CN111859645B - Improved MUSL format material dot method for shock wave solving - Google Patents
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Abstract
The invention provides an improved MUSL format object particle method for solving shock waves, which comprises the steps of firstly improving the momentum correction step in the traditional MUSL format; then establishing a shock wave flow field model, and carrying out discretization treatment on the shock wave flow field; then giving initial conditions and boundary conditions of a shock wave flow field, and setting a total calculation length; then calculating the time step of the current calculation step based on the stability condition; solving a shock wave flow field by an improved MUSL format material dot method based on shock wave solving; and finally, performing visual processing on the impact wave flow field. Compared with the traditional MUSL format, the method can effectively reduce the noise when the object particles pass through the grid, has lower requirement on the precision of a shape function, stronger robustness and stability, higher calculation efficiency and less calculation energy consumption, and can provide a new method for solving shock waves in engineering.
Description
Technical Field
The invention belongs to the technical field of shock wave flow field solving methods, and particularly relates to an improved MUSL-format material dot method for solving shock waves.
Background
Shock waves are discrete peaks that propagate in a fluid medium, resulting in a step-like change in physical properties of the flow field, such as pressure, temperature, density, etc. It is widely existed in engineering, such as train shock wave, mine blasting, explosion-proof vehicle, etc. when high-speed train runs, and may cause casualties and equipment damage. With the development of computer technology, the CFD technology is gradually applied to the simulation of shock waves, and the disturbance effect of the shock waves on the flow field is obtained by numerically calculating the propagation process of the shock waves, so that calculation parameters can be provided for the calculation of the aerodynamic load of an acting object of the shock waves. In addition, the shock wave problem is one of the classical examples of CFD, the parameters such as density, pressure and the like of a wave front flow field and a wave rear flow field are discontinuous, and the solution of the discontinuous problems is always a difficult point and a core problem of CFD development. Therefore, the method for exploring the novel shock wave solving method has wide engineering value.
The physical point method is a numerical calculation method mixing the lagrange method and the eulerian method, and has been widely applied to solving the high-speed impact problem and the explosion problem in the solid structure. The method uses material points to disperse objects, related physical parameters are carried by the material points, and a background grid is only used for integrating a momentum equation and solving a gradient, so that a structural grid covering a calculation domain is only needed to be simply generated. The material point method gets rid of the constraint of the grid to a certain extent, avoids the grid distortion problem caused by large deformation, can well treat the phenomena of material fracture failure, structural breakage and the like, does not cause mass loss, and is suitable for solving extreme working conditions. The MUSL format is a calculation format with momentum correction, in the traditional MUSL format in the material point method, the material point momentum of the next time step is mapped to the grid nodes based on the shape function of the current time step to obtain the corrected grid node momentum of the next time, and the grid node speed is calculated by using the corrected grid node momentum and the grid node quality before correction. Since the position of the material point is changed at this time, the shape function of the material point should be changed accordingly, but the change is not considered in the conventional MUSL format, when the material point passes through the grid, the correction method generates a large error, which affects the calculation accuracy and even causes the calculation to be not converged.
Disclosure of Invention
The invention aims to provide an improved MUSL format matter dot method for solving shock waves, and provides a method for predicting the influence of the shock waves in practical engineering.
The technical solution for realizing the purpose of the invention is as follows:
an improved MUSL-format material dot method for solving shock waves comprises the following steps:
step 2, establishing a shock wave flow field model, and carrying out discretization treatment on the shock wave flow field: dividing a background grid and distributing material points;
step 3, setting initial conditions and boundary conditions of a shock wave flow field, and setting a total calculation length;
step 4, calculating the time step length of the current calculation step based on the stability condition;
step 5, solving a shock wave flow field based on an improved MUSL format material dot method for solving shock waves;
step 6, performing visualization processing on the impact wave flow field: and outputting the density and pressure parameters of the flow field.
