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Commit 30d765d1 authored by Christophe Geuzaine's avatar Christophe Geuzaine
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Elasticity/wrench2D.pro
+ 88
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View file @ 30d765d1
......@@ -13,41 +13,34 @@
/* Linear elasticity with GetDP:
GetDP has a peculiar way to deal with linear elasticity.
Instead of vector field "u = Vector[ ux, uy, uz ]",
the displacement field is regarded as two (2D case) or 3 (3D case) scalar fields.
Unlike conventional formulations, GetDP's formulation is written
in terms of the gradient "Grad u" of the displacement field,
which is a non-symmetric tensor, and the needed symmetrization
(to define the strain tensor and relate it to the stress tensor)
is done through the constitutive relationship (Hooke law).
The reason for this unusual formulation is to be able to use also for elastic problems
the powerful geometrical and homological kernel of GetDP,
which relies on the operators Grad, Curl and Div.
The "Grad u" formulation entails a small increase of assembly work
but makes in counterpart lots of geometrical features
implemented in GetDP (change of coordinates, function spaces, etc...)
applicable to elastic problems out-of-the-box,
since the scalar fields { ux, uy, uz } have exactly the same geometrical properties
as, e.g. a scalar electrip potential or a temperature field.
*/
GetDP has a peculiar way to deal with linear elasticity. Instead of a vector
field "u = Vector[ ux, uy, uz ]", the displacement field is regarded as two
(2D case) or 3 (3D case) scalar fields. Unlike conventional formulations
then, GetDP's formulation is written in terms of the gradient "Grad u" of the
displacement field, which is a non-symmetric tensor, and the needed
symmetrization (to define the strain tensor and relate it to the stress
tensor) is done through the constitutive relationship (Hooke law). The
reason for this unusual formulation is to be able to use also for elastic
problems the powerful geometrical and homological kernel of GetDP, which
relies on the operators Grad, Curl and Div.
The "Grad u" formulation entails a small increase of assembly work but makes
in counterpart lots of geometrical features implemented in GetDP (change of
coordinates, function spaces, etc...) applicable to elastic problems
out-of-the-box, since the scalar fields { ux, uy, uz } have exactly the same
geometrical properties as, e.g. a scalar electric potential or a temperature
field. */
Include "wrench2D_common.pro";
Young = 1e9 *
DefineNumber[ 200, Name "Material/Young modulus [GPa]"];
Poisson =
DefineNumber[ 0.3, Name "Material/Poisson coefficient []"];
AppliedForce =
DefineNumber[ 100, Name "Material/Applied force [N]"];
Young = 1e9 * DefineNumber[ 200, Name "Material/Young modulus [GPa]"];
Poisson = DefineNumber[ 0.3, Name "Material/Poisson coefficient []"];
AppliedForce = DefineNumber[ 100, Name "Material/Applied force [N]"];
// Approximation of the maximum deflection by an analytical model:
// Deflection = PL^3/(3EI) with I = Width^3*Thickness/12
Deflection =
DefineNumber[4*AppliedForce*((LLength-0.018)/Width)^3/(Young*Thickness)*1e3,
Name "Solution/Deflection (analytical) [mm]", ReadOnly 1];
Deflection = DefineNumber[4*AppliedForce*((LLength-0.018)/Width)^3/(Young*Thickness)*1e3,
Name "Solution/Deflection (analytical) [mm]", ReadOnly 1];
Group {
// Physical regions: Give explicit labels to the regions defined in the .geo file
......@@ -56,17 +49,16 @@ Group {
Force = Region[ 3 ];
// Abstract regions:
Vol_Elast_Mec = Region[ { Wrench } ];
Vol_Force_Mec = Region[ { Wrench } ];
Sur_Clamp_Mec = Region[ { Grip } ];
Sur_Force_Mec = Region[ { Force } ];
/*
Signification of the abstract regions:
Vol_Elast_Mec Elastic domain
Vol_Force_Mec Region with imposed volumic force
Sur_Force_Mec Surface with imposed surface traction
Sur_Clamp_Mec Surface with imposed zero displacements (all components)
*/
Vol_Mec = Region[ Wrench ];
Vol_Force_Mec = Region[ Wrench ];
Sur_Clamp_Mec = Region[ Grip ];
Sur_Force_Mec = Region[ Force ];
/* Signification of the abstract regions:
- Vol_Mec : Elastic domain
- Vol_Force_Mec : Region with imposed volumic force
- Sur_Force_Mec : Surface with imposed surface traction
- Sur_Clamp_Mec : Surface with imposed zero displacements (all components) */
}
Function {
......@@ -84,18 +76,17 @@ Function {
pressure_y[] = -AppliedForce/(SurfaceArea[]*Thickness); // downward vertical force
}
/* Hooke law
/* Hooke's law
The material law
sigma_ij = C_ijkl epsilon_ij
is represented in 2D by 4 2x2 tensors C_ij[], i,j=1,2
depending on the Lamé coefficients of the isotropic linear material,
is represented in 2D by 4 2x2 tensors C_ij[], i,j=1,2, depending on the Lamé
coefficients of the isotropic linear material,
lambda = E[]*nu[]/(1.+nu[])/(1.-2.*nu[]);
mu = E[]/2./(1.+nu[]);
lambda = E[]*nu[]/(1.+nu[])/(1.-2.*nu[])
mu = E[]/2./(1.+nu[])
as follows
......@@ -103,8 +94,7 @@ Function {
EPD: a[] = lambda + 2 mu b[] = mu c[] = lambda
3D: a[] = lambda + 2 mu b[] = mu c[] = lambda
respectively for the 2D plane strain (EPD), 2D plane stress (EPS) and 3D cases.
