions, we obtain the analytical expression for the energy density eigenfunctions of an arbitrary-order polynomial.
[0034] In this step, the cohesive eigenfunction of the cohesion law of an arbitrary-order polynomial (Equation 15) is substituted into the expression for the energy density eigenfunction (Equation 12) to obtain the analytical expression for the energy density eigenfunction of an arbitrary-order polynomial: (16) in, (17) In the formula, These are basis functions.
[0035] In some implementations, the step "solving the expression for the energy density eigenfunctions of the target-order polynomial and its first two derivatives" includes steps H1-H8: H1 determines the target order expression of the basis functions in the analytical expression of the energy density eigenfunction of an arbitrary-order polynomial. Combining the constraints of the fitting coefficients in the parametric equation of the cohesion law of the arbitrary-order polynomial and the values of the fitting coefficients at each order, the target-order polynomial energy density eigenfunction expression used as input to the solver is obtained.
[0036] In this embodiment, taking a target order of 6 as an example, the expression for the 6th order basis function is: (18) in, For meaningless parameters used to simplify expressions, This is the inverse hyperbolic tangent operator.
[0037] Substituting formula (18) into the analytical expression of the energy density eigenfunction of an arbitrary-order polynomial (formula 16), and using the constraint condition of the fitting coefficients in the parametric equation of the cohesion law of an arbitrary-order polynomial (formula 14), the expression of the energy density eigenfunction of the 6th-order polynomial used as input to the solver is calculated: (19) in, In the formula, The energy density eigenfunction is a 6th-order polynomial. Represents the damage state of the material. and For meaningless parameters used to simplify expressions, It is a parameter related to the fitting coefficient. represents the values of the fitting coefficients at each order.
[0038] H2 introduces the first, second, third, and fourth intermediate variables into the expression for the energy density eigenfunction of the target polynomial to obtain a simplified expression for the energy density eigenfunction of the target polynomial.
[0039] To numerically implement the fracture phase-field method, the first two derivatives of the energy density function must be solved. The techniques required for these derivatives are more complex. Therefore, we will continue to derive the derivative of Equation 19, first simplifying it by using intermediate variables: (20) in, As the fourth intermediate variable, Represents the damage state of the material. These are the first, second, and third intermediate variables, respectively. It is a parameter related to the fitting coefficient. and For use as a meaningless parameter to simplify expressions.
[0040] Therefore, the simplified expression for the energy density eigenfunction of the target polynomial (Equation 19) can be written as: (twenty one) H3, by taking the first two derivatives of the simplified expression of the energy density eigenfunction of the target polynomial, we obtain the first-order and second-order simplified expressions of the energy density eigenfunction of the target polynomial.
[0041] In this step, using This yields the simplified form of the first two derivatives of the eigenfunctions of the sixth-order polynomial energy density: (twenty two) (twenty three) In the formula, and These are the first and second derivatives of the energy density eigenfunctions of the 6th-order polynomial.
[0042] H4, with respect to the first, second, and third intermediate variables, takes the first two derivatives to determine the first-order expression of the fourth intermediate variable.
[0043] In this step, because Therefore, the third intermediate variable The derivative with respect to a is: (twenty four) First intermediate variable The first two derivatives with respect to s and the second intermediate variable The first two derivatives with respect to a are relatively simple and are given directly here: (25) And because of the chain rule Therefore, the fourth intermediate variable is given first. The derivative with respect to 'a', followed by an inverse transformation, determines the first-order expression of the fourth intermediate variable: (26) H5. Substitute the first two derivative expressions of the first, second, and third intermediate variables, as well as the first-order expression of the fourth intermediate variable, into the first-order simplified expression of the target polynomial energy density eigenfunction to obtain the first-order expression of the target polynomial energy density eigenfunction used as input to the solver.
[0044] In this step, Equations 20 and 24-26 are substituted into the first-order simplified form of the target-order polynomial energy density eigenfunction (Equation 22) to obtain the first-order expression of the target-order polynomial energy density eigenfunction used as input to the solver.
[0045] H6 introduces a fifth and a sixth intermediate variable into the first-order expression of the fourth intermediate variable to simplify the second-order simplified expression of the fourth intermediate variable.