A shock wave solving system based on an improved MUSL format material dot method comprises a material dot momentum correction module, a shock wave flow field model establishing module, a total calculation length setting module, a time step calculation module, a shock wave flow field calculation module and a visualization processing module;
the material point momentum correction module is used for mapping the material point momentum of the next time step to the grid nodes based on the shape function of the current time step to obtain the corrected grid node momentum of the next time, canceling the momentum correction step in the traditional MUSL format, calculating the shape function based on the material point position of the next time step, and simultaneously mapping the material point mass and the momentum of the next time step to the grid nodes; the shock wave flow field model establishing module is used for establishing a shock wave flow field model and carrying out discretization processing on a shock wave flow field; the total length calculating setting module is used for setting the total length of calculation under the initial condition and the boundary condition of a given shock wave flow field; the time step calculation module is used for calculating the time step of the current calculation step; the shock wave flow field calculation module is used for solving a shock wave flow field based on an improved MUSL format material particle method for solving shock waves; the visual processing module is used for carrying out visual processing on the impact wave flow field: and outputting the density and pressure parameters of the flow field.
Compared with the prior art, the invention has the following remarkable advantages:
(1) The method solves the problem of shock waves by using a material point method, and generates a regular structure grid covering a calculation domain, thereby greatly reducing the workload and time of grid division; the mass of a single material point is always unchanged in calculation, the mass conservation law is automatically met, and a mass equation does not need to be solved; physical parameters such as pressure, density and the like are stored on the material points, so that the numerical dissipation is reduced; the method adopts the Lagrange method to solve in a single calculation step, avoids the problem of difficult processing of the convection term in the Eulerian method, abandons a deformed background grid between two calculation steps, reestablishes the mapping relation between the object points and the grid, and avoids the grid distortion problem in the Lagrange method.
(2) The correction method improves the momentum correction step in the traditional MUSL format, avoids the defect that the shape function lags behind the node momentum by one time step in the traditional MUSL format, corrects the grid node momentum, corrects the grid node quality, reduces the error caused by the material point crossing the grid, can more accurately capture the discontinuous characteristics before and after the wave surface of the shock wave, can well inhibit the oscillation, and can improve the calculation convergence speed.
Drawings
FIG. 1 is a flow chart of a method for solving shock wave material points according to the present invention.
Fig. 2 is a schematic of the Riemann problem.
Fig. 3 is a diagram of the results of solving the Riemann problem in accordance with the present invention and the conventional MUSL format.
Fig. 4 is a diagram comparing the analytic solutions of the Riemann problem of the present invention and the conventional MUSL format.
Detailed Description
The invention is further described with reference to the following figures and embodiments.
whereinFor the quality of the grid node i in the next calculation step, ->For the momentum of grid node i in the next computation step, m p Mass of a substance point, based on the mass of a substance point>For the speed of the substance point p in the next calculation step>Is the mapping function between the mesh node i and the material point p in the next calculation step.
Step 2, establishing a shock wave flow field model, carrying out discretization treatment on the shock wave flow field, carrying out geometric cleaning on the boundary of the flow field and establishing topology, covering the whole calculation domain with a uniform structural grid to generate a background grid, generating at least one layer of grid outside the boundary as a virtual grid, and uniformly distributing object points in the calculation range of the shock wave.
Step 3, giving initial conditions and boundary conditions of a shock wave flow field, setting and calculating total length, and giving density of each material point at initial timePressure>Speed->The volume of each substance point is calculated at the initial moment>Mass m p And internal energy->
Wherein V Ω Is the volume of the entire shock wave flow field, n p The total number of material points, gamma is the gas specific heat ratio, and the air is 1.4.
And applying a boundary condition on the background grid, wherein for a solid wall boundary, the speed, momentum and node force of the grid at the boundary and the virtual grid outside the boundary are always 0.
Step 4, calculating the time step of the current calculation step based on the stability condition, and setting the number of the database, wherein the time step of the current calculation step is calculated as follows:
where Δ t t For the time step of the current calculation step, C CFL The number of the chips is the number of the Kuran,is the sound velocity of the material point p at the present moment,is the speed of the substance point p in the x, y, z direction in the current calculation step, respectively->Is the density of the substance point p in the current calculation step, based on the value of the value point p in the current calculation step>L is the grid length, which is the pressure of the material point p in the current calculation step.