*/
respectively for the 2D plane strain (EPD), 2D plane stress (EPS) and 3D cases. */
Function {
Flag_EPC = 1;
......@@ -138,45 +128,40 @@ Constraint {
}
}
/* As explained above, the displacement field is discretized
as two scalar fields "ux" and "uy", which are the spatial components
of the vector field "u" in a fixed Cartesian coordinate system.
/* As explained above, the displacement field is discretized as two scalar
fields "ux" and "uy", which are the spatial components of the vector field
"u" in a fixed Cartesian coordinate system.
Boundary conditions like
ux = ... ;
uy = ... ;
translate naturally into Dirichlet constraints
on the scalar "ux" and "uy" FunctionSpaces.
Conditions like, however,
translate naturally into Dirichlet constraints on the scalar "ux" and "uy"
FunctionSpaces. More exotic conditions like
u . n = ux Cos [th] + uy Sin [th] = ... ;
are less naturally accounted for within the "Grad u" formulation.
Fortunately, they are rather uncommon.
are less naturally accounted for within the "Grad u" formulation; but they
could be easily implemented with a Lagrange multplier.
Finite element shape (triangles or quadrangles) makes no difference
in the definition of the FunctionSpaces. The appropriate shape functions to be used
Finite element shape (triangles or quadrangles) makes no difference in the
definition of the FunctionSpaces. The appropriate shape functions to be used
are determined by GetDP at a much lower level on basis of the information
contained in the *.msh file.
Second order elements, on the other hand, are implemented in the hierarchical fashion
by adding to the first order node-based shape functions
a set of second order edge-based functions
to complete a basis for 2d order polynomials on the reference element.
*/
Second order elements, on the other hand, are implemented in the hierarchical
fashion by adding to the first order node-based shape functions a set of
second order edge-based functions to complete a basis for 2d order
polynomials on the reference element. */
// Domain of definition of the "ux" and "uy" FunctionSpaces
Group {
Dom_H_u_Mec = Region[ { Vol_Elast_Mec,
Sur_Force_Mec,
Sur_Clamp_Mec} ];
Dom_H_u_Mec = Region[ { Vol_Mec, Sur_Force_Mec, Sur_Clamp_Mec} ];
}
Flag_Degree =
DefineNumber[ 0, Name "Geometry/Use degree 2 (hierarch.)", Choices{0,1}, Visible 1];
Flag_Degree = DefineNumber[ 0, Name "Geometry/Use degree 2 (hierarch.)",
Choices{0,1}, Visible 1];
FE_Degree = ( Flag_Degree == 0 ) ? 1 : 2; // Convert flag value into polynomial degree
FunctionSpace {
......@@ -233,36 +218,30 @@ Jacobian {
}
/* Adapt the number of Gauss points to the polynomial degree of the finite elements
is as simple as this: */
If (FE_Degree == 1)
is as simple as this: */
Integration {
{ Name Gauss_v;
Case {
{Type Gauss;
If (FE_Degree == 1)
{ Type Gauss;
Case {
