A method to improve computational efficiency and stability of hypervelocity collision simulation

Abstract

The present invention provides a method for improving the computational efficiency and stability of hypervelocity collision simulation. In order to improve the simulation efficiency, a hypervelocity collision model is established, symmetry constraints are set, rigid-flexible conversion technology and parallel processing are used, thereby significantly accelerating the solution process. In addition, in order to enhance the stability of the solution, static memory allocation is used to ensure the stability of the solution process and reduce unnecessary fluctuations. The present invention uses these strategies in combination to effectively improve the simulation efficiency of hypervelocity collision and provide more efficient computational support for hypervelocity collision research.

Description

Technical Field

[0001] The present invention relates to the technical field of hypervelocity collision, and in particular to a method for improving the computational efficiency and stability of hypervelocity collision simulation. Background Art

[0002] Hypervelocity collisions are widely found in many fields, such as aerospace, automotive engineering, materials science, and protective engineering. Research in these fields not only involves the behavior of objects under extreme conditions, but also the strength, toughness, and deformation capacity of materials. With the development of science and technology, the need to understand hypervelocity collisions is becoming more and more urgent, which has prompted researchers to explore more accurate and efficient simulation methods.

[0003] Traditional experimental methods are often limited by experimental equipment and safety factors when studying hypervelocity collisions, making it difficult to systematically obtain the required data. In addition, the experiments are costly and time-consuming, and cannot meet the needs of rapid iteration and real-time feedback. Therefore, computer simulation has become an important tool for studying hypervelocity collisions. Through numerical simulation, researchers can test the performance of different materials and designs in a virtual environment and quickly evaluate their responses under extreme conditions.

[0004] Among the many numerical simulation methods, the FEM-SPH method, which combines FEM (finite element method) and SPH (smoothed particle hydrodynamics), has attracted widespread attention. This method combines the advantages of FEM in dealing with the behavior of solid materials with the advantages of SPH in fluid dynamics problems, and can more accurately describe the complex physical phenomena in the collision process. Although the FEM-SPH method has significant theoretical advantages, its computational efficiency is still an issue that needs to be addressed, especially in the SPH part, where the solution time is long, limiting large-scale applications.

[0005] In the simulation of hypervelocity collisions, the FEM-SPH method achieves an accurate description of complex physical phenomena by coupling the finite element method (FEM) with smoothed particle hydrodynamics (SPH). However, the solution speed of this method is slow, mainly because the SPH solution process is too complicated.

[0006] To sum up, it is necessary to make further innovations on existing technologies. Summary of the invention

[0007] In view of the technical problems existing in the above-mentioned background technology, the present invention proposes a method for improving the computational efficiency and stability of hypervelocity collision simulation. The method has a reasonable concept and a simple process, can effectively improve the simulation efficiency of hypervelocity collision, and provide more efficient computational support for hypervelocity collision research.

[0008] In order to solve the above technical problems, the present invention provides a method for improving the computational efficiency and stability of hypervelocity collision simulation, which specifically includes the following steps:

[0009] (1) Establishing a hypervelocity collision model

[0010] Prepare the projectile and target plate in advance, and then use finite element units to model both the projectile and the target plate, such as Figure 2 As shown in the figure, when the material undergoes obvious deformation, fracture or breakage during the collision between the projectile and the target plate, the projectile and the target plate are transformed from finite element units to SPH particles;

[0011] (2) Set symmetry constraints

[0012] Set up symmetry constraints for nodes and smooth particles in the hypervelocity collision model;

[0013] (3) Rigid-Flexible Conversion

[0014] The projectile and the first layer of target plates are set as soft bodies, and the remaining target plates are defined as rigid bodies. The rigid body and soft body conversion of the target plate is determined based on the moment of collision to determine whether the target plate needs to be converted from a rigid body to a soft body;

[0015] (4) Parallel processing

[0016] First, all the elements of the projectile and the target board are combined into a set, and then according to the number of computer CPU cores The difference will be collected Divide into sub-regions and ensure that the total number of elements in each sub-region is evenly distributed in each region. Each CPU is responsible for processing a sub-region and performs corresponding computing tasks according to the sub-region assigned to it during the processing;

[0017] (5) Static memory allocation

[0018] The required memory resources are pre-allocated during the program initialization phase, a fixed-size memory space is allocated to each SPH particle, and its memory usage is ensured to be consistent throughout the entire processing process.

