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rAKA akantu
geometry_utils.cc
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/**
* @file geometry_utils.cc
*
* @author Mohit Pundir <mohit.pundir@epfl.ch>
*
* @date creation: Mon Mmay 20 2019
* @date last modification: Mon May 20 2019
*
* @brief Implementation of various utilities needed for contact geometry
*
* @section LICENSE
*
* Copyright (©) 2010-2018 EPFL (Ecole Polytechnique Fédérale de Lausanne)
* Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)
*
* Akantu is free software: you can redistribute it and/or modify it under the
* terms of the GNU Lesser General Public License as published by the Free
* Software Foundation, either version 3 of the License, or (at your option) any
* later version.
*
* Akantu is distributed in the hope that it will be useful, but WITHOUT ANY
* WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
* A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
* details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with Akantu. If not, see <http://www.gnu.org/licenses/>.
*
*/
/* -------------------------------------------------------------------------- */
#include "geometry_utils.hh"
#include "element_class_helper.hh"
/* -------------------------------------------------------------------------- */
namespace
akantu
{
/* -------------------------------------------------------------------------- */
void
GeometryUtils
::
normal
(
const
Mesh
&
mesh
,
const
Array
<
Real
>
&
positions
,
const
Element
&
element
,
Vector
<
Real
>
&
normal
,
bool
outward
)
{
UInt
spatial_dimension
=
mesh
.
getSpatialDimension
();
UInt
surface_dimension
=
spatial_dimension
-
1
;
UInt
nb_nodes_per_element
=
Mesh
::
getNbNodesPerElement
(
element
.
type
);
Matrix
<
Real
>
coords
(
spatial_dimension
,
nb_nodes_per_element
);
UInt
*
elem_val
=
mesh
.
getConnectivity
(
element
.
type
,
_not_ghost
).
storage
();
mesh
.
extractNodalValuesFromElement
(
positions
,
coords
.
storage
(),
elem_val
+
element
.
element
*
nb_nodes_per_element
,
nb_nodes_per_element
,
spatial_dimension
);
Matrix
<
Real
>
vectors
(
spatial_dimension
,
surface_dimension
);
switch
(
spatial_dimension
)
{
case
1
:
{
normal
[
0
]
=
1
;
break
;
}
case
2
:
{
vectors
(
0
)
=
Vector
<
Real
>
(
coords
(
1
))
-
Vector
<
Real
>
(
coords
(
0
));
Math
::
normal2
(
vectors
.
storage
(),
normal
.
storage
());
break
;
}
case
3
:
{
vectors
(
0
)
=
Vector
<
Real
>
(
coords
(
1
))
-
Vector
<
Real
>
(
coords
(
0
));
vectors
(
1
)
=
Vector
<
Real
>
(
coords
(
2
))
-
Vector
<
Real
>
(
coords
(
0
));
Math
::
normal3
(
vectors
(
0
).
storage
(),
vectors
(
1
).
storage
(),
normal
.
storage
());
break
;
}
default
:
{
AKANTU_ERROR
(
"Unknown dimension : "
<<
spatial_dimension
);
}
}
// to ensure that normal is always outwards from master surface
if
(
outward
)
{
const
auto
&
element_to_subelement
=
mesh
.
getElementToSubelement
(
element
.
type
)(
element
.
element
);
Vector
<
Real
>
outside
(
spatial_dimension
);
mesh
.
getBarycenter
(
element
,
outside
);
// check if mesh facets exists for cohesive elements contact
Vector
<
Real
>
inside
(
spatial_dimension
);
if
(
mesh
.
isMeshFacets
())
{
mesh
.
getMeshParent
().
getBarycenter
(
element_to_subelement
[
0
],
inside
);
}
else
{
mesh
.
getBarycenter
(
element_to_subelement
[
0
],
inside
);
}
Vector
<
Real
>
inside_to_outside
=
outside
-
inside
;
auto
projection
=
inside_to_outside
.
