— Variable: *REFERENCE-VERTEX*

The reference vertex.

— Class: <BOUNDARY-CELL>

This cell is only fuzzily defined. Its use is mostly for defining domains by their boundary. The slot midpoint can be useful for the graphical output of the cell, the slot holes contains a list of points lying inside holes. This is intended as help for triangulation programs.

Superclasses: <STANDARD-CELL>

Direct slots:

• DIMENSION: NIL
• MIDPOINT: NIL
• HOLES: NIL

— Class: <CELL>

The basic cell class.

— Class: <DOMAIN>

A <domain> is a special <skeleton>. We call its cells patches, and the properties of a patch carries geometric information. Properties supported up to now are:

• IDENTIFIED: list of identified patches
• EXTENSION: extender
• METRIC: metric tensor function
• VOLUME: volume function

Metric and volume should be functions depending on keyword arguments like :LOCAL and :GLOBAL and allowing arbitrary other keys.

Superclasses: <SKELETON> PROPERTY-MIXIN

Direct slots:

• BOUNDARY: A skeleton containing all boundary patches which is used for classification of patches.
• CLASSIFIERS: A list of functions of two arguments -patch and classifications so far- which are called from the right to classify the patch.

— Class: <HIERARCHICAL-MESH>

Hierarchical-meshes are those meshes which will be used most often, because they remember the refinement history and therefore allow for refinement and coarsening. The slot levels is an array of skeletons containing the cells for different levels.

Superclasses: <MESH>

Direct slots:

• LEVELS: NIL

— Class: <MAPPED-CELL>

A mixin which distinguishes cells which are mapped by a special mapping.

Direct slots:

• MAPPING: NIL

— Class: <MESH>

A <mesh> is a special <skeleton> mapping cells to property lists with properties of the cell. The most important property of a cell is its patch in the domain. Another one could be a list of possibly identified cells. The slot parametric determines which kind of cell mappings are used for approximating the domain. These can be the nonlinear mappings used in the domain definition, but also arbitrary approximations, to those mappings, e.g. isoparametric mappings. The special value NIL means that multilinear mappings are used for all cells outside the boundaries.

Superclasses: <SKELETON>

Direct slots:

• DOMAIN: The domain of the mesh.
• PARAMETRIC: NIL

— Class: <PRODUCT-CELL>

A mixin for simplex-product cells.

— Class: <SIMPLEX>

A mixin for simplicial cells.

— Class: <SKELETON>

A skeleton is a vector of hash-tables containing the cells of a certain dimension as keys. The information stored in the values is different depending on the subclass derived from skeleton.

Direct slots:

• DIM: NIL
• ETABLES: NIL

— Class: <VERTEX>

The vertex class.

Superclasses: <CELL>

Direct slots:

• POSITION: NIL

— Function: BOUNDARY-IDENTIFICATIONS IDENTIFICATIONS

Returns an identification list for the boundaries of the cells in identifications.

— Function: BOX-DOMAIN DIMENSIONS

Generates a box domain for the given dimensions. Here, dimensions is expected to be a list of 2-element lists denoting the interval along the respective axis. The algorithm works by copying the unit cube and modifying the coordinates of the vertices of the copy.

— Function: CELL->CUBE CELL

Transforms a product-cell into a degenerated cube with the same vertices.

— Function: CELL-IDENTIFICATION CELL SKEL

Returns cell's identification in skel or NIL.

— Function: CELLS-OF-DIM SKEL DIM

Returns the cells of skel of dimension dim in form of a list.

— Function: CELLS-OF-HIGHEST-DIM SKEL

Returns the cells of skel of highest dimension in form of a list.

— Function: COMBINE-IDENTIFICATIONS SETS

Reduces identifications to maximally connected sets.

— Function: COMPARE-LEXICOGRAPHICALLY &KEY (FUZZY 1.e-12) DIRECTION

Returns a function which compares two vectors lexicographically.

— Function: COPY-MESH MESH

Copies a mesh. Properties copied are only patch and identification. If necessary, one might add further properties to be copied as a keyword argument.

— Function: CORNERS CELL

Returns a list of corners of the cell, i.e. the global positions of the cell's vertices.

— Function: CUBE-EXTENDER DOMAIN-CUBE DIRECTION

Makes domain-cube —which should be a cube in a domain— extensible in the given direction.

— Function: CUBE-P CELL

Returns T iff CELL is a cube.

