Netgen网格划分程序采用的推进前线法算法实现步骤

叙述二重网格迭代的主要步骤
二重网格迭代是一种常用的求解偏微分方程数值解的算法。

它主要包含以下步骤：
1. 网格划分：将求解区域划分为不同大小的网格，在二重网格迭代中通常采用两层网格，即粗网格和细网格。

2. 初始化：在粗网格上初始化初始值或初始猜测解。

3. 粗网格迭代：在粗网格上进行迭代求解，通常使用一种迭代方法（如Jacobi、Gauss-Seidel 等）来逐步更新方程组的解。

4. 限制：将粗网格上的解限制到细网格上，通常使用插值技术（如线性插值、加权插值等）实现。

5. 细网格迭代：在细网格上进行迭代求解，使用与粗网格相同的迭代方法。

6. 平滑：将细网格上的解平滑处理，可采用与粗网格迭代方法相同的技术。

7. 拉回：将平滑后的解拉回到粗网格上，通常使用差值技术实现。

8. 重复进行第3-7步，直到达到收敛条件或迭代次数达到预设值。

通过以上步骤，二重网格迭代可以逐步提高数值解的精度，并在每个粗网格和细网格上进行迭代求解，从而有效地加快计算速度。

复杂网络分析法的实施步骤1. 理解复杂网络分析法的概念和原理•复杂网络分析法是一种将网络结构和节点之间的关系转化为数据模型进行分析和研究的方法。

•复杂网络分析法的原理是基于图论和复杂系统理论，通过构建网络模型来揭示网络中存在的规律和特征。

2. 收集和整理网络数据•收集需要进行复杂网络分析的数据，可以是社交网络、生物网络、交通网络等各种类型的网络数据。

•对收集到的数据进行整理和预处理，确保数据的准确性和完整性。

3. 构建网络模型•使用合适的图论工具，将收集到的网络数据转化为网络模型。

•根据数据的特点选择合适的网络模型，常用的网络模型包括随机网络模型、小世界网络模型和无标度网络模型等。

4. 分析网络结构的特征和指标•使用复杂网络分析的工具和算法，计算网络结构的各种特征和指标。

•常用的网络特征和指标包括节点度中心性、介数中心性、连通性和聚类系数等。

5. 解读和分析网络特征•根据计算得到的网络特征和指标，分析网络的特点和规律。

•探索网络中的核心节点、社区结构、信息传播路径等关键特征。

6. 实施复杂网络分析的相关方法•根据分析的需求和目标，选择合适的复杂网络分析方法。

•常用的复杂网络分析方法包括节点重要性排序、社区发现、信息传播模型等。

7. 验证和评估结果的可靠性•对分析得到的结果进行验证和评估，确保结果的可靠性和有效性。

•使用相关的统计方法和指标，对结果进行检验和比较。

8. 结果的解释和呈现•将分析得到的结果进行解释和呈现，以便对结果进行理解和应用。

•使用可视化工具和技术，将结果以图表、图形等形式展示出来。

9. 结论和进一步研究•根据分析的结果，总结出结论，并提出进一步研究的方向和问题。

•推进复杂网络分析方法的应用和发展。

以上是复杂网络分析法的实施步骤，通过理解和运用这些步骤，可以更好地应用复杂网络分析法来揭示网络中的规律和特征，为实际问题的解决提供参考和支持。

NETGEN-4.3 Joachim Sch¨o berlMay7,20032Contents1Getting Started51.1What is NETGEN (5)1.2The history of NETGEN (5)1.3How to receive NETGEN (5)1.4Installing NETGEN (6)1.4.1Installing NETGEN for Unix/Linux (6)1.4.2Installing NETGEN for Windows98/NT (6)1.4.3Testing Netgen (7)2Constructive Solid Geometry(CSG)92.1Available Primitives (11)2.2Known problems and work-arounds (12)2.2.1Interfaces (12)2.2.2Degenerated edges (13)3Other Geometry Formats153.1Using STL Geometries (15)3.22D Spline Geometry (15)4Mesh and Solution Formats174.1Neutral Format (17)4.2Fepp Format2D (17)4.3Surface triangulatonfile (18)4.4Solution File Format (18)5Netgen operations215.1Command line arguments (21)6Using the Graphical User Interface236.1The Netgen menu items (25)6.1.1The menu item File (25)6.1.2The menu item Geometry (25)6.1.3The menu item Mesh (26)34CONTENTS6.1.4The menu item View (26)6.1.5The menu item Refinement (27)6.2Meshing Options (27)6.3Visualization Options (28)7The Algorithms of Netgen29 8Programming Interfaces318.1The nginterface (31)8.2The nglib (31)8.2.1Introduction (31)8.2.2The Header File (31)8.2.3Types and Constants (32)8.2.4Initialization (33)8.2.5Mesh access (33)8.2.6STL Geometry (34)8.2.7Programming Example (35)Chapter1Getting Started1.1What is NETGENNETGEN is an automatic mesh generation tool for two and three dimensions. Netgen is open source under the conditions of the LGPL.It comes as stand alone programme with graphical user interface,or as C++library to be linked into an other gen is available for Unix/Linux and Windows98/-gen generates triangular or quadrilateral meshes in2D,and tetrahedral meshes in 3D.The input for2D is described by spline curves,and the input for3D problems is either defined by constructive solid geometries(CSG),see Chapter2,or by the standard STLfile GEN contains modules for mesh optimization and hierarchical mesh refinement.Curved elements are supported of arbitrary order.1.2The history of NETGENThe NETGEN project was started1994in the master’s programme of Joachim Sch¨o berl,under supervision of Prof.Ulrich Langer,at the Department of Compu-tational Mathematics and Optimization,University Linz,Austria.Its further de-velopment was supported by the Austrian science Fund“Fonds zur F¨o rderung der wissenschaftlichen Forschung”(http://www.fwf.ac.at)under projects P10643-TEC and SFB1306.The current home of Netgen is the Start project“hp-FEM”(http://www.hpfem.jku.at)granted by the FWF.Special thanks go to•Robert Gaisbauer:High order curved elements•Hannes Gerstmayr:Meshing of STL geometry1.3How to receive NETGENNETGEN is available from the WEB at56CHAPTER1.GETTING STARTEDhttp://www.hpfem.jku.at/netgenYoufind there source code releases for Linux/Unix/Windows,as well as compiledversions for Windows.You can use CVS access to receive the most up to dateversion.1.4Installing NETGEN1.4.1Installing NETGEN for Unix/LinuxTo install NETGEN on Unix/Linux you will download the source code and com-pile it yourself.You need the following libraries:•The3D visualization library OpenGL.It comes with most systems withhardware graphics.The free software version mesagl is available from.