{ GeoElement Line ; NumberOfPoints 3; }
{ GeoElement Line ; NumberOfPoints 3; }
{ GeoElement Triangle ; NumberOfPoints 3; }
{ GeoElement Quadrangle ; NumberOfPoints 4; }
}
}
}
}
}
ElseIf (FE_Degree == 2)
Integration {
{ Name Gauss_v;
Case {
{Type Gauss;
Else
{ Type Gauss;
Case {
{ GeoElement Line ; NumberOfPoints 5; }
{ GeoElement Triangle ; NumberOfPoints 7; }
{ GeoElement Quadrangle ; NumberOfPoints 7; }
}
}
EndIf
}
}
}
EndIf
Formulation {
{ Name Elast_u ; Type FemEquation ;
......@@ -272,13 +251,13 @@ Formulation {
}
Equation {
Integral { [ -C_xx[] * Dof{d ux}, {d ux} ] ;
In Vol_Elast_Mec ; Jacobian Vol ; Integration Gauss_v ; }
In Vol_Mec ; Jacobian Vol ; Integration Gauss_v ; }
Integral { [ -C_xy[] * Dof{d uy}, {d ux} ] ;
In Vol_Elast_Mec ; Jacobian Vol ; Integration Gauss_v ; }
In Vol_Mec ; Jacobian Vol ; Integration Gauss_v ; }
Integral { [ -C_yx[] * Dof{d ux}, {d uy} ] ;
In Vol_Elast_Mec ; Jacobian Vol ; Integration Gauss_v ; }
In Vol_Mec ; Jacobian Vol ; Integration Gauss_v ; }
Integral { [ -C_yy[] * Dof{d uy}, {d uy} ] ;
In Vol_Elast_Mec ; Jacobian Vol ; Integration Gauss_v ; }
In Vol_Mec ; Jacobian Vol ; Integration Gauss_v ; }
Integral { [ force_x[] , {ux} ];
In Vol_Force_Mec ; Jacobian Vol ; Integration Gauss_v ; }
......@@ -310,29 +289,36 @@ Resolution {
PostProcessing {
{ Name Elast_u ; NameOfFormulation Elast_u ;
PostQuantity {
{ Name u ; Value { Term { [ Vector[ {ux}, {uy}, 0 ]];
In Vol_Elast_Mec ; Jacobian Vol ; } } }
{ Name uy ; Value { Term { [ 1e3*{uy} ];
In Vol_Elast_Mec ; Jacobian Vol ; } } }
{ Name sig_xx ; Value { Term {
[ CompX[ C_xx[]*{d ux} + C_xy[]*{d uy} ] ];
In Vol_Elast_Mec ; Jacobian Vol ; } } }
{ Name sig_xy ; Value { Term {
[ CompY[ C_xx[]*{d ux} + C_xy[]*{d uy} ] ];
In Vol_Elast_Mec ; Jacobian Vol ; } } }
{ Name sig_yy ; Value { Term {
[ CompY [ C_yx[]*{d ux} + C_yy[]*{d uy} ] ];
In Vol_Elast_Mec ; Jacobian Vol ; } } }
{ Name u ; Value {
Term { [ Vector[ {ux}, {uy}, 0 ]]; In Vol_Mec ; Jacobian Vol ; }
}
}
{ Name uy ; Value {
Term { [ 1e3*{uy} ]; In Vol_Mec ; Jacobian Vol ; }
}
}
{ Name sig_xx ; Value {
Term { [ CompX[ C_xx[]*{d ux} + C_xy[]*{d uy} ] ];
In Vol_Mec ; Jacobian Vol ; }
}
}
{ Name sig_xy ; Value {
Term { [ CompY[ C_xx[]*{d ux} + C_xy[]*{d uy} ] ];
In Vol_Mec ; Jacobian Vol ; }
}
}
{ Name sig_yy ; Value {
Term { [ CompY [ C_yx[]*{d ux} + C_yy[]*{d uy} ] ];
In Vol_Mec ; Jacobian Vol ; }
}
}
}
}
}
PostOperation {
{ Name pos; NameOfPostProcessing Elast_u;
Operation {
If(FE_Degree == 1)
Print[ sig_xx, OnElementsOf Wrench, File "sigxx.pos" ];
Print[ u, OnElementsOf Wrench, File "u.pos" ];
......@@ -356,7 +342,6 @@ PostOperation {
}
}
// Tell Gmsh which GetDP commands to execute when running the model.
DefineConstant[
R_ = {"Elast_u", Name "GetDP/1ResolutionChoices", Visible 0},
......
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