[0046] The fourth intermediate variable is then solved. With respect to the second derivative of a, we introduce a fifth and a sixth intermediate variable: Therefore, the fourth intermediate variable The simplified form of the second derivative with respect to a is: (27) H7 determines the first derivatives of the fifth and sixth intermediate variables to obtain the second-order expression of the fourth intermediate variable.
[0047] In step H7, the fifth and sixth intermediate variables need to be solved separately. First derivative : (28) The derivation is divided into two steps. (29) (30) therefore The expression is: (31) Substituting equation 28-31 into equation 27 yields the second-order expression for the fourth intermediate variable: (32) H8. Substitute the first-order and second-order expressions of the fourth intermediate variable, and the first two derivative expressions of the first, second, and third intermediate variables into the second-order simplified form of the target polynomial energy density eigenfunction to obtain the second-order expression of the target polynomial energy density eigenfunction used as input to the solver.
[0048] In this step, substitute Equations 20 and 24-32 into Equation 23 to obtain the second-order expression for the target-order polynomial energy density eigenfunction used as input to the solver, which will not be shown here.
[0049] At this point, the theoretical formulas required for fitting cohesion laws to polynomials of arbitrary order have been largely derived. The process of removing endpoint singularities will now be explained.
[0050] Retrospective (12), when At this time, the upper and lower limits of the integration interval will coincide or the denominator of the integrand will be 0. Specifically, observe equations 22 and 23, when In the denominator When it is 0, In the denominator of time The value is 0. To address this issue, this invention further proposes a numerical solution method for the endpoints of the Maclaurin expansion. The Taylor series obtained by taking the derivative of the function at the zero point of the independent variable is also called the Maclaurin series.
[0051] In some implementations, the step "using McLaurin series to determine the eigenfunction expression of the target order polynomial energy density and the expansion expression of its first two derivatives at zeros" includes K1-K3: K1 is used to obtain a simplified expression for the eigenfunction of the target order polynomial energy density, as well as the first, second, third, and fourth intermediate variables.
[0052] In this step, obtain the required expression: (33) These expressions have all been defined in the previous text and will not be repeated here. They are only compiled here for the convenience of subsequent expansion at the zero point.
[0053] K2, determine the Maclaurin expansions of the first, second, third, and fourth intermediate variables respectively, in order to determine the expansion expression of the eigenfunctions of the target order polynomial energy density.
[0054] First of all Expand (34) Based on the expansion of the power function, we can obtain the following equation: (35) in, This represents a higher-order infinitesimal operator. The third intermediate variable is the inverse hyperbolic tangent function. The expansion expression: (36) Substitute formulas 35 and 36 into the first intermediate variable. The expansion expression: (37) Next processing ,right Expand the expansion to the first level: (38) in This is the 0th order expansion expression. This is the first-order expansion expression. The meaning is right The derivative of The value of time. Substituting equations 36 and 38 into equation 33, we get The expansion: (39) Substituting equations 37 and 39 into equation 33, we obtain the fourth intermediate variable. Expand expression: (40) (41) (42) in, These are introduced intermediate variables and have no practical significance. Next, we expand the denominator terms of the target-order polynomial energy density eigenfunction expression: (43) Combining equations 33-43, we can obtain the expansion expression for the eigenfunctions of the sixth-order polynomial energy density: (44) in, Intermediate variables introduced to simplify calculations have no real meaning.
[0055] K3, perform first-order and second-order differentiation on the expansion expression of the energy density eigenfunction of the target order polynomial to obtain the expression of the energy density eigenfunction of the target order polynomial and the expansion expression of its first two derivatives at zero.
[0056] In this step, the first and second derivatives of the expansion expression of the target order polynomial energy density eigenfunction (Equation 44) are performed respectively: (45) In practice, it has been found that singularity only occurs when... Since this occurs at a certain point, we give the expression for the eigenfunction of the 6th-order polynomial energy density and the expansion of its first two derivatives at zeros: (46) Regarding step 104: The expression for the energy density eigenfunction of the target polynomial (Equation 19) and its first two derivatives (Equations 22 and 23), as well as the expansion expression of the energy density eigenfunction of the target polynomial and its first two derivatives at zero (Equation 46), are input into the solver used for the integral transform fracture phase field model to obtain the simulation calculation results of the integral transform fracture phase field model of the target polynomial cohesion law.