•The graphical toolkit TclTk developed by John Ousterhout(available from/)and its extension Tix available from ) by Iam gen has been tested with version TclTk8.0-TclTk8.4and Tix4.6.-Tix8.2You can also download these packages from the Netgen site.To install NETGEN please move into the directory ng4.You set the Unix-variable MACHINE according to your machine/operating system,e.g.setenv MACHINE LINUX(in bash shell you type export MACHINE=LINUX).The Makefile includes the makefile-includelibsrc/makefile.mach.$(MACHINE)Please create/modify the accordingfile,(e.g.copy makefile.mach.LINUX tomakefile.mach.SUN).Then you enter make to build the executable.To make NETGEN globally available you just copy the binary“ng”to theglobal bin-directory.In difference to earlier versions,it needs no additionalfiles.1.4.2Installing NETGEN for Windows98/NTNETGEN is available now for Windows in binary form.You download the zip-archive ng4win.zip.After unpacking it with winzip,you can start the executable“ng4.exe”.1.4.INSTALLING NETGEN7 1.4.3Testing NetgenPlease start Netgen by entering“ng”or clicking the“ng.exe”icon.A black window with menu items should appear.Please load a geometryfile by selecting”File->Load Geometry”,choose e.g.examples/cube.geo.Then press the button”Generate Mesh”.By keeping pressed the left,middle or right button of your mouse you can rotate,move or zoom the object.With“File-> Export Mesh”you can save the meshfile.8CHAPTER1.GETTING STARTEDChapter2Constructive Solid Geometry (CSG)The CSG input format is a useful geometry format for small and medium size geometries.One defines the geometry by writing an ASCIIfile in a text editor.The geometry is defined by the Eulerian operations(union,intersection and complement)from primitives.A complete list of availabe primitives is given in Section2.1.The following input describes a cube:#A cubealgebraic3dsolid cube=orthobrick(0,0,0;1,1,1);tlo cube;Lines starting with#are comment lines.Every CSGfile must contain the keyword algebraic3d before any non-comment line.The keyword solid defines a named solid,here the solid cube is defined.A solid is defined by the Eulerian operations applied to primitives.Here,the solid is just the primitve defined by orthobrick.This is a brick parallel to the axis,specified by the minimal x,y,and z coordinates,and the maximal x,y,and z coordinates.The present definition gives the cube[0,1]3.Finally,the definition tlo cube declares the solid cube as a top-level-object,what is necessary for meshing.Please start netgen with the geometryfile above by enteringng cube.geoInstead,you can also load the geometry from thefile menu.You will see a blue cube,which you can rotate by keeping the left mouse button pressed.Pressing the big generate mesh button will result in a(very coarse)mesh of that cube.Instead of using the primitive orthobrick,one can also specify a cube by intersecting six halfspaces(called planes).Each primitive plane is given by an910CHAPTER2.CONSTRUCTIVE SOLID GEOMETRY(CSG) arbitrary point in the plane,and a outward vector,not necessarily a unit vector. The six halfspaces are intersected by the keyword and.The following input gives an equivalent result:#A cubealgebraic3dsolid cube=plane(0,0,0;0,0,-1)and plane(0,0,0;0,-1,0)and plane(0,0,0;-1,0,0)and plane(1,1,1;0,0,1)and plane(1,1,1;0,1,0)and plane(1,1,1;1,0,0);tlo cube;To drill a hole though the cube,we will intersect the cube and the comple-ment of a cylinder.A cylinder is defined by two points on the central axis,and the radius.Please note,a cylinder is understood as an infinitely long cylinder (although the visualization may suggest afinite cylinder):#cube with holealgebraic3dsolid cubehole=orthobrick(0,0,0;1,1,1)and not cylinder(0.5,0.5,0;0.5,0.5,1;0.1);tlo cubehole;Like and denotes the intersection,or denotes the union:solid cubeball=orthobrick(0,0,0;1,1,1)or sphere(0,0,0;0.5)-maxh=0.2;Theflag-maxh=0.2assignes the maximal mesh size of0.2to the solid.The current version,NG4.1,uses the mesh size assigned to the main solid of the top-level-object for the domain.Future version will contain more possibilities to define mesh-sizes for parts of a domain.It is possible to define geometries with several sub-domains,simply by declar-ing several tlos:algebraic3dsolid cube=orthobrick(0,0,0;1,1,1);solid