Requirement :

The MESH MUST BE CONFORM to obtain CONTINUOUS solutions <=>

The INTERSECTION of 2 FINITE ELEMENTS may be one FACE, one EDGE, one VERTEX or EMPTY, but NOT only a PART of them.

The Mefisto TOP-DOWN ANALYSIS to construct the mesh:

=> a list of Volumes, Surfaces, Lines, Points and the algorithms to construct them.

The ANALYSIS has been done, on a rough paper, NOT in front of a computer, from Volumes to Surfaces, to Lines, to Points. This is a TOP-DOWN analysis, from the whole complex object to its simplest geometric entities.

The Mefisto BOTTOM-UP CONSTRUCTION of the mesh:

The construction is now done in front of a computer by typing commands.

- Execution of chosen algorithms to construct the POINTS;
- Execution of chosen algorithms to construct the LINES;
- Execution of chosen algorithms to construct the SURFACES;
- Execution of chosen algorithms to construct the VOLUMES;

A demonstration to construct a mesh of a crank shaft

The Mefisto-MESHER data file of test/demo: The POINTS

The Mefisto-MESHER data file of test/demo: The LINES

The Mefisto-MESHER data file of test/demo: The SURFACES

The Mefisto-MESHER data file of test/demo: The VOLUMES 1

The Mefisto-MESHER data file of test/demo: The VOLUMES 2

The Mefisto-MESHER data file of test/demo: The OBJECT

The Mefisto-HEATER data file of test/demo: The HEAT PROBLEM

The Mefisto-ELASTICER data file of test/demo: The ELASTICITY PROBLEM

The Mefisto-ELASTICER data file of test/demo: The EIGENFREQUENCY PROBLEM

Remark:

Mefisto stores the coordinates of the finite element vertices and the vertice number of each finite element.

But, in fact, if the object has "curved" lines or surfaces and if the mesh algorithm permits it, the tangents of edges of finite elements are also stored to take in account the P3-Hermite curve, the reduced Hsieh-Clough-Tocher triangle and the reduced de Veubeke-Sander quadrangle, in order to construct G1-continuous meshes on surfaces.

G1 Finite Elements

P3 HERMITE's segment and triangle

P3 HERMITE's segment example

HSIEH CLOUGH TOCHER triangle in 2D

HSIEH CLOUGH TOCHER triangle in 3D

HSIEH CLOUGH TOCHER triangle 2D example

HSIEH CLOUGH TOCHER triangle 3D example 1

HSIEH CLOUGH TOCHER triangle 3D example 2

de VEUBEKE SANDER quadrangle in 2D

de VEUBEKE SANDER quadrangle 2D example 1

de VEUBEKE SANDER quadrangle 2D example 2

de VEUBEKE SANDER quadrangle 2D example 3

de VEUBEKE SANDER quadrangle 3D example 1

de VEUBEKE SANDER quadrangle 3D example 2

The Mefisto ALGORITHMS to construct a mesh of an object:

- A mesh may be structured or unstructured .
- A structured mesh is defined by an implicit numbering of vertices from some integers.
- An unstructured mesh is defined by the list of numbers of each vertex of each finite element.
- Each point, line, surface, volume, object, is named up to 24 characters.

- A volume (resp. a surface, a line, a point) may be the result of a mapping of an other volume (resp. surface, line, point). $MEFISTO/td/da/a_transfo__definition
- A volume (resp. a surface, a line, a point) may be a collection of others volumes (resp. surfaces, lines, points), with identification of the common vertices.
- The edge length of finite elements may be adapted by the using of a function
Edge_Length(x,y,z) given by the user.

- A POINT may be : Cf $MEFISTO/td/da/a_point__definition
- picked with a mouse from a drawing of 2d or 3d space;
- defined by typing of the coordinates: x y z or cylindrical r θ z or spherical r θ φ ;

- A LINE may be : Cf $MEFISTO/td/da/a_ligne__definition
- the list of 3 coordinates of vertices of the edges;
- the list of names of points from first to last vertex;
- a segment of a structured straight forward line defined by its extremities;
- a circle arch defined by 3 points;
- a circle defined by either 3 points,or its centre and a point of the plane;
- a B-spline curve defined by interpolation or control points, polynomial or rational, opened or closed;
- a list of extracted edges from an other line or surface;
- an intersection of 2 cut cones;
- an intersection of 2 cylinders;
- a line obtained by a mapping of an other line;
- an union of lines;
- ...