dot
(
normal
);
if
(
projection
<
0
)
{
normal
*=
-
1.0
;
}
}
}
/* -------------------------------------------------------------------------- */
void
GeometryUtils
::
normal
(
const
Mesh
&
mesh
,
const
Element
&
element
,
Matrix
<
Real
>
&
tangents
,
Vector
<
Real
>
&
normal
,
bool
outward
)
{
UInt
spatial_dimension
=
mesh
.
getSpatialDimension
();
// to ensure that normal is always outwards from master surface we
// compute a direction vector form inside of element attached to the
// suurface element
Vector
<
Real
>
inside_to_outside
(
spatial_dimension
);
if
(
outward
)
{
const
auto
&
element_to_subelement
=
mesh
.
getElementToSubelement
(
element
.
type
)(
element
.
element
);
Vector
<
Real
>
outside
(
spatial_dimension
);
mesh
.
getBarycenter
(
element
,
outside
);
// check if mesh facets exists for cohesive elements contact
Vector
<
Real
>
inside
(
spatial_dimension
);
if
(
mesh
.
isMeshFacets
())
{
mesh
.
getMeshParent
().
getBarycenter
(
element_to_subelement
[
0
],
inside
);
}
else
{
mesh
.
getBarycenter
(
element_to_subelement
[
0
],
inside
);
}
inside_to_outside
=
outside
-
inside
;
// auto projection = inside_to_outside.dot(normal);
// if (projection < 0) {
// normal *= -1.0;
//}
}
// to ensure that direction of tangents are correct, cross product
// of tangents should give be in the same direction as of inside to outside
switch
(
spatial_dimension
)
{
case
2
:
{
normal
[
0
]
=
-
tangents
(
0
,
1
);
normal
[
1
]
=
tangents
(
0
,
0
);
auto
ddot
=
inside_to_outside
.
dot
(
normal
);
if
(
ddot
<
0
)
{
tangents
*=
-
1.0
;
normal
*=
-
1.0
;
}
break
;
}
case
3
:
{
auto
tang_trans
=
tangents
.
transpose
();
auto
tang1
=
Vector
<
Real
>
(
tang_trans
(
0
));
auto
tang2
=
Vector
<
Real
>
(
tang_trans
(
1
));
auto
tang1_cross_tang2
=
tang1
.
crossProduct
(
tang2
);
normal
=
tang1_cross_tang2
/
tang1_cross_tang2
.
norm
();
auto
ddot
=
inside_to_outside
.
dot
(
normal
);
if
(
ddot
<
0
)
{
tang_trans
(
1
)
*=
-
1.0
;
normal
*=
-
1.0
;
}
tangents
=
tang_trans
.
transpose
();
break
;
}
default
:
break
;
}
}
/* -------------------------------------------------------------------------- */
void
GeometryUtils
::
covariantBasis
(
const
Mesh
&
mesh
,
const
Array
<
Real
>
&
positions
,
const
Element
&
element
,
const
Vector
<
Real
>
&
normal
,
Vector
<
Real
>
&
natural_coord
,
Matrix
<
Real
>
&
tangents
)
{
UInt
spatial_dimension
=
mesh
.
getSpatialDimension
();
const
ElementType
type
=
element
.
type
;
UInt
nb_nodes_per_element
=
mesh
.
getNbNodesPerElement
(
type
);
UInt
*
elem_val
=
mesh
.
getConnectivity
(
type
,
_not_ghost
).
storage
();
Matrix
<
Real
>
nodes_coord
(
spatial_dimension
,
nb_nodes_per_element
);
mesh
.
extractNodalValuesFromElement
(
positions
,
nodes_coord
.
storage
(),
elem_val
+
element
.
element
*
nb_nodes_per_element
,
nb_nodes_per_element
,
spatial_dimension
);
auto
&&
dnds
=
ElementClassHelper
<
_ek_regular
>::
getDNDS
(
natural_coord
,
type
);
tangents
.