cyl=cylinder(0.5,0.5,0;0.5,0.5,1;0.1);solid dom1=cube and not cyl;solid dom2=cube and cyl;tlo dom1-col=[0,0,1]-transparent;tlo dom2-col=[1,0,0];2.1.AVAILABLE PRIMITIVES11 This example show also solid trees involving previously defined named solids. Top-level-objects can be assigned a color specified by the amount of red,green and blue(RGB)values.Theflag-transparent makes the solid appear transparent.It is possible to specify bounday condition numbers for individual surfaces of a solid.Theflag-bc assignes the bc to all surfaces of that solid-tree.If several flags are given the one closest to the leaves of the tree dominates.The following file defines a cube,with bc=1at the bottom,bc=2at the top,and bc=3for all other surfaces:algebraic3dsolid bottom=plane(0,0,0;0,0,-1)-bc=1;solid top=plane(1,1,1;0,0,1)-bc=2;solid cube=bottm and topand plane(0,0,0;0,-1,0)and plane(0,0,0;-1,0,0)and plane(1,1,1;0,1,0)and plane(1,1,1;1,0,0)-bc=3;tlo cube;2.1Available PrimitivesNetgen4.1supports the following primitives:1.A halfspace,i.e.,a plane and everything on one side of it,given by anarbitrary point p=(p x,p y,p z)in the plane and an outside normal vec-tor n=(n x,n y,n z),not necessarily a unit vector:plane(p x,p y,p z;n x,n y,n z)2.A cylinder of infinite length,given by two points a=(a x,a y,a z)and b=(b x,b y,b z)on the central axis and the radius r:cylinder(a x,a y,a z;b x,b y,b z;r)3.A sphere,given by the center c=(c x,c y,c z)and the radius r:sphere(c x,c y,c z;r)4.An elliptic cylinder,given by the point c=(c x,c y,c z)on the main axis,andthe vectors v and w of the long and short axis of the ellipse,respectively: ellipticcylinder(c x,c y,c z;v x,v y,v z;w x,w y,w z)12CHAPTER2.CONSTRUCTIVE SOLID GEOMETRY(CSG)5.An ellipsoid,given by the center c=(c x,c y,c z),and the vectors u,v and wof the main axis of the ellipsoid:ellipsoid(c x,c y,c z;u x,u y,u z;v x,v y,v z;w x,w y,w z)6.A cone is given by two points on the central axis and the two correspondingradii.It is not possible to mesh the top of the cone yet,it must be cut off.cone(a x,a y,a z;r a;b x,b y,b z;r b)7.A orthobrick is a brick parallel to the coordinate axis.It is specified by twoopposite corner points a=(a x,a y,a z)and b=(b x,b y,b z):orthobrick(a x,a y,a z;b x,b y,b z)8.A polyhedron is defined by a set of triangles forming a closed polyhedron.First,a set of points is defined,then the triangles are given by point indices.The triangles must be oriented counter-clockwise when looking onto the object.The following polyhedron describes a tetrahedron:algebraic3dsolid poly=polyhedron(0,0,0;1,0,0;0,1,0;0,0,1;;1,3,2;1,4,3;1,2,4;2,3,4);tlo poly;2.2Known problems and work-arounds2.2.1InterfacesA airdomain with two connected interior parts may be described byalgebraic3dsolid cube=orthobrick(0,0,0;1,1,1);solid part1=orthobrick(0.2,0.2,0.2;0.5,0.8,0.8);solid part2=orthobrick(0.5,0.2,0.2;0.8,0.8,0.8);solid air=cube and not part1and not part2;tlo air;tlo part1;tlo part2;The problem is,that a domain is an open domain.Thus,the domain air is not only the outer part,but also the interface between part1and part2.The result2.2.KNOWN PROBLEMS AND WORK-AROUNDS13is unspecified.Tofix this problem,one can define the air-domain by cutting outone big brick:solid air=cube and not othrobrick(0.2,0.2,0.2;0.8,0.8,0.8);2.2.2Degenerated edgesDegenerated edges are found sometimes,but some still cause troubles.A sphereon top of a cylinder my be described by:solid cyl=cylinder(0,0,0;1,0,0;0.5)and plane(0,0,0;-1,0,0)and plane(1,0,0;1,0,0);solid main=cyl or sphere(1,0,0;0.5);tlo main;The edge is a degenerated one.A work-around is to split the domain(artificially)into two non-degenerated parts:solid cyl=cylinder(0,0,0;1,0,0;0.5)and plane(0,0,0;-1,0,0)and plane(1,0,0;1,0,0);solid hemisphere=sphere(1,0,0;0.5)and not plane(1,0,0;-1,0,0);tlo cyl;tlo hemisphere;14CHAPTER2.CONSTRUCTIVE SOLID GEOMETRY(CSG)Chapter3Other Geometry Formats3.1Using STL GeometriesSTL is a standardizedfile format to describe(approximate)geometies by trian-gulated surfaces.It is useful to describe complicated parts which are modeled with some CAD programmes.Also,some users have written their own(C)pro-grammes to define STL geometries,where was not so easy to use the CSG format. The syntac of STLfiles is as follos(not available yet.pleasefigure out the syntax from the examples)We found that many STL geometries