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A Remeshing Procedure for Numerical Simulation of Forming Processes in Three Dimensions

A Remeshing Procedure for Numerical Simulation of Forming Processes in Three Dimensions
A Remeshing Procedure for Numerical Simulation of Forming Processes in Three Dimensions

A Remeshing Procedure for Numerical Simulation

of Forming Processes in Three Dimensions

L. Giraud-Moreau, H. Borouchaki and A. Cherouat

Institut Charles Delaunay (FRE CNRS 2848)

University of Technology of Troyes

12 rue Marie Curie BP2060

10010 Troyes Cedex - France

laurence.moreau@utt.fr

Abstract.This article presents a remeshing procedure of thin sheets for numerical simulation of metal forming process in three dimensions. D uring simulation of metal forming processes, where large plastic deformations are possible, severe mesh distortion occur after a few incremental steps. Hence an automatic mesh generation with remeshing capabilities is essential to carry out the finite element analysis. This paper gives the necessary steps to remesh a structure in finite ele-ment simulation of forming processes. The proposed remeshing technique based on geometrical criteria includes adaptive refinement and coarsening procedures. It has been implemented with triangular and quadrilateral elements. The proposed method has been integrated in a computational environment using the ABAQUS solver. Numerical examples show the efficiency of the proposed approach. Keywords: Adaptive remeshing, forming process, geometrical error estimator, thin sheet.

1 Introduction

The finite element method has been very successful in the numerical simulation of metal forming processes like deep-drawing, hydro-forming or forging [1-2]. However, due to the imposition of large

128L. Giraud-Moreau et al.

plastic strains and friction, the finite element mesh representing the workpiece undergoes severe distortion and hence it is necessary to generate a new mesh for the deformed domain [3]. It is therefore convenient to update the mesh in such a way that it conforms to the new deformed geometry and becomes dense enough in the critical region while remaining reasonably coarse in the rest of the domain. The remeshing procedure must be automatic and robust. Several remeshing methods have been proposed during the last years. Glob-ally, three different types of adaptive remeshing strategies can be distinguished: r-adaptivity, p-adaptivity and h-adaptivity [4-5]. Strategies based on r-adaptivity consist of keeping the number of special grid points fixed, but allowing them to move into regions where a finer spatial discretization is needed. This type of adaptation is particulary powerful on problems where a large domain is needed to capture a time varying solution which has steep slopes over only a small fraction of that domain [6]. The remeshing techniques pre-sented by Zienkiewicz et al [7], Fourment et al [8], Coupez [9], Borouchaki et al [10] are based on the computation of a size map to govern a global remeshing of the part at each iteration. Strategies based on p-adaptivity consist of changing the degree of the interpo-lating polynomials in appropriate regions of the mesh. This method is preferred for (linear) smooth solutions or over subregions where the solution is smooth [11]. Strategies based on h-adaptivity consist of adapting the number of grid points and changing the mesh con-nectivity. Grid points are added to areas where more accuracy is demanded (the interpolation will be enriched) and can be deleted where the solution is accurate enough. As part of these methods, remeshing techniques based on the computation of a size map to govern a global remeshing of the part at each iteration have been proposed [7-10]. Cho and Yang [12] have proposed a refinement al-gorithm based on h-adaptivity which consists in splitting each de-formed element in two elements along an edge. This procedure drags to the creation of small edges and consequently degenerates

A Remeshing Procedure for Numerical Simulation 129 elements during repetitive refinement iterations. Moreover, all simi-lar refinement methods only based on the break of edges lead to the formation of small edges or poor shaped elements.

This paper presents a new remeshing technique for the numerical simulation of thin sheet metal forming processes. This method is based on a geometrical criterion. It is applied to computational do-main after each small displacement step of forming tools. It allows, in particular to refine the current mesh of the part under the numeri-cal simulation of the forming process in the curved area with pre-serving shape quality element and to coarsen this mesh in the flat area.

The mesh refinement is necessary to avoid large element distor-tions during the deformation. It ensures the convergence of the com-putation and allows an adequate representation of the geometry of the deformed domain. The mechanical fields are simply induced from the old mesh into the new mesh.

The proposed remeshing method looks like to the remeshing method presented by Meinders [13] in the case of a triangular mesh. Compared to Meinders method, the proposed remeshing technique generates a smaller number of elements, it has been implemented with triangular and quadrilateral elements and a coarsening tech-nique is considered, in addition to the refinement technique.

