fem

The EasyFEA/fem/ module in EasyFEA provides essential tools for creating and managing finite element meshes, which are crucial for numerical simulations using the Finite Element Method (FEM).

What is a mesh in EasyFEA ?

A Mesh object in EasyFEA represents a collection of ElemType used to define the geometry and structure for finite element analysis. It contains multiple _GroupElem instances, which are groups of ElemType that collectively define the spatial discretization of the domain for numerical simulations.

For example, a HEXA8 mesh includes the following element types:

• POINT (0D element)

• SEG2 (1D element)

• QUAD4 (2D element)

• HEXA8 (3D element)

All implemented element types, along with their corresponding shape functions and derivatives, are defined in the EasyFEA/fem/elems/ directory. The Gauss point quadratures are implemented in the EasyFEA/fem/_gauss.py module.

Creating or importing a Mesh

To construct a Mesh using the Mesher, you must first create _Geom objects (see geoms for some examples). The Mesher class serves as an interface to Gmsh, a powerful meshing tool, and includes the following primary functions for mesh generation:

• Mesh_2D(): Generates a 2D mesh.

• Mesh_Extrude(): Creates a mesh by extruding a 2D shape.

• Mesh_Revolve(): Generates a mesh by revolving a 2D shape around an axis.

• Mesh_Import_part(): Imports a cad (e.g. .stp) part to create a mesh.

• Mesh_Import_mesh(): Imports an existing gmsh mesh. EasyFEA is also linked to meshio and can be used threw the

• Medit_to_EasyFEA(): Imports medit mesh.

• Gmsh_to_EasyFEA(): Imports gmsh mesh.

• PyVista_to_EasyFEA(): Imports pyvista mesh (UnstructuredGrid or MultiBlock).

• Ensight_to_EasyFEA(): Imports ensight mesh.

Several examples are available in Meshes.

Detailed fem API

class EasyFEA.fem.BiLinearForm(form)

Bases: _Form

Assemble(field)

Assemble de form with the field.

Parameters:

field (Field) – field

Returns:

the assembled (Nn * dof_n, Nn * dof_n) sparse matrix.

Return type:

csr_matrix

Integrate_e(field)

Integrates de form with the field on elements.

Parameters:

field (Field) – field

Returns:

the integrated (Ne, …) numpy array

Return type:

np.ndarray

class EasyFEA.fem.BoundaryCondition(problemType, nodes, dofs, unknowns, dofsValues, description)

Bases: object

static Get_dofs(problemType, list_Bc_Condition)

Returns the degrees of freedom for the problem type.

Parameters:

• problemType (str) – Problem type.

• list_Bc_Condition (list[BoundaryCondition]) – List of boundary conditions.

Returns:

degrees of freedom.

Return type:

_types.IntArray

static Get_dofs_nodes(availableUnknowns, nodes, unknowns)

Retrieves degrees of freedom (dofs) associated with the nodes.

Parameters:

• availableUnknowns (list[str]) – Available dofs as a list of strings. Must be a unique string list.

• nodes (_types.IntArray) – Nodes for which dofs are calculated.

• unknowns (list[str]) – unknowns.

Returns:

degrees of freedom.

Return type:

_types.IntArray

static Get_nBc(problemType, list_Bc_Condition)

Returns the number of conditions for the problem type.

Parameters:

• problemType (str) – Problem type.

• list_Bc_Condition (list[BoundaryCondition]) – List of boundary conditions.

Returns:

Number of boundary conditions (nBc).

Return type:

int

static Get_values(problemType, list_Bc_Condition)

Returns the dofs values for problem type.

Parameters:

• problemType (str) – Problem type.

• list_Bc_Condition (list[BoundaryCondition]) – List of boundary condition.

Returns:

dofs values.

Return type:

_types.FloatArray

property dofs: ndarray[tuple[Any, ...], dtype[IntType]]

degrees of freedom associated with the nodes and unknowns

property dofsValues: ndarray[tuple[Any, ...], dtype[floating]]

values applied

property nodes: ndarray[tuple[Any, ...], dtype[IntType]]

nodes on which the condition is applied

property problemType: str

type of problem

property unknowns: list[str]

dofs unknowns

EasyFEA.fem.Calc_projector(oldMesh, newMesh)

Get the matrix used to project the solution from the old mesh to the new mesh such that:

newU = proj * oldU

(newNn) = (newNn x oldNn) (oldNn)

Parameters:

• oldMesh (Mesh) – old mesh

• newMesh (Mesh) – new mesh

Returns:

projection matrix (newMesh.Nn, oldMesh.Nn)

Return type:

sp.csr_matrix

EasyFEA.fem.Det(mat)

Computes det(mat)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

class EasyFEA.fem.ElemType(value)

Bases: str, Enum

Implemented Lagrange isoparametric element types.

static Get_1D()

Returns 1D element types.

Return type:

list[ElemType]

static Get_2D()

Returns 2D element types.

Return type:

list[ElemType]

static Get_3D()

Returns 3D element types.

