Dense Optical Flow Prediction From a Static Image

Dense Optical Flow Prediction from a Static Image

Jacob Walker,Abhinav Gupta,and Martial Hebert Robotics Institute,Carnegie Mellon University

{jcwalker,abhinavg,hebert }@https://www.wendangku.net/doc/035865202.html,

Abstract

Given a scene,what is going to move,and in what di-rection will it move?Such a question could be considered a non-semantic form of action prediction.In this work,we present predictive convolutional neural networks (P-CNN).Given a static image,P-CNN predicts the future motion of each and every pixel in the image in terms of optical ?ow.Our P-CNN model leverages the data in tens of thousands of realistic videos to train our model.Our method relies on absolutely no human labeling and is able to predict motion based on the context of the scene.Since P-CNNs make no assumptions about the underlying scene they can predict fu-ture optical ?ow on a diverse set of scenarios.In terms of quantitative performance,P-CNN outperforms all previous approaches by large margins.

1.Introduction

Consider the images shown in Figure 1.Given the girl in front of the cake,we humans can easily predict that her head will move downward to extinguish the candle.The man with the discus is in a position to twist his body strongly to the right,and the squatting man on the bottom has nowhere to move but up.Humans have an amazing ability to not only recognize what is present in the image but also predict what is going to happen next.Prediction is an important com-ponent of visual understanding and cognition.In order for computers to react to their environment,simple activity de-tection is not always suf?cient.For successful interactions,robots need to predict the future and plan accordingly.

While prediction is still a relatively new problem,there has been some recent work that has focused on this task.The most common approach to this prediction problem is to use a planning-based agent-centric approach:an object [10]or a patch [24]is modeled as an agent that performs actions based on its current state and the goal state.Each action is decided based on compatibility with the environment and how this actions helps the agent move closer to the goal state.The priors on actions are modeled via transition ma-trices.Such an approach has been shown to produce

im-(a)Input Image (b)Prediction

Figure 1:Motion Prediction.Con-sider single,static input images (a).Our method can ?rst identify what these actions are and predict (b)correct motion based on the pose and stage of the action without any video information.We use the color coding from [1]shown on the

right.

pressive results:predicting trajectories of humans in park-ing lots [10]or hallucinating car movements on streets [24].There are two main problems with this approach.First,the predictions are still sparse,and the motion is still modeled as a trajectory.Second,and more importantly,these ap-proaches have always been shown to perform in restrictive domains such as parking lots or streets.

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a r X i v :1505.00295v 1 [c s .C V ] 2 M a y 2015

In this paper,we take the next step towards generalized prediction—a framework that can be learned from tens of thousands of realistic videos.The framework can work in indoor and outdoor environments if the agent is an animal, a human,or even a car.It can account for one or multiple agents.Speci?cally,we look at the task of motion predic-tion—given a static image we predict the dense expected optical?ow as if this image was part of a video.This op-tical?ow represents how and where each and every pixel in the image is going to move in the future.Of course,we can see that motion prediction is a highly-context dependent problem.The future motion not only depends on what is active in the scene but also its context.For example,some-one’s entire body may move up or down if they are jump-roping,but most of the body will be stationary if they are playing the?ute.Instead of modeling agents and its context separately under restrictive assumptions,we use a learning based approach for motion prediction.Speci?cally,we train a deep network that can incorporate all of this contextual in-formation to make accurate predictions of future motion in a wide variety of scenes.We train our model from thou-sands of realistic video datasets,namely the UCF-101[21] and the HMDB-51[13].

Contributions:Our paper makes threefold contributions. First,we present a predictive-CNN(P-CNN)model for mo-tion prediction.Given a static image,our P-CNN model predicts expected motion in terms of optical?ow.Our CNN-based model is agent-free and makes almost no as-sumptions about the underlying scene.Therefore,we show experimental results on diverse set of scenes.Second,P-CNN gives state of the art performance on prediction com-pared to contemporary approaches.Finally,we also present an proof of concept extension of P-CNN which makes long-range prediction about future motion.Our preliminary re-sults indicate the P-CNN might indeed be promising even on the task of long-range prediction.

2.Background

Prediction has caught the interest of the vision commu-nity in recent years.Most of research in this area has looked at different aspects of the problem.The?rst aspect of inter-est is what should be predicted:what is the output space of prediction.Some of the initial work in this area focused on predicting the optical?ow for the given input image[28]. Others have looked at more semantic forms of prediction: that is,predict the action class of what is going to happen next[5,14].But one of the issues with semantic predic-tion is that it tells us nothing about the future action beyond the category.One of our goals in prediction is to go beyond classi?cation and predict the spatial layout of future actions. For example,in case of agents such as humans,the output space of prediction can be trajectories themselves[10].But recent approaches have argued for much richer form of pre-dictions even in terms of pixels[24]or the features of the next frame[7,19].

The other aspect of research in visual prediction looks at the question:what is the right approach for prediction? There have been two classes of approaches for the tempo-ral prediction.The?rst is data-driven,non-parametric ap-proach.In case of non-parameteric approaches,they do not make any assumptions about the underlying scene.For ex-ample,[28]simply retrieves videos visually similar to the static scene,allowing a warping[16]of the matched ac-tion into the scene.The other end of the spectrum is para-metric and domain-speci?c approaches.Here,we make assumptions of what are the active elements in the scene whether they may be cars or people.Once the assumption is made,then a model is developed to predict agent behav-ior.This includes forecasting pedestrian trajectories[10], human-human interactions[7],[14],human expressions through SOSVM[5],and human-object interaction through graphical models[11],[3].

Some of the recent work in this area has looked at a more hybrid approach.For example,Walker et al.[24] builds a data-derived dictionary of rigid objects given a video domain and then makes long-term motion and ap-pearance predictions using a transition and context model. Recent approaches such as[19]and[22]have even looked at training convolutional neural networks for predicting one future frame in a clip[19]or motion of handwritten charac-ters[22].

We make multiple advances over previous work in this paper.First,our unsupervised method can generalize across a large number of diverse domains.While[24]does not explicitly require video labels,it is still domain dependent, requiring a human-given distinction between videos in and outside the domain.In addition,[24]focused only on birds-eye domains where scene depth was limited or non existent, while our method is able to generalize to scenes with per-spective.[18]also uses unsupervised methods to train a Structured Random Forest for motion prediction.However, the authors only learn a model from the simple KTH[15] dataset.We show that our method is able to learn from a set of videos that is far more diverse across scenes and actions. In addition,we demonstrate much better generalization can be obtained as compared to the nearest-neighbor approach of Yuen et al.[28].

