A space-time fractional phase-field fracture simulation method based on fast fourier transform
By using a spatiotemporal fractional phase field fracture simulation method based on fast Fourier transform, the limitations of traditional models in the fracture analysis of complex materials are overcome, and efficient and accurate simulation of the material fracture process is achieved, especially in scenarios with significant nonlocal interactions and memory effects.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CNOOC ENERGY TECHNOLOGY & SERVICES LTD
- Filing Date
- 2026-05-09
- Publication Date
- 2026-07-31
AI Technical Summary
Existing technologies are insufficient to effectively describe and efficiently solve the fracture behavior of complex materials under load, especially when the particle distribution does not satisfy a Gaussian distribution or there are long-distance jumps. Traditional phase-field fracture models cannot accurately characterize nonlocal interactions and memory effects.
A spatiotemporal fractional phase field fracture simulation method based on fast Fourier transform is adopted. By constructing the spatiotemporal fractional phase field control equations and discrete time fractional derivatives, the calculation of spatial nonlocal terms is accelerated by fast Fourier transform. An iterative coupling algorithm is used to update the phase field until the convergence condition is met, and the macroscopic stress field and crack propagation path are output.
It enables refined analysis and prediction of fracture behavior in complex materials, improves computational efficiency, and ensures accurate simulation of crack propagation paths and stress distribution.
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Figure CN122494076A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of materials mechanics and numerical simulation of mechanics, and in particular relates to a spatiotemporal fractional phase field fracture simulation method based on fast Fourier transform. Background Technology
[0002] In physics, the fractional-order Laplace operator is called the fractional-order diffusion flux, used to characterize the anomalous diffusion process of particles hopping long distances under Leviian flight. Traditional phase-field breaking models are based on integer-order derivative theory, and the integer-order Laplace operator can characterize the normal diffusion process in isotropic media without considering long-distance hopping. In other words, when the particle distribution follows a heavy-tailed distribution, or when the hop distance of random particles is no longer finite, or when the diffusion process of many complex systems typically no longer follows a Gaussian distribution, the classical convection-diffusion equations no longer apply.
[0003] Fractional derivative theory can effectively characterize memory effects and spatial nonlocal interactions in physical processes, but its application in phase-field fracture models is not yet systematic, and efficient numerical solution algorithms are lacking. Therefore, how to construct a spatiotemporal fractional phase-field fracture model and achieve its efficient numerical solution has become a key challenge in the field of materials failure analysis. Summary of the Invention
[0004] The problem to be solved by this invention is to provide a spatiotemporal fractional phase field fracture simulation method based on fast Fourier transform. This method is suitable for the refined analysis and prediction of fracture behavior of complex materials under load. It describes the nonlocal evolution characteristics of the fracture phase field through fractional-order theory and achieves efficient calculation by combining fast Fourier transform (FFT).
[0005] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a method for simulating spatiotemporal fractional-order phase field fracture based on fast Fourier transform, comprising the following steps: S1: Construct the spatiotemporal fractional phase field control equations; S2: Discrete-time fractional derivatives, and establish a global system of linear equations; S3: Accelerate the calculation of spatial nonlocal terms using Fast Fourier Transform; S4: Update the phase field d using an iterative coupling algorithm until the convergence condition is met; S5: Outputs the macroscopic stress field and crack propagation path.
[0006] Furthermore, S1 includes the following steps: S11: In phase field fracture theory, the total potential energy control equation is obtained by using the total potential energy of the material, which includes elastic free energy and crack surface potential energy. S12: In the spatiotemporal fractional phase field fracture theory, the fracture density function is generalized to a fractional form, as shown in the following formula: in, These are phase-field variables used to characterize the damage state and crack evolution of materials. This indicates that the material has not been damaged. The interval between these two points indicates a state of damage transition. This indicates that the material is completely damaged; l0 is a characteristic length parameter. The fractional derivative of the phase field variable; S13: The expression for the total potential energy Ψ is as follows: Where Ψe is the elastic potential energy; Ψc is the surface potential energy of the crack; Gc is the critical energy release rate; and ψ e + (ε) is the elastic strain energy density, and k is a constant; S14: By performing variational analysis on the total potential energy, the fractional-order governing equations are obtained as follows: Where σ is the stress tensor, δε is the variation of strain, and H is the historical maximum strain energy density. It is a fractional Laplace operator; S15: In the spatiotemporal fractional phase-field fracture theory, the time fractional derivative is introduced to describe the dynamic process of crack propagation. The formula for the spatiotemporal fractional phase-field governing equation is as follows. .