This paper gives the different steps of the proposed remeshing method. Some application examples are presented in order to show the pertinence of our approach.

2 General Remeshing Scheme

The simulation of the forming process is based on an iterative proc-ess. At first, a coarse initial mesh of the part is generated with trian-gular or quadrilateral elements. At each iteration, a finite element computation is then realized in order to simulate numerically the forming process for a small displacement step of forming tools. This displacement step must be sufficiently small with respect to the specified minimal size of mesh elements.

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Then, remeshing is applied after each deformation increment, if nec-essary, according to the following scheme:

x coarsening procedure applied to elements which are in flat area, x iterative refinement to restore mesh conformity,

x refinement procedure applied to elements which are in curved area (the refinement is applied in the vicinity of nodes for which the shape of the surface is changed and only if the minimal element size is not reached),

x iterative refinement to restore mesh conformity.

This process (simulation of the forming process for a small dis-placement step of forming tools, remeshing of the part) is repeated until the final tool displacement is reached.

The computation convergence is principally based on the mesh refinement and coarsening procedures. The applied refinement must in particularly not introduce a mesh distortion, which could increase during iterations and stop the forming process simulation.

During the remeshing procedure, a geometrical criterion is used to refine the current mesh of the part in the curved area, and to coarsen this mesh in the flat area. For a given element, this geometrical crite-rion represents the maximal angular gap between the normal to the element and the normals at its vertices. An element is thus consid-ered to be “curved” (resp. “flat”) if the corresponding angular gap is greater (resp. smaller) than a given threshold (for example 8 de-grees). The geometrical refinement and coarsening methods based on the same geometrical criterion are thus consistent. The normal vector Q G at node P can be defined as the weighted av-erage of the unit normal vectors i N G

(i =1,..m) to elements sharing node P:

2.1 Geometrical Criterion

131

|| m i i

i m

i i i N N 00G G G D D Q (1)

where i D is the angle at P of the ith element sharing P.

The computation of normal vector to the element depends on the element shape (triangle or quadrilateral). The normal vector N G to a triangle P 1P 2P 3

is the unit normal vector to its supporting plane :

N G (2)

The normal vector to a quadrilateral element is the average of the normal vectors to the four triangles defined by joining its barycentre to its edges.

The geometrical criterion applied to an element can be written as:

g i Z t max (3)

Where i Q G is the normal at vertex i of the element, N G is its normal

and g Z is an angular gap threshold. In this case, the element must be refined.

The adaptive remeshing technique consists in improving the mesh in order to conform to the geometry of the current part surface during deformation. In the following, the mesh refinement and coarsening methods are detailed.

A Remeshing Procedure for Numerical Simulation 2.2 Mesh Refinement and Coarsening Methods

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The refinement technique consists in subdividing mesh elements. An element is refined if it is a “curved” element (geometrical crite-rion). There is only one element subdivision which allows to pre-serve the element shape quality: the uniform subdivision into four new elements. In the case of a triangle, three new nodes are added : one in the middle of each edge. In the case of a quadrilateral, five nodes are added : one in the middle of each edge and one in the ele-ment barycentre. Figure 1 shows the triangular and quadrilateral element refinements.

After each refinement procedure, an iterative refinement to restore mesh conformity is necessary. Indeed, after applying the subdivision according to the geometrical criterion, adjacent elements to subdi-vided elements must be modified. As the edges of the subdivided elements are divided in two, there is a node in the middle of the edges common to the subdivided element and its adjacent elements. The mesh is then not conforming. To retrieve the mesh conformity, adjacent elements to subdivided elements must be also subdivided. This last subdivision can not be a homothetic subdivision in four elements because it would result in the systematic homothetic subdi-vision of all mesh elements.

There are three different configurations for adjacent elements which must be subdivided in order to ensure the mesh conformity: x no edge is saturated (i.e. containing a new added node),

x only one edge is saturated,

x

at least two edges are saturated.