Convolutional Neural Networks:We show in this paper that a convolutional neural network can be trained for the task of motion prediction in terms of optical?ow.Current work on CNNs have largely focused on recognition tasks both in images and video[12,9,4,23,29,20,22].There has been some initial work where CNNs have been com-bined with recurrent models for prediction.For example, [19]uses a LSTM[6]to predict the immediate next frame given a video input.[22]uses a recurrent architecture to

(a)Input Image(b)Prediction(c)Ground Truth

Figure3:Consider the images on the left.Is the man squat-

ting up or down?The bottom is near completion(or just

starting),and the top image is right in the middle of the

action.Our dataset contains a large number of ambiguous

images such as these.In our evaluation we consider the

underlying distribution of movements predicted by our net-

work.It is highly likely that this man is going to move up or

down,but unlikely that he will veer off to the left or right.

predict motions of handwritten characters from a video.On

the other hand,our approach predicts motion for each and

every pixel from a static image for any generic scene.

3.Methods

Our goal is to learn a mapping between the input RGB

image and the output space which corresponds to the pre-

dicted motion of each and every pixel in terms of optical

?ow.We propose to use CNNs as the underlying learning

algorithm for this task.However,there are few questions

that need to be answered:what is a good output space and

what is a good loss function?Should we model optical?ow

prediction as a regression or a classi?cation problem?What

is a good architecture to solve this problem?We now dis-

cuss these issues below.

3.1.Regression as Classi?cation

Intuitively,motion estimation can be posed as a regres-

sion problem since the space is continuous.Indeed,this is

exactly the approach used in[18],where the authors used

structured random forests to regress the magnitude and di-

rection of the optical?ow.However,such an approach has

one drawback:such an output space tends to smoothen re-

sults as the ambiguity is handled by averaging out the?ow.

Interestingly,in a related problem of surface normal pre-

diction,researchers have proposed reformulating structured

regression as a classi?cation problem[26,17].Speci?-

cally,they quantize the surface normal vectors into a code-

book of clusters and then output space becomes predicting

the cluster membership.In our work,we take a similar ap-

proach.We quantize optical?ow vectors into40clusters by

k-means.We can then treat the problem in a manner similar

to semantic segmentation,where we classify each region as

the image as a particular cluster of optical?ow.We use a

soft-max loss layer at the output for computing gradients.

However,at test time,we create a soft output by consid-

ering the underlying distribution of all the clusters,taking

a weighted-probability sum over all the classes in a given

pixel for the?nal output.Transforming the problem into

classi?cation also leads directly to a discrete probability

distribution over vector directions and magnitudes.As the

problem of motion prediction can be ambiguous depending

on the image(see Figure3),we can utilize this probability

distribution over directions to measure how informative our

predictions are.We may be unsure if the man in Figure3is

sitting down or standing up given only the image,but we can

be quite sure he will not turn right or left.In the same way,

our network can rank upward and downward facing clusters

much higher than other directions.Even if the ground truth

is upward,and the highest ranked cluster is downward,it

may be that the second-highest cluster is also upward.A

discrete probability distribution,through classi?cation,al-

lows an easier understanding of how well our network may

be performing.In addition,we can simply compute the en-

tropy of the distribution,allowing us to compute the con?-

dence of our motion prediction and retrieve images that are

more likely to be correct.

https://www.wendangku.net/doc/035865202.html,work Design

Our model is similar to the standard seven-layer archi-

tecture from[12].To simplify the description,we denote

the convolutional layer as C(k,s),which indicates the there

are k kernels,each having the size of s×s.During convo-

lution,we set all the strides to1except for the?rst layer,

which is4.We also denote the local response normaliza-

tion layer as LRN,and the max-pooling layer as MP.The

stride for pooling is2and we set the pooling operator size

as3×3.Finally,F(n)denotes fully connected layer with

n neurons.Our network architecture can be described as:

This can be described as:C(96,11)→LRN→P→

C(256,5)→LRN→P→C(384,3)→C(384,3)→

C(256,3)→P→F(4096)→F(4096).We used a modi-

?ed version of the popular Caffe toolbox[8]for our imple-

mentation.For computational simplicity,we use200x200

windows as input.We used a learning rate of0.0001and

a stepsize of50000iterations.Other network parameters

were set to default.The only exception is that we used

Xavier initialization of parameters.Instead of using the de-

fault softmax output,we used a spatial softmax loss func-

tion from[26]to classify every region in the image.This

leads to a M×N×K softmax layer,where M is the number

of rows,N is the number of columns,and C is the number

of clusters in our codebook.We used M=20,N=20,

and K=40for a softmax layer of16,000neurons.Our

Figure2:Overview.Our network is similar to the standard7-layer architecture[12]used for many recognition tasks.We take a200x200image as input.However,we use a spatial softmax as the?nal output.For every pixel in the image we predict a distribution of various motions with various directions and magnitudes.We can combine a weighted average of these vectors to produce the?nal output for each pixel.For computational reasons,we predict a coarse20x20output. softmax loss is spatial,summing over all the individual re-

gion losses.Let I represent the image and Y be the ground

truth optical?ow labels represented as quantized clusters.

Then our spatial loss function L(I,Y)is following:

L(I,Y)=?M×N

i=1

K

r=1

(1(y i=r)log F i,r(I))(1)

F i,r(I)represents the probability that the i th pixel will move according to cluster r.1(y i=r)is an indicator func-tion.

Data Augmentation:For many deep networks,datasets which are insuf?ciently diverse or too small will lead di-rectly to over?tting.[20]and[9]show that training di-rectly on datasets such as the UCF-101for action classi?ca-tion leads to over?tting,as there are only data on the order of tens of thousands of videos.However,our problem of single-frame prediction is different from this task.We?nd that we are able to build a generalizable representation for prediction by training our model over350,000frames from the UCF-101dataset as well as over150,000frames from the HMDB-51dataset.We bene?t additionally from data augmentation techniques.We randomly?ip each image as well as use randomly cropped windows.For each input,we also change our labels according to our spatial transforma-tion.In this way we are able to avoid spatial biases(such as humans always appearing in the middle of the image)and train a general model on a far smaller set of videos than for recognition tasks.

