diff --git a/Elasticity/wrench2D.pro b/Elasticity/wrench2D.pro
index 76147ed1b51383cea809f24f305debff127c7ec0..cd70662385c43098b5a1786396c2d3849de74fa9 100644
--- a/Elasticity/wrench2D.pro
+++ b/Elasticity/wrench2D.pro
@@ -1,35 +1,38 @@
/* -------------------------------------------------------------------
- Tutorial 3 : linear elastic model of a wrench
+ Tutorial 3 : linear elastic model of a wrench
- Features:
- - "grad u" GetDP specific formulation for linear elasticity
- - first and second order elements
- - triangular and quadrangular elements
+ Features:
+ - "grad u" GetDP specific formulation for linear elasticity
+ - first and second order elements
+ - triangular and quadrangular elements
- To compute the solution interactively from the Gmsh GUI:
+ To compute the solution in a terminal:
+ getdp wrench2D.pro -solve Elast_u -pos pos
+
+ To compute the solution interactively from the Gmsh GUI:
File > Open > wrench.pro
Run (button at the bottom of the left panel)
------------------------------------------------------------------- */
-/* Linear elasticity with GetDP:
-
- 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. */
+/*
+ Particularities of linear elasticity in GetDP:
+
+ Instead of a vector field "u = Vector[ ux, uy, uz ]", the displacement field
+ is regarded as two (2D case) or three (3D case) scalar fields.
+ Unlike conventional formulations, GetDP formulation is then 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 via the constitutive relationship (Hooke law).
+ This unusual formulation allows to take advantage of the powerful geometrical
+ and homological GetDP kernel, 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. an electric scalar potential or a temperature
+ field.
+*/
Include "wrench2D_common.pro";
@@ -54,24 +57,30 @@ Group {
Sur_Clamp_Mec = Region[ Grip ];
Sur_Force_Mec = Region[ Force ];
- /* Signification of the abstract regions:
+ /* Meaning of abstract regions:
- Vol_Mec : Elastic domain
- - Vol_Force_Mec : Region with imposed volumic force
+ - Vol_Force_Mec : Region with imposed volume force
- Sur_Force_Mec : Surface with imposed surface traction
- Sur_Clamp_Mec : Surface with imposed zero displacements (all components) */
}
Function {
- /* Material coefficients.
- No need to define them regionwise here ( E[{Wrench}] = ... ; )
- as there is only one region in this model. */
+ /*
+ Material coefficients
+ No need of regionwise definition ( E[{Wrench}] = ... ; )
+ as this model comprises only one region.
+ If there is more than one region and coefficients are not particularised,
+ the same value holds for all of them.
+ */
E[] = Young;
nu[] = Poisson;
- /* Components of the volumic force applied to the region "Vol_Force_Mec"
- Gravity could be defined here ( force_y[] = 7000*9.81; ) ; */
+ /*
+ Volume force components applied to the region "Vol_Force_Mec"
+ Gravity could be defined here as well ( force_y[] = 7000*9.81; ) ;
+ */
force_x[] = 0;
force_y[] = 0;
- /* Components of the surface traction force applied to the region "Sur_Force_Mec" */
+ /* Surface traction force components applied to the region "Sur_Force_Mec" */
pressure_x[] = 0;
pressure_y[] = -AppliedForce/(SurfaceArea[]*Thickness); // downward vertical force
}
@@ -82,7 +91,7 @@ Function {
sigma_ij = C_ijkl epsilon_ij
- is represented in 2D by 4 2x2 tensors C_ij[], i,j=1,2, depending on the Lamé
+ is represented in 2D by four 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[])
@@ -94,7 +103,8 @@ 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;
@@ -128,7 +138,7 @@ Constraint {
}
}
-/* As explained above, the displacement field is discretized as two scalar
+/* As mentioned 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.
@@ -143,17 +153,17 @@ Constraint {
u . n = ux Cos [th] + uy Sin [th] = ... ;
are less naturally accounted for within the "grad u" formulation; but they
- could be easily implemented with a Lagrange multplier.
+ could be easily implemented with e.g. a Lagrange multiplier.
- Finite element shape (triangles or quadrangles) makes no difference in the
- definition of the FunctionSpaces. The appropriate shape functions to be used
+ The finite element shape (e.g. triangles or quadrangles in 2D) has no influence
+ in the definition of the FunctionSpaces. The appropriate shape functions
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 are hierarchically implemented 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 {
@@ -217,8 +227,8 @@ Jacobian {
}
}
-/* Adapt the number of Gauss points to the polynomial degree of the finite elements
- is as simple as this: */
+/* Adapting the number of Gauss points to the polynomial degree of the finite elements
+ is simple: */
Integration {
{ Name Gauss_v;
Case {