mul
<
false
,
true
>
(
dnds
,
nodes_coord
);
auto
temp_tangents
=
tangents
.
transpose
();
for
(
UInt
i
=
0
;
i
<
spatial_dimension
-
1
;
++
i
)
{
auto
temp
=
Vector
<
Real
>
(
temp_tangents
(
i
));
temp_tangents
(
i
)
=
temp
.
normalize
();
}
tangents
=
temp_tangents
.
transpose
();
// to ensure that direction of tangents are correct, cross product
// of tangents should give the normal vector computed earlier
switch
(
spatial_dimension
)
{
case
2
:
{
Vector
<
Real
>
e_z
(
3
);
e_z
[
0
]
=
0.
;
e_z
[
1
]
=
0.
;
e_z
[
2
]
=
1.
;
Vector
<
Real
>
tangent
(
3
);
tangent
[
0
]
=
tangents
(
0
,
0
);
tangent
[
1
]
=
tangents
(
0
,
1
);
tangent
[
2
]
=
0.
;
auto
exp_normal
=
e_z
.
crossProduct
(
tangent
);
auto
ddot
=
normal
.
dot
(
exp_normal
);
if
(
ddot
<
0
)
{
tangents
*=
-
1.0
;
}
break
;
}
case
3
:
{
auto
tang_trans
=
tangents
.
transpose
();
auto
tang1
=
Vector
<
Real
>
(
tang_trans
(
0
));
auto
tang2
=
Vector
<
Real
>
(
tang_trans
(
1
));
auto
tang1_cross_tang2
=
tang1
.
crossProduct
(
tang2
);
auto
exp_normal
=
tang1_cross_tang2
/
tang1_cross_tang2
.
norm
();
auto
ddot
=
normal
.
dot
(
exp_normal
);
if
(
ddot
<
0
)
{
tang_trans
(
1
)
*=
-
1.0
;
}
tangents
=
tang_trans
.
transpose
();
break
;
}
default
:
break
;
}
}
/* -------------------------------------------------------------------------- */
void
GeometryUtils
::
covariantBasis
(
const
Mesh
&
mesh
,
const
Array
<
Real
>
&
positions
,
const
Element
&
element
,
Vector
<
Real
>
&
natural_coord
,
Matrix
<
Real
>
&
tangents
)
{
UInt
spatial_dimension
=
mesh
.
getSpatialDimension
();
const
ElementType
&
type
=
element
.
type
;
UInt
nb_nodes_per_element
=
mesh
.
getNbNodesPerElement
(
type
);
UInt
*
elem_val
=
mesh
.
getConnectivity
(
type
,
_not_ghost
).
storage
();
Matrix
<
Real
>
nodes_coord
(
spatial_dimension
,
nb_nodes_per_element
);
mesh
.
extractNodalValuesFromElement
(
positions
,
nodes_coord
.
storage
(),
elem_val
+
element
.
element
*
nb_nodes_per_element
,
nb_nodes_per_element
,
spatial_dimension
);
auto
&&
dnds
=
ElementClassHelper
<
_ek_regular
>::
getDNDS
(
natural_coord
,
type
);
tangents
.
mul
<
false
,
true
>
(
dnds
,
nodes_coord
);
auto
temp_tangents
=
tangents
.
transpose
();
for
(
UInt
i
=
0
;
i
<
spatial_dimension
-
1
;
++
i
)
{
auto
temp
=
Vector
<
Real
>
(
temp_tangents
(
i
));
temp_tangents
(
i
)
=
temp
.
normalize
();
}
tangents
=
temp_tangents
.
transpose
();
}
/* -------------------------------------------------------------------------- */
void
GeometryUtils
::
curvature
(
const
Mesh
&
mesh
,
const
Array
<
Real
>
&
positions
,
const
Element
&
element
,
const
Vector
<
Real
>
&
natural_coord
,
Matrix
<
Real
>
&
curvature
)
{
UInt
spatial_dimension
=
mesh
.