[0007] Furthermore, S11 includes the following steps: S111: In phase-field fracture theory, the expression for the fracture density function is as follows: in, These are phase-field variables used to characterize the damage state and crack evolution of materials. This indicates that the material has not been damaged. The interval between these two points indicates a state of damage transition. This indicates that the material is completely damaged; l0 is a characteristic length parameter. It is the gradient of the phase field variables; S112: The relationship between stress and strain is as follows: in, It is the stress tensor, and its unit is 1 / 2. ; C is the strain tensor, a dimensionless quantity; C is the elastic modulus tensor, with units of 1000 kilometres per second (kJ / m²). k is a constant, usually taken as 10. -9 ; S113: The total potential energy Ψ includes the elastic potential energy Ψe and the crack surface potential energy Ψc, and its expression is as follows: Where Gc is the critical energy release rate, ψ e + (ε) is the elastic strain energy density, and k is a constant. It is the fracture density function; S114: By performing variational analysis on the total potential energy, the governing equations are obtained as follows: Where σ is the stress tensor; δε is the variation of strain; and H is the historical maximum strain energy density.
[0008] Furthermore, in S2, the time fractional derivative is discretized using the weighted difference method. The discretization formula for the time fractional derivative is as follows: Where Γ() is the gamma function; Δt is the time step.
[0009] Furthermore, step S3 includes the following steps: S31: The evolution equation of the crack damage field is as follows: Among them, coefficients A and B are driven by material parameters and energy release rate; S32: Introducing a homogeneous variable A0, FFT accelerates the calculation of nonlocal terms: ; S33: Transform the spatial fractional terms to the frequency domain using Fast Fourier Transform: in, Let be the fractional kernel function in the frequency domain, and s be the fractional order in the spatial domain.
[0010] Furthermore, S4 includes the following steps: S41: Initialization parameters, including time step Δt n+1 , strain field ε(t n,x), damage field d(t) n (x) and energy increment; S42: Initialize the damage field; S43: The iterative coupling algorithm uses the Steffensen accelerated method to solve the damage field. Until the convergence condition is met: .
[0011] Furthermore, in S5, the formula for calculating the macroscopic stress field is: in, It represents the volume of the computation space.
[0012] Furthermore, the present invention also provides an apparatus for performing the above-described data processing method.
[0013] Furthermore, the present invention also provides an apparatus including a memory, a processor, and an algorithm stored in the memory and executable on the processor, wherein the processor implements the above-described data processing method when executing the computer program.
[0014] Furthermore, the present invention also provides a computer-readable storage medium storing a computer algorithm, which, when executed by a processor, performs the above-described data processing.
[0015] The advantages and positive effects of this invention are: This invention constructs a nonlocal phase field evolution equation by introducing temporal and spatial fractional derivatives, accurately describing the history dependence and spatial nonlocal interactions during material fracture. It utilizes the Fast Fourier Transform to convert spatial differential operators into frequency domain multiplication operations, significantly reducing the computational complexity and improving efficiency for three-dimensional heterogeneous materials. Efficient convergence is achieved through the Steffensen accelerated iteration method, ensuring accurate simulation of crack propagation paths and stress distribution. Attached Figure Description
[0016] Figure 1 This is a schematic diagram of the overall process of an embodiment of the present invention.
[0017] Figure 2 These are crack evolution diagrams corresponding to different fractional orders in embodiments of the present invention.