[0019] The method for improving the computational efficiency and stability of hypervelocity collision simulation, wherein the specific process of setting the symmetry constraints of the nodes and smooth particles of the hypervelocity collision model in step (2) is:

[0020] (2.1) Set the symmetry constraints of the nodes

[0021] Establish a Cartesian coordinate system, including Symmetrical plane and Symmetrical plane, The nodes on the symmetry plane have no Displacement in direction , that is, formula (1):

[0022] ;

[0023] To avoid Nodes on the symmetry plane or The rotation of the axis, that is, formula (2) and formula (3):

[0024] ;

[0025] ;

[0026] In the above formula (2) and formula (3), Indicates winding The rotation angle of the axis, Indicates winding The rotation angle of the axis;

[0027] exist The nodes on the symmetry plane have no Displacement in direction , that is, formula (4):

[0028] ;

[0029] To avoid Nodes on the symmetry plane or The rotation of the axis, that is, formula (5) and formula (6):

[0030] ;

[0031] ;

[0032] In the above formula (5) and formula (6), Indicates winding The rotation angle of the axis, Indicates winding The rotation angle of the axis;

[0033] (2.2) Setting symmetry constraints for smooth particles

[0034] Select Symmetrical plane and For SPH particles on the symmetry plane, create the same number of virtual SPH particles as the boundary particles on the symmetry plane of the SPH particles to ensure that the SPH particles and the virtual SPH particles are consistent in mass, pressure and absolute velocity, so that the virtual particles are in the calculation neighborhood of the real particles;

[0035] If the velocity of the boundary SPH particle is:

[0036] ;

[0037] for The speed of the virtual SPH particle on the symmetry plane is:

[0038] ;

[0039] for The speed of the virtual SPH particle on the symmetry plane is:

[0040] ;

[0041] The interaction between virtual particles and actual particles can be calculated by the following formula:

[0042] ;

[0043] In the above formula (10), The actual particle The physical quantity of and The actual particles Mass and physical quantities; They are the mass and physical quantity of the virtual SPH particles respectively; and are the kernel functions between real particles and other real particles or virtual particles, respectively; is the relative distance between particles; is the smoothing length, which is used to control the support range of the kernel function.

[0044] The method for improving the computational efficiency and stability of hypervelocity collision simulation, wherein the collision moment between the projectile and the subsequent target plate in step (3) is determined by the following formula (11):

[0045] ;

[0046] In formula (11), n is the total number of target plates; Indicates that the projectile flies from the starting point to the The time of the target board, ; Indicates the position of the target plate, , Indicates the projectile position; For the projectile and The residual velocity after the target plate hits, ; , represents the initial velocity of the projectile;

[0047] The residual velocity of the projectile can be calculated using the conservation of momentum:

[0048] ;

[0049] ;

[0050] In the above formula (12) and formula (13), Indicates the mass of the projectile; Indicates The mass of the part of the target plate that collides with the projectile; With The residual velocity after the target plate hits.

[0051] The method for improving the computational efficiency and stability of hypervelocity collision simulation, wherein the specific process of step (4) is as follows:

[0052] Suppose the projectile and the target plate together form a set , which contains all elements of all projectiles and targets:

[0053] ;

[0054] in, Indicates element of a projectile or target, ;

[0055] Define each sub-region as ,in , so that the set is represented as the union of these subregions:

[0056] ;

[0057] Assume that the size of each sub-region is , then we get:

[0058] ;

[0059] Each CPU core , , responsible for processing a sub-area , that is, the number of tasks processed by each CPU core is , so each CPU is responsible for processing the entire model Part; Each CPU will process according to its own assigned sub-area Execute the corresponding computing tasks.