getSpatialDimension
();
const
ElementType
&
type
=
element
.
type
;
UInt
nb_nodes_per_element
=
mesh
.
getNbNodesPerElement
(
type
);
UInt
*
elem_val
=
mesh
.
getConnectivity
(
type
,
_not_ghost
).
storage
();
auto
&&
d2nds2
=
ElementClassHelper
<
_ek_regular
>::
getD2NDS2
(
natural_coord
,
type
);
Matrix
<
Real
>
coords
(
spatial_dimension
,
nb_nodes_per_element
);
mesh
.
extractNodalValuesFromElement
(
positions
,
coords
.
storage
(),
elem_val
+
element
.
element
*
nb_nodes_per_element
,
nb_nodes_per_element
,
spatial_dimension
);
curvature
.
mul
<
false
,
true
>
(
coords
,
d2nds2
);
}
/* -------------------------------------------------------------------------- */
UInt
GeometryUtils
::
orthogonalProjection
(
const
Mesh
&
mesh
,
const
Array
<
Real
>
&
positions
,
const
Vector
<
Real
>
&
slave
,
const
Array
<
Element
>
&
elements
,
Real
&
gap
,
Vector
<
Real
>
&
natural_projection
,
Vector
<
Real
>
&
normal
,
Real
alpha
,
UInt
max_iterations
,
Real
projection_tolerance
,
Real
extension_tolerance
)
{
UInt
index
=
UInt
(
-
1
);
Real
min_gap
=
std
::
numeric_limits
<
Real
>::
max
();
UInt
spatial_dimension
=
mesh
.
getSpatialDimension
();
UInt
surface_dimension
=
spatial_dimension
-
1
;
UInt
nb_same_sides
{
0
};
UInt
nb_boundary_elements
{
0
};
UInt
counter
{
0
};
const
auto
&
contact_group
=
mesh
.
getElementGroup
(
"contact_surface"
);
for
(
const
auto
&
element
:
elements
)
{
// filter out elements which are not there in the element group
// contact surface created by the surface selector and is stored
// in the mesh or mesh_facet, if a element is not there it
// returnas UInt(-1)
const
auto
&
elements_of_type
=
contact_group
.
getElements
(
element
.
type
);
if
(
elements_of_type
.
find
(
element
.
element
)
==
UInt
(
-
1
))
{
continue
;
}
nb_boundary_elements
++
;
// find the natural coordinate corresponding to the minimum gap
// between slave node and master element
Vector
<
Real
>
master
(
spatial_dimension
);
Vector
<
Real
>
xi
(
natural_projection
.
size
());
GeometryUtils
::
naturalProjection
(
mesh
,
positions
,
element
,
slave
,
master
,
xi
,
max_iterations
,
projection_tolerance
);
Matrix
<
Real
>
tangent_ele
(
surface_dimension
,
spatial_dimension
);
GeometryUtils
::
covariantBasis
(
mesh
,
positions
,
element
,
xi
,
tangent_ele
);
Vector
<
Real
>
normal_ele
(
spatial_dimension
);
GeometryUtils
::
normal
(
mesh
,
element
,
tangent_ele
,
normal_ele
);
// if gap between master projection and slave point is zero, then
// it means that slave point lies on the master element, hence the
// normal from master to slave cannot be computed in that case
auto
master_to_slave
=
slave
-
master
;
Real
temp_gap
=
master_to_slave
.
norm
();
if
(
temp_gap
!=
0
)
{
master_to_slave
/=
temp_gap
;
}
// also the slave point should lie inside the master surface in
// case of explicit or outside in case of implicit, one way to
// check that is by checking the dot product of normal at each
// master element, if the values of all dot product is same then
// the slave point lies on the same side of each master element
// A alpha parameter is introduced which is 1 in case of explicit
// and -1 in case of implicit, therefor the variation (dot product
// + alpha) should be close to zero (within tolerance) for both
// cases
Real
direction_tolerance
=
1e-8
;
auto
product
=
master_to_slave
.
