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zFS*^!Rt`d_u!_KbQ`MHFp}>X0(`BMgw4tNIE=j>aPw-y5P6D}ILjfnF%^RGFUX6>+ z;*n6YwxjVMPb&3HlIG}LmGC`v9mx(7mTSgCOEaLpo>-~+I**ZlIBeT&C#)g}hwAZJ ztyHQia7-!LVMR9Ftkewz6V#=v*4`nIfV4Bn+pRX$bO^l#-6R6d-}BGRO(YXnXZD5hH+PdT)H z5Bx;I`76iVY|jOYmT@X@DeEpgYn2PLKY`)l0*yh5{ku&6U)WtmVXj=IS$k8E8TtQ- zhyVEt5tAT0`+xauiRt~`RDs maxSpeed) + { + vel2[index] = vel2[index] / glm::length(vel2[index])*maxSpeed; + } + } + //Why not vel1? because we need the former velocity to compute new velocity in rule3. } /** @@ -289,6 +370,19 @@ __global__ void kernComputeIndices(int N, int gridResolution, // - Label each boid with the index of its grid cell. // - Set up a parallel array of integer indices as pointers to the actual // boid data in pos and vel1/vel2 + int id = (blockIdx.x * blockDim.x) + threadIdx.x; + if (id < N) + { + int x = (pos[id].x - gridMin.x)*inverseCellWidth; + int y = (pos[id].y - gridMin.y)*inverseCellWidth; + int z = (pos[id].z - gridMin.z)*inverseCellWidth; + gridIndices[id] = gridIndex3Dto1D(x, y, z, gridResolution); + indices[id] = id; + } + else + { + return; + } } // LOOK-2.1 Consider how this could be useful for indicating that a cell @@ -306,6 +400,27 @@ __global__ void kernIdentifyCellStartEnd(int N, int *particleGridIndices, // Identify the start point of each cell in the gridIndices array. // This is basically a parallel unrolling of a loop that goes // "this index doesn't match the one before it, must be a new cell!" + int index = (blockIdx.x * blockDim.x) + threadIdx.x; + if (index < N) + { + if (index == 0) + { + gridCellStartIndices[particleGridIndices[0]] = 0; + } + else if (index == N - 1) + { + gridCellEndIndices[particleGridIndices[N - 1]] = N - 1; + } + else if (particleGridIndices[index] != particleGridIndices[index - 1]) + { + gridCellStartIndices[particleGridIndices[index]] = index; + gridCellEndIndices[particleGridIndices[index-1]] = index-1; + } + } + else + { + return; + } } __global__ void kernUpdateVelNeighborSearchScattered( @@ -322,8 +437,132 @@ __global__ void kernUpdateVelNeighborSearchScattered( // - Access each boid in the cell and compute velocity change from // the boids rules, if this boid is within the neighborhood distance. // - Clamp the speed change before putting the new speed in vel2 + + + int index = (blockIdx.x * blockDim.x) + threadIdx.x; + if (index >= N) + { + return; + } + int particleIndex = particleArrayIndices[index]; + float x = (pos[particleIndex].x - gridMin.x) * inverseCellWidth; + float y = (pos[particleIndex].y - gridMin.y) * inverseCellWidth; + float z = (pos[particleIndex].z - gridMin.z) * inverseCellWidth; + + int dx = (x - (int)x) > 0.5f ? 1 : -1; + int dy = (y - (int)y) > 0.5f ? 1 : -1; + int dz = (x - (int)z) > 0.5f ? 1 : -1; + + int neighborsIndex[8]; + neighborsIndex[0] = gridIndex3Dto1D(x, y, z, gridResolution); + neighborsIndex[1] = gridIndex3Dto1D(x + dx, y, z, gridResolution); + neighborsIndex[2] = gridIndex3Dto1D(x + dx, y + dy, z, gridResolution); + neighborsIndex[3] = gridIndex3Dto1D(x + dx, y, z + dz, gridResolution); + neighborsIndex[4] = gridIndex3Dto1D(x + dx, y + dy, z + dz, gridResolution); + neighborsIndex[5] = gridIndex3Dto1D(x, y + dy, z, gridResolution); + neighborsIndex[6] = gridIndex3Dto1D(x, y + dy, z + dz, gridResolution); + neighborsIndex[7] = gridIndex3Dto1D(x, y, z + dz, gridResolution); + + glm::vec3 v1(0.0f, 0.0f, 0.0f); + glm::vec3 v2(0.0f, 0.0f, 0.0f); + glm::vec3 v3(0.0f, 0.0f, 0.0f); + + glm::vec3 c(0.0f, 0.0f, 0.0f); + glm::vec3 perceived_center(0.0f,0.0f,0.0f); + glm::vec3 perceived_velocity(0.0f, 0.0f, 0.0f); + + int gridCount = gridResolution * gridResolution * gridResolution; + + int r1 = 0 , r3 = 0; + + for (int i = 0; i < 8; i++) + { + if (neighborsIndex[i] < 0 || neighborsIndex[i] >= gridCount) + { + continue; + } + if (gridCellStartIndices[neighborsIndex[i]] != -1 && gridCellEndIndices[neighborsIndex[i]] != -1) + { + for (int j = gridCellStartIndices[neighborsIndex[i]]; j <= gridCellEndIndices[neighborsIndex[i]]; j++) + { + int k = particleArrayIndices[j]; + //if the particle is not itself ***see the rule definition in 1.2 + if (particleIndex != k) + { + float distance = glm::length(pos[k] - pos[particleIndex]); + + if (distance < rule1Distance) + { + + perceived_center += pos[k]; + //printf("I'm here\n"); + r1 ++; + } + if (distance < rule2Distance) + { + if (distance < 100) + { + c -= (pos[k] - pos[particleIndex]); + } + } + if (distance < rule3Distance) + { + perceived_velocity += vel1[k]; + r3 ++; + } + } + } + } + } + + if (r1 > 0) + { + perceived_center /= r1; + v1 = (perceived_center - pos[particleIndex])* rule1Scale; + } + + v2 = c * rule2Scale; + + if (r3 > 0) + { + perceived_velocity /= r3; + v3 = perceived_velocity * rule3Scale; + } + + + glm::vec3 finVel = vel1[particleIndex] + v1 +v2 +v3; + + + + //clamp + if (glm::length(finVel) > maxSpeed) + { + finVel = glm::normalize(finVel) * maxSpeed; + } + + vel2[particleIndex] = finVel; + } + +__global__ void kernRearrange( + int N, int* particleArrayIndices, glm::vec3* pos, glm::vec3* posAf, + glm::vec3* vel, glm::vec3* velAf) +{ + int index = (blockIdx.x * blockDim.x) + threadIdx.x; + if (index >= N) { + return; + } + posAf[index] = pos[particleArrayIndices[index]]; + velAf[index] = vel[particleArrayIndices[index]]; + + + +} + + + + __global__ void kernUpdateVelNeighborSearchCoherent( int N, int gridResolution, glm::vec3 gridMin, float inverseCellWidth, float cellWidth, @@ -341,6 +580,110 @@ __global__ void kernUpdateVelNeighborSearchCoherent( // - Access each boid in the cell and compute velocity change from // the boids rules, if this boid is within the neighborhood distance. // - Clamp the speed change before putting the new speed in vel2 + + + int index = (blockIdx.x * blockDim.x) + threadIdx.x; + if (index >= N) + { + return; + } + + float x = (pos[index].x - gridMin.x) * inverseCellWidth; + float y = (pos[index].y - gridMin.y) * inverseCellWidth; + float z = (pos[index].z - gridMin.z) * inverseCellWidth; + + int dx = ((x - (int)x) >= 0.5f) ? 1 : -1; + int dy = ((y - (int)y) >= 0.5f) ? 1 : -1; + int dz = ((z - (int)z) >= 0.5f) ? 1 : -1; + + int neighborsIndex[8]; + neighborsIndex[0] = gridIndex3Dto1D(x, y, z, gridResolution); + neighborsIndex[1] = gridIndex3Dto1D(x + dx, y, z, gridResolution); + neighborsIndex[2] = gridIndex3Dto1D(x + dx, y + dy, z, gridResolution); + neighborsIndex[3] = gridIndex3Dto1D(x + dx, y, z + dz, gridResolution); + neighborsIndex[4] = gridIndex3Dto1D(x + dx, y + dy, z + dz, gridResolution); + neighborsIndex[5] = gridIndex3Dto1D(x, y + dy, z, gridResolution); + neighborsIndex[6] = gridIndex3Dto1D(x, y + dy, z + dz, gridResolution); + neighborsIndex[7] = gridIndex3Dto1D(x, y, z + dz, gridResolution); + + glm::vec3 v1(0.0f, 0.0f, 0.0f); + glm::vec3 v2(0.0f, 0.0f, 0.0f); + glm::vec3 v3(0.0f, 0.0f, 0.0f); + + glm::vec3 c(0.f); + glm::vec3 perceived_center(0.f); + glm::vec3 perceived_velocity(0.f); + + int r1=0, r3 = 0; + + int gridCount = gridResolution * gridResolution * gridResolution; + + for (int i = 0; i < 8; i++) + { + if (neighborsIndex[i] < 0 || neighborsIndex[i] >= gridCount) + { + continue; + } + if (gridCellStartIndices[neighborsIndex[i]] != -1 && gridCellEndIndices[neighborsIndex[i]] != -1) + { + for (int j = gridCellStartIndices[neighborsIndex[i]]; j <= gridCellEndIndices[neighborsIndex[i]]; j++) + { + //if the particle is not itself ***see the rule definition in 1.2 + if (index != j) + { + float distance = glm::length(pos[j] - pos[index]); + + if (distance < rule1Distance) + { + + perceived_center += pos[j]; + //printf("I'm here\n"); + r1++; + } + if (distance < rule2Distance) + { + if (distance < 100) + { + c -= (pos[j] - pos[index]); + } + } + if (distance < rule3Distance) + { + perceived_velocity += vel1[j]; + r3++; + } + } + } + } + } + + if (r1 > 0) + { + perceived_center /= r1; + v1 = (perceived_center - pos[index])* rule1Scale; + } + + v2 = c * rule2Scale; + + if (r3 > 0) + { + perceived_velocity /= r3; + v3 = perceived_velocity * rule3Scale; + } + + + glm::vec3 finVel = vel1[index] + v1 + v2 + v3; + + + + //clamp + if (glm::length(finVel) > maxSpeed) + { + finVel = glm::normalize(finVel) * maxSpeed; + } + + vel2[index] = finVel; + } /** @@ -349,6 +692,12 @@ __global__ void kernUpdateVelNeighborSearchCoherent( void Boids::stepSimulationNaive(float dt) { // TODO-1.2 - use the kernels you wrote to step the simulation forward in time. // TODO-1.2 ping-pong the velocity buffers + int N = numObjects; + dim3 fullBlocksPerGrid((N + blockSize - 1) / blockSize); + kernUpdateVelocityBruteForce << > > (N, dev_pos, dev_vel1, dev_vel2); + kernUpdatePos << > > (N, dt, dev_pos, dev_vel2); + + std::swap(dev_vel1, dev_vel2); } void Boids::stepSimulationScatteredGrid(float dt) { @@ -364,6 +713,29 @@ void Boids::stepSimulationScatteredGrid(float dt) { // - Perform velocity updates using neighbor search // - Update positions // - Ping-pong buffers as needed + + dim3 fullBlocksPerGrid((numObjects + blockSize - 1) / blockSize); + + kernComputeIndices << > > (numObjects, gridSideCount, gridMinimum, gridInverseCellWidth, dev_pos, dev_particleArrayIndices, dev_particleGridIndices); + + dev_thrust_particleArrayIndices = thrust::device_ptr(dev_particleGridIndices); + dev_thrust_particleGridIndices = thrust::device_ptr(dev_particleArrayIndices); + thrust::sort_by_key(dev_thrust_particleGridIndices, dev_thrust_particleGridIndices + numObjects, dev_thrust_particleArrayIndices); + + dim3 BlocksPerGridCell = (gridCellCount + blockSize - 1) / blockSize; + + kernResetIntBuffer << > > (gridCellCount, dev_gridCellStartIndices, -1); + kernResetIntBuffer << > > (gridCellCount, dev_gridCellEndIndices, -1); + + + + kernIdentifyCellStartEnd << > > (numObjects, dev_particleGridIndices, dev_gridCellStartIndices, dev_gridCellEndIndices); + + kernUpdateVelNeighborSearchScattered << > > (numObjects, gridSideCount, gridMinimum, gridInverseCellWidth, gridCellWidth, dev_gridCellStartIndices, dev_gridCellEndIndices, dev_particleArrayIndices, dev_pos, dev_vel1, dev_vel2); + + kernUpdatePos << > >(numObjects, dt, dev_pos, dev_vel2); + + std::swap(dev_vel1, dev_vel2); } void Boids::stepSimulationCoherentGrid(float dt) { @@ -382,6 +754,38 @@ void Boids::stepSimulationCoherentGrid(float dt) { // - Perform velocity updates using neighbor search // - Update positions // - Ping-pong buffers as needed. THIS MAY BE DIFFERENT FROM BEFORE. + + + dim3 fullBlocksPerGrid((numObjects + blockSize - 1) / blockSize); + + kernComputeIndices << > > (numObjects, gridSideCount, gridMinimum, gridInverseCellWidth, dev_pos, dev_particleArrayIndices, dev_particleGridIndices); + + dev_thrust_particleArrayIndices = thrust::device_ptr(dev_particleGridIndices); + dev_thrust_particleGridIndices = thrust::device_ptr(dev_particleArrayIndices); + thrust::sort_by_key(dev_thrust_particleGridIndices, dev_thrust_particleGridIndices + numObjects, dev_thrust_particleArrayIndices); + + dim3 BlocksPerGridCell = (gridCellCount + blockSize - 