SPH-WENO numerical simulation method for solid strong impact problems

Abstract

The present invention discloses an SPH-WENO numerical simulation method applied to the problem of strong impact of solids, and relates to the technical field of SPH numerical simulation for the strong impact process of elastoplastic materials, including high-precision WENO reconstruction of the left and right states of the contact particle pair based on the SPH framework, and interpolation of the missing template points in the WENO template using the second-order Taylor expansion to reconstruct the left and right states of the particle pair; constructing a coordinate rotation-independent contact solution framework, substituting the reconstructed left and right states into the framework structure for iterative solution, and obtaining the contact solution of each component of the physical parameters of the particle itself. Therefore, by adopting the above-mentioned SPH-WENO numerical simulation method applied to the problem of strong impact of solids, it is possible to construct a WENO high-order format under the SPH framework, obtain the left and right states containing the density velocity vector and the stress vector, thereby obtaining a high-precision contact solution, and realizing low dissipation and high-precision numerical simulation for the problem of strong impact of solids.

Description

Technical Field

[0001] The invention relates to the technical field of SPH numerical simulation for strong impact process of elastoplastic materials, and in particular to an SPH-WENO numerical simulation method applied to the strong impact problem of solids. Background Art

[0002] Smoothed particle dynamics (SPH) is a meshless numerical solution method for simulating fluid mechanics problems. With the continuous improvement and development of the SPH method, various particle formats continue to emerge, thereby improving its calculation accuracy.

[0003] At present, many studies have proposed a weighted essential non-oscillation (WENO) format for inviscid fluids, which aims to effectively overcome the numerical pseudo-oscillation near the discontinuous solution through the nonlinear weighted combination of different templates, while achieving high-resolution capture of discontinuous solutions such as shock waves and high-precision solution of smooth solutions. However, the existing WENO method generally has the problem of being time-consuming and is only applicable to inviscid liquids.

[0004] The traditional approach is to use artificial viscosity to eliminate stretching instability and use kernel functions to discretize the particle pair information. However, it is necessary to manually adjust the empirical parameters according to the calculation conditions to suppress non-physical numerical oscillations, thereby damaging the reliability of the calculation. Based on this, Parshikov et al. proposed a contact algorithm under the SPH framework, which alleviated this problem by introducing implicit numerical viscosity through the Riemann solution. However, the contact variables use first-order approximation, so the accuracy is still first-order accuracy, and the implicit viscosity still introduces dissipation in the particle pair sparse process, making it difficult to accurately approximate the sparse wave. In addition, Parshikov et al. proposed that the contact algorithm framework cannot guarantee rotation independence in the three-dimensional case, and there is a potential lack of consistency in the reconstruction of variables under strong rotational motion.

[0005] During the solid impact process, as the elastic-plastic characteristics of the material differ, a complex evolutionary wave system will be generated accompanied by deformation movement. Therefore, developing a numerical SPH algorithm that abandons artificial empirical parameters and has high solution accuracy is conducive to improving the calculation accuracy of SPH in solid impact problems and providing a reference for the development of high-precision numerical simulation methods in solid problems. Summary of the invention

[0006] The purpose of the present invention is to provide an SPH-WENO numerical simulation method for solid strong impact problems. By constructing a solid WENO construction format under the SPH framework, its numerical theoretical accuracy can be improved to between the second and third order.

[0007] To achieve the above object, the present invention provides a SPH-WENO numerical simulation method applied to the solid strong impact problem, comprising:

[0008] The left and right states of the contact particle pairs are reconstructed by WENO based on the SPH framework, and the missing template points in the WENO template are interpolated using the second-order Taylor expansion to obtain the left and right states of the particle pairs.

[0009] Preferably, based on the reconstructed left state and right state, the contact solution of each component of the particle's own physical parameters is obtained by iteratively solving the rotation-independent coordinate system.