have some difficulties.Some of them can be corrected(removed)by the STL-Doctor.Please see the corresponding manual pages(not available yet).3.22D Spline GeometryThe extension for2D spline geometry is“.in2d”.The boundary is given in terms of straight lines and of quadratic rational spline patches.A line is given by the two endpoints,a spline patch by3d’Boor points.The patch is an elliptic arc from point1to point3,such that the lines 1-2and2-3are tangents.It is possible to use different subdomains with this format.Thisfile format also supports a priori mesh grading.To the spline point i one adds a local refinement factor rp i.Close to this point the mesh-size h(x) is h Glob/rp i.The global parameter grading describes how fast the mesh-size decreases.The gradient of the local mesh-size function h(x)is bounded by |∇x h(x)|≤grading−1Also a refinement by a factor rs i¿1along the whole segment i is possible.Thefile looks like:1516CHAPTER3.OTHER GEOMETRY FORMATS splinecurves2dgradingnpx1y1rp1...x np y np rp npnsdil1dir1[2|3]pi1,1pi1,2[pi1,3]rs1...dil nl dir nl[2|3]pi nl,1pi nl,2[pi nl,3]rs nlnp is the number of points,ns the number of spline segments.Every segment starts with the domain numbers at the left and at the right sides of the segment. Domain number0is reserved for the exterior.Then the number of points and two or three point indices follow.Finally,the refinement factor along the line follows.Chapter4Mesh and Solution FormatsYou can export meshes to a couple offile formats.Some are self-defined,some other are standard formats.The self-defined are the followings:4.1Neutral FormatThe neutral volume mesh format contains the following sections:1.nodesAfter the number of nodes there follows a list of x,y,and z-coordinates of the mesh-nodes.2.volume elementsAfter the number of volume elements there follows the list of tetrahedra.Each element is specified by the sub-domain number,and4node indices.The node indices start with1.3.surface elementsAfter the number of surface elements there follows the list of triangles.Each element is specified by the boundary condition number,and3node indices.The node indices start with1.4.2Fepp Format2DThe Fepp2D format contains the following sections:1.boundary segmetnsAfter the number of boundary segments there follows a list of segments.Each segment is specified by the spline-patch number,and the two node indices.Counting starts with11718CHAPTER4.MESH AND SOLUTION FORMATS2.domain elementsAfter the number of domain elements there follows the list of elements.Each element is specified by the sub-domain number,the number of nodes (3or4)and the node indices.Counting starts with13.nodesAfter the number of nodes there follows a list of x and y-coordinates of the mesh-nodes.4.geometric informationAfter the number of spline patches there follows a list of spline specifi-cations.Each spline patch is given by the6coefficients of the descibing quadratic polynomial equationc1x2+c2y2+c3xy+c4x+c5y+c6=04.3Surface triangulatonfileOne can export to and import from a surface meshfile.Its structure is as follows:1.surfacemeshstarts with that keyword2.number of pointspoint coordinates(x,y,z).3.number of surface triangles,surface triangles oriented counter-clock wise when looking at the object, index starts from1.4.4Solution File FormatThe Netgen software includes also a simple visualizer forfinite element gridfunc-tions.It supports scalarfields(e.g.temperature),and vector valuedfields(flow velocities,mechanical deformations).The solutionfield is imported by the menu item File−>Import Solution.It is important to load the corresponding mesh in advance.The format of the solutionfile is as follows.It consists of an arbitrary number of blocks of this structure:1.solution function-nameflagssolution is the keyword,function-name is a string to refer to that func-tions.The supportedflags are4.4.SOLUTION FILE FORMAT19(a)-size=snumber of entries(default:number of mesh-points)(b)-components=cnumber of components(default:1).Mechanical deformations have3components.(c)-type=[nodal,element,surfaceelement]the grid-funciton has nodal values,or one value per volume element,or one value per surface element(default:nodal)2.block of size×components valuesPlease try out to import the solutionfile’tutorials/cube.sol’fitting to the mesh’tutorials/cube.vol’.20CHAPTER4.MESH AND SOLUTION FORMATSChapter5Netgen operationsYou can use netgen in interactive mode using its menus,or,you can run netgen in batch-mode using command line arguments.5.1Command line argumentsCommand line arguments are specified as-flag=value.