Step 5, solving the shock wave flow field based on an improved MUSL format substance particle method for solving shock waves, which comprises the following steps:
5.1: and mapping the material point parameters to the grid nodes to obtain the mass and momentum of the grid nodes as follows:
whereinFor the quality of the grid node i in the current calculation step, < >>For the momentum of grid node i in the current calculation step, < >>Is the speed of the substance point p in the current calculation step->Is a mapping function between the grid node i and the material point p in the current calculation step.
5.2: the grid node forces are calculated as:
wherein f is i t For the force value of grid node i in the current computation step,the derivative of the mapping function between the grid node i and the material point p in the current computation step.
5.3: the integral momentum equation is:
whereinFor the momentum of grid node i in the current calculation step, <' >>The momentum in the next computation step for mesh node i.
5.4: calculating the position and the speed of the material point at the next moment as follows:
whereinIs the speed of the substance point p in the current calculation step->For the velocity of the material point p in the next calculation step, n i For the number of nodes in the grid, is counted and/or selected>For the position of substance point p in the current calculation step>The position of the material point p in the next calculation step is calculated.
5.5: the step of correcting the momentum of the grid nodes in the traditional MUSL format is cancelled, the mass and the momentum of the object points are mapped to the grid nodes again, and the mass and the momentum of the grid nodes are corrected as follows:
5.6, the density of the updated material points is:
whereinThe strain increment of the material point p in the next calculation step, which is a square matrix, with rows j and k,is the density of the substance point p in the current calculation step, based on the value of the value point p in the current calculation step>The density of the material point p in the next calculation step is calculated.
5.7: the pressure of the mass point is updated using the ideal gas equation of state as:
5.8: and judging whether the end point moment is calculated or not, if not, enabling t = t +1, turning to the step 4, calculating the time step again, and if so, ending the step and entering the next step.
And 6, visualizing the shock wave flow field, storing time-course data of parameters such as density on grid nodes and the like, and obtaining the propagation process of the shock wave in the flow field.
The invention also provides an improved MUSL-format object particle method and a computer program based on the shock wave solving, and the shock wave solving system also comprises an object particle momentum correction module, a shock wave flow field model establishing module, a total length calculating module, a time step calculating module, a shock wave flow field calculating module and a visualization processing module;
the material point momentum correction module is used for mapping the material point momentum of the next time step to the grid nodes based on the shape function of the current time step to obtain the corrected grid node momentum of the next time, canceling the momentum correction step in the traditional MUSL format, calculating the shape function based on the material point position of the next time step, and simultaneously mapping the material point mass and the momentum of the next time step to the grid nodes, wherein the specific process is shown in the step 1; the shock wave flow field model establishing module is used for establishing a shock wave flow field model and carrying out discretization treatment on the shock wave flow field, and the specific process is shown in the step 2; the total length calculating setting module is used for setting the total length calculation according to the initial condition and the boundary condition of the given shock wave flow field, and the specific process is shown in the step 3; the time step calculation module is used for calculating the time step of the current calculation step, and the specific process is shown in the step 4; the shock wave flow field calculation module is used for solving the shock wave flow field based on an improved MUSL format material dot method for solving shock waves, and the specific process is shown in the step 5; the visual processing module is used for carrying out visual processing on the impact wave flow field: outputting the density and pressure parameters of the flow field, wherein the specific process is shown in the step 6; the program processing process is the same as the processing process of the method, so that the method is not described in detail.
Example 1
The process of the above shockwave solving method does not specify any specific object, that is to say, is applicable to solving any shockwave propagation problem in engineering. In order to explain the operation flow and application effect of the method in detail, the application of the invention is explained in detail by taking a classic Riemann problem as an example. The Riemann problem, namely the decomposition problem of the initial discontinuity, is one of the classical examples of CFD, and comprises various discontinuities in CFD, and the solution of the discontinuity problems is always a difficult and core problem in the development of CFD. The Riemann problem has an analytic solution, and can be used for checking the correctness and the precision of a numerical simulation method, and therefore almost all flow field solving methods are established on the basis of solving the Riemann problem and then are expanded towards multiple dimensions.
