High-order explicit compact difference method for incompressible flow problem in irregular region
By combining high-order explicit compact difference with the submerged boundary method, the problem of fully explicit solution of incompressible flow in irregular regions is solved, achieving high-precision flow simulation applicable to various engineering application scenarios.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-02
- Publication Date
- 2026-03-10
AI Technical Summary
Existing technologies struggle to achieve high-precision, fully explicit solutions to incompressible flow problems in irregular regions. Traditional methods suffer from high computational complexity and low efficiency, and cannot effectively capture complex flow structures.
By employing a high-order explicit compact difference method combined with the submerged boundary method, and by constructing a rectangular region and arranging Lagrange points on the boundary of the irregular region, an entropy-damped artificial compressibility equation is established, thereby achieving a fully explicit solution to the incompressible flow problem in the irregular region.
It achieves high-precision, fully explicit solutions to flow problems in irregular regions, avoiding the solution step of the pressure Poisson equation, ensuring the stability of the pressure field and the simplicity of calculation, and can accurately capture complex flow field structures.
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Figure CN121638121A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The application belongs to the technical field of solving the problem of incompressible flow in irregular regions, and particularly relates to a high-order explicit compact difference method for solving the problem of incompressible flow in irregular regions. BACKGROUND
[0002] The problem of incompressible flow in irregular regions is widely used in key fields such as ocean engineering, biomedical engineering and advanced manufacturing. For example, the flow around and scouring of irregular structures such as deep-sea drilling platform legs and underwater vehicle shells can affect the stability of equipment structures. The blood flow in irregular flow channels formed by human arterial branches and atherosclerotic plaques and the flow around implanted devices such as artificial blood vessels and blood vessel lesions are closely related to the optimization of devices. The filling of molten polymers in irregular cavities during the injection molding of automobile parts and the spreading of composite material slurry in irregular paths during 3D printing also belong to this type of flow problem. The flow in these scenarios not only faces the problem of irregular or dynamic changes in geometric boundaries, but also often involves complex phenomena such as unsteady effects and vortex evolution. However, traditional numerical methods have always struggled to balance grid adaptation, precision retention and computational efficiency. This problem has become a key bottleneck restricting engineering design and technological breakthroughs in related fields. Therefore, the research on high-precision explicit methods for incompressible flow in irregular regions can not only improve the safety and reliability of structures, but also provide theoretical support for the development of new materials and new technologies.
[0003] The incompressible Navier-Stokes equation is the core control equation for describing incompressible flow and can comprehensively describe the laws of fluid momentum transfer and mass conservation. However, when solving with original variables, mainstream methods such as the projection method require solving a pressure Poisson equation to achieve velocity and pressure decoupling, which not only increases computational complexity, but also becomes a key bottleneck hindering full explicit calculation of incompressible flow. While the current known artificial compressibility method can achieve "explicit" solving, it needs to be combined with a double time step technique to ensure stability, which significantly offsets the efficiency advantage and easily causes pressure oscillation, reducing the physical authenticity of simulation results. Explicit calculation is sensitive to the capture of transient changes in unsteady complex flow, and its application is limited, which seriously affects the efficiency and applicability of simulation.
[0004] In the numerical simulation of incompressible flow in irregular regions, the traditional body-fitted grid method is the mainstream way to deal with complex boundaries. This method needs to strictly tailor the grid according to the geometric profile of the irregular region, so that the grid boundary accurately fits the physical boundary of the flow region. However, when facing highly complex geometric scenarios such as deep-sea equipment flow channels, curved blood vessels in the human body, and injection mold cavities, not only does the grid generation process require a large amount of manual adjustment costs, such as repeated adjustments of grid density to avoid local distortion, but it is also prone to quality problems such as distortion and poor orthogonality of the generated grid due to complex structures such as protrusions, recesses, and branches in the region. Such low-quality grids can directly cause numerical oscillation in subsequent simulations, reducing the calculation accuracy of key physical quantities such as flow velocity and pressure, and in severe cases, even causing computational divergence, greatly affecting the reliability and efficiency of the simulation.
[0005] In high-precision numerical simulation of incompressible flow, accurately capturing fine flow structures such as vortex evolution and shear layer separation is a core requirement, but mainstream numerical methods all have defects that restrict the realization of high-order accuracy. Although the traditional finite difference method is intuitive and easy to program, its high-order format requires an expanded calculation template, which will produce significant numerical dissipation and dispersion in regions with steep flow gradients, leading to calculation bias of key physical quantities such as flow velocity and pressure, making it difficult to restore the fine flow field; the finite element method has flexible element subdivision, but high-order accuracy requires complex high-order basis functions, which will sharply increase the number of grid nodes and significantly increase the cost of calculation and storage, and also easily cause numerical oscillation at the junction of elements, affecting the stability of simulation results such as pressure field; the finite volume method is widely used, but its high-order format needs to be implemented through complex reconstruction, not only does the construction difficulty increase exponentially with the increase of accuracy, but inappropriate interpolation strategies can also destroy the local conservation of flow, so high-order improvement is always limited.