•-helpPrints the availabel command line arguments•-geofile=filenameSpecifies geometryfile.Is equivalent tofilename,i.e.,you can scip-geofile=.•-meshfile=filenameMeshfile will be stored infilefilename.•-batchmodeExit after mesh generation.Otherwise,the GUI will be started•-VVerbose mode.Prints some additional information•-verycoarse,-coarse,-moderate,-fine,-veryfineMesh size control21GEN OPERATIONSChapter6Using the Graphical User InterfaceThe Netgen main window looks like:It consists of the menuline and the button line at the top,the status line at the bottom,and the large drawing window.The menu items will be explained in6.1. The button line provides shot-cuts for common opteration:•QuitTerminate Netgen23ING THE GRAPHICAL USER INTERFACE •Generate meshPerforme mesh generation•Stop MeshingStop mesh generation•Geometry/Edges/Mesh/SolutionSwitch between operation modes of visualization.•Zoom allZooms such that the whole visualization scenefits into the window.•CenterCenter rotation and scaling at marked point,available only in mesh-vi-suailzation mode.•Rotate/Move/Zoom Left mouse drag rotates/moves/zooms object.The status line shows information,namely•PointsNumber of points in the mesh•ElementsNumber of volume elements(3D)in the mesh•Surf ElementsNumber of surface elements(3D)or inner elements(2d)in the mesh.•MemUsed memory in the large memory arena•Meshing Job,percentage Douriing mesh generation,the current job as well as the progress is displayed on the right side of the statu line.The drawing window displays the geometry or the mesh.The view can be changed with the mouse:•drag with left button pressed rotates the object,•drag with middle button pressed moves the object,•drag with right button pressed zooms the object.The view can also be changed with the keyboard:•cursor keys rotate the object•shift+cursor keys move the object6.1.THE NETGEN MENU ITEMS25•control+cursor keys zoom the objectWhen in Mesh-visualization scene,double clicking on triangles mark the surface.The point cursor is set.6.1The Netgen menu items6.1.1The menu item File6.1.2The menu item GeometryING THE GRAPHICAL USER INTERFACE 6.1.3The menu item Mesh6.1.4The menu item View6.2.MESHING OPTIONS27 6.1.5The menu item Refinement6.2Meshing OptionsING THE GRAPHICAL USER INTERFACE 6.3Visualization OptionsChapter7The Algorithms of NetgenNetgen follows a top down strategy.It starts from computing the corner points (CSG only).Then,the edges are defined and meshed into segments(CSG and STL).Next,the faces are meshed by an advancing front surface mesh gener-ator.After meshing,the faces meshes are optimized.Finally,the individual sub-domains arefilled with tets.Therefore,a fast Delaunay algorithm generates most of the elements(about98percent).But often it fails for mesh the whole domain,then the slower back-tracking rule-base algorithm takes over.Finally, the volume is optimized by the usual node-movement,element swapping and splitting algorithms.2930CHAPTER7.THE ALGORITHMS OF NETGENChapter8Programming Interfaces8.1The nginterfaceBy means of the nginterface one’s own simulation code can be included into the netgen environment.This is particular useful for FEM(FVM,BEM)code devel-opers,since they may profit from the netgen preprocessing and postprocessing possibilities.Please download the example Netgen-add-on module demoapp and follow the instructions therein8.2The nglib8.2.1IntroductionThe NETGEN mesh generation library nglib is available in C++source code and can be compiled for Unix/Linux as well as Win95/98/NT and linked to one libraryfile.The interface to the application programme is by the C language headerfile nglib.h.The functionality of nglib is volume mesh generation by a domain given by a surface triangulation,and surface mesh generation from a domain described by an STLfile(standardfile format for geometries defined by triangle approximation). It can do mesh optimization as well as mesh refinement.It can generate4node tetrahedra and10node tetrahedrons(with quadratic shape functions).The local mesh size can be defined automatically by geometric features and/or by user specification.8.2.2The Header FileThe interfacefile