Return type:

list[ElemType]

HEXA20 = 'HEXA20'

HEXA27 = 'HEXA27'

HEXA8 = 'HEXA8'

POINT = 'POINT'

PRISM15 = 'PRISM15'

PRISM18 = 'PRISM18'

PRISM6 = 'PRISM6'

QUAD4 = 'QUAD4'

QUAD8 = 'QUAD8'

QUAD9 = 'QUAD9'

SEG2 = 'SEG2'

SEG3 = 'SEG3'

SEG4 = 'SEG4'

SEG5 = 'SEG5'

TETRA10 = 'TETRA10'

TETRA4 = 'TETRA4'

TRI10 = 'TRI10'

TRI15 = 'TRI15'

TRI3 = 'TRI3'

TRI6 = 'TRI6'

class EasyFEA.fem.FeArray(input_array, broadcastFeArrays=False)

Bases: ndarray[tuple[Any, …], dtype[Any]]

Finite Element array.

A finite element array has at least two dimensions.

FeArrayALike

alias of FeArray | ndarray[tuple[Any, …], dtype[Any]]

property T: FeArray | ndarray[tuple[Any, ...], dtype[Any]]

View of the transposed array.

Same as self.transpose().

Examples

>>> import numpy as np
>>> a = np.array([[1, 2], [3, 4]])
>>> a
array([[1, 2],
       [3, 4]])
>>> a.T
array([[1, 3],
       [2, 4]])

>>> a = np.array([1, 2, 3, 4])
>>> a
array([1, 2, 3, 4])
>>> a.T
array([1, 2, 3, 4])

See also

transpose

static asfearray(array, broadcastFeArrays=False)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

ddot(other)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

dot(other)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

max(*args, **kwargs)

np.max() wrapper.

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

min(*args, **kwargs)

np.min() wrapper.

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

static ones(*shape, dtype=None)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

sum(*args, **kwargs)

np.sum() wrapper.

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

static zeros(*shape, dtype=None)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

class EasyFEA.fem.Field(groupElem, dof_n, thickness=1.0, matrixType=MatrixType.mass)

Bases: object

Field class.

Get_coords(concatenate=False)

Returns integration point coordinates (x,y,z) for each element.

copy()

Return type:

Field

ddot(other)

property dof_n: int

degrees of freedom per node.

dot(other)

property grad: FeArray | ndarray[tuple[Any, ...], dtype[Any]]

Returns the gradient of the field.

property groupElem: _GroupElem

Group of elements.

property matrixType: MatrixType

class EasyFEA.fem.Gauss(elemType, matrixType)

Bases: object

property coord: ndarray[tuple[Any, ...], dtype[floating]]

integration point coordinates

property nPg: int

number of integration points

property weights: ndarray[tuple[Any, ...], dtype[floating]]

integration point weights

class EasyFEA.fem.GroupElemFactory

Bases: object

static Create(elemType, connect, coordGlob)

Creates an element group

Parameters:

• elemType (ElemType) – element type

• connect (_types.IntArray) – connection matrix storing nodes for each element (Ne, nPe)

• coordGlob (_types.FloatArray) – nodes coordinates

Returns:

the element group

Return type:

GroupeElem

DICT_ELEMTYPE: dict[ElemType, tuple[int, int, int, int, int, int, int, int]] = {ElemType.HEXA20: (17, 20, 3, 2, 8, 12, 0, 0), ElemType.HEXA27: (12, 27, 3, 2, 8, 12, 6, 1), ElemType.HEXA8: (5, 8, 3, 1, 8, 0, 0, 0), ElemType.POINT: (15, 1, 0, 0, 0, 0, 0, 0), ElemType.PRISM15: (18, 15, 3, 2, 6, 9, 0, 0), ElemType.PRISM18: (13, 18, 3, 2, 6, 9, 3, 0), ElemType.PRISM6: (6, 6, 3, 1, 6, 0, 0, 0), ElemType.QUAD4: (3, 4, 2, 1, 4, 0, 0, 0), ElemType.QUAD8: (16, 8, 2, 2, 4, 4, 0, 0), ElemType.QUAD9: (10, 9, 2, 2, 4, 4, 1, 0), ElemType.SEG2: (1, 2, 1, 1, 2, 0, 0, 0), ElemType.SEG3: (8, 3, 1, 2, 2, 1, 0, 0), ElemType.SEG4: (26, 4, 1, 3, 2, 2, 0, 0), ElemType.SEG5: (27, 5, 1, 4, 2, 3, 0, 0), ElemType.TETRA10: (11, 10, 3, 2, 4, 6, 0, 0), ElemType.TETRA4: (4, 4, 3, 1, 4, 0, 0, 0), ElemType.TRI10: (21, 10, 2, 3, 3, 6, 1, 0), ElemType.TRI15: (23, 15, 2, 4, 3, 9, 3, 0), ElemType.TRI3: (2, 3, 2, 1, 3, 0, 0, 0), ElemType.TRI6: (9, 6, 2, 2, 3, 3, 0, 0)}

(gmshId, nPe, dim, order, Nvertex, Nedge, Nface, Nvolume)

Type:

ElemType

DICT_GMSH_DATA: dict[int, tuple[ElemType, int, int, int, int, int, int, int]] = {1: (ElemType.SEG2, 2, 1, 1, 2, 0, 0, 0), 2: (ElemType.TRI3, 3, 2, 1, 3, 0, 0, 0), 3: (ElemType.QUAD4, 4, 2, 1, 4, 0, 0, 0), 4: (ElemType.TETRA4, 4, 3, 1, 4, 0, 0, 0), 5: (ElemType.HEXA8, 8, 3, 1, 8, 0, 0, 0), 6: (ElemType.PRISM6, 6, 3, 1, 6, 0, 0, 0), 8: (ElemType.SEG3, 3, 1, 2, 2, 1, 0, 0), 9: (ElemType.TRI6, 6, 2, 2, 3, 3, 0, 0), 10: (ElemType.QUAD9, 9, 2, 2, 4, 4, 1, 0), 11: (ElemType.TETRA10, 10, 3, 2, 4, 6, 0, 0), 12: (ElemType.HEXA27, 27, 3, 2, 8, 12, 6, 1), 13: (ElemType.PRISM18, 18, 3, 2, 6, 9, 3, 0), 15: (ElemType.POINT, 1, 0, 0, 0, 0, 0, 0), 16: (ElemType.QUAD8, 8, 2, 2, 4, 4, 0, 0), 17: (ElemType.HEXA20, 20, 3, 2, 8, 12, 0, 0), 18: (ElemType.PRISM15, 15, 3, 2, 6, 9, 0, 0), 21: (ElemType.TRI10, 10, 2, 3, 3, 6, 1, 0), 23: (ElemType.TRI15, 15, 2, 4, 3, 9, 3, 0), 26: (ElemType.SEG4, 4, 1, 3, 2, 2, 0, 0), 27: (ElemType.SEG5, 5, 1, 4, 2, 3, 0, 0)}

(ElemType, nPe, dim, order, Nvertex, Nedge, Nface, Nvolume)

Type:

gmshId

static Get_ElemInFos(gmshId)

return elemType, nPe, dim, order, Nvertex, Nedge, Nface, Nvolume

associated with the gmsh id.

Return type:

tuple[ElemType, int, int, int, int, int, int, int]

EasyFEA.fem.Inv(mat)

Computes inv(mat)

class EasyFEA.fem.LagrangeCondition(problemType, nodes, dofs, unknowns, dofsValues, lagrangeCoefs, description='')

Bases: BoundaryCondition

property lagrangeCoefs: ndarray[tuple[Any, ...], dtype[floating]]

Lagrange coefficients.

class EasyFEA.fem.LinearForm(form)

Bases: _Form

Assemble(field)

Assemble de form with the field.

Parameters:

field (Field) – field

Returns:

the assembled (Nn * dof_n, 1) sparse vector.

Return type:

csr_matrix

Integrate_e(field)

Integrates de form with the field on elements.

Parameters:

field (Field) – field

Returns:

the integrated (Ne, …) numpy array

Return type:

np.ndarray

class EasyFEA.fem.MatrixType(value)

Bases: str, Enum

Order used for integration over elements, which determines the number of integration points.

static Get_types()

Return type:

list[MatrixType]

beam = 'beam'

int_Ω ddNv • ddNv dΩ type

mass = 'mass'

int_Ω N • N dΩ type

rigi = 'rigi'

int_Ω dN • dN dΩ type

class EasyFEA.fem.Mesh(dict_groupElem, verbosity=False)

Bases: Observable

Mesh class that contains several _GroupElem instances.

Elements_Nodes(nodes, exclusively=True, neighborLayer=1)

Returns elements that exclusively or not use the specified nodes.

Return type:

ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]

Elements_Tags(tags)

Returns elements associated with the tag.

Return type:

ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]

Evaluate_dofsValues_at_coordinates(coordinates_n, dofsValues, elements=None)

Evaluates dofsValues with shape (Nn*dof_n, ) at the specified coordinates.

Parameters:

• coordinates_n (_types.FloatArray) – coordinates that must be a (Nnodes, 3) array.

• dofsValues (_types.FloatArray) – dofs values that must be a (Nn * dof_n) array.

• elements (Optional[_types.IntArray], optional) – elements that may contain the specified coordinates to speed up evaluation, by default None

Returns:

The interpolated values as a (Nnodes, dof_n) array.

Return type:

_types.FloatArray

Get_B_e_pg(matrixType)

Get the matrix used to calculate deformations from displacements.

WARNING: Use Kelvin Mandel Notation

[N1,x 0 … Nn,x 0

0 N1,y … 0 Nn,y

N1,y N1,x … N3,y N3,x]

(Ne, nPg, (3 or 6), nPe*dim)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

Get_DiffusePart_e_pg(matrixType)

Get the part that builds the diffusion term (scalar).