1) / blockSize; + + kernResetIntBuffer << > > (gridCellCount, dev_gridCellStartIndices, -1); + kernResetIntBuffer << > > (gridCellCount, dev_gridCellEndIndices, -1); + + + + kernIdentifyCellStartEnd << > > (numObjects, dev_particleGridIndices, dev_gridCellStartIndices, dev_gridCellEndIndices); + + kernRearrange<< > >(numObjects, + dev_particleArrayIndices, dev_pos, dev_posAf, dev_vel1, dev_velAf); + + kernUpdateVelNeighborSearchCoherent << > >(numObjects, gridSideCount, gridMinimum, gridInverseCellWidth, gridCellWidth, + dev_gridCellStartIndices, dev_gridCellEndIndices, dev_posAf, dev_velAf, dev_vel2); + + std::swap(dev_pos, dev_posAf); + + kernUpdatePos << > >(numObjects, dt, dev_pos, dev_vel2); + + std::swap(dev_vel1, dev_vel2); + + } void Boids::endSimulation() { @@ -390,6 +794,14 @@ void Boids::endSimulation() { cudaFree(dev_pos); // TODO-2.1 TODO-2.3 - Free any additional buffers here. + + cudaFree(dev_gridCellStartIndices); + cudaFree(dev_gridCellEndIndices); + cudaFree(dev_particleArrayIndices); + cudaFree(dev_particleGridIndices); + + cudaFree(dev_posAf); + cudaFree(dev_velAf); } void Boids::unitTest() { diff --git a/src/main.cpp b/src/main.cpp index a29471d..a7c2c99 100644 --- a/src/main.cpp +++ b/src/main.cpp @@ -14,8 +14,8 @@ // LOOK-2.1 LOOK-2.3 - toggles for UNIFORM_GRID and COHERENT_GRID #define VISUALIZE 1 -#define UNIFORM_GRID 0 -#define COHERENT_GRID 0 +#define UNIFORM_GRID 1 +#define COHERENT_GRID 1 // LOOK-1.2 - change this to adjust particle count in the simulation const int N_FOR_VIS = 5000; From 734d00adec5a25904b556cd74190d21525e18baf Mon Sep 17 00:00:00 2001 From: XinCastle Date: Tue, 12 Sep 2017 00:36:08 -0400 Subject: [PATCH 2/5] Readme file update --- README.md | 74 +++++++++++++++++++++++++++++++++++++++++++++++++++---- 1 file changed, 69 insertions(+), 5 deletions(-) diff --git a/README.md b/README.md index 98dd9a8..ea13a32 100644 --- a/README.md +++ b/README.md @@ -1,10 +1,74 @@ **University of Pennsylvania, CIS 565: GPU Programming and Architecture, Project 1 - Flocking** -* (TODO) YOUR NAME HERE -* Tested on: (TODO) Windows 22, i7-2222 @ 2.22GHz 22GB, GTX 222 222MB (Moore 2222 Lab) +* Xincheng Zhang +* Tested on: Windows 10, i7-4702HQ @ 2.20GHz 8GB, GTX 870M 3072MB (Personal Laptop) -### (TODO: Your README) -Include screenshots, analysis, etc. (Remember, this is public, so don't put -anything here that you don't want to share with the world.) + +## ScreenShot + +___ + + +![](https://github.com/XinCastle/Project1-CUDA-Flocking/blob/master/images/2017.09.11.23.34.16.gif) + + +This result is tested under the following settings: + number of particles: 5000 + blocksize: 128 + coherent-uniform + rule1Distance 5.0f + rule2Distance 3.0f + rule3Distance 5.0f + rule1Scale 0.01f + rule2Scale 0.1f + rule3Scale 0.1f + maxSpeed 1.0f + + +## Performance Analysis + +___ + +The performance of **Naive Boids**, **Uniform Boids** and **Coherent Uniform Boids** is measured by FPS. +The following is the diagram comparing these three methods. + +https://github.com/XinCastle/Project1-CUDA-Flocking/blob/master/images/diagram.png + +**Note:I find the movement of mine looks the same as the example. However, the frame rates of uniform boids + and coherent uniform boids are much slower than I expect. I checked my algorithm but wasn't able to tell why + it happens. + + In conclusion, when the boids number is more than 1000, the performance order is: Coherent Uniform Boids > Uniform Boids > Naive Boids + + +### Questions + +___ +*For each implementation, how does changing the number of boids affect performance? Why do you think this is? + +1. As the number of boids increases, the performance becomes worse