[0018] Figure 3 These are stress-strain diagrams of different fractional order from embodiments of the present invention. Detailed Implementation
[0019] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. 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.
[0020] The embodiments of the present invention will be further described below with reference to the accompanying drawings: like Figure 1 As shown, a spatiotemporal fractional-order phase field fracture simulation method based on Fast Fourier Transform (FFT) is presented. This method describes the nonlocal evolution characteristics of the fracture phase field using fractional-order theory and achieves efficient computation by combining it with FFT. Specifically, it includes the following steps: S1: Construct the spatiotemporal fractional phase field control equations.
[0021] Specifically, reviewing the basic principles of the phase-field method, the elastic solid Ω considers the induced damage exhibited due to the evolution of microcracks. As the load increases, the density of microcracks reaches a critical state, and therefore, cracks begin to appear as microcracks coalesce into narrow bands. The discontinuities in these cracks should be considered. In the phase-field method, this is approximated by introducing an auxiliary damage field and then by using a regularized field for calculation. This is considered an internal variable representing the state of damage. The material is intact. And completely collapsed .
[0022] The origins of the spacetime fractional-order phase-field fracture model can be traced back to the classical phase-field fracture model. Its core idea is to view the fracture process as a competition between volume energy and surface energy. Traditional phase-field fracture models are based on integer-order calculus, but they have limitations in dealing with complex fracture problems. The introduction of fractional-order calculus provides a new tool for describing the nonlocality and memory effects in the fracture process.
[0023] In phase-field fracture theory, the crack density function describes the distribution and density of cracks in a material, and its expression is: in, These are phase-field variables used to characterize the damage state and crack evolution of materials. This indicates that the material has not been damaged. The interval between these two points indicates a state of damage transition. This indicates that the material is completely damaged; l0 is a characteristic length parameter. It is the gradient of the phase field variable.
[0024] The relationship between stress and strain is as follows: in, It is the stress tensor, and its unit is 1 / 2. ; C is the strain tensor, a dimensionless quantity; C is the elastic modulus tensor, with units of 1000 kilometres per second (kJ / m²). k is a constant, usually taken as 10. -9 .
[0025] The total potential energy Ψ includes the elastic potential energy Ψe and the crack surface potential energy Ψc, and its expression is as follows: Where Gc is the critical energy release rate, ψ e + (ε) is the elastic strain energy density, and k is a constant. It is the fracture density function.
[0026] Variationally applying the total potential energy, we obtain the governing equations as follows: Where σ is the stress tensor; δε is the variation of strain; and H is the historical maximum strain energy density.
[0027] In the spatiotemporal fractional phase field fracture theory, the fracture density function is generalized to a fractional form, as shown in the following formula: in, These are phase-field variables used to characterize the damage state and crack evolution of materials. This indicates that the material has not been damaged. The interval between these two points indicates a state of damage transition. This indicates that the material is completely damaged; It represents the fractional derivative of the phase field variable.
[0028] S13: The expression for the total potential energy is similar to that in the phase-field fracture theory, but the fracture density function is replaced with a fractional form. The expression for the total potential energy Ψ is as follows: Where Ψe is the elastic potential energy; Ψc is the surface potential energy of the crack; Gc is the critical energy release rate; and ψ e + (ε) is the elastic strain energy density, and k is a constant, usually taken as 10. -9 .
[0029] By performing variational analysis on the total potential energy, the fractional-order governing equations are obtained as follows: in: σ is the stress tensor, δε is the variation of strain, and H is the historical maximum strain energy density. It is a fractional Laplace operator.
[0030] In the spatiotemporal fractional phase-field fracture theory, the time fractional derivative is introduced to describe the dynamic process of crack propagation. The formula for the spatiotemporal fractional phase-field governing equation is as follows. .
[0031] S2: Discrete the time fractional derivative and establish a global linear equation system. Specifically, the time fractional derivative is discretized using the weighted difference method. The discretization formula for the time fractional derivative is as follows: in: Γ() is the gamma function; Δt is the time step.