Labeling:We automatically label our training dataset with an optical?ow algorithm.With a publically available implementation,we chose DeepFlow[27]to compute op-tical?ow.The UCF-101and the HMDB-51dataset use realistic,sometimes low-quality videos from a wide vari-ety of sources.They often suffer from compression arti-facts.Thus,we aim to make our labels somewhat less noisy by taking the average optical?ow of?ve future frames for each image.The videos in these datasets are also unstabi-lized.[25]showed that action recognition can be greatly improved with camera stabilization.In order to further de-noise our labels,we wish to focus on the motion of objects inside the image,not the camera motion.We thus use the stabilization portion of the implementation of[25]to auto-matically stabilize videos using an estimated homography.

4.Experiments

For our experiments,we focused on two datasets,the UCF-101and HMDB-51,which have been popular for action recognition.For both of these datasets,we com-pared against baselines using3-fold cross validation with the splits speci?ed by the dataset organizers.For training, we subsampled frames by a factor of5.For testing,we sampled26,000frames per split.For our comparison with AlexNet?netuning,we used a split which incorporated a larger portion of the training data.We will release this split publicly.We used three baselines for evaluation.First we used the technique of[18],a SRF approach to motion pre-diction.We took their publicly available implementation and trained a model according to their default parameters. Because of the much larger size of our datasets,we had to sample SIFT-patches less densely.We also use a Nearest-Neighbor baseline using both fc7features from the pre-trained AlexNet network as well as pooled-5features.Fi-nally,we compare unsupervised training from scratch with ?netuning on the supervised AlexNet network.

(a)Input Image(b)Prediction(c)Ground Truth(a)Input Image(b)Prediction(c)Ground Truth Figure4:Qualitative results from our method for the single frame model.Our network can?nd the active

elements in the scene and correctly predict future motion based on the context in a wide variety and scenes

and actions.The color coding is on the

right.

4.1.Evaluation Metrics

Because of the complexity and sometimes high level of

label ambiguity in motion prediction,we use a variety of

metrics to evaluate our method and baselines.Following

from[18],we use traditional End-Point-Error,measuring

the Euclidean distance of the estimated optical?ow vector

from the ground truth vector.In addition,given vectors x1

and x2,we also measure direction similarity using the co-

sine similarity distance:x T1x2

x1 x2

and orientation similarity

(angle taken on half-circle):|x T1x2|

x1 x2

The orientation similarity measures how parallel is pre-

dicted optical?ow vector with respect to given ground truth

optical?ow vector.Some motions may be strictly left-right

or up-down,but the exact direction may be ambiguous.This

measure accounts for this situation.

We choose these metrics established by earlier work.

However,we also add some additional metrics to account

for the level of ambiguity in many of the test images.As

[18]notes,EPE is a poor metric in the case where motion

is small and may reasonably proceed in more than one pos-

sible direction.We thus additionally look at the underlying

distribution of the predicted classes to understand how well

the algorithm accounts for this ambiguity.For instance,if

we are shown an image as in Figure3,it is unknown if the

man will move up or down.It is certainly the case,however,

that he will not move right or left.Given the probability dis-

tribution over the quantized?ow clusters,we check to see

if the ground truth is within the top probable clusters.For

the implementation of[18],we create an estimated proba-

bility distribution by quantizing the regression output from

all the trees and then,for each pixel,we bin count the clus-

ImageNet-Pretrained vs.Trained from Scratch

Method EPE EPE-Canny EPE-NZ

Pretrained 1.19 1.12 3.12

From Scratch 1.28 1.21 3.21

—Orient Orient-Canny Orient-NZ

Pretrained.661.659.692

From Scratch.659.658.691

—Top-5Top-5-Canny Top-5-NZ

Pretrained91.0%91.1%65.8%

From Scratch89.9%90.3%65.1%

Table1:Here we compare our unsupervised network,trained from scratch,to the same network?ne-tuned from supervised AlexNet features on UCF-101.The Canny suf?x represents pixels on the Canny edges,and the NZ suf?x represents moving pixels according to the ground-truth.NN represents a nearest-neighbor approach.Dir and Orient represent direction and orientation met-rics respectively.For EPE,less is better,and for other metrics, higher is better.

ters over the trees.For Nearest-Neighbor we take the top-N matched frames and use the matched clusters in each pixel as our top-N ranking.We evaluate over the mean rank of all pixels in the image.

Following[18],we also evaluate over the Canny edges. Because of the simplicity of the datasets in[18],Canny edges were a good approximation for measuring the error of pixels of moving objects in the scene.However,our data includes highly cluttered scenes that incorporate multiple non-moving objects.In addition,we?nd that our network is very effective at identifying moving vs non-moving el-ements in the scene.We?nd that the difference between overall pixel mean and Canny edges is very small across all metrics and baselines.Thus,we also evaluate over the moving pixels according to the ground-truth.Moving pixels in this case includes all clusters in our codebook except for the vector of smallest magnitude.While unfortunately this metric depends on the choice of codebook,we?nd that the greatest variation in performance and ambiguity lies in pre-dicting the direction and magnitude of the active elements in the scene.

4.2.Qualitative Results

Figure4shows some of our qualitative results.For sin-gle frame prediction,our network is able to predict motion in many different contexts.Although most of our datasets consist of human actions,our model can generalize beyond simply detecting general motion on humans.Our method is able to successfully predict the falling of the ocean wave in the second row,and it predicts the motion of the entire horse in the?rst row.Furthermore,our network can specify mo-tion depending on the action being performed.For the man playing guitar and the man writing on the wall,the arm is

UCF-101

Method EPE EPE-Canny EPE-NZ

SRF[18] 1.30 1.23 3.24

NN pooled-5 2.31 2.20 4.40

NN fc7 2.24 2.16 4.27

Ours 1.27 1.17 3.19

—Dir Dir-Canny Dir-NZ

SRF[18].004.000-.013

NN pooled-5-.001-.001-.067

NN fc7-.005-.006-.060

Ours.045.025.092

—Orient Orient-Canny Orient-NZ

SRF[18].492.600.515

NN pooled-5.650.650.677

NN fc7.649.649.651

Ours.659.657.688

—Top-5Top-5-Canny Top-5-NZ

SRF[18]79.4%81.7%10.0%

NN pooled-577.8%79.5%20.0%

NN fc778.3%79.9%18.8%

Ours89.7%90.5%65.0%

—Top-10Top-10-Canny Top-10-NZ

SRF[18]82.2%84.4%17.2%

NN pooled-583.2%85.3%32.9%

NN fc784.0%85.4%32.3%

Ours96.5%96.7%90.9%

Table2:Single-image evaluation using the3-fold split on UCF-101.The Canny suf?x represents pixels on the Canny edges,and the NZ suf?x represents moving pixels according to the ground-truth.NN represents a nearest-neighbor approach.Dir and Orient represent direction and orientation metrics respectively.For EPE, less is better,and for other metrics,higher is better.

the most salient part to be moved.For the man walking the dog and the man doing a pushup,the entire body will move according to the action.