for all the three methods. As for how much they are affected, the order +is: Naive Boids > Uniform Boids ~=(almost equal to) Cohereent Uniform Boids. I think the reason is because as we add more boids in the system, +there are more neighbors around a particle so that the whole algorithm requires more time for calculation. The naive approach is the least +efficient, so it decreases faster than the other two methods. + + +*For each implementation, how does changing the block count and block size affect performance? Why do you think this is? + +2. As the blocksize becomes larger, the frame rate slightly drops. I think the reason is the following: +It's less efficient to fetch and access data in larger memory. Therefore, as the blocksize becomes larger, it takes more and more time for +the system to access the memory. + + +*For the coherent uniform grid: did you experience any performance improvements with the more coherent uniform grid? Was this the outcome you expected? Why or why not? + +3. Yes, the coherent uniform grid has better performance than the uniform grid because reading data which is stored in contiguous memory is faster. +Therefore, the main reason of the improvement is because of contiguous memory. + + +*Did changing cell width and checking 27 vs 8 neighboring cells affect performance? Why or why not? + +4. Yes, when the number of neighboring cells becomes 27 rather than 8, the performance becomes better. I think it's because as cell width decreases, +the total particles defined as "neighboring particles" significantly decreases. Therfore, the time cost decreases although we have more cells to loop +in the function. \ No newline at end of file From 857c881eb49a3b335f718b4074f7c1e26fb254c2 Mon Sep 17 00:00:00 2001 From: XinCastle Date: Tue, 12 Sep 2017 00:40:58 -0400 Subject: [PATCH 3/5] README --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index ea13a32..462a914 100644 --- a/README.md +++ b/README.md @@ -34,7 +34,7 @@ ___ The performance of **Naive Boids**, **Uniform Boids** and **Coherent Uniform Boids** is measured by FPS. The following is the diagram comparing these three methods. -https://github.com/XinCastle/Project1-CUDA-Flocking/blob/master/images/diagram.png +![](https://github.com/XinCastle/Project1-CUDA-Flocking/blob/master/images/diagram.png) **Note:I find the movement of mine looks the same as the example. However, the frame rates of uniform boids and coherent uniform boids are much slower than I expect. I checked my algorithm but wasn't able to tell why From 4324ff5ba74889afa50dded9417e05e35f4c82cf Mon Sep 17 00:00:00 2001 From: XinCastle Date: Tue, 12 Sep 2017 00:43:43 -0400 Subject: [PATCH 4/5] README modify --- README.md | 21 ++++++++++----------- 1 file changed, 10 insertions(+), 11 deletions(-) diff --git a/README.md b/README.md index 462a914..0595972 100644 --- a/README.md +++ b/README.md @@ -14,17 +14,16 @@ ___ ![](https://github.com/XinCastle/Project1-CUDA-Flocking/blob/master/images/2017.09.11.23.34.16.gif) -This result is tested under the following settings: - number of particles: 5000 - blocksize: 128 - coherent-uniform - rule1Distance 5.0f - rule2Distance 3.0f - rule3Distance 5.0f - rule1Scale 0.01f - rule2Scale 0.1f - rule3Scale 0.1f - maxSpeed 1.0f +*This result is tested under the following settings: +number of particles: 5000 +blocksize: 128 +coherent-uniform +rule1Distance 5.0f +rule2Distance 3.0f +rule3Distance 5.0f +rule1Scale 0.01f +rule2Scale 0.1f +maxSpeed 1.0f ## Performance Analysis From eb06e8f879b7f91a86092d9ef4a3f16230b11f4b Mon Sep 17 00:00:00 2001 From: XinCastle Date: Tue, 12 Sep 2017 00:46:38 -0400 Subject: [PATCH 5/5] README MODIFY --- README.md | 30 +++++++++++++++--------------- 1 file changed, 15 insertions(+), 15 deletions(-) diff --git a/README.md b/README.md index 0595972..fca931f 100644 --- a/README.md +++ b/README.md @@ -14,16 +14,16 @@ ___ ![](https://github.com/XinCastle/Project1-CUDA-Flocking/blob/master/images/2017.09.11.23.34.16.gif) -*This result is tested under the following settings: -number of particles: 5000 -blocksize: 128 -coherent-uniform -rule1Distance 5.0f -rule2Distance 3.0f -rule3Distance 5.0f -rule1Scale 0.01f -rule2Scale 0.1f -maxSpeed 1.0f +* This result is tested under the following settings: +* number of particles: 5000 +* blocksize: 128 +* coherent-uniform +* rule1Distance 5.0f +* rule2Distance 3.0f +* rule3Distance 5.0f +* rule1Scale 0.01f +* rule2Scale 0.1f +* maxSpeed 1.0f ## Performance Analysis @@ -35,7 +35,7 @@ The following is the diagram comparing these three methods. ![](https://github.com/XinCastle/Project1-CUDA-Flocking/blob/master/images/diagram.png) -**Note:I find the movement of mine looks the same as the example. However, the frame rates of uniform boids +** Note:I find the movement of mine looks the same as the example. However, the frame rates of uniform boids and coherent uniform boids are much slower than I expect. I checked my algorithm but wasn't able to tell why it happens. @@ -45,7 +45,7 @@ The following is the diagram comparing these three methods. ### Questions ___ -*For each implementation, how does changing the number of boids affect performance? Why do you think this is? +* For each implementation, how does changing the number of boids affect performance? Why do you think this is? 1. As the number of boids increases, the performance becomes worse for all the three methods. As for how much they are affected, the order is: Naive Boids > Uniform Boids ~=(almost equal to) Cohereent Uniform Boids. I think the reason is because as we add more boids in the system, @@ -53,20 +53,20 @@ there are more neighbors around a particle so that the whole algorithm requires efficient, so it decreases faster than the other two methods. -*For each implementation, how does changing the block count and block size affect performance? Why do you think this is? +* For each implementation, how does changing the block count and block size affect performance? Why do you think this is? 2. As the blocksize becomes larger, the frame rate slightly drops. I think the reason is the following: It's less efficient to fetch and access data in larger memory. Therefore, as the blocksize becomes larger, it takes more and more time for the system to access the memory. -*For the coherent uniform grid: did you experience any performance improvements with the more coherent uniform grid? Was this the outcome you expected? Why or why not? +* For the coherent uniform grid: did you experience any performance improvements with the more coherent uniform grid? Was this the outcome you expected? Why or why not? 3. Yes, the coherent uniform grid has better performance than the uniform grid because reading data which is stored in contiguous memory is faster. Therefore, the main reason of the improvement is because of contiguous memory. -*Did changing cell width and checking 27 vs 8 neighboring cells affect performance? Why or why not? +* Did changing cell width and checking 27 vs 8 neighboring cells affect performance? Why or why not? 4. Yes, when the number of neighboring cells becomes 27 rather than 8, the performance becomes better. I think it's because as cell width decreases, the total particles defined as "neighboring particles" significantly decreases. Therfore, the time cost decreases although we have more cells to loop