DiffusePart_e_pg = k_e_pg * jacobian_e_pg * weight_pg * dN_e_pg’ @ A @ dN_e_pg

Returns (epij) -> jacobian_e_pg * weight_pg * dN_e_pg’

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

Get_Gradient_e_pg(u, matrixType=MatrixType.rigi)

Returns the gradient of the discretized displacement field u as a matrix

Parameters:

• u (_types.FloatArray) – discretized displacement field [ux1, uy1, uz1, …, uxN, uyN, uzN] of size Nn * dim

• matrixType (MatrixType, optional) – matrix type, by default MatrixType.rigi

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

Returns:

• FeArray – grad(u) of shape (Ne, nPg, 3, 3)

• dim = 1

• ——-

• dxux 0 0

• 0 0 0

• 0 0 0

• dim = 2

• ——-

• dxux dyux 0

• dxuy dyuy 0

• 0 0 0

• dim = 3

• ——-

• dxux dyux dzux

• dxuy dyuy dzuy

• dxuz dyuz dzuz

Get_N_pg(matrixType)

Evaluates shape functions in (ξ, η, ζ) coordinates.

[N1, … , Nn]

(nPg, 1, nPe)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

Get_N_vector_pg(matrixType)

Returns shape functions matrix in (ξ, η, ζ) coordinates

[N1 0 … Nn 0

0 N1 … 0 Nn]

(nPg, dim, npe*dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

Get_New_meshSize_n(error_e, coef=0.5)

Returns the scalar field (at nodes) used to refine the mesh.

meshSize = (coef - 1) / error_e.max() * error_e + 1

Parameters:

• error_e (_types.FloatArray) – error evaluated on elements

• coef (float, optional) – mesh size division ratio, by default 1/2

Returns:

meshSize_n, new mesh size at nodes (Nn)

Return type:

_types.FloatArray

Get_Node_Values(result_e)

Get node values from element values.

The value of a node is calculated by averaging the values of the surrounding elements.

Parameters:

• mesh (Mesh) – mesh

• result_e (_types.FloatArray) – element values (Ne, i)

Returns:

nodes values (Nn, i)

Return type:

_types.FloatArray

Get_Paired_Nodes(corners, plot=False)

Get the paired nodes used to construct periodic boundary conditions.

Parameters:

• corners (_types.FloatArray) – Either nodes or nodes coordinates.

• plot (bool, optional) – Set whether to plot the link between nodes; defaults to False.

Returns:

Paired nodes, a 2-column matrix storing the paired nodes (n, 2).

Return type:

_types.IntArray

Get_Quality(criteria='aspect', nodeValues=False)

Calculates mesh quality [0, 1] (bad, good).

Parameters:

• criteria (str, optional) –

  criterion used, by default ‘aspect’

  • ”aspect”: hMin / hMax, ratio between minimum and maximum element length

  • ”angular”: angleMin / angleMax, ratio between the minimum and maximum angle of an element

  • ”gamma”: 2 rci/rcc, ratio between the radius of the inscribed circle and the circumscribed circle multiplied by 2. Useful for triangular elements.

  • ”jacobian”: jMax / jMin, ratio between the maximum jacobian and the minimum jacobian. Useful for higher-order elements.

• nodeValues (bool, optional) – calculates values on nodes, by default False

Returns:

mesh quality

Return type:

_types.FloatArray

Get_ReactionPart_e_pg(matrixType)

Get the part that builds the reaction term (scalar).

ReactionPart_e_pg = r_e_pg * jacobian_e_pg * weight_pg * N_pg’ @ N_pg

Returns (epij) -> jacobian_e_pg * weight_pg * N_pg’ @ N_pg

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

Get_SourcePart_e_pg(matrixType)

Get the part that builds the source term (scalar).

SourcePart_e_pg = f_e_pg * jacobian_e_pg * weight_pg * N_pg’

Returns (epij) -> jacobian_e_pg * weight_pg * N_pg’

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

Get_assembly_e(dof_n)

Returns assembly matrix for specified dof_n (Ne, nPe*dof_n)

Return type:

ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]

Get_connect_n_e()

Sparse matrix (Nn, Ne) of zeros and ones with ones when the node has the element such that:

values_n = connect_n_e * values_e

(Nn,1) = (Nn,Ne) * (Ne,1)

Return type:

csr_matrix

Get_dN_e_pg(matrixType)

Evaluates the first-order derivatives of shape functions in (x,y,z) coordinates.

[Ni,x … Nn,x

Ni,y … Nn,y]

(e, pg, dim, nPe)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

Get_ddN_e_pg(matrixType)

Evaluates the first-order derivatives of shape functions in (x, y, z) coordinates.

[Ni,x … Nn,x

Ni,y … Nn,y

Ni,z … Nn,z]

(Ne, nPg, dim, nPe)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

Get_jacobian_e_pg(matrixType, absoluteValues=True)

Returns the jacobians

variation in size (length, area or volume) between the reference element and the real element

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

Get_leftDispPart(matrixType)

Get the left side of local displacement matrices.

Ku_e = jacobian_e_pg * weight_pg * B_e_pg’ @ c_e_pg @ B_e_pg

Returns (epij) -> jacobian_e_pg * weight_pg * B_e_pg’

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

Get_list_groupElem(dim=None)

Returns the list of mesh element groups.

Parameters:

dim (int, optional) – The dimension of elements to retrieve, by default None (uses the main mesh dimension).

Returns:

A list of _GroupElem objects with the specified dimension.

Return type:

list[_GroupElem]

Get_meshSize(doMean=True)

Returns the mesh size of the mesh.

returns meshSize_e if doMean else meshSize_e_s

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

Get_nPg(matrixType)

Returns integration points according to the matrix type.