Fig. 1. Triangular and quadrilateral element refinements

133

D epending on the configuration, a subdivision is applied if neces-sary. In the first case (no saturated edge), the element is not subdi-vided and is not modified. In the second case (one saturated edge), a triangular element is subdivided in two triangles and a quadrilateral element in three triangles (see figure 2). This subdivision allows to stop the propagation of the homothetic subdivision. In the third case, if all the edges are saturated the element is subdivided in four homo-thetic elements. Otherwise, in the case of triangular elements (hav-ing two saturated edges), all possible subdivisions lead to the forma-tion of poor shaped elements (stretched elements). It is then necessary to add a new node in order to subdivide also this element into four homothetic elements (see figure

3). In the quadrilateral A Remeshing Procedure for Numerical Simulation case, when only two edges are saturated and are adjacent, the quadri-lateral is subdivided in four triangles (see figure 4). This subdivision allows to stop the propagation of the homothetic subdivisions. In the other cases, the element is subdividing into four homothetic quadri-lateral elements (see figure 5) by adding nodes in the middle of no-saturated edges and in the barycentre of the element.

This refinement procedure is iteratively applied until no new node is added.

Fig. 2. Subdivision of elements with one saturated edge

Fig. 3. Subdivision of triangle with two saturated edges

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From an algorithmic point of view, the mesh is composed of two types of element: ordinary and extraordinary. An ordinary element is a triangle or a quadrilateral without saturated edges (see figure 6). An extraordinary triangle is a triangle with one and only one satu-rated edge. An extraordinary quadrilateral is a quadrilateral with only one saturated edge or two adjacent saturated edges. Figure 7 shows extraordinary triangle and quadrilaterals. The remeshing algo-rithm must take into account these two element types. During the re-finement operation, the geometrical criterion is applied to elements of both types. An ordinary or extraordinary element which is curved is then subdivided into four ordinary elements. After this operation, all ordinary elements with at least two saturated edges, except the case of two adjacent edges for quadrilateral elements, are iteratively subdivided into four ordinary elements. Then, all the elements with at least one saturated edge are transformed to extraordinary elements and the other elements remain unchanged.

At the end of the refinement operation, for the mechanical com-putational purpose, the extraordinary elements of the resulting mesh are transformed : an extraordinary triangle is divided in two trian-gles, an extraordinary quadrilateral with one middle node is divided in three triangles and an extraordinary quadrilateral with two middle

nodes is divided in four triangles.

Fig. 4. Subdivision of quadrilateral element with two adjacent saturated edges

Fig. 5. Other cases of subdivision of quadrilateral element

Fig. 6.

Ordinary elements

135

A Remeshing Procedure for Numerical Simulation Fig. 7. Extraordinary elements

The coarsening technique is the reciprocal operation of the refine-ment procedure. It can only be applied to a set of four ordinary ele-ments, called associated elements, obtained during a homothetic element refinement. Thanks to the coarsening technique, the initial element is restored when the area in which this element belongs, be-comes flat (see figure 8).

Fig. 8. Triangular and quadrilateral elements coarsening

A quad tree structure can be considered to coarsening the mesh of the part. This structure allows to quickly localize associated ele-ments. Each root of the tree is an element of the initial mesh and each edge is an intermediate element created during the refinement procedures. The leafs of the tree are elements of the current mesh and are brought together if they are associated. Figure 10 presents the quad tree structure associated to the mesh of the part on figure 9 at iteration 3. Elements whose number is underlined, are extraordi-nary elements.

Fig. 9.

Example of adaptive remeshing during three iterations

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The coarsening technique is only applied to the leafs of a same level which are brought together. In the above example, associated ele-ments which could be coarsened are: (6, 7, 8, 9) or (10, 11, 12, 13) or (18, 19, 20, 21). As in the refinement procedure, to ensure the mesh conformity, some coarsened elements could be refined if nec-essary. This last operation can only be applied when all the flat areas have been coarsened by the coarsening technique.

2.3 D uring the refinement procedure, the mechanical fields are simply induced from the current mesh to the new mesh. During the coarsen-ing procedure, the mechanical fields associated to four associated elements are averaged and the result is associated to the new ele-ment. D uring the refinement procedure, the mechanical fields of curved elements are simply associated to the four new created ele-ments from the subdivision.