[0006] Compact difference method exactly makes up for some of the shortcomings of traditional methods. It can achieve high-order accuracy with a small calculation template through special format construction, and has low numerical dissipation and dispersion, which can accurately capture the subtle changes of incompressible flow. It has excellent simulation results for complex flows such as turbulence and boundary layers. However, most of the current mainstream compact difference methods are implicit, and the velocity and pressure of incompressible flow have strong coupling characteristics. Using this format to solve will inevitably form a large sparse linear equation system. Solving this type of equation requires repeated iteration, which is slow and consumes a lot of memory and computing power. Especially for incompressible flow simulation, it often needs to encrypt the grid to ensure accuracy, and the grid size expands, making the computational efficiency problem more prominent, which greatly limits its large-scale application in engineering practice. SUMMARY
[0007] The purpose of the present application is to solve the core bottleneck that the incompressible Navier-Stokes equation is decoupled from the pressure Poisson equation in the existing incompressible flow solving, which leads to the difficulty of full explicit solving of flow in irregular regions, and the problem of high computational complexity of traditional flow solving method in irregular regions, and a high-order explicit compact difference method for incompressible flow problem in irregular regions is proposed.
[0008] The technical scheme of the present application is: a high-order explicit compact difference method for incompressible flow problem in irregular regions, comprising the following steps: Construct the smallest rectangular region containing the irregular region; Grid division is performed on the rectangular region, and the immersed boundary method is used to arrange Lagrange points on the boundary of the irregular region; Based on the grid division of the regular region, the entropy-damped artificial compressibility equation of incompressible flow containing spatial first-order derivative, spatial second-order derivative and time first-order derivative is established; High-order explicit upwind compact difference format is used to calculate high-order spatial first-order derivative, and based on high-order spatial first-order derivative, high-order spatial second-order derivative is obtained through explicit compact relationship, and then combined with third-order high-precision explicit time discretization format, the entropy-damped artificial compressibility equation is fully discretely solved to obtain high-precision flow field information of the whole rectangular region; Based on the high-precision flow field information of the whole rectangular region, the force on the rectangular region grid points participating in the calculation on the boundary of the irregular region is solved by the immersed boundary method; Repeat the calculation of high-precision flow field information of the whole rectangular region, and solve the force on the rectangular region grid points participating in the calculation on the boundary of the irregular region, until the preset termination condition is met, and the high-order explicit compact difference solving of incompressible flow problem in irregular regions is completed.
[0009] The beneficial effects of the present application are: 1. The present application thoroughly avoids the solving link of the pressure Poisson equation by the deep integration of the explicit high-order compact difference format and the immersed boundary method, and the solving framework of the entropy-damped artificial compressibility equation, and realizes the full explicit solving of incompressible flow in irregular regions. With the design of thermodynamic compatible damping term, the stability and physical consistency of the pressure field are guaranteed, and the simplicity and solving accuracy of explicit calculation are also considered.
[0010] 2. The present application relies on the characteristics of the immersed boundary method that does not need to distort, reconstruct or regionally decompose the grid according to the shape of the irregular region, avoids the damage of traditional adaptive mode to the accuracy of high-order compact difference format. By optimizing the coupling logic of the two methods, it is ensured that the high-order compact difference has the advantage of spectral resolution, which is still effective in irregular boundary regions, and can accurately capture the fine structure of complex flow field such as vortex shedding and flow separation.
[0011] 3. This invention does not require adjustments to the core solution logic for specific irregular regions, and can flexibly adapt to various irregular boundary flow scenarios in many engineering application fields. This invention provides a generalized and efficient solution scheme, offering reliable numerical calculation support for equipment optimization, product development, and process improvement in related fields. It has a wide range of applications and strong practicality. Attached Figure Description
[0012] Figure 1 The diagram shows a flowchart of a high-order explicit compact difference method for incompressible flow problems in irregular regions.
[0013] Figure 2 The diagram shows a rectangular area containing an irregular region.
[0014] Figure 3 The figure shown is a diagram of the vorticity results in a regular region when the Reynolds number Re=1000.
[0015] Figure 4 The figure shown is a diagram of the vorticity results in a regular region when the Reynolds number Re=3200.
[0016] Figure 5 The figure shown is a diagram of the vorticity results in a regular region when the Reynolds number Re=5000.
[0017] Figure 6 The figure shown is a diagram of the vorticity results in a regular region when the Reynolds number Re=7500.
[0018] Figure 7 The figure shown is the vorticity result of the central circular obstacle when the Reynolds number Re=400.
[0019] Figure 8 The figure shown is the vorticity result of the central circular obstacle when the Reynolds number Re=1000.
[0020] Figure 9 The figure shown is the vorticity result of the central circular obstacle when the Reynolds number Re=3200.