contains the following type definitions and function calls.All Netgen types and functions start with Ng.Types and functions have capital3132CHAPTER8.PROGRAMMING INTERFACES initial letters,constants are in capital letters.8.2.3Types and Constants///Data type for NETGEN meshtypedef void*Ng_Mesh;///Data type for NETGEN STL geomtytypedef void*Ng_STL_Geometry;//max number of nodes per element#define NG_VOLUME_ELEMENT_MAXPOINTS10//implemented element types:enum Ng_Volume_Element_Type{NG_TET=1,NG_PYRAMID=2,NG_PRISM=3,NG_TET10=4};//max number of nodes per surface element#define NG_SURFACE_ELEMENT_MAXPOINTS6//implemented element types:enum Ng_Surface_Element_Type{NG_TRIG=1,NG_QUAD=2,NG_TRIG6=3};struct Ng_Meshing_Parameters{double maxh;double fineness;//0..coarse,1..fineint secondorder;};enum Ng_Result{NG_OK=0,NG_SURFACE_INPUT_ERROR=1,NG_VOLUME_FAILURE=2,NG_STL_INPUT_ERROR=3,NG_SURFACE_FAILURE=4};Ng Mesh is a data type representing a Netgen mesh.Ng STL Geometry rep-resents an STL geometry.One can operate on these data structures by the functions defined gen can(now and/or in future)work with various8.2.THE NGLIB33 element types defined by generic constants.Several parameters can be speci-fied in the Ng Meshing Parameters structure for volume and/or surface mesh generation.The result of Netgen functions is of type Ng Result.8.2.4InitializationPlease call these functions before using netgen functions and after using netgen functions,respectively://initialize,deconstruct Netgen library:void Ng_Init();void Ng_Exit();8.2.5Mesh accessNetgen meshes can be processed by the means of the following functions.A mesh contains nodes,surface elements and volume elements.Counting starts from1.//Generates new mesh structureNg_Mesh*Ng_NewMesh();void Ng_DeleteMesh(Ng_Mesh*mesh);//feeds points,surface elements and volume elements to the meshvoid Ng_AddPoint(Ng_Mesh*mesh,double*x);void Ng_AddSurfaceElement(Ng_Mesh*mesh,Ng_Surface_Element_Type et,int*pi);void Ng_AddVolumeElement(Ng_Mesh*mesh,Ng_Volume_Element_Type et,int*pi);//ask for number of points,surface and volume elementsint Ng_GetNP(Ng_Mesh*mesh);int Ng_GetNSE(Ng_Mesh*mesh);int Ng_GetNE(Ng_Mesh*mesh);//return point coordinatesvoid Ng_GetPoint(Ng_Mesh*mesh,int num,double*x);//return surface and volume element in piNg_Surface_Element_TypeNg_GetSurfaceElement(Ng_Mesh*mesh,int num,int*pi);Ng_Volume_Element_TypeNg_GetVolumeElement(Ng_Mesh*mesh,int num,int*pi);34CHAPTER8.PROGRAMMING INTERFACESMesh GenerationThe user can specify the mesh size function by the global parameter maximalmesh size,and can additionally restrict the mesh size in points or cubes.Thefunction Ng GenerateVolumeMesh generates the volume mesh starting from thesurface.//Defines MeshSize Functionsvoid Ng_RestrictMeshSizeGlobal(Ng_Mesh*mesh,double h);void Ng_RestrictMeshSizePoint(Ng_Mesh*mesh,double*p,double h);void Ng_RestrictMeshSizeBox(Ng_Mesh*mesh,double*pmin,double*pmax,double //generates volume mesh from surface meshNg_Result Ng_GenerateVolumeMesh(Ng_Mesh*mesh,Ng_Meshing_Parameters*mp);8.2.6STL GeometryA STL geometry can be read from a STLfile(ASCII or binary),or can beassembled by providing triangle by triangle.Either,the user can specify theedges of the geometry,or netgen can define edges by Ng STL MakeEdges by usingan angle criterium.//loads geometry from STL fileNg_STL_Geometry*Ng_STL_LoadGeometry(char*filename,int binary=0);//generate new STL GeometryNg_STL_Geometry*Ng_STL_NewGeometry();//fills STL Geometry//positive orientation//normal vector may be null-pointervoid Ng_STL_AddTriangle(Ng_STL_Geometry*geom,double*p1,double*p2,double*p3,double*nv);//add(optional)edges:void Ng_STL_AddEdge(Ng_STL_Geometry*geom,double*p1,double*p2);//after adding triangles(and edges)initializeNg_Result Ng_STL_InitSTLGeometry(Ng_STL_Geometry*geom);//automatically generates edges:void Ng_STL_MakeEdges(Ng_STL_Geometry*geom);8.2.THE NGLIB35//generates mesh,empty mesh be already created.Ng_Result Ng_STL_GenerateSurfaceMesh(Ng_STL_Geometry*geom,Ng_Mesh*mesh,Ng_Meshing_Parameters*mp);8.2.7Programming ExampleThe File ,see Appendix A,is a simple application using the netgen volume mesh generator.First,the surface mesh is read from afile containing point coordinates and surface triangles(see e.g.file cube.surf).The volume mesh generate is called,and the volume mesh is written to the standard output,see file cube.vol.。