The Riemann problem describes a closed tube with different gases on the left and right sides, the gases on the two sides being separated by a thin film in the middle, and at the beginning of the calculation, the diaphragm in the middle is suddenly removed, the gases on the two sides will be mixed with each other under the action of pressure, and finally, an equilibrium state is reached, which is also called a shock tube problem. The total calculation time of the calculation example is t =0.2, all parameters in the calculation example are dimensionless parameters, a virtual grid needs to be generated at a boundary when a B spline mapping function is used, the total number of background grids generated in the present example is 1002, 1 virtual grid is respectively arranged at the outer sides of the left end and the right end of a shock tube, each grid body at the initial moment except the virtual grid contains 2 object points, so that the number of the object points is 2000, the number of the library is set to be 0.8, the gas specific heat ratio gamma =1.4, the calculation domain is x e (0, 1), and the intermittent distribution at the initial moment is as follows:
in order to verify the robustness of the method, a 2-time B spline basis function (3-point interpolation) with lower precision and a 3-time B spline basis function (4-point interpolation) with higher precision are respectively adopted as shape functions between the material point and the grid nodes, and the calculation result of the method is compared with the analysis solution of the traditional MUSL format and Riemann problem.
whereinFor the quality of the grid node i in the next calculation step, ->For the momentum of grid node i in the next computation step, m p Is the mass of the substance point, is based on>For the speed of the substance point p in the next calculation step>Is the mapping function between the mesh node i and the material point p in the next calculation step.
Step 2, establishing a shock tube model, regarding the shock tube model as a one-dimensional line model with the length of 1, carrying out discretization treatment on a flow field in the shock tube, uniformly generating 1000 grids in the tube, respectively arranging 1 virtual grid outside the left end point and the right end point of the tube, setting the total number of background grids to be 1002, and setting 2 material points in each grid except the virtual grid, wherein the total number of the material points is 2000.
Step 3, giving initial conditions and boundary conditions of the shock tube, and setting the density, the speed and the pressure of material points in the tube at the initial moment as follows:
the volume, mass and internal energy of each material point are calculated as follows:
here V Ω =1,n p =2000。
And applying boundary conditions to the left end and the right end of the shock tube to enable the speed, momentum and force of the end points and the grid nodes outside the end points to be 0 all the time.
And 4, calculating the time step according to the stability condition as follows:
wherein Δ t t For the time step of the current calculation step, C CFL =0.8,Is the sound speed of the present moment of the substance point p, < >>Is the speed of the substance point p in the x-direction in the current calculation step->Is the density of the substance point p in the current calculation step, based on the value of the value point p in the current calculation step>Is the pressure of the mass point p in the current calculation step.
Step 5, solving the shock wave flow field based on an improved MUSL format substance particle method for solving shock waves, which comprises the following steps:
5.1: mapping the material point parameters to grid nodes to obtain the mass and momentum of the grid nodes as follows:
whereinFor the quality of the grid node i in the current calculation step, < >>For the momentum of grid node i in the current computation step, m p Is the mass of substance point p>Is the speed of the substance point p in the current calculation step->Is a mapping function between the grid node i and the material point p in the current calculation step.
5.2: the grid node forces are calculated as:
wherein f is i t For the force value of grid node i in the current computation step,the derivative of the mapping function between the grid node i and the material point p in the current computation step.
5.3: the integral momentum equation is:
whereinFor the momentum of grid node i in the current calculation step, < >>The momentum in the next computation step for mesh node i.
5.4: calculating the position and the speed of the material point at the next moment as follows:
whereinIs the speed of the substance point p in the current calculation step->For the velocity of the material point p in the next calculation step, n i For the number of nodes in the grid, is counted and/or selected>Is the position of the substance point p in the current calculation step, based on the evaluation value>The position of the material point p in the next calculation step is taken.
5.5: the step of correcting the momentum of the grid nodes in the traditional MUSL format is cancelled, the mass and the momentum of the object points are mapped to the grid nodes again, and the mass and the momentum of the grid nodes are corrected as follows:
5.6, the density of the updated material points is:
whereinThe strain increment of the material point p in the next calculation step, which is a square matrix, with rows j and k,is the density of the substance point p in the current calculation step, based on the value of the value point p in the current calculation step>The density of the material point p in the next calculation step is calculated.
5.7: the pressure of the mass point is updated using the ideal gas equation of state as:
5.8: and judging whether the calculation time reaches 0.2, if not, turning to the step 4, and if so, ending the next step.
