## B. Finite-element Mesh generator: MOSAIQUE

MOSAIQUE is a preprocessor of the ATILA code. It is an automatic finite-element mesh generatorprogram that helps in building complex meshes. From the description of themesh using "super-elements", and instructions on how to divide them, MOSAIQUE generates all the necessary element and node datafor ATILA. A super-element can be a one-dimensional,two-dimensional or three-dimensional element. It is possible to assigndifferent types of elements (triangle, quadrilateral, hexahedron, and prism) toa super-element.

Data for MOSAIQUE is provided by aninput file, generally of extension MOS, written by theuser. This file has the same characteristics as an ATILA data file: same header type, same entries, equivalent boundary conditions, thusthe user may refer to the Sections III.D and III.E for entries reference. Compared to an ATILA data file, the ELEMENTS entrydefines "super-elements" instead of regular elements. A super-element entry isa regular element entry completed with element's division information. So,almost all entries are duplicated to the resulting output file, except the NODES and ELEMENTS entries thatreflect the elements divisions, as new nodes and elements are created. Whennecessary, node numbers in entries are changed to reflect the node generation.The output file is named with an .ATI extension and isdirectly usable by the graphics pre-processor MDES orby the solver GA. A listing file (.LST)is also output, which enables the user to rapidly check and validate thegeneration.

The entries used for the automatic mesh generation that arespecific to MOSAIQUE are described below:

### 1. ELEMENTS

NAMEL1 MATER1 NUMGEO1
N11 N12 N13 ... N1p
-1 ITYP NN NM NL NK IBIAIS
[ XP1 XP1 ... XPNN ] if IBIAIS = 1
[ YP1 YP1 ... YPNM ] if IBIAIS = 1 and NM not null
[ ZP1 ZP1 ... ZPNL ] if IBIAIS = 1 and NL not null
[ TP1 TP2 ... TPNK ] if IBIAIS = 1 and NK not null
BLANK line
...
...
NAMELN MATERN NUMGEON
N11 N12 N13 ... N1q
-1 ITYP NN NM NL NK IBIAIS
[ XP1... XPNN ] if IBIAIS = 1
[ YP1... YPNM ] if IBIAIS = 1 and NM not null
[ ZP1... ZPNL ] if IBIAIS = 1 and NL not null
[ TP1 TP2 ... TPNK ] if IBIAIS = 1 and NK not null
BLANK line
BLANK line

This entry enables the user to introduce the super-elementtopology in accordance with the rules of Chapter IV. The -1 after the element topology triggers the super-element supplemental information. If one or several super-element sides are straight, the middle nodes can be omitted. The maximal value of NN, NM, NL, or NK is 40.

MATER and NUMGEO are given or not. They depend upon theelement type used and they are specified in the element directives. They referto the MATERIALS and GEOMETRY entries (or GEOMETRY POLARIZATION).

In generation mode, the line following the topology descriptionstarts with -1 and gives the splitting rules. If there is no such line, thesuper-element is considered as a simple element. On this line, ITYP refers toa given super-element frame; NN, NM, NL and NK give the number of generatedelements along the natural axes of the super-element; and IBIAIS is a controlinteger that must be equal to 0 if the splitting along each axis is regular and1 if the splitting along at least one axis is irregular. If IBIAIS is equal to0, the four lines that are devoted to the description of the splitting must notappear. If IBIAIS is equal to 1, these one to four lines must be provided, evenif one or two of them correspond to a regular splitting.

XP1, XP2,... XPI,...,XPNN are integers defining thesplitting along the natural Ox axis of the super-element. The correspondingside is divided into NN different parts, the lengths of which are,respectively, equal to: XP1/XL, XP2/XL,..., XPI/XL,..., XPNN/XL, withXL=XP1+XP2+...+XPNN. XPI is always an integer.

A blank line indicates the end of a super-element datainput. A blank line is necessary to terminate the ELEMENT entries.

### 2. Merge
XMERGE

This entry allows the user to change the merge point valuethat is, for each element, the ratio of the tolerance circle diameter to the shortestdistance between two nodes. Default is XMERGE = 1.E-01

### 3. Nogen

This entry suppresses automatic mesh generation by disablingthe division of super-elements. This is useful for visualizing thesuper-elements. Mid-side nodes will be generated for each super-element. Thus,the node numbering will be changed from the element entry.

### 4. SORTfluid
NCA

This entry enables the user to organize the fluid-structuremeshes so as to place the radiating nodes at the end of the array.

NCA is the node number of the acoustic center. Bydefault, the acoustic center is at the origin of the global system. Therefore,this entry is not required.