[0060] The method for improving the computational efficiency and stability of hypervelocity collision simulation, wherein the specific process of step (5) is as follows:

[0061] Assume that each SPH particle is , the amount of memory required is , in static memory allocation, each particle The memory size is a preset constant ,Right now ;

[0062] Assume particle In the neighborhood of particles, assuming that each particle's interaction with its neighbors requires a fixed amount of memory to be allocated in advance , then the particle The total memory required is:

[0063] ;

[0064] By reasonably estimating the number of particles and the number of neighboring particles , calculate the total amount of memory required when the program starts:

[0065] ;

[0066] In the above formula (18), is the average number of particles in the particle neighborhood.

[0067] By adopting the above technical solution, the present invention has the following beneficial effects:

[0068] The method for improving the computational efficiency and stability of hypervelocity collision simulation in the present invention has a reasonable concept and a simple process, adopts a symmetry method, a rigid-flexible conversion method, and a parallel technology to improve the computational efficiency, and adopts static memory allocation to improve the stability of SPH.

[0069] In order to improve the simulation efficiency, the present invention adopts the symmetry method, the rigid-flexible conversion method and the parallel technology, thereby significantly accelerating the solution process; in addition, in order to enhance the stability of the solution, static memory allocation is used to ensure the stability of the solution process and reduce unnecessary fluctuations; the comprehensive use of these strategies can effectively improve the simulation efficiency of hypervelocity collisions and provide more efficient computing support for hypervelocity collision research.

[0070] The main goal of the present invention is to significantly improve the computational efficiency and stability of ultra-high-speed collision simulations. By introducing symmetry constraints and rigid-flexible conversion, the present invention can quickly switch between different deformation modes of an object in a complex collision simulation, thereby efficiently processing the dynamic response of the object and avoiding the problems of long computation time and uneven processing in traditional methods. At the same time, by using parallel computing technology, computational tasks can be effectively distributed to multiple processing units, further improving the computational speed, especially in large-scale simulations, and giving full play to the advantages of modern computing hardware.

[0071] In addition, the static memory allocation method of the present invention ensures the uniformity and stability of memory resources throughout the solution process, avoiding unnecessary fluctuations caused by dynamic memory allocation in traditional methods. This memory management strategy is particularly suitable for the case of a large number of particles in the SPH method, avoiding frequent memory allocation and release operations, thereby improving the stability and efficiency of the calculation.

[0072] Compared with the existing technology, the present invention has significant advantages. First, by using symmetry constraints and rigid-flexible conversion, the calculation scale can be reduced and the solution speed can be improved. Second, by combining parallel computing and static memory allocation, the present invention greatly improves the calculation speed and memory usage efficiency, especially when facing a large number of model grids, it shows better computing power and higher stability than the existing methods. Finally, the comprehensive application of these technologies not only improves the simulation accuracy and calculation speed, but also provides stronger computing support for the research of hypervelocity collisions, and promotes progress in related fields. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] In order to more clearly illustrate the specific implementation methods of the present invention or the technical solutions in the prior art, the drawings required for use in the specific implementation methods or the description of the prior art will be briefly introduced below. Obviously, the drawings in the following description are some implementation methods of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0074] Figure 1 A flow chart of a method for improving the computational efficiency and stability of hypervelocity collision simulation according to the present invention;

[0075] Figure 2 A state diagram of a projectile and a target plate involved in a method of improving the computational efficiency and stability of a hypervelocity collision simulation according to the present invention;

[0076] Figure 3 A schematic diagram of symmetry constraints involved in the method of the present invention for improving the computational efficiency and stability of hypervelocity collision simulation. DETAILED DESCRIPTION

[0077] The technical solution of the present invention will be described clearly and completely below in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0078] The present invention is further explained below in conjunction with specific implementation modes.

[0079] like Figure 1As shown, the present embodiment provides a method for improving the computational efficiency and stability of hypervelocity collision simulation, which specifically includes the following steps:

[0080] S100, establishing a hypervelocity collision model

[0081] The hypervelocity collision model consists of two parts: a high-speed projectile and a target plate, such as Figure 1 . In order to simulate the high-speed collision process more accurately, a numerical model combining the finite element method (FEM) and the smoothed particle hydrodynamics method (SPH), namely the FEM-SPH conversion method, is usually used. Specifically, both the projectile and the target plate are modeled using finite element (FEM) units; however, as the collision progresses, the strain of the projectile and the target plate gradually increases. When the strain in the local area exceeds the failure condition of the material, the material of the projectile and the target plate will undergo obvious deformation, fracture or shattering; at this time, the projectile and target plate in the model will be transformed from FEM units to SPH particles; through this FEM-SPH coupling method, the complex physical phenomena in hypervelocity collisions can be captured more realistically, especially the accurate description of material destruction and flow behavior.