基恩士算法-概述说明以及解释1.引言1.1 概述基恩士算法是一种常用于解决最短路径问题的算法，广泛应用于网络路由、地图导航等领域。

该算法由荷兰计算机科学家艾兹赫尔·基恩士（Edsger W. Dijkstra）在1956年提出，并于1962年发表。

基恩士算法通过不断更新节点的最短路径信息来找到从起点到所有其他节点的最短路径。

其核心思想是每次选择未加入最短路径集合中的最短路径节点，更新其邻居节点的最短路径信息。

基恩士算法的时间复杂度为O(V^2)，其中V为节点数。

在实际应用中，可以利用优先队列等数据结构将时间复杂度降低至O((V+E)logV)，其中E为边数。

基恩士算法的简洁高效使其成为解决最短路径问题的首选算法之一。

1.2 文章结构文章结构部分主要介绍整篇文章的组织结构，帮助读者更好地理解文章的内容和脉络。

本文主要包括引言、正文和结论三个部分。

- 引言部分：介绍了文章的背景和基本概念，包括基恩士算法的概述、文章结构和目的。

- 正文部分：主要包括基恩士算法的简介、应用领域以及其优缺点。

详细介绍了基恩士算法的原理和特点，以及在实际应用中的具体情况和表现。

- 结论部分：总结了文章的主要内容和观点，展望了基恩士算法的未来发展，并提出了一些结论性的观点和见解，为整篇文章做一个简洁的总结。

通过这样的文章结构，读者可以清晰地了解基恩士算法的概念和应用，深入探讨其优缺点，并对未来的发展做出一些展望和思考。

1.3 目的:基恩士算法作为一种经典的优化算法，在实际应用中具有广泛的用途。

本文旨在对基恩士算法进行深入的探讨和解析，旨在帮助读者更好地理解算法的原理和应用领域。

通过对基恩士算法的优缺点进行分析，可以帮助读者更好地评估在实际问题中选择该算法的可行性。

同时，结合对基恩士算法的未来展望，可以帮助读者更好地把握算法的发展方向，从而更好地应用于实际问题中。

通过本文的阐述，期望读者能够对基恩士算法有一个更加全面和深入的认识，为其在实际问题中的应用提供参考和指导。

格栅算法的原理和应用实例1. 格栅算法的原理格栅算法是一种解决二维空间中多个对象之间的冲突和碰撞问题的计算方法。

其基本原理是将二维空间划分为大小相等的网格，并将对象放置到对应的网格中。

通过在每个网格内对对象的位置进行计算和比较，可以有效地检测出碰撞和冲突。

格栅算法的原理可以分为以下几个步骤：•步骤一：划分格栅将二维空间划分为大小相等的网格，每个网格可以表示为一个矩形区域。

•步骤二：放置对象将需要进行碰撞检测的对象放置到对应的网格中。

•步骤三：检测碰撞在每个网格内，对对象进行碰撞检测。

通过比较对象的位置和网格的边界，可以判断对象是否与其他对象发生碰撞。

•步骤四：处理碰撞如果发现碰撞，则根据具体的应用需求进行相应的处理，比如进行位置调整、速度变化等。

格栅算法的优点是可以在一定程度上提高碰撞检测的效率，尤其是在处理大量对象的情况下，可以减少冲突的判断次数和计算量。

2. 格栅算法的应用实例格栅算法在许多领域都有广泛的应用，下面列举了几个常见的实例。

2.1 2D游戏碰撞检测在2D游戏中，碰撞检测是非常重要的一部分。

游戏中的角色、障碍物、道具等都需要进行碰撞检测，以实现各种交互效果。

格栅算法可以在游戏中快速检测出碰撞的对象，实现角色的碰撞效果、道具的捡取等功能。

2.2 交通流量分析在城市交通管理中，交通流量的分析是非常重要的一项任务。

格栅算法可以利用城市的地理数据和实时的交通信息，对交通流量进行实时监测和分析。

通过将城市划分为网格，并将车辆的位置信息映射到对应的网格中，可以快速计算出每个网格的交通流量，为交通管理部门提供参考。

2.3 图像处理在图像处理领域，格栅算法可以应用于图像的分割和对象的检测。

通过将图像划分为网格，可以对每个网格进行特征提取和对象识别，从而实现图像的分割和对象的检测。

这对于图像识别、目标跟踪等应用有着重要的意义。

2.4 自动驾驶在自动驾驶领域，格栅算法可以用于路径规划和避障。

通过将地图划分为网格，并将车辆的位置和障碍物的位置放置到对应的网格中，可以快速计算出车辆的行驶路径和避障方案。

掌握分支界限法的基本步骤分支界限法（Branch and Bound）是一种常用的数学规划方法，主要用于解决优化问题，特别是问题规模较大且无法通过穷举法求解的情况。

本文将介绍分支界限法的基本步骤，帮助读者掌握这一重要的问题求解方法。

一、问题描述在深入了解分支界限法之前，首先需要明确问题的描述及目标。

一般来说，分支界限法适用于具有以下特点的问题：1. 问题具有离散性质，即决策变量只能取离散的值；2. 问题的解空间较大，无法通过穷举法遍历所有可能的解；3. 问题具有最优解的性质，需要找到某个最优解或最优值。

二、基本步骤分支界限法的基本思想是通过对问题的解空间进行划分，同时利用界限函数（bound function）对每个划分区域进行界定，从而剪枝（pruning）不可能得到最优解的区域，减少问题的规模，提高求解效率。

下面是分支界限法的基本步骤：1. 确定初始界限函数初始界限函数用于对整个问题的解空间进行初始界定。

根据问题的具体情况，可以选择合适的界限函数，如目标函数下界、松弛问题的最优值等。

初始界限函数的选择至关重要，它影响了问题的规模、求解效率以及最终得到的解的质量。

2. 划分解空间根据问题的约束条件和初始界限函数，将问题的解空间划分成若干个区域。

划分的方式可以是二叉树、多叉树或者其他适合问题特点的结构。

划分过程中需要考虑以下几个因素：- 如何选择划分的决策变量及其取值，以使得每个区域的解空间较小；- 如何保证每个区域的解都满足问题的约束条件。

3. 界定每个区域的界限函数对于每个划分区域，根据问题的约束条件和初始界限函数，计算该区域的界限函数值。

界限函数可以在划分过程中不断更新，以提高求解效率。

4. 剪枝和下界更新根据界限函数的值，对划分区域进行剪枝。

具体来说，如果某个划分区域的界限函数值小于当前已获得的最优解，可以将该区域剪枝。

同时，根据已求得的最优解可以更新下界，以进一步缩小解空间。

5. 更新最优解在搜索过程中，不断更新已求得的最优解。

“第一款真正的任意多物理场直接耦合分析软件”COMSOL Multiphysics V4.x操作手册丛书网格剖分用户指南中仿科技公司(CnTech Co., Ltd.)2010年10月前言COMSOL Multiphysics是一款大型的高级数值仿真软件，由瑞典的COMSOL公司开发，广泛应用于各个领域的科学研究以及工程计算，被当今世界科学家誉为“第一款真正的任意多物理场直接耦合分析软件”，适用于模拟科学和工程领域的各种物理过程。

作为一款大型的高级数值仿真软件，COMSOL Multiphysics以有限元法为基础，通过求解偏微分方程（单场）或偏微分方程组（多场）来实现真实物理现象的仿真。