dot
(
normal_ele
);
auto
variation
=
std
::
abs
(
product
+
alpha
);
if
(
variation
<=
direction_tolerance
and
temp_gap
<=
min_gap
and
GeometryUtils
::
isValidProjection
(
xi
,
extension_tolerance
))
{
gap
=
-
temp_gap
;
min_gap
=
temp_gap
;
index
=
counter
;
natural_projection
=
xi
;
normal
=
normal_ele
;
}
if
(
temp_gap
==
0
or
variation
<=
direction_tolerance
)
{
nb_same_sides
++
;
}
counter
++
;
}
// if point is not on the same side of all the boundary elements
// than it is not consider even if the closet master element is
// found
if
(
nb_same_sides
!=
nb_boundary_elements
)
{
index
=
UInt
(
-
1
);
}
return
index
;
}
/* -------------------------------------------------------------------------- */
UInt
GeometryUtils
::
orthogonalProjection
(
const
Mesh
&
mesh
,
const
Array
<
Real
>
&
positions
,
const
Vector
<
Real
>
&
slave
,
const
Array
<
Element
>
&
elements
,
Real
&
gap
,
Vector
<
Real
>
&
natural_projection
,
Vector
<
Real
>
&
normal
,
Matrix
<
Real
>
&
tangent
,
Real
/*alpha*/
,
UInt
max_iterations
,
Real
projection_tolerance
,
Real
extension_tolerance
)
{
UInt
index
=
UInt
(
-
1
);
Real
min_gap
=
std
::
numeric_limits
<
Real
>::
max
();
UInt
spatial_dimension
=
mesh
.
getSpatialDimension
();
UInt
surface_dimension
=
spatial_dimension
-
1
;
const
auto
&
contact_group
=
mesh
.
getElementGroup
(
"contact_surface"
);
for
(
auto
&&
tuple
:
enumerate
(
elements
))
{
auto
&
counter
=
std
::
get
<
0
>
(
tuple
);
const
auto
&
element
=
std
::
get
<
1
>
(
tuple
);
// filter out elements which are not there in the element group
// contact surface created by the surface selector and is stored
// in the mesh or mesh_facet, if a element is not there it
// returnas UInt(-1)
const
auto
&
elements_of_type
=
contact_group
.
getElements
(
element
.
type
);
if
(
elements_of_type
.
find
(
element
.
element
)
==
UInt
(
-
1
))
{
continue
;
}
Vector
<
Real
>
master
(
spatial_dimension
);
Vector
<
Real
>
xi_ele
(
natural_projection
.
size
());
GeometryUtils
::
naturalProjection
(
mesh
,
positions
,
element
,
slave
,
master
,
xi_ele
,
max_iterations
,
projection_tolerance
);
Matrix
<
Real
>
tangent_ele
(
surface_dimension
,
spatial_dimension
);
GeometryUtils
::
covariantBasis
(
mesh
,
positions
,
element
,
xi_ele
,
tangent_ele
);
Vector
<
Real
>
normal_ele
(
spatial_dimension
);
GeometryUtils
::
normal
(
mesh
,
element
,
tangent_ele
,
normal_ele
);
// if gap between master projection and slave point is zero, then
// it means that slave point lies on the master element, hence the
// normal from master to slave cannot be computed in that case
auto
master_to_slave
=
slave
-
master
;
Real
temp_gap
=
master_to_slave
.
norm
();
if
(
temp_gap
!=
0
)
{
master_to_slave
/=
temp_gap
;
}
// A alpha parameter is introduced which is 1 in case of explicit
// and -1 in case of implicit, therefor the variation (dot product
// + alpha) should be close to zero (within tolerance) for both
// cases
auto
product
=
master_to_slave
.