[0082] S200, set symmetry constraints

[0083] When the mesh of the hypervelocity collision model is small, the computational scale of the full-size hypervelocity collision model is large. In this case, a 1 / 2 model or a 1 / 4 model can be used, and then symmetry constraints can be set to reduce the computational scale. Next, the 1 / 4 model is used as an example. Figure 2 ;It is necessary to set the symmetry constraints of nodes and smooth particles separately;

[0084] S201. Set the symmetry constraint of the node

[0085] Establish a Cartesian coordinate system, including Symmetrical plane and Symmetrical plane, such as Figure 3 As shown, in The nodes on the symmetry plane have no Displacement in direction , that is, formula (1):

[0086] ;

[0087] To avoid Nodes on the symmetry plane or The rotation of the axis, that is, equation (2) and equation (3):

[0088] ;

[0089] ;

[0090] In formula (2) and formula (3), Indicates winding The rotation angle of the axis, Indicates winding The rotation angle of the axis;

[0091] exist The nodes on the symmetry plane have no Displacement in direction , that is, formula (4):

[0092] ;

[0093] To avoid Nodes around the symmetry plane or The rotation of the axis, that is, equation (5) and equation (6):

[0094] ;

[0095] ;

[0096] In formula (5) and formula (6), Indicates winding The rotation angle of the axis, Indicates winding The rotation angle of the axis.

[0097] S202. Set symmetry constraints for smooth particles

[0098] Select Symmetrical plane and For SPH particles on the symmetry plane, create the same number of virtual SPH particles as the boundary particles on the symmetry plane of the SPH particles to ensure that the SPH particles and the virtual SPH particles are consistent in mass, pressure and absolute velocity, so that the virtual particles are in the calculation neighborhood of the real particles;

[0099] If the velocity of the boundary SPH particle is:

[0100] ;

[0101] for The speed of the virtual SPH particle on the symmetry plane is:

[0102] ;

[0103] for The speed of the virtual SPH particle on the symmetry plane is:

[0104] ;

[0105] The interaction between virtual particles and actual particles can be calculated by the following formula:

[0106] ;

[0107] In formula (10), The actual particle Physical quantities (such as density, velocity, pressure, etc.); and The actual particle Mass and physical quantities; and They are the mass and physical quantity of the virtual SPH particles respectively; and is the kernel function between real particles and other real particles or virtual particles, is the relative distance between particles, It is the smoothing length (or smoothing radius), which is used to control the support range of the kernel function.

[0108] S300, rigid-flexible conversion

[0109] First, the projectile and the first layer of target plates are set as soft bodies, that is, they can deform during the collision process, and the remaining target plates are defined as rigid bodies, that is, they do not participate in the numerical calculation. The rigid-flexible conversion of the target plate is determined by the moment of collision to determine whether the target plate needs to be transformed from a rigid body to a soft body. The collision moment between the projectile and the subsequent target plates is determined by the following formula:

[0110] ;

[0111] In formula (11), n is the total number of target plates, Indicates that the projectile flies from the starting point to the Target board time , Indicates the position of the target plate ( , Indicates the projectile position), For projectiles and The residual velocity after the target plate hits ( ; , represents the initial velocity of the projectile);

[0112] Next, the residual velocity of the projectile can be determined from the conservation of momentum:

[0113] ;

[0114] ;

[0115] In formula (12) and formula (13), represents the mass of the projectile, Indicates The mass of the part where the target plate collides with the projectile, With The residual velocity after the collision of the target plate;

[0116] By introducing this rigid-flexible body conversion mechanism, not only can the collision process between the projectile and the target be accurately simulated, but also the calculation scale is significantly reduced and the calculation efficiency is improved while ensuring the calculation accuracy.