[0021] Figure 10 The figure shown is the vorticity result of the central circular obstacle when the Reynolds number Re=5000.
[0022] Figure 11 The figure shown is the vorticity result of the central elliptical obstacle when the Reynolds number Re=3200.
[0023] Figure 12 The figure shown is the vorticity result of the central triangular obstacle when the Reynolds number Re=3200.
[0024] Figure 13 The figure shown is the vorticity result of the central hexagonal obstacle when the Reynolds number Re=3200.
[0025] Figure 14 The figure shown is the vorticity result of the central octagonal obstacle when the Reynolds number Re=3200. Detailed Implementation
[0026] Exemplary embodiments of the present invention will now be described in detail with reference to the accompanying drawings. It should be understood that the embodiments shown and described in the drawings are merely exemplary and are intended to illustrate the principles and spirit of the invention, and are not intended to limit the scope of the invention.
[0027] Example 1: The basic idea of this invention is to provide a rectangular region to encompass an irregular region, use the submerged boundary method to describe the boundary of the irregular region, and then employ explicit high-order compact difference calculus to solve the problem on the rectangular region. This, combined with the submerged boundary method, jointly describes the incompressible flow problem on the irregular region. Providing the submerged boundary method to establish the boundary of the irregular region allows for highly flexible generation of Cartesian meshes without requiring body-fit mesh design for the irregular region, enhancing adaptability to complex geometries. Furthermore, it allows for the use of explicit high-order compact difference calculus to solve the incompressible flow problem, providing high-precision numerical simulations to capture richer flow details. Figure 1 As shown, a high-order explicit compact difference method for incompressible flow problems in irregular regions includes the following steps: S1. Construct the smallest rectangular region containing the irregular area; Set the maximum size of the irregular region, and determine the size of the rectangular region based on the maximum size of the irregular region. The specific formula is as follows:
[0028] in, Indicates the length of the rectangular region. Indicates the width of the rectangular area. The left boundary of the irregular region, Indicates the right boundary of the irregular region. Indicates the upper boundary of the irregular region. This represents the lower boundary of the irregular region; subsequently, the rectangular region is calculated. Boundary range This ensures that the rectangular area completely encompasses the irregular area. A diagram illustrating the rectangular area encompassing the irregular area is shown below. Figure 2 As shown.
[0029] S2. Divide the rectangular region into grids and use the submerged boundary method to arrange Lagrange points on the boundary of the irregular region; The specific method for meshing a rectangular region is as follows: An Eulerian background structure mesh is generated on the rectangular region with a preset fixed mesh spacing, and the mesh nodes are arranged in the order of the x and y directions, i.e.:
[0030] in, This represents the coordinates of the i-th grid node. Indicates the left boundary of the rectangular region. This indicates the preset fixed grid spacing. This indicates the number of grid nodes in the x-direction. This represents the j-th node in the y-direction. Indicates the lower boundary of the rectangular region. This indicates the number of grid nodes in the y-direction; the generated grid covers irregular areas, ensuring the integrity of the computational domain. It depends on the required computational accuracy and efficiency of the problem.
[0031] The method for constructing irregular region boundaries using the submerged boundary method is as follows: A corresponding function is generated based on the composition information of the irregular region boundary to evenly distribute the Lagrange point coordinates; that is, for each approximately straight boundary segment… , The number of straight boundary segments can be represented by their starting coordinates. and endpoint coordinates Determine the coordinates of the Lagrange points on the straight boundary. and straight boundary segments Number of Upper Lagrange Points The specific formula is as follows:
[0032]
[0033] For each curve segment , To represent the number of curve segments, three characteristic points are selected: the starting point... ,end and midpoint These three points can be used to determine the curve segment. curvature The specific expression is:
[0034] in, , , , It consists of three vertices. , and The circumcenter coordinates of the triangle formed The solution formula is:
[0035] in, Curved segment Angle range for:
[0036] in, This is a two-parameter arctangent function. It maps the coordinates on the numerical curve to the curve segment. The actual boundary is compared, and the specific formula is as follows:
[0037]
[0038] Curve segment Number of Upper Lagrange Points Similarly determined by the Lagrange point spacing Confirmed, the calculation formula is:
[0039] In summary, the total number of Lagrange points on the boundaries of irregular regions The sum of the number of Lagrange points on the straight boundary segment and the number of Lagrange points on the curved segment is:
[0040] This method can ensure the Lagrange point. The points are evenly distributed along the boundary of the irregular region, and the spacing between the points remains constant. Even if the irregular region has a complex shape, it will not cause deviation in the calculation results.