### 5. Super-Element Types (Available Values of ITYP)

In the following library description, the super-elementshape is represented by a thick line, while the generated elements arerepresented by a thin line.

#### a) ITYP = 1: Linear 1D elements

Two-node linear element generation (element SPRI02E). Control integers NM, NL, NK are null.

#### b) ITYP = 2: Quadratic 1D elements

Three-node quadratic element generation (element LINE03R). Super-node 3 is needed only if the super-element is to be curved. Controlintegers NM, NL, NK are null. This super-element is used to generate dampingelements.

#### c) ITYP = 3: 6-node interface elements

Six-node fluid-structure interface element generation(element LINE06I). Only the solid super-nodes are specified, that is, only two or three nodes must be given; all generatednodes are doubled to realize the fluid-structure interface. Super-node 3 isneeded only if the super-element is to be curved. Control integers NM, NL, NKare null. The orientation of the element is automatically inverted by the MOSAIQUE program to get the orientation required in the ATILA code.

#### d) ITYP = 4: Quadrilateral elements in a quadrilateral super-element

This super-element enables the user to generate 8-nodeelements (QUAD-type geometry). The topology is always read on 8 nodes. Mid-side super-nodes (5-8) are required only for those super-element edges thatare to be curved. Special attention must be given to the data entry format inthis case; one or more zero-valued place holders may be required (see Chapter IV). NN, NM, respectively, define the number of elements on the 1-2, 1-3 sides. Control integers NL, NK are null.

#### e) ITYP = 5: Triangular elements in a quadrilateral super-element

This super-element enables the user to generate 6-nodeelements. Due to the super-element shape, the element entry in the .MOS file must refer to an element of QUAD-type geometry. Thetopology is always read on 20 nodes. Mid-side super-nodes (5-8) are requiredonly for those super-element edges that are to be curved. Special attentionmust be given to the data entry format in this case; one or more zero-valuedplace holders may be required (see Chapter IV). NN, NM, respectively, define the number of elements on the 1-2, 1-3 sides. Control integers NL, NK are null.

#### f) ITYP = 6: Triangular elements in a triangular super-element

This super-element enables the user to generate 6-nodeelements (TRIA-type geometry). The topology is always read on 6 nodes. Mid-side super-nodes (4-6) are required only for those super-element edges thatare to be curved. Special attention must be given to the data entry format inthis case; one or more zero-valued place holders may be required (see Chapter IV). NN, NM, respectively, define the number of elements on sides 1-2 and 1-3. NM must be equal to NN. Control integers NL, NK are null. An irregular splitting can be applied for this super-element, provided thatsplitting along the sides 1-2 and 1-3 are similar but in reverse order, i.e.: 1 2 4 and 4 2 1. Any other irregular splitting willlead to unpredictable meshes.

#### g) ITYP = 7: Triangular and quadrilateral elements in a triangularsuper-element

This super-element enables the user to generate 6-node(TRIA-type geometry) and 8-node elements (QUAD-type geometry). The topology isalways read on 6 nodes. Mid-side super-nodes (4-6) are required only for thosesuper-element edges that are to be curved. Special attention must be given tothe data entry format in this case; one or more zero-valued place holders maybe required (see Chapter IV). NN, NM define the number of elements on side 1-2 and 1-3, respectively. If IBIAIS=1, the spacing has to be indicated on the following line.

#### h) ITYP = 8: 3-D triangular interface element

This super-element is similar to element type 7. Allgenerated nodes are doubled to build the fluid-structure interface (elementTRIA12I).

#### i) ITYP = 9: 3-D quadrilateral interface elements

This super-element is similar to element type 4. Allgenerated nodes are doubled to build the fluid-structure interface (elementQUAD16I).

#### j) ITYP = 10: Hexahedron elements in an hexahedron

This super-element enables the user to generate 20-nodeelements (HEXA-type geometry). The topology is always read on 20 nodes. Mid-side super-nodes are only required for those super-element edges which areto be curved. Special attention must be given to the data entry format in thiscase; one or more zero-valued place holders may be required (see Chapter IV). NN is the number of elements on side 1-2, NM is the number of elements on side 1-3 and NL is the number of elements on side 1-5. If IBIAIS=1, the spacing has to be indicated first for side 1-2, then for side 1-3 and finally for side 1-5.