[0057] Finally, the accuracy of the invention is verified through numerical examples.
[0058] Consider a one-dimensional tension member of length L = 200 mm, fixed at the left end and subjected to a displacement load u at the right end. Material parameters are as follows: fracture energy... Tensile strength Young's modulus Poisson's ratio .
[0059] Figure 3The normalized cohesion curves of the model's predicted members are shown, with a mesh size of 0.25 mm. From the load-displacement curves, the model's predictions are independent of the intrinsic characteristic length l of the phase field. The normalized cohesion curves also show good agreement between the model's predictions and the analytical solutions. The method proposed in this invention achieves highly accurate calculation results.
[0060] Figure 4 The normalized cohesion curves of the model's predicted members are shown, with a grid size of 0.1 mm. It can also be seen that the model's predictions are independent of the intrinsic characteristic length *l* of the phase field and agree well with the analytical solution. Through further analysis... Figure 3 and Figure 4 Comparing the results, it was found that the method established by the present invention does not depend on the mesh size and achieves the most important objective of the fractured phase field.
[0061] Please refer to Figure 5 This invention provides a device for implementing an integral transform fracture phase field model of an arbitrary-order polynomial cohesion law, which is installed in finite element analysis software. The device includes: The acquisition unit 501 is used to acquire a subroutine for solving the energy density eigenfunction expression of an arbitrary-order polynomial and its first two derivatives, and inputs the parametric equation of the cohesion law of the arbitrary-order polynomial and the target order into the subroutine. Solver 502 is used to derive the cohesive eigenfunction of the cohesion law of an arbitrary-order polynomial using subroutines, and obtain the analytical expression of the energy density eigenfunction of the arbitrary-order polynomial conjugate with the cohesive eigenfunction based on integral transform. Then, it solves the expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives, and uses the Maclaurin series to determine the expansion expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives at the zero point to remove endpoint singularities. Simulation unit 503 is used to input the expression of the energy density eigenfunction of the target polynomial and its first two derivatives, as well as the expansion expression of the energy density eigenfunction of the target polynomial and its first two derivatives at zero points, into the solver for the integral transform fracture phase field model, so as to obtain the simulation calculation results of the integral transform fracture phase field model of the cohesion law of the target polynomial.
[0062] It should be noted that the apparatus for implementing the integral transform fracture phase-field model of arbitrary-order polynomial cohesion law provided in the above embodiments is only an example of the division of the above functional modules. In practical applications, the above functions can be assigned to different functional modules as needed, that is, the internal structure of the apparatus can be divided into different functional modules to complete all or part of the functions described above. In addition, the apparatus for implementing the integral transform fracture phase-field model of arbitrary-order polynomial cohesion law provided in the above embodiments and the method embodiment for implementing the integral transform fracture phase-field model of arbitrary-order polynomial cohesion law belong to the same concept. The specific implementation process is detailed in the method embodiment and will not be repeated here.
[0063] Embodiments of this application also provide a computer device, please refer to... Figure 6 The computer device includes a processor and a memory, the memory storing at least one instruction, at least one program, code set or instruction set, the at least one instruction, at least one program, code set or instruction set being loaded and executed by the processor to implement the integral transform fracture phase field model implementation method of arbitrary order polynomial cohesion law provided in the above method embodiments.
[0064] Embodiments of this application also provide a computer-readable storage medium storing at least one instruction, at least one program, code set, or instruction set, wherein the at least one instruction, at least one program, code set, or instruction set is loaded and executed by a processor to implement the integral transform fracture phase field model implementation method of arbitrary order polynomial cohesion law provided in the above-described method embodiments.
[0065] Embodiments of this application also provide a computer program product, which includes a computer program. A processor of a computer device reads the computer program from a computer-readable storage medium and executes the computer program, causing the computer device to execute any of the integral transform fracture phase field model implementation methods of arbitrary-order polynomial cohesion laws in the above embodiments.
[0066] For ease of description, the above systems or devices are described separately as various modules or units based on their functions. Of course, in implementing this application, the functions of each unit can be implemented in one or more software and / or hardware components.