— Function: DIAMETER CELL

Returns the diameter of a cell.

— Function: DOMAIN-CHARACTERISTICS DOMAIN

Returns a property list of characteristics. The property curved means that curved patches exist. The property exact is set to t if the domain mappings are exact. Otherwise, only the boundary of the domain should be assumed to be provided in an exact form.

— Macro: DOSKEL (LOOPING-VAR SKEL &KEY (DIRECTION UP) WHERE DIMENSION) &BODY BODY

Loop through a skeleton. If looping-var is an atom, it loops through all cells, otherwise it loops through cells and properties.

— Function: EMBEDDED-DIMENSION SKEL

Dimension of the embedding space for object.

— Function: ENSURE-DOMAIN DOMAIN

If domain is an integer, return the corresponding n-cube-domain, if domain is a domain return it unchanged, otherwise signal an error.

— Function: ENSURE-SIMPLEX-PRODUCT FACTOR-DIMS

Returns the reference product-cell for the given factor dimensions.

— Function: EXTEND MESH &KEY (TEST (CONSTANTLY T))

Extends a mesh on all extensible cells for which test —if provided— yields T.

— Function: FACTOR-SIMPLICES CELL

Returns the factor-simplices.

— Function: FIND-CELLS TEST SKEL &KEY DIMENSION WITH-PROPERTIES WHERE

Returns a list of cells contained in skel and satisfying test.

— Function: FOR-EACH-CELL-OF-HIGHEST-DIMENSION-ON-SURFACE FUNC H-MESH

Calls func for each cell on the hierarchical-mesh surface.

— Function: GET-CELL-PROPERTY CELL SKEL PROPERTY

Returns the value of the property.

— Function: GET-REFINEMENT-RULE CELL ID

Finds the refinement rule for cell defined by the id. This id can be a number (position of the rule, T (meaning 0), or some symbol which is contained in the names of some rule. Two values are returned: the rule and its position in the refinement-rule vector.

— Function: GLOBAL->EMBEDDED-LOCAL CELL GLOBAL-POS

This computes a local coordinate which solves the Ausgleichsproblem of mapping to a point as near as possible to global-pos. It involves more computational work than global->local. As a second value, the distance to global-pos is returned.

— Function: GLOBAL->LOCAL VTX GLOBAL-POS

Mainly useful for finite element evaluation: from the local position, the value of a fe function can be obtained by interpolation. This is done by a Newton iteration, which converges in one step for linear mappings.

— Function: HIERARCHICALLY-ORDERED-CELLS H-MESH &KEY LEVEL

Sorts the cells up to the given level (defaulting to the last level) hierarchically for use in something similar to the nested disection method. Returns a list of the sorted cells.

— Function: IDENTIFIED-CELLS CELL SKEL

Returns a list of cells in skel which are identified with cell.

— Function: IDENTIFIED-P CELL SKEL

Returns cell's identification in skel or NIL.

— Function: IDENTIFY IDENTIFIED-CELLS SKEL

Identifies all cells in identified-cells within skel.

— Function: IDENTIFY-UNIT-CELL-FACES SKEL &KEY (INDICES ALL)

This routines identifies boundary cells in skel which correspond to boundary cells in the unit cube. Warning: exact arithmetic is used to recognize identified cells. This should work for skeletons derived from the unit cell, but may create problems in other situations.

— Function: INNER-REFCELL-CHILDREN REFCELL RULE

Returns the children of refcell.

— Function: INSERT-CELL! SKEL CELL

Inserts a cell and its boundary into a skeleton.

— Function: INSERT-CELL-FROM-CORNERS MESH CORNERS->CELL CELL-CLASS CORNERS PROPERTIES &KEY (CREATE-SUBCELLS T)

Creates a cell of type cell-class with corners given by corners. corners->cell has to be an equalp hash-table mapping corners to the corresponding cell. It is updated by this function.

— Function: INSERT-CELLS! SKEL CELLS

Inserts a list of cells into a skeleton.

— Function: INSIDE-CELL? CELL POS

Checks if global-pos is inside the interior of the cell. It calls coordinates-inside? which is defined for every cell class.

— Function: ITERATE-IDENTIFICATIONS INITIAL-IDENTIFICATIONS

Generates all identifications of the skeleton from the identifications of some higher-dimensional cells.

— Function: KEY-IS-SUBCELL-P KEY1 KEY2

Checks if key1 occurs as subcell of key2. The keys can be either cells or identifications.