[0010] Preferably, the second-order Taylor expansion is used to interpolate the missing template points in the WENO template as follows:

[0011] ;

[0012] In the formula, , Represents the template points , The physical parameters of Represents particles The derivative of its own physical parameter, Represents the distance between particles.

[0013] Preferably, the physical parameters of the particle itself include components of the velocity vector and the stress vector.

[0014] Preferably, the coordinate rotation-independent contact solution framework includes reconstructing each basis vector of the rotation-independent coordinate system, and the rotation-independent coordinate system will ensure the consistency of the variable reconstruction when the system has strong rotational motion, as follows:

[0015] First, the direction of the particle pair connection is Direction, take the basis vector as ;

[0016] Then construct based on the variables of the particle itself , Basis vectors in the direction , :

[0017] Get the maximum eigenvalue of the strain rate tensor matrix, and its corresponding eigenvector and basis vector The direction of the vector after the cross product, that is, the basis vector The direction of the basis vector Since the basis vector With basis vectors Obtained by cross multiplication.

[0018] Preferably, the basis vector The calculation method is as follows:

[0019] First, the strain rate tensor of the particle pair is obtained ,in , Particles , The strain rate tensor of

[0020] Then, check Is it a zero matrix: When When it is a zero matrix, the basis vector Solved using the Parshikov framework ,in is the basis vector on the z-axis; otherwise, first calculate the matrix The eigenvector corresponding to the largest eigenvalue , then the basis vector .

[0021] Preferably, the contact solutions of the components of the physical parameters of the particle itself are as follows:

[0022] ;

[0023] ;

[0024] ;

[0025] ;

[0026] ;

[0027] ;

[0028] In the formula, , , Respectively represent the particles in , , Contact solution for stress components in the direction, , , Respectively expressed in , , The contact solution for the velocity component in the direction, , , , denote the density, wave speed, velocity and stress vector of the particle, respectively. , , Respectively represent the components of the corresponding scalar, subscript , Represent the left state and the right state respectively.

[0029] Therefore, the present invention adopts the above-mentioned SPH-WENO numerical simulation method applied to the solid strong impact problem, which has the following technical effects:

[0030] (1) Based on the SPH framework, the WENO high-order format is constructed to realize high-order numerical simulation of solid strong impact problems and improve its accuracy to between the second and third order.

[0031] (2) Use the second-order Taylor expansion interpolation to obtain the missing template points in the WENO template, which are used to complete the components of the particle stress tensor and realize WENO reconstruction.

[0032] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 It is a schematic diagram of the left and right states of a particle pair in the framework of the traditional Parshikov contact solution in an embodiment of the SPH-WENO numerical simulation method applied to the solid strong impact problem;

[0034] Figure 2 It is a schematic diagram of WENO reconstruction of the left and right states of a particle pair in an embodiment of the SPH-WENO numerical simulation method applied to the solid strong impact problem;

[0035] Figure 3 is a result accuracy diagram of different algorithms for calculating solid shock waves in an embodiment of the SPH-WENO numerical simulation method applied to the solid strong impact problem;

[0036] Figure 4 are parameter curves obtained by calculating solid shock waves using different methods in an embodiment of the SPH-WENO numerical simulation method applied to the solid strong impact problem, wherein: Figure 4 (a) is the density curve. Figure 4 (b) is the pressure curve. Figure 4 (c) is the speed curve. Figure 4 Middle (d) is the deviatoric stress curve;

[0037] Figure 5 is a density diagram of a two-dimensional example in an embodiment of the SPH-WENO numerical simulation method applied to the solid strong impact problem. Figure 5 (a) is the two-dimensional example density map of each algorithm. Figure 5 (b) 2D example shear diagram of the traditional SPH algorithm. Figure 5 (c) is a shear diagram of a two-dimensional example of the contact SPH algorithm. Figure 5 (d) is a shear diagram of a two-dimensional example of the WENO SPH algorithm;