[0041] S3. Based on the regular region after grid division, establish an entropy-damped artificial compressibility equation for incompressible flow that includes the first spatial derivative, the second spatial derivative, and the first temporal derivative; The entropy-damped artificial compressibility equation is solved using an explicit compact difference scheme. The following detailed explanation uses an incompressible flow problem as an example. The dimensionless form of the entropy-damped artificial compressibility equation for incompressible flow is:
[0042] in, These are the velocities in the x and y directions, respectively. For pressure, These are the Mach number and the Reynolds number, respectively. These are the submerged boundary forces in the x and y directions, respectively, meaning they exist only at the boundaries of the irregular region. The unknown quantity to be solved is the velocity in the flow field. and pressure , The velocity is in the x-direction. The velocity is in the y-direction.
[0043] S4. The first derivative of the higher-order space is calculated using a higher-order explicit upwind compact difference scheme. Based on the first derivative of the higher-order space, the second derivative of the higher-order space is obtained through an explicit compact relation. Then, combined with a third-order high-precision explicit time discretization scheme, the entropy-damped artificial compressibility equation is solved by full explicit discretization to obtain high-precision flow field information of the entire rectangular region. Using known variables Represents the velocity in the flow field and pressure Using higher-order explicit compact difference to solve for known variables First derivative and second derivative ,in This represents the grid node index in the x or y direction. For the discretization of the first derivative, this embodiment of the invention employs the average third-order upwind compact difference scheme (AUCD3), which requires first solving for the upwind and headwind derivatives of the first derivative. The higher-order discretization at the interior points is as follows:
[0044] in, Represents grid nodes The approximate value of the first derivative at the top windward position (such as the first derivative of pressure and velocity variables), Represents grid nodes The approximate value of the first derivative at the downwind direction (such as the first derivative of pressure and velocity variables), Represents grid nodes The approximate value of the first derivative at the top windward position (such as the first derivative of pressure and velocity variables), Represents grid nodes The function values at that location (such as pressure and velocity variables), Represents grid nodes The function values at that location (such as pressure and velocity variables), Represents grid nodes The function values at that location (such as pressure and velocity variables), Represents grid nodes The approximate value of the first derivative at the downwind direction (such as the first derivative of pressure and velocity variables), Indicates the preset fixed grid spacing; according to and The solution yields the value of the first derivative in the AUCD3 scheme with high-order precision. :
[0045] The higher-order discretization formula for the first derivative at the boundary points is:
[0046] in, Represents grid nodes The approximate value of the first derivative at a given point (such as pressure and velocity variables), Represents grid nodes The approximate value of the first derivative at a given point (such as the first derivative of pressure and velocity variables). , , , , , , , , and Represents grid nodes , , , , , , , , and The function values at the location (such as pressure and velocity variables); Based on the above formula, we can use known variables... By solving for the first derivative at the boundary points and then substituting it into the solution for the first derivative at the interior points, an explicit solution can be achieved.
[0047] For higher-order discrete solutions of the second derivative at interior points, using variables The solution is obtained explicitly using the first derivative information, and the specific formula is as follows:
[0048] The higher-order discrete solution of the second derivative at the boundary points requires the use of the second derivative at the interior points, specifically:
[0049] in, , , , , , Represents grid nodes , , , , , , and The approximate value of the second derivative at a given point (such as the second derivative of pressure and velocity variables).
[0050] Based on the formulas for solving the second derivative at interior points and boundary points, it can be seen that solving for the second derivative is also an explicit method; that is, first solve for the second derivative at interior points, and finally solve for the second derivative at boundary points. The above completes the solution process for higher-order explicit compact differences of the first and second derivatives, requiring only the known variables... Replace with speed and pressure This allows us to solve for the first and second derivatives of velocity and pressure.
[0051] For convection terms ( , ), pressure gradient term ( , and divergence term Regarding the first derivative ( , , , , Spatial discretization, embodiments of the present invention employ variable-specific methods. Solving for the first derivative, taking These are the indices of the coordinate points in the x and y directions, respectively. First, regarding... The first derivative at the boundary point is obtained as follows:
[0052] Next, the discretization scheme for the interior points is as follows:
[0053] The first derivative at each grid point in the rectangular region is obtained by solving. And targeting The first derivative at the boundary can be obtained as:
[0054] Next, the discretization scheme for the interior points is as follows:
[0055] The first derivative at each grid point in the rectangular region is obtained by solving. And for , , Simply change the variables in the formula Replace with That completes the explicit solution for the convection, pressure gradient, and divergence terms.
[0056] Regarding diffusion terms and viscosity , Discrete solutions can be obtained using the results of the first derivative, for example, for... Discrete solution, using step S3 for known variables To find the second derivative, first calculate the value of the second derivative at the interior point:
[0057] Then, based on the calculation of the second derivative at the boundary point, we obtain:
[0058] And targeting Discrete solution, the second derivative value at the interior point is:
[0059] Based on the calculation of the second derivative at the boundary point:
[0060] And for Simply change the variable Replace with That completes the explicit high-order discrete solution for the diffusion and viscosity terms.