— Function: L-DOMAIN DIM

Creates an L-domain by cutting out a small cube of the uniform refinement of the unit cube.

— Function: L2DG SIMPLEX LOCAL-POS

Computes the gradient for a multilinear interpolation from the vertices.

— Function: L2G VTX LOCAL-POS

Computes the global position by interpolation from the vertices.

— Function: LEVEL-OF-CELL CELL H-MESH

Returns the level of cell in the hirearchical mesh h-mesh.

— Function: LOCAL->DGLOBAL VTX LOCAL-POS

local->Dglobal checks if a mapping is given for the cell. If yes, then the gradient of this mapping is evaluated (if available). If no, then the function l2Dg is called which gives the gradient for a multilinear interpolation from the cell's corners.

— Function: LOCAL->GLOBAL CELL LOCAL-POS

local->global checks if a mapping is given for the cell. If yes, then this mapping is evaluated. If no, then the function l2g is called which should do a multilinear interpolation from the cell's corners.

— Function: LOCAL-COORDINATES-OF-MIDPOINT CELL

Returns the local coordinates of the cell midpoint.

— Function: MAKE-CELL-FROM-CORNERS CELL-CLASS CORNERS

Creates a cell of class CELL-CLASS having the given CORNERS.

— Function: MAKE-CELL-FROM-VERTICES CELL-CLASS VERTICES

Creates a cell of class CELL-CLASS having the given VERTICES.

— Function: MAKE-HIERARCHICAL-MESH-FROM-DOMAIN DOMAIN &KEY PARAMETRIC &ALLOW-OTHER-KEYS

Construct a hierarchical-mesh from a domain.

— Function: MAKE-LINE FROM-VTX TO-VTX &KEY (CHECK T) MAPPING

Creates a line given its endpoints.

— Function: MAKE-MESH-FROM-DOMAIN DOMAIN &KEY PARAMETRIC &ALLOW-OTHER-KEYS

Transforms a domain which is specified sufficiently well into a mesh.

— Function: MAKE-SIMPLEX BOUNDARY &KEY (CHECK T) MAPPING

Short form of creating a simplex given its boundary. An alternative is creating it from its vertices, see the functions MAKE-CELL-FROM-VERTICES and INSERT-CELL-FROM-CORNERS.

— Function: MAKE-VERTEX POSITION &OPTIONAL MAPPING

General vertex constructor.

— Function: MAPPED-CELL-CLASS CLASS &OPTIONAL DISTORTED

Constructs a cell class with <mapped-cell> mixin.

— Function: MEMBER-OF-SKELETON? CELL SKEL

Returns T if cell is in skel, NIL otherwise.

— Function: MESHSIZE MESH

Computes a meshsize. Please refer to the method documentations for the exact definition.

— Function: MIDPOINT <BOUNDARY-CELL>

Returns cell midpoint in global coordinates.

— Function: N-BALL-DOMAIN DIM

Generates an n-dimensional ball domain with 2^n simplex patches.

— Function: N-CELL-DOMAIN DIM

Generates an n-dimensional cell domain which is a n-dimensional unit cube with its opposite sides identified.

— Function: N-CUBE DIM

Returns the reference cube of dimension dim.

— Function: N-SIMPLEX DIM

Returns the reference simplex of the given dimension.

— Function: NR-OF-CELLS SKEL &OPTIONAL DIMENSION

Returns number of cells in a skeleton.

— Function: NR-OF-SIDES CELL

Returns the number of boundary faces.

— Function: NR-OF-SUBCELLS CELL

Returns the number of subcells.

— Function: NR-OF-VERTICES CELL

Returns the number of vertices.

— Function: ORIGIN CELL

Returns cell origin in global coordinates.

— Function: PATCH-CLASSIFICATION PATCH DOMAIN

Returns a list of classifications for patch in domain.

— Function: REFCELL-CHILDREN REFCELL RULE

Returns the children for refcell and subcells.

— Function: REFCELL-REFINEMENT-SKELETON REFCELL &OPTIONAL (LEVEL 1) (RULE 0) REINIT

Returns an LEVEL times refined skeleton of REFCELL. It is partially memoized, see the documentation of *REFCELL-REFINEMENT-MEMOIZE-DEPTH*.

— Function: REFERENCE-CELL-P CELL

Tests if a cell is a reference cell.

— Function: REFINE H-MESH &KEY (INDICATOR (CONSTANTLY T))

Refines skel either locally or globally depending on the indicator.