[0038] Figure 6These are the parameter curves obtained by applying different methods to a two-dimensional example in the embodiment of the SPH-WENO numerical simulation method for solid strong impact problems. Figure 6 (a) is the density curve. Figure 6 (b) is the pressure curve. Figure 6 (c) is the tangential velocity curve. Figure 6 Middle (d) is the xy component curve of deviatoric stress;

[0039] Figure 7 It is the density cloud map and effective plastic strain map of the three-dimensional example in the embodiment of the SPH-WENO numerical simulation method applied to the solid strong impact problem. Figure 7 (a) is the density cloud map of the traditional SPH algorithm. Figure 7 (b) is the density cloud diagram of the contact SPH algorithm. Figure 7 (c) is the density cloud map of WENO SPH algorithm. Figure 7 (d) is the effective plastic strain diagram of the traditional algorithm. Figure 7 (e) is the effective plastic strain diagram of the contact algorithm. Figure 7 (f) is the effective plastic strain diagram of WENO SPH algorithm;

[0040] Figure 8 These are the parameter curves obtained by applying different methods to three-dimensional examples in the implementation of the SPH-WENO numerical simulation method for solid strong impact problems. Figure 8 (a) is the density curve. Figure 8 (b) is the radial velocity curve. Figure 8 (c) is the radial stress curve. Figure 8 (d) is the pressure curve. Figure 8 (e) is the radial deviator stress curve. Figure 8 (f) is the effective plastic strain curve. DETAILED DESCRIPTION

[0041] The present invention can be explained in more detail by the following examples. The purpose of disclosing the present invention is to protect all changes and improvements within the scope of the present invention. The present invention is not limited to the following examples.

[0042] Embodiment 1

[0043] The present invention provides an SPH-WENO numerical simulation method for solid strong impact problems, and performs SPH numerical simulation on complex elastic-plastic wave systems and deformation evolution during strong impact of elastic-plastic materials. The SPH numerical simulation method in traditional solid problems usually uses an artificial viscosity method that requires empirical parameters, which is essentially a first-order accuracy calculation format, and there has been no higher-precision SPH framework application in solid calculations. In addition, the existing SPH-WENO framework is only used in fluid calculations, and its reconstructed quantity is limited to the scalar velocity and scalar pressure of the particle pair direction, while its application in solids requires consideration of WENO reconstruction with more components such as stress tensors. Therefore, in this embodiment, a Riemann contact solution framework that is independent of coordinate rotation is proposed for complex three-dimensional rotation states, that is, a contact solution framework for rotation-independent coordinate systems; based on this framework, the reconstructed left and right states of the contact particle pair are substituted into a time-iterative solution to obtain the contact solutions of the components of the particle's own physical parameters, as follows:

[0044] like Figure 1 As shown, each particle pair (particle and particles ) constitutes a containing density ,speed , stress vector The left and right states of the same parameters and , where the subscript Indicates left state, subscript Indicates right state, superscript , , Represents the corresponding components of the three basis vectors in the Cartesian coordinate system. Corresponding particles direction, Direction and The z-axis direction is perpendicular to the The direction corresponds to , The direction is perpendicular. Through the contact algorithm, the velocity and stress vectors of the left and right particles are solved as follows:

[0045] ;

[0046] Where the subscript * represents the contact solution of the corresponding component, Represents the wave speed of the particle.

[0047] in, is the direction of the particle connection, and the corresponding basis vector is ; and the tangent direction of the coordinate system , In general three-dimensional conditions, it can be in any direction, but it is difficult to ensure the independence of coordinate rotation in strong rotation motion. Therefore, it is necessary to construct a new coordinate system basis vector based on the variables of the particle itself. and .