[0061] For time-discrete problems, the time term is... Unified use This indicates that, in the dimensionless form of the entropy-damped artificial compressibility equation for incompressible flow, all other terms are placed on the right-hand side. In, that is:
[0062] Furthermore, the dimensionless form of the entropy-damped artificial compressibility equation for incompressible flow is rewritten as:
[0063] Choosing the explicit third-order Runge-Kutta method, the discretization scheme is as follows:
[0064] in, This represents the variable (e.g., velocity and pressure) at the first moment between time n and time n+1. This represents the variable (e.g., velocity and pressure) at the second time point between time n and time n+1. Represents the variable at time n+1 (e.g., velocity and pressure). Represents the variable at time n (e.g., velocity and pressure). Indicates the time step.
[0065] Based on the above time and space discretization, the high-order explicit compact difference solution for the incompressible flow problem represented by the entropy-damped artificial compressibility equation is completed.
[0066] S5. Based on the high-precision flow field information of the entire rectangular region, the force on the grid points of the rectangular region participating in the calculation on the boundary of the irregular region is solved by the submerged boundary method; Firstly, through The number of rectangular grid cells involved in the calculation near the Lagrange point on the boundary of the irregular region is determined as follows: and the corresponding grid point coordinates are Here we choose to use X= ( X 1 ,X 2 () represents the coordinates of a Lagrange point. x k = ( x k 1 ,x k 2 () represents the rectangular grid coordinates near the Lagrange point. The weights of the rectangular grid points participating in the calculation near the boundary of each irregular region are determined using the following formula:
[0067] in, This represents the interpolation weights between grid points and Lagrange points. This represents the distance between grid points and Lagrange points; Calculate the interpolation rate on the boundary of an irregular region The specific calculation formula is as follows:
[0068] in, This represents the interpolated velocity function at the Lagrange points in the x-direction. This represents the interpolated velocity function at the Lagrange points in the y-direction. This represents the number of rectangular grid cells participating in the calculation near the Lagrange point on the boundary of the irregular region. Represents the rectangular region near the boundary of the irregular region. k grid points Directional velocity, Represents the rectangular region near the boundary of the irregular region. k The coordinates of each grid point Represents the coordinates of a Lagrange point on the boundary of an irregular region. Represents the rectangular region near the boundary of the irregular region. k grid points Directional velocity, This indicates the preset fixed grid spacing.
[0069] Because the boundary conditions on the boundary of the irregular region are set, its actual velocity is... Therefore, by using the interpolated velocity and its actual velocity, the force at the Lagrange point on the boundary of the irregular region can be calculated as follows:
[0070] in, This represents the force at the Lagrange point in the x-direction. This represents the force at the Lagrange point in the y-direction. This represents the actual velocity at the Lagrange point in the x-direction. This represents the interpolation velocity of the Lagrange points in the x-direction. Indicates the time step. This represents the actual velocity at the Lagrange point in the y-direction. This represents the interpolation velocity of the Lagrange points in the y-direction. This parameter represents the thickness of the boundary of the irregular region. It can enhance the boundary conditions at the boundary and achieve streamline non-penetration. Therefore, it yields the forces acting on the grid points of the rectangular region near the boundary of the irregular region. for:
[0071] in, This represents the submerged boundary force at grid points in a rectangular region along the x-direction. This represents the submerged boundary force at grid points in a rectangular region along the y-direction. This represents the number of Lagrange points on the boundary of an irregular region. This represents the force at the Lagrange point in the x-direction. This represents the coordinates of the k-th Lagrange point. This represents the force at the Lagrange point in the y-direction. This represents the Lagrange point spacing.
[0072] S6. Determine whether the preset termination condition has been met. If the preset termination condition has been met, the process ends and the high-order explicit compact difference solution of the incompressible flow problem in the irregular region is completed. If the preset termination condition has not been met, return to step S4.
[0073] The core bottleneck in existing incompressible flow solutions lies in the dependence of the incompressible Navier-Stokes equations on the pressure Poisson equation for decoupling, making it difficult to achieve fully explicit solutions for irregular flow regions. Existing explicit methods, such as traditional artificial compressibility methods, are prone to pressure oscillations and require dual-time-step techniques to offset their efficiency advantages, failing to balance explicit characteristics with solution reliability and thus failing to meet the fundamental requirements for simulating irregular flow regions. Furthermore, traditional methods for solving irregular flow regions suffer from high computational complexity. Traditional methods, such as arbitrary Lagrange-Eulerian methods, require frequent mesh updates or reconstructions, increasing computation time and introducing additional numerical errors, resulting in cumbersome simulation processes and insufficient adaptability to large-scale complex scenarios. The core objective of this invention is to provide a solution scheme that combines explicit high-order compact difference and submerged boundary methods, overcoming the core limitation of the inability to achieve fully explicit, high-precision solutions for incompressible flows in irregular regions. By combining the advantages of two methods and relying on the improved approach of entropy-damped artificial compressibility equations, the solution to the pressure Poisson equation is avoided, achieving a fully explicit solution for incompressible flows while ensuring consistency between the solution results and actual physical laws. A general and efficient solution adapted to flow simulation in irregular regions is provided, offering reliable numerical computation support for equipment optimization and product development in related fields. Figure 3 , Figure 4 , Figure 5 and Figure 6 The graph shows the results of solving the regular region at different Reynolds numbers using the method proposed in this invention. Figure 7 , Figure 8 , Figure 9 and Figure 10 The figures show the results of the method proposed in this invention for solving irregular regions of the same shape at different Reynolds numbers. Figure 11 , Figure 12 , Figure 13 and Figure 14 The figure shows the results of the method proposed in this invention for solving irregular regions of different shapes at the same Reynolds number.