21

Transfer of Mechanical Fields

Fig. 10. Quad tree structure for the mesh of the part of figure 9

137

3.1 An early application of adaptive mesh refinement was the simulation of 3-D sheet metal stamping example (Benchmark square box of Numisheet’93). According to Onate et al. [14], the geometric data of the square cup are: drawing depth 40 mm, sheet dimensions 150u 150 mm 2, thickness h 0= 0.78 mm, friction coefficient between the sheet and rigid tools is assumed to be P = 0.144 and the blank-holder force F = 19600 N. The material model used is an isotropic elastoplastic von-Mises model with multi-linear isotropic hardening approximating a power law yield stress curve defined as 2637.0)07127.0(29.567p

H V . The punch velocity is 20 mm/s and its stroke is 80 mm. The tools (punch, die and blank-holder) are sup-posed rigid and modeled by discrete rigid surfaces. Two examples are presented: the first example concerns the stamping of square sheet in which the angle T between initial sheet plane frame (X,Y) and the tools orientation (x,y,z) is T = 0° (see Figure 11a) and the second concerns the stamping of square sheet with T = 45° (see Fig-ure 11b). In these two cases, the solver 3D ABAQUS/EXPLICIT has been used. The element size adaptive discretization of the de-formable sheet uses h min = 0.75 mm, geometrical criterion = 8°.

Meshes adapted to the part curvature corresponding to different punch displacement (u = 6, 15, 24, 30, 36, 45 and 48 mm) are shown in Figures 12 to 18 for T = 0° and T = 45°. We can note that, the ini-tial blank sheet is computed using an initial coarse mesh (100 quad-rilateral elements), the mesh is again refined uniformly and the adap-tive mesh refinement procedure is activated where elements are created automatically in regions of large curvature to even more ac-curately represent the complex material flow (large stretching) around the punch and die radii. The final contour of the sheet for 60 mm of punch displacement is presented in Figure 22 for T = 0° and in Figure 23 for T = 45°. We can note the final shape of the sheet is completely different due the initial sheet orientation. A Remeshing Procedure for Numerical Simulation 3 Numerical Examples

Sheet Metal Stamping

138L. Giraud-Moreau et al.

(a)T = 0° (b) T = 45°

Fig. 11. Tools and initial sheet orientation

Fig. 12. Displacement u = 6 mm

Fig. 13. Displacement u = 15 mm

Fig. 14.

Displacement u = 24 mm

Y Y

X

139 A Remeshing Procedure for Numerical Simulation

Fig. 15. Displacement u = 30 mm

Fig. 16. Displacement u = 36 mm

Fig. 17. Displacement u = 45 mm

140L. Giraud-Moreau et al.

Fig. 18. Displacement u = 48 mm

Fig. 19. Final shape for T = 0°

Fig. 20. Final shape for T = 45°

141 3.2

The second example is the crushing of a thin cylinder. The cylinder blank has initially 140 mm length, 44 mm diameter and 0.5 mm thickness. The initial mesh of the cylinder is constituted by 2048 quadrilateral sheet finite elements. Two concentrate loads diametri-cally opposite was prescribed using a linear ramp to simulate the crushing operation. The deformation evolution of the blank is illus-trated in Figure 21. Here, the mesh refinement is localized on large deformed blank areas. The final mesh of the blank contains 30301

quadrilateral and 26714 triangular elements.

A Remeshing Procedure for Numerical Simulation

Crushing of a Thin Cylinder

142L. Giraud-Moreau et al.

Fig. 21. Deformed cylinder for different crushing step

4 Conclusions

The different steps necessary to the remeshing of the computation domain in large elastoplastic deformations in three dimensions have been presented. The proposed adaptive remeshing technique is based on refinement and coarsening procedures using ageometrical crite-rion. This approach has been implemented with triangular and quad-rilateral elements in the ABAQUS code. Numerical simulations of thin sheet metal forming process in three dimensions have validated the proposed approach and proved its efficiency. The extension in three dimensions for massive structure metal forming is currently under progress.

References

1. H. Borouchaki, A. Cherouat, K. Saanouni, A. Cherouat and P. Laug,

“Remaillage en grandes deformation. Applications à la mise en forme de

structures 2D”, Revue Européenne des éléments finis (to appear), 2002.

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2. K. Saanouni, A. Cherouat and Y. Hammi, “Optimization of hydroforming

processes with respect to fracture”, Esaform, Belguim, 1, 361-364, 2001.