4.3.Quantitative Results

We show in tables2and3that our method strongly out-performs both the Nearest-Neighbor and SRF-based base-lines by a large margin by most metrics.This holds true for both datasets.Interestingly,the SRF-based approach seems to come close to ours based on End-Point-Error,but is heav-ily outperformed on all other metrics.This is largely a prod-uct of the End-Point-Error metric,as we?nd that the SRF tends to output the mean(optical?ow with very small mag-nitude).This is consistent with the results found in[18], where actions with low,bidirectional motion can result in higher EPE than predicting no motion at all.When we ac-count for this ambiguity in motion in the top-N metric,how-ever,the difference in performance is https://www.wendangku.net/doc/035865202.html,pared to unsupervised training from scratch,we?nd that?netuning

HMDB-51

Method EPE EPE-Canny EPE-NZ

SRF[18] 1.23 1.20 3.46

NN pooled-5 2.51 2.49 4.89

NN fc7 2.43 2.43 4.69

Ours 1.21 1.17 3.45

—Dir Dir-Canny Dir-NZ

SRF[18].000.000-.010

NN pooled-5-.008-.007-.061

NN fc7-.007-.005-.061

Ours.019.012.030

—Orient Orient-Canny Orient-NZ

SRF[18].461.557.495

NN pooled-5.631.631.644

NN fc7.630.631.655

Ours.636.636.667

—Top-5Top-5-Canny Top-5-NZ

SRF[18]81.9%83.6%13.5%

NN pooled-576.3%77.8%14.0%

NN fc777.3%78.7%13.5%

Ours90.2%90.5%61.0%

—Top-10Top-10-Canny Top-10-NZ

SRF[18]84.4%86.1%22.1%

NN pooled-582.9%84.0%23.9%

NN fc783.6%84.4%23.2%

Ours95.9%95.9%87.5%

Table3:Single-image evaluation using the3-fold split on HMDB-51.The Canny suf?x represents pixels on the Canny edges,and the NZ suf?x represents moving pixels according to the ground-truth.NN represents a nearest-neighbor approach.Dir and Orient represent direction and orientation metrics respectively. For EPE,less is better,and for other metrics,higher is better. from supervised,pretrained features yield only a very small improvement in performance.Looking over all pixels,the difference in performance between approaches is small.On absolute levels,the orientation and top-N metrics also tend to be high.This is due to the fact that most pixels in the im-age are not going to move.Outputting low or zero-motion over the entire image can thus lead to good performance for many metrics.Canny edge pixels yield similar results, as our natural images often include background clutter with objects that do not move.The most dramatic differences appear over the non-zero pixels.The direction metric is for our method is low at.09because of direction ambiguity, but orientation similarity is much larger.The largest perfor-mance gains come at the top-N rankings.For40clusters, random chance for top-5is12.5%,and for top-10it is25%. Nearest-neighbor does slightly better than chance,but SRF actually performs slightly worse.This is most likely due to the SRF tendency to output the overall mean?ow,which is of low magnitude.Our method performs much better,with the ground truth direction and magnitude vector coming in 65%of the time in the top-5ranking,and to a very high 90.9%of the time in the top ten.In spite of exposure to more data and use of semantic labels,we see in Table1 that?netuning using the ImageNet network only leads to a small increase in performance compared to training from scratch.

5.Multi-Frame Prediction

Until now we have described an architecture for predict-ing optical?ow given a static image as input.However,it would be interesting to predict not just the next frame but a few seconds into future.How should we design such a network?

We present a proof-of-concept network to predict6fu-ture frames.In order to predict multiple frames into the future,we take our pre-trained single frame network and output the seventh feature layer into a”temporally deep”network,using the implementation of[2].This network architecture is the same as an unrolled recurrent neural net-work with some important differences.On a high level,our network is similar to the unfactored architecture in[2],with each sequence having access to the image features and the previous hidden state in order to predict the next state.We replace the LSTM module with a fully connected layer as in a RNN.However,we also do not use a true recurrent net-work.The weights for each sequence layer are not shared, and each sequence has access to all the past hidden states. We used2000hidden states in our network,but we predict at most six future sequences.We attempted to use recurrent architectures with the publicly available LSTM implemen-tation from[2].However,in our experiments they always regressed to a mean trajectory across the data.Our fully connected network has much higher number of parameters than a RNN and therefore highlights the inherent dif?culty of this task.Due to the much larger size of the state space, we do not predict optical?ow for each and every pixel.In-stead,we use kmeans to created a codebook of1000pos-sible optical?ow frames,and we predict one of1000class as output as each time step.This can be thought of as anal-ogous to a sequential prediction problem similar to caption generation.Instead of a sequence of words,our”words”are clusters of optical?ow frames,and our”sentence”is an entire trajectory.We used a set number of sequences,six,in our experiments with each frame representing the average optical?ow of one-sixth of a second.

6.Conclusion

In this paper we have presented an approach to general-ized event prediction in static https://www.wendangku.net/doc/035865202.html,ly,our frame-work focuses on motion prediction as a non-semantic form of action prediction.By using an optical?ow algorithm

Figure 5:Overview .For our multiframe prediction,we predict entire clustered frames of optical ?ow as a sequence of frames.We take the learned features for our single frame model as our input,and we input them to a series of six fully connected layers,with each layer having access to the states of the past

layers.

(a)Input Image (b)Frame 1(c)Frame 2(c)Frame 3(d)Frame 4(e)Frame 5

Figure 6:Qualitative results for multi-frame prediction.The four rows represent predictions from our multi-frame model for future frames.Our extension can predict optical ?ow over multiple

frames.to label the data,we can train this model on a large num-ber of unsupervised videos.Furthermore,our framework utilizes the success of deep networks to outperform con-temporary approaches to motion prediction.We ?nd that our network successfully predicts motion based on the con-text of the scene and the stage of the action taking place.Possible work includes incorporating this motion model to predict semantic action labels in images and video.Another possible direction is to utilize the predicted optical ?ow to

predict in raw pixel space,synthesizing a video from a sin-gle image.Acknowedgements:We thank Xiaolong Wang for many helpful discussions.We thank the NVIDIA Cor-poration for the donation of Tesla K40GPUs for this re-search.In addition,this work was supported by NSF grant IIS1227495.