[0048] Since it is necessary to make the tangential direction of the transformed coordinate system satisfy the coordinate rotation independence under three-dimensional working conditions, and the traditional Parshikov coordinate system architecture cannot guarantee independence from the coordinate system rotation, therefore, in this embodiment, the Parshikov coordinate system framework is improved and the basis vectors of the coordinate system are reconstructed. and :First, find the maximum eigenvalue of the strain rate tensor matrix, and its corresponding eigenvector is related to the particle direction The vector direction obtained by the cross product of is the first tangent vector direction. Then we can get Cross product get , the steps are as follows:

[0049] First, the strain rate tensor of the particle pair is obtained The matrix is ​​in the particle pair It remains unchanged under exchange conditions, so the symmetric structure of particle pairs can ensure the conservation of momentum and energy during the calculation process.

[0050] Then, check Is it a zero matrix? If so, then the basis vectors Solved using the Parshikov framework ,in is the basis vector on the z-axis. If not, first calculate the matrix The eigenvector corresponding to the largest eigenvalue , then the basis vector Finally, we get the basis vector .

[0051] On this basis, the left and right states are reconstructed by WENO, such as Figure 2 The traditional approach is to use first-order or nearest point interpolation, which is time-consuming and has low precision. In order to obtain a low-dissipation, high-precision WENO format, in this embodiment, the missing template points in the WENO template are interpolated by second-order Taylor expansion, as follows:

[0052] A complete WENO third-order reconstruction framework requires four equally spaced template points. , , , in , Corresponding particles , The physical parameters of the template point are missing in the WENO template. (particle Left side), (particle The parameter values ​​of the right side) are expanded by the second-order Taylor expansion and the particle ,particle The physical parameters of and its spatial derivative The expression is as follows:

[0053] ;

[0054] In the formula, represents the distance between particles, the superscript represents the derivative of a scalar.

[0055] Physical parameters Including the components of the velocity vector and the stress vector, its derivatives in each direction are:

[0056] ;

[0057] ;

[0058] ;

[0059] ;

[0060] ;

[0061] ;

[0062] In the formula, , denote velocity gradient and stress vector gradient respectively, , , Respectively , , The basis vectors of the direction, represents the Kronecker product.

[0063] In summary, we get a complete WENO template to reconstruct the left state of the particle pair and right status In the WENO high-order format reconstruction process, the template point Particle Since the reconstruction process of the left state and the right state is symmetrical, this embodiment only describes the construction process of the left state. , , The physical parameters of the left state can be obtained through two candidate templates and Weighted to get:

[0064] ;

[0065] in,

[0066] .

[0067] In the formula, , They represent the weight functions of the corresponding templates respectively.

[0068] The weight function Determined by the smoothness of the numerical solution, the optimal order weight function is obtained under the WENO framework , calculated as follows:

[0069] ;

[0070] Among them, the parameters is a small quantity that avoids the denominator being zero, The parameters adopt the optimal weights of the traditional WENO method:

[0071] , ;

[0072] and and is the smoothness index:

[0073] ;

[0074] .

[0075] According to the reconstructed states of the left and right particles and , the corresponding contact solution is obtained and solved iteratively using the control equations.

[0076] like Figure 3 As shown in the figure, by constructing the solid WENO construction format under the SPH framework, namely the new convergence analysis algorithm WENOSPH, the theoretical numerical accuracy has been successfully improved to between the second and third order; while the traditional SPH algorithm and contact SPH algorithm can only achieve first-order accuracy. This shows that the SPH-WENO numerical simulation method has great advantages when applied to solid strong impact problems.

[0077] Embodiment 2

[0078] The SPH-WENO numerical simulation method is used to calculate the solid shock wave, such as Figure 4 As shown in the figure, it can be seen that for the elastic shock wave and plastic shock wave on the right, the WENO SPH algorithm effectively improves the calculation accuracy, and it greatly reduces the shock wave dissipation compared with other methods. In the traditional SPH algorithm and contact SPH algorithm, the plastic wave and elastic wave are almost merged and lose resolution due to high dissipation.

[0079] Embodiment 3

[0080] The WENO SPH algorithm is applied to high-dimensional examples and compared with the traditional SPH algorithm and the contact SPH algorithm. The WENO SPH algorithm greatly reduces the dissipation of shock waves and greatly suppresses the non-physical numerical oscillation near the contact discontinuity compared with the traditional SPH algorithm.