3. P. Ladeveze and J.P. Pelle, “La ma?trise du calcul en mécanique linéaire et

non linéaire, études en mécanique des matériaux et des structures”, Hermès, Paris, France, 2001.

4. D. Djokovic, “Splines for approximating solutions of partial differential equa-

tions”, Ph.D. Thesis, University of Twente, Enschede, ISBN 90-36510511, 1998.

5. A. Huerta, P. Diez, A. Rodriguez-Ferran, “Adaptivity and error estimation”,

Proceedings of the 6th International Conference on Numerical Methods in In-dustrial Forming Processes, J. Huétink and F.P.T. Baaijens (eds), Balkema, Rotterdam, 63-74, 1998.

6. R. Drake, V.S. Manoranjan, “A method of dynamic mesh adaptation”, Int. J.

Num. Meth. Eng., 39, 939-949, 1996.

7. O. C. Zienkiewicz and J. Z. Zhu, “Adaptivity and mesh generation”, in Int. J.

Numer. Methods Eng. 32, 783-810, 1991.

8. L. Fourment and J.-L. Chenot, “Adaptive remeshing and error control for

forming processes”, Revue européenne des éléments finis 3, 2, 247-279, 1994.

9. T. Coupez, “Génération de maillage et adaptation de maillage par

optimisation locale”, Revue européenne des éléments finis 9, 4, 403-422, 2000.

10. H. Borouchaki, A. Cherouat, P. Laug and K. Saanouni, “Adaptive remeshing

for ductile fracture prediction in metal forming”, C.R. Mecanique 330, 709-716, 2002.

11. Y. Li, I. Babuska, “A convergence analysis of a p-version finite element

method for one-dimensional elastoplasticity problem with constitutive laws based on the gauge function method”, J. Num. Anal., 33, 2, 809-842, 1996. 12. J.-W. Cho and D.-Y. Yang, “A mesh refinement scheme for sheet metal form-

ing analysis”, Proc. of the 5th International Conference, NUMISHEET’02, 307-312, 2002.

13. T. Meinders, “Developmsents in numerical simulations of the real life deep

drawing process, Ph.D. Thesis, University of Twente, Enschede, ISBN 90-36514002, 2000.

14. E. Onate, J. Rojek, C. Garcia Garino, Numistamp: "A research project for as-

sessment of finite element models for stamping process", Journal of Materials Processing Technology 50, 17-38, 1995.

英语选修六课文翻译Unit5 The power of nature An exciting job的课文原文和翻译

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英语选修六课文翻译第五单元word版本

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人教版英语选修六Unit5 the power of nature(An exciting Job)

高二英语教学设计 Book6 Unit 5 Reading An Exciting Job 1.教学目标(Teaching Goals): a. To know how to read some words and phrases. b. To grasp and remember the detailed information of the reading material . c. To understand the general idea of the passage. d. To develop some basic reading skills. 2.教学重难点: a.. To understand the general idea of the passage. b. To develop some basic reading skills. Step I Lead-in and Pre-reading Let’s share a movie T: What’s happened in the movie? S: A volcano was erupting. All of them felt frightened/surprised/astonished/scared…… T: What do you think of volcano eruption and what can we do about it? S: A volcano eruption can do great damage to human beings. It seems that we human beings are powerless in front of these natural forces. But it can be predicted and damage can be reduced. T: Who will do this kind of job and what do you think of the job? S: volcanologist. It’s dangerous. T: I think it’s exciting. Ok, this class, let’s learn An Exciting Job. At first, I want to show you the goals of this class Step ⅡPre-reading Let the students take out their papers and check them in groups, and then write their answers on the blackboard (Self-learning) some words and phrases:volcano, erupt, alongside, appoint, equipment, volcanologist, database, evaluate, excite, fantastic, fountain, absolutely, unfortunately, potential, be compared with..., protect...from..., be appointed as, burn to the ground, be about to do sth., make one’s way. Check their answers and then let them lead the reading. Step III Fast-reading 这是一篇记叙文,一位火山学家的自述。作者首先介绍了他的工作性质,说明他热爱该项工作的主要原因是能帮助人们免遭火山袭击。然后,作者介绍了和另外二位科学家一道来到火山口的经历。最后,作者表达了他对自己工作的热情。许多年后,火山对他的吸引力依然不减。 Skimming Ⅰ.Read the passage and answer: (Group4) 1. Does the writer like his job?( Yes.) 2. Where is Mount Kilauea? (It is in Hawaii) 3. What is the volcanologist wearing when getting close to the crater? (He is wearing white protective suits that covered his whole body, helmets, big boots and