Return type:

int

Get_normals(nodes=None, displacementMatrix=None)

Returns normal vectors and nodes belonging to the edge of the mesh.

returns normals, nodes.

Return type:

tuple[ndarray[tuple[Any, ...], dtype[floating]], ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]]

Get_weight_pg(matrixType)

Returns integration points according to the matrix type.

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

Get_weightedJacobian_e_pg(matrixType)

Returns jacobian_e_pg * weight_pg.

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

Locates_sol_e(sol, dof_n=None, asFeArray=False)

Locates solution on elements.

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

property Ne: int

number of elements in the mesh

property Nn: int

number of nodes in the mesh

Nodes_Circle(circle, onlyOnEdge=True)

Returns the nodes in the circle.

Return type:

ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]

Nodes_Conditions(func)

Returns nodes that meet the specified conditions.

Parameters:

func (function) –

Function using the x, y and z nodes coordinates and returning boolean values.

examples :

lambda x, y, z: (x < 40) & (x > 20) & (y<10)

lambda x, y, z: (x == 40) | (x == 50)

lambda x, y, z: x >= 0

Returns:

nodes that meet the specified conditions.

Return type:

_types.IntArray

Nodes_Cylinder(circle, direction=[0, 0, 1], onlyOnEdge=False)

Returns the nodes in the cylinder.

Return type:

ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]

Nodes_Domain(domain)

Returns nodes in the domain.

Return type:

ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]

Nodes_Line(line)

Returns the nodes on the line.

Return type:

ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]

Nodes_Point(point)

Returns nodes on the point.

Return type:

ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]

Nodes_Points(points)

Returns nodes on points.

Return type:

ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]

Nodes_Tags(tags)

Returns nodes associated with the tags.

Return type:

ndarray[tuple[Any, ...], dtype[TypeVar(IntType, bound= Union[integer, int])]]

Rotate(theta, center=(0, 0, 0), direction=(0, 0, 1))

Rotates the mesh coordinates around an axis.

Parameters:

• theta (float) – rotation angle [deg]

• center (tuple, optional) – rotation center, by default (0,0,0)

• direction (tuple, optional) – rotation direction, by default (0,0,1)

Return type:

None

Set_Tag(nodes, tag)

Set a tag on the nodes and elements belonging to each group of elements in the mesh.

Symmetry(point=(0, 0, 0), n=(1, 0, 0))

Symmetrizes the mesh coordinates with respect to a specified plane.

Parameters:

• point (tuple, optional) – a point belonging to the plane, by default (0,0,0)

• n (tuple, optional) – normal to the plane, by default (1,0,0)

Return type:

None

Translate(dx=0.0, dy=0.0, dz=0.0)

Translates the mesh coordinates.

Return type:

None

property area: float

total area of the mesh.

property assembly_e: ndarray[tuple[Any, ...], dtype[IntType]]

assembly matrix (Ne, nPe*dim)

Used to position the rigi matrix in the global matrix.

property center: ndarray[tuple[Any, ...], dtype[floating]]

center of mass / barycenter / inertia center

property columnsScalar_e: ndarray[tuple[Any, ...], dtype[IntType]]

columns to fill the assembly matrix in scalar form (damage or thermal problems)

property columnsVector_e: ndarray[tuple[Any, ...], dtype[IntType]]

columns to fill the assembly matrix in vector (e.g elastic problem)

property connect: ndarray[tuple[Any, ...], dtype[IntType]]

connectivity matrix (Ne, nPe)

property coord: ndarray[tuple[Any, ...], dtype[floating]]

nodes coordinates matrix (Nn,3) for the main groupElem

property coordGlob: ndarray[tuple[Any, ...], dtype[floating]]

global nodes coordinates matrix (Nn, 3)

Contains all nodes coordinates

copy()

property dict_groupElem: dict[ElemType, _GroupElem]

dictionary containing all the element groups in the mesh

property dim

mesh dimension

property elemType: ElemType

mesh element type

property groupElem: _GroupElem

main group element

property inDim

dimension in which the mesh lies.

A 2D mesh can be oriented in a 3D space.

property length: float

total length of the mesh.

property nPe: int

nodes per element

property nodes: ndarray[tuple[Any, ...], dtype[IntType]]

mesh nodes

property orphanNodes: list[int]

nodes not connected to any mesh elements

property rowsScalar_e: ndarray[tuple[Any, ...], dtype[IntType]]

rows to fill the assembly matrix in scalar form (damage or thermal problems)

property rowsVector_e: ndarray[tuple[Any, ...], dtype[IntType]]

rows to fill the assembly matrix in vector (e.g elastic problem)

property verbosity: bool

the mesh can write in the terminal

property volume: float

total volume of the mesh.

EasyFEA.fem.Mesh_Optim(DoMesh, folder, criteria='aspect', quality=0.8, ratio=0.7, iterMax=20, coef=0.5)

Optimize the mesh using the given criterion.

Parameters:

• DoMesh (Callable[[str], Mesh]) –

  Function that constructs the mesh and takes a .pos file as argument for mesh optimization.

  The function must return a Mesh.

• folder (str) – Folder in which .pos files are created and then deleted.