[0117] S400, parallel processing

[0118] First, assume that the projectile and the target plate together form a set , which contains all elements of all projectiles and targets:

[0119] ;

[0120] in, Indicates element of a projectile or target, ;

[0121] Next, according to the number of CPU cores in the computer The difference will be collected Divide into sub-regions, and the total number of elements in these sub-regions is distributed as evenly as possible in each region; to simplify the expression, we define each sub-region as , so that the set can be represented as the union of these subregions:

[0122] ;

[0123] Assume that the size of each sub-region is , then we can conclude that:

[0124] ;

[0125] Thus, through this division, the set is evenly distributed sub-areas; each CPU core ,

[0126] Responsible for handling a sub-area , that is, the number of tasks processed by each CPU core is , so each CPU is responsible for processing the entire model part;

[0127] During the processing, each CPU will Execute the corresponding computing tasks; in this way, the tasks of the entire model can be processed in parallel. The process is efficiently distributed and completed between CPU cores.

[0128] S500, set static memory allocation

[0129] In the simulation of hypervelocity collisions, the interaction between SPH particles often requires a large amount of memory resources; the size of the computational domain is usually restricted by the size of the computer memory; in order to ensure the efficiency and stability of the computational process, the present invention adopts a static memory allocation method, that is, the required memory resources are pre-allocated during the program initialization phase, rather than being dynamically adjusted during operation; this strategy can effectively avoid memory fragmentation and frequent allocation / recycling operations caused by dynamic memory allocation, reduce computational interruptions or errors caused by insufficient memory, and improve the stability of the simulation;

[0130] Specifically, the key to static memory allocation is to allocate a fixed size of memory space to each SPH particle and ensure that its memory usage is consistent throughout the entire calculation process; suppose each SPH particle is , the amount of memory required is , in static memory allocation, each particle The memory size is a preset constant ,Right now ;

[0131] In addition, in order to consider the interaction between particles, the SPH method usually needs to record the relationship between each particle and its neighboring particles; In the neighborhood of The interaction between these particles also requires a certain amount of memory during the calculation process; to simplify the model, it can be assumed that the interaction between each particle and its neighboring particles requires a fixed amount of memory to be allocated in advance. , then the particle The total memory required is:

[0132] ;

[0133] Due to the use of static memory allocation, this amount of memory It will be pre-calculated and allocated to each particle at the beginning of the simulation, ensuring that the memory requirement of each particle is constant during the calculation process; therefore, the amount of memory required for each particle remains consistent throughout the simulation, avoiding potential risks brought about by dynamic adjustments.

[0134] In addition, static memory allocation helps improve the locality of memory access, reduce memory fragmentation, and improve cache hit rate; by reasonably estimating the number of particles and the number of neighboring particles , the total amount of memory required can be calculated when the program starts:

[0135] ;

[0136] in, The average number of particles in a particle's neighborhood; by allocating and locking memory in advance, frequent memory application and recycling operations during the calculation process are avoided, thereby improving calculation efficiency and reducing memory allocation overhead.

[0137] The invention has a reasonable concept and a simple process, can effectively improve the simulation efficiency of hypervelocity collisions, and provide more efficient computing support for hypervelocity collision research.

[0138] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for improving the computational efficiency and stability of hypervelocity collision simulation, characterized in that: The specific steps include: (1) Establishing a hypervelocity collision model Prepare the projectile and target plate in advance, and then use finite element units to model both the projectile and the target plate. When the material undergoes obvious deformation, fracture or breakage during the collision between the projectile and the target plate, the projectile and the target plate are transformed from finite element units to SPH particles. (2) Set symmetry constraints Set up symmetry constraints for nodes and smooth particles in the hypervelocity collision model; (3) Rigid-Flexible Conversion The projectile and the first layer of target plates are set as soft bodies, and the remaining target plates are defined as rigid bodies. The rigid body and soft body conversion of the target plate is determined based on the moment of collision to determine whether the target plate needs to be converted from a rigid body to a soft body; (4) Parallel processing First, all the elements of the projectile and the target are combined into a set, and then the set is divided into Divide into sub-regions and ensure that the total number of elements in each sub-region is evenly distributed in each region. Each CPU is responsible for processing a sub-region and performs corresponding computing tasks according to the sub-region assigned to it during the processing; (5) Static memory allocation The required memory resources are pre-allocated during the program initialization phase, a fixed-size memory space is allocated to each SPH particle, and its memory usage is ensured to be consistent throughout the entire processing process.