[0074] Example 2: Based on Example 1, this embodiment of the invention provides a high-order explicit compact difference system for incompressible flow problems in irregular regions, which can be used to implement the high-order explicit compact difference method for incompressible flow problems in irregular regions as described in the foregoing embodiments. The system includes: The first module is used to construct the smallest rectangular region containing irregular areas; The second module is used to mesh the rectangular area and use the submerged boundary method to arrange Lagrange points on the boundary of the irregular area. The third module is used to establish an entropy-damped artificial compressibility equation for incompressible flows based on the regular regions after grid division, which includes the first spatial derivative, the second spatial derivative, and the first temporal derivative. The fourth module is used to calculate the first derivative of the higher-order space using a higher-order explicit upwind compact difference scheme. Based on the first derivative of the higher-order space, the second derivative of the higher-order space is obtained through an explicit compact relation. Then, combined with a third-order high-precision explicit time discretization scheme, the entropy-damped artificial compressibility equation is solved in a fully explicit discretization manner to obtain high-precision flow field information of the entire rectangular region. The fifth module is used to solve the forces on the grid points of the rectangular region participating in the calculation on the boundary of the irregular region by using the submerged boundary method based on the high-precision flow field information of the entire rectangular region. The sixth module is used to repeatedly calculate the high-precision flow field information of the entire rectangular region, solve the forces on the grid points of the rectangular region participating in the calculation on the boundary of the irregular region, until the preset termination condition is met, and complete the high-order explicit compact difference solution of the incompressible flow problem in the irregular region.
[0075] According to embodiments of the present invention, the present invention also provides an electronic device, a readable storage medium, and a computer program product.
[0076] In an exemplary embodiment, an electronic device includes: at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores instructions executable by the at least one processor, the instructions being executed by the at least one processor to enable the at least one processor to perform a high-order explicit compact difference method for incompressible flow problems on irregular regions as described in Embodiment 1 above.
[0077] In an exemplary embodiment, the readable storage medium may be a non-transitory computer-readable storage medium storing computer instructions for causing a computer to execute a high-order explicit compact difference method for incompressible flow problems on irregular regions as described in Embodiment 1 above.
[0078] In an exemplary embodiment, the computer program product includes a computer program that, when executed by a processor, implements a high-order explicit compact difference method for incompressible flow problems in irregular regions as described in Embodiment 1 above.
[0079] The program code used to implement the methods of the present invention can be written in any combination of one or more programming languages. This program code can be provided to a processor or controller of a general-purpose computer, special-purpose computer, or other programmable data processing device, such that when executed by the processor or controller, the program code causes the functions / operations specified in the flowcharts and / or block diagrams to be implemented. The program code can be executed entirely on the machine, partially on the machine, as a standalone software package partially on the machine and partially on a remote machine, or entirely on a remote machine or server.
[0080] In the context of this invention, a machine-readable medium can be a tangible medium that may contain or store a program for use by or in conjunction with an instruction execution system, apparatus, or device. A machine-readable medium can be a machine-readable signal medium or a machine-readable storage medium. Machine-readable media can include, but are not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatus, or devices, or any suitable combination of the foregoing. More specific examples of machine-readable storage media include electrical connections based on one or more wires, portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fibers, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination of the foregoing.
[0081] To provide interaction with a user, the systems and techniques described herein can be implemented on a computer having: a display device for displaying information to the user (e.g., a CRT (cathode ray tube) or LCD (liquid crystal display) monitor); and a keyboard and pointing device (e.g., a mouse or trackball) through which the user provides input to the computer. Other types of devices can also be used to provide interaction with the user; for example, feedback provided to the user can be any form of sensory feedback (e.g., visual feedback, auditory feedback, or tactile feedback); and input from the user can be received in any form (including sound input, voice input, or tactile input).
[0082] The systems and technologies described herein can be implemented in computing systems that include backend components (e.g., as a data server), or computing systems that include middleware components (e.g., an application server), or computing systems that include frontend components (e.g., a user computer with a graphical user interface or web browser through which a user can interact with embodiments of the systems and technologies described herein), or any combination of such backend, middleware, or frontend components. The components of the system can be interconnected via digital data communication of any form or medium (e.g., a communication network). Examples of communication networks include local area networks (LANs), wide area networks (WANs), and the Internet.