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scifinder使用介绍

6.6.1 内容简介 SciFinder Scholar是美国化学学会所属的化学文摘服务社CAS（Chemical Abstract Service）出版的化学资料电子数据库学术版。它是全世界最大、最全面的化学和科学信息数据库。 《化学文摘》(CA)是涉及学科领域最广、收集文献类型最全、提供检索途径最多、部卷也最为庞大的一部著名的世界性检索工具。CA报道了世界上150多个国家、56种文字出版的20000多种科技期刊、科技报告、会议论文、学位论文、资料汇编、技术报告、新书及视听资料，摘录了世界范围约98%的化学化工文献，所报道的内容几乎涉及化学家感兴趣的所有领域。 CA网络版SciFinder Scholar，整合了Medline医学数据库、欧洲和美国等30几家专利机构的全文专利资料、以及化学文摘1907年至今的所有内容。涵盖的学科包括应用化学、化学工程、普通化学、物理、生物学、生命科学、医学、聚合体学、材料学、地质学、食品科学和农学等诸多领域。 SciFinder Scholar 收集由CAS 出版的数据库的内容以及MEDLINE?数据库，所有的记录都为英文（但如果MEDLINE 没有英文标题的则以出版的文字显示）。 6.6.2 通过 SciFinder Scholar 可以得到的信息：

6.6.3 SciFinder? Scholar? 使用的简单介绍 主要分为Explore 和Browse。如图6.6.1 一、Explore Explore Tool 可获取化学相关的所有信息及结构等，有如下： 1、Chemical Substance or Reaction – Retrieve the corresponding literature 2、By chemical structure 3、By substance identifier 4、By molecular formula

有限元网格划分心得

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_基于ANSYS的有限元法网格划分浅析

文章编号:1003-0794(2005)01-0038-02 基于ANSYS的有限元法网格划分浅析 杨小兰,刘极峰,陈 旋 (南京工程学院,南京210013) 摘要:为提高有限元数值的计算精度和对复杂结构力学分析的准确性,针对不同分析类型采用了不同的网格划分方法,结合实例阐述了ANSYS有限元网格划分的方法和技巧,指出了采用ANSYS有限元软件在网格划分时应注意的技术问题。 关键词:ANSYS;有限元;网格;计算精度 中图号:O241 82;TP391 7文献标识码:A 1 引言 ANSYS有限元分析程序是著名的C AE供应商美国ANSYS公司的产品,主要用于结构、热、流体和电磁四大物理场独立或耦合分析的CAE应用,功能强大,应用广泛,是一个便于学习和使用的优秀有限元分析程序。在ANSYS得到广泛应用的同时,许多技术人员对ANSYS程序的了解和认识还不够系统全面,在工作和研究中存在许多隐患和障碍,尤为突出的是有限元网格划分技术。本文结合工程实例,就如何合理地进行网格划分作一浅析。 2 网格划分对有限元法求解的影响 有限元法的基本思想是把复杂的形体拆分为若干个形状简单的单元,利用单元节点变量对单元内部变量进行插值来实现对总体结构的分析,将连续体进行离散化即称网格划分,离散而成的有限元集合将替代原来的弹性连续体,所有的计算分析都将在这个模型上进行。因此,网格划分将关系到有限元分析的规模、速度和精度以及计算的成败。实验表明:随着网格数量的增加,计算精确度逐渐提高,计算时间增加不多;但当网格数量增加到一定程度后,再继续增加网格数量,计算精确度提高甚微,而计算时间却大大增加。在进行网格划分时,应注意网格划分的有效性和合理性。 3 网格划分的有效性和合理性 (1)根据分析数据的类型选择合理的网格划分数量 在决定网格数量时应考虑分析数据的类型。在静力分析时,如果仅仅是计算结构的变形,网格数量可以少一些。如果需要计算应力,则在精度要求相同的情况下取相对较多的网格。同样在响应计算中,计算应力响应所取的网格数应比计算位移响应多。在计算结构固有动力特性时,若仅仅是计算少数低阶模态,可以选择较少的网格。如果计算的模态阶次较高,则应选择较多的网格。在热分析中,结构内部的温度梯度不大,不需要大量的内部单元,可划分较少的网格。 (2)根据分析数据的分布特点选择合理的网格疏密度 在决定网格疏密度时应考虑计算数据的分布特点,在计算固有特性时,因为固有频率和振型主要取决于结构质量分布和刚度分布,采用均匀网格可使结构刚度矩阵和质量矩阵的元素不致相差很大,可减小数值计算误差。同样,在结构温度场计算中也趋于采用均匀的网格形式。在计算数据变化梯度较大的部位时,为了更好地反映数据变化规律,需要采用比较密集的网格,而在计算数据变化梯度较小的部位,为了减小模型规模,则应划分相对稀疏的网格,这样整个结构就表现出疏密不同的网格划分形式。 以齿轮轮齿的有限元分析模型为例,由于分析的目的是求出齿轮啮合传动过程中齿根部分的弯曲应力,因此,分析计算时并不需要对整个齿轮进行计算,可根据圣文男原理将整个区域缩小到直接参与啮合的轮齿。虽然实际上参与啮合的齿数总大于1,但考虑到真正起作用的是单齿,通常只取一个轮齿作为分析对象,这样作可以大大节省计算机内存。考虑到轮齿应力在齿根过渡圆角和靠近齿面处变化较大,网格可划分得密一些。在进行疏密不同网格划分操作时可采用ANSYS提供的网格细化工具调整网格的疏密,也可采用分块建模法设置网格疏密度。 图1所示即为采用分块建模法进行网格划分。图1(a)为内燃机中重要运动零件连杆的有限元应力分析图,由于连杆结构对称于其摆动的中间平面,其厚度方向的尺寸远小于长度方向的尺寸,且载荷沿厚度方向近似均匀分布,故可按平面应力分析处 38 煤 矿 机 械 2005年第1期