[0067] As can be seen from the above description of the embodiments, those skilled in the art can clearly understand that this application can be implemented by means of software plus necessary general-purpose hardware platforms. Based on this understanding, the technical solution of this application, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods of various embodiments or some parts of the embodiments of this application.
[0068] Finally, it should be noted that in this document, relational terms such as first, second, third, and fourth are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes the element.
[0069] The above are merely preferred embodiments of this application. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this application, and these improvements and modifications should also be considered within the scope of protection of this application.
Claims
1. A method for implementing an integral transform fracture phase field model of an arbitrary-order polynomial cohesion law, applied to finite element analysis software, characterized in that... The method includes: Obtain a subroutine for solving the energy density eigenfunction expression of an arbitrary-order polynomial and its first two derivatives, and input the parametric equation of the cohesion law of the arbitrary-order polynomial and the target order into the subroutine; The cohesive eigenfunctions of the cohesion law of an arbitrary-order polynomial are derived using the subroutine. After obtaining the analytical expression of the energy density eigenfunction of the arbitrary-order polynomial conjugate with the cohesive eigenfunction based on the integral transform, the expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives are solved. The Maclaurin series is then used to determine the expansion expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives at the zero point to remove endpoint singularities. The expression for the energy density eigenfunction of the target polynomial and its first two derivatives, as well as the expansion expression of the energy density eigenfunction of the target polynomial and its first two derivatives at zero, are input into the solver used for the integral transform fracture phase-field model to obtain the simulation calculation results of the integral transform fracture phase-field model of the target polynomial cohesion law.
2. The method according to claim 1, characterized in that, The process of deriving the cohesive eigenfunction of the cohesion law of an arbitrary-order polynomial using the subroutine, and obtaining the analytical expression of the energy density eigenfunction of the arbitrary-order polynomial conjugate with the cohesive eigenfunction based on integral transform, includes: Obtain the definitions of equivalent crack opening displacement and equivalent cohesive stress to define the parametric equations of the cohesive eigenvalue law; Based on the regularized variational method, the expression for the Griffith surface energy is determined, so as to determine the energy functional of the quasi-elastic body. The analytical derivation of the one-dimensional rod is performed. Substituting the parametric equation of the cohesive eigenfunction law into the energy functional of the quasi-elastic body, the conjugate relationship between the cohesive eigenfunction and the energy density eigenfunction is obtained. By performing an integral transform on the conjugate relation, the expression for the energy density eigenfunction is obtained; Obtain the parametric equation of the cohesion law of an arbitrary-order polynomial, and substitute it into the parametric equation of the cohesion eigenvalue law to obtain the cohesion eigenfunction of the cohesion law of an arbitrary-order polynomial. Substituting the cohesive eigenfunction of the cohesion law of an arbitrary-order polynomial into the expression of the energy density eigenfunction, we obtain the analytical expression of the energy density eigenfunction of an arbitrary-order polynomial.
3. The method according to claim 1, characterized in that, The process of solving for the eigenfunction expression of the target-order polynomial energy density and its first two derivatives includes: The target order expression of the basis functions in the analytical expression of the energy density eigenfunction of the arbitrary order polynomial is determined. Combined with the constraints of the fitting coefficients in the parametric equation of the cohesion law of the arbitrary order polynomial and the values of the fitting coefficients at each order, the expression of the target order polynomial energy density eigenfunction used as input to the solver is obtained. Introducing a first intermediate variable, a second intermediate variable, a third intermediate variable, and a fourth intermediate variable into the expression for the energy density eigenfunction of the target order polynomial, a simplified expression for the energy density eigenfunction of the target order polynomial is obtained. By taking the first two derivatives of the simplified expression of the target order polynomial energy density eigenfunction, we obtain the first-order and second-order simplified expressions of the target order polynomial energy density eigenfunction. The first two derivatives of the first intermediate variable, the second intermediate variable, and the third intermediate variable are respectively calculated to determine the first-order expression of the fourth intermediate variable; Substituting the first two derivative expressions of the first intermediate variable, the second intermediate variable, and the third intermediate variable, as well as the first-order expression of the fourth intermediate variable, into the first-order simplified expression of the target-order polynomial energy density eigenfunction, we obtain the first-order expression of the target-order polynomial energy density eigenfunction used as input to the solver. A fifth and a sixth intermediate variable are introduced into the first-order expression of the fourth intermediate variable to simplify the second-order simplified expression of the fourth intermediate variable; The first derivatives of the fifth and sixth intermediate variables are determined respectively to obtain the second-order expression of the fourth intermediate variable; Substituting the first-order and second-order expressions of the fourth intermediate variable, and the first two derivative expressions of the first, second, and third intermediate variables into the second-order simplified form of the target-order polynomial energy density eigenfunction, we obtain the second-order expression of the target-order polynomial energy density eigenfunction used as input to the solver.