[0083] Computer systems can include clients and servers. Clients and servers are generally located far apart and typically interact via communication networks. Client-server relationships are created by computer programs running on the respective computers and having a client-server relationship with each other. Servers can be cloud servers, servers in distributed systems, or servers incorporating blockchain technology.
[0084] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.
Claims
1. A high-order explicit compact difference method for incompressible flow problems on irregular domains, characterized in that, The method comprises the following steps: constructing a minimum rectangular region containing an irregular region; grid dividing the rectangular region and arranging Lagrange points on the boundary of the irregular region by using the immersed boundary method; based on the grid divided regular region, establishing an entropy-damped artificial compressibility equation of incompressible flow containing spatial first-order derivative, spatial second-order derivative and time first-order derivative; calculating high-order spatial first-order derivative by using high-order explicit upwind compact difference format, obtaining high-order spatial second-order derivative by using explicit compact relation based on the high-order spatial first-order derivative, and then combining with third-order high-precision explicit time discrete format, discretely solving the entropy-damped artificial compressibility equation of incompressible flow in full explicit mode to obtain high-precision flow field information of the whole rectangular region; based on the high-precision flow field information of the whole rectangular region, solving the force on the grid points of the rectangular region participating in calculation on the boundary of the irregular region by using the immersed boundary method; repeating the calculation of the high-precision flow field information of the whole rectangular region and the force on the grid points of the rectangular region participating in calculation on the boundary of the irregular region until a preset termination condition is met, and completing the high-order explicit compact difference solution of the incompressible flow problem on the irregular region.
2. The high order explicit compact difference method for the incompressible flow problem on irregular region according to claim 1, wherein, The method for grid dividing the rectangular region is specifically as follows: Euler background structure grid is generated on the rectangular region at a preset fixed grid spacing, and the grid nodes are arranged in sequence in the x direction and the y direction, namely: wherein, represents represents the left boundary of the rectangular region, represents the number of grid nodes in the direction, represents the left boundary of the rectangular region, represents a preset fixed grid spacing, represents represents the number of grid nodes in the direction, represents represents the number of grid nodes in the direction, represents the lower boundary of the rectangular region, represents the lower boundary of the rectangular region, represents represents the number of grid nodes in the direction.
3. The high order explicit compact difference method for the non-hy p erbolic flow problem on irregular region according to claim 1, wherein, The specific method for arranging the Lagrange points on the boundary of the irregular region by using the immersed boundary method is as follows: For a straight boundary segment on the boundary of an irregular region, the starting coordinates of the straight boundary segment are used. and endpoint coordinates Determine the coordinates of the Lagrange points on the boundary segment of the straight line. and the number of Lagrange points on the straight boundary segment; For a curve boundary segment on the irregular region boundary, the start point , the end point and the midpoint are selected ; the curvature radius and the circumcenter coordinates of the curve boundary segment are determined Determining the angular extent of a curve boundary segment in terms of excenter coordinates is: wherein is the two-parameter arctangent function; Then, the number of Lagrange points on the curved boundary segment is determined according to the curvature radius and the angle range of the curved boundary segment, and the arrangement of the Lagrange points on the boundary of the irregular region is completed.
4. The high order explicit compact difference method for the non-hy- perbolic flow problem on irregular region according to claim 3, characterized in that, The specific formula for calculating the number of Lagrange points on the straight boundary segment is as follows: wherein represents the number of Lagrangian points on the straight border segment, represents the Lagrangian point spacing.
5. The high order explicit compact difference method for the non-hy- perbolic flow problem on irregular region according to claim 3, characterized in that, radius of curvature The formula for calculating the radius of curvature is: wherein , , and are auxiliary variables, with , , , ; The formula for calculating the coordinates of the circumcenter is as follows: wherein are auxiliary variables, and ; The number of Lagrange points on a curved boundary segment The formula for calculating is: wherein denotes the maximum of the angle range denotes the minimum of the angle range denotes the minimum of the angle range denotes the minimum of the angle range denotes the Lagrange point distance.
6. The high order explicit compact difference method for the non-hy p erbolic flow problem on irregular region according to claim 1, wherein, The dimensionless form of the entropy-damped artificial compressibility equation of incompressible flow is as follows: wherein, denotes partial derivative symbol, denotes directional velocity, denotes time, denotes directional velocity, denotes pressure, denotes directional immersed boundary force, denotes directional immersed boundary force, denotes Reynolds number, denotes Mach number; spatial first order derivative is convection term , ), pressure gradient term , and divergence term , spatial second order derivative is diffusion term , viscous term and viscous term , time first order derivative is .