2. The method for improving the computational efficiency and stability of hypervelocity collision simulation according to claim 1, characterized in that: The specific process of setting the symmetry constraints of the nodes and smooth particles of the hypervelocity collision model in step (2) is as follows: (2.1) Set the symmetry constraints of the nodes Establish a Cartesian coordinate system, including Symmetrical plane and Symmetrical plane, The nodes on the symmetry plane have no displacement along the direction , that is, formula (1): ； To avoid Nodes on the symmetry plane or The rotation of the axis, that is, formula (2) and formula (3): ； ； In the above formula (2) and formula (3), Indicates winding The rotation angle of the axis, Indicates winding The rotation angle of the axis; exist The nodes on the symmetry plane have no Displacement in direction , that is, formula (4): ； To avoid Nodes on the symmetry plane or The rotation of the axis, that is, formula (5) and formula (6): ； ； In the above formula (5) and formula (6), represents the rotation angle around the y-axis, Represents the rotation angle around the z-axis; (2.2) Setting symmetry constraints for smooth particles Select Symmetrical plane and on symmetric plane SPH particles, create the same number of virtual SPH particles as boundary particles on the symmetry plane of the SPH particles, ensure that the SPH particles and virtual SPH particles are consistent in mass, pressure and absolute velocity, so that the virtual particles are in the calculation neighborhood of the real particles; If the velocity of the boundary SPH particle is: ； for The speed of the virtual SPH particle on the symmetry plane is: ； for The speed of the virtual SPH particle on the symmetry plane is: ； The interaction between virtual particles and actual particles is calculated by the following formula: ； In the above formula (10), The actual particle The physical quantity of and The actual particles Mass and physical quantities; and They are the mass and physical quantity of the virtual SPH particles; and are the kernel functions between real particles and other real particles or virtual particles, respectively; is the relative distance between particles; is the smoothing length, which is used to control the support range of the kernel function.

3. The method for improving the computational efficiency and stability of hypervelocity collision simulation according to claim 1, characterized in that: The collision moment between the projectile and the subsequent target plate in step (3) is determined by the following formula (11): ； In formula (11), n is the total number of target plates, Indicates that the projectile flies from the starting point to the The time of the target board, ; Indicates the position of the target plate, , Indicates the projectile position; For projectiles and The residual velocity after the collision of the target plate is ; , represents the initial velocity of the projectile; Calculate the residual velocity of the projectile using the conservation of momentum: ； ； In the above formula (12) and formula (13), Indicates the mass of the projectile; Indicates The mass of the part of the target plate that collides with the projectile; For the The residual velocity after the target plate hits.

4. The method for improving the computational efficiency and stability of hypervelocity collision simulation according to claim 1, characterized in that: The specific process of step (4) is as follows: Suppose the projectile and the target plate together form a set , which contains all elements of all projectiles and targets: ； in, Indicates element of a projectile or target, ; Define each sub-region as ,in , so that the set is represented as the union of these subregions: ； Assume that the size of each sub-region is , then we get: ； Each CPU core , , responsible for processing a sub-area , that is, the number of tasks processed by each CPU core is , so each CPU is responsible for processing the entire model Part; Each CPU will process according to its own assigned sub-area Execute the corresponding computing tasks.

5. The method for improving the computational efficiency and stability of hypervelocity collision simulation according to claim 1, characterized in that: The specific process of step (5) is as follows: Assume that each SPH particle is , the amount of memory required is , in static memory allocation, each particle The memory size is a preset constant ,Right now ; Assume particle In the neighborhood of particles, assuming that each particle's interaction with its neighbors requires a fixed amount of memory to be allocated in advance , then the particle The total memory required is: ； By reasonably estimating the number of particles and the number of neighboring particles , calculate the total amount of memory required when the program starts: ； In the above formula (18), is the average number of particles in the particle neighborhood.

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