• criteria (str, optional) –

  criterion used, by default ‘aspect’

  • ”aspect”: hMin / hMax, ratio between minimum and maximum element length

  • ”angular”: angleMin / angleMax, ratio between the minimum and maximum angle of an element

  • ”gamma”: 2 rci/rcc, ratio between the radius of the inscribed circle and the circumscribed circle multiplied by 2. Useful for triangular elements.

  • ”jacobian”: jMax / jMin, ratio between the maximum jacobian and the minimum jacobian. Useful for higher-order elements.

• quality (float, optional) – quality target, by default .8

• ratio (float, optional) – target ratio of mesh elements that must respect the specified quality, by default 0.7 (must be in [0,1])

• iterMax (int, optional) – Maximum number of iterations, by default 20

• coef (float, optional) – mesh size division ratio, by default 1/2

Returns:

optimized mesh size and ratio

Return type:

tuple[Mesh, float]

class EasyFEA.fem.Mesher(openGmsh=False, gmshVerbosity=False, verbosity=False)

Bases: object

Mesher class used to construct and generate the mesh via gmsh.

Create_posFile(coord, values, folder, filename='data')

Creates of a .pos file that can be used to refine a mesh in a zone.

Parameters:

• coord (_types.FloatArray) – coordinates of values

• values (_types.FloatArray) – scalar nodes values

• folder (str) – save folder

• filename (str, optional) – .pos file name, by default “data”.

Returns:

Returns the path to the created .pos file

Return type:

str

static Get_Entities(points=[], lines=[], surfaces=[], volumes=[])

Get entities from from points, lines, surfaces and volumes tags

Return type:

list[tuple[int, int]]

Mesh_2D(contour, inclusions=[], elemType=ElemType.TRI3, cracks=[], refineGeoms=[], isOrganised=False, additionalSurfaces=[], additionalLines=[], additionalPoints=[], folder='')

Creates a 2D mesh from a contour and inclusions that must form a closed plane surface.

Parameters:

• contour (Domain | Circle | Points | Contour) – geom object

• inclusions (list[Domain | Circle | Points | Contour], optional) – list of hollow and filled geom objects inside the domain

• elemType (ElemType, optional) – element type, by default “TRI3” [“TRI3”, “TRI6”, “TRI10”, “TRI15”, “QUAD4”, “QUAD8”, “QUAD9”]

• cracks (list[Line | Points | Contour | CircleArc]) – list of geom object used to create open or closed cracks

• refineGeoms (list[Domain|Circle|str], optional) – list of geom object for mesh refinement, by default []

• isOrganised (bool, optional) – mesh is organized, by default False

• additionalSurfaces (list[Domain | Circle | Points | Contour]) – additional surfaces that will be added to or removed from the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). Tip: if the mesh is not well generated, you can also give the inclusions.

• additionalLines (list[Union[Line,CircleArc]]) – additional lines that will be added to the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). WARNING: lines must be within the domain.

• additionalPoints (list[Point]) – additional points that will be added to the surfaces created by the contour and the inclusions. WARNING: points must be within the domain.

• folder (str, optional) – default mesh.msh folder, by default “” does not save the mesh

Returns:

Created mesh

Return type:

Mesh

Mesh_Beams(beams, elemType=ElemType.SEG2, additionalPoints=[], folder='')

Creates a beam mesh.

Parameters:

• beams (list[_Beam]) – list of Beams

• elemType (ElemType, optional) – element type, by default “SEG2” [“SEG2”, “SEG3”, “SEG4”]

• folder (str, optional) – default mesh.msh folder, by default “” does not save the mesh

• additionalPoints (list[Point]) – additional points that will be added to the mesh. WARNING: points must be within the domain.

Returns:

Created mesh

Return type:

Mesh

Mesh_Extrude(contour, inclusions=[], extrude=(0, 0, 1), layers=[], elemType=ElemType.TETRA4, cracks=[], refineGeoms=[], isOrganised=False, additionalSurfaces=[], additionalLines=[], additionalPoints=[], folder='')

Creates a 3D mesh by extruding a surface constructed from a contour and inclusions.

Parameters:

• contour (Domain | Circle | Points | Contour) – geom object

• inclusions (list[Domain | Circle | Points | Contour], optional) – list of hollow and filled geom objects inside the domain

• extrude (Coords, optional) – extrusion vector, by default [0,0,1]

• layers (list[int], optional) – layers in the extrusion, by default []

• elemType (ElemType, optional) – element type, by default “TETRA4” [“TETRA4”, “TETRA10”, “HEXA8”, “HEXA20”, “HEXA27”, “PRISM6”, “PRISM15”, “PRISM18”]

• cracks (list[Line | Points | Contour | CircleArc]) – list of geom object used to create open or closed cracks

• refineGeoms (list[Domain|Circle|str], optional) – list of geom object for mesh refinement, by default []

• isOrganised (bool, optional) – mesh is organized, by default False

• additionalSurfaces (list[Domain | Circle | Points | Contour]) – additional surfaces that will be added to or removed from the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). Tip: if the mesh is not well generated, you can also give the inclusions.