dot
(
normal_ele
);
if
(
product
<
0
and
temp_gap
<=
min_gap
and
GeometryUtils
::
isValidProjection
(
xi_ele
,
extension_tolerance
))
{
gap
=
-
temp_gap
;
min_gap
=
temp_gap
;
index
=
counter
;
natural_projection
=
xi_ele
;
normal
=
normal_ele
;
tangent
=
tangent_ele
;
}
}
return
index
;
}
/* -------------------------------------------------------------------------- */
void
GeometryUtils
::
realProjection
(
const
Mesh
&
mesh
,
const
Array
<
Real
>
&
positions
,
const
Vector
<
Real
>
&
slave
,
const
Element
&
element
,
const
Vector
<
Real
>
&
normal
,
Vector
<
Real
>
&
projection
)
{
UInt
spatial_dimension
=
mesh
.
getSpatialDimension
();
const
ElementType
&
type
=
element
.
type
;
UInt
nb_nodes_per_element
=
Mesh
::
getNbNodesPerElement
(
element
.
type
);
UInt
*
elem_val
=
mesh
.
getConnectivity
(
type
,
_not_ghost
).
storage
();
Matrix
<
Real
>
nodes_coord
(
spatial_dimension
,
nb_nodes_per_element
);
mesh
.
extractNodalValuesFromElement
(
positions
,
nodes_coord
.
storage
(),
elem_val
+
element
.
element
*
nb_nodes_per_element
,
nb_nodes_per_element
,
spatial_dimension
);
Vector
<
Real
>
point
(
nodes_coord
(
0
));
Real
alpha
=
(
slave
-
point
).
dot
(
normal
);
projection
=
slave
-
alpha
*
normal
;
}
/* -------------------------------------------------------------------------- */
void
GeometryUtils
::
realProjection
(
const
Mesh
&
mesh
,
const
Array
<
Real
>
&
positions
,
const
Element
&
element
,
const
Vector
<
Real
>
&
natural_coord
,
Vector
<
Real
>
&
projection
)
{
auto
spatial_dimension
=
mesh
.
getSpatialDimension
();
const
auto
&
type
=
element
.
type
;
auto
nb_nodes_per_element
=
Mesh
::
getNbNodesPerElement
(
element
.
type
);
auto
shapes
=
ElementClassHelper
<
_ek_regular
>::
getN
(
natural_coord
,
element
.
type
);
Matrix
<
Real
>
nodes_coord
(
spatial_dimension
,
nb_nodes_per_element
);
UInt
*
elem_val
=
mesh
.
getConnectivity
(
type
,
_not_ghost
).
storage
();
mesh
.
extractNodalValuesFromElement
(
positions
,
nodes_coord
.
storage
(),
elem_val
+
element
.
element
*
nb_nodes_per_element
,
nb_nodes_per_element
,
spatial_dimension
);
projection
.
mul
<
false
>
(
nodes_coord
,
shapes
);
}
/* -------------------------------------------------------------------------- */
void
GeometryUtils
::
naturalProjection
(
const
Mesh
&
mesh
,
const
Array
<
Real
>
&
positions
,
const
Element
&
element
,
const
Vector
<
Real
>
&
slave_coords
,
Vector
<
Real
>
&
master_coords
,
Vector
<
Real
>
&
natural_projection
,
UInt
max_iterations
,
Real
projection_tolerance
)
{
UInt
spatial_dimension
=
mesh
.
getSpatialDimension
();
UInt
surface_dimension
=
spatial_dimension
-
1
;
const
ElementType
&
type
=
element
.
type
;
UInt
nb_nodes_per_element
=
mesh
.
getNbNodesPerElement
(
type
);
UInt
*
elem_val
=
mesh
.
getConnectivity
(
type
,
_not_ghost
).
storage
();
Matrix
<
Real
>
nodes_coord
(
spatial_dimension
,
nb_nodes_per_element
);
mesh
.
extractNodalValuesFromElement
(
positions
,
nodes_coord
.