[0032] S3: Accelerate the calculation of spatial nonlocal terms using Fast Fourier Transform.
[0033] Specifically, for the nonlocal fracture behavior of heterogeneous materials under dynamic loading, an efficient numerical solution framework based on Fast Fourier Transform (FFT) is adopted. The crack phase-field evolution equation is transformed into a nonlocal differential equation in Fourier space, and is solved efficiently through the following steps: The evolution equation of the crack damage field is as follows: Among them, coefficients A and B are driven by material parameters and energy release rate.
[0034] Introducing a homogeneous variable A0, FFT accelerates the calculation of nonlocal terms: .
[0035] Transform the spatial fractional terms to the frequency domain using the Fast Fourier Transform: in, Let be the fractional kernel function in the frequency domain, and s be the fractional order in the spatial domain.
[0036] S4: Update the phase field d using an iterative coupling algorithm until the convergence condition is met.
[0037] Specifically, initialization parameters include the time step Δt. n+1 , strain field ε(tn ,x), damage field d(t) n ,x) and energy increment.
[0038] Initialize the damage field.
[0039] The damage field is updated iteratively until the convergence condition is met.
[0040] The iterative coupling algorithm uses the Steffensen accelerated method to solve for the damage field, and the iterative formula is as follows: Until the convergence condition is met: .
[0041] S5: Outputs the macroscopic stress field and crack propagation path.
[0042] The formula for calculating the macroscopic stress field is: in, It represents the volume of the computation space.
[0043] The following example uses a uniaxial tensile test of a single-hole structure to illustrate the principle, neglecting damage history memory effect and the elastic modulus of the material. Poisson's ratio Critical fracture energy of materials Phase field canonical length The applied strain increment Following the steps outlined above, the crack evolution law in a uniaxial tensile test of a single-hole structure is as follows: Figure 2 As shown, stress-strain diagrams of different fractional orders are as follows: Figure 3 As shown.
[0044] The advantages and positive effects of this invention are: This invention constructs a nonlocal phase field evolution equation by introducing temporal and spatial fractional derivatives, accurately describing the history dependence and spatial nonlocal interactions during material fracture. It utilizes the Fast Fourier Transform to convert spatial differential operators into frequency domain multiplication operations, significantly reducing the computational complexity and improving efficiency for three-dimensional heterogeneous materials. Efficient convergence is achieved through the Steffensen accelerated iteration method, ensuring accurate simulation of crack propagation paths and stress distribution.
[0045] The foregoing has provided a detailed description of one embodiment of the present invention, but this description is merely a preferred embodiment and should not be construed as limiting the scope of the invention. All equivalent variations and modifications made within the scope of the claims of this invention should still fall within the patent coverage of this invention.
Claims
1. A method for simulating spatiotemporal fractional-order phase field fracture based on Fast Fourier Transform, characterized in that: Includes the following steps, S1: Construct the spatiotemporal fractional phase field control equations; S2: Discrete-time fractional derivatives, and establish a global system of linear equations; S3: Accelerate the calculation of spatial nonlocal terms using Fast Fourier Transform; S4: Update the phase field d using an iterative coupling algorithm until the convergence condition is met; S5: Outputs the macroscopic stress field and crack propagation path.