SciFinder使用说明

SciFinder使用说明 SciFinder简介 SciFinder?由美国化学会（American Chemical Society, ACS）旗下的美国化学文摘社（Chemical Abstracts Service, CAS）出品，是一个研发应用平台，提供全球最大、最权威的化学及相关学科文献、物质和反应信息。SciFinder涵盖了化学及相关领域如化学、生物、医药、工程、农学、物理等多学科、跨学科的科技信息。SciFinder收录的文献类型包括期刊、专利、会议论文、学位论文、图书、技术报告、评论和网络资源等。 通过SciFinder，可以： ?访问由CAS全球科学家构建的全球最大并每日更新的化学物质、反应、专利和期刊数据库，帮助您做出更加明智的决策。 ?获取一系列检索和筛选选项，便于检索、筛选、分析和规划，迅速获得您研究所需的最佳结果，从而节省宝贵的研究时间。 无需担心遗漏关键研究信息，SciFinder收录所有已公开披露、高质量且来自可靠信息源的信息。 通过SciFinder可以获得、检索以下数据库信息：CAplus SM（文献数据库）、CAS REGISTRY SM (物质信息数据库)、CASREACT? （化学反应数据库）、MARPAT?（马库什结构专利信息数据库）、CHEMLIST? (管控化学品信息数据库)、CHEMCATS?(化学品商业信息数据库)、MEDLINE?(美国国家医学图书馆数据库)。 专利工作流程解决方案PatentPak TM已在SciFinder上线，帮助用户在专利全文中快速定位难以查找的化学信息。 SciFinder 注册须知： 读者在使用SciFinder之前必须用学校的email邮箱地址注册，注册后系统将自动发送一个链接到您所填写的email邮箱中，激活此链接即可完成注册。参考“SciFinder注册说明”。

CATIA有限元高级划分网格教程

CATIA有限元高级网格划分教程 盛选禹李明志 1.1进入高级网格划分工作台 （1）打开例题中的文件Sample01.CATPart。 （2）点击主菜单中的【开始】→【分析与模拟】→【Advanced Meshing Tools】(高级网格划分工具），就进入【Advanced Meshing Tools】（高级网格划分工具）工作台，如图1－1所示。进入工作台后，生成一个新的分析文件，并且显示一个【New Analysis Case】（新分析算题）对话框，如图1－2所示。 图1－1【开始】→【分析与模拟】→【Advanced Meshing Tools】(高级网格划分工具）（3）在【New Analysis Case】（新分析算题）对话框内选择【Static Analysis】（静力分析）选项。如果以后打开该对话框的时候均希望是计算静力分析，可以把对话框内的【Keep as default starting analysis case】（在开始时保持为默认选项）勾选。这样，下次进入本工作台时，将自动选择静力分析。 （4）点击【新分析算题】对话框内的【确定】按钮，关闭对话框。 1.2定义曲面网格划分参数 本节说明如何定义一个曲面零件的网格类型和全局参数。 （1）点击【Meshing Method】(网格划分方法)工具栏内的【高级曲面划分】按钮

，如图1－3所示。需要在【Meshing Method】(网格划分方法)工具栏内点击中间按钮的下拉箭头才能够显示出【高级曲 面划分】按钮。 图1－2【New Analysis Case】（新分析算题）对话框图1－3【高级曲面划分】按钮

有限元网格划分

有限元网格划分 摘要:总结近十年有限元网格划分技术发展状况。首先,研究和分析有限元网格划分的基本原则；其次,对当前典型网格划分方法进行科学地分类,结合实例,系统地分析各种网格划分方法的机理、特点及其适用范围,如映射法、基于栅格法、节点连元法、拓扑分解法、几何分解法和扫描法等；再次,阐述当前网格划分的研究热点,综述六面体网格和曲面网格划分技术；最后,展望有限元网格划分的发展趋势。 关键词:有限元网格划分；映射法；节点连元法;拓扑分解法；几何分解法；扫描法；六面体网格 1 引言 有限元网格划分是进行有限元数值模拟分析至关重要的一步，它直接影响着后续数值计算分析结果的精确性。网格划分涉及单元的形状及其拓扑类型、单元类型、网格生成器的选择、网格的密度、单元的编号以及几何体素。在有限元数值求解中，单元的等效节点力、刚度矩阵、质量矩阵等均用数值积分生成，连续体单元以及壳、板、梁单元的面内均采用高斯（Gauss）积分，而壳、板、梁单元的厚度方向采用辛普生（Simpson）积分。 2 有限元网格划分的基本原则 有限元方法的基本思想是将结构离散化,即对连续体进行离散化,利用简化几何单元来近似逼近连续体,然后根据变形协调条件综合求解。所以有限元网格的划分一方面要考虑对各物体几何形状的准确描述,另一方面也要考虑变形梯度的准确描述。为正确、合理地建立有限元模型,这里介绍划分网格时应考虑的一些基本原则。 2.1 网格数量 网格数量直接影响计算精度和计算时耗,网格数量增加会提高计

算精度,但同时计算时耗也会增加。当网格数量较少时增加网格，计算精度可明显提高,但计算时耗不会有明显增加;当网格数量增加到一定程度后,再继续增加网格时精度提高就很小,而计算时耗却大幅度增加。所以在确定网格数量时应权衡这两个因素综合考虑。 2.2 网格密度 为了适应应力等计算数据的分布特点,在结构不同部位需要采用大小不同的网格。在孔的附近有集中应力,因此网格需要加密;周边应力梯度相对较小,网格划分较稀。由此反映了疏密不同的网格划分原则:在计算数据变化梯度较大的部位,为了较好地反映数据变化规律,需要采用比较密集的网格;而在计算数据变化梯度较小的部位,为减小模型规模,网格则应相对稀疏。 2.3 单元阶次 单元阶次与有限元的计算精度有着密切的关联,单元一般具有线性、二次和三次等形式,其中二次和三次形式的单元称为高阶单元。高阶单元的曲线或曲面边界能够更好地逼近结构的曲线和曲面边界,且高次插值函数可更高精度地逼近复杂场函数,所以增加单元阶次可提高计算精度。但增加单元阶次的同时网格的节点数也会随之增加,在网格数量相同的情况下由高阶单元组成的模型规模相对较大,因此在使用时应权衡考虑计算精度和时耗。 2.4 单元形状 网格单元形状的好坏对计算精度有着很大的影响,单元形状太差的网格甚至会中止计算。单元形状评价一般有以下几个指标: (1)单元的边长比、面积比或体积比以正三角形、正四面体、正六面体为参考基准。 (2)扭曲度:单元面内的扭转和面外的翘曲程度。 (3)节点编号:节点编号对于求解过程中总刚矩阵的带宽和波前因数有较大的影响,从而影响计算时耗和存储容量的大小 2.5 单元协调性 单元协调是指单元上的力和力矩能够通过节点传递给相邻单元。为保证单元协调,必须满足的条件是: (1)一个单元的节点必须同时也是相邻点,而不应是内点或边界