• additionalLines (list[Union[Line,CircleArc]]) – additional lines that will be added to the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). WARNING: lines must be within the domain.

• additionalPoints (list[Point]) – additional points that will be added to the surfaces created by the contour and the inclusions. WARNING: points must be within the domain.

• folder (str, optional) – default mesh.msh folder, by default “” does not save the mesh

Returns:

Created mesh

Return type:

Mesh

Mesh_Import_mesh(mesh, setPhysicalGroups=False, coef=1.0)

Creates the mesh from an .msh file.

Parameters:

• mesh (str) – .msh file

• setPhysicalGroups (bool, optional) – creates physical groups, by default False

• coef (float, optional) – coef applied to the node coordinates, by default 1.0

Returns:

Created mesh

Return type:

Mesh

Mesh_Import_part(file, dim, meshSize=0.0, elemType=None, refineGeoms=[None], folder='')

Creates the mesh from .stp or .igs files.

You can only use triangles or tetrahedrons.

Parameters:

• file (str) –

  .stp or .igs files.

  Note that for igs files, entities cannot be recovered.

• meshSize (float, optional) – mesh size, by default 0.0

• elemType (ElemType, optional) – element type, by default “TRI3” or “TETRA4” depending on dim.

• refineGeoms (list[Domain|Circle|str]) – geometric objects to refine the mesh

• folder (str, optional) – default mesh.msh folder, by default “” does not save the mesh

Returns:

Created mesh

Return type:

Mesh

Mesh_Revolve(contour, inclusions=[], axis=<EasyFEA.geoms._line.Line object>, angle=360, layers=[30], elemType=ElemType.TETRA4, cracks=[], refineGeoms=[], isOrganised=False, additionalSurfaces=[], additionalLines=[], additionalPoints=[], folder='')

Creates a 3D mesh by rotating a surface along an axis.

Parameters:

• contour (Domain | Circle | Points | Contour) – geometry that builds the contour

• inclusions (list[Domain | Circle | Points | Contour], optional) – list of hollow and filled geom objects inside the domain

• axis (Line, optional) – revolution axis, by default Line(Point(), Point(0,1))

• angle (float|int, optional) – revolution angle in [deg], by default 360

• layers (list[int], optional) – layers in extrusion, by default [30]

• elemType (ElemType, optional) – element type, by default “TETRA4” [“TETRA4”, “TETRA10”, “HEXA8”, “HEXA20”, “HEXA27”, “PRISM6”, “PRISM15”, “PRISM18”]

• cracks (list[Line | Points | Contour | CircleArc]) – list of geom object used to create open or closed cracks

• refineGeoms (list[Domain|Circle|str], optional) – list of geom object for mesh refinement, by default []

• isOrganised (bool, optional) – mesh is organized, by default False

• additionalSurfaces (list[Domain | Circle | Points | Contour]) – additional surfaces that will be added to or removed from the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). Tip: if the mesh is not well generated, you can also give the inclusions.

• additionalLines (list[Union[Line,CircleArc]]) – additional lines that will be added to the surfaces created by the contour and the inclusions. (e.g Domain, Circle, Contour, Points). WARNING: lines must be within the domain.

• additionalPoints (list[Point]) – additional points that will be added to the surfaces created by the contour and the inclusions. WARNING: points must be within the domain.

• folder (str, optional) – default mesh.msh folder, by default “” does not save the mesh

Returns:

Created mesh

Return type:

Mesh

Save_Simu(simu, results=[], details=False, edgeColor='black', plotMesh=True, showAxes=False, folder='')

Save the simulation in gmsh.pos format using gmsh.view

Parameters:

• simu (_Simu) – simulation

• results (list[str], optional) – list of result you want to plot, by default []

• details (bool, optional) – get default result values with details or not see simu.Results_nodesField_elementsField(details), by default False

• edgeColor (str, optional) – color used to plot the edges, by default ‘black’

• plotMesh (bool, optional) – plot the mesh, by default True

• showAxes (bool, optional) – show the axes, by default False

• folder (str, optional) – folder used to save .pos file, by default “”

Return type:

None

Set_meshSize(meshSize)

Sets the mesh size

Return type:

None

EasyFEA.fem.Norm(array, **kwargs)

np.linalg.norm() wrapper.

see https://numpy.org/doc/stable/reference/generated/numpy.linalg.norm.html

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

EasyFEA.fem.Sym_Grad(u)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

EasyFEA.fem.TensorProd(A, B, symmetric=False, ndim=None)

Computes tensor product.

Parameters:

• A (FeArray.FeArrayALike) – array A

• B (FeArray.FeArrayALike) – array B

• symmetric (bool, optional) – do symmetric product, by default False

• ndim (int, optional) – ndim=1 -> vect or ndim=2 -> matrix, by default None

Returns:

the calculated tensor product

Return type:

FeArray.FeArrayALike

EasyFEA.fem.Trace(mat)

Computes trace(mat)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

EasyFEA.fem.Transpose(mat)

Computes transpose(mat)

Return type:

Union[FeArray, ndarray[tuple[Any, ...], dtype[Any]]]

class EasyFEA.fem.elems.HEXA20(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property segments: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to construct segments (for display purposes).