storage
(),
elem_val
+
element
.
element
*
nb_nodes_per_element
,
nb_nodes_per_element
,
spatial_dimension
);
// initial guess
natural_projection
.
zero
();
// obhjective function computed on the natural_guess
Matrix
<
Real
>
f
(
surface_dimension
,
1
);
// jacobian matrix computed on the natural_guess
Matrix
<
Real
>
J
(
surface_dimension
,
surface_dimension
);
// Jinv = J^{-1}
Matrix
<
Real
>
Jinv
(
surface_dimension
,
surface_dimension
);
// dxi = \xi_{k+1} - \xi_{k} in the iterative process
Matrix
<
Real
>
dxi
(
surface_dimension
,
1
);
// gradient at natural projection
Matrix
<
Real
>
gradient
(
surface_dimension
,
spatial_dimension
);
// second derivative at natural peojection
Matrix
<
Real
>
double_gradient
(
surface_dimension
,
surface_dimension
);
// second derivative of shape function at natural projection
Matrix
<
Real
>
d2nds2
(
surface_dimension
*
surface_dimension
,
nb_nodes_per_element
);
auto
compute_double_gradient
=
[
&
d2nds2
,
&
nodes_coord
,
surface_dimension
,
spatial_dimension
](
UInt
&
alpha
,
UInt
&
beta
)
{
auto
index
=
alpha
*
surface_dimension
+
beta
;
Vector
<
Real
>
d_alpha_beta
(
spatial_dimension
);
auto
d2nds2_transpose
=
d2nds2
.
transpose
();
Vector
<
Real
>
d2nds2_alpha_beta
(
d2nds2_transpose
(
index
));
d_alpha_beta
.
mul
<
false
>
(
nodes_coord
,
d2nds2_alpha_beta
);
return
d_alpha_beta
;
};
/* --------------------------- */
/* init before iteration loop */
/* --------------------------- */
// do interpolation
auto
update_f
=
[
&
f
,
&
master_coords
,
&
natural_projection
,
&
nodes_coord
,
&
slave_coords
,
&
gradient
,
surface_dimension
,
spatial_dimension
,
type
]()
{
// compute real coords on natural projection
auto
&&
shapes
=
ElementClassHelper
<
_ek_regular
>::
getN
(
natural_projection
,
type
);
master_coords
.
mul
<
false
>
(
nodes_coord
,
shapes
);
auto
distance
=
slave_coords
-
master_coords
;
// first derivative of shape function at natural projection
auto
&&
dnds
=
ElementClassHelper
<
_ek_regular
>::
getDNDS
(
natural_projection
,
type
);
gradient
.
mul
<
false
,
true
>
(
dnds
,
nodes_coord
);
// gradient transpose at natural projection
Matrix
<
Real
>
gradient_transpose
(
surface_dimension
,
spatial_dimension
);
gradient_transpose
=
gradient
.
transpose
();
// loop over surface dimensions
for
(
auto
alpha
:
arange
(
surface_dimension
))
{
Vector
<
Real
>
gradient_alpha
(
gradient_transpose
(
alpha
));
f
(
alpha
,
0
)
=
-
2.
*
gradient_alpha
.
dot
(
distance
);
}
// compute initial error
auto
error
=
f
.
norm
<
L_2
>
();
return
error
;
};
auto
projection_error
=
update_f
();
/* --------------------------- */
/* iteration loop */
/* --------------------------- */
UInt
iterations
{
0
};
while
(
projection_tolerance
<
projection_error
and
iterations
<
max_iterations
)
{
// compute covariant components of metric tensor
auto
a
=
GeometryUtils
::
covariantMetricTensor
(
gradient
);
// computing second derivative at natural projection
d2nds2
=
ElementClassHelper
<
_ek_regular
>::
getD2NDS2
(
natural_projection
,
type
);
// real coord - physical guess
auto
distance
=
slave_coords
-
master_coords
;
// computing Jacobian J
for
(
auto
alpha
:
arange
(
surface_dimension
))
{
for
(
auto
beta
:
arange
(
surface_dimension
))
{
auto
dgrad_alpha_beta
=
compute_double_gradient
(
alpha
,
beta
);
J
(
alpha
,
beta
)
=
2.