#### k) ITYP = 11: Prismatic elements in a triangular base prism

This super-element enables the user to generate 15-nodeprismatic elements (PRIS-type geometry). The topology is always read on 15nodes. Mid-side super-nodes are required only for those super-element edgesthat are to be curved. Special attention must be given to the data entryformat in this case; one or more zero-valued place holders may be required (seeChapter IV). NN is the number of elements on side 1-2, NM is the number of elements on side 1-3 and NL is the number of elements on side 1-4. An irregular spacing can be applied for this super-element, provided that spacing along the sides 1-2 and 1-3 are similar but in reverse order, i.e.: 1 2 4 and 4 2 1. Any otherirregular spacing will lead to unpredictable meshes.

#### l) ITYP = 12: Triangular base prismatic elements in a hexahedron

This super-element enables the user to generate 15-nodeelements in a cube. Due to the super-element shape, the element entry in thefile JOB.MOS must refer to an element of HEXA-typegeometry. The topology is always read on 20 nodes. Mid-side super-nodes arerequired only for those super-element edges that are to be curved. Specialattention must be given to the data entry format in this case; one or morezero-valued place holders may be required (see Chapter IV). NN is the number of elements on side 1-2, NM is the number of elements on side 1-3 and NL is the number of elements on side 1-5. If IBIAIS=1, the spacing has to be indicated first for side 1-2, then for side 1-3 and finally for side 1-5.

#### m) ITYP = 13: Hexahedron and pyramid elements in a pyramid

This super-element enables the user to generate 13-node(PYRA-type geometry) and 20-node (HEXA-type geometry) elements. The topologyis always read on 13 nodes. Mid-side super-nodes are required only for thosesuper-element edges that are to be curved. Special attention must be given to thedata entry format in this case; one or more zero-valued place holders may berequired (see Chapter IV). NN is the number of elements on side 1-2, NM is the number of elements on side 1-3 and NL is the number of elements on side 1-5. If IBIAIS=1, the spacing has to be indicated first for side 1-2, then for side 1-3 and finally for side 1-5.

#### n) ITYP = 14: Shell element

This super-element enables the user to generate 3-node shellelements (element SHEL03E). It is used in the same way as the type 2super-element. Only the node numbering is different in the output file.

#### o) ITYP = 15: Triangular base prismatic elements in a prism

This super-element enables the user to generate 15-node(PRIS-type geometry) and 20-node (HEXA-type geometry) elements. The topologyis always read on 15 nodes. Mid-side super-nodes are required only for thosesuper-element edges that are to be curved. Special attention must be given tothe data entry format in this case; one or more zero-valued place holders maybe required (see Chapter IV). NN is the number of elements on side 1-2, NM is the number of elements on side 1-3 and NL is the number of elements on side 1-4. If IBIAIS=1, the spacing has to be indicated first for side 1-2, then for side 1-3 and finally for side 1-4.

#### p) ITYP = 16: Triangular and quadrilateral elements in a quadrilateralsuper-element

This super-element enables the user to generate 6- and8-node elements. (TRIA- and QUAD-type geometries). Due to the super-elementshape, the element entry in the .MOS file must refer toan element of QUAD-type geometry. The topology is always read on 8 nodes. Mid-side super-nodes are required only for those super-element edges that areto be curved. Special attention must be given to the data entry format in thiscase; one or more zero-valued place holders may be required (see Chapter IV). NN, NM, NL, NK respectively define the number of elements on the 1-2, 1-3, 3-4 and 2-4. If IBIAIS=1, the spacing has to be indicated in this order.

#### q) ITYP= 17: Triangular and quadrilateral thin-film magnetostrictive elements in aquadrilateral super-element

This super-element enables the user to generate 12- and16-node thin film magnetostrictive elements. (TRIA- and QUAD-type geometry). It is equivalent to the type 16 super-element, except that it generates twiceas many nodes to build the double layer needed by such elements.

### 6. Technical Information: Node and Element Generation Method

ATILA node and element generation isbased upon the division, or splitting of triangular or quadrilateralsuper-elements in 2-D, and cubic or prismatic super-elements in 3-D. Thesuper-element sides or edges are defined by two or three points. Linear orquadratic interpolation is used for the generation. Attention must be paid tothe super-element orientation that defines the orientation of the generatedelements.

For ATILA node coordinategeneration, each super-element, defined in the global Oxyz axis system is firsttransformed into a local O'rst system, in which it is represented by a regularfigure. Inside this reduced super-element, the different node coordinates aregenerated in the local system, according to the number of elements to generate,their spacing (regular or irregular) on each side, and their type. Thegenerated node coordinates are then transformed into global coordinates usingthe same types of shape functions as those of the ATILAcode. After all nodes and all elements generation, the redundant nodes areeliminated (except the fluid-structure interface nodes). Two nodes are assumedidentical if they are inside the same tolerance circle. In this case, the lastone is eliminated and the node numbering is readjusted in the topology.