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.HEXA27(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property segments: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to construct segments (for display purposes).

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.HEXA8(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property segments: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to construct segments (for display purposes).

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.POINT(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.PRISM15(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property segments: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to construct segments (for display purposes).

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.PRISM18(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property segments: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to construct segments (for display purposes).

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.PRISM6(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property segments: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to construct segments (for display purposes).

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.QUAD4(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.QUAD8(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.QUAD9(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.SEG2(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_EulerBernoulli_N()

Euler-Bernoulli beam shape functions in the (ξ, η, ζ) coordinates.

[phi_i psi_i … phi_n psi_n]

(nPe*2, 1)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_EulerBernoulli_dN()

Euler-Bernoulli beam shape functions first derivatives in the (ξ, η, ζ) coordinates.

[phi_i,ξ psi_i,ξ … phi_n,ξ psi_n,ξ]

(nPe*2, 1)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_EulerBernoulli_ddN()

Euler-Bernoulli beam shape functions second derivatives in the (ξ, η, ζ) coordinates.

[phi_i,ξ psi_i,ξ … phi_n,ξ psi_n,ξ]

(nPe*2, 2)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.SEG3(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_EulerBernoulli_N()

Euler-Bernoulli beam shape functions in the (ξ, η, ζ) coordinates.

[phi_i psi_i … phi_n psi_n]

(nPe*2, 1)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_EulerBernoulli_dN()

Euler-Bernoulli beam shape functions first derivatives in the (ξ, η, ζ) coordinates.

[phi_i,ξ psi_i,ξ … phi_n,ξ psi_n,ξ]

(nPe*2, 1)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_EulerBernoulli_ddN()

Euler-Bernoulli beam shape functions second derivatives in the (ξ, η, ζ) coordinates.

[phi_i,ξ psi_i,ξ … phi_n,ξ psi_n,ξ]

(nPe*2, 2)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.SEG4(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_EulerBernoulli_N()

Euler-Bernoulli beam shape functions in the (ξ, η, ζ) coordinates.

[phi_i psi_i … phi_n psi_n]

(nPe*2, 1)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_EulerBernoulli_dN()

Euler-Bernoulli beam shape functions first derivatives in the (ξ, η, ζ) coordinates.

[phi_i,ξ psi_i,ξ … phi_n,ξ psi_n,ξ]

(nPe*2, 1)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_EulerBernoulli_ddN()

Euler-Bernoulli beam shape functions second derivatives in the (ξ, η, ζ) coordinates.

[phi_i,ξ psi_i,ξ … phi_n,ξ psi_n,ξ]

(nPe*2, 2)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.SEG5(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_EulerBernoulli_N()

Euler-Bernoulli beam shape functions in the (ξ, η, ζ) coordinates.

[phi_i psi_i … phi_n psi_n]

(nPe*2, 1)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_EulerBernoulli_dN()

Euler-Bernoulli beam shape functions first derivatives in the (ξ, η, ζ) coordinates.

[phi_i,ξ psi_i,ξ … phi_n,ξ psi_n,ξ]

(nPe*2, 1)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_EulerBernoulli_ddN()

Euler-Bernoulli beam shape functions second derivatives in the (ξ, η, ζ) coordinates.

[phi_i,ξ psi_i,ξ … phi_n,ξ psi_n,ξ]

(nPe*2, 2)

Return type:

ndarray[tuple[Any, ...], dtype[Any]]

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.TETRA10(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property segments: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to construct segments (for display purposes).

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.TETRA4(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property segments: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to construct segments (for display purposes).

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.TRI10(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.TRI15(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.TRI3(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function

class EasyFEA.fem.elems.TRI6(gmshId, connect, coordGlob, nodes)

Bases: _GroupElem

Get_Local_Coords()

Get local ξ, η, ζ coordinates as a (nPe, dim) numpy array

_N()

Shape functions in (ξ, η, ζ) coordinates.

N1

⋮

Nn

(nPe, 1)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_abc_impl = <_abc._abc_data object>

_dN()

Shape functions first derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ Ni,η Ni,ζ

⋮

Nn,ξ Nn,η Nn,ζ

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddN()

Shape functions second derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ2 Ni,η2 Ni,ζ2

⋮

Nn,ξ2 Nn,η2 Nn,ζ2

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_dddN()

Shape functions third derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ3 Ni,η3 Ni,ζ3

⋮

Nn,ξ3 Nn,η3 Nn,ζ3

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

_ddddN()

Shape functions fourth derivatives in the (ξ, η, ζ) coordinates.

Ni,ξ4 Ni,η4 Ni,ζ4

⋮

Nn,ξ4 Nn,η4 Nn,ζ4

(nPe, dim)

Return type:

ndarray[tuple[Any, ...], dtype[floating]]

property faces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the element faces (for FEM purposes).

property origin: list[int]

reference element origin coordinates

property surfaces: ndarray[tuple[Any, ...], dtype[IntType]]

array of indices used to form the contour of the surfaces that make up the element (for display purposes). WARNING: When adding new 3D elements, ensure that the resulting surface normals point inward the element.

property triangles: list[int]

list of index used to form the triangles of an element that will be used for the 2D trisurf function