4. The method according to claim 3, characterized in that, When the target order is 6, the expression for the target order polynomial energy density eigenfunction used as input to the solver is: in, In the formula, The energy density eigenfunction is a 6th-order polynomial. Represents the damage state of the material. and For meaningless parameters used to simplify expressions, It is a parameter related to the fitting coefficient. represents the values of the fitting coefficients at each order.
5. The method according to claim 3, characterized in that, When the target order is 6, the first-order and second-order abbreviated forms of the target order polynomial energy density eigenfunctions are as follows: in, In the formula, and These are the first and second derivatives of the energy density eigenfunctions of the 6th-order polynomial. For the fourth intermediate variable, Represents the damage state of the material. These are the first intermediate variable, the second intermediate variable, and the third intermediate variable, respectively. It is a parameter related to the fitting coefficient. and This is a meaningless parameter used to simplify expressions.
6. The method according to claim 3, characterized in that, The method of using McLaurin series to determine the eigenfunction expression of the target-order polynomial energy density and the expansion expression of its first two derivatives at zero points includes: Obtain the simplified expression of the target order polynomial energy density eigenfunction, as well as the first intermediate variable, the second intermediate variable, the third intermediate variable, and the fourth intermediate variable; The Maclaurin expansions of the first intermediate variable, the second intermediate variable, the third intermediate variable, and the fourth intermediate variable are determined respectively to determine the expansion expression of the eigenfunction of the target order polynomial energy density. The expansion expression of the target order polynomial energy density eigenfunction is subjected to first-order and second-order differentiation to obtain the expression of the target order polynomial energy density eigenfunction and the expansion expression of its first two derivatives at zero.
7. A device for implementing an integral transform fracture phase field model of an arbitrary-order polynomial cohesion law, configured in finite element analysis software, for implementing the steps of the method described in any one of claims 1-6, characterized in that, The device includes: The acquisition unit is used to acquire a subroutine for solving the energy density eigenfunction expression of an arbitrary-order polynomial and its first two derivatives, and inputs the parametric equation of the cohesion law of the arbitrary-order polynomial and the target order into the subroutine. The solution unit is used to derive the cohesive eigenfunction of the cohesion law of an arbitrary-order polynomial using the subroutine, and after obtaining the analytical expression of the energy density eigenfunction of the arbitrary-order polynomial conjugate with the cohesive eigenfunction based on the integral transform, solve the expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives, and use the Maclaurin series to determine the expansion expression of the energy density eigenfunction of the target-order polynomial and its first two derivatives at the zero point to remove endpoint singularities. The simulation unit is used to input the expression of the energy density eigenfunction of the target polynomial and its first two derivatives, as well as the expansion expression of the energy density eigenfunction of the target polynomial and its first two derivatives at zero points, into the solver for the integral transform fracture phase field model, so as to obtain the simulation calculation results of the integral transform fracture phase field model of the target polynomial cohesion law.
8. A computer device, characterized in that, The computer device includes a memory and a processor. The memory is used to store computer programs, and the processor is used to execute the computer programs stored in the memory to implement the steps of the method according to any one of claims 1-6.
9. A computer-readable storage medium, characterized in that, The storage medium stores a computer program, which, when executed by a processor, implements the steps of the method described in any one of claims 1-6.
10. A computer program product, characterized in that, Includes a computer program, which, when executed by a processor, implements the steps of the method according to any one of claims 1-6.