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Unit5 Reading An Exciting Job 说课稿 Liu Baowei Part 1 My understanding of this lesson The analysis of the teaching material:This lesson is a reading passage. It plays a very important part in the English teaching of this unit. It tells us the writer’s exciting job as a volcanologist. From studying the passage, students can know the basic knowledge of volcano, and enjoy the occupation as a volcanologist. So here are my teaching goals: volcanologist 1. Ability goal: Enable the students to learn about the powerful natural force-volcano and the work as a volcanologist. 2. Learning ability goal: Help the students learn how to analyze the way the writer describes his exciting job. 3. Emotional goal: Make the Students love the nature and love their jobs. Learn how to express fear and anxiety Teaching important points: sentence structures 1. I was about to go back to sleep when suddenly my bedroom became as bright as day. 2. Having studied volcanoes now for more than twenty years, I am still amazed at their beauty as well as their potential to cause great damage. Teaching difficult points: 1. Use your own words to retell the text. 2. Discuss the natural disasters and their love to future jobs. Something about the S tudents: 1. The Students have known something about volcano but they don’t know the detailed information. 2. They are lack of vocabulary. 3. They don’t often use English to express themselves and communicate with others.

an exciting job 翻译

我的工作是世界上最伟大的工作。我跑的地方是稀罕奇特的地方,我见到的是世界各地有趣味的人们,有时在室外工作,有时在办公室里,有时工作中要用科学仪器,有时要会见当地百姓和旅游人士。但是我从不感到厌烦。虽然我的工作偶尔也有危险,但是我并不在乎,因为危险能激励我,使我感到有活力。然而,最重要的是,通过我的工作能保护人们免遭世界最大的自然威力之一,也就是火山的威胁。 我是一名火山学家,在夏威夷火山观测站(HVO)工作。我的主要任务是收集有关基拉韦厄火山的信息,这是夏威夷最活跃的火山之一。收集和评估了这些信息之后,我就帮助其他科学家一起预测下次火山熔岩将往何处流,流速是多少。我们的工作拯救了许多人的生命,因为熔岩要流经之地,老百姓都可以得到离开家园的通知。遗憾的是,我们不可能把他们的家搬离岩浆流过的地方,因此,许多房屋被熔岩淹没,或者焚烧殆尽。当滚烫沸腾的岩石从火山喷发出来并撞回地面时,它所造成的损失比想象的要小些,这是因为在岩石下落的基拉韦厄火山顶附近无人居住。而顺着山坡下流的火山熔岩造成的损失却大得多,这是因为火山岩浆所流经的地方,一切东西都被掩埋在熔岩下面了。然而火山喷发本身的确是很壮观的,我永远也忘不了我第一次看见火山喷发时的情景。那是在我到达夏威夷后的第二个星期。那天辛辛苦苦地干了一整天,我很早就上床睡觉。我在熟睡中突然感到床铺在摇晃,接着我听到一阵奇怪的声音,就好像一列火车从我的窗外行驶一样。因为我在夏威夷曾经经历过多次地震,所以对这种声音我并不在意。我刚要再睡,突然我的卧室亮如白昼。我赶紧跑出房间,来到后花园,在那儿我能远远地看见基拉韦厄火山。在山坡上,火山爆发了,红色发烫的岩浆像喷泉一样,朝天上喷射达几百米高。真是绝妙的奇景! 就在这次火山喷发的第二天,我有幸做了一次近距离的观察。我和另外两位科学被送到山顶,在离火山爆发期间形成的火山口最靠近的地方才下车。早先从观测站出发时,就带了一些特制的安全服,于是我们穿上安全服再走近火山口。我们三个人看上去就像宇航员一样,我们都穿着白色的防护服遮住全身,戴上了头盔和特别的手套,还穿了一双大靴子。穿着这些衣服走起路来实在不容易,但我们还是缓缓往火山口的边缘走去,并且向下看到了红红的沸腾的中心。另外,两人攀下火山口,去收集供日后研究用的岩浆,我是第一次经历这样的事,所以留在山顶上观察他们

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