And 6, performing visual processing on the flow field in the shock tube, and outputting the density of each grid node to obtain the density distribution characteristic of the flow field.
When a 2-order B spline function is taken as a shape function, the calculation result is shown in FIG. 3, and it can be seen that the difference between the calculation result of the traditional MUSL format and the analytic solution is large, the dispersion and dissipation of the density curve are large, and the result is severely oscillated; the calculation result and the analytic solution of the method of the invention are very close, and the traditional MUSL format needs about 1220 steps of calculation to complete the calculation, and the method of the invention only needs 1040 steps. When a 3-order B spline function is taken as a shape function, the calculation result is shown in figure 4, along with the improvement of the precision of a mapping function, the calculation precision of the MUSL format and the traditional MUSL format is improved, but the dispersion and the dissipation of the MUSL format are smaller, particularly at the discontinuous part, the density curve of the MUSL format is closer to an analytical solution, and the convergence speed is higher.
In summary, under the same shape function, the calculation speed and the calculation accuracy of the method are higher than those of the traditional MUSL format; the method has better robustness and lower requirement on the precision of the shape function, can obtain better calculation results under a 3-point interpolation format, and can better capture shock wave characteristics under a 4-point interpolation format in the traditional MUSL format, thereby indicating that the method has lower calculation energy consumption.
Claims (6)
1. An improved MUSL-format material dot method for solving shock waves is characterized by comprising the following steps of:
step 1, improving the momentum correction step in the traditional MUSL format: mapping the material point momentum of the next time step onto the grid nodes based on the shape function of the current time step to obtain the corrected grid node momentum of the next time, canceling the momentum correction step in the traditional MUSL format, calculating the shape function based on the material point position of the next time step, and mapping the material point mass and the momentum of the next time step onto the grid nodes;
step 2, establishing a shock wave flow field model, and carrying out discretization treatment on the shock wave flow field: dividing a background grid and distributing material points;
step 3, setting initial conditions and boundary conditions of a shock wave flow field, and setting a total calculation length;
step 4, calculating the time step length of the current calculation step based on the stability condition; and (3) calculating the time step length of the current calculation step by the set Curan number as follows:
where Δ t t For the time step of the current calculation step, C CFL Is the number of the Kuran number,is the sound velocity of the material point p at the present moment,respectively the speed of the substance point p in the current calculation step in the x, y, z direction, in conjunction with a selection of the number of key points in the key block, in the x, y, z direction>Is the density of the substance point p in the current calculation step, based on the value of the value point p in the current calculation step>Is the pressure of the material point p in the current calculation step, gamma is the gas specific heat ratio, L is the grid length, n p Is the total number of material points;
step 5, solving a shock wave flow field by an improved MUSL format object particle method based on shock wave solving; the method comprises the following steps:
step 5.1, mapping the material point parameters to grid nodes to obtain the quality and momentum of the grid nodes as follows:
whereinFor the quality of the grid node i in the current calculation step, < >>For the momentum of mesh node i in the current computation step,is the speed of substance point p in the current calculation step>For a mapping function between grid node i and material point p in the current computation step, m p Is the mass of a material point;
step 5.2, calculating the grid node force as follows:
wherein f is i t For the force value of grid node i in the current computation step,the derivative of the mapping function between the grid node i and the material point p in the current calculation step is obtained;
and 5.3, an integral momentum equation is as follows:
whereinFor the momentum of grid node i in the current calculation step, <' >>The momentum of the grid node i in the next calculation step;
and 5.4, calculating the position and the speed of the material point at the next moment:
whereinIs the speed of the substance point p in the current calculation step->For the velocity of the material point p in the next calculation step, n i For the number of grid nodes, <' >>Is the position of the substance point p in the current calculation step, based on the evaluation value>The position of the material point p in the next calculation step;
and 5.5, canceling the step of correcting the momentum of the grid nodes in the traditional MUSL format, and mapping the mass and the momentum of the object points to the grid nodes again, wherein the step of correcting the mass and the momentum of the grid nodes is as follows:
and 5.6, updating the density of the object points as follows:
whereinThe increment of the strain in the next calculation step for substance point p is a square matrix with rows j and k, which is based on the value of the strain in the next calculation step>Is the density of the substance point p in the current calculation step, based on the value of the value point p in the current calculation step>The density of the material point p in the next calculation step;
and 5.7, updating the pressure of the material point by using an ideal gas state equation to be as follows:
whereinThe internal energy of the material point p in the next calculation step is shown, and gamma is the specific heat ratio of the gas;
step 5.8, judging whether the terminal time is calculated or not, if not, making t = t +1 and turning to the step 4, if so, ending the step and entering the next step;
step 6, performing visualization processing on the impact wave flow field: and outputting the density and pressure parameters of the flow field.