2. The method for simulating spatiotemporal fractional-order phase field fracture based on fast Fourier transform according to claim 1, characterized in that: S1 includes the following steps: S11: In phase field fracture theory, the total potential energy control equation is obtained by using the total potential energy of the material, which includes elastic free energy and crack surface potential energy. S12: In the spatiotemporal fractional phase field fracture theory, the fracture density function is generalized to a fractional form, as shown in the following formula: in, These are phase-field variables used to characterize the damage state and crack evolution of materials. This indicates that the material has not been damaged. The interval between these two points indicates a state of damage transition. This indicates that the material is completely damaged; l0 is a characteristic length parameter. The fractional derivative of the phase field variable; S13: The expression for the total potential energy Ψ is as follows: where Ψe is the elastic potential energy; Ψc is the crack surface potential energy; Gc is the critical energy release rate, ψ e + (ε) is the elastic strain energy density, k is a constant; S14: By performing variational analysis on the total potential energy, the fractional-order governing equations are obtained as follows: Where σ is the stress tensor, δε is the variation of strain, and H is the historical maximum strain energy density. It is a fractional Laplace operator; S15: In the spatiotemporal fractional phase-field fracture theory, the time fractional derivative is introduced to describe the dynamic process of crack propagation. The formula for the spatiotemporal fractional phase-field governing equation is as follows. 。 3. The method for simulating spatiotemporal fractional-order phase field fracture based on fast Fourier transform according to claim 2, characterized in that: S11 includes the following steps: S111: In phase-field fracture theory, the expression for the fracture density function is as follows: in, These are phase-field variables used to characterize the damage state and crack evolution of materials. This indicates that the material has not been damaged. The interval between these two points indicates a state of damage transition. This indicates that the material is completely damaged; l0 is a characteristic length parameter. It is the gradient of the phase field variables; S112: The relationship between stress and strain is as follows: in, It is the stress tensor, and its unit is 1 / 2. ; C is the strain tensor, a dimensionless quantity; C is the elastic modulus tensor, with units of 1000 kilometres per second (kJ / m²). k is a constant, usually taken as 10. -9 ; S113: The total potential energy Ψ includes the elastic potential energy Ψe and the crack surface potential energy Ψc, and its expression is as follows: Where Gc is the critical energy release rate, ψ e + (ε) is the elastic strain energy density, and k is a constant. It is the fracture density function; S114: By performing variational analysis on the total potential energy, the governing equations are obtained as follows: Where σ is the stress tensor; δε is the variation of strain; and H is the historical maximum strain energy density.
4. A method for simulating spatiotemporal fractional-order phase field fragmentation based on fast Fourier transform according to any one of claims 1 to 3, characterized in that: In S2, the weighted difference method is used to discretize the fractional time derivative. The discretization formula for the fractional time derivative is as follows: Where Γ() is the gamma function; Δt is the time step.
5. A method for simulating spatiotemporal fractional-order phase field fracture based on fast Fourier transform according to any one of claims 1 to 3, characterized in that: S3 includes the following steps: S31: The evolution equation of the crack damage field is as follows: Among them, coefficients A and B are driven by material parameters and energy release rate; S32: Introducing a homogeneous variable A0, FFT accelerates the calculation of nonlocal terms: ； S33: Transform the spatial fractional terms to the frequency domain using Fast Fourier Transform: in, Let be the fractional kernel function in the frequency domain, and s be the fractional order in the spatial domain.
6. A method for simulating spatiotemporal fractional-order phase field fracture based on fast Fourier transform according to any one of claims 1 to 3, characterized in that: S4 includes the following steps: S41: Initialization parameters, including time step Δt n+1 , strain field ε(t n ,x), damage field d(t) n (x) and energy increment; S42: Initialize the damage field; S43: The iterative coupling algorithm uses the Steffensen accelerated method to solve the damage field. Until the convergence condition is met: 。 7. A method for simulating spatiotemporal fractional-order phase field fracture based on fast Fourier transform according to any one of claims 1 to 3, characterized in that: In S5, the formula for calculating the macroscopic stress field is: in, It represents the volume of the computation space.
8. An apparatus, characterized in that: The data processing method described in any one of claims 1 to 7 is executed.
9. An apparatus comprising a memory, a processor, and an algorithm stored in the memory and executable on the processor, characterized in that: When the processor executes the computer program, it implements the data processing method as described in any one of claims 1 to 7.
10. A computer-readable storage medium storing a computer algorithm, characterized in that, When the computer algorithm is executed by the processor, it performs the data processing as described in any one of claims 1 to 7.