*
(
a
(
alpha
,
beta
)
-
dgrad_alpha_beta
.
dot
(
distance
));
}
}
Jinv
.
inverse
(
J
);
// compute increment
dxi
.
mul
<
false
,
false
>
(
Jinv
,
f
,
-
1.0
);
// update our guess
natural_projection
+=
Vector
<
Real
>
(
dxi
(
0
));
projection_error
=
update_f
();
iterations
++
;
}
}
/* -------------------------------------------------------------------------- */
void
GeometryUtils
::
contravariantBasis
(
const
Matrix
<
Real
>
&
covariant
,
Matrix
<
Real
>
&
contravariant
)
{
auto
inv_A
=
GeometryUtils
::
contravariantMetricTensor
(
covariant
);
contravariant
.
mul
<
false
,
false
>
(
inv_A
,
covariant
);
}
/* -------------------------------------------------------------------------- */
Matrix
<
Real
>
GeometryUtils
::
covariantMetricTensor
(
const
Matrix
<
Real
>
&
covariant_bases
)
{
Matrix
<
Real
>
A
(
covariant_bases
.
rows
(),
covariant_bases
.
rows
());
A
.
mul
<
false
,
true
>
(
covariant_bases
,
covariant_bases
);
return
A
;
}
/* -------------------------------------------------------------------------- */
Matrix
<
Real
>
GeometryUtils
::
contravariantMetricTensor
(
const
Matrix
<
Real
>
&
covariant_bases
)
{
auto
A
=
GeometryUtils
::
covariantMetricTensor
(
covariant_bases
);
auto
inv_A
=
A
.
inverse
();
return
inv_A
;
}
/* -------------------------------------------------------------------------- */
Matrix
<
Real
>
GeometryUtils
::
covariantCurvatureTensor
(
const
Mesh
&
mesh
,
const
Array
<
Real
>
&
positions
,
const
Element
&
element
,
const
Vector
<
Real
>
&
natural_coord
,
const
Vector
<
Real
>
&
normal
)
{
UInt
spatial_dimension
=
mesh
.
getSpatialDimension
();
auto
surface_dimension
=
spatial_dimension
-
1
;
const
ElementType
&
type
=
element
.
type
;
UInt
nb_nodes_per_element
=
Mesh
::
getNbNodesPerElement
(
type
);
UInt
*
elem_val
=
mesh
.
getConnectivity
(
type
,
_not_ghost
).
storage
();
auto
&&
d2nds2
=
ElementClassHelper
<
_ek_regular
>::
getD2NDS2
(
natural_coord
,
type
);
Matrix
<
Real
>
coords
(
spatial_dimension
,
nb_nodes_per_element
);
mesh
.
extractNodalValuesFromElement
(
positions
,
coords
.
storage
(),
elem_val
+
element
.
element
*
nb_nodes_per_element
,
nb_nodes_per_element
,
spatial_dimension
);
Matrix
<
Real
>
curvature
(
spatial_dimension
,
surface_dimension
*
surface_dimension
);
curvature
.
mul
<
false
,
true
>
(
coords
,
d2nds2
);
Matrix
<
Real
>
H
(
surface_dimension
,
surface_dimension
);
UInt
i
=
0
;
for
(
auto
alpha
:
arange
(
surface_dimension
))
{
for
(
auto
beta
:
arange
(
surface_dimension
))
{
Vector
<
Real
>
temp
(
curvature
(
i
));
H
(
alpha
,
beta
)
=
temp
.
dot
(
normal
);
i
++
;
}
}
return
H
;
}
}
// namespace akantu
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