7. The high order explicit compact difference method for the non-hy p erbolic flow problem on irregular region according to claim 6, wherein, The specific method for discretely solving the entropy-damped artificial compressibility equation of incompressible flow in full explicit mode is as follows: Utilizing variables representing velocity and pressure on the flow field for the variable The first derivative gives the convection term , , the pressure gradient term , and the divergence term ; on the variable a second order derivative of the variable , a diffusion term , and a viscosity term ; wherein the second order derivative of the variable is obtained from a first order derivative of the variable . For time-discretization, the time term is uniformly denoted by , the dimensionless form of the entropy-damped artificial compressibility equation for incompressible flow is rewritten as wherein are auxiliary variables, and have Then, the explicit third-order Runge-Kutta method is selected for explicit time discretization, and the entropy-damped artificial compressibility equation of incompressible flow is discretely solved in explicit mode.
8. The high order explicit compact difference method for the non-hy p erbolic flow problem on irregular region according to claim 7, wherein, On the variables The first derivative is solved, in particular: Performing a first derivative solution at the grid interior point Performing a first derivative solution at the grid interior point wherein, denotes an approximation of the first derivative at a point within the grid, denotes an approximation of the upper upwind first derivative at a grid node denotes an approximation of the lower upwind first derivative at a grid node denotes an approximation of the upper upwind first derivative at a grid node denotes an approximation of the lower upwind first derivative at a grid node denotes an approximation of the upper upwind first derivative at a grid node denotes an approximation of the lower upwind first derivative at a grid node denotes a value of the function at a grid node denotes a value of the function at a grid node denotes a value of the function at a grid node denotes a value of the function at a grid node denotes a value of the function at a grid node denotes a value of the function at a grid node denotes a value of the function at a grid node denotes an approximation of the lower upwind first derivative at a grid node denotes a predetermined fixed grid spacing; The variables are evaluated at the boundary points The first derivative is solved for a high order approximation of the first derivative at the grid nodes: wherein denotes an approximation of the first derivative at the grid node , denotes an approximation of the first derivative at the grid node , , , , , , , , , and denotes a function value at the grid node , , , , , , , , and On the variables The second derivative is solved, in particular: Variables at points inside the grid By solving for the second derivative, we obtain higher-order approximations of the second derivative at points within the grid: wherein, represents a second order derivative high order approximation of the function f at the point within the grid. At the boundary points the variables The second order derivative is solved to obtain a high order approximation of the second order derivative at the grid nodes: wherein , , , , , denotes the second derivative approximation at the grid node , , , , , , and .
9. The high order explicit compact difference method for the non-hy p erbolic flow problem on irregular region according to claim 1, wherein, The specific method for solving the force on the grid points of the rectangular region participating in calculation on the boundary of the irregular region by using the immersed boundary method is as follows: determining the weight of the grid points on the rectangular region near each irregular region boundary participating in calculation; calculating the interpolation velocity on the irregular region boundary based on the weight of the grid points; calculating the force on the Lagrange points on the irregular region boundary according to the interpolation velocity and the actual velocity; calculating the force on the grid points of the rectangular region near the irregular region boundary participating in calculation according to the force on the Lagrange points on the irregular region boundary.
10. The high order explicit compact difference method for the non-hy p erbolic flow problem on irregular region according to claim 9, wherein, The formula for calculating the weight of the grid points is as follows: wherein, represents an interpolation weight value between the grid point and the Lagrange point, represents a distance between the grid point and the Lagrange point; The formula for calculating the interpolation velocity on the irregular region boundary is as follows: wherein, represents an interpolation velocity function of the x-direction Lagrange point, represents an interpolation velocity function of the y-direction Lagrange point, represents the number of rectangular region grids participating in the calculation near the Lagrange point on the boundary of the irregular region, represents the x-direction velocity of the i-th grid point near the boundary of the irregular region, k represents the i-th grid point coordinate near the boundary of the irregular region, k represents the x-direction velocity of the i-th grid point near the boundary of the irregular region, k represents a preset fixed grid spacing; The formula for calculating the force on the Lagrange points on the irregular region boundary is as follows: wherein, represents the force on the x-direction Lagrange point, represents the force on the y-direction Lagrange point, represents the actual velocity of the x-direction Lagrange point, represents the interpolated velocity of the x-direction Lagrange point, represents the time step, represents the actual velocity of the y-direction Lagrange point, represents the interpolated velocity of the y-direction Lagrange point, represents the irregular region boundary thickness; The formula for calculating the force on the grid points of the rectangular region near the irregular region boundary participating in calculation is as follows: wherein, represents the immersed boundary force of the rectangular region grid point in the x direction, represents the immersed boundary force of the rectangular region grid point in the y direction, represents the number of Lagrangian points on the boundary of the irregular region, represents the force of the Lagrangian point in the x direction, represents the coordinates of the kth Lagrangian point, represents the force of the Lagrangian point in the y direction, represents the Lagrangian point spacing.