- A SURFACE may be : Cf $MEFISTO/td/da/a_surface__definition
- a structured straight triangle subdivided in n
^{2}sub-triangles; - a structured straight rectangle subdivided in mxn sub-squares with an edge lenght of 1;
- a structured transfinite triangle limited by 3 meshed lines which constitute its boundary;
Cf Transfinite interpolations

- a structured transfinite quadrangle limited by 4 meshed lines which constitute its boundary;
Cf Transfinite interpolations

- a structured elliptic quadrangle limited by 4 meshed lines which constitute its boundary;
Cf Elliptic quadrangulation

- a plane polygon limited by lines and triangulated by 2 Delaunay-Voronoi's algorithms on a grid of regular triangles;
Cf Delaunay-Voronoi 2d triangulations

- a regular triangulation of a sphere;
- a surface B-spline interpolated or controlled by points;
- an extruded surface from a line;
- a rotated surface from a line around an axis;
- an extracted surface from a structured hexahedron by reduction of its indexes;
- an extracted surface from a volume and a function to define a criterion of extraction;
- an intersection, addition or substraction of 2 plane triangulations;
- a triangulation of a quadrangulation;
- an improved triangulation;
- a subdivided triangulation and/or quadrangulation, each initial finite element subdivided into 4 sub elements of the same type;
- a list of extracted triangles or quadrangles from an other surface or volume
- a surface, mapping of an other surface;
- an union of surfaces;
- ...

- a structured straight triangle subdivided in n
- A VOLUME may be : Cf $MEFISTO/td/da/a_volume__definition
- a structured hexahedron subdivided in n
^{3}sub-hexahedra by an elliptic method and defined by its 6 structured quadrangulated faces; Cf Elliptic hexahedrization by the Winslow's algorithm with corrector of the mesh quality near the boundary;

- a structured tetrahedron (resp. pentahedron or hexahedron) subdivided in n
^{3}sub-tetrahedra (resp. pentahedra or hexahedra) by a transfinite algorithm and defined by its structured triangulated faces; Cf Transfinite interpolations

- a structured straight tetrahedron (resp. pentahedron or hexahedron) subdivided in n
^{3}regular sub-tetrahedra (resp. pentahedra or hexahedra) by a regular subdivision; - a cone or an half cone subdivided with regularity;
- a cylinder or an half cylinder subdivided with regularity;
- a tetrahedrization obtained by 2 Delaunay-Voronoi's algorithms from the triangulated boundary of volumes;
Cf Delaunay-Voronoi 3d tetrahedrizations

- a tetrahedrization of an initial pentahedrization and/or hexahedrization;
- an extruded volume from an initial meshed surface;
- a subdivided tetrahedrization and/or pentahedrization and/or hexahedrization, each initial finite element subdivided into 8 sub elements of the same type;
- a list of extracted tetrahedra or pentahedra or hexahedra from an other volume;
- a volume, mapping of an other volume;
- an union of volumes;
- ...

- a structured hexahedron subdivided in n
- An OBJECT is :

- a set of points, lines, surfaces, volumes and objects.
- each element of the object is given in order to associate on it a physical characteristic
or a boundary condition or an initial condition.

Cf $MEFISTO/td/da/a_objet__definition

- a choice of a finite element type through the degree of polynomials (1 or 2 or 3)

Cf the topology in $MEFISTO/td/da/a_objet__topologie

Cf the choice of FE types in $MEFISTO/td/da/a___npef

Cf the coordinates of FE nodes in $MEFISTO/td/da/a___xyznoeud

Cf the coordinates of FE points in $MEFISTO/td/da/a___xyzpoint

This file is created in the mefistox project directory to be used in other softwares.

The fortran subroutine $MEFISTO/mail/xyzsef.f constructs it.

This file contains the 3 coordinates of FE vertices and the number of vertices of every FE of a PLSV.

This file is created in the mefistox project directory to be used in other softwares.

The fortran subroutine $MEFISTO/mail/xyznpe.f constructs it.

This file contains the 3 coordinates of FE nodes and the number of nodes of every FE of every FE types of the interpolation of an OBJECT.

A fortran subroutine $MEFISTO/mail/ReadObjMesh.f recuperates the data.

Department of Mathematics, Tsing Hua University TAIWAN

- How to construct a mesh and solve an heat transfer problem with Mefisto
- An example to construct a mesh of a volume

Some meshes obtained with Mefisto-Mesher:

A tetrahedrization of a 3-axis piece

The rainbow colors show the quality from blue of the best quality 1 to red of the worst quality 0.

The Mefisto-MESHER data file of test/triax

A surface mesh of a prototype car

The rough draft 1 of algorithms of the test/bm

The rough draft 2 of algorithms of the test/bm

The Mefisto-MESHER data file of the test/bm

A surface mesh of a prototype car with the function Taille_Ideale(x,y,z) or Edge_Length(x,y,z)

The Mefisto-MESHER data file of the test/bm

A triangulation of the YF22 fighter

The rough draft 1 of algorithms of the test/yf

The rough draft 2 of algorithms of the test/yf

The points, lines and surfaces of the test/yf

The Mefisto-MESHER data file of the test/yf

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