[0081] like Figure 5 and Figure 6 As shown in the figure, in the results of the two-dimensional shear example, for plastic shear waves and contact discontinuity, WENO SPH effectively suppresses the numerical oscillation caused by tensile instability and other reasons compared with the traditional SPH algorithm. At the same time, compared with the contact SPH algorithm, it significantly improves the calculation accuracy of plastic shear waves and realizes the accurate capture of left and right elastic shear waves.

[0082] Embodiment 4

[0083] Apply the WENO SPH algorithm to three-dimensional examples, such as Figure 7 and Figure 8 As shown, it can be seen that the WENO SPH algorithm is suitable for three-dimensional working conditions, and compared with the traditional SPH algorithm and the contact SPH algorithm, it effectively improves the accuracy of shock wave capture. At the same time, compared with the traditional SPH algorithm, it effectively suppresses the oscillation caused by contact discontinuity, and greatly alleviates the non-physical oscillation problem caused by the tensile instability of the sparse wave of the traditional SPH algorithm.

[0084] Therefore, the present invention adopts the above-mentioned SPH-WENO numerical simulation method applied to the strong impact problem of solids, which can construct the WENO high-order format under the SPH framework, obtain the left and right states including the density velocity vector and the stress vector, thereby obtaining a high-precision contact solution and realizing high-order numerical simulation of the strong impact problem of solids.

[0085] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. The SPH-WENO numerical simulation method applied to the solid strong impact problem is characterized by: include: Based on the SPH framework, the left and right states of the contact particle pair are reconstructed by WENO, and the missing template points in the WENO template are interpolated using the second-order Taylor expansion to obtain the left and right states of the particle pair. Based on the reconstructed left and right states, the contact solution framework of the rotation-independent coordinate system is iteratively solved to obtain the contact solution of each component of the particle's own physical parameters; Among them, the contact solution framework of the rotation-independent coordinate system includes reconstructing the basis vectors of the rotation-independent coordinate system as follows: First, the direction of the particle pair connection is Direction, take the basis vector as ; Then construct based on the variables of the particle itself , Basis vectors in the direction , : Get the maximum eigenvalue of the strain rate tensor matrix, and its corresponding eigenvector and basis vector The direction of the vector after the cross product, that is, the basis vector The direction of the basis vector Since the basis vector With basis vectors Obtained by cross multiplication.

2. The SPH-WENO numerical simulation method for solid strong impact problems according to claim 1 is characterized in that: The second-order Taylor expansion interpolates the missing template points in the WENO template as follows: ； In the formula, , Represents the template points and The physical parameters of Represents particles The derivative of its own physical parameter, Represents the distance between particles.

3. The SPH-WENO numerical simulation method for solid strong impact problems according to claim 1 is characterized in that: The physical parameters of the particle itself include the components of the velocity vector and the stress vector.

4. The SPH-WENO numerical simulation method for solid strong impact problems according to claim 1 is characterized in that: Basis Vectors The calculation method is as follows: First, the strain rate tensor of the particle pair is obtained ,in , Particles , The strain rate tensor of Then, check Is it a zero matrix: When When it is a zero matrix, the basis vector Solved using the Parshikov framework ,in is the basis vector on the z-axis; otherwise, first calculate the matrix The eigenvector corresponding to the largest eigenvalue , then the basis vector .

5. The SPH-WENO numerical simulation method for solid strong impact problems according to claim 1 is characterized in that: The contact solutions of the components of the particle's own physical parameters are as follows: ； ； ； ； ； ； In the formula, , , Respectively represent the particles in , , Contact solution for stress components in the direction, , , Respectively expressed in , , The contact solution for the velocity component in the direction, , , , denote the particle density, wave speed, velocity and stress vector, respectively. , , Respectively represent the components of the corresponding scalar, subscript , Represent the left state and the right state respectively.

Images