浙江大学scifinder使用教程

浙江大学scifinder使用教程 1、输入网址：https://www.wendangku.net/doc/035865202.html,/ 如图1，点击继续浏览 图1 2、进入浙大的入口，输入用户名密码（卖家提供） 图2 3、登陆进去是图3这个页面。注意此时会自动安装插件，切记要一路放行。登陆成功的标志是屏幕右上角有个蓝框绿蓝S的LOGO！ 如果未出现S，那么请根据图2的手动安装组件，下载安装组件！

图3 4、登陆页面不要覆盖，新标签页打开浙江大学图书馆 https://www.wendangku.net/doc/035865202.html,/libweb/点数据库导航，找到scifinder页面，进入 图4

5、点击图5红框中的链接https://www.wendangku.net/doc/035865202.html,/cgi-bin/casip，看下IP是不是浙大的IP，一般是61或者210开头，确定是，那就可以输入链接https://https://www.wendangku.net/doc/035865202.html,/登陆了 图5 6、scifinder登陆页面输入用户名密码（卖家提供），您就可以使用scifinder啦 图6

常见问题以及解决方法 1.浏览器不支持JavaScript，提示“您的浏览器不支持JavaScript（或它被 禁止了）请确认您的浏览器能支持JavaScript”，请启用“工具- >Internet选项->安全->自定义级别->活动脚本”选项。 2.浏览器不支持Cookie，提示“你的浏览器禁止了Cookie，必须设置为允许 才可以继续使用”，请在“工具->Internet选项->隐私->高级”启用 Cookie支持 3.浏览不支持BHO，提示:"您的浏览器没有启用第三方扩展"，关闭IE时无法 自动注销用户。请在 "工具->Internet选项->高级->启用第三方浏览器扩展”前打勾启用! 4.APP服务不可用,可能是控件不是最新的，请您关闭IE，重新登陆VPN 5.IP服务不可用,可能你安装的IP服务控件不是最新的。请点击“程序- >SINFOR SSL VPN->卸载CS应用支持”和“程序->SINFOR SSL VPN->卸载SSL VNIC”，手动卸载IP服务控件，然后再重新登陆VPN。 6.IP服务可能与某些杀毒软件冲突。请在杀毒软件中放行IP服务的客户端程 序，或者在使用时暂时禁用杀毒软件。

有限元网格划分和收敛性

一、基本有限元网格概念 1．单元概述?几何体划分网格之前需要确定单元类型.单元类型的选择应该根据分析类型、形状特征、计算数据特点、精度要求和计算的硬件条件等因素综合考虑。为适应特殊的分析对象和边界条件,一些问题需要采用多种单元进行组合建模。? 2.单元分类选择单元首先需要明确单元的类型，在结构有限元分析中主要有以下一些单元类型：平面应力单元、平面应变单元、轴对称实体单元、空间实体单元、板单元、壳单元、轴对称壳单元、杆单元、梁单元、弹簧单元、间隙单元、质量单元、摩擦单元、刚体单元和约束单元等。根据不同的分类方法,上述单元可以分成以下不同的形式。?3。按照维度进行单元分类 根据单元的维数特征，单元可以分为一维单元、二维单元和三维单元。?一维单元的网格为一条直线或者曲线。直线表示由两个节点确定的线性单元。曲线代表由两个以上的节点确定的高次单元,或者由具有确定形状的线性单元。杆单元、梁单元和轴对称壳单元属于一维单元,如图1~图3所示。 ?二维单元的网 格是一个平面或者曲面，它没有厚度方向的尺寸.这类单元包括平面单元、轴对称实体单元、板单元、壳单元和复合材料壳单元等,如图4所示。二维单元的形状通常具有三角形和四边形两种，在使用自动网格剖分时,这类单元要求的几何形状是表面模型或者实体模型的边界面。采用薄壳单元通常具有相当好的计算效率。

??三维单元的网格具有空间三个方向的尺寸，其形状具有四面体、五面体和六面体,这类单元包括空间实体单元和厚壳单元，如图5所示．在自动网格划分时，它要求的是几何模型是实体模型(厚壳单元是曲面也可以）。 ? 4.按照插值函数进行单元分类 根据单元插值函数多项式的最高阶数多少，单元可以分为线性单元、二次单元、三次单元和更高次的单元。 线性单元具有线性形式的插值函数,其网格通常只具有角节点而无边节点，网格边界为直线或者平面.这类单元的优点是节点数量少，在精度要求不高或者结果数据梯度不太大的情况下，采用线性单元可以得到较小的模型规模.但是由于单元位移函数是线性的，单元内的位移呈线性变化,而应力是常数,因此会造成单元间的应力不连续,单元边界上存在着应力突变，如图6所示。

比较PageRank算法和HITS算法的优缺点

题目：请比较PageRank算法和HITS算法的优缺点，除此之外，请再介绍2种用于搜索引擎检索结果的排序算法，并举例说明。 答： 1998年，Sergey Brin和Lawrence Page[1]提出了PageRank算法。该算法基于“从许多优质的网页链接过来的网页，必定还是优质网页”的回归关系，来判定网页的重要性。该算法认为从网页A导向网页B的链接可以看作是页面A对页面B的支持投票，根据这个投票数来判断页面的重要性。当然，不仅仅只看投票数，还要对投票的页面进行重要性分析，越是重要的页面所投票的评价也就越高。根据这样的分析，得到了高评价的重要页面会被给予较高的PageRank值，在检索结果内的名次也会提高。PageRank是基于对“使用复杂的算法而得到的链接构造”的分析，从而得出的各网页本身的特性。 HITS 算法是由康奈尔大学( Cornell University ) 的JonKleinberg 博士于1998 年首先提出。Kleinberg认为既然搜索是开始于用户的检索提问，那么每个页面的重要性也就依赖于用户的检索提问。他将用户检索提问分为如下三种：特指主题检索提问（specific queries，也称窄主题检索提问）、泛指主题检索提问（Broad-topic queries，也称宽主题检索提问）和相似网页检索提问（Similar-page queries）。HITS 算法专注于改善泛指主题检索的结果。 Kleinberg将网页（或网站）分为两类，即hubs和authorities，而且每个页面也有两个级别，即hubs（中心级别）和authorities（权威级别）。Authorities 是具有较高价值的网页，依赖于指向它的页面；hubs为指向较多authorities的网页，依赖于它指向的页面。HITS算法的目标就是通过迭代计算得到针对某个检索提问的排名最高的authority的网页。 通常HITS算法是作用在一定范围的，例如一个以程序开发为主题的网页，指向另一个以程序开发为主题的网页，则另一个网页的重要性就可能比较高，但是指向另一个购物类的网页则不一定。在限定范围之后根据网页的出度和入度建立一个矩阵，通过矩阵的迭代运算和定义收敛的阈值不断对两个向量authority 和hub值进行更新直至收敛。 从上面的分析可见，PageRank算法和HITS算法都是基于链接分析的搜索引擎排序算法，并且在算法中两者都利用了特征向量作为理论基础和收敛性依据。