COMSOL Multiphysics以高效的计算性能和杰出的多场直接耦合分析能力实现了任意多物理场的高度精确的数值仿真，在全球领先的数值仿真领域里广泛应用于声学、生物科学、化学反应、电磁学、流体动力学、燃料电池、地球科学、热传导、微系统、微波工程、光学、光子学、多孔介质、量子力学、射频、半导体、结构力学、传动现象、波的传播等领域得到了广泛的应用。

在全球各著名高校，COMSOL Multiphysics已经成为讲授有限元方法以及多物理场耦合分析的标准工具；在全球500强企业中，COMSOL Multiphysics被视作提升核心竞争力，增强创新能力，加速研发的重要工具。

COMSOL Multiphysics多次被NASA技术杂志选为“本年度最佳上榜产品”，NASA技术杂志主编点评到，“当选为NASA科学家所选出的年度最佳CAE产品的优胜者，表明COMSOL Multiphysics是对工程领域最有价值和意义的产品”。

COMSOL Multiphysics 提供大量预定义的物理应用模式，涵盖声学、化工、流体流动、热传导、结构力学、电磁分析等多种物理场，模型中的材料属性、源项、以及边界条件等都可以是常数、任意变量的函数、逻辑表达式、或者直接是一个代表实测数据的插值函数等。

CosmosWorks Designer 2005 Training Manual（网格划分、求解器、提示与技巧）（1）网格划分策略网格划分，更精确地说应该称为离散化，就是将一数学模型转化为有限元模型以准备求解。

作为一种有限元方法，网格划分完成两项任务。

第一，它用一离散的模型替代连续模型。

因此，网格划分将问题简化为一系列有限多个未知域，而这些未知域符合由近似数值技术的求解结果。

第二，它用一组单元各自定义的简单多项式函数来描述我们渴望得到的解 (e.g位移或温度)。

对于使用者来说，网格划分是求解问题必不可少的一步。

许多FEA 初学者急切盼望格划分为全自动过程而几乎不需要自己输入什么。

随着经验的增加，就会意识到这样一个现实：网格划分常常是要求非常苛刻的任务。

商用FEA 软件的发展历史见证了网格划分对FEA 用户透明的诸多尝试，然它并不是一条成功的途径。

而当网格划分过程既简单又自动执行时，它也仍旧不是一个“非手工干涉”而仅靠后台运行的任务。

作为FEA 用户，我们想要有一种可以和网格划分过程交互的方法。

COSMOSWorks 通过将用户从那些纯粹网格细节意义上的问题中解脱出来，找到了良好的平衡点；并使我们在需要时可以控制网格划分。

几何体准备理想情况下，我们用 SolidWorks 的几何体，联入 COSMOSWorks环境。

在这里，我们定义分析和材料的类型，施加载荷与约束，然后为几何体划分网格并得到求解。

这种方法在简单模型下能起作用。

对于更为复杂的几何体，则要求在网格划分前作些准备。

在FEA 的几何体准备过程中，我们从特定制造， CAD 几何体出发，为分析而特地构造几何体。

我们称这个几何体为FEA 几何体。

基于两者的不同要求，我们对CAD 几何体和FEA 几何体作一区别：CAD 几何体FEA 几何体必须包含机械制造所需的所有信息必须可划分网格必须允许创建能正确模拟所关心资料的网格必须允许创建能在合理时间内可求解的网格通常， CAD 几何体不能满足FEA 几何体的要求。

1 综述Netgen 为奥地利科学家Joachim Schoeberl负责编写的格网（曲面和实体）剖分程序。

是格网划分技术中极为先进与完善的，在3D格网划分领域更是具有极大的优势。

2 Netgen5.0.0下载地址相关下载列表页包括：/projects/netgen-mesher/files/netgen-mesher/5.0/Netgen-5.0.0_x64.exe64位安装程序Netgen-5.0.0_Win32.exe32位安装程序netgen-5.0.0.zip netgen源码netgen-5.0.0.tar.gzNetgen-5.0-dev_x64.exeNetgen-5.0-dev_Win32.exe3 Netgen5.0.0安装安装Netgen-5.0.0_Win32.exe和普通软件安装一下，双击Netgen-5.0.0_Win32.exe,点点点就安装上了。

4 源码编译4.1 环境配置和所需三方库文件源码使用的第三方库1) POSIX Thread Library多线程库。

编译netgen必须有，编译nglib必须有2) Tcl库、Tk库、Tix库、Togl库 生成NETGEN的图形用户界面不可缺少的库。

编译netgen必须有3) OpenCascade Library OpenCascade Library。

OpenCascade 是一个非常有名的开源几何图形软件，在这里用来为NETGEN提供STEP、IGES、BREP三种格式的输入功能。

编译netgen时可以不使用该库4.2 三方库文件下载地址MSVC2010_libspthreads-w32_x64.zipTclTkTixTogl_x64.zip包括Tcl库、Tk库、Tix库、Togl库与界面相关的库。

pthreads-w32_Win32.zipTclTkTixTogl_Win32.zipMSVC2008_libspthread-w64.zipTclTkTixTogl-w64.zippthread-w32.zipTclTkTixTogl-w32.zipOpenCascade Library OpenCascade LibraryOpen CASCADE Technology, 3D modeling & numerical simulation注意：请下载6.2或6.3版本，因为NETGEN已经为这两个版本做过测试，用别的可能会出错。