— Function: REFINE-INFO CELL

Returns refinement information for the cell.

— Function: REFINEMENT CELL SKELETON

Returns the refinement of cell in skeleton as two values: the rule and the children.

— Function: REFINEMENT-INTERFACE H-MESH &KEY LEVEL

Returns the refined boundary subcells of unrefined cells in a skeleton. Those cells are found as all refined cells which are not part of the domain boundary. At the moment, this is a global operation. Later on, it should probably be localized.

— Function: REFINEMENT-RULE CELL SKEL

Returns the refinement rule of cell in skel.

— Class: REFINEMENT-RULE

Rule for refining reference cells. Those rules are stored in the refine-info slot of the cell class.

Direct slots:

• NAMES: Names identifying the rule.
• REFCELL: Reference cell for this refinement rule.
• BOUNDARY-REFINEMENT-RULES: Refinement rules for the sides required by this rule.
• REFINEMENT-INFO: Vector of refinement information for the children.

— Function: REPRESENTATIVE OBJ

Returns a representative for this object.

— Function: SIMPLEX-PRODUCT-DOMAIN DIMS

Generates a product-cell domain for the given factor dimensions.

— Function: SKEL-ADD! SKEL-1 SKEL-2 &KEY (OVERRIDE NIL) ACTIVE-SKEL-1

Adds skel-2 to skel-1 destructively for skel-1. Overlaying cells are identified. override is a list of properties which are copied from skel-2 on the overlap. active-skel-1 is used for hierarchical-meshes for selecting a level to which skel-2 is added. This function returns three values: the first is skel-1, the second is skel-2, the third is a hash-table mapping overlapping cells from skel-2 to their counterpart in skel-1.

— Function: SKEL-FOR-EACH FUNC SKEL &KEY DIRECTION DIMENSION WHERE WITH-PROPERTIES

Loops through a skeleton applying func. When direction is :down then loops with dimension of the cells decreasing, otherwise increasing.

— Function: SKEL-REF SKEL CELL

Returns the properties of cell in skel.

— Function: SKELETON CELL

Returns a skeleton for the given cell or the given cells.

— Function: SKELETON-WITHOUT-CELL SKEL CELL-TO-REMOVE

Removes a cell from a skeleton such that the rest remains a skeleton. Warning: does not handle identifications yet.

— Function: SORT-LEXICOGRAPHICALLY ELIST &KEY (FUZZY 1.e-12)

Sorts a cell list lexicographically by the coordinates of their midpoint.

— Function: SPECIAL-MESH-ON-BOX-DOMAIN DOMAIN PATCH->MESH-SIZES

Creates a uniform mesh consisting of N_1 x ... x N_dim cells on a box domain.

— Function: STANDARD-EXTENDER ORIGINAL-CELL REPLACEMENT

Extension function replacing an original-cell with a replacement.

— Function: STRUCTURED-SKELETON N H &KEY CORNERS->CELL

Create a uniform box skeleton consisting of N_1 x ... x N_dim cubes of dimensions h_1 x ... x h_dim.

— Function: SUBCELL-CHILDREN CELL SKELETON

Returns a vector of all children of the subcells of cell in skeleton.

— Function: SUBCELLS CELL

Returns a vector containing all subcells of a given cell. The code is special to each class and often automatically generated by generate-subcell-access-code.

— Function: SURFACE-CELLS-OF-DIM H-MESH DIM

This function returns the surface cells of a locally refined hierarchical-mesh structure.

— Function: SURFACE-CELLS-OF-HIGHEST-DIM H-MESH

This function returns the surface cells of highest dimension of a locally refined hierarchical-mesh structure.

— Function: TRIANGULATE DOMAIN &REST ARGS &KEY PARAMETRIC &ALLOW-OTHER-KEYS

Triangulate domain by successively building a mesh on the domain skeleton starting from the 0-dimensional patches.

— Function: TRIANGULIZE MESH

Transforms a product-cell mesh into a simplex mesh.

— Function: UNIFORM-MESH-ON-BOX-DOMAIN DOMAIN N

Creates a uniform mesh consisting of N_1 x ... x N_dim cells on a box domain.

— Function: UNIFORMLY-REFINED-MESH DOMAIN N &KEY PARAMETRIC

Generates a mesh by refining the domain partition uniformly.

— Function: VERTICES CELL

Returns a list of all vertices of the cell.