ANSYS有限元网格划分的基本要点

ANSYS有限元网格划分的基本要点 1引言 ANSYS有限元网格划分是进行数值模拟分析至关重要的一步，它直接影响着后续数值计算分析结果的精确性。网格划分涉及单元的形状及其拓扑类型、单元类型、网格生成器的选择、网格的密度、单元的编号以及几何体素。从几何表达上讲，梁和杆是相同的，从物理和数值求解上讲则是有区别的。同理，平面应力和平面应变情况设计的单元求解方程也不相同。在有限元数值求解中，单元的等效节点力、刚度矩阵、质量矩阵等均用数值积分生成，连续体单元以及壳、板、梁单元的面内均采用高斯（Gauss）积分，而壳、板、梁单元的厚度方向采用辛普生（Simpson）积分。辛普生积分点的间隔是一定的，沿厚度分成奇数积分点。由于不同单元的刚度矩阵不同，采用数值积分的求解方式不同，因此实际应用中，一定要采用合理的单元来模拟求解。 2ANSYS网格划分的指导思想 ANSYS网格划分的指导思想是首先进行总体模型规划，包括物理模型的构造、单元类型的选择、网格密度的确定等多方面的内容。在网格划分和初步求解时，做到先简单后复杂，先粗后精，2D单元和3D单元合理搭配使用。为提高求解的效率要充分利用重复与对称等特征，由于工程结构一般具有重复对称或轴对称、镜象对称等特点，采用子结构或对称模型可以提高求解的效率和精度。利用轴对称或子结构时要注意场合，如在进行模态分析、屈曲分析整体求解时，则应采用整体模型，同时选择合理的起点并设置合理的坐标系，可以提高求解的精度和效率，例如，轴对称场合多采用柱坐标系。有限元分析的精度和效率与单元的密度和几何形状有着密切的关系，按照相应的误差准则和网格疏密程度，避免网格的畸形。在网格重划分过程中常采用曲率控制、单元尺寸与数量控制、穿透控制等控制准则。在选用单元时要注意剪力自锁、沙漏和网格扭曲、不可压缩材料的体积自锁等问题 ANSYS软件平台提供了网格映射划分和自由适应划分的策略。映射划分用于曲线、曲面、实体的网格划分方法，可使用三角形、四边形、四面体、五面体和六面体，通过指定单元边长、网格数量等参数对网格进行严格控制，映射划分只用于规则的几何图素，对于裁剪曲面或者空间自由曲面等复杂几何体则难以

ANSYS有限元分析中的网格划分

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三、SciFinder ? Scholar? 使用的简单介绍： 主要分为Explore 和Browse ： 3．1、Explore Explore Tool 可获取化学相关的所有信息及结构等，有如下方式：（如上图所示） ? Chemical Substance or Reaction – Retrieve the corresponding literature ? By chemical structure ? By substance identifier ? By molecular formula ? Research Topic – to find literature relevant to a topic of interest. ? Author Name – to locate literature written by a specific author. ? Document Identifier – to find literature for a specific CA Accession Number or Patent Number. ? Company Name / Organization – to locate literature for a specific company, university, governmental agency, or other organization. 3．2、Browse Journal Table of Contents 可直接浏览1800多核心期刊的摘要及其引文等编目内容，如果带有 则可直接点击，就会通过ChemPort ? Connection .SM 获取全文（in-house ） Explore Browse 菜单

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一、基本有限元网格概念 1.单元概述 几何体划分网格之前需要确定单元类型。单元类型的选择应该根据分析类型、形状特征、计算数据特点、精度要求和计算的硬件条件等因素综合考虑。为适应特殊的分析对象和边界条件，一些问题需要采用多种单元进行组合建模。 2.单元分类 选择单元首先需要明确单元的类型，在结构有限元分析中主要有以下一些单元类型：平面应力单元、平面应变单元、轴对称实体单元、空间实体单元、板单元、壳单元、轴对称壳单元、杆单元、梁单元、弹簧单元、间隙单元、质量单元、摩擦单元、刚体单元和约束单元等。根据不同的分类方法，上述单元可以分成以下不同的形式。 3.按照维度进行单元分类 根据单元的维数特征，单元可以分为一维单元、二维单元和三维单元。 一维单元的网格为一条直线或者曲线。直线表示由两个节点确定的线性单元。曲线代表由两个以上的节点确定的高次单元，或者由具有确定形状的线性单元。杆单元、梁单元和轴对称壳单元属于一维单元，如图1～图3所示。 二维单元的网格是一个平面或者曲面，它没有厚度方向的尺寸。这类单元包括平面单元、轴对称实体单元、板单元、壳单元和复合材料壳单元等，如图4所示。二维单元的形状通常具有三角形和四边形两种，在使用自动网格剖分时，这类单元要求的几何形状是表面模型或者实体模型的边界面。采用薄壳单元通常具有相当好的计算效率。

三维单元的网格具有空间三个方向的尺寸，其形状具有四面体、五面体和六面体，这类单元包括空间实体单元和厚壳单元，如图5所示。在自动网格划分时，它要求的是几何模型是实体模型(厚壳单元是曲面也可以)。 4.按照插值函数进行单元分类 根据单元插值函数多项式的最高阶数多少，单元可以分为线性单元、二次单元、三次单元和更高次的单元。 线性单元具有线性形式的插值函数，其网格通常只具有角节点而无边节点，网格边界为直线或者平面。这类单元的优点是节点数量少，在精度要求不高或者结果数据梯度不太大的情况下，采用线性单元可以得到较小的模型规模。但是由于单元位移函数是线性的，单元内的位移呈线性变化，而应力是常数，因此会造成单元间的应力不连续，单元边界上存在着应力突变，如图6所示。