2. The improved MUSL format object point method for shockwave resolution as recited in claim 1, wherein step 1 improves the momentum correction step in the conventional MUSL format, and the obtained grid node quality and momentum after correction are:
whereinFor the quality of the grid node i in the next calculation step>For the momentum of grid node i in the next computation step, m p Is the mass of the substance point, is based on>Is the speed of the substance point p in the next calculation step>Is the mapping function between the mesh node i and the material point p in the next calculation step.
3. The improved MUSL-format object point method for shock wave solving as recited in claim 1, wherein step 2 is to create a shock wave flow field model, discretize the shock wave flow field, geometrically clean the boundaries of the flow field and create topology, cover the whole computational domain with a uniform structural grid to create a background grid, create at least one layer of grid outside the boundaries as a virtual grid, and uniformly lay object points within the computational range of the shock wave.
4. The method of claim 1, wherein step 3 provides initial conditions and boundary conditions of the shockwave flow field, sets the total length of the calculation, and provides the density of each material point at the initial timePressure->Speed->The volume of each substance point is calculated at the initial moment>Mass m of matter point p And internal energy->
Wherein V Ω Is the volume of the entire shock wave flow field, n p Is the total number of material points, and gamma is the specific heat ratio of gas;
and applying boundary conditions on the background grids, wherein for the solid wall boundary, the speed, momentum and nodal force of the grids at the boundary and the virtual grids outside the boundary are always 0.
5. The improved MUSL format material point method for shock wave solution according to claim 1, wherein step 6 is performed to visualize the shock wave flow field, and time-course data of density parameters on grid nodes is stored to obtain the propagation process of the shock wave in the flow field.
6. A shock wave solving system based on an improved MUSL format material point method is characterized by comprising a material point momentum correction module, a shock wave flow field model establishing module, a total length calculation setting module, a time step length calculation module, a shock wave flow field calculation module and a visualization processing module;
the material point momentum correction module is used for mapping the material point momentum of the next time step to the grid nodes based on the shape function of the current time step to obtain the corrected grid node momentum of the next time, canceling the momentum correction step in the traditional MUSL format, calculating the shape function based on the material point position of the next time step, and simultaneously mapping the material point mass and the momentum of the next time step to the grid nodes; the shock wave flow field model establishing module is used for establishing a shock wave flow field model and carrying out discretization processing on a shock wave flow field; the total length calculating setting module is used for setting the total length of calculation under the initial condition and the boundary condition of a given shock wave flow field; the time step calculation module is used for calculating the time step of the current calculation step; the shock wave flow field calculation module is used for solving a shock wave flow field based on an improved MUSL format material particle method for solving shock waves; the visualization processing module is used for performing visualization processing on the impact wave flow field: outputting density and pressure parameters of a flow field;
the mass and momentum of the grid nodes after being corrected by the material point momentum correction module are as follows:
whereinFor the quality of the grid node i in the next calculation step>For the momentum of grid node i in the next computation step, m p Is the mass of the substance point, is based on>Is the speed of the substance point p in the next calculation step>A mapping function between the grid node i and the material point p in the next calculation step is obtained;
the time step length of the current calculation step obtained by the time step length calculation module is:
where Δ t t For the time step of the current calculation step, C CFL Is the number of the Kuran number,is the sound velocity of the material point p at the present moment,is the speed of the substance point p in the x, y, z direction in the current calculation step, respectively->Is the density of the substance point p in the current calculation step, based on the value of the value point p in the current calculation step>Is the pressure of the material point p in the current calculation step, gamma is the gas specific heat ratio, L is